Keywords: glass transition, Voronoi tessellations, molecular dynamic simulations, dynamical arrest, fragility, mode coupling theory, potential energy landscape, locally favoured structures Céline RUSCHER

Ce manuscrit clôture mon cycle universitaire à Strasbourg ainsi que trois ans d'un long et passionnant travail de recherche. Cette thèse a été, sans nul doute, une prise de risque, mais aujourd'hui je constate avec plaisir que tout ce travail a permis de faire émerger un tout nouveau système qui, je l'espère, aura l'attention qu'il mérite. Malgré la solitude qui va de paire avec la nouveauté, j'ai eu l'opportunité d'avoir un grand nombre d'échanges enrichissants avec des personnes de tous horizons et je voudrais les en remercier.

En premier lieu, je tiens à exprimer ma gratitude aux deux personnes qui ont grandement contribuer à façonner la jeune scientifique que je suis devenue. D'abord Jean Farago, mon directeur de thèse, qui a eu la patience de m'encadrer tout au long de cette thèse (et même bien avant !). Ta présence et ton aide tout au long de ce travail m'ont été plus que précieuses. Mais je voudrais surtout te dire merci pour ton exigence (même parfois too much). En me demandant constamment de repousser mes limites, d'aller toujours plus en profondeur, j'ai progressé bien plus que je n'aurais pu l'imaginer. Ensuite, je voudrais remercier tout particulièrement Jörg Baschnagel pour la confiance qu'il m'a témoigné en soutenant de façon toujours enthousiaste mes projets de la licence au doctorat. Son aide précieuse ainsi que ses conseils avisés m'ont permis de vivre des expériences enrichissantes et d'avancer dans mon travail de la meilleure façon qui soit. Je tiens aussi à remercier Paddy Royall et Daniele Coslovich pour l'intérêt qu'ils ont témoigné envers mon travail au cours des différentes conférences dans lesquelles nous nous sommes croisés. De plus, je leur suis très reconnaissante d'avoir bien accepté d'être les rapporteurs de cette thèse et d'avoir pris le temps de lire et commenter mon travail. Je remercie aussi les examinateurs de mon jury, Liesbeth Janssen et Pascal Hébraud, pour leurs remarques constructives. Un grand merci à Olivier Benzerara pour son aide avec les simulations numériques et bien plus ! Ton amitié a été précieuse pour moi durant ces dernières années et je te suis très reconnaissante de m'avoir soutenue même dans les moments les plus difficiles. Je te remercie aussi pour ta formation express au premier secours numérique et j'ai une pensée pour nos indispensables acolytes : remus, kalliope, romulus, est33, data5 et surtout data8 que j'ai si souvent maltraité.

Merci à tous mes collègues de l'équipe de simulations et théories ainsi qu'à toutes les personnes de l'Institut Charles Sadron avec qui j'ai eu le plaisir de discuter de science ou de tout autre chose pendant ces 3 ans.

Je voudrais remercier aussi l'IRTG SoMaS pour m'avoir permis de participer à un grand nombre de conférences. Je suis très reconnaissante envers Jana Husse et Birgitta Zovko pour leur aide au cours de ces trois dernières années. J'ai eu le privilège de partager mes joies et mes galères avec des personnes qui vont me manquer. D'abord mon co-bureau, Julien Fierling alias Julian Fielding, avec qui j'ai passé deux super années. Merci pour ton amitié, ton soutien et pour tous nos fou-rires.

On se rappellera toujours

Résumé

Lorsqu'un liquide est refroidi, il se peut que ce dernier évite la cristallisation. Il entre alors dans un état métastable par rapport au cristal, qu'on appelle régime surfondu. Ce régime se caractérise par un accroissement important du temps de relaxation et de la viscosité du système étudié. Lorsqu'on poursuit le refroidissement, il existe une température, appelée température de transition vitreuse, en deçà de laquelle, aux échelles de temps accessibles à l'expérimentation, le liquide se retrouve dans un état hors équilibre et forme un verre.

La température de transition vitreuse est définie expérimentalement lorsque la viscosité atteint 10 13 Poise ou que le temps de relaxation est de l'ordre de 100 -1000s. Bien que le ralentissement de la dynamique semble indiquer la présence d'une transition, la nature thermodynamique de celle-ci n'a jamais été clairement identifiée [START_REF] Cavagna | Supercooled liquids for pedestrians[END_REF].

Le temps de relaxation dans le liquide surfondu est lié à des réarrangements structuraux impliquant un nombre fini de particules [START_REF] Cavagna | Supercooled liquids for pedestrians[END_REF][START_REF] Adam | On the temperature dependence of cooperative relaxation properties in glass forming liquids[END_REF]. L'augmentation du temps de relaxation suggère donc qu'un nombre croissant de particules est impliqué dans des réorganisations.

Ces dernières ne sont pas des évènements isolés les uns des autres mais au contraire résultent de processus coopératifs. Ces observations suggèrent l'existence d'une longueur de corrélation croissante au voisinage de la transition vitreuse, réminiscente du scénario ordinaire des phénomènes critiques. Cependant la difficulté principale réside dans l'application de la méthodologie classique des transitions de phases. Quel est le paramètre d'ordre pertinent ? Quelle fonction de corrélation utiliser ? De nombreux travaux se sont attelés à la recherche de cette longueur en étudiant les fonctions de corrélations mettant en jeu les fluctuations de densité. Cependant aucune longueur pertinente n'a pu être extraite [START_REF] Cavagna | Supercooled liquids for pedestrians[END_REF]. À ce jour, trouver cette longueur de corrélation reste donc un défi ainsi qu'une quête de première importance pour clarifier la nature de la transition vitreuse. En partant du constat que les réarrangements structuraux jouent un rôle clé, il est intéressant d'étudier le voisinage des particules composant le liquide et de voir comment ce dernier change avec la température. Un outil intensivement utilisé en physique des liquides pour obtenir des informations à la fois sur le voisinage d'une particule et sur sa topologie sont les tessellations de Voronoï. Ces dernières correspondent à un découpage mathématique de l'espace en cellules qui sont construites à partir des points de l'espace les plus proches i d'une particule donnée que de n'importe quelle autre particule appartenant à ce même espace.

Nous avons récemment proposé, non pas de se servir des tessellations de Voronoï comme un simple outil, mais pour la première fois d'utiliser leurs propriétés géométriques afin de définir le champ de force d'un nouveau type de liquide dit liquide de Voronoï [START_REF] Ruscher | The voronoi liquid[END_REF][START_REF] Ruscher | Anomalous sound attenuation in voronoi liquid[END_REF].

Ce dernier se distingue de la majorité des liquides étudiés dans le cadre de la transition vitreuse car d'une part ses interactions ne sont pas de paires mais intrinsèquement à plusieurs corps. D'autre part, la nature des interactions impliquent que le liquide de Voronoï est constamment sous tension dans le sens où deux particules voisines s'attirent en permanence. Une première étude théorique et numérique menée sur la forme monodisperse du liquide de Voronoï a permis de constater que ce système présente des similitudes avec les fluides usuels tels que ceux de Lennard-Jones, mais a également sa propre phénoménologie : des propriétés de scaling anormales des fonctions thermodynamiques, un scaling particulier de l'atténuation des ondes sonores dans le régime mésocopique [START_REF] Ruscher | Anomalous sound attenuation in voronoi liquid[END_REF],

ainsi qu'une importante mobilité dans la phase cristalline. Dans sa forme monodisperse, le système est néanmoins incapable d'éviter la cristallisation. L'objet de cette thèse est d'étudier une extension de ce modèle en introduisant de la polydispersité de façon à empêcher la cristallisation et de permettre ainsi de sonder le régime surfondu. La question fondamentale à laquelle nous cherchons à répondre est la suivante : de quelle façon les interactions à plusieurs corps modifient-elles le scénario habituel de la transition vitreuse ?

Afin d'essayer d'apporter une réponse à cette question, nous avons étudié un mélange binaire du liquide de Voronoï par le biais de la théorie et des simulations numériques.

Considérations théoriques

L'introduction de la polydispersité a été possible grâce à l'utilisation des tessellations de Voronoï-Laguerre qui permettent d'associer à des particules ponctuelles j un rayon intrinsèque R j . Dans le cas des tessellations de Voronoï classiques, la cellule d'une particule j est construite à partir de l'intersection des plans médiateurs entre la particule j et ses plus proches voisins i. Les tessellations de Voronoï-Laguerre, quant à elles, prennent en compte la présence des rayons intrinsèques R j et R i associés à chaque type de particules. De ce fait, les plans délimitants les bords de la cellule ne sont plus forcément médians.

L'énergie potentielle du liquide de Voronoï polydisperse est définie de la façon suivante :

E P = γ 2 j v j d 3 r[r 2 -R 2 j + R 2 ]
ii ou r est un vecteur partant de la particule et parcourant toute la cellule j. R correspond au rayon moyen des particules et γ est une constante arbitraire. Le champ de forces dérive de l'énergie potentielle de telle sorte que f j = -∇ j E P , avec

f j = γτ j = γM j G j
La force s'exerçant sur la particule j est donc directement proportionnelle au vecteur allant de la position de la particule vers le barycentre de la cellule j comme illustré sur la Figure 1. Comme mentionné précédemment, notre étude s'est limité à la caractérisation d'un mélange binaire (A : B) avec R A > R B . Le mélange de référence étudié est équimolaire, N A = N B = 500 particules. Nous avons travaillé à densité constante de telle sorte que v = V /N = 1. Les simulations ont été réalisée avec une version modifiée de LAMMPS [START_REF] Plimpton | Fast parallel algorithms for short-range molecular dynamics[END_REF] qui permet l'interfaçage avec la librairie Voro++ [START_REF] Rycroft | Voro++: A three-dimensional voronoi cell library in c++[END_REF].

Au cours de ce travail, nous avons remarqué que la bidispersité est gouvernée par un 1/3 . En effet, lorsque R A = R B , η = 0 et on retrouve la limite monodisperse du liquide de Voronoï [START_REF] Ruscher | The voronoi liquid[END_REF]. Nous avons montré que l'aptitude du liquide de Voronoï bidisperse à éviter la cristallisation est directement lié à la valeur de η. Ainsi des expériences de refroidissement ont montré que lorsque η ≥ 0.25, la disparité de taille entre les deux types de rayons est suffisamment importante pour empêcher la cristallisation. Nous avons choisi de travailler avec η = 0.375 pour être certain, d'une part, d'être dans le régime où la cristallisation est évitée. D'autre part cette valeur de η est suffisamment faible pour considérer que le système se trouve dans une limite de iii faible polydispersité. Pour η = 0.375, nous avons estimé que la température de transition vitreuse T G ≈ 0.66.

seul paramètre η = R 2 A -R 2 B /v

Thermodynamique

Nous avons débuté l'analyse du liquide de Voronoï bidisperse en s'intéressant aux propriétés thermodynamiques. Nous avons formulé analytiquement l'expression des potentiels d'excès tels que l'énergie libre ou l'entropie. Comme attendu, ces potentiels thermodynamiques se sont révélés être peu sensibles au voisinage de la transition vitreuse, nous nous sommes donc par la suite intéressés à leurs dérivés premières qui sont connues pour manifester des changements notables aux alentours de T G . Nous avons eu un intérêt tout particulier pour la chaleur spécifique à volume constant C e v qui présente un accroissement relativement fort au voisinage de la transition vitreuse comparativement à d'autres systèmes vitreux. En effet, nous avons comparé l'évolution de la chaleur spécifique du liquide de Voronoï avec celles calculées pour 3 liquides de Lennard-Jones (LJ) :

1. Le mélange binaire de Kob-Andersen (KA) [START_REF] Kob | Testing made-coupling theory for a supercooled binary lennard-jones mixture: The van hove correlation function[END_REF][START_REF] Kob | Testing mode-coupling theory for a supercooled binary lennard-jones mixture. ii. intermediate scattering function and dynamic susceptibility[END_REF].

2. Le modèle de Wahnström (WAHN) [START_REF] Wahnström | Molecular-dynamics study of a supercooled two-component lennard-jones system[END_REF].

3. Un mélange binaire (AMLJ-0.80) pour lequel les interactions sont additives et le ratio σ BB /σ AA = 0.80 [START_REF] Coslovich | Understanding fragility in supercooled lennard-jones mixtures. i. locally preferred structures[END_REF].

Sur la Figure 2, on remarque très clairement l'augmentation de C e v /N , lorsque T diminue est plus importante pour le liquide de Voronoï. Pour expliquer ce comportement, nous nous sommes intéressés aux paysages d'énergie potentielle (PEL) et nous avons calculé l'énergie moyenne e IS des structures inhérentes IS (ou minima) pour le liquide de Voronoï et les 3 systèmes LJ (voir Figure 3). Nous avons ainsi pu montrer que la forte décroissance de e IS pour le liquide de Voronoï est responsable de l'accroissement important de C e v /N au voisinage de la transition vitreuse.

Observables microscopiques structurales

Nous avons ensuite étudié les observables microscopiques du liquide de Voronoï bidisperse et nous nous sommes dans un premier temps intéressé aux observables structurales. Nous v avons calculé les fonctions de paires partielles g ab (r) ainsi que les facteurs de structures statiques partiels S ab (k). Comme tous les systèmes vitreux, ces grandeurs structurales sont peu sensibles au refroidissement (voir Figure 4) [START_REF] Cavagna | Supercooled liquids for pedestrians[END_REF][START_REF] Kob | Testing mode-coupling theory for a supercooled binary lennard-jones mixture. ii. intermediate scattering function and dynamic susceptibility[END_REF][START_REF] Berthier | Theoretical perspective on the glass transition and amorphous materials[END_REF].

Ensuite, pour le liquide de Voronoï bidipserse, dans la limite de faible bidispersité, la position des extrema des fonctions de paires partielles et des facteurs de structures partiels peuvent être exprimés de manière analytique, et ce quelque soit la nature de la composition. Ainsi, nous avons pu montrer que pour la fonction de paire, la position des extrema r * AB , associée à l'interaction entre grandes et petites particules, est donnée par :

r * AB = 1 2 (r * AA + r * BB )
Cette dernière relation implique que les positions des extrema de g AB (r) sont toujours localisés au milieu de ceux de g AA (r) et g BB (r) comme montré sur la Figure 5.

Observables microscopiques dynamiques

Dans un second temps, nous nous sommes intéressés aux observables microscopiques dynamiques qui sont connues pour être très sensibles au voisinage de la transition vitreuse.

Nous avons calculé le déplacement carré moyen (MSD) le facteur de structure dynamique cohérent F (k, t), ainsi que sa partie incohérente F s (k, t). Nous avons pu observer le scenario typique de relaxation en deux étapes [START_REF] Cavagna | Supercooled liquids for pedestrians[END_REF][START_REF] Binder | Glassy Materials And Disordered Solids[END_REF]. La connaissance de

Fragilité

Cette deviation du comportement Arrhénien est observée pour un grand nombre de matériaux surfondus et l'explication de son origine est toujours manquante. Pour tenter de comparer différents systèmes vitreux entre-eux, Angell a proposé de représenter graphiquement l'évolution des quantités de transport : viscosité, coefficient de diffusion, temps de relaxation en fonction de T G /T [START_REF] Angell | Perspectives on the glass transition[END_REF]. Cette renormalisation de la température permet de comparer différents liquides surfondus en fonction d'un seul paramètre, la fragilité, qui quantifie le degré de déviation de la loi Arrhénienne. Afin de déterminer la fragilité vii du liquide de Voronoï, et à l'instar de ce qui a été proposé par Coslovich et Pastore [START_REF] Coslovich | Understanding fragility in supercooled lennard-jones mixtures. i. locally preferred structures[END_REF],

nous avons utilisé un fit permettant de passer de façon continue du régime Arrhénien au régime super-Arrhénien. Nous avons pu montrer que des changements dans la composition du mélange influencent la fragilité. Ainsi, plus le nombre de grandes particules A est augmentée, moins la déviation du régime Arrhénien est marquée (voir Figure 6). 1. Le travail microscopique E µ exercé sur une particule lors de son déplacement sur l'échelle de longueur caractéristique v 1/3 : E µ = F 2 (0)

2. Les fluctuations de l'énergie potentielle δE 2 P . Dans l'ensemble canonique, ces dernières sont directement reliées à la chaleur spécifique à volume constant C e v par la relation

δE 2 P = T k B C e v .
3. Le module de cisaillement G ∞ calculé au temps t = 0. Son utilisation a déjà était proposée dans le cadre du Shoving Model développé par Dyre [START_REF] Dyre | Elastic models for the non-arrhenius viscosity of glass-fromings liquids[END_REF][START_REF] Dyre | Local elastic expansion model for viscous-flow activation energies of glass-forming molecular liquids[END_REF][START_REF] Dyre | The instantaneous shear modulus in the shoving model[END_REF]. Dans ce cas, l'énergie caractéristique est donnée par E(T ) = vG ∞ (T ). xi Sur la Figure 8 est représentée l'évolution de Ω 0 τ α en fonction de l'inverse de la température adimensionnée par chacune des trois propositions précédentes pour l'échelle d'énergie microscopique. De façon très générale, on remarque que quelque soit la représentation choisie, le liquide de Voronoï se distingue des liquides de Lennard-Jones.

Cette distinction est directement liée aux différences qui existent entre le potentiel de LJ qui est de coeur dur et le potentiel du liquide de Voronoï qui est lui de coeur mou. À l'exception de l'adimensionnement par E µ , ces représentations mettent en avant les similitudes entre les différents modèles appartenant à la famille des potentiels LJ.

Relaxations collectives et individuelles du temps de relaxation

Nous nous sommes également intéressés à l'évolution des relaxations individuelles et collectives à différentes échelles (voir Figure 9). xii La présence d'une courbe maitresse pour k > k * suggère que, à grands vecteurs d'ondes, relaxations individuelles et collectives sont pilotées par τ α et k * . Pour k < k * , dans chacun des deux cas, un départ de la courbe maitresse ainsi qu'une transition vers un régime en loi de puissance sont observés. On note également une décroissance importante du temps de relaxation collectif dans le régime mésoscopique pour k ∼ 4.90. Ceci signifie qu'à l'échelle de longueur ∼ 2π/4.90, le temps de relaxation collectif est découplé du temps de relaxation structural, suggérant très certainement la présence d'un état de transition qui n'a pas été identifié pour le moment.

Analyse avec la théorie du couplage de mode (MCT)

Nous avons également testé les prédictions de la théorie de couplage de mode (MCT) dans sa version idéale [START_REF] Reichman | Mode-coupling theory[END_REF][START_REF] Baschnagel | Computer simulations of supercooled polymer melts in the bulk and in confined geometry[END_REF]. Nous avons ainsi vérifié que le principe de superposition temps-température (TTSP) était valide pour les températures les plus basses accessibles dans notre système. De la même façon nous avons testé le principe de factorisation qui s'est avéré valide pour tous vecteurs d'onde q 6.00. En effet, la violation du principe de factorisation est liée à la présence, dans le régime mésoscopique et pour le facteur de structure cohérent, d'oscillations amorties très importantes dans le régime β. Ceci est visible dans la Figure 10 (haut) où on observe très clairement les oscillations pour le plus petit q. Cette phénoménologie n'étant pas prise en compte dans une description MCT idéale, il n'est donc pas surprenant d'observer une violation des prédictions de la MCT dans ce régime de vecteurs d'ondes.

Nous avons ensuite poursuivi l'analyse MCT en déterminant l'exposant b à partir d'un fit de von Schweidler :

φ q (t) = f c q -hq (t/t ′ σ ) b + hq Bq (t/t ′ σ ) 2b
Afin de vérifier la valeur obtenue pour b ainsi que la qualité du fit à q * , nous avons fixé les paramètres t ′ σ , f c q et b puis nous avons essayé le fit de von Schweidler pour le facteur de structure cohérent calculé pour différents vecteurs d'ondes (voir Figure 10). Pour q 6.00, nous avons remarqué, d'une part, que le fit de von Schzeilder contraint ne permet pas de décrire correctement le régime mésoscopique que ce soit pour le facteur de structure cohérent ou pour l'incohérent. D'autre part, en ne contraignant pas le fit de von Schweidler, nous avons remarqué que l'exposant b augmente lorsque q diminue. Cette apparente violation des prédictions de la MCT dans le régime mésoscopique tend à suggérer la présence de phénomènes physiques sous-jacents non pris en compte par la MCT. φ s q (t) q = 3.00 q = 6.75 q = 8.35 q = 11.15 q = 14.45 q = 16.55

A particles Toutefois, afin de déterminer la température de couplage de mode T c , nous avons fait l'hypothèse que la MCT pouvait s'appliquer à notre système et nous avons ainsi pu trouver que T c = 0.7989.

Etude du paysage d'énergie potentielle

Afin de comprendre plus en profondeur la phénoménologie de la transition vitreuse, nous avons étudié le paysage d'énergie potentielle (PEL) dans lequel coexistent deux types de xiv points stationnaires : les minima appelés également structures inhérentes et les points selles.

Analyse des structures inhérentes

Dans un premier temps, nous avons analysé les structures inhérentes (IS). Pour se faire, à l'instar de Stillinger et Weber [START_REF] Stillinger | Hidden structure in liquids[END_REF], nous avons minimisé l'énergie potentielle E P de configurations thermalisées en utilisant l'algorithme FIRE [START_REF] Bitzek | Structural relaxation made simple[END_REF]. La minimisation de E P permet un découpage clair du PEL sous forme de bassins contenant chacun une IS.

En représentant l'énergie moyenne des IS, e IS , en fonction de l'inverse de la température 1/T , on a remarqué (voir Figure 11) qu'il existait une relation linéaire entre ces deux grandeurs à basse température : Cette relation linéaire provient du fait que la distribution des IS, Ω(e IS ), dans notre système a été identifiée comme étant Gaussienne. On s'est intéressé ensuite à la valeur xv prise par σ 2 dans l'équation précédente. De manière intéressante, on a remarqué que cette valeur est dix fois plus grande pour le liquide de Voronoï bidisperse que pour les liquides LJ.

e IS = E max - σ 2 k B T 0.
Nous avons également voulu tester la validité de l'approximation harmonique. Nous avons remarqué que cette dernière semble valide à très basses températures. A plus haute température, mais toujours en dessous de la température de crossover T * , nous avons noté la présence d'anharmonicité relativement importantes qui ne peuvent pas être envisagées simplement comme une faible perturbation du régime harmonique.

Test de la théorie de Adam-Gibbs

En faisant l'hypothèse que l'approximation harmonique est valide, nous avons calculé l'entropie configurationnelle comme étant la différence entre l'entropie du liquide et l'entropie du solide désordonné, S c = S liq -S sol .

Nous avons ensuite montré qu'à basse température, τ α ≈ exp(B/(T S c )) (voir Figure 12).

La théorie de Adam-Gibbs semble ainsi être vérifiée pour le liquide de Voronoï bidisperse.

Analyse des points selles

Nous avons eu un intérêt tout particulier pour l'analyse des points-selles qui sont des points stationnaires caractérisés par un nombre K de directions négatives. Nous avons calculé les points selles de deux façons différentes à partir d'une configuration thermalisée. La première façon consiste à minimiser la fonction W = ∇E 2 P tandis que la seconde méthode, nommée Eigenvector Following (EV), consiste en une maximisation de E P le xvi long des K directions montantes et une minimisation le long des 3N -K directions descendantes. Alors que la première méthode converge vers des quasi-point-selles (QS), pour lesquelles W ≈ 10 -2 , la seconde méthode permet de trouver de vrais points-selles W ≈ 0.

Dans un premier temps, nous nous sommes intéressés à l'évolution de la densité de direction négative k = K/(3N ) en fonction de la densité énergétique des points-selles u = U/N . Nous avons ainsi observé que k(u) est une fonction croissante monotone. Au cours de cette analyse nous avons observé que la méthode EV ne nous permettait pas d'avoir accès aux faibles densités de k.

Nous avons aussi évalué les barrières entre points-selles de deux façons différentes en considérant des analyses uniquement sur les QS.

La première consiste à évaluer ∆U b ≈ du/dk qui correspond à la barrière énergétique entre un point-selle ayant K directions négatives et un autre point-selle ayant K + 1 directions négatives. Nous avons remarqué que la barrière énergétique entre deux points selles adjacents augmente de façon significative lorsque la température diminue (voir Figure 13: Haut : évolution de la hauteur de barrière énergétique entre deux pointsselles adjacents en fonction de la température. Bas : évolution de l'écart énergétique entre un point-selle ayant K directions négatives et l'IS sous-jacent en fonction de la densité de directions négatives k = K/(3N ).

Analyse des structures localement favorables (LFS) Identification des LFS

Dans une dernière partie, nous nous sommes intéressés aux structures localement favorables (LFS) qui minimisent localement l'énergie potentielle. Différentes techniques permettent de déterminer ces structures [START_REF] Coslovich | Understanding fragility in supercooled lennard-jones mixtures. i. locally preferred structures[END_REF][START_REF] Malins | Identification of long-lived clusters and their link to slow dynamics in a model glass former[END_REF][START_REF] Doye | The favored cluster structures of model glass formers[END_REF], nous avons retenu la méthode proposée par Coslovich et Pastore [START_REF] Coslovich | Understanding fragility in supercooled lennard-jones mixtures. i. locally preferred structures[END_REF]. Cette dernière se base sur une analyse des cellules (polyèdres) de Voronoï, les polyèdres qui apparaissent le plus souvent sont ceux permettant de remonter à la structure la plus favorable. Ainsi nous avons montré que le liquide de Voronoï bidisperse a tendance à favoriser des arrangements d'icosaèdres.

xviii Cette tendance est d'autant plus prononcée que la température est basse (voir Figure 14). (0, 0, 12) (0, 2, 8, 2) (0, 3, 6, 3) (0, 3, 6, 4)

Présence de domaines

Nous nous sommes ensuite intéressés à l'extension spatial de ces icosaèdres, et pour cela nous avons cherché si ces derniers formaient des clusters. Nous avons noté que lorsque la température était abaissée, le système avait tendance à former de grand clusters pouvant contenir plus de 100 icosaèdres (voir Figure 16).

Ces clusters ont un impact direct sur la dynamique. En effet, nous avons calculé les temps de relaxation des particules au centre des icosaèdres et nous avons remarqué que ces temps de relaxation sont bien plus importants que ceux des particules n'appartenant pas du tout à des icosaèdres.

Cristallisation

A très basse température, nous avons observé que le système formait un pseudo-cristal.

L'énergie potentielle transite vers une valeur de plus basse énergie, comme montrée sur la

Introduction

When a liquid is cooled down, crystallization can be avoided and the liquid enters into the metastable supercooled regime. This regime is characterized by an important slowing down of the dynamics. Upon cooling, a temperature is reached below which, on experimental time-scales, the system falls out of equilibrium and forms a glass. This temperature, named glass transition temperature, is defined arbitrarily as the temperature for which the viscosity reaches 10 13 Poise or for which the relaxation time is of the order of 10-1000s [START_REF] Cavagna | Supercooled liquids for pedestrians[END_REF][START_REF] Ediger | Supercooled liquids and glasses[END_REF][START_REF] Ediger | Spatially heterogeneous dynamics in supercooled liquids[END_REF]. Although the slowing down of the dynamics points toward the existence of a transition, the thermodynamic nature of this latter has never been clearly identified and is still today subjected to debates [START_REF] Leuzzi | Thermodynamics of the Glassy State[END_REF].

Understanding the microscopic origin of dynamical arrest is still one of the fundamental problems of condensed matter physics. The increase of the relaxation is a common trend to all glass formers, however it changes from a material to another one. For some materials, such as SiO 2 , the relaxation time can be described by an Arrhenian law, whereas some other species like organic glasses, exhibits a super-Arrhenius relaxation [START_REF] Angell | Perspectives on the glass transition[END_REF][START_REF] Ediger | Supercooled liquids and glasses[END_REF][START_REF] Ngai | Relaxation and Diffusion in Complex Systems[END_REF]. These observations lead to the famous Angell's classification of strong/fragile glass formers [START_REF] Angell | Perspectives on the glass transition[END_REF][START_REF] Angell | Relaxation in liquids, polymers and plastic crystals -strong/fragile parrterns and problems[END_REF]. The deviation from the Arrhenian behavior is associated with the presence of free energy barriers which increase with temperature. But nowadays, the origin of these free energy barrier is still not well understood.

It is well-accepted that the relaxation time in supercooled liquids is related to structural rearrangements implying only a finite number of particles and that the increase of the relaxation time results from an increasing number of particles that move in a cooperative way to reorganize. In the 60's, Adam and Gibbs defined the concept of cooperative rearranging region (CRR), and proposed a theory where the free energy barrier is considered to be directly proportional to the number of particles involved in these regions [START_REF] Adam | On the temperature dependence of cooperative relaxation properties in glass forming liquids[END_REF]. Moreover, upon cooling, the size of the CRR are assumed to grow. All these observations point toward the existence of an increasing correlation length which would be reminiscent of classical critical phenomena. Nevertheless, the main issue comes from difficulties to apply that classical methodology of critical phenomena. What is a reliable order parameter ? Which correlation function have to be used ? Numerous work have tried to identify this growing correlation length [START_REF] Dasgupta | Is there a growing correlation length near the glass transition?[END_REF][START_REF] Hanakata | Interfacial mobility scale determines the scale of collective motion and relaxation rate in polymer films[END_REF][START_REF] Betancourt | String model for the dynamics of glass-forming liquids[END_REF] but no proper definition of this correlation length have been proposed. Moreover other studies point toward a dynamical correlation length instead of a structural one [START_REF] Dasgupta | Is there a growing correlation length near the glass transition?[END_REF][START_REF] Karmakar | Growing length and time scales in glass-forming liquids[END_REF][START_REF] Berthier | Direct experimental evidence of a growing length scale accompanying the glass transition[END_REF][START_REF] Berthier | Dynamical Heterogeneities in Glasses, Collloids and Granular Media[END_REF].

Considering that structural rearrangements play a key role in the phenomenology of glass transition, numerous studies have focused on characterizing the local neighborhood of particles [START_REF] Coslovich | Understanding fragility in supercooled lennard-jones mixtures. i. locally preferred structures[END_REF][START_REF] Malins | Identification of long-lived clusters and their link to slow dynamics in a model glass former[END_REF] and investigated how this environment evolves upon cooling. The existence of local preferred structures minimizing the local energy has been found [START_REF] Coslovich | Understanding fragility in supercooled lennard-jones mixtures. i. locally preferred structures[END_REF][START_REF] Frank | Complex alloy structures regarded as sphere packings. i. definitions and basic principles[END_REF][START_REF] Royall | The role of local structure in dynamical arrest[END_REF][START_REF] Dzugutov | Decoupling of diffusion from structural relaxation and spatial heterogeneity in a supercooled simple liquid[END_REF]. It was also shown that upon cooling, this local favored structures form domains where the dynamics is significantly slower than in the surrounding liquid [START_REF] Coslovich | Understanding fragility in supercooled lennard-jones mixtures. i. locally preferred structures[END_REF][START_REF] Malins | Identification of long-lived clusters and their link to slow dynamics in a model glass former[END_REF][START_REF] Dzugutov | Decoupling of diffusion from structural relaxation and spatial heterogeneity in a supercooled simple liquid[END_REF].

When looking for such structures, a tool intensively used in physics to obtain information on the neighborhood of particles, are Voronoi tessellations that tile space into cells where one cell is built from point of space closer from a given particle than from the other ones.

We recently proposed to consider Voronoi tessellations, not only as a tool, but for the first time, we use their intrinsic geometrical properties to define the force field of a new liquid called The Voronoi Liquid [START_REF] Ruscher | The voronoi liquid[END_REF]. As the interactions in this liquid are not pairwise but intrinsically manybody and put the system always under tension (in a way neighboring particles are constantly attracting each other) the Voronoi liquid is different from most of the liquids used nowadays to investigate glass transition. A first theoretical and numerical study of the monodisperse form of the Voronoi liquid revealed that this system shares similarities with usual liquids, as for instance Lennard-Jones liquids. However the Voronoi liquid has also its own phenomenology as for instance:

a characteristic scaling of thermodynamical functions, a specific scaling of the sound attenuation in the mesoscopic regime and an relative mobility within the crystalline phase [START_REF] Ruscher | The voronoi liquid[END_REF][START_REF] Ruscher | Anomalous sound attenuation in voronoi liquid[END_REF]. In its monodisperse form, the Voronoi liquid crystallizes.

The aim of this thesis is to study an extension of the monodisperse Voronoi liquid, where polydispersity has been introduced to prevent crystallization. Therefore with the polydisperse Voronoi liquid, we want to probe the neighborhood of glass transition by trying to answer the following question: How do these exotic interactions disrupt the classical scenario of the glass transition ? The idea is that this strongly uncommon interactions, which preserve the thermodynamic stability of the liquid, would modify the classical scenario of glass transition in a significative way. In this way this new model would act as a probe of glass transition.

In order to answer this question, throughout this work, we will investigate theoretically and numerically (with molecular dynamic simulations) the properties of a binary mixture of the Voronoi liquid. To quantify the differences between the bidisperse Voronoi liquid

ϕ = {x 1 , • • • , x j , • • • , x N } where x j ∈ R d , to each x j ∈ ϕ there is an associated Voronoi cell C(x j , ϕ) defined by C(x j , ϕ) = {y ∈ R d : ||y -x j || ≤ ||y -x i || ∀x i ∈ ϕ} (1.1)
Each cell is obtained from the intersection of half-spaces and it results that Voronoi cells are convex d-polytopes (polygons in 2d and polyhedra in 3d). Note that the strict equality in equation 1.1 defines the equation of the limiting hyperplane between two cells.

In other words equation 1.1 means that the d-polytope associated to the j th particle is defined from the point of space which are closer from the particle j than from any other particles. In figure 1.1 is represented an example of 2-d Voronoi tessellation. The polygon associated to the particle j is built from the nearest neighbors of j. Each segment composing the polygon corresponds to the median plane between j and one of its nearest neighbors. The intersection of these median planes defines the polygon.

We can define scalar and vectorial observables related to the tessellations. We consider the j th polytope and we denote its volume v j . As the total volume is conserved we can write V = j v j . Then the particle j is located at M j and we denote G j the centroid of the j th polytope. The origin O of the coordinate system is arbitrarily defined, therefore the position of particle j is given by r j = OM j .

We define now a vectorial observable named the geometrical polarization, denoted τ j , and given by :

τ j = v j M j G j (1.2)
This quantity is similar to the electric dipole moment where charge q is multiplied by the vector starting from the barycenter of negative charges and going to the barycenter of positive charges. In our case, v j would play the role of the charge, the centroid G j would correspond to the barycenter of the positive charge uniformly distributed in the polygon and M j would be the local negative point charge.

Chapter 1. A brand new class of liquids 6

An important property related to the geometrical polarization is that the total polarization is conserved for any given configuration of particles :

j τ j = 0 (1.3)
Later, the conservation of the total geometrical polarization will be of huge interest when we will define the interaction among particles. Especially we will see that the force exerts on particle j is directly proportional to τ j . This force field derives from a potential energy. A demonstration of equation 1.3 based only on geometrical argument is not obvious and more details can be found in [START_REF] Farago | New conserved structural fields for supercooled liquids new conserved structural fields for supercooled liquids[END_REF].

With classical Voronoi tessellations all the particles are identical regarding the partitioning of the space [START_REF] Aurenhammer | An optimal algorithm for constructing the weighted voronoi diagram in the plane[END_REF]. However when disparities are introduced between particles we could be interested to find them back in the tessellation. A way to process is to deal with weighted Voronoi diagrams which enable to take into account disparities in the strength of influence among particles. In these tessellations cells are defined in term of a distance function which are different from the usual Euclidian distance. For most of weighted Voronoi diagrams resulting cells have curved edges but it exists one particular tessellation which captures the idea of influence without dealing with nonlinear edges:

the Voronoi-Laguerre tessellations. Additionally we will see that this extension of the classical Voronoi tessellation enables also the conservation of the geometrical polarization as seen in equation 1.3.

Voronoi-Laguerre tessellations

As for the classical Voronoi tessellations, N point particles (1 ≤ j ≤ N ) are contained into a cubic box of volume V = L d with L the length of the box and d the dimension of the space. For weighted point particles we associated to each of them an intrinsic length that we call radius for convenience and denote R j . This intrinsic length plays a role in the construction of the polytopes as in this case the plane between two nearest neighbors is not necessarily median. For building the Laguerre tessellations we define first the distance between particle j located at M J and a point M by the power of M with respect to the hypersphere of center M J and radius R j as

pow(M, (M j , R j )) = M j M 2 -R 2 j (1.4)
This relation is illustrated in figure 1.2. We immediately see that depending on the position of M the power could be :

• pow(M, (M j , R j )) < 0, M is inside the hypersphere

• pow(M, (M j , R j )) = 0, M is on the hypersphere

• pow(M, (M j , R j )) > 0, M is outside the hypersphere and in this case, if H is the intersection between the hypersphere and its tangent hyperplane we have pow(M, (M j , R j )) = HM j 2 Given the set ϕ = {(r j , R j )}, where r j is the location of the point particle and R j its intrinsic length. To each couple (r j , R j ) there is an associated Voronoi-Laguerre cell C((r j , R)j), ϕ) defined by :

C((x j , R j ), ϕ) = {y ∈ R d : ||y -r j || 2 -R 2 j ≤ ||y -r i || 2 -R 2 i ∀(r i , R i ) ∈ ϕ} (1.5)
One can show that Voronoi-Laguerre tessellations is a mathematical partitioning of the space into convex polytopes where the polytope associated to particle j is built from the point of space which are closer to the particle j (in term of power distance) than to any other particles.

Points y ∈ R d satisfying the strict equality in equation 1.5 form the plane P (r i , r j ) between two point-particles i and j. Therefore the plane is given by the following equation :

P (r i , r j ) = {y ∈ R d : y, 2(r i -r j ) = ||r i || 2 -||r j || 2 -R 2 i + R 2 j } (1.6)
The distance from the plane P (r i , r j ) to the particle located in r i is

d r i ,P = r i , 2(r i -r j ) -(||r i || 2 -||r j || 2 -R 2 i + R 2 j ) 2||r i -r j || = r 2 ji + R 2 i -R 2 j 2r ji = r ji 2 + R 2 i -R 2 j 2r ji (1.7)
where r ji = r ir j and r ji = ||r ji ||. From equation 1.7 we observe that for Voronoi-Laguerre tessellation the plane is no more located at r ji /2 but its position depends on the intrinsic length associated to each point particle. In the case where all these intrinsic lengths are equal we find back the result expected for classical Voronoi tessellation that is d r i ,P = r ji /2. In Figure 1.3 is represented different possibilities for the plane position regarding the placement of two particles with different intrinsic lengths. The distance is kept fixed between the two particles and the size of the large particle is increased. We notice that as the biggest particle grows, the position of the plane is more and more shifted to the right. At a certain size of the big particle, the plane is not located between the two particles. It happens when the small particle is included in the largest one. It immediately raises the question of the possibility that a particle is not contained into its Voronoi cell. In Figure 1.4 is represented an example of Voronoi-Laguerre tessellation for particles with various intrinsic length. It is important to mention that :

• A Laguerre cell can be empty. In Figure 1.4 the particle C 3 has no cell at all.

• It may not contain its point-particle. It is for instance the case of the cell V (C 2 )

associated to particle C 2 .
• It may contain other point-particles even those whose cell is empty. In Figure 1.5 is represented an exemple of a two dimensional Voronoi-Laguerre tessellation where to each point particle is associated a Voronoi cell. Similarly to the classical tessellation, the total volume is conserved V = j v j , the centroid of the j th cell is located in G j and the particle j of radius R j is in M j . The position of the particle j is given by r j = OM j . As in equation 1.2 the geometrical polarisation is given by τ j = v i M j G j .

We have presented some geometrical characteristics of the Voronoi tessellation and of one of its possible extension the Voronoi-Laguerre tessellation. Whereas the first one is used to deal with systems where particles play identical role the second tessellation is more suited to work with polydisperse systems. In both cases we have observed that the geometrical polarization is conserved. In what follows we will give a mechanical role to this vectorial quantity. As a matter of fact the force exerted on each particle will be directly proportional to the geometrical polarization. In next section we will first focus on the potential energy as its derivation will naturally lead to an expression of the force field. Then a link with the geometrical polarization will be provided.

Energy and force field : general expressions

For the moment we have considered the tessellations from a geometrical point of view.

Numerous recent studies [START_REF] Farago | New conserved structural fields for supercooled liquids new conserved structural fields for supercooled liquids[END_REF][START_REF] Hentschel | Statistical mechanics of the glass transition as revealed by a voronoi tesselation[END_REF] have used them as a tool to recover information from the local neighborhood of particles in liquids. In what follows we propose to use the geometrical properties of Voronoi tessellation to define a brand-new class of liquids.

It has been shown for granular materials that an analogous Hamiltonian can be defined to describe the microstate of jammed matter in term of volume function. This volume function is equivalent to the Voronoi volume associated to each particle [START_REF] Song | Jamming i: A volume function for jammed matter[END_REF]. Starting from this observation we propose a new Hamiltonian to describe a system of N identical particles.

If we consider a function f which depends only on r = ||r|| where r is the vector starting from the particle i to a running point spanning the Voronoi cell as represented on figure

1.6, we can define the potential energy:

E P = γ 2 N i=1 v i d d rf (r) (1.8)
where v i is the volume of the voronoi cell associated to particle i and γ is an arbitrary constant. We can extend the formulation of the Hamiltonian for taking into account the disparity between particles. To do so, we work in the framework of the Voronoi-Laguerre tessellation where the distance function is not Euclidian. We consider the function r 2 -R 2 i :

E P = γ 2 N i=1 v i d d r(r 2 -R 2 i ) (1.9)
From equations 1.8 and 1.9 we see that the energy is the sum over a local term which is not a pairwise function. These new Hamiltonians are describing a system where interactions among particles are intrinsically many-body. We need to know the information on the positions of particle i and of all its nearest neighbors in order to define the Voronoi cell of particle i and its volume v i over which the integration is performed.

For both equations 1.8 and 1.9 we can show that -∇ j E p = F j where F j is the force acting on particle j. We find that for the monodisperse system the force is given by :

F j = γ 2 v j d d r∇f (r) (1.10)
Note that passing from equation 1.8 to equation 1.10 is not trivial. For the polydisperse case we find:

F j = γ 2 v j d d r∇(r 2 -R 2 j ) (1.11)
We have presented general equations for the potential energy and for the forces for a system with identical particles and also for a system where size disparities could be introduced. In what follows we give a specific form of the general function f defined in the monodisperse case. It enables on the one hand to deal with a quite simple expression of the potential energy and on the other hand to make a link between the geometrical polarization and the forces.

The monodisperse Voronoi liquid

In this section we focus our attention on a system, named the monodisperse Voronoi

E p = γ 2 N i=1 v i d d rr 2 (1.12)
where γ has the dimension of the energy divided by the length L d+2 . The potential energy E p is a sum of local, positive and non pairwise elements that can be seen as the moments of inertia of the polytopes with respect to the particles. Moreover the potential is ultrasoft, there is no excluded volume. Consequently superposing two particles has a finite cost in energy.

As the force derives from the potential energy, following 1.10 we can write F j = γ v j d d rr.

In this case the force is proportional to geometrical polarization as v j = d d r1 and

M j G j = d d rr/ d d r1.
Consequently the expression for the force is given by :

F j = γτ j = γ v j d d rr (1.13)
The interactions among particles are thus local, invariant by translation or rotation and non pairwise. They are also directly related to the inherent geometrical properties of the Voronoi tessellations.

The monodisperse Voronoi liquid has been described in details in [START_REF] Ruscher | The voronoi liquid[END_REF], in what follows we sum up its main properties. For sake of generality most of theoretical expressions are given for any dimension d. However all results coming from molecular dynamic simulations are performed in 3 dimensions with a modified version of LAMMPS that takes into account elements from the library Voro++ [START_REF] Rycroft | Voro++: A three-dimensional voronoi cell library in c++[END_REF]. The system is composed of

N =

Scaling relation for the potential energy and thermodynamics

The potential energy obeys a scaling relation as shown in equation 1.14. This scaling is similar to the one derived for liquids with inverse power law potential [START_REF] Schrøder | Hidden scale invariance in molecular van der waals liquids: A simulation study[END_REF][START_REF] Bailey | Statistical mechanics of roskilde liquids : Configurational adiabats specific heat contours, and density dependence of the scaling exponent[END_REF] at the difference that the power is positive for the monodisperse Voronoi liquid.

E p (λr 1 • • • , λr N ) = λ d+2 E p (r 1 • • • , r N ) (1.14)
This scaling relation has two direct consequences. The first one is that we can express easily the variation of the partition function in the case of a small dilatation and consequently find the pressure P which is given by

P = N k B T V - d + 2 d E p V (1.15)
A second consequence is that the partition function relative to the potential energy Z pot inherits the scaling properties. As a result all the thermodynamic observables can be expressed in term of a scaling function φ which depends on only one parameter the scaling variable x given by

x = k B T γ v -d+2 d (1.16)
We have the following expression for the excess free energy and the excess chemical potential :

F e (N, V, T ) = N k B T φ(x) = N v d+2 d γxφ(x) (1.17) µ e (N, V, T ) = ∂F e ∂N T,V = k B T φ(x) + d + 2 d xφ ′ (x) (1.18)
For a system in 3D, these two thermodynamic functions are represented in Figure 1.7. These scaling relations make the investigation of the space of state variable easier as when thermodynamic observables are known as a function of one of the state variable, the scaling variable x enables to express the result as a function of the other state variables.

For this system a unique relation between the excess chemical potential µ e and the pair correlation function g(r) which measures the probability that two particles are separated by distance r. In the limit where r → 0 it has been found that :

µ e = k B T ln(g(0)) = γv 5 3 x ln(g(0)) (1.19)
This relation, illustrated in Figure 1.7 (Right), is a direct consequence of the ultrasoft character of the potential energy since the superposition of two particles has a finite cost in energy. More precisely the energy of a system with N + 1 particles is the same as the one for a system with N particles in the specific case where r N +1 → r N .

lim

r N +1 →r N E p (r 1 , • • • , r N +1 ) = E p (r 1 , • • • , r N ) (1.20)
This property is only valid for this system and cannot be found for a system with pairwise interactions. However, equation 1.19 is reminiscent of the zero separation theorem derived for cavities in hard spheres [START_REF] Meeron | Statistical mechanics of hard particle systems[END_REF].

The thermodynamics of the monodisperse Voronoi liquid has been described. An unexpected link between the excess chemical potential and a microscopic observable, the pair correlation function, has been found. In what follows results on microscopic observables such as the structure factor or the mean square displacement are provided.

Microscopic observables: similarities with usual simple liquids

We look now at structural quantities such as the pair correlation function g(r) or the static structure factor S(k) to see to what extent the monodisperse Voronoi liquid is similar usual simple liquids. We recall first the definitions of these two quantities. The pair correlation function measures the probability that two distinct particles are separated by a distance r:

g(r) = V N 2 i,j/i =j δ(r -r ij ) with r ij = r j -r i (1.21)
Assuming the invariance by translation, the isotropy implies that the pair correlation depends only on r = |r|. Liquids are characterized by a pair correlation function structured at short range which results from a local order. Contrary to crystals the order is not preserved at larger distance and consequently g(r) → 1. Structure factor and pair correlation complement one another. In the reciprocal space the static structure factor measures the fluctuations of density:

S(k) = N -1 ρ(-k)ρ(k) with ρ(k) = N j=1 exp(-ik • r j ) (1.22)
With k the wavevector defined as k = 2π L (n x , n y , n z ). One of the main advantages of the structure factor is that it is experimentally accessible from X-Ray or neutron experiments [START_REF] Umberto | Dynamics of the Liquid State[END_REF][START_REF] Hansen | Theory of Simple Liquids[END_REF] and it is linked to the pair correlation function by the relation :

S(k) = 1 + N V drg(r) exp(-ik • r) (1.23)
Once again assuming isotropy in the system we can express the static structure factor as a function of k = |k|:

S(k) = 1 + 4π N V ∞ 0 dr r 2 g(r) sin(kr) kr (1.24)
Denoting σ the diameter of particles, for k ≫ 2π/σ the structure factor probes distances lower than the interparticle distance for which the pair correlation is close to zero and consequently S(k) → 1 when k → ∞. On the opposite, looking at very small values of the wavector, k ≪ 2π/σ, means probing the hydrodynamic limit. It can been shown

that the value of S(k) when k → 0 is related to the compressibility χ T [START_REF] Hansen | Theory of Simple Liquids[END_REF]:

S(k → 0) = N V k B T χ T (1.25)
The small values of S(k) when k → 0 generally observed for simple liquids are representative of the difficulties encountered for compressing the liquid. When k ∼ σ the structure factor brings information on the local environment of particles.

In Figure 1.8 is represented for several temperatures the pair correlation function (left)

and the static structure factor (right). The first peak of the pair correlation function informs on the location of the nearest neighbors. In this case they are located at r ∼ 1.1.

When temperature is decreased the pair correlation gets more and more structured but remains decorrelated at long distances. The different peaks represent the successive shells of neighbors. The distance between two of these shells is ∼ 0.9. The period of oscillations in the pair correlation function can be found in the position of the first peak of the static structure factor. This later is located at k max ∼ 7 which in real space gives a distance 2π/7.0 ∼ 0.9. For k → 0 the static structure factor has very small values, meaning that in this range of temperature the monodisperse Voronoi liquid is difficult to compress as it is often the case for usual simple liquids. Looking at the distance σ r below which g(r) ∼ 0 we observe that σ r is decreasing when the temperature increases.

This behavior results from the ultrasoft nature of interactions and is also observed in soft-core model such as Gaussian Core Model (GCM) [START_REF] Zachary | Gaussian core model phase diagram and pair correlations in high euclidean dimensions[END_REF]. Regarding all the observations on structural quantities the monodisperse Voronoi liquid behaves like usual simple liquids.

We now investigate the dynamical properties of this system by looking for instance at the mean square displacement which measures the correlation between the position of a tagged particle at time t = 0 and its position at time t. The mean square displacement (MSD) is given by

δr 2 (t) = [r i (t) -r i (0)] 2 (1.26)
At short time scale, the mean free path of the tagged particle can be seen as a ballistic motion an consequently δr 2 (t) ∝ t 2 . At larger time scale, the movement of the tagged particle appears as erratic and it can be described by a diffusive process. Consequently δr 2 (t) = 6Dt in 3 dimensions and therefore we can extract from long-time limit the diffusion coefficient D. In Figure 1.9 (left) is represented the scaled MSD for different temperatures.

We observed that at high temperature the ballistic regime is immediately followed by the diffusive regime and we can describe the MSD with the Enskog theory. Assuming that successive collisions are decorrelated one from each other leading to a Markovian process. Thus the velocity autocorrelation is given by an exponential law v(t)v(0) = (3k B T /m) exp(Γt) where Γ = T /D and consequently the MSD which is also given by the integral of the velocity autocorrelation becomes:

δr 2 (t) = 6D[exp(-Γt) -1 + Γt]/Γ (1.27)
In Figure 1.9 (left) the good collapse between points (circle in the graph) evaluated with equation 1.27 and the mean square displacement directly calculated with equation 1.26 (straight line in graph) suggests that at high enough temperature the Enskog description seems to apply to our system. However when looking at the asymptotic behavior of the diffusion coefficient a difference arises with the Enskog prediction which gives D ∼ (T ).

In the case of the monodisperse Voronoi liquid we find D ∼ T 5/2 (see insert in Figure 1.9 (right)). To understand this difference we use a kinetic description of the diffusion coefficient that is D ∼ v T ℓ mf p where v T is the thermal velocity and ℓ mf p the mean free path. We know that v T ∼ √ T and ℓ mf p = V /(N σ c ) ∼ v/R 2 with σ c = πR 2 the collision cross section and R the effective radius of particles. As a result D ∼ v (T )/R 2 .

Due to the soft core character of particles an expansion of effective radius in the high temperature limit gives R ∼ 1/T and thus as a result D ∼ T 5/2 .

When the temperature is lowered the emergence of an intermediate regime in the MSD, characterized by the presence of a plateau whose length increases as temperature decreases, suggests that the tagged particle is trapped into its cage of nearest neighbor.

The cage picture proposed by Eyring implies that the trapped particle has to overcome an energy barrier E A in order to escape the first shell of neighbors. The dynamic is thus activated and the diffusion coefficient can be expressed as the Arrhenian relation

D ∝ exp(-E A /k B T ).
This assumption can be verified when looking at the evolution of diffusion coefficient with temperature. In Figure 1.9 (right) is represented the logarithm of the scaled diffusion coefficient ln(Dγ -1/2 v -7/6 ) vs. 1/x. We observe the presence of an Arrhenian behavior for large value of 1/x (which means small values of T according to relation 1. [START_REF] Hanakata | Interfacial mobility scale determines the scale of collective motion and relaxation rate in polymer films[END_REF]) and an estimation of the energy barrier gives E A ∼ 4.95. As we mentioned before the mean square displacement describes different regimes regarding the time scale or the length scale observed. In the study of liquids it is the case for numerous time-dependent and we can defined these regimes by comparing length and time to respectively mean free path ℓ mf p and collision time τ c . Generally observables are defined in the space of wavelength k and frequency ω. Therefore by analyzing the value of products kℓ mf p and ωτ c we identify three types of regimes as illustrated in Figure 1.10.

1. The hydrodynamic regime when kℓ mf p ≪ 1 and ωτ c ≪ 1. The behavior of the liquid can be described by the equations of hydrodynamics.

2. The kinetic regime when kℓ mf p ≈ 1 and ωτ c ≈ 1. The distances probe at this scale are the ones of the molecular structure and the dynamic should be described by microscopic equations of motion.

3. The ballistic regime when kℓ mf p ≫ 1 and ωτ c ≫ 1. In this regime distances and time are so small that the particles move as if they were alone in the medium.

There are no correlation among them. In what follow our attention is restricted to the hydrodynamic regime which is characterized by slow variations of local properties on microscopic time and length scales.

Anomalous sound attenuation

When investigating the MSD, we notice that a relation exists between a microscopic quantity, the velocity autocorrelation, and a transport coefficient, the diffusion coefficient. This is one of the most famous example of Green-Kubo relations which in the context of linear response theory make the link between transport coefficient and integral of time correlation functions.

Time correlations play a central role in the description of liquids and it is therefore interesting to consider a general theoretical framework to express time correlation functions for different wavelengths and frequencies.

In what follow we investigate the hydrodynamic limit of the intermediate scattering function F (k, t) which is defined as the time correlation of the fluctuation of the density δn.

F (k, t) = 1 N δn(-k, 0)δn(k, t) (1.28)
In the hydrodynamic limit this time correlation function is given by:

F (k, t) F (k, 0) = c V c P ε exp(-D T k 2 t) + exp(-k 2 Γt)[cos(v s kt) + bk sin(v s kt)] (1.29)
where c P = C P /N and c V = C V /N are the heat capacity at respectively constant pressure and constant volume, ε is related to both of heat capacities by the following relationship:

ε = c P c V -1 (1.30)
The thermal diffusivity D T implied in the first exponential in equation 1.29 is related to the thermal conductivity via :

D T = κ ρc P (1.31)
In the second exponential appears the sound attenuation Γ which is linked to the thermal diffusivity and longitudinal viscosity η L :

Γ = D T ε 2 + η L 2ρ with η L = 4 3 η + ξ (1.32)
where η is the shear viscosity and ξ is the bulk viscosity. Finally the parameter b in equation 1.29 is given by :

b = 3Γ -η L /ρ (ε + 1)v s (1.33)
In equation 1.29 we observe that relaxation of density fluctuation is made through :

1. diffusion of energy as it is suggested by the presence of the thermal diffusivity D T 2. propagation of acoustic waves which are damped by the combined effect of viscosity and thermal conduction (see equation 1.32)

Assuming that we can neglect the thermal contributions, we can rewrite 1.29 as :

F (k, t) F (k, 0) ≃ exp(-k 2 Γt)[cos(v s kt) + Γ v s k sin(v s kt)] (1.34)
where Γ = η L /(2ρ). Consequently we define the spectrum of density fluctuations:

S(k, ω) S(0) ≈ 1 2π Γk 2 (ω + v s k) 2 + (Γk 2 ) 2 + Γk 2 (ω -v s k) 2 + (Γk 2 ) 2
(1.35)

In the hydrodynamic limit, Γ is expected to be constant, i.e it will not depend on the wave vector k. To test the validity of this hypothesis for the Voronoi liquid, we try to express S(k, ω)/S(k) as a scaling function. For the purpose of the scaling we define

δω = ω -ω max = ω -v s k. S(k, ω) S(k) = 1 2π Γk 2 (ω -vsk) 2 + (Γk 2 ) 2 = 1 2π Γk 2 (δω) 2 + (Γk 2 ) 2 = 1 2π 1 k 2 Γ (δω/k 2 ) 2 + Γ 2 (1.

S(k, ω)

S(k) = 1 2π Γk 2 (ω -vsk) 2 + (Γk 2 ) 2 = 1 2π Γ 0 k (ω -vsk) 2 + Γ 2 0 k 2 = 1 2π 1 k Γ (ω/k -vs) 2 + Γ 2 (1.37)
Thus in this case kS(k, ω)/S(k) is simply a function of ω/k and it is represented for different k in Figure 1.11. We notice the good collapse of the different curves meaning that for the Voronoi liquid Γ is apparently linear in k.

We gave evidences that the sound attenuation Γ -1 scales as k and not as k 2 as expected.

The explanation of this behavior is rather complicated and a detailed analysis can be found in [START_REF] Ruscher | Anomalous sound attenuation in voronoi liquid[END_REF].

In what follow descriptions of the metastable zone and of the crystalline phase, which appears below T = 1.05, are provided.

k 2 S(k, ω)/S(k) vs. [ω -ω max ]/k 2 1.3.

Metastability and crystalline phase

The presence of a liquid/crystal transition is revealed by a discontinuity in the first derivatives of the thermodynamic potentials. Consequently to probe its existence the system has been progressively cooled down than heated up at two different cooling rates denoted as fast and slow and having for values k f = dT /dt = 2.5 • 10 -4 and k s = dT /dt = 2.5 • 10 -5 . Then the potential energy per particle has been calculated as a function of temperature as represented in Figure 1.12. In both cases (fast in grey and slow in orange) jumps in the value of E p /N informs on the presence of a first order transition and marks boundaries of the metastable regime. In the same figure are represented mean values of the potential energy per particle evaluated on isotherms.

The black and blue crosses represent E P /N for systems of respectively N = 8000 and N = 8192 particles. The choice of N = 8192 is motivated by the fact that this number allows a matching between the box size and the b.c.c crystal which is the crystalline structure of the monodisperse Voronoi liquid.

It is interesting to notice that the slower the process the larger the hysteresis and the greater the value of E P /N . However the slow process (in orange in the graph) seems to be well equilibrated as the data points from isotherms collapse on the curve. This result is surprising because one could imagine that by leaving more time to relax, the system would have pass from the liquid to the solid phase at higher temperature. An explanation for this phenomenon is still lacking. Metastability affects both structural and dynamical quantities. In what follows we briefly compare results obtained for isotherms in the liquid phase (black crosses) with isotherms in the crystalline phase (blue crosses). We first look at the pair correlation function which is represented for three different temperatures in Figure 1.13. While in the liquid phase (in blue) the pair correlation function is slightly more structured upon cooling, in the crystalline phase (dashed red lines) it get is already well structured at T = 1.85 and when T is lowered, peaks are getting sharper and long-range correlation are increasing. The MSD is also sensitive to metastability as it is observed in Figure 1.14. When looking at data taken from isotherms in the crystalline phase, the ballistic regime is followed by a regime, characterized by a plateau, where particles vibrates around their equilibrium position. Moreover we notice that the size of the plateau increases upon cooling until reaching a temperature where no diffusion takes place during the time observed. When investigating the mobility within the crystalline phase in the metastable region we notice that particles jumps from site to site leaving the whole structure unchanged.

The jumping process is Arrhenian and particles need to overcome an activation energy E A in order to move.

Cooling then heating the system at different rates enabled to reveal a metastable zone in a range of temperature 1.05 ≤ T ≤ 1.85. The precise value of the melting temperature has been found by thermodynamic integration and is equal to T m ∼ 1.65.

Whereas many monodispere Lennard-Jones liquids crystallize into face centered cubic (FCC) crystals [START_REF] Sosso | Crystal nucleation in liquids: Open questions and future challenges in molecular dynamics simulations[END_REF][START_REF] Jungblut | Crystallization on prestructured seeds[END_REF], the crystalline phase of the monodisperse Voronoi liquid is a body centered cubic (BCC) lattice. This can be explained by the fact that the sphere minimizes the local potential energy e

(i) p = (γ/2) v i d 3 r 2 .
The Voronoi cell associated to the FCC lattice is the rhombic dodecahedron (12 faces) whereas the Voronoi cell associated to the BCC lattice is the truncated octahedron (14 faces). By comparing these two structures (see Figure 1.15) we see that the truncated octahedron has more faces than de rhombic dodecahedron and consequently is closer to a spherical structure.

The energy is therefore minimized by the truncated octahedron structure. To sum up, the monodisperse Voronoi liquid is a system of point particles whose interactions directly expressed via the inherent geometrical properties of classical Voronoi tessellation are intrinsically manybody. This system presents a scaling of the potential energy whom partition function inherits. As a result all the thermodynamic observables in the system can be expressed as a function of a scaling variable. The ultrasoft character of the potential energy enables to derive an unusual relation between the excess chemical potential and the pair correlation function thus providing a link between microscopic structural quantities and macroscopic observables. On the structural point of view, the Voronoi liquid shares similarities with soft-core potential liquids such as GCM.

The dynamics of the system is also similar to what has been observed for simple liquids.

However in the range of temperature 1.05 ≤ T ≤ 2.00 the relaxation of the correlation of density fluctuation in the hydrodynamic limits shows that the sound attenuation does not scale as k -2 but as k -1 . In the same range of temperature, metastability is observed and depending on the initial configuration the system is either liquid or crystal. At enough high temperature the crystalline phase is relatively mobile and it is interesting to notice that jumps of particles occur also below the melting point T m ∼ 1.65 Finally below T = 1.05 the Voronoi liquids crystallizes into a BCC crystal.

The monodisperse Voronoi liquid is thus a model with exotic features that could reveal some interesting information when probing glass transition. Therefore in order to avoid crystallization, we propose an extension of this model by using Voronoi-Laguerre tessellations where size disparities are introduced among particles. Firstly a general description of the polydisperse Voronoi liquid is proposed in next section, then we introduce the selected model, a binary mixture, whose study is the aim of this work.

The polydisperse Voronoi liquid

We define now an extension of the monodisperse Voronoi liquid, named the polydisperse Voronoi liquid, where size disparity among particles is introduced. As mentioned before to take into account the polydispersity in the system, Voronoi-Laguerre tessellations are required. Assuming a box of length L such as V = L d we consider N point particles to which is associated an intrinsic length R i that we call radius for convenience. Following equation 1.9, the potential energy is

E P = γ 2 N i=1 v i d d r(r 2 -R 2 i + R 2 ) (1.38)
where R 2 is the square of the mean radius and is given by

R 2 = N -1 N i=1 R 2 i .
Adding the square value of the mean radius ensures to recover the monodisperse Voronoi liquid when all the radii are equal. It is interesting to notice that the potential energy can be decomposed into the sum of two terms

E P = E (m) P + E (a)
P where E (m) P is defined as:

E (m) P = γ 2 N i=1 v i d d rr 2 (1.39)
and the second term can be written as

E (a) P = γ 2 N i=1 v i d d r(R 2 -R 2 i ) = γ 2 N i=1 v i (R 2 -R 2 i ) (1.40)
Equation 1.11 gives the expression of the force exerted on particle j which is defined as:

F j = γτ j = γ v j d d rr (1.41)

The bidisperse Voronoi liquid

The polydispersity could be described either by discrete values or by a continuous distribution of the radii sizes. In what follows and all through this work we deal with a binary mixture meaning that intrinsic lengths R A and R B are associated to respectively N A and N B particles. We chose arbitrarily that R A > R B . The choice of the size ratio x = R B /R A and the fraction of A and B particles (respectively N A /N and N B /N ) are of great importance regarding the ability of the system to be a reasonably good glass former. It has been shown for hard spheres that for a sufficiently marked size disparity, typically a size ratio x 0.6, an increase of the percentage of small particles leads to a speed up of the dynamic. On the contrary when the size disparity is not so pronounced x 0.8, increasing the fraction of small particles leads to a slowing down of the dynamic [START_REF] Foffi | Mixing effects for the structure relaxation in binary hard-sphere liquids[END_REF]. As this last effect is the key feature of glass transition we decide to follow what has been found for hard spheres and consequently the size ratio is fixed at x = 0.83 and the fractions of small and large particles are taken equal, N A /N = N B /N = 0.5.

We notice that polydispersity is governed by only one parameter denoted η and given by:

η = R 2 A -R 2 B v 1/3 (1.42) If R A = R B , η = 0
R A = ηv 1/3 √ 1 -x 2 and R B = xR A (1.43)
To test which numerical values of η are appropriate to define properly the system in the range of temperature of interest, i.e T ≤ 2.00, we perform cooling processes for different values of η, 0.20 ≤ η ≤ 0.70. More details about the molecular dynamic simulation will be given in next subsection but for the cooling processes, we worked with N = 1000

particles at v = V /N = 1.
We start from equilibrate configurations at T start = 2.00 and we progressively decrease the temperature until we reach the final temperature

T stop = 0.1.
The cooling rate is dT /dt = 8 • 10 -5 . We measure the value of the energy per particle E P /N as a function of temperature T for 0.1 ≤ T ≤ 2.00.

Results are presented in Figure 1. [START_REF] Hanakata | Interfacial mobility scale determines the scale of collective motion and relaxation rate in polymer films[END_REF] where we observe that for :

• η = 0.20 a marked discontinuity in the curve of Ep/N (T ) is present. A first order transition occurs, the system crystallizes for T ∼ 0.95

• η ∈ [0.25; 0.60] no discontinuities are present in the curve of Ep/N meaning that first order transition is prevented. In this range of η, systems falls out of equilibrium for T < T G where T G is the glass transition temperature.

• When the value of η is increased, the value of the total energy per particle E P /N decreases. We notice however that for η ∈ [0.20; 0.30] the values of E P /N are closed from the one measured for the monodisperse system as observed in Figure 1.12 meaning that the polydispersity can be seen as a small perturbation of the monodisperse system. • For η = 0.70 the cooling process has been done between T start = 4.00 and T stop = 0.1 as it appears that crystallization occurs for T ∼ 2.60. Thus it exists a limiting value of η upper which systems crystallizes again. This can be explain by the fact that when η is increased, the intrinsic lengths of particles A and B are also increased. By keeping the density constant the probability of having small particles sharing Voronoi cell of large particles is also increased as it confirms by the representation in Figure 1.17 (Top) of the partial pair correlation function g AB (r) which measures the distance between particles A and B. In the inset the partial pair correlation is represented for a system at η = 0.70 in the liquid phase at T = 4.00. We notice that there is a nonzero probability for finding particle in close "contact" as it appears that g AB (r) → 0.45 when r → 0. Even in the crystalline phase, at T = 2.00, the pair correlation tends to a nonzero value when r → 0. On the contrary at the same temperature, for systems with η ≤ 0.60 which are in the liquid phase, we observe that particles A and B are well separated.

To go a step further, for η = 0.70, we have computed the distribution of volumes of the Voronoi cells v B and v A of respectively small and large particles. Results are shown in Figure 1.17 (Bottom). We notice that the distribution associated to large particle is a gaussian well-centered around its mean value v A ∼ 1.91

whereas the distribution associated to small particles shows a huge probability for having small volumes. In the inset we see that the distribution is well described by a power law P (v) ∼ v -0.6 when v → 0. Therefore a substantial fraction of small particles have a volume v B ∼ 0. The absence of cells for these particles leads to a zero contribution in term of potential energy and forces. Consequently these particles are "ghost" and the system behaves as there were only one type of particles and we recover the monodisperse Voronoi liquid. Regarding the previous results, we chose η = 0.375 as when looking at the potential energy of the cooling process, it remains on the first hand sufficiently far from the one of the monodisperse liquids for expecting that crystallization would be avoided even at slower cooling rate. On the other hand E P /N is sufficiently close from the one of the monodisperse to see the polydispersity as a weak perturbation. This point of view is reinforced by the fact the partial pair correlation function g AB (r) is not strongly varying when η goes from 0.

Molecular dynamics (MD) simulations

In molecular dynamic simulations, the motion of particles is described by the Verlet algorithm [START_REF] Verlet | Computer "experiments" on classical fluids. i. thermodynamical properties of lennard-jones molecules[END_REF]. The position r j (t + ∆t) of particle j at time t + ∆t is given by :

r j (t + ∆t) = 2r j (t) -r j (t -∆t) + F j m ∆t 2 + O(∆t 4 ) (1.44)
where F j is the force exerted on particle j and it corresponds to the one defined in equation 1.41. It requires the knowledge of geometrical polarization τ (see equation 1.2) meaning that we need to have access to the centroid and the volume of each Voronoi cell. To this purpose we use a modified version of LAMMPS [START_REF] Plimpton | Fast parallel algorithms for short-range molecular dynamics[END_REF] interfacing with the Voro++ library [START_REF] Rycroft | Voro++: A three-dimensional voronoi cell library in c++[END_REF]. This library enables to perform efficiently Voronoi tessellation on 3D systems and so far to get access to plenty of geometrical elements relative to Voronoi tessellation.

The dynamic is correctly describe by the Verlet algorithm if the term ∆t 4 is small enough to be considered as negligible. The choice of the timestep ∆t in MD is therefore not so trivial. It has to be chosen by taking into account that a too large timestep could imply difficulties for well-equilibrated the system whereas a timestep too small would lead to an important increase of calculation time. A way to determine the limit upper which ∆t would be too big to describe accurately the dynamic, is to search for the characteristic time of the system τ c by nondimensionalizing equation 1. [START_REF] Foffi | Mixing effects for the structure relaxation in binary hard-sphere liquids[END_REF]. By assuming the the characteristic length in the system is v 1/d we can define r * = r/v 1/d . We need now to nondimensionalize the force. To this purpose we consider equation 1.41:

F j = γ v j d d rr r=v 1/d r * = γv d+1 d v * j d d r * r * = γv d+1 d F * j (1.45)
Now we can rewrite equation 1.44 by dividing each term by v 1/d :

r * j (t + ∆t) = 2r * j (t) -r * j (t -∆t) + γv F * j m ∆t 2 + O(∆t 4 ) (1.46)
As the last term in equation 1.46, γvF * j /m∆t 2 should have no dimension, we conclude that the characteristic time is given by :

τ c = m γv (1.47)
As the definition of the force field is the same for monodisperse and polydisperse systems (see equation 1.13 and 1.41), the characteristic time remains the same for both models.

We arbitrarily chose to work with γ = 1000, m = 1 whatever the type of the particle and at v = 1. In this case it implies that τ c = 1/1000 ≈ 0.03.

The term of order 4 in equation 1.44 can be neglected if ∆t/τ c ≪ 1. For all MD simulations done in this work we chose the timestep ∆t = 0.001. Thus the ratio ∆t/τ c ≈ 0.03 is small enough to say that terms of order 4 are negligible and that the Verlet algorithm may describe accurately the dynamic of particles.

In Chapter 4 we will see that the reliable characteristic time Ω -1 0 is associated to the Einstein frequency and is represented in Figure 4.15. As Ω -1 0 ≈ 0.05, we see that the choice of our timestep is well-adapted as ∆t/Ω -1 0 ≈ 0.02

Up to now we have defined a characteristic length v 1/d , a characteristic time τ c , for completing the description we need to provide a characteristic potential energy for the polydisperse Voronoi liquid. Starting for instance from equation 1.38 we can nondimensionalize the potential energy:

E P = γ 2 N j=1 v j d d r(r 2 -R 2 j + R 2 ) = γ 2 v d+2 d N j=1 v * j d d r * (r * ) 2 -(R * j ) 2 + (R * ) 2 = γ 2 v d+2 d E * P (1.48)
We notice that γv

d+2 d
has the dimension of an energy. Furthermore we recall that we consider the polydispersity as a perturbation of the monodisperse system that's why as for as the monodisperse Voronoi liquid, assuming that γ = 1000, the characteristic potential energy E v is defined as

E v = 0.001γv d+2 d (1.49)
where the factor 10 -3 was chosen to work with temperatures of the order of the unit.

To sum up we propose to study a reference binary mixture such as:

• The total number of particles is N = 1000 and N A = N B = 500

• The density is kept constant meaning v = V /N = 1

• The system is designed in such a way that crystallization should be avoided : η = 0.37 ⇒R A = 0.6729 and R B = 0.5585

• Most of the work is done in the NVT ensemble and a wide range of temperatures is investigated via MD simulations with Nosé-Hoover thermostat.

We defined the bidisperse Voronoi liquid. In Figure 1. [START_REF] Hanakata | Interfacial mobility scale determines the scale of collective motion and relaxation rate in polymer films[END_REF] we have seen that the system seems to be able to avoid crystallization at least for a cooling rate of dT /dt = 8 • 10 -5 . In Chapter 2, we will test slower cooling rates to check the ability of the bidisperse Voronoi liquid to avoid nucleation. Then, we will investigate its thermodynamic properties. When a material is cooled down, under certain conditions, it may avoid crystallization and enters into a metastable state named the supercooled regime. If one keeps on cooling, a temperature is reached below which the liquid falls out of equilibrium and becomes an amorphous solid [START_REF] Leuzzi | Thermodynamics of the Glassy State[END_REF]. This temperature at which this occurs, the ergodicity breaking, is defined as the glass transition temperature T G . It is important to stress that this temperature does not correspond to a universal critical temperature of a thermodynamic phase transition. Actually T G depends on the cooling rate and on the preparation protocol [START_REF] Leuzzi | Thermodynamics of the Glassy State[END_REF]. The glass transition temperature, defined as the temperature for which relaxation times are in the range of 100 -1000s, reflects the time limitation of experiments. Indeed below T G the system needs a very long period of time to relax and therefore the material appears as solid on timescale accessible experimentally. However if one would have an infinite time, one could observe the flow of the material.

For a given material, its ability to form a glass depends on the probability to avoid the onset of crystallization upon cooling, i.e to avoid nucleation [START_REF] Das | Statistical Physics of Liquids at Freezing and Beyond[END_REF]. Whatever the technique used to manufacture the glassy material, the underlying principle is to quench sufficiently rapidly the material from its liquid state to be sure that crystallization is avoided [START_REF] Fecht | Thermodynamic properties of amorphous solids -glass formation and glass transition (overview)[END_REF][START_REF] Zhang | Thermochemistry of glasses along the 2ndalo3-3sio2 join[END_REF][START_REF] Skinner | Novel behaviour and structure of new glasses of the type ba-al-o and ba-al-ti-o produced by aerodynamic levitation and laser heating[END_REF][START_REF] Ranasinghe | Containerless processing of a lithium disilicate glass[END_REF]. However it was reported that the cooling protocol directly influences the physical properties of the resulting amorphous material [START_REF] Zhang | Thermochemistry of glasses along the 2ndalo3-3sio2 join[END_REF][START_REF] Skinner | Novel behaviour and structure of new glasses of the type ba-al-o and ba-al-ti-o produced by aerodynamic levitation and laser heating[END_REF].

Chapter 2. Thermodynamic properties

33

The influence of the cooling rate is directly visible on thermodynamics accessible from experiments, whereas the signature of the ergodicity breaking is visible on derivative of thermodynamics observables, such as the heat capacity [START_REF] Fecht | Thermodynamic properties of amorphous solids -glass formation and glass transition (overview)[END_REF][START_REF] Zhang | Thermochemistry of glasses along the 2ndalo3-3sio2 join[END_REF][START_REF] Angell | Thermodynamics of the glass transition: empirical aspects[END_REF][START_REF] Angell | Thermodynamic aspects of the glass transition in liquids and plastic crystals[END_REF]. Thermodynamic observables are thus of huge importance to probe the neighborhood of glass transition and to get more physical insight about what happens when the material is cooled towards

T G .
In Chapter 1, we have seen that depending on the value of η (defined in equation 1.42) the system is able to avoid crystallization. This latter is indeed avoided for η ∈ [0.25, 0.60].

Therefore the choice of η = 0.375 has appeared reasonable to study the supercooled regime of the bidisperse Voronoi liquid. On the one hand as one should not be concerned by crystallization. On the other hand η = 0.375 should be enough small to enable to work in the limit of weak bidispersity meaning that the polydispersity might be seen as a perturbation of the monodisperse case that would thereby remain a state of reference.

In this chapter we will first check the ability of the bidisperse Voronoi liquid to avoid crystallization by performing cooling processes at very slow cooling rates. This will also gives us a rough estimation of T G . In a second time, we will focus on the study of the thermodynamics of the system. Excess quantities with respect to the ideal gas state will be computed in a range of temperature starting from T = 2.00 and going to the smallest equilibrated temperature T = 0.83. A special interest will be given to the pressure as we will see that it is strongly negative revealing the fact that the bidisperse Voronoi liquid is always under tension. However we will see that the nature of the interactions keeps the system stable preventing cavitation phenomenon to occur.

Cooling processes

In order to test the ability of the bidisperse Voronoi liquid to avoid crystallization, we perform cooling processes at different cooling rates k = dT /dt. For η = 0.375 the cooling process has been done at the rate k = 10 -4 and the crystallization is prevented.

Therefore the cooling rates are progressively decreased to see whether nucleation is avoided. As a matter of fact, by leaving more time to the system to relax upon cooling it might be able to rearrange and find a crystalline order.

Before giving more details on the cooling procedure, it is important here to highlight the fact that our cooling processes are performed in the N V T ensemble, whereas most of simulations focusing on cooling processes have been done in the N pT ensemble to reproduce the same protocols as in experiments performed at constant pressure. [START_REF] Fox | Molecular dynamics simulations of a supercooled monatomic liquid and glass[END_REF][START_REF] Vollmayr | How do the properties of a glass depend on the cooling rate? a computer simulation study of a lennardjones system[END_REF][START_REF] Buchholz | Cooling rate dependence of the glass transition temperature of polymer melts: Molecular dynamics study[END_REF][START_REF] Kreck | Characterization of the glass transition of water predicted by molecular dynamics simulations using nonpolarizable intermolecular potentials[END_REF]. Most of time glass transition is experimentally tracked by dilatometry which follows the variation of specific volume as temperature is decreased, or by differential scanning calorimetry (DSC) which measures the difference of the amount of heat between the sample analyzed and a sample of reference. This method enables to determine the enthalpy and thereafter the heat capacity at constant pressure [START_REF] Angell | Viscosity-temperature function for sorbitol from combined viscosity and differential scanning calorimetry studies[END_REF][START_REF] Hutchinson | Determination of the glass transition temperature -methods correlation and structural heterogeneity[END_REF]. In our case, we monitor the evolution of the internal energy and more precisely we focus on the evolution with temperature of the potential energy per particle

E P /N k = 2 • 10 -4 k = 8 • 10 -5 k = 5 • 10 -5 k = 2 • 10 -5 k = 8 • 10 -6 k = 5
E P /N for different cooling rates k = dT /dt ∈ {5 • 10 -6 ; 2 • 10 -4 }.
The choice of the slowest cooling rate comes from "experimental" time limitation as it requires approximately one month of simulation on 16 cores (Intel(R) Xeon(R) CPU E5-2680 v4, 2.40GHz) to obtain the MD trajectory. Going to slower cooling rates would be too time consuming regarding the purpose of the simulation.

In Figure 2.1 are represented the different results for E P /N vs. T where we have performed moving averages on E P /N in order to suppress fluctuations.

• The main observation is the absence of discontinuity in the different curves of E P /N vs. T meaning that crystallization is avoided even for the slower cooling rate.

• We also remark the superimposition on the same curve of all the liquid branches.

This can be explained by the fact that the cooling rates investigated are slow enough to represent the liquid equilibrium. This superimposition has also been observed for the enthalpy as noticed by Vollmayr et .al who investigated the effect of cooling on the properties of the Kob-Andersen mixture [START_REF] Vollmayr | How do the properties of a glass depend on the cooling rate? a computer simulation study of a lennardjones system[END_REF].

• Below a given temperature, the glass transition temperature T G , a change in a slope occurs and this change is related to the fact that below T G the system falls out of equilibrium and becomes a glass. The inset of Figure 2.1 shows that the slower the cooling rate, the lower the potential energy in the glassy branch. An explanation can be found by considering the free energy landscape of the system.

When it is rapidly quenched, system has no time to relax and thus to explore the free energy landscape. It can only visit shallow minima. However when more time is given to relax, the system is able to rearrange and to explore lower minima of the free energy landscape.

As mentioned above, if one searches numerically for a precise value of T G , one should consider working in the N pT ensemble. However in our case performing N pT simulations was not possible at this stage of the work as the expression of the pressure tensor, which relies on the knowledge of the stress tensor, is rather complicated as we are going to see in the last section of this chapter.

Nevertheless in the rest of this work, we will consider thermalized configurations obtained from N V T simulations therefore having an idea about the localization of T G , even roughly, is helpful to have, in a first phase, one temperature of reference with which one can rationalize our observations. Of course as observed, T G depends on the cooling rate, and we need to investigate its evolution as function of the cooling rate k. To this purpose we can first notice that when T → T G we can fit linearly both liquid and glassy branches.

The intersection of the two lines gives the value of T G . This method is illustrated in Figure 2.2 where the glass transition temperature is estimated for the slowest cooling rate.

In Table 2.1 are recorded the values of T G for the other cooling rates where we notice that T G decreases when the cooling rate becomes slower. Vollmayr et .al also observed that the glass transition temperature of the Kob-Andersen mixture decreases while decreasing the cooling rate. However they observe that below a certain value of k, the decrease is much more slower [START_REF] Vollmayr | How do the properties of a glass depend on the cooling rate? a computer simulation study of a lennardjones system[END_REF]. We notice the same as in the range of cooling rates investigated, the decrease is relatively weak as the value of T G changes only by 3% when the cooling Chapter 2. Thermodynamic properties 36 rate is lowered by a factor 10. This means that in the range of cooling rates accessible, the glass transition temperature is weakly sensitive to the speed of the cooling processes.

Thermalized equilibrated configurations would correspond to a k = 0 cooling rate.

Therefore it makes sense to select the value of T G obtained with the lowest value of k and thus for the remaining work we will consider that T G ∼ 0.66. We have confirmed that the bidisperse Voronoi liquid is able to avoid crystallization and is consequently a good model to probe the behavior of observables near the neighborhood of glass transition. We estimated the value of the glass transition temperature to be T G ∼ 0.66 for the range of cooling rates accessible numerically.

In what follow we investigate the principal thermodynamic observables of the bidisperse Voronoi liquid to see how they behave upon cooling.

Thermodynamic observables 2.2.1 General expressions

For the monodisperse Voronoi liquid the scaling relation for the partition function has enabled us to express easily all the thermodynamic observables. Such a scaling relation does not exist for the bidisperse Voronoi liquid making thus more complicated to describe the system for the whole space of state variables. Before computing the thermodynamic observables, we define them and we focus on their excess part with respect to the ideal gas.

For the bidisperse Voronoi liquid it is possible to write the free energy

F e (V, T, N, N 1, R 2 A - R 2 B )
as the function of only 3 dimensionless intensive parameters:

F e (V, T, N, N 1, R 2 A -R 2 B ) = F e (x, η, α A ) = N v 5/3 γxφ(x, η, α A ) (2.1)
where the φ is non-dimensional function of variables x, η and α A defined as follow:

x = T γv 5/3 η = R 2 A -R 2 B v 1/3 = η v 1/3 α A = N A N (2.2)
Consequently by taking the first two derivatives of the excess free energy F e it is possible to have access to the excess part of the usual thermodynamic observables.

Excess entropy S e

The excess entropy corresponds to the first derivative of the excess free energy with respect to temperature.

S e (x, η, α A ) = - ∂F e (x, η, α A ) ∂T N,η,α A = - ∂F e (x, η, α A ) ∂x ∂x ∂T η,α A = -N φ(x, η, α A ) -x ∂φ(x, η, α A ) ∂x η,α A (2.3)

Excess internal energy U e

Excess internal energy is directly accessible from simulations as it is directly related to the average of the potential energy. On the other hand, to compute U e from F e we use the Helmholtz formula :

U e (x, η, α A ) = E P (x, η, α A ) = F e (x, η, α A ) -T ∂F e (x, η, α A ) ∂T = -N T x ∂φ(x, η, α A ) ∂x η,α A (2.4)
This last equation is interesting as it provides a direct link between the first derivative with respect to x of the unknown function φ and the numerical data E P .

Therefore one should be able to determine from the integration of E P , the x dependence of φ.

Excess pressure P e

The excess pressure is the first derivative of the free energy with respect to the volume:

P e = - ∂F e ∂V = - ∂F e ∂x ∂x ∂V - ∂F e ∂η ∂η ∂V (2.5) = - 5 3
U e N + T 3 η ∂φ ∂η (2.6)
The second term of the R.H.S of equation 2.6 describing the excess pressure can be rewritten as function of η defined in equation 2.2:

η ∂φ ∂η = η ∂φ ∂ η ∂ η ∂η = η ∂φ ∂ η (2.7)
The function φ is given by φ

= -1 N ln Zpot V N
where Z pot is the part of the partition function relative to the potential energy and is defined by:

Z pot (T, η, α A ) = d 3N r exp -E P (r; η) k B T (2.8)
Here, and more generally all through this work, k B = 1.

Thus the derivative of φ versus η is related to the average value of the derivative of potential energy versus η assuming that the positions of particles are kept constant.

∂φ ∂ η = 1 N T 1 Z pot d 3N r ∂E P (r; η) ∂ η exp -E P (r; η) k B T = 1 N T ∂E P ∂ η r (2.9)
It can be shown that, when the positions of particles are kept constant, the derivative of the potential energy with respect to η is expressed as:

∂E P ∂ η r = 2E (a) P η = γ η (V B α A -V A α B ) (2.10)
Therefore equation 2.10 provides a rather simple expression for E (a)

P :

E (a) P = γ 2 η2 (V B α A -V A α B ) (2.11)
Finally using equation 2.10 in equation 2.9 and inserting the final result in equation 2.6 allows to express the excess pressure P e , as a function of the average values of the two components of the potential energy:

P e v = - 5 3 E (m) P N - E (a) P N (2.12)

Excess chemical potential relative to A particles µ e,A

Finally the excess chemical potential is obtained as follow:

µ e,A (x, η, α A ) = ∂F e ∂N A T,V,N 2 = ∂F e ∂x ∂x ∂N A T,V,N 2 + ∂F e ∂η ∂η ∂N A T,V,N 2 + ∂F e ∂α A ∂α A ∂N A T,V,N 2 = 5 3 T x ∂φ ∂x + η 3 T ∂φ ∂η + T φ + α B T ∂φ ∂α A = P e v + T φ + α B T ∂φ ∂α A = µ e,B + T ∂φ ∂α A (2.13)
As for the monodisperse Voronoi liquid, a "zero-separation" theorem exists for the bidisperse system. For instance we can obtain a configuration of N A -1 particles of type A by superimposing two particles A on top of each other. On top of this possibility, two other situations must be take into considerations: i) superimposing two particles of type B and ii) superimposing particle of type A with a particle of type B.

(i) In the case where the two superimposed particles are of the same type:

µ e,α = k B T ln(g aa (r = 0)) where a ∈ {A, B} (2.14)
g αα (r) is the partial pair correlation function evaluated on particles of the same type defined in Chapter 3 via equation 3.1.

(ii) When particles A and B are on top of each other, the smaller particle acts as a ghost particle and has an empty cell as the Voronoi-Laguerre tessellation is entirely determined by the larger particle. We recall that R A > R B leading to a configuration with N B -1 particles and as a result:

µ e,B = k B T ln(g AB (r = 0)) (2.15)
The two previous relationships are interesting while working at high temperature.

In our case as we limit our attention to low temperatures they are of little use the repulsion between particles prevents from being in contact, i.e having cases where

g ab (r = 0) = 0.
Chapter 2. Thermodynamic properties 40

We derived analytical expression for the main excess thermodynamical observables.

Their computation will be done in what follow but prior to that we verify that the mixing is homogeneous, i.e there is no phase separation between the small and the large particles.

Strong mixing

As we are working with a binary mixture the question of the demixing naturally arises.

We wonder if there is a possibility that upon cooling the two components of the mixture separate to form two distinct phases at it is illustrated in the left picture of Figure 2.3.

In this case, apart from the interface (grey dashed line) each phases are identical to a monodisperse configurations. Indeed particles are of the same type and Voronoi-Laguerre tessellation are equivalent to classical Voronoi tessellation where radii are equal. The energetic contribution of the interface is ∝ L 2 and is thus negligible in the thermodynamics limit where bulk contributions are predominant. is F e when η = 0 and thus corresponds to the excess free energy of the monodisperse system. Equations 2.9 and 2.10 play a central role as the polydispersity contribution is carried by the term E (a)

P in the potential energy. Therefore we can write:

∆F mix e = N T η 0 dη ′ ∂φ ∂η ′ = -γv 1/3 η 0 dη ′ η ′ V A α B -V B α A = -γv 1/3 η 0 dη ′ η ′ ( v A -v) (2.16)
As the average volume of large particle is larger than the average volume per particle, i.e v A > v the integral in equation 2.16 is always positive leading to ∆F mix e < 0 and as a result the mixing state is always more stable than the state where two phases may coexist. We conclude that demixing is always avoided in the bidisperse Voronoi liquid.

Computing thermodynamical observables

Theoretical expression of the thermodynamic observable have been derived above and in this section we compute the different excess quantities to how they behave upon cooling and see how the neighborhood of glass transition may affect them.

Excess free energy

We start logically by computing the excess free energy as it allows the determination of all the other observables. We restrict our attention to the temperature interval T ∈ [0.83; 2.00] and we perform a thermodynamic integration.

The thermodynamics of the monodisperse Voronoi liquid has been studied previously during my master thesis and we thus consider it as the state of reference for thermodynamic integration. We start from the reference point (T = 2.00, η = 0.00) and we integrate, with respect to η, the excess free energy along the T = 2.00 isotherm (or equivalently x = 0.002)

until reaching the values of η = 0.375v 1/3 . To this purpose we use the thermodynamic identity:

F e (x, η, α A ) = F e (x, η = 0.0) + η=0.375v 1/3 η=0.0 ∂F e (x, η, α A ) ∂ η dη (2.17) = F e (x, η = 0.0) + η=0.375v 1/3 η=0.0 2 E (a) P η dη (2.18)
Once F e (T = 2.00, η = 0.375v 1/3 ) is determined we can perform an integration along the "iso-η" path v = 1 from T 0 = 2.00 to any temperature T . To this aim we use the Helmholtz formula U e = -T 2 ∂(F e /T )/∂T which links the excess free energy F e to the excess internal energy U e as U e = E P is directly accessible numerically. Therefore the excess free energy is given by the following thermodynamic relation:

F e (T, η, α A ) = T T 0 F e (T 0 , η = 0.0) + η=0.375v 1/3 η=0.0 2 E (a) P η dη -T T T 0 dT ′ U e (T ′ ) T ′2 (2.19)
The last equation shows that the computation of the excess free energy requires the knowledge of both integrals on η and T . These integrals can be found numerically as the functions to integrate are directly related to the potential energy and consequently are directly accessible from MD simulations. Different fitting processes were used to describe the functions to integrate:

• The function E (a)
P /η is well described by a cubic polynomial expression as observed in Figure 2.5 (Upper graph) where we see that the fit represented by a straight line is in good agreement with the data points. The integration of the polynomial expression is easily performed numerically.

• Regarding U e /T 2 we notice that U e (T ) = E P can be fitted by a rational expression U e (T ) = aT 2 + bT + c T 2 + dT + e as observed in Figure 2.5 (Lower graph) where we observe that U e (T )/N is given by a solid line that corresponds to the fit and it goes well through all the data points E P /N . As consequence U e /T 2 is given by the formula U e (T )

T 2 = aT 2 + bT + c T 4 + dT 3 + eT 2 .
The integration of this rational function has been done numerically. p /η as a function of η. The red crosses are the data points coming from simulations whereas the blue solid line corresponds to a cubic polynomial fit. Lower graph: Representation of the average potential energy per particle E P /N as a function of temperature (black crosses). The solid blue line represents U e /N vs T which has been found by fitting E P /N with a rational expression.

Finally we can compute 2.19 numerically and the result is presented in Figure 2.6 where we observe that F e is increasing with temperature.

Excess entropy

We have determined the excess internal energy and the excess free energy, we can thus compute easily the excess entropy by considering the following thermodynamic relation- Result is presented in Figure 2.7 where we observe that:

• the entropy is negative. This is related to the fact that F e is monotonously increasing as F e = -∂S e /∂T .

• the entropy increases with temperature. This observation allows the conclusion that c e V /T (where c e V is the excess heat capacity at constant volume) is positive.

The computation of the heat capacity at constant volume allows to test the reliability of S e (T ) and as a consequence of the whole fitting procedure. c e V = C e V /N can be determined from three different routes. Firstly we use the usual expression which links c e V to U e via the equation:

c e V = 1 N ∂U e ∂T (2.21)
An alternative expression valid in the NVT ensemble enables to make a link between c e V and the fluctuations of the potential energy E P :

c e V = E 2 P -E P 2 N k B T 2 (2.22)
Finally we can also take the first derivative of the entropy versus the temperature to The heat capacity is increasing while the temperature is decreasing and we notice that the phenomenon starts to be more marked for temperature below T ≤ 1.10. This increase of heat capacity at constant volume upon cooling is a typical feature of systems close to glass transition [START_REF] Kreck | Characterization of the glass transition of water predicted by molecular dynamics simulations using nonpolarizable intermolecular potentials[END_REF]. Explanation can be found by considering what happens in the potential energy landscape (whose framework will be described in details in Chapter 5).

determine c e V c e V = T N ∂S e ∂T ( 2
Indeed it is possible to rewrite the potential energy as the energy of the minima (inherent structure), to which is added a contribution which corresponds to the vibration around minima, E P = e IS + e vib [START_REF] Sciortino | Potential energy landscape description of supercooled liquids and glasses[END_REF]. The square of fluctuations of E p , that we denote ∆E 2 P is expressed as:

∆E 2 P = ∆e 2 IS + ∆e 2 vib + ∆e IS ∆e vib (2.24)
If we assume that the cross-term is negligible, the heat capacity c e V is now given by:

c e v = c IS v + c vib v (2.25)
When the temperature is decreased, c vib v → 3k B /2 which is the harmonic limit determined by Dulong and Petit [START_REF] Sciortino | Potential energy landscape description of supercooled liquids and glasses[END_REF]. It has been shown that when the temperature is decreased below a certain temperature that we denote T * , a rapid decrease of e IS is observed [START_REF] Stillinger | Energy Landscapes, Inherent Structures, and Condesend Matter Phenomena[END_REF][START_REF] Debenedetti | Supercooled liquids and the glass transition[END_REF]. This strong variation leads to an increase of contribution of d e IS /dT to the c IS v and thus to c e v as c vib v remains approximately constant.

We will see in Chapter 4, that the temperature T * corresponds actually to a dynamical crossover between Arrhenian and super-Arrhenian regime.

Excess pressure

The excess pressure defined in equation 2.12, is the easiest thermodynamic observable to compute as it requires only the knowledge of the two components of the potential energy which are represented in Figure 2.9. We notice that regarding the absolute values of the two components, E (m) P is 10 times greater than E (a)

P , the part relative to polydispersity. This observation is in agreement with the fact that we work in a regime of weak polydispersity. We can now focus on the excess pressure described by equation 2.12 and by quickly analyzing the order of magnitude of the different terms we find

-5/(3v) E (m) P /N ∼ 5/3 • 125 ∼ 210 -(1/v) E (a) P /N ∼ -10
As a result P e ∼ -200. The excess pressure is thus strongly negative for T ∈ [0.83; 2.00] and this implies that the total pressure P is also negative as the kinetic term T /v is of the order of 1 and cannot counterbalance P e in the range of temperature of interest.

Having a negative pressure means that the bidisperse Voronoi liquid (it is also the case for the monodisperse system) is always under tension. Usual liquids can experience negative pressure in a certain range of volume and in this case they are in a metastable state [START_REF] Herbert | Cavitation pressure in water[END_REF]. However when pressure is too negative, cavitation phenomenon occurs and positive pressure is recovered. Therefore the system is at thermodynamic equilibrium and a coexistence between liquid and gas phases is observed. We focus on a tagged particle labelled M j close to a fictive cavity. Most of its nearest neighbors (with a black contour in the graph) are located on the same side of the cavity as the tagged particle. Only one nearest neighbor M k is at the opposite side of the cavity leading to a larger distance between M j and M k than the average distance between M j and the other nearest neighbors. Despite this larger distance, the particle M k is essential to build the Voronoi cell (in green in the graph). The Voronoi cell is therefore enlarged in the direction of M k , the centroid G is located inside the fictive cavity and as a result the force exerted on M j tends to push the particle inside the cavity decreasing therefore its volume. Same scenarii apply on all particles close to the cavity, making this latter unstable. This sketch shows thus that the force field prevents from the formation of cavitation in the Voronoi liquid and that is why the liquid can have a strong negative pressure while remaining stable.

In the next section we will compute the excess pressure first by using equation 2.12, and then by mean of the Virial theorem which links pressure to the stress tensor when wavevector k = 0. This latter requires the use of pair forces. We will see that we can define pseudo pair forces for the bidisperse Voronoi liquid that enable to compute correctly the stress tensor in the case of periodic boundary conditions.

A stable fluid under tension

As just mentioned there are two routes to determine the excess pressure, the first one is to use equation 2.12:

P e v = - 5 3
E (m) P N - E (a) P N (2.26)
The second route is to use the Virial theorem [START_REF] Hansen | Theory of Simple Liquids[END_REF]:

P e v = 1 N σ e zz (k = 0) (2.27)
where we have chosen on purpose to limit our attention to the excess part of the Virial relationship. σ e zz corresponds to the last diagonal components of the microscopic stress tensor and the e superscript is here to mention that we focus on the potential contribution to the stress tensor.

However in the case of system with periodic boundary conditions (P.B.C), the definition of the stress tensor could be problematic.

σ e zz (k = 0) = i F i,z r i,z -L z ∂E P ∂L z (2.28)
(iii) Fortunately for pair forces the problem is circumvented [START_REF] Louwerse | Calculation of pressure in case of periodic boundary conditions[END_REF] and the usual formulation for the "2-body" stress tensor applies :

σ e zz (k = 0) = i j =i F ji,z rji,z (2.29)
where rji,z =

r ij • e z -L z [(r ij • e z /L z ] the z component of r ij is compliant with
the minimum image convention.

(iv) However for manybody force field can be a real challenge to find a pair decomposition that satisfies equation 2.29.

For the bidisperse Voronoi liquid it can be shown that a pair decomposition satisfying equation 2.29 is given by:

F ji = r ij γ 2r ij S ij dS(r 2 -R 2 i + R 2 ) (2.30)
where F ji is the force that particle j exerts on particle i. It is interesting to mention that this is not the only pair decomposition compliant with equation 2.29. There should be other possible formulations and for instance, we could use the general procedure developed by Admal and Tadmor that enables to find a decomposition in pair forces which always satisfied 2.29 [START_REF] Admal | A unified interpretation of stress in molecular systems[END_REF].

Regarding equation 2.30 as r ij = r jr i we immediately see that the force F ji points toward the j particle meaning that i is attracted by j. Consequently neighboring particles in the Voronoi liquids are attracting each other which is coherent with the picture of fluid always under tension. On the range of temperature represented, the total pressure P remains negative. Therefore we know that

∂P ∂T V = - ∂P ∂V T ∂V ∂T P = χ -1 T α (2.31)
where χ T is the thermal compressibility and α the volumetric coefficient of thermal expansion. As we have shown that the system is always stable, χ T > 0 and consequently we deduce that α < 0. This means that, at constant pressure, under heating the system would contract.

In this chapter, we have observed that the bidisperse Voronoi liquid is able to avoid crystallization even at the slower cooling rates.

We were able to define and compute the main thermodynamic observables such as the excess free energy or the excess entropy. This latter will play an important role in Chapter 5 where we will focus on the potential energy landscape of the system. By looking at minima of the bidisperse Voronoi liquid (inherent structures) we will be able to compute the vibrational entropy S vib . As a result, both knowledge of S liq = S id + S e Chapter 2. Thermodynamic properties

51

(where S id is the ideal gas contribution) and S vib will enable to have access to the configurational entropy S c which is a measure of the number of the different minima accessible by the system [START_REF] Cavagna | Supercooled liquids for pedestrians[END_REF].

Finally we have observed the peculiarity that the bidisperse Voronoi liquid is a stable fluid always under tension.

In what follow, we will mainly focus on microscopic quantities and probe their sensitivity upon cooling. First we investigate the behavior of structural quantities in Chapter 3 and Chapter 4 is devoted to the analysis of dynamical observables. Finding the structure is a key step in the characterization of materials. Experimentally, structures of supercooled liquids or amorphous materials are generally investigated by mean of X-rays or neutron scattering. These methods allow to determine the static structure factor S(k) and thereafter to find the pair correlation function g(r), the static structure factor counterpart in real space [START_REF] Royall | The role of local structure in dynamical arrest[END_REF][START_REF] Higgings | Polymers and Neutron Scattering[END_REF].

It is well established experimentally [START_REF] Menelle | A neutron scattering study of supercooled liquid tellurium[END_REF][START_REF] Leheny | Structural studies of an organic liquid through the glass transition[END_REF] and numerically [START_REF] Kob | Testing mode-coupling theory for a supercooled binary lennard-jones mixture. ii. intermediate scattering function and dynamic susceptibility[END_REF][START_REF] Wahnström | Molecular-dynamics study of a supercooled two-component lennard-jones system[END_REF] that structural changes upon cooling are weak for glass formers. One cannot distinguish on a qualitative level the liquid phase above the glass transition temperature T G from the amorphous solid below T G [START_REF] Cavagna | Supercooled liquids for pedestrians[END_REF]. Liquids and amorphous solids do not show long range order contrary to crystalline structures but they rather exhibit a short range order. Therefore the macroscopic physical properties of these systems depend mostly on the short range order as they are mostly determined by the density and the strength of the forces between nearest neighbors [START_REF] Binder | Glassy Materials And Disordered Solids[END_REF].

Therefore if one wants to get more insight into the understanding of physical properties of the bidisperse Voronoi liquid, it is essential to characterize its structure. To this purpose we mainly focus in this chapter on pair correlation functions and structure factors that are accessible numerically.

Pair correlation function

For an isotropic liquid, the pair correlation function g(r) measures the probability that two distinct particles are separated by distance r. The pair correlation function as defined in equation 1.21 considers that all particles in the mixture are identical. While working with a bidisperse mixture, it is interesting to introduce partial pair correlation function g ab (r) which would take into account the nature of species in the liquid. Partial pair correlation function g ab (r) where the particles are of type a or b such as {a, b} ∈ {A, B}, is defined as follow [START_REF] Binder | Glassy Materials And Disordered Solids[END_REF] :

g aa (r) = V N 2 a Na i=1 Na j =i δ(r + r i -r j ) (i, j) ∈ a g ab (r) = V N a N b Na i=1 N b j=1 δ(r + r i -r j ) with i ∈ a, j ∈ b and ∀a = b (3.1)
The link between the total pair correlation function g(r) and the three different partial pair correlation functions is provided via the following equation:

g(r) = α 2 A g AA (r) + (1 -α A ) 2 g BB (r) + 2α A (1 -α A )g AB (r) (3.2)
where α A = N A /N . This last relation can be understood by considering the expression 3.3 of the total pair correlation function g(r) and by considering the different possibilities for i and j.

g(r) = V N 2 i =j δ(r + r i -r j ) (3.3)
1. We focus only on A particles, thus i = A and j = A.

2. Only B particles are considered, i = B and j = B.

3. Both particles A and B are considered. In this case i = A and j = B for instance, but as i and j are interchangeable, one can also write i = B and i = A.

These observations imply that equation 3.3 can be rewritten as:

g(r) = V N 2 N 2 A V δ(r + r A i -r A j ) + N 2 B V δ(r + r B i -r B j ) + 2 N A N B V δ(r + r A i -r B j ) (3.4)
From this last equation we immediately find the expression of equation 3.2.

Knowing equation 3.1, it is thus possible to compute numerically g ab (r) and to investigate its properties. The following paragraph is devoted to the impact of temperature on this observable.

Evolution with temperature

The partial pair correlation functions have been computed for temperatures T ∈ [0.83; 2.00]

and N = 1000. They are represented in Figure 3.1 where, for T decreasing, each g αb (r)

has been shifted upward for sake of clarity.

We notice first that whatever the partial pair correlation function when the temperature is lowered, g ab (r) gets more and more structured as the decrease of temperature leads to the decrease of fluctuations around average values. However the system remains liquid as g ab (r) → 1 when r → ∞. One might argue that for the lowest T , g α,b (r) exhibits some order at L/2 but this order is weak and the effect of correlations would be negligible. It is interesting to notice that the partial pair correlation function computed for a larger system size of N = 8000 at T = 0.85 shows that total decorrelation is observed for r ∼ 6

(see Figure 3.8).

Then we see in Figure 3.1 that for T ≤ 1.00 the second peak of g a,b (r) starts to split.

This phenomenon is related to the fact that, when the temperature is decreased locally, favored structures emerge and affect the local order upon cooling. First shell of neighbors organized into different preferred types of polyhedra and it was shown for metallic glasses that the splitting comes from the different possible connections between these preferred structures [START_REF] Ding | Second-nearest-neighbor correlations from connection of atomic packing motifs in metallic glasses and liquids[END_REF]. In order to characterize the local underlying structure for at least the first shell of neighbors we investigate the cosine distribution P (R c , cos(θ)) which informs on the angular distribution of neighbors around a tagged particle.

Cosine distribution

The cosine distribution P (R c , cos(θ)) informs on the angular distribution of the neighbors of a tagged particle located at the center of a sphere of radius R c . It is computed as follows [START_REF] Balucani | Evolution of bond-angle distribtuion from liquid to glassy states[END_REF]. A particle i is located at the center of a sphere of radius R c and two of its neighbors particles j and k are also inside the sphere. This means that distances from neighbors to the central particle are

r ij = |r j -r i | ≤ R c and r ik ≤ R c . In Figure 3.2 is
represented the typical sketch of the situation in 2d. The angle θ which measures the angle formed by the triplet is defined as: As the local order is prevailing at the first peak of the partial pair correlation function, the cutoff radius R c is generally chosen to correspond to the position of the first minima of g ab [START_REF] Balucani | Evolution of bond-angle distribtuion from liquid to glassy states[END_REF]. To investigate the structure of the neighborhood of A and B we respectively fix R c to be equal to R c,A = 1.50 and R c,B = 1.25. We rename P (R c , cos(θ)) as P jik (R c , cos(θ))

cos(θ) = r ij • r ik r ij r ik (3.
with (i, j, k) ∈ {A, B}.
Here i is the central particle and j et k are the neighboring particles.

In Figure 3.3 is represented the cosine distribution at T = 0.85 when central particle is of A type (Top) and B type (Bottom). We first note the presence of three marked peaks whatever the nature of P jik (R c , cos(θ)). The values of the cosine taken at these three different maxima are converted into degrees and presented in Table 3 Firstly we notice that in average, when a small particle is central, the 1st peak corresponds to an angle θ 1 ∼ 60 • which is very close to the typical angle ϕ ∼ 63 • between particles in icosahedral configuration [START_REF] Umberto | Dynamics of the Liquid State[END_REF].

The left part in Figure 3. We can therefore also find the value of Γ ≈ 58.5 • which is not so far from the values of θ 1 found when the central particle is a large one. In both cases we observe that the second peak is in average located at θ 2 = 2θ 1 . Other typical angles of the icosahedron are probed and it corresponds to cases illustrated in Figure 3.5 (Left). For the third peak there is a small difference as θ 3 ∼ 3θ 1 (see Figure 3.5 (Right)) when the central particle is large whereas when the central particle is small this proportionality seems to break as we do not observe θ 3 ∼ 180 • . As this would have suggested that three particles are aligned, it seems thus that the alignment of large particles is not favored. Therefore, the central particle of the icosahedron is likely to be a small one, and if we consider that the icosahedron has the conformation shown in Figure 3.4 (Left) thus the two particles, that do not belong to pentagons, should also be small particles to ensure the existence of angle ∼ 180 • .

When looking in deeper details to the relative position of the maxima of P jAk (R c , cos(θ))

(resp. P jBk (R c , cos(θ))) a small decrease (resp. increase) in the values of θ 1 , θ 2 and θ 3 is observed when large (resp. small) particles are progressively replaced by small (resp. large) ones. This behavior can easily be explained and is illustrated in Figure 3.6 where we have arbitrary chosen to focus on configurations with a central small particle. Nonetheless we observe that r * BB < r * AB < r * AA . Thus the distance between j and k is increasing when small particles are substituted for larger ones and as a result the angle θ jik is also increasing. A similar argument (by considering a large particle at center of the triplet) would lead to a decrease of angle θ jik . While the previous reasoning enables to explain the relative positions of the maxima of P jik (R c , cos(θ)) it is also interesting to notice that θ 3 remained unchanged when a small particle is at the center of the triplet. In Figure 3.7 the evolution of P AAA (R c , cos(θ)) and P BBB (R c , cos(θ)) are represented upon cooling. For both distributions we observe for T ≤ 1.00 that the emergence of marked peaks is related to the decrease of fluctuations around the average position also observed for partial pair correlation functions. In average particles tend to locate on preferred position relatively to a tagged central particle and this preferred position is characterized by the angular values given in Table 3.1. The analyze of the neighborhood of a central particle brings informations on the local structures that emerge and reveal the presence of icosahedra where center might be occupied by small particles. In Chapter 6 when we will investigate the locally favored structures, we will observe that the bidisperse Voronoi liquid has a strong tendency to favor icosahedra-like structures. Moreover an analysis of the local structures by mean of Voronoi tessellation will reveal that small particles are at the center of icosahedra.

We have characterized geometrical properties of the first shell of neighbors. In the next paragraph we will focus on the extrema of the partial pair correlation functions and we will see that, in the limit of weak polydispersity, it is possible to define analytically the position of these extrema. At first sight, one could be surprised by the relative position of r * AA and r * BB as one could have imagined that first peaks associated to identical particle interactions would have superimposed. Because with Voronoi-Laguerre tessellation the position of the plane between two identical particles remains median. Therefore one may wonder why the first peaks of g AA (r) and g BB (r) split as observed in Figure 3.8. (i) Assuming that we start from a monodisperse configuration with only small particles, the plane between identical particles is median.

(ii) To generate a bidisperse configuration the radius of the right small particles is enlarged. Now as there is a size disparity between central and right particles, the Voronoi-Laguerre tessellation acts by shifting the plane between the particles to the left. As a consequence the Voronoi cell associated to the central particle is now smaller and the central particle does not coincide with its centroid anymore.

A force is exerted on it and points toward the new position of the centroid of the central cell.

(iii) As a result the neutral position of the central particle shifts to the left, which induces a smaller average inter-particle distances between small particles. Therefore the plane between the two small particles is also shifted to the left to remain median.

A similar argument on big particles leads to the fact that big particles are farther apart in the bidisperse Voronoi liquid than in the monodisperse one. This simple argument in one dimension gives good indication to understand the relative position at first peak of g AA and g BB . In what follow we will show that in the limit of weak polydispersity, the relative position of first peaks can be computed.

Computation of first peak position for weak polydispersity

In Chapter 1 we explained that as η → 0, R A ∼ R B and as a consequence we recover the monodisperse system. In what follow we propose to see how the polydispersity affects the positions r * ab of the first peaks of the partial pair correlation functions. To this purpose we chose to work with values of η that avoid crystallization but are small enough to consider the system as weakly polydisperse.

When looking at the position of r * ab the position of the first maxima of g ab (r) in Figure 3.8, we observe that r * AB seems to be located at half the distance between r * AA and r * BB . More generally we are going to demonstrate that whatever the nature of the mixture:

r * AB = 1 2 (r * AA + r * BB ) (3.7)
When η → 0, r * ab with (a, b) ∈ {A, B} tends to the position r * 0 of the first peak of the pair correlation function g 0 evaluated for the monodisperse liquid. As a consequence if we assume a small value of η we can write the position r * ab as r * ab = r * 0 + δr * ab where δr * ab is the deviation from the monodisperse case of the position of first peaks. The partial pair correlation functions can be expressed as g ab (r) = g 0 (r) + δg ab (r) where δg ab (r) also corresponds to the deviation from the monodisperse system.

We are interested in the values of δr * ab which measure for each interactions the distance to the monodisperse position r * 0 . At first peak, and this is also true for the other extrema, the first derivative of g ab (r) is zero and therefore we can write:

g ′ ab (r * ab ) = 0 = g ′ ab (r * 0 + δr * ab ) = g ′ ab (r * 0 ) + δr * ab g ′′ ab (r * 0 ) = g ′ 0 (r * 0 ) + δg ′ ab (r * 0 ) + δr * ab g ′′ 0 (r * 0 ) + δr * ab δg ′′ ab (r * 0 ) (3.8)
By keeping terms of the lowest order and by noticing that by definition g ′ 0 (r * 0 ) = 0 and g ′ ab (r * ab ) = 0, equation 3.8 enables to find an expression for δr * ab , that is:

δr * ab = - δg ′ ab (r * 0 ) g ′′ 0 (r * 0 ) (3.9)
We look now for an expression of δg ab (r) = g ab (r) -g 0 (r) as it would enable to express analytically the equation 3.9.

We start by evaluating g ab (r) which is defined as follow:

g ab (r) = V N α N β Nα i=1 N β i=1 1 Z N dr i dr j δ(r -r ij ) d N -2 r k exp(-βE P (r 1 , • • • , r N ; η)) (3.10)
Chapter 3 Structural observables 64 with the partition function

Z N = d N r exp(-βE P (r 1 , • • • , r N ; η)).
As η is considered as small, an expansion of the potential energy can be performed leading to :

E P (r 1 , • • • , r N ; η) = E P (r 1 , • • • , r N ) + η ∂E P (η = 0) ∂η (3.11)
The partition function can also be expanded Z N = Z 0 1βη ∂E P (η = 0) ∂η 0 where the subscript "0" means that the average is performed on thermally equilibrated monodisperse configurations. Therefore by using the last two previous results in equation 3.10, g ab (r) becomes when η → 0:

g ab (r) = V δ(r -r ij ) 0 + V βη δ(r -r ij ) ∂E P ∂η 0 - ∂E P (η = 0) ∂η 0 (3.12)
with the precision that particles i if of type a and particle j is of type b.

In Chapter 2, we have seen that for the potential energy only the component relative to the polydispersity 1.40 depends on η. As a consequence by rewriting equation 2.11 we find that

∂E P ∂η = γv 2/3 η(V α A -V A )
and consequently g ab (r) becomes:

g ab (r) = V δ(r -r ij ) 0 + γv 2/3 V βη 2 δ(r -r ij ) (V A -V A 0 ) 0 with i ∈ a and j ∈ b (3.13)
Similarly we can express the pair correlation function for the monodisperse system and we have g 0 (r) = V δ(rr ij ) 0 . Combining with equation 3.13 δg ab (r) is of the form :

δg ab (r) = γv 2/3 V βη 2 δ(r -r ij ) (V A -V A 0 ) 0 with i ∈ a and j ∈ b (3.14)
At first sight the last formulation seems strange as it refers to type A particles in a monodisperse average. We propose to rewrite equation 3.14 by removing the explicit dependence on the particle type in the term relative to the average. There are 3 different cases.

• 1st case i ∈ A, j ∈ A
The average term in equation 3.14 can be rewritten as:

δ(r -r ij ) (V A -V A 0 ) 0 = 2 δ(r -r ij )[v i -v] 0 + (N A -2) δ(r -r ij )[v k -v] 0 (3.15)
where k ∈ A.

Chapter 3 Structural observables 65 We notice that δ(r -r ij ) [V -V 0 ] 0 = 0 and V = v i + v j + k( =(i,j) v k ,
where k represents now a particle without label A or B. This is possible as we perform the average on a thermally equilibrated monodisperse configuration. Therefore we deduce that the second average of the R.H.S is given by:

δ(r -r ij )[v k -v] 0 = - 2 N -2 δ(r -r ij )[v i -v] 0 (3.16)
We can replace by equation 3.16 in equation 3.15 and by finally taking the thermodynamic limit, we find:

δ(r -r ij ) (V A -V A 0 ) 0 = 2α B δ(r -r ij )[v i -v] 0 (3.17)
Thus we can express δg AA as

δg AA (r) = 2α B V γv 2/3 βη 2 δ(r -r ij )[v i -v] 0 (3.18) • 2nd case i ∈ B, j ∈ B
This approach is similar to the previous one. Equation 3.14 becomes in the case where both i and j are particles B:

δg BB (r) = -γv 2/3 V βη 2 δ(r -r ij ) (V B -V B 0 ) 0 (3.19)
Using the same procedure as previously, the average in the R.H.S can be expressed as:

δ(r -r ij ) (V B -V B 0 ) 0 = 2α A δ(r -r ij )[v i -v] 0 (3.20)
Consequently g BB (r) is given by:

δg BB (r) = -2α A V γv 2/3 βη 2 δ(r -r ij )[v i -v] 0 (3.21) • 3rd case i ∈ A, j ∈ B When i ∈ A, j ∈ B, equation 3.
14 can be expressed as:

δg ab (r) = γv 2/3 V βη 2 δ(r -r ij ) [α B (V A -V A 0 ) -α A (V B -V B 0 )] 0 (3.22)
The last equation 3.22 becomes:

δ(r -r ij ) [α B (V A -V A 0 ) -α A (V B -V B 0 )] 0 = δ(r -r ij )[v i -v](α B -α A ) 0 + 2 δ(r -r ij )[v i -v] α B N A -1 N -2 -α A N B -1 N -2 0 (3.23)
We immediately see that the second term of the R.H.S vanishes in the thermodynamic

limit as it is ∝ (α B α A -α A α B ).
Thus we obtain:

δg AB (r) = (α B -α A )V γv 2/3 βη 2 δ(r -r ij )[v i -v] 0 (3.24)
By using equation 3.9 and the fact that g ′′ 0 (r * 0 ) < 0 at the maximum, we have access to the expression of δr * ab :

δr * AA = 2γv 2/3 βη 2 Q g |g ′′ 0 (r * 0 )| α B (3.25)
where Q g correspond to the following derivative:

Q g = d dr   j/j =i δ(r -r ij )[v i -v] 0   r=r * (3.26)
Similarly we can express δr * BB and δr * AB :

δr * BB = - 2γv 2/3 βη 2 Q g |g ′′ 0 (r * 0 )| α A (3.27) δr * AB = 2γv 2/3 βη 2 Q g |g ′′ 0 (r * 0 )| α B -α A 2 (3.28)
We define

∆ g = 2γv 2/3 βη 2 Q g /|2g ′′ 0 (r * 0 )
| and we immediately see that:

• δr * AA -δr * AB = δr * AB -δr * BB = ∆ g . • ∆ g is independent
of the nature of the composition, i.e independent of α A . The two previous points are clearly visible in Table 3.2 where we have recorded the position of the two first peaks of g ab . We see that ∆ g = 0.1 whatever the mixture.

• As we mentioned above, these results are not only valid for the first peaks but applied for all extrema as it is for instance shown in Figure 3.11 where we have zoomed on the two first maxima and minima of the three partial pair correlation functions. We observe furthermore that while increasing the number of large (resp. small) particles r * AA (resp. r * BB ) is shifted toward the left (resp. the right) and is closer from r * 0 ∼ 1.1.

• As g ′′ 0 < 0 we have assumed for the moment that when looking at maxima Q g > 0 as δr * AA (resp. δr * BB ) should be shifted in the positive (resp. negative) direction with respect to the position r * 0 of the monodisperse system. On the opposite, we expect that Q g < 0 for the minima as g ′′ 0 > 0. To check this assumption we have computed δ(rr ij ) (v i -v) 0 to see the variation of the function. It is represented in Figure 3.10 and we notice that indeed near maxima of g 0 (r) the function is increasing meaning that the derivative is positive whereas near mimima the derivative is negative. We have characterized the properties of the pair correlation function, we propose in the following to investigate the properties of its Fourier transform, namely the structure factor.

g 0 (r) δ(r -r ij )[v i -v] 0
Chapter 3 Structural observables 69

Static structure factor

As for the monodisperse system, it is useful to compute the static structure factor S(k)

and the related partial structure factors S ab (k) defined by Ashcroft and Langreth [START_REF] Ashcroft | Structure of binary liquid mixtures[END_REF] formalism by :

S ab (k) = 1 N α N β ρ α (-k)ρ β (k) where ρ β (k) = N β j=1 exp(-ik • r j ) (3.29)
The link between total static structure factor S(k) and partial structure factors is provided by the following equation:

S(k) = α A S AA (k) + (1 -α A )S BB (k) + 2 α A (1 -α A )S AB (k) (3.30)
The partial structure factors S ab (k) can also be expressed as the Fourier transform of the partial pair correlation functions g ab (r):

S ab (k) = δ ab + 4π N V α A (1 -α A ) ∞ 0 drr 2 sin(kr) kr (g ab (r) -1) (3.31)
where the large wavevector limit, i.e k → ∞ of S ab is given by

lim k→∞ S ab (k) = δ ab (3.32)

Evolution with temperature

As for the partial pair correlation functions we observed in Figure 3.12 on the first hand, that when the temperature is lowered, due to the decrease of fluctuations, partial structure factors are more and more structured. On the other hand, there is also for each S ab (k) a marked shouldering of the second peak. In Figure 3.13 is shown the three different partial structure factor and the total structure factor S(k). We denote k * and k * α,β the value of the wavevector corresponding respectively to the first peak of the total structure factor and of each of partial structure factors. Their are presented in Table 3 Regarding the results in Table 3.3 it seems that, as for the partial pair correlation functions, the position of the first peak of S AB is located at half the distance of S AA and

S BB .
In what follow we will show that it is indeed the case.

Relative position of the first peaks of S ab (k)

As for the partial pair correlation functions we are interested in the position k * ab of the first peaks of the partial structure factors S ab (k) when the polydispersity is weak η → 0. As we assume a small deviation from the monodisperse value, we can write

k * ab = k * 0 + δk * ab
where k * 0 is the position of the first peak in the monodisperse case and δk * ab is the functional deviation that we are looking for.

In the case of a weak polydispersity the partial structure factor given by equation 3.29 can be expressed as S ab (k) = S 0 ab (k) + δS ab (k) where S 0 ab (k) corresponds to structure factor of a monodisperse configuration where particles have been arbitrarily labelled as α or β and δS ab (k) is the deviation from the monodisperse case.

As for the partial pair correlation functions we assume an expansion at first order of S ′ ab (k * 0 + δk * ab ) which leads to the following result:

δk * ab = - δS ′ ab (k * 0 ) S ′′0 ab (k * 0 ) (3.33)
Consequently we have to evaluate δS ab (k) to compare the different shifts δk * ab . The partial structure factors can be written as:

S ab (k) = 1 N α N β 1 Z N d N rρ α (-k)ρ β (k) exp(-βE P (r 1 , • • • , r N ; η)) (3.34)
where the partition function is given by

Z N = d N r exp(-βE P (r 1 , • • • , r N ; η)).
Assuming small values of η we perform the same expansion for the potential energy as we did in the case of the partial pair correlation functions, this lead to the following result for S ab (k):

S ab (k) = 1 N α N β ρ α (-k)ρ β (k) 0 + γv 2/3 βη 2 N α N β ρ α (-k)ρ β (k) (V A -V A 0 ) 0 (3.35)
δS ab is defined as δS ab = S ab (k) -S 0 ab (k). With this definition we immediately see that by subtracting the first term of the R.H.S to the L.H.S we find δS ab and consequently:

δS ab (k) = γv 2/3 βη 2 N α N β ρ α (-k)ρ β (k) (V A -V A 0 ) 0 (3.36)
To find δk * ab defined by equation 3.33 we need to treat separately the different possible combinations of (a, b) ∈ {A, B}.

• 1st case α = β = A
We start by defining the partial structure factor S 0 AA for a monodisperse configuration. We recall that it corresponds to a configuration where N A identical particles have been arbitrarily labelled as A. Consequently we have:

S 0 AA (k) = 1 N A ρ A (-k)ρ A (k) 0 = 1 N A N A + N A (N A -1) e ik•r ij 0 (3.37)
The average e ik•r ij 0 is related to the total structure factor of the monodisperse configuration by e ik•r ij 0 = S(k) -1 N -1 and we finally find that:

S 0 AA (k) = 1 + α A (S(k) -1) = α A S(k) + α B (3.38)
This last equation enables to have access to the denominator of equation 3.33. To express the numerator we focus now on δS AA (k) and equation 3.36 can be written :

δS AA (k) = γv 2/3 βη 2 N A ρ A (-k)ρ A (k) (V A -V A 0 ) 0 = γv 2/3 βη 2 (N A -1) e ik•r ij (V A -V A 0 ) 0 with (i, j) ∈ A = γv 2/3 βη 2 (N A -1) 2 e ik•r ij (v i -v i 0 ) 0 + (N A -2) e ik•r ij (v k -v k 0 ) 0
where k ∈ A. To find a more convenient expression for the average over v k we use the fact that we are working at constant volume V meaning that exp(k

• r ij ) (V -V 0 ) 0 = 0
and as a result:

e ik•r ij (v k -v k 0 ) 0 = - 2 N -2 e ik•r ij (v i -v i 0 ) 0 (3.39)
The last equation 3.39 reminds equation 3.16 derived in case of the partial pair correlation functions. The idea is similar. In equation 3.39, k is no more labeled A or B. This possible due to the fact that we perform the average on monodisperse configuration.

By using this last equation in the equation for δS AA (k) we finally find that:

δS AA (k) = 2γv 2/3 βη 2 α B α A N e ik•r ij (v i -v i 0 ) 0 (3.40)
Therefore equations 3.38 and 3.40 enable to express δk

* AA δk * AA = - δS ′ AA (k * 0 ) S 0 ′′ AA (k * 0 ) = -2γv 2/3 βη 2 Q S |S ′′ (k * 0 )| α B (3.41) Where S ′′ (k * 0 ) < 0 and Q S is the derivative of N e ik•r ij (v i -v i 0 )
0 and is given by:

Q S = d dk j e ik•r ij (v i -v i 0 ) 0 k=k * 0 (3.42)
We have thus find an expression for δk * AA in what follows we use the same procedure to find δk * BB and δk * AB .

• 2nd case α = β = B

As the previous case we can express the partial structure factor S 0 BB (k) for a monodisperse configuration, it leads to

S 0 BB (k) = α B S(k) + α A (3.43)
and δS BB (k) is given by:

δS BB (k) = -2γv 2/3 βη 2 α B α A N e ik•r ij (v i -v i 0 ) 0 (3.44)
Thus we find that:

δk * BB = - δS ′ BB (k * 0 ) S 0 ′′ BB (k * 0 ) = 2γv 2/3 βη 2 Q S |S ′′ (k * 0 )| α A (3.45)
where Q S is given by 3.42 and S 0 ′′ BB < 0.

• 3rd case α = A and β = B

We use again the same procedure to express S 0 AB (k)

S 0 AB (k) = √ α A α B (S(k) -1) (3.46)
and δS AB (k):

δS AB (k) = γv 2/3 βη 2 √ α A α B (α B -α A ) N e ik•r ij (v i -v i 0 ) 0 (3.47)
We find therefore that δk * AB is given by:

δk * AB = - δS ′ AB (k * 0 ) S 0 ′′ AB (k * 0 ) = 2γv 2/3 βη 2 Q S |S ′′ (k * 0 )| α A -α B 2 (3.48)
where Q S is given by 3.42 and S 0 ′′ AB < 0.

Expressions of the different δk * ab have been derived and we remark that the first peak associated to the cross interaction δk * AB = 1 2 (δk * AA + δk * BB ). This can be observed in Figure 3.15.

We define now ∆ S = 2γv 2/3 βη 2 Q S /|2S ′′ (k * 0 )| and we notice the following points:

• δk * BB -δk * AB = δk * AB -δk * AA = ∆ S
and in Table 3.3 we notice that ∆ g = 0.20 for the first peaks.

• As ∆ g for the partial pair correlation functions, we observe that ∆ S is independent of the composition A : B of the mixture.

• These results do not concern only the first peak of the partial structure factors but are also valid for any extrema of δS ab (k).

• As S ′′ (k * 0 ) < 0 near maxima we have assumed that Q S < 0 as δk * AA (resp. δk * BB ) should be shifted negatively (resp. positively) with respect to the position k * 0 of the first peak of the monodisperse structure factor S(k). We have computed

j e ik•r ij (v i -v i 0 ) 0
to study the variation of the function. We have represented it in Figure 3.14 where we observed that the function is decreasing near minima meaning that the derivative Q S is negative. S ′′ (k * 0 ) > 0 for minima, and we find that as expected Q S > 0 around minima. for instance [START_REF] Wahnström | Molecular-dynamics study of a supercooled two-component lennard-jones system[END_REF], this characteristic positions of r * ab at low temperature is associated with the presence of additivity. For the bidisperse Voronoi liquid additivity cannot be defined. However when we will investigate locally favored structures (LFS) in Chapter 6, the analysis of LFS will reveal that the polydisperse Voronoi liquid forms icosahedra and shares similarities with additive glass formers. The main issue in the physics of supercooled liquids is to understand the origin of the important slowing down of the dynamics while approaching the glass transition temperature whereas there is no obvious structural changes. Dynamical arrest is well visible when probing molecular motion as for instance the typical relaxation time reaching 10 -10 3 s which is 14 orders of magnitude larger than the typical picosecond relaxation observed above the melting point [START_REF] Ediger | Supercooled liquids and glasses[END_REF][START_REF] Berthier | Theoretical perspective on the glass transition and amorphous materials[END_REF]. Moreover viscosity, which measures the resistance of liquid to flow, may increase by 12 orders of magnitude whereas slight changes in the local packing are experimentally observed [START_REF] Ediger | Supercooled liquids and glasses[END_REF].

Depending on the nature of the species, viscosity or relaxation time does not slow down the same way near the glass transition. For instance materials such as SiO 2 exhibit an Arrhenian behavior, i.e the relaxation time τ ∼ exp(-E A /k B T ). For these systems, an activation energy E A can be extracted. Therefore the typical relaxation time corresponds to the time required before a microscopic event occurs that allows to overcome the energy barrier. In the case of SiO 2 it corresponds to the energy required to break the bond Si-O [START_REF] Ediger | Spatially heterogeneous dynamics in supercooled liquids[END_REF][START_REF] Berthier | Theoretical perspective on the glass transition and amorphous materials[END_REF]. Most of glass-formers deviate more or less strongly from the Arrhenian behavior when the temperature is coming closer to T G , implying an increase of the energy barriers as the temperature is lowered. The molecular origin of this "super-Arrhenian" behavior remains an open issue of the physics of supercooled liquids. [START_REF] Baschnagel | Polymer Glasses, chapter Glass transition and relaxation behavior of supercooled melts: An introduction to modeling approaches by molecular dynamics simulations and to comparison with mode-coupling theory[END_REF].

From the statistical mechanics point of view, viscosity is a transport coefficient and is linked via the Green-Kubo relation to the shear correlation function. Viscosity is an example among others, and for instance we can also mention the diffusion coefficient which is linked to the velocity autocorrelation [START_REF] Hansen | Theory of Simple Liquids[END_REF]. Consequently as transport coefficients seem to be very sensitive to the neighborhood of the glass transition, it appears reasonable to investigate the properties of their related correlation functions as they must convey information about what occurs near glass transition.

In this chapter, we first briefly describe the typical two-step relaxation processes that arise upon cooling. To this purpose, intermediate scattering functions and mean-square displacement of the bidisperse Voronoi liquid are computed. Therefore we extract respectively the relaxation time and the diffusion coefficient. Their evolution with temperature is then discussed in the framework of different approaches such as fragility or mode-coupling theory. Moreover a systematic comparison with usual glass-formers such as Kob-Andersen and Wahnström models is proposed.

Two-step relaxation processes

Assuming a structure-sensitive random observable A(t), as for instance the density fluctuations, the correlation function φ A = A ⋆ (t)A(0) (where A ⋆ (t) is the complex conjugate of A(t)) [START_REF] Baschnagel | Polymer Glasses, chapter Glass transition and relaxation behavior of supercooled melts: An introduction to modeling approaches by molecular dynamics simulations and to comparison with mode-coupling theory[END_REF] measures the decay of correlations with time. [START_REF] Cavagna | Supercooled liquids for pedestrians[END_REF]. Typical decays at high and low temperature are shown in Figure 4.1.

• At high temperature in the short-time ballistic regime, a tagged particle moves by being weakly sensible to interactions with its surrounding neighbors. Then the particle experiences collisions with their surrounding environment and after a certain number of collisions the tagged particle has forgotten about all the initial information. Therefore at large-time scale the associated stochastic process is Chapter 4 Dynamical observables 80 Markovian meaning that the relaxation is exponential φ A (t) = φ 0 exp(-t/τ ) where τ is the characteristic relaxation time of the system.

• When the temperature is lowered while remaining greater than the glass transition temperature T G , the correlation function is more complex. Following the ballistic regime C(t) decays and exhibits a crossover to a plateau associated with the presence of an intermediate regime. This intermediate regime is reminiscent of the behavior found for crystals where the correlation function does not decay in the long-time limit as the motion of particles consists only on vibrations around their equilibrium position [START_REF] Binder | Glassy Materials And Disordered Solids[END_REF]. For supercooled liquids, a vibrational motion is also observed for intemediate time-scale. As a matter of fact particles remain trapped in the cage composed by their surrounding neighbors and consequently large displacements are hindered. Thus a certain time is required to enable particles to escape the cage [START_REF] Cavagna | Supercooled liquids for pedestrians[END_REF]. Finally in the long-time relaxation also named α-relaxation the correlation function does not decay exponentially but the final relaxation is well described by the Kohlrausch-Williams-Watt (KWW) function given by φ(t) = φ 0 exp(-(t/τ ) β ) where β < 1.00 is called the stretched exponent [START_REF] Binder | Glassy Materials And Disordered Solids[END_REF].

Intermediate scattering function

As mentioned early a relevant structure-sensitive observable A is the fluctuations of density defined as follows:

δρ(k, t) = N i=1 exp(ik • r j (t)) (4.1)
The associated correlation function is the total intermediate scattering function

F (k, t)
given by the following relationship [START_REF] Hansen | Theory of Simple Liquids[END_REF]:

F (k, t) = 1 N δρ(-k, 0)δρ(k, t) = 1 N N j=1 N k=1 exp(ik • (r j (t) -r k (0))) (4.2)
F (k, t) is also known as the coherent intermediate scattering function. It correlates the position of a tagged particle i at time t = 0 to the position of another tagged particle j at time t [START_REF] Frey | Propriétés viscoélastiques des fondus de polymères vitrifiables[END_REF]. It can be decomposed into self and distinct parts where we denote F s the self part also known as the incoherent intermediate scattering function and defined as:

F s (k, t) = 1 N N j=1 exp(ik • (r j (t) -r j (0))) (4.3)
It can be useful to study the incoherent intermediate scattering only for one specific type of particles, in either case we define: We notice that they are both compliant with the two-step relaxationscheme when lowering the temperature such as the coherent intermediate scattering function which is also presented in Figure 4.2. For temperatures T > 1.00 the system exhibits the classical exponential decay and when the temperature is lowered below T = 1.00, we notice the emergence of a plateau whose size is increasing while T is decreased. As expected we observe the drop associated to the boson peak before the plateau but other oscillations appear on the plateau which are relatively pronounced for the incoherent intermediate scattering functions.

F a s (k, t) = 1 N a δρ a (-k, 0)δρ a (k, t) = 1 N a Na j=1 exp(ik • (r j (t) -r j (0))) (4.
As the temperature is lowered, the separation of time scales is more and more pronounced. The structural relaxation time associated to the α-relaxation can be extracted from the long-time limit of the incoherent intermediate scattering function. We define arbitrarily the structural relaxation time τ α as the value of the time t when

F a s (k * aa , t = τ a α ) = 0.1.
This relaxation time corresponds to the individual relaxation of particles.

It is important to stress that one could also have chosen to define the relaxation time from the coherent intermediate scattering function F (k, t). The information conveyed is a little bit different as it would correspond to a collective relaxation time which informs on the time particles need to rearrange cooperatively. In both cases the behavior of τ α vs. T is qualitatively the same as it is shown in Figure 4.3 (Lower graph), however the value of τ α are a bit larger in the collective case than in the individual.

In Figure 4.3 (Upper graph) we have represented the evolution of τ A α and τ B α as functions of the temperature T . We notice first that small particles relax a bit faster than the larger ones. We have deliberately chosen a semilogy representation to emphasize the Arrhenian behavior of the system for temperature T ≤ 1.10 and the departure from this regime for temperature below T ∼ 1.10.

Mean-square displacement

As already mentioned in Chapter 1, the mean-square displacement (MSD) measures the correlation between the position of a tagged particle i at time t = 0 and its position at time t and is related to velocity autocorrelation of the tagged particle via the following equation [START_REF] Umberto | Dynamics of the Liquid State[END_REF]:

δr 2 (t) = [r 2 i (t) -r 2 i (0)] = 2 t 0 dτ (t -τ ) v i (0) • v i (τ ) (4.5)
In the case of a binary mixture the MSD is more precisely computed by focusing on just one of the species.

The MSD also captures the two-step relaxation footprint of supercooling as we observe in Figure 4.4 that the short-time ballistic regime and the long-time diffusive regime are separated by a plateau. In the case of the MSD the increase of the size of the plateau when the temperature is lowered is directly related to the time needed by the tagged particle to escape its first shell of neighbors. In the long-time limit the MSD describes a purely diffusive behavior and as a result the diffusion can be computed knowing that δr 2 (t) ∼ 6Dt. The results are presented in As for the structural relaxation time we have chosen a semilogy representation to stress the Arrhenian behavior at high temperature and the departure from this regime at lower temperature. We also notice the higher diffusivity of small particles compared to larger ones as it is generally observed in binary mixtures [START_REF] Kob | Scaling behavior in the beta-relaxation regime of a supercooled lennard-jones mixture[END_REF].

Fragility

While comparing the behavior of the viscosity or the relaxation time as a function of temperature for different glass-formers obtained under the same experimental conditions, one may represent the evolution of these transport quantities as a function of 1/T .

However some insights into the relative behavior of the different materials can be gained by comparing the variation of η or τ α with respect to the same corresponding-states variables, as the T G scaled temperature in the case of an isosbaric cooling experiment for instance [START_REF] Ngai | Relaxation and Diffusion in Complex Systems[END_REF].

m = d log(η) d(T /T G ) T =T G (4.6)
A material with a low value of m is considered as strong whereas materials with a large m value is named fragile. These two terms strong and fragile were initially chosen by Angell to differentiate respectively species that have the ability to preserve a short and medium range order above and below T G from species where the local order is submitted to substantial changes upon cooling [START_REF] Angell | Perspectives on the glass transition[END_REF][START_REF] Angell | Relaxation in liquids, polymers and plastic crystals -strong/fragile parrterns and problems[END_REF]. For instance materials like SiO 2 , which are classified as strong liquids, preserve a tetrahedral structure upon cooling. On the opposite fragile liquids usually do not present directional bonds and correspond to materials with ionic or aromatic character [START_REF] Angell | Relaxation in liquids, polymers and plastic crystals -strong/fragile parrterns and problems[END_REF]. However one has to be careful as this link between fragility and underlying structures of liquids is purely qualitative and some Tamman (VFT) equation [START_REF] Angell | Relaxation in liquids, polymers and plastic crystals -strong/fragile parrterns and problems[END_REF]:

η = η 0 exp[DT 0 /(T -T 0 )] (4.7)
D is called the fragility parameter and is proportional to m [START_REF] Böhmer | Correlations of the nonexponentiality and state dependence of mechanical with bond connectivity in ge-as-se supercooled liquids[END_REF]. Equation 4.7 implies a divergence of the transport quantity under interest, for T = T 0 , supporting the idea of a phase change at T = T 0 [START_REF] Kivelson | Fitting viscosity: Distinguishing the temperature dependences predicted by various models of supercooled liquids[END_REF]. The possible nature of this phase change and its link with thermodynamics will be discussed in Chapter 5 when we will deal with configurational entropy and the Kauzmann paradox. Regarding the fragility, generally speaking we can say that if T 0 is close from T = 0 the system investigated is more likely to have a strong behavior whereas if T 0 ≤ T G the system is fragile [START_REF] Cavagna | Supercooled liquids for pedestrians[END_REF].

As explained by Kivelson et al. [START_REF] Kivelson | Fitting viscosity: Distinguishing the temperature dependences predicted by various models of supercooled liquids[END_REF] the VFT equation 4.7 can be used to fit the transport quantities at low and high temperatures. However it results in performing two distinct fits. On the one hand in the high temperature regime, a fit has to be performed above the crossover temperature T * , which separates the Arrhenian from the super-Arrhenian Chapter 4 Dynamical observables 89 regime. On the other hand a second fit has to be done for T < T * in order to characterize the super-Arrhenian regime.

It is important to stress that using the VFT equation for fitting the whole range of temperatures would lead to inaccurate results. Moreover for high-temperature there is no peculiar interest to use the VFT instead of the usual Arrhenian law with all the more reason that the divergence of the transport quantity would have no physical meaning in the high temperature regime.

Recently in a systematic study of the isobaric fragility of various Lennard-Jones glass formers, Coslovich and Pastore [START_REF] Coslovich | Understanding fragility in supercooled lennard-jones mixtures. i. locally preferred structures[END_REF] proposed to use an equation that enables to pass continuously from the Arrhenian to the super-Arrhenian regime:

τ (T ) =            τ ∞ exp E ∞ T T > T * τ ′ ∞ exp 1 K(T /T 0 -1) T < T * (4.8)
where for continuity one must have:

τ ′ ∞ = τ ∞ exp E ∞ T * - 1 K(T * /T 0 -1) (4.9)
In equation 4.8 the expression used for T < T * corresponds to the VFT equation 4.7

where D = 1/K.

Contrary to Coslovich and Pastore, we also use the same set of equations to fit the diffusion coefficient.

D(T ) =            D ∞ exp - E ∞ T T > T * D ′ ∞ exp - 1 K(T /T 0 -1) T < T * (4.10)
where

D ′ ∞ = D ∞ exp - E ∞ T * + 1 K(T * /T 0 -1) (4.11)
These continuous descriptions of both relaxation time and diffusion coefficient involve the determination in each case of 5 parameters. We look for the values of these parameters not only for the bidisperse Voronoi liquids but also for 3 Lennard-Jones glass formers as our main motivation is to compare our system with well-known models of glass formers.

Chapter 4 Dynamical observables 90

Influence of the mixture composition

For the Kob-Andersen mixture, it has been shown [START_REF] Crowther | The nature of geometric frustration in the kob-andersen mixture[END_REF] that the composition of the mixture has a direct impact on fragility. As a matter of fact, the more the composition tends to be equimolar, the more fragile the mixture.

We test whether this observation is also valid in the case of the bidisperse Voronoi liquid.

To this purpose in addition of the equimolar mixture, we have simulated three other mixtures where the percentage of large A particles has been systematically increased from 50% to 80%. We use equations 4.8 and 4.10 to fit the data.

As mentioned above, it is useful to compare the evolution of relaxation times with respect to the same corresponding states. In experiments the corresponding state is chosen when We can make the following observations:

T = T G .
• As for the Kob-Andersen mixture, in the case of the bidisperse Voronoi liquid, an increase of the number of large particles will lead to a less marked fragility. This is suggested by the decrease of the parameter K when N A is increased for both relaxation time and diffusion coefficient.

• The fragility parameter K is a bit stronger in the case of the diffusion coefficient than for the structural relaxation time.

• When looking at the evolution of T 0 we notice, that T 0 < T r but close from T r , which is qualitatively in agreement with the fact that the bidisperse Voronoi has a fragile behavior. The gap between T 0 and T r increases when K decreases supporting the idea that for strong mixture the value of T 0 is shifted toward T = 0 We decided to also investigate these two previous systems and an additional one, an additive mixture of Lennard-Jones particles that is described below. We firstly characterize the fragility of these models and see to what extent our results are consistent with the ones found by Coslovich and Pastore. Then a great attention is given to the comparison of the bidisperse Voronoi liquid with this LJ glass formers. A specific attention is given to define proper states of references for comparing efficiently these different models.

Comparing models: Lennard-Jones glass formers

To investigate the isochoric fragility we performed NVT simulations, using the Nosé-Hoover thermostat, on 3 different LJ glass-formers which are all binary mixtures (A : B) composed of N = 1000 particles and interacting via the LJ potential:

U ab (r) = 4ǫ ab σ ab r 12 - σ ab r 6 with (a, b) ∈ {A, B} (4.12)
To ensure the continuity of the energy, the potential is shifted to zero at its cutoff r c = 2.5. We run the simulation using a timestep δt = 0.001 and the total time of the equilibrated trajectories correspond to at least 10τ α .

The glass-formers under interest are the following:

1. The classical Kob-Andersen (KA) mixture [START_REF] Kob | Scaling behavior in the beta-relaxation regime of a supercooled lennard-jones mixture[END_REF]. It is a non-additive mixture where the concentration of large particles N A /N = 0.8.

The Wahnström model (WAHN) introduced by Wahnström [30] is an additive

mixture where the mass ratio m 2 /m 1 = 0.5 and the concentration of large particles is N A /N = 0.5.

3. An additive mixture of a LJ particles (AMLJ-0.80) with a diameter σ BB = 0.8

and N A /N = 0.5.

Those peculiar models were chosen to span a wide range of isobaric fragility. Considering the work of Coslovich and Pastore, we can sort these systems by ascending fragility parameters, that is KA < WAHN < AMLJ-0.80. Despite the fact that the isochoric fragility has been experimentally [START_REF] Huang | Dynamic fragility in polymers: A comparison in isobaric and isochoric conditions[END_REF] and numerically [START_REF] Coslovich | Understanding fragility in supercooled lennard-jones mixtures. i. locally preferred structures[END_REF] reported weaker than the isobaric one, we can expect that the general tendency would be preserved.

All the parameters used to define these models are presented in As for the bidisperse Voronoi liquid, the value of τ α is chosen such that F s (k * , t = τ α ) = 0.1 and the diffusion coefficient is extracted from the long-time limit of MSD. We have used equations 4.8 and 4.10 to fit respectively the relaxation time and the diffusion coefficient of the 3 LJ systems.

Results for τ α are presented in Table 4.4 and the fitting curves are represented in dashed lines in Figure 4.9.

τ ∞ E ∞ T * K T 0 T r KA 0.
(T = T r ) = 4 • 10 -4 .
The values of the fitting parameters are slightly different from the ones found by Coslovich and Pastore [START_REF] Coslovich | Understanding fragility in supercooled lennard-jones mixtures. i. locally preferred structures[END_REF]. This could be due to the fact that the crossover temperature estimated by our fitting procedure is a bit lower than the one found in [START_REF] Coslovich | Understanding fragility in supercooled lennard-jones mixtures. i. locally preferred structures[END_REF]. However values are qualitatively in good agreement. As expected the fragility of the KA mixture is weaker than the one of WAHN and AMLJ-0.80. AMLJ-0.80 has the more marked fragility as it is suggested by the value of K or the localisation of the graph of AMLJ-0.80 in the Angell-plot 4.9. While fitting the diffusion coefficient we found again values of K that are slightly lower than the one found in [START_REF] Coslovich | Understanding fragility in supercooled lennard-jones mixtures. i. locally preferred structures[END_REF]. However as for the relaxation time, the fitting parameters for D are in good agreement with the ones found for the relaxation times. In the Angellplot 4.10 and in Table 4.5 we observe that AMLJ-0.80 remains the more fragile mixture whereas KA is the strongest of the 3 LJ glass formers. We also note that the fragility index K is bigger than the one found for τ α which was also observed in [START_REF] Coslovich | Understanding fragility in supercooled lennard-jones mixtures. i. locally preferred structures[END_REF].

D ∞ E ∞ T * K T 0 KA 0.

Comparing models : the problem of units

Previously we compared the fragility of the Voronoi liquid while changing the composition of the mixture. As all these systems were expressed in the same units the comparison of the fragility was straightforward and we didn't need to pay attention to a rescaling of the relaxation time or diffusion coefficient. In the same way, we didn't rescale these transport quantities when we looked at the fragility of the Lennard-Jones glass formers as all these different systems were expressed in the terms of LJ units.

Now, if we want to compare the bidisperse Voronoi liquid with these different glassformers, we have to render the different physical observables dimensionless. To this end, we need characteristic time and length scales which would share a common physical basis, because the scales given by the "natural" units of the models (σ AA and τ AA = m AA σ 2 AA /ǫ AA for the LJ-inspired models, v 1/3 and (γv) -1/2 for the Voronoi models) are not physically fully justified, and would not allow a fair comparison (see 4.2.4.1).

The more straightforward approach here to rescale the temperature is to use again the extrapolated temperature T r . For making the relaxation time dimensionless, we can use the typical time extracted from the Arrhenian fit. In this way we can see how the different systems behave regarding their fragility between two points of references, one taken in the high temperature limit when τ α /τ ∞ = 1.0 and the other one in the low temperature limit when T r /T = 1.0.

In Figure 4.11 represents the scaled relaxation time as a function of T r /T . We observe that:

• The KA system appears to be the stronger of the four models whereas AMLJ-0.80 is visibly the more fragile glass formers.

• For temperatures where data are available for the Wahnström model, WAHN and the bidisperse Voronoi liquid seem to have a similar behavior. However this observation is based on extrapolation for T → T r and it is not obvious that this trend would be confirmed when T → T r which corresponds to the range of temperature where the fragility is defined.

• We notice that when T → T r the fragility of the bidisperse Voronoi liquid is closer to the one of AMLJ-0.80 than from KA, meaning that our model might be considered relatively fragile. Contrary to the relaxation time, it is less obvious that for T = T r all the curves would collapse on the same point. This is not a surprise, as it is well-known for supercooled liquids that there is not only one intrinsic time scale which would be given by the structural relaxation time τ α computed at k = k * . Therefore other observable may relax on shorter time-scale and this may be the case for the diffusion coefficient. A better choice for T r would be for instance, the one defined arbitrarily when D(T = T D r ) = 10 -5 where the values of D(T r ) would be extrapolated from equation 4.10. • The Kob-Andersen mixture is still the stronger glass formers whereas AMLJ-0.80 is the more fragile one. The Wahnstróm system has a fragility intermediate between those last two systems.

Voro

• Regarding the diffusion coefficient, when T → T r the fragility of the bidisperse Voronoi liquid seems to be closer from KA. This is the opposite behavior of the one found for the relaxation time.

The last point suggests that there is a difference of behavior depending on the scale observed. When we are probing the microscopic scales, that is when we are looking at the relaxation time, the bidisperse Voronoi liquid has a fragility similar to the one of AMLJ-0.80, a LJ glass formers known to be relatively fragile. On the other hand when we are investigating hydrodynamic limit by looking at the diffusion coefficient, we notice that our system becomes less fragile and is comparable to the KA model. This difference of behavior for the two transport quantities points to the weakness of a description in terms of fragility and reminds that Angell-plots are firstly a tool to sort systems and see how they behave qualitatively with respect to each other.

However the idea of representing transport quantities with respect to corresponding states, as for instance the glass transition temperature, is crucial to ensure efficient comparison among different classes of systems. In the case of the comparison of the Voronoi liquid with the LJ glass-formers, the breakdown of the Angell-plot representation suggests that the corresponding states chosen were probably not adequate.

One immediate objection to the classical Angell representation is that we compare systems with respect to a unique corresponding state that carries information of what happens at low temperature. Moreover, the corresponding state is chosen arbitrary, with an "anthropic" criterion τ /τ ∞ ∼ 10 4 which cannot be satisfactorily. In computer simulation studies, we can benefit from the fact that many different observables are easily computed, and try to use for each temperature probed microscopic length and time scales which would be well adapted to each temperature probed. In any case an nondimensionalization of the axis 1/T and τ α must be provided, otherwise the purpose of comparing different models cannot be undertaken.

In what follow we look for such quantities and we focus particularly on finding microscopic time and energy scales that would have a common definition to all the different systems of glass-formers. At first sight, one could say that the stronger the departure from the master curve, the more fragile the glass former. If this appears to be true for the LJ systems, it is not the case for the bidisperse Voronoi liquid and the order found in Figure 4.11 is apparently not conserved in this representation. This raises question about the relevance of the Angell plot, which seems not sufficient to capture all the phenomenology that occurs upon cooling. One could also a argue that rescaling T by E ∞ could make no sense at lower temperature as the structure of the liquids may have substantially changed in comparison to high temperature states.

In any cases, it could be safer to do a step backward and to try to make dimensionless quantities from 1) the fastest characteristic time of the system: the inverse of the Einstein frequency, and 2) a characteristic energy that would be shared by all the glass formers.

The Einstein-frequency a common microscopic time scale

In the short times limit when t → 0, the velocity autocorrelation of a tagged particle can be expanded at the second order in t in the following way [START_REF] Umberto | Dynamics of the Liquid State[END_REF]:

v i (t)v i (0) = v 2 i (0) -v2 i (0) (t 2 /2) + O(t 4 ) (4.13)
where v 2 i (0) = 3k B T /m corresponds to the square of the thermal velocity and v2 i (0) = (3k B T /m)Ω 2 0 with Ω 0 is the Einstein frequency. Ω -1 0 corresponds to the typical time at Chapter 4 Dynamical observables 101 which particles quit the ballistic regime and experience "collisions" with their neighbors [START_REF] Umberto | Dynamics of the Liquid State[END_REF]. Consequently it is the microscopic time for which a tagged particle starts to feel the presence of its surrounding environment. This microscopic time is accessible for all glass-formers and more important the physical meaning is preserved whatever the mixture under interest. Therefore it appears as a good candidate for rescaling the time.

Ω -1 0 can easily be computed from the knowledge of the forces:

Ω -1 0 = 3k B T F 2 i (0) /m i (4.14)
Ω -1 0 has been computed for the Voronoi liquid and for the LJ glass-formers. Results are shown in Figure 4.14 where Ω -1 0 is rescaled on purpose by the characteristic time units τ voro and τ LJ defined above. above as we see that Ω -1 /τ voro ∼ 30Ω -1 /τ LJ . We immediately see that, as the proper intrinsic timescale is Ω -1 0 , the choice of τ LJ and τ voro to nondimentionalize the temporal observables would introduce a spurious large discrepancy factor.

Another representation of Ω -1 0 without rescaling is shown in Figure 4.15. We observe that in this case the values of Ω -1 0 expressed in the system units are of the same order. This is strongly supporting the choice of the inverse of the Einstein frequency as a microscopic time of reference that would be common to all the glass-formers. When looking into the qualitative behavior of Ω -1 0 we notice that it is increasing with temperature for the bidisperse Voronoi liquid whereas we observe the strict opposite for the LJ models. This is related to the interplay between kinetic contribution and the different nature of potentials, i.e soft-core vs. hard-core potentials.

To better understand how these potentials act, one can imagine the case where the nearest neighbors of a tagged particle are frozen. As a result, the tagged particle can only oscillate in its cage and it experiences a local effective potential whose form is represented in Figure 4.16. Apart from the immediate neighborhood of the minimum, both hard-core and soft-core potentials show anharmonicity. In the case of an hard-core potential (left), the potential is steeper than quadratic whereas the soft-core potential (right) is milder than the quadratic one.

As a result, for the hard-core potential, the period of the oscillations is decreasing when the temperature is increased, whereas for soft-core potential period is increasing with temperature.

Figure 4.16: Sketch of the difference in the effective potential felt by a particle whose neighborhood has been frozen (upper schema). At left the effective potential is hardcore (solid line) whereas at right it is a soft-core potential (solid line). In both cases, the dashed line represents the harmonic approximation.

The inverse of the Einstein frequency can reasonably play the role of the microscopic time scale that we were looking for. Now it remains to define the microscopic energy length scale. Three plausible candidates have been identified: the typical microscopic work, the fluctuations of the potential energy and the high-frequency shear modulus.

The choice of these quantities, their efficiency as energy length scale and the associated results are discussed in more details in what follows.

Microscopic energy length scale: Typical microscopic work

One of the easiest energy scale E µ possible is related to the work exerted on a tagged particle to observe a displacement on a characteristic length ℓ, that is

E µ ∼ F 2 (0) ℓ c
where F is the force exerted on a particle. There is no obvious choice of ℓ and one could use for instance chose σ AA in the case of LJ systems. However as mentioned above a meaningful characteristic distance would be the one associated with the mean interparticle distance ℓ = v 1/3 which approximately corresponds to the distance a particle may travel before experiencing a "collision" with one of the neighbors.

As a consequence the characteristic energy scale E µ could be written as:

E µ = F 2 (0) v 1/3 (4.15)
Chapter 4 Dynamical observables 104

In Figure 4.17 we have represented the evolution of F 2 (0) v 1/3 as a function of the temperature for the Voronoi system and for the 3 other LJ glass formers. We observe that the energy required to displace a particles on the average inter-particle distance is decreasing when the temperature is lowered. The evolution of the typical work while increasing the temperature is easily explained by the shape of the potentials shown in Figure 4.16. As a matter of fact, at low temperature, the particles vibrate around their equilibrium position and consequently explore a part of the potential U (r) which is nearly flat or slightly curved. As a result the force which is the derivative of U (r) is relatively weak. As the temperature is increased, the force increases as particles are able to explore part of U (r) where the local slope of U (r) is relatively important. Obviously this slope is more pronounced for hardcore potentials explaining why the increase of F 2 (0) v 1/3 is larger for LJ glass-formers than for the bidisperse Voronoi liquid.

We thus propose to rescale the temperature by using E µ /k B . In Figure 4.18 is represented the scaled relaxation time as a function of the scaled temperature. We observe that:

• There is a clear distinction between the bidisperse Voronoi liquid and the LJ systems.

• However the curve of the Wahnström is somehow different as its high temperature regime does not collapse on the corresponding regime of KA and AMLJ-0.80.

• The marked increase of Ω 0 τ α for given values of E µ /(k B T ) is enhanced by the decrease of E µ with decreasing T . However for the LJ systems, it is reduced by the simultaneous decrease of Ω -1 0 . The effect is the opposite for the bidisperse Voronoi liquid.

In conclusion we notice that Figure 4.18 shows that there is no universality among different classes of glass formers at the neighborhood of glass transition. Moreover we see that the sole microscopic parameters Ω -1 0 and E µ are not sufficient to convey the whole phenomenology of what happens upon cooling. However, from a physical point of view, the representation in Figure 4.18 is a more reasonable comparison of widely different systems that those previously shown (Angell plot or (τ ∞ , E ∞ ) scaling). As T k B C e v is a reasonable energy scale, we investigate the behavior of Ω 0 τ α as a function of δE 2 P /(k B T ) which is can also be rewritten as C e v /(N k B ) where we have introduced the number of particles for questions of intensivity. Results presented in Figure 4.20 suggest that there is a clear separation between the different classes of system. Moreover we observe that among the LJ glass-formers, the high temperature behavior (equivalent to the low C e v /(N k B ) behavior) appears similar for the different systems.

We notice also the presence of an exponential regime for the four glass-formers. This regime is for instance well-visible, in Figure 4.20, for the bidisperse Voronoi liquid for

C e v /(N k B ) ∈ [1.40, 1.70]. This implies that Ω 0 τ α ∼ exp(A C e v /(N k B )
). Therefore we propose to investigate this regime by testing the following fitting function:

ln(Ω 0 τ α ) = A C e v /(N k B ) -B ≡ A T δE 2 P - B A T (4.16)
where A and B are fitting parameters. A can be interpreted as the average numbers of particles concerned by the fluctuation of energy δE 2 P . The introduction of the parameter B can be seen as a renormalization of Ω -1 0 if B < 0 and |B| ≪ 1. In the other cases, we must consider the entropic contribution B/A. For sake of clarity we do not show the results for KA and AMLJ-0. • For both Voro and KA we observe that the typical energy fluctuation involves ≈ 11 particles. This number drops to ≈ 8 for WAHN and ≈ 7 for AMLJ-0.80.

In

• We notice that parameter B cannot be considered here as a rescaling of Ω -1 0 as it would lead to times ≪ Ω -1 0 . This would have no physical meaning as Ω -1 0 corresponds the smallest accessible time scale of systems.

• Therefore we consider the ratio B/A which corresponds to a positive entropic contribution ≈ 1 or 2 for the different glass-formers.

Taking into account the previous observations, we could imagine the following scenario regarding the crossing of energy barrier. The initial configuration corresponds to a local minimum of the free energy landscape. Fluctuations of the potential energy would locally affect ≈ 10 particles. In the range of temperatures considered, this local supply in energy would enable those particles to rearrange locally and thus to have access to other new configurations of higher energy. As a result entropy is thus increased.

This scenario remains hypothetical but it could be tested by looking at the correlation between the fluctuations of the potential energy and the local rearrangements that occur in the liquid.

Microscopic energy length scale: Instantaneous shear modulus

The instantaneous shear modulus G ∞ corresponds to the shear modulus measured at t = 0 when the liquid does not have time to flow [START_REF] Dyre | Elastic models for the non-arrhenius viscosity of glass-fromings liquids[END_REF]. G ∞ is defined from the nondiagonal elements of the stress-tensor as [START_REF] Zwanzig | Highfrequency elastic moduli of simple fluids[END_REF]:

G ∞ = σ xy (0) 2 V k B T (4.17)
Chapter

As this quantity is measured on very short-time scale, it is a good candidate to define a microscopic energy scale as G ∞ v.

The idea of rescaling the temperature with respect to G ∞ was already suggested twenty years ago by Dyre when he introduced the shoving model [START_REF] Dyre | Local elastic expansion model for viscous-flow activation energies of glass-forming molecular liquids[END_REF], an elastic model used to describe the activated flow events that could occur for highly viscous liquids. The main idea of this model is that the appearance of flow events is related to the creation of an extra volume. To create this volume, particles shove aside the surrounding liquid to increase the available volume [START_REF] Dyre | Local elastic expansion model for viscous-flow activation energies of glass-forming molecular liquids[END_REF]. The work associated to this process corresponds to the activation energy [START_REF] Dyre | Elastic models for the non-arrhenius viscosity of glass-fromings liquids[END_REF]. As the flow event occurs on fast time scale, the surrounding liquid behaves like a solid and solid-state elasticity could be used to calculate the shoving work which is related to the shear modulus [START_REF] Dyre | Elastic models for the non-arrhenius viscosity of glass-fromings liquids[END_REF].

As a consequence of these observations Dyre proposed to express this T -dependent activation energy as a function of G ∞ [START_REF] Dyre | Elastic models for the non-arrhenius viscosity of glass-fromings liquids[END_REF]:

E(T ) = V C G ∞ (T ) (4.18)
where V C is referred as the characteristic volume and it is assumed to be temperature independent [START_REF] Dyre | Local elastic expansion model for viscous-flow activation energies of glass-forming molecular liquids[END_REF].

In We observe that all the systems exhibit the same decreasing behavior with temperature however values of G ∞ are weaker for the bidisperse Voronoi liquid than for the LJ glassformers. We notice moreover a collapse onto the same master curve for the three LJ systems.

Shoving model has already been verified by experimental measurements as for instance

on metallic glasses and molecular liquids [START_REF] Dyre | Local elastic expansion model for viscous-flow activation energies of glass-forming molecular liquids[END_REF][START_REF] Xu | Evaluation of the dyre shoving model using dynamic data near the glass temperature[END_REF][START_REF] Dyre | The instantaneous shear modulus in the shoving model[END_REF]. Therefore we test numerically the shoving model for the bidisperse Voronoi liquid and the LJ glass-formers. In Figure 4.23 is represented the scaled relaxation time as a function of G ∞ v/T .

We note that once again there is a clear disparity between the Voronoi liquid and the LJ systems. We recall that the shoving model predicts that the relaxation time is given by This observation is in agreement with experiments for which the shortest times accessible correspond to plateau values [START_REF] Dyre | The instantaneous shear modulus in the shoving model[END_REF]. As a consequence further investigations could be done by computing (V k B T ) -1 σ xy (t)σ xy (0) and then rescaling the temperature by the plateau value G ∞,p (T ).

τ α ∼ exp(G ∞ (T )/k B T )
We attempted to rescale the transport quantity and the temperature in order to compare the different glass-formers on the one hand. On the other hand the idea was to find a set of parameters (time, temperature) that would make arise a common framework for all the systems. However we notice that for each of the representation that we used points towards the different natures that exist between the bidisperse Voronoi liquid and models built from the LJ potential. In this sense we face the fundamental open question related to glass-transition problem, is there a fundamental mechanism responsible for the dynamical arrest that would be common to all the glassforming liquids ?

In what follow we pursue our description and comparison of the Voronoi liquid by having a look to the Stokes Einstein relation which is known to be a sensitive marker of dynamical heterogeneties.

Stokes-Einstein relation

As mentioned before, both structural relaxation time and diffusion coefficient depart from the Arrhenian temperature below T * . Regarding the results on fragility we already know that the deviation of both quantities is likely to differ one from each other. A way to quantify this difference is to use the Stokes-Einstein relation which links the diffusion coefficient D to the viscosity η through the following relation:

D = k B T 6πRη (4.19)
where R is the radius of the particle [START_REF] Hansen | Theory of Simple Liquids[END_REF]. Physically this relation implies that two different measurements of the relaxation time R • When the temperature is decreased below T * we observe the departure from the plateau meaning that the Stokes-Einstein breaks down.

• When looking at the deviation we notice that the more fragile the mixture, the more important the breakdown. Indeed it is particularly true for AMLJ-0.80 which has the highest values of the Stokes-Einstein ratio, 10 times more important than the other glass-formers.

• In comparison with the Kob-Andersen mixture and the Wahnström model, the bidisperse Voronoi shows also a marked deviation from its plateau value. The breakdown of the Stokes Einstein relation is associated with the presence of dynamical heterogeneities [START_REF] Cavagna | Supercooled liquids for pedestrians[END_REF]. Liquid and glassy systems present disordered structures and even if they are composed of identical particles, two tagged particles may experience two different local environments [START_REF] Berthier | Dynamical Heterogeneities in Glasses, Collloids and Granular Media[END_REF]. At high temperature, these differences are negligible and a tagged particle taken at random is representative of the other one. However upon cooling a tagged particle chosen at random may have a behavior completely different Chapter 4 Dynamical observables 113 from another one [START_REF] Berthier | Dynamical Heterogeneities in Glasses, Collloids and Granular Media[END_REF]. This is for instance the case when investigating the mobility of particles. It has been shown, experimentally for molecular liquids [START_REF] Ediger | Spatially heterogeneous dynamics in supercooled liquids[END_REF][START_REF] Sillescu | Heterogeneity at the glass transition: a review[END_REF], colloids and granular medial [START_REF] Weeks | Threedimensional direct imaging of structural relaxation near the colloidal glass transition[END_REF][START_REF] Dauchot | Dynamical heterogeneity close to the jamming transition in a sheared granular material[END_REF] and also numerically [START_REF] Widmer-Cooper | On the study of collective dynamics in supercooled liquids through the statistics of the isoconfigurational ensemble[END_REF][START_REF] Candelier | Spatiotemporal hierarchy of relaxation events, dynamical heterogeneities, and structural reorganization in a supercooled liquid[END_REF], that domains where particles are relatively mobile coexist with domains where particles only vibrate around their equilibrium position.

A typical sketch that would represent the different environments in a supercooled liquid is presented in Figure 4.25. Fast, intermediate and slow domains coexist and when one compute the usual microscopic quantities, more importance can given to a particular type of domain. It is for instance the case of the diffusion coefficient which is dominated by the diffusion that occurs in the fast domains whereas the relaxation time is associated to the relaxation of the slowest domains [START_REF] Cavagna | Supercooled liquids for pedestrians[END_REF][START_REF] Berthier | Dynamical Heterogeneities in Glasses, Collloids and Granular Media[END_REF]. Therefore the breakdown of the Stokes Einstein occurs as different environments are probed [START_REF] Cavagna | Supercooled liquids for pedestrians[END_REF].

Individual and collective relaxations

Up to now, we only focussed on the relaxation time extracted from the density correlators computed for wavevector k * corresponding to the maximum of the structure factor. • For both individual and collective relaxations, we observe that for k ≥ k * all the relaxation times rescaled by the structural relaxation time fall onto the same master curve. This suggests that the whole phenomenology that occurs at large wavevectors is governed by parameters τ α and k * .

For k ≤ k * , we note the presence of an apparent power-law regime for which the power increases with the temperature. By comparing our results with the ones obtained for a polydisperse system of quasihard spheres studied by Weysser et al. [START_REF] Weysser | Structural relaxation of polydisperse hard spheres: Comparison of the mode-coupling theory to a langevin dynamics simulation[END_REF] we notice a strong similarity between the individual relaxation of two systems as it is visible in Figures 4. [START_REF] Kob | Testing mode-coupling theory for a supercooled binary lennard-jones mixture. ii. intermediate scattering function and dynamic susceptibility[END_REF]

Ideal Mode Coupling Analysis

Previously the slowing down of the dynamics was described by mean of phenomenological theories. Evolution of the structural relaxation time has been characterized by mean of simple physical approaches which mainly focused on the presence of temperaturedependent energy barriers. These qualitative approaches are leading to equations that can be used to describe experimental or numerical data [START_REF] Binder | Glassy Materials And Disordered Solids[END_REF].

The quest for a theory that would enable to make comparison with experimental results and predict (possible new) non trivial phenomena is still nowadays a challenge as for the moment a theory that would describe the whole phenomenology of the glass-transition is still missing [START_REF] Binder | Glassy Materials And Disordered Solids[END_REF]. However there is one theory, the Mode-Coupling Theory (MCT) that enables quantitative calculations and which provides predictions that can be tested.

The MCT takes into account two main observations:

1. the weak temperature dependence of the structural properties 2. at low temperature the separation of time-scales, i.e α-relaxation is much more larger than the microscopic one As a result a description in term of slow varying variables is possible. By identifying the slow variable of the structural relaxation, the Mori-Zwanzig projection operator formalism [START_REF] Mori | Transport, collective motion and brownian motion[END_REF][START_REF] Zwanzig | Memory effects in irreversible thermodynamics[END_REF] can be applied and an exact equation for the motion of F (q, t) can be found [START_REF] Reichman | Mode-coupling theory[END_REF]:

d 2 F (q, t) dt 2 + q 2 k B T mS(q) F (q, t) + m N k B T t 0 dτ M (q, τ ) dF (q, t -τ ) dt (4.20)
where M (q, t), the memory kernel, is related to the fluctuating force acting on the density field [START_REF] Berthier | Theoretical perspective on the glass transition and amorphous materials[END_REF][START_REF] Reichman | Mode-coupling theory[END_REF].

In the present form equation 4.20 is impossible to solve [START_REF] Reichman | Mode-coupling theory[END_REF]. Nevertheless in order to close the equation some approximations can be done by noticing that the fluctuating force contains products of the density fluctuations: In other words M (q, t) contains slow modes and it is possible to project the fluctuating force onto the slowest mode. After projections the memory kernel contains four-point density terms but it is possible to factorize them into products of two-point density functions F (q, t) [START_REF] Berthier | Theoretical perspective on the glass transition and amorphous materials[END_REF][START_REF] Reichman | Mode-coupling theory[END_REF].

The previous approximations enable to derive the MCT self-consistent equations [START_REF] Reichman | Mode-coupling theory[END_REF]:

d 2 F (q, t) dt 2 + q 2 k B T mS(q) F (q, t) + t 0 dτ M (q, t -τ ) dF (q, t) dt (4.21)
with the memory kernel expressed as

M (q, t) = ρk B T 16π 3 m dk| Ṽq-k,k | 2 F (k, t)F (|k -q|, t) (4.22)
where

Ṽq-k,k = {( q • k)c(k) + q(q -k)c(|q -k|)} with c(q) = (1 -1/S(q))/ρ (4.23)
Equation 4.21 is thus an approximate equation of motion for F (q, t) at wavevector q. The expression 4.22 of the memory kernel shows that the mode q is coupled to the product of mode k by mode |q -k| [START_REF] Baschnagel | Computer simulations of supercooled polymer melts in the bulk and in confined geometry[END_REF]. The coupling coefficients c(q) are fully determined by the static structure factor of the glass former [START_REF] Baschnagel | Computer simulations of supercooled polymer melts in the bulk and in confined geometry[END_REF].

It is important to stress that the previous equations have been derived for a single component liquid. Generalization to binary mixture has been proposed by Barrat and Latz [START_REF] Barrat | Mode-coupling theory for the glass transition in a simple binary mixture[END_REF] and a general discussion can be found in the review written by Das [START_REF] Das | Mode-coupling theory and the glass transition in supercooled liquids[END_REF].

Solutions of equations 4.21 have been discussed by Leutheusser [START_REF] Leutheusser | Dynamical model of the liquid-glass transition[END_REF] and Bengtzelius et. al. [START_REF] Bengtzelius | Dynamics of supercooled liquids and the glass transition[END_REF] and in what follow we propose to test for the bidisperse Voronoi liquid the predictions of the MCT.

Theoretical predictions of MCT

In its ideal form MCT predicts for the correlator of density fluctuations φ q (t) 1 a transition from an ergodic to a non-ergodic phase at a critical temperature T c , called the mode-coupling temperature [START_REF] Reichman | Mode-coupling theory[END_REF]:

lim t→∞ φ q (t) =      0 for T > T c f q (T ) for T ≤ T c (4.24)
This transition means that for temperature above T c , the intermediate structure factor φ q (t) may relax. In the long-time limit a tagged particle would transit through different cages of neighbors and would forgot about its initial position. This loss of memory is characteristic of the liquid phase. Below T c the scenario is different, there is no complete decay of the density fluctuations [START_REF] Baschnagel | Computer simulations of supercooled polymer melts in the bulk and in confined geometry[END_REF]. A particle remains trapped in its surrounding environment and vibrates around its initial position. Consequently the particle does not lost completely the memory of its initial state and f q (T ) also called the non-ergodicity parameter is a measure of the number of density fluctuations that becomes frozen at T c [START_REF] Berthier | Theoretical perspective on the glass transition and amorphous materials[END_REF][START_REF] Baschnagel | Computer simulations of supercooled polymer melts in the bulk and in confined geometry[END_REF].

1. Only for this section we denote the coherent intermediate scattering function φq(t) (and no more F (q, t)/S(q)) as we decided to adopt the same notations as in [START_REF] Baschnagel | Computer simulations of supercooled polymer melts in the bulk and in confined geometry[END_REF].

Chapter 4 Dynamical observables 119

The presence of the critical temperature T c enables to introduce the "separation parameter σ" which measures the relative distance to T c :

σ = C T c -T T c (4.25)
where C is a constant that depends on the system under interest [START_REF] Baschnagel | Computer simulations of supercooled polymer melts in the bulk and in confined geometry[END_REF]. The separation parameter plays the role of a small parameter used to compute the asymptotic expansions around T c .

The MCT predicts that φ q (t) relaxes in two steps when T → T + c . The first step corresponds to the relaxation toward the plateau value given by f c q = f q (T c ) whereas in a second time, which corresponds to the α-relaxation, φ q (t) departs from the plateau and decays to zero. The intermediate time regime including the plateau corresponds to the β-relaxation [START_REF] Binder | Glassy Materials And Disordered Solids[END_REF][START_REF] Baschnagel | Computer simulations of supercooled polymer melts in the bulk and in confined geometry[END_REF]. In what follows we describe the prediction of the MCT for the β and α regimes.

Predictions for the β regime

This time scale corresponding to the location of the plateau is given by [START_REF] Binder | Glassy Materials And Disordered Solids[END_REF]

t σ = t 0 |σ| 1/2a (4.26)
where t 0 is a microscopic time-scale and a is an exponent whose calculation will be discussed below.

A interesting prediction of the MCT in the β-regime is the so-called factorization theorem which states that the correlator φ q (t) can be expanded for t ∼ t σ up to the first order [START_REF] Binder | Glassy Materials And Disordered Solids[END_REF][START_REF] Baschnagel | Computer simulations of supercooled polymer melts in the bulk and in confined geometry[END_REF]:

φ q (t) = f c q + h q |σ|g( t) where t = t/t σ (4.27)
h q is the critical amplitude. By defining G(t) = |σ|g( t) the previous equation may be rewritten as

φ q (t) = f c q + h q G(t) (4.28)
where we immediately see that the time-dependence of φ q (t) is given by the sole function G(t), also referred as the β correlator which is independent of q. As a consequence we remark that φ q (t) -f c q corresponds to the product of a time-independent function h q and a time-dependent function G(t) [START_REF] Binder | Glassy Materials And Disordered Solids[END_REF].

Predictions for the α regime

For the α-relaxation the relevant time-scale is given by the following relationship:

t ′ σ = t 0 |σ| γ (4.29)
where γ is an exponent related to the a exponent through the equation:

γ = 1 2a + 1 2b (4.30)
Here b is the called the von Schweidler exponent. a and b are related one to each other by the exponent parameter λ:

λ = Γ(1 -a) 2 Γ(1 -2a) = Γ(1 + b) 2 Γ(1 + 2b) (4.31)
It has been shown that these exponents do not depend on T and can only take values in a certain range that are given by the equilibrium properties of the glass formers at T c . Thus we expect from a system compliant with MCT that γ > 1.765, 0 < a < 0.3953

and 1/2 ≤ λ ≤ 1 [98].
An important prediction of the ideal MCT is that the α process must satisfy the timetemperature superposition principle (TTSP) when T → T + c which means that assuming a rescaling of the time t by the structural relaxation time t ′ σ , correlators φ q (t, T ) taken at different temperatures T should collapse onto the same master curve φq . This is expressed by the following analytical relation:

φ q (t, T ) = φq (t/t ′ σ (T )) (t ≥ t σ ) (4.32)
This equation has two direct consequences: φq is independent of the temperature and there is only one time-scale t ′ σ associated with the α process.

In the late β process and the early α regime the behavior of the correlator φ q (t) is well-reproduced by the von Schweidler equation:

φ q (t) = f c q -hq (t/t ′ σ ) b + hq Bq (t/t ′ σ ) 2b (t σ ≤ t) (4.33)
where hq = h q B and Bq = BB q , h q is the critical amplitude mentioned above, B is a constant and B q is a q-dependent constant. Note here that we use an expression of the von Schweidler law with a second order corrective term [START_REF] Baschnagel | Computer simulations of supercooled polymer melts in the bulk and in confined geometry[END_REF].

Chapter 4 Dynamical observables 121

Finding T c

The MCT predicts that the structural relaxation time τ α (T ) and the diffusion coefficient D(T ) follow a power law while approaching sufficiently close to T c :

τ α (T ) = C τ (T -T c ) γ and D(T ) = C D (T -T c ) -γ (4.34)
where C τ and C D are constants.

Consequently if one could have access to the exponent γ, the mode-coupling temperature The following procedure is applied to determine all the exponents and T c :

T c could
1. In order to know in which range of temperature the ideal MCT is valid, the TTSP will be checked for temperatures T ∈ [0.84, 0.88].

2.

After defining the time interval of the β process, the factorization theorem will be tested. This step serves as a control to see if this prediction of MCT applies.

3. We use the von Schweidler equation 4.33 to determine the exponent b.

4. Finally a, γ and T c can be found.

We performed the MCT analysis on the coherent intermediate scattering function and to test the reliability of our analysis we check then the validity of the results on the incoherent intermediate scattering function separately for the A and B particles .

Time-temperature superposition principle

As mentioned above the ideal MCT predicts that in the α regime, the TTSP should apply when T → T + c . Thus a first test to check whether or not the ideal MCT applies for the bidisperse Voronoi liquid is to see if there exists an interval of temperatures into which the TTSP is satisfied.

We have thus computed the coherent intermediate scattering function φ q (t) at first peak of the total structure factor S(q). Three main points should be verified in order to confirm that the TTSP applies [START_REF] Frey | Propriétés viscoélastiques des fondus de polymères vitrifiables[END_REF]:

• There should be an overlap of all φ q (t, T ) in the α regime when T is sufficiently close to T c . Chapter 4 Dynamical observables 122

• Assume T 1 < T 2 , φ q (t, T 1 ) should not intersect with φ q (t, T 2 ) or with any φ q (t, T ) computed for higher temperatures.

• The higher the temperature the earlier the departure of φ q (t, T ) from the master curve is observed.

In Figure 4.31 φ q (t) is represented as a function of t/τ α . We observe that in the range of temperatures selected, T ∈ [0.84, 0.88], the different curves of φ q (t) seem to collapse on a master curve. However one may wonder if T = 0.88 is satisfying the TTSP as it apparently does not collapse perfectly on the master curve. As a consequence of these observations the interval of temperatures in which the ideal MCT is valid is T ∈ [0.84; 0.86].

Factorization theorem

Now that we have determined the temperature interval into which the MCT should be valid, we test another of its prediction, that is the factorization theorem. If we assume two times belonging to the β regime, t 1 and t 2 , such as t 1 < t 2 , the factorization theorem Chapter 4 Dynamical observables 123 states that:

R q (t) = φ ( t) -φ q (t 2 ) φ ( t 1 ) -φ q (t 2 ) = G(t) -G(t 2 ) G(t 1 ) -G(t 2 ) (4.35)
where we have used relation 4.27 as an expression for φ q (t). We immediately see that the R.H.S of the equation is q independent. This observation implies that if the factorization theorem applies, different ratios R q computed for different values of q should collapse onto the same master curve. R q (t)

t 1
t 2 q = 3.00 q = 6.85 q = 9.55 q = 11.75 q = 15.35 At T = 0.84, we have computed R q (t) for different wavevectors q which, apart from q = 4.00, have been chosen in such way that they correspond to extrema of S(k). The time t 1 and t 2 have been chosen in order to cover a large part of the β regime while avoiding the huge oscillations at the beginning of the plateau.

In Figure 4.32 we have represented R q (t) and we notice that the different curves collapse onto a master curve for a specific time interval t ∈ [t 1 = 6.0, t 2 = 16.0]. Moreover the ordering of the different curves when entering the plateau is preserved when they leave it. However whereas the collapse seems to remain valid for a short range of time t > t 2 , we clearly see that for t < t 1 the factorization theorem does not hold anymore due to the presence of strong oscillations, in particular for q = 3.00.

Chapter 4 Dynamical observables 124

von Schweidler fit

The main step in this analysis is to fit our data using the von Schweidler equation 4.33 in order to determined the exponent b and the other fitting parameters. To have the best precision, this fit must cover the late β process and the α process. The choice of the fit interval is thus primordial and will depend on the specie under interest. We have decided to fit the data of the lowest temperature that satisfies the TTSP, that is T = 0.84. This choice is motivated by the fact that the lower the temperature the longer the relaxation and consequently the time interval would be larger than for other temperature leading to better results [START_REF] Frey | Propriétés viscoélastiques des fondus de polymères vitrifiables[END_REF]. In what follow we detail the procedure and we then explain the different procedures we use to verify that the results obtained from the fit are in agreement with ideal MCT.

Fit procedure

As mentioned above, the quality of the fit will strongly depend on the choice of the time interval. Our aim is to cover a large part of the late β relaxation and to take into account the α process. We thus need to define a left time-interval associated with the β process and a right-time interval associated with the α for which the fit would give good results.

For this purpose as a first step we started by fixing the right border in the late α process at t = 1500, then we change progressively the value of the left border from t β1 = 6.0 to t β2 = 12.0. The choice of t β1 is based on the fact that we want to avoid the oscillation regime that occurs at the beginning of the plateau. t β2 = 12.0 is a limiting value. It corresponds more or less to the end of the plateau and we also noticed numerically that a choice of t β2 > 12.0 would degrade the quality of the fit. Thus the time interval into which the left border may vary is t lef t ∈ [6.0, 12.0].

In a second step, we fix the value of left border to be equal to t β1 = 6.0 and we vary the value of time in the late α regime from t α1 = 1500 to t α2 = 2500. For t < t α1 the fit does not work well. This might be due to the fact that for t < t α1 a relatively large part of the α process is neglected. For t > t α2 the fit becomes deficient, we maybe cover a time interval which is too large to be suitable for a description with the von Schweidler equation.

To check the reliability of the time-interval for the α regime, we change at the same time the value of the left border to t β2 . We found that this change affects very weakly the fitting parameters and as a result the right border may vary into a time interval

t right ∈ [1500, 2500].

Chapter 4 Dynamical observables 125

The time intervals for the left and right border for the fit with the von Schweidler equation are illustrated in Figure 4.33. One could assume that the best total time interval would have been [t lef t = 6, t right = 2500] as it covers the largest part of φ q (t).

We would have certainly obtained a coherent result but it would have been difficult to appreciate the accuracy. So as we want to obtain the most representative values of the fitting parameters, we decide to increase the statistics by performing the calculation as follow:

1. We start by taking t lef t = 0.6 and t right = 600. We perform the von Schweidler fit and collect the fitting parameters. A representation of the von Schweidler fit is proposed in Figure 4.34.

2. We repeat the step 1 by increasing t right := t right +200 until it reaches t right = 2500.

3. We increase the value of t lef t := t lef t + 1 and we repeat steps 1-2.

4. Steps 1-3 are repeated until t lef t = 12.0 is reached . Average and standards deviation are computed for each of the fitting parameters. This values can be found in Table 4.8. 4.8.

f c q t ′ σ b B q h q average 0.

Other q values

This step consists on checking the reliability of the values of the fitting parameters found in the previous series of fits. The aim is to fit φ q (t) for other values of the wavevector q by reducing the number of fitting parameters. For this purpose we will consider that the values of both exponent b and relaxation time t ′ σ are the ones found previously. It is possible to define the crossover-time t co as the time for which φ q=q * (t co ) = f c q [START_REF] Baschnagel | Computer simulations of supercooled polymer melts in the bulk and in confined geometry[END_REF]. Knowing this time we can thus determine f c q for the other values of q. As a consequence in the following only B q and h q will be varying parameters. This time, t co corresponds logically to the left border of the time interval for the fit. The right border is adjusted for each values of q in such a way that t right < τ α , the relaxation time found when φ q (t = τ α ) = 0.1.

The results are presented in Figure 4.35. We notice that for high-q values, there is a quite good agreement between the constrained fits and the data points which suggests that the fixed values of b, t ′ σ and f c s are consistent. For q = 9.55, the von Schweidler appears less good than for higher q-values. This is related to the fact that there are relatively marked oscillations at the beginning of the plateau and the crossover-time, which defines the left border of the fit interval, is located right in the middle of one of these oscillations leading to a biased fit. For this wavevector results are significantly However the result of the constrained fit is very bad for the lowest q represented. It is not very surprising as marked damped oscillations are visible on the plateau. We have already observed this phenomenon when looking at the important decrease of collective relaxation in the mesoscopic regime. We proceed the same way as for q = 9.55 to see to what extent the fit is improved when we shift the left border to t lef t = 10. Contrary to q = 9.55 the late β regime and the beginning of the α process are not well described by the fit. Test of the von Schweidler fit on φ q (t) for several wavevectors q. The data points are represented by crosses (+). The dot grey line marks the value of the crossover time t co = 0.37. The solid lines represent the von Schweidler fits where the left border of the time interval is given by the crossover time. The dashed black line is the von Schweidler fit for q = 9.35 but this time the left border has been shifted to larger time t lef t = 1.5. The blue dashed line is von Schweidler fit for q = 3.00 where the left border has been shifted to t lef t = 10. The right border is by t right = 100 apart from q = 3.00 where t right = 1500.

This first test of the fitting results are quite convincing apart from the results obtained for the smallest q value. We proceed now to another verification by fitting the incoherent scattering function F s (q, t) that we denote φ s q (t) for the purpose of the following analysis.

Chapter 4 Dynamical observables 128

Incoherent intermediate scattering function

The procedure is similar to the previous one: b and t ′ σ are taken equal to their average value shown in Table 4.8, f sc q , the parameter of non-ergodicity of φ s q (t) is given by the crossover-time t co defined in the same way as before. Thus only B s q and h s q are varying parameters. φ s q (t) q = 3.00 q = 6.75 q = 8.35 q = 11.15 q = 14.45 q = 16.55

We test the von

A particles

Figure 4.36: Test of the von Schweidler fit on φ s q (t) of the particles A for several wavevectors q. The data points are represented by crosses (+). The dot grey line marks the value of the crossover time t co = 0.37. The solid lines represent the von Schweidler fit where the left border of the time interval is given by the crossover time. The right border changes for each q value and is given by t right = [1800, 1000, 700, 300, 200, 100] (from top to bottom).

• For the A particles, the von Schweidler fit appears to be in good agreement with the data for values of q ≥ 6.75.

• For the B particles, the fit seems to underestimate a bit the value of the plateau but the early α process appears well fitted.

• For q = 3.00 we notice that in both cases the fits poorly describe φ s q (t). We recover the same issue that we had for the coherent intermediate scattering function.

Chapter 4 Dynamical observables 129

The last point is not really suprising as the memory kernel involved in the MCTequations of the incoherent intermediate scattering function is not quadratic in F s (q, t), but is in fact a bi-linear function of F s (q, t) and F (q, t) [START_REF] Binder | Glassy Materials And Disordered Solids[END_REF]. This implies that to some extent the relaxation dynamics of a single tagged particle depends on the one of the whole ensemble of particles. This might explain why we recover the same issues for q < q * . Behaviors are qualitatively the same for φ(q, t) and φ s q (t). φ s q (t) q = 3.00 q = 7.15 q = 10.75 q = 12.65 q = 14.65 q = 17.05

B particles

Figure 4.37: Test of the von Schweidler fit on φ s q (t) of the particles B for several wavevectors q. The data points are represented by crosses (+). The dashed grey line marks the value of the crossover time t co = 0.37. The solid lines represent the von Schweidler fit where the left border of the time interval is given by the crossover time. The right border changes for each q values and is given by t right = [1200, 800, 500, 300, 200, 50] (from top to bottom).

To understand what occurs at low q, we proceeded to other investigations by testing first the constrained fit on φ s q (t) for other mesoscopic wavevectors. The main advantage of working with φ s q (t) is the lack of damped oscillations on the plateau. That allows for a better fitting process. However as for q = 3.00, it leads to very bad collapse between the fit and the data points. So we then decided to remove some constraints and to proceed to a fit by leaving the b exponent as a free parameter. Surprisingly we notice that the exponent b is more important in the mesoscopic regime than for high q values and that the lower q the higher b. It is indeed what we observe in Table 4.9 where we have collected the b exponent for several q. These observations suggest that for wavevectors in the mesoscopic range there is an apparent violation of the MCT predictions for the coherent and the incoherent intermediate scattering function. Apart from phenomena that occur in the mesoscopic regime we have observed that predictions of the MCT are well-verified. Therefore we assume that the ideal MCT applies and that we can compute the mode-coupling temperature T c . The procedure is detailed below.

Determining T c

From the fitting procedure, we have access to the average value of the exponent b, Once γ is found, we can use the equation 4.29 that gives the expression of t ′ σ as a function of the separation parameter σ. As we explained, σ measures the distance from T c thus it can be rewritten as σ ∼ T -T c and consequently the expression for t ′ σ is now given by [START_REF] Frey | Propriétés viscoélastiques des fondus de polymères vitrifiables[END_REF]:

consequently
(t ′ σ ) -1/γ = a(T -T c ) (4.36)
By assuming that τ α ∼ τ ′ σ we can easily have access to T c with a linear fit as it can be seen in Figure 4.38. The value of T c is found when linear fit intersects the T axis. We find that T c = 0.7989.

T c is a characteristic temperature of the glass transition. By considering equation 4.29 we see that T c corresponds to the temperature at which the relaxation time would diverge revealing the transition from the ergodic to the non-ergodic behavior. Experimentally this is not observed and the relaxation time increases when cooling systems towards the glass transition temperature [START_REF] Baschnagel | Computer simulations of supercooled polymer melts in the bulk and in confined geometry[END_REF]. This divergence is thus an artefact of the theory which suggests that some other relaxation mechanisms should be taken into account and extension of the ideal MCT have been proposed [START_REF] Fuchs | A theory for the beta-relaxation process near the liquid-to-glass crossover[END_REF].

Among the effects neglected in ideal MCT one may think about the hopping processes between minima that prevail at low temperature [START_REF] Goldstein | Viscous liquids and the glass transition: A potential energy barrier picture[END_REF]. Therefore some useful informations on the slowing down of the dynamics are likely to be found in the underlying potential energy landscape whose study is the scope of next chapter. In 1969, Goldstein [START_REF] Goldstein | Viscous liquids and the glass transition: A potential energy barrier picture[END_REF] proposed an explanation of the huge increase of the viscosity based on the evolution of a system in its phase space. It corresponds to the space of all the configurational degrees of freedom [START_REF] Cavagna | Supercooled liquids for pedestrians[END_REF]. Each configuration of the system is represented by a point in phase space and the dynamics of the system consists on motion of the point on the potential energy landscape (PEL) associated to the phase space [START_REF] Cavagna | Supercooled liquids for pedestrians[END_REF].

Goldstein's ideas were the following [START_REF] Goldstein | Viscous liquids and the glass transition: A potential energy barrier picture[END_REF] :

(i) At low temperature, as crystals, glassy systems are close to a minimum.

(ii) However, as the thermodynamic properties are sensitive to the cooling process, this implies that potential energy surface is not smooth but contains plenty of minima of various depth.

(iii) A system spends most of its time vibrating in amorphous minima but thanks to thermal energy it manages to overcome energy barriers and to visit other minima ensuring thus the ergodicity. As a consequence there is a clear separation of time scales: the thermal relaxation which corresponds to the vibration into a minimum which is fast and the hopping between the several minima which require much more time to occur.

(iv) Overcome a potential energy barrier requires a local rearrangement of the system meaning that most of the particles would remain at the same position. Only a small number of particles are going to travel on a significative distance. More important these rearrangements are localized in limited regions of the space.

(v) This potential energy barriers description is not sufficient to describe the hightemperature behavior when thermal fluctuations are comparable to heights of the energy barriers. This description suggests that upon cooling there is a crossover in the dynamics of a liquid from continuous fluid-type motion to an activated dynamics which consists on hopping over energy barriers [START_REF] Das | Statistical Physics of Liquids at Freezing and Beyond[END_REF]. As ideal MCT, which does not take activation into account, manages to describe the high-temperature phenomenology of supercooled liquids, it is generally assumed that the mode-coupling temperature T c corresponds to the crossover temperature between the non-activated and the activated regime.

Goldstein gave a qualitative description of the dynamics near minima and we owe to Stillinger and Weber [START_REF] Stillinger | Hidden structure in liquids[END_REF], the development of a formalism, still used today, that enables a thermodynamic description of the inherent structures (minima of E P ).

The aim of this chapter is to describe the PEL of the bidisperse Voronoi liquid. In a first section devoted to the study of inherent structures, we recall the formalism developed by Stillinger and Weber. It enables to derive an expression for the configurational entropy and therefore to test the Adam-Gibbs theory for our system.

In a second part we focus on saddles which are stationary points of the PEL with an arbitrary number of unstable directions. They play a significative role to understand the link between PEL and dynamics in real space when probing temperatures for which the hopping between minima could be neglected.

Inherent structures

In 1982, Stillinger and Weber published an article in which they introduced the concepts of what we call now basins and also presented the procedure that can be used to partition the PEL into disjoint basins [START_REF] Stillinger | Hidden structure in liquids[END_REF].

Stillinger and Weber noticed that there exists a set of points in the configurational space that are connected to the local inherent structure (IS) via an energy minimization protocol, as for instance the steepest-descent [START_REF] Heyes | Chemical Modelling Applications and Theory, volume 2, chapter Simulation of the Liquid State[END_REF][START_REF] Stillinger | Hidden structure in liquids[END_REF]. A typical sketch of the situation is represented in Figure 5.1 where we observe that each thermalized state, is uniquely associated, via quenching, to an IS. It appears thus that the configurational space can be partitioned into the different contributions coming from the different basins. This last observation has enabled the derivation of a theoretical framework that we examine in the following paragraph.

Theoretical framework

Let consider a binary mixture of N = N A + N B particles contained into a volume V .

The multidimensional vector r N = (r 1 , • • • , r N ) contains the configurational coordinates of the N particles. As in [START_REF] Das | Statistical Physics of Liquids at Freezing and Beyond[END_REF] we denote V (r N ) the potential of this system. In these conditions, the partition function Z N is given by:

Z N = 1 N A !N B !λ 3N V d N r exp(-βV (r N )) (5.1)
where λ = 2πβ 2 /m is the thermal de Broglie wavelength. Now if we assume that the configurational can be decomposes as the sum over all distinct basins, it is possible to rewrite the integral in the partition function as:

V dr N exp(-βV (r N )) = α exp(-βe α IS ) Bα dr N exp(-β Ṽα (r N )) (5.2)
where B α means that the integral is computed for the set of points of the αth basin [START_REF] Das | Statistical Physics of Liquids at Freezing and Beyond[END_REF].

The potential energy of the system has been decomposed into a sum of two elements V (r N ) = e IS + Ṽ (r N ), the potential energy of the inherent structure e IS and Ṽ is a non-negative quantity that measures, at a point r N ∈ B α , the potential energy relative to the minimum [START_REF] Stillinger | Hidden structure in liquids[END_REF][START_REF] Sciortino | Thermodynamics of supercooled liquids in the inherent-structure formalism: a case study[END_REF].

Minima can be classified according to their IS energy and as a consequence the sum over the basins can be split into a sum over all the possible values of e IS and the sum over all basins α ′ at the same energy e IS [START_REF] Sciortino | Thermodynamics of supercooled liquids in the inherent-structure formalism: a case study[END_REF]. This implies the following expression for the partition function:

Z N = 1 λ 3N e IS exp(-βe IS ) α ′ B α ′ dr N exp(-β Ṽα ′ (r N )) (5.3)
We notice here that N A !N B ! vanishes as the sum over all distinct basins and the permutation of particles does not change the basin [START_REF] Das | Statistical Physics of Liquids at Freezing and Beyond[END_REF]. Henceforth, we consider only distinct IS with respect to trivial particle permutations.

As Stillinger and Weber, we define Ω(e IS ) the IS density of states which counts the number of distinct basins of energy between e IS and e IS + δe IS [START_REF] Sciortino | Thermodynamics of supercooled liquids in the inherent-structure formalism: a case study[END_REF]. It is then possible to define an average basin free energy f (β, e IS ) as:

-βf (β, e IS ) = ln 1 λ 3N δe IS Ω(e IS ) α ′ B α ′ dr N exp(-β Ṽα ′ (r N )) (5.4)
We can therefore rewrite equation 5.3 as a function of the energy of the IS:

Z N = de IS Ω(e IS ) exp(-βe IS -βf (β, e IS )) (5.5)
The free energy F of the system is minimal when the integrand in equation 5.5 is maximal. Hence in the thermodynamic limit, a maximum-integrand evaluation enables to determine an expression for F :

F = e * IS -T S c (e * IS ) + f (β, e * IS ) (5.6)
where e * IS is the value that minimizes F . S c is the configurational entropy which measures the number of distinct IS and is defined as:

S c (e IS ) = k B ln(Ω(e IS )δe IS ) (5.7)
Considering the basin free energy f (β, e * IS ) in equation 5.6, it can be decomposed into its energetic term u b and an entropic contribution s b , which implies that the configurational entropy S c corresponds to the difference between the liquid entropy S liq and the basin entropy s b , i.e the entropy of the system remaining in an IS with energy e * IS [START_REF] Sciortino | Thermodynamics of supercooled liquids in the inherent-structure formalism: a case study[END_REF].

Low-temperature approximations regarding the form of f (β, e * IS ) can be done [START_REF] Sciortino | Thermodynamics of supercooled liquids in the inherent-structure formalism: a case study[END_REF]. • βf (β, e * IS ) ≈ f (β): basins are likely to have the same shape in configuration space.

• βf (β, e * IS ) ≈ g(β) + h(e IS ): at different T , system visits basins that are always harmonic but have different densities of states. This is this factorization approximation that we consider thereafter.

For both forms, it is possible to factorize the partition function of equation 5.5 as

Z N ≈ Z IS Z b .
In the range of temperature where the factorization holds, the system can be decomposed into one subsystem corresponding to the IS which is weakly coupled to the vibrational degrees of freedom. The coupling between the two subsystems, essential for the process of equilibration, is possible via the weak T -dependence of the e IS which is neglected in the factorization approximation. We look in deeper details to the result of the factorization approximation under the harmonic approximation which yields:βf (β, e * IS ) ≈ g(β) + h(e IS ).

Starting from equation 5.5, in the harmonic approximation, the Hessian of the potential energy is denoted H. The term exp(-βf (β, e * IS )) can be written as:

exp(-βf (β, e * IS )) = 1 λ (3N -3) d (3N -3) r exp - β 2 r ⊺ Hr = 1 λ 3N 3N -3 i=1 dy i exp - β 2 m i y 2 i ω 2 i = 3N -3 i=1 2πT m i ω 2 i λ 2 1/2 (5.8)
where ω i are the eigenfrequencies of the Hessian of the potential energy and m i are the masses (m i = 1). Thus by taking the logarithm of equation 5.8 we find:

-βf (β, e * IS ) = 3N -3 i=1 1 2 ln T 2 ω 2 i 2 (5.9)
Therefore in the case of the bidisperse Voronoi liquid, equation 5.6 can be expressed as:

F = e * IS -T S c (e * IS ) -T 3N -3 i=1 1 2 ln T 2 ω 2 i 2 (5.10) = e * IS -T S c (e * IS ) - 3N -3 i=1 ln ω i γv 5/3 = Ŝc (e * IS ) -T 3N -3 i=1 ln T γv 5/3
(5.11)

where we define Ŝc (e IS ) as the effective configurational entropy.

As mentioned by Sciortino et. al in [START_REF] Sciortino | Thermodynamics of supercooled liquids in the inherent-structure formalism: a case study[END_REF], it is possible to look at a range of temperatures where the factorization holds by mean of numerical simulations. Assuming an equilibrated system at temperature T , we can define the probability density for finding a configuration belonging to a basin with IS energy e IS :

P (e IS |T ) = Ω(e IS ) exp[-βe IS -βf (β, e IS )] Z N (T ) (5.12)
By taking the logarithm of previous expression, we find: The aim of this section is to determine the configurational entropy S c and the effective one Ŝc and then to use it to test the Adam-Gibbs scenario that we will explain later.

ln[P (e IS |T )δe IS ] + βe IS = k -1 B Ŝc (e IS ) -T 3N -3 i=1 ln T γv 5/3 -ln[Z N (T )] (5
But prior to that we will briefly explain how IS are computed.

Computing IS and Gaussian landscape

As mentioned above any algorithm that minimizes the potential energy E P is a good candidate to compute inherent structures. As there is a clear splitting of the PEL in terms of IS and related basin, the results should not differ from an algorithm to another one. In practice we selected the Fast Inertial Relaxation Engine algorithm (better known as FIRE) as it was shown that this algorithm is faster and has a better convergence than usual conjugate gradient or Newton methods [START_REF] Bitzek | Structural relaxation made simple[END_REF].

We have computed the IS for the bidisperse Voronoi liquid and the LJ glass-formers.

In Figure 5.2 we have represented the evolution of (e IS -e IS,∞ )/(k B T * ) as a function of T * /T . e IS,∞ is the average of the potential energy of the IS computed from hightemperature thermalized configurations. For all the systems we noticed that for T > T * , the value of e IS is almost constant, thus we have evaluated e IS -e IS,∞ in order to measure the relative decrease of e IS upon cooling. Results are presented in Figure 5.2.

We observe that the bidisperse Voronoi liquid has the more pronounced decrease of e IS upon cooling. Regarding the LJ glass-formers, it seems that the more fragile the mixture, the more marked the decrease. We also notice the presence of a linear regime at low temperature which is more visible in Figure 5.3 where we have represented e IS as a function of 1/T . This observation is in agreement with the predictions of the random energy model (REM) [START_REF] Derrida | Random-energy model: An exactly solvable model of disordered systems[END_REF][START_REF] Keyes | Entropy, dynamics, and instantaneous normal modes in a random energy mode[END_REF]. For the REM the number of distinct basins of energy between e IS and e IS + de IS , which is given by Ω(e IS )de IS , can be described by a Gaussian distribution [START_REF] Sciortino | Potential energy landscape description of supercooled liquids and glasses[END_REF]:

Ω(e IS )de IS = e αN exp[-(e IS -E 0 ) 2 )/(2σ 2 )] √ 2πσ 2 de IS (5.14)
where the e αN accounts for the total number of basins, E 0 is an energy scale and σ 2 measures the width of the distribution. By using the REM (or Gaussian harmonic approximation), one assumes that each IS can be decomposed into independent subsystems which have their own values of e IS . Therefore, in the thermodynamic limit, the central limit theorem applies and the IS energy follows the gaussian distribution 5.14.

The knowledge of Ω(e IS ) enables to rewrite all the thermodynamic observables described above. It can be shown [START_REF] Sciortino | Potential energy landscape description of supercooled liquids and glasses[END_REF][START_REF] Heuer | Why is the density of inherent structures of a lennard-jones-type system gaussian?[END_REF] that the average value of the IS energy obeys the following relationship:

e IS (T ) = E max - σ 2 k B T (5.15)
where E max depends on E 0 and σ 2 . We immediately see that equation 5.15 implies a linear behavior of e IS (T ) when represented as a function of 1/T . It has been shown

that theoretical predictions of the Gaussian harmonic approximation are only valid in a range of temperature where the two-step relaxation behavior is already well-developed [START_REF] Sciortino | Potential energy landscape description of supercooled liquids and glasses[END_REF]. Therefore it is not a surprise to see that the linear regime is only seen at low temperature in Figure 5.3. We performed a linear fit on the range of temperature where equation 5.15 holds. We found thus values of σ 2 that are shown in Table 5 The relatively big large value of σ 2 could be correlated to the important values of C e v for the bidisperse Voronoi liquid. Indeed if we assume that E P = e IS + 3N T /2, we find that:

C e V = ∂ e IS ∂T + 3N 2 = σ 2 T 2 + 3N 2 (5.16)
From equation 5.16, we notice that the larger σ 2 , the more important is C e V .

Previous numerical studies have shown that fragile glass-formers generally satisfy the predictions of the Gaussian harmonic approximation. Our observations are thus consistent with the fact that both the bidisperse Voronoi liquid and the LJ glass formers exhibit a certain degree of fragility.

The Gaussian harmonic approximation provides also an expression which can be numerically tested for the probability density P (e IS , T ) defined in equation 5.12. We will see this in more details when computing the configurational entropy from the IS.

Configurational entropy

In what follow we compute the configuration entropy in two different ways. On the one hand, we use the fact that S c is given by the difference between the liquid and the basin entropies. On the other hand, we will see that the low-T factorization approximation holds and consequently S c may be expressed as a function of e IS . We will test the consistency between the two approaches.

Temperature dependence

In Chapter 2 we computed the excess part of the entropy S e from the knowledge of the excess part of the free energy F e defined in equation 2. [START_REF] Berthier | Direct experimental evidence of a growing length scale accompanying the glass transition[END_REF]. The total entropy of the liquid is S liq = S id + S e where S id is the ideal part defined in the case of a binary mixture as [START_REF] Shi | Structure, dynamics, and thermodynamics of a family of potentials with tunable softness[END_REF]:

S id (T, V, N ) N k B = - N A N ln N A N - N B N ln N B N + 3 2 ln mV 2/3 β 2 2π -ln(N ) + 5 2 (5.17)
It can be assumed that for the lowest T studied, the term f (β, e IS ) in equation 5.13 can be approximated by the harmonic free energy of a disordered solid whose eigenfrequencies are calculated from the IS at T [START_REF] Sciortino | Thermodynamics of supercooled liquids in the inherent-structure formalism: a case study[END_REF]. Thus the basin entropy corresponds to the one of the disordered solid S sol (T, V ) and is given by:

S sol (T, V ) = 3N -3 j=1 [1 -ln(β ω j )] (5.18)
where ω j is the eigenfrequency of the jth normal mode.

Both liquid entropy S liq (T ) and entropy of the disordered solid S sol (T ) are represented in Figure 5.4 where we observe that as expected they both decrease upon cooling. From the lowest accessible temperatures, we can extrapolate the values of S liq (T ) and S sol (T )

to lower temperatures. If the extrapolation are valid, we observe that at a certain temperature, the entropy of the liquid is equal to the entropy of the disordered solid.

This temperature, that we denote T K , is analogous to the Kauzmann temperature [START_REF] Kauzmann | The nature of the glassy state and the behavior of liquids at low temperatures[END_REF],

which is defined as the temperature for which the liquid entropy equals the crystal entropy. Thus in our case, T K corresponds to the Kauzmann temperature, only if the crystal vibrational entropy is a good approximation of the vibrational entropy of the basin [START_REF] Sciortino | Thermodynamics of supercooled liquids in the inherent-structure formalism: a case study[END_REF].

0.5 1 1.5 2 T -7 -6 -5 -4 -3 -2 -1
S liq /N S sol /N T K = 0.687 The knowledge of S liq (T ) and S sol (T ) allows the determination of S c (T ) = S liq (T ) -S sol (T ) which is represented in Figure 5.5. We observe that the configurational entropy decreases upon cooling and that if we extrapolate its value to the temperature T K for which S c (T K ) = 0, we find T K = 0.687. We have computed S liq and S sol and deduced the value of S c . However we need to test whether the harmonic approximation is valid to see to what extent the configurational entropy is reliable. Therefore we evaluate the energy of the anharmonicity by using the following relation [START_REF] Sciortino | Potential energy landscape description of supercooled liquids and glasses[END_REF]:

E anh (T, V ) = E P (T, V ) -e IS (T, V ) - 3 2 N k B T (5.19)
One possible way to describe anharmonicities is by assuming that E anh is only a function of the temperature and can be described by polynomial whose lower order is T 2 [START_REF] Sciortino | Potential energy landscape description of supercooled liquids and glasses[END_REF]:

E anh (T, V ) = imax i=2 c i (V )T i (5.20)
where c i is a coefficient that only depends on the volume V . The corresponding anharmonic entropy is given by:

S anh (T, V ) = imax i=2 ic i (V ) i -1 T i-1 (5.21)
In the upper part of Figure 5.6 the energy per particle of anharmonicity for the bidisperse Voronoi liquid is represented as a function of the temperature. As expected the anharmonicities become more important when the temperature is increased. In the range of temperature investigated, the values of E anh /N are of the order of 10 -1 and therefore are only 10 times smaller than the vibrations 3/2k B T what means that for the bidisperse Voronoi liquid, anharmonicities might play a role at some point.

Below T * , we can describe E anh by using the polynomial expressed in equation 5.20.

However we notice that we need to start with a term in T 4 to obtain a good fit. Indeed, polynomials in T 2 and T 3 were not able to fit the data. We have thus described E anh with a polynomial c 6 T 6 +c 5 T 5 +c 4 T 4 . We also observed that if we limit the investigation to T < 1.00, E anh for the Voronoi liquid can be well described by only c 4 T 4 .

As it is surprising that the starting term of the polynomial is in T 4 , we computed

Effective configurational entropy

We have determined the configurational entropy S c (T ) in the harmonic approximation as S c = S liq -S sol . We look now for the value of the effective configurational entropy Ŝc (e IS )

which is linked to S c via equation 5.11. The aim is to see first whether the factorization approximation holds, meaning that in equation 5.13 only the term Ŝc would depend on e IS . Then assuming it is the case, we would have access to Ŝc up to a constant and therefore we could determine whether the harmonic approximation is valid.

The first step to test the factorization approximation is to compute the probability density P (e IS |T ) defined in equation 5.12. As P (e IS |T ) corresponds to probability to find from a thermalized configuration an IS with energy e IS knowing the temperature T , it is easily accessible numerically. In Figure 5. that the Gaussian harmonic approximation (see equation 5.14) holds [START_REF] Saika | Distributions of inherent structure energies during aging[END_REF]:

P (e IS |T ) = 1 2πσ 2 P exp - (e IS -e IS (T, V ) ) 2 2σ 2 P (5.22)
Here the width of the gaussian σ P and the average value e IS (T, V ) are both functions of variables E 0 and σ defined in equation 5.14 [START_REF] Saika | Distributions of inherent structure energies during aging[END_REF]. We fitted the data of P (e IS |T ) with equation 5.22 and we notice a good agreement between the data points and the fit as observed in Figure 5.8. In practice, we determine Ŝc directly from S liq as it avoids the determination of the eigenfrequencies of the Hessian of E P . Thus if the factorization approximations works: Ŝc = S liq -3(N -1) 1 + ln T γv 5/3 (5.23)

In equation 5.23, Ŝc depends on T . To determine its e IS -dependence, for T ≤ 1.00,

we fit e IS (T ) with a polynomial expression of order 4. This enable us to determine Ŝc (e IS ) and therefore it can be compared to the polynomial of order 2 that represents the master curve in Figure 5.10. This comparison is shown in Figure 5.11 where we observed that there is a good agreement between the two methods for values of e IS ∈ This observation immediately raises the question of the validity of the harmonic approximation. We have seen previously that the anharmonic contribution are non-negligible and therefore we should take them into account. We have noticed that for T ≤ 1.00,

E anh ≈ N c 4 T 4
where c 4 is a constant. Thus with equation 5.21 we find that S anh ≈ 4 3 c 4 T 4 (where c 4 = 0.17). By taking into account anharmonicities, equation 5.23 is now given by:

Ŝc = S liq -3(N -1) 1 + ln T γv 5/3 - 4 3 c 4 T 3 (5.24)
In Figure 5.12, we compare Ŝc from equation 5.24 with the polynomial of order 2 that represents the master curve in Figure 5.10. We notice first that adding the anharmonic contribution seems to improve a bit the results, especially for e IS ≥ 1.1375•10 5 . However for e IS ≥ 1.1370 • 10 5 the agreement between equation 5.24 and the fit in Figure 5.10 is less good than in the harmonic approximation. Ŝc fit master curve Ŝc anharmonic approx These mixed results point toward the fact that formulation 5.20 of E anh is likely to be not adapted to our model. We have already stressed that for the bidisperse Voronoi liquid E anh = c 4 T 4 whereas E anh = c 2 T 2 is expected.

In [START_REF] Sciortino | Potential energy landscape description of supercooled liquids and glasses[END_REF], Sciortino proposed another formulation for the energy relative to anharmonicity, which comes from the perturbative development at order 3 of the Hessian of E P , where the T 2 dependence is assumed to be always valid but this time c 2 is more a constant but depends on e IS :

E anh = c 2 (e IS )T 2 . ( 5.25)
As a consequence the corresponding anharmonic entropy is:

S anh = -2c 2 (e IS )T (5.26)
We can therefore rewrite equation 5.24 as:

Ŝc = S liq -3(N -1) 1 + ln T γv 5/3 -2c 2 (e IS )T (5.27)
Again we compare the results of equation 5.27 with the fit obtained from Figure 5.10 as shown in Figure 5.13. We notice that within this approximation of the anharmonicity, the agreement between equation 5.27 and the polynomial fit of order 2 is worst pointed toward the fact that this description of anharmonicity does not apply for our system.

Adam-Gibbs theory

The knowledge of the configurational entropy S c (T ) allows to test the Adam-Gibbs theory which links the relaxation time through the configurational entropy via relation [START_REF] Adam | On the temperature dependence of cooperative relaxation properties in glass forming liquids[END_REF]:

τ (T ) = τ 0 exp B T S c (T ) (5.28)
where B is a constant. The activation energy in this model is thus ∆E ∝ 1/S c (T ).

Relation 5.28 comes from the fact that Adam and Gibbs postulated that structural relaxation is due to rearrangement of regions of correlated particles that they called cooperative rearranging region (CRR). These rearrangements could be due to a local fluctuation of the enthalpy that promote collective motion. The Adam-Gibbs model assumes the following ideas [START_REF] Adam | On the temperature dependence of cooperative relaxation properties in glass forming liquids[END_REF]:

(i) The CRR is defined as the smallest region that can rearrange independently from their surrounding environment.

(ii) A typical CRR has a small number Ω of accessible states. The minimal number of accessible states to a CRR is Ω = 2: the state before the transition and the state after the rearrangement.

(iii) The configurational entropy S c is directly related to Ω and the number of particles involved in CRR.

(iv) The energy barrier ∆E is proportional to the number of particle involved in the CRR and as consequence the size of CRR increases when S c decreases.

We also immediately see from equation 5.28 that whether the configurational entropy vanishes for T K > 0 (T K corresponds to the Kauzmann temperature), the structural relaxation time would diverge suggesting the presence of a phase transition. In 1958, prior to the theory of Adam and Gibbs, Gibbs and Di Marzio demonstrated that the vanishing S c leads to a second order phase transition at T K , also called ideal glass transition, and the resulting phase corresponds to a disordered solid to which is associated only one peculiar configuration [START_REF] Gibbs | Nature of the glass transition and the glassy state[END_REF][START_REF] Stillinger | Supercooled liquids, glass transitions, and the kauzmann paradox[END_REF]. Equation 5.28 provides also an interesting explanation for the fragility. By assuming that configurational entropy is given by S c = S liq -S sol , we easily see that the variation of configurational entropy ∆S c (T ) is proportional to ∆C p the variation of heat capacity at constant pressure. Adam and Gibbs assumed that ∆C p is T -independent. Therefore if one considers that the configurational entropy vanishes at the Kauzmann temperature [START_REF] Cavagna | Supercooled liquids for pedestrians[END_REF][START_REF] Adam | On the temperature dependence of cooperative relaxation properties in glass forming liquids[END_REF][START_REF] Binder | Glassy Materials And Disordered Solids[END_REF]. Thus equation 5.28 becomes:

T K , it results that for T close to T K , S c (T ) ∼ ∆C p (T -T K )/T K
τ (T ) = τ 0 exp BT K ∆C p T (T -T K ) T →T K ≈ τ 0 exp B ∆C p (T -T K ) (5.29)
We recognize that the last term in equation 5.29 corresponds to the VFT law and by comparing with equation 4.7, two observations can be made:

• The fragility is proportional to the variation of the heat capacity ∆C p at glass transition [START_REF] Cavagna | Supercooled liquids for pedestrians[END_REF].

• The temperature T 0 at which the VFT law diverges, corresponds to the Kauzmann temperature T K . In Figure 5.5 we have shown that the temperature at which S c (T )

vanishes for the bidisperse Voronoi liquid is given by T K = 0.687. In Chapter 4 by mean of the VFT equation 4.8, we were able to extract T 0 which was equal to T 0 = 0.691 (see Table 4.1). Thus we notice that T K ≈ T 0 .

We test the Adam-Gibbs theory for the bidisperse Voronoi liquid. In Figure 5.14 we have represented in a semi-log plot the structural relaxation time τ α as function of 1/(T S c ).

If the Adam-Gibbs theory holds, we should observe a linear relation between the two quantities and it is indeed the case for low temperatures (T < 0.92). This observation tends to suggest that at enough low temperature, the structural relaxation is partly driven by cooperative rearrangements. Despite the success of the Adam-Gibbs theory, it failed to win unanimous support for several reasons:

• The predictions of Adam-Gibbs theory regarding the fact that T K /T 0 ∼ 1 is generally observed for fragile liquids [START_REF] Angell | Entropy and fragility in supercooling liquids[END_REF] but it is not satisfied when liquids tends to have an intermediate behavior or are considered as strong [START_REF] Tanaka | Relation between thermodynamics and kinetics of glass-forming liquids[END_REF].

• The ideal glass transition is controversial, firstly because this range of temperature cannot be reached experimentally and secondly one can wonder about the nature of the "new phase" reached at T K [START_REF] Dyre | A brief critique of the adam-gibbs entropy model[END_REF]. Moreover whether a transition occurs, there should be a diverging length scale associated to it. However Adam and Gibbs didn't provide a clear definition of what should be this diverging length-scale.

• The Adam-Gibbs theory has been verified numerically for LJ glass formers as KA and WAHN systems [START_REF] Sengupta | Adam-gibbs relation for glass-forming liquids in two, three, and four dimensions[END_REF][START_REF] Mossa | Dynamics and configurational entropy in the lewis-wahnström model for supercooled orthoterphenyl[END_REF]. However this validity is also directly linked to the way configurational entropy is computed. Recently Berthier and Coslovich proposed a novel approach to determine the configurational entropy [START_REF] Berthier | Novel approach to numerical measurements of the configurational entropy in supercooled liquids[END_REF] where no evaluation of the liquid entropy neither the vibrational entropy is required. In particular they showed for the KA model and for a binary mixture of harmonic spheres that the Adam-Gibbs theory does not apply anymore for T > T c .

With this relatively good verification of the Adam-Gibbs theory, we end our study devoted to the IS. We focus now on saddles which are other existing stationary points of the PEL. They can be relevant to get a better understanding of the dynamics when Goldstein's description breaks at sufficiently high temperature.

Saddles

Goldstein's description of the dynamical evolution of a supercooled liquids in terms of the motion of the state point in the PEL is valid at low temperature where there is a clear separation of time scales between the thermal relaxation into basins (intra-basin relaxation) and the relaxation related to the hoppings from basin to basin by overcoming energy barriers (inter-basins relaxation).

However when the temperature is increased, there exists a crossover temperature T x for which these two time scales are of the same order, and therefore Goldstein's description breaks down. Angell [START_REF] Angell | Perspectives on the glass transition[END_REF] and then Sokolov [START_REF] Sokolov | The glass transition: general scenario and crossover temperature[END_REF] suggested that T x corresponds to the mode-coupling temperature T c as for temperature above T x , experimental results satisfy MCT predictions, whereas for T < T x it is no more the case.

In 2001, Cavagna published a letter [START_REF] Cavagna | Fragile vs. strong liquids: a saddles-rules scenario[END_REF] in which he proposed a generalization for T > T x of Goldstein's landscape picture. His central idea is that for high temperatures the relevant stationary points of the PEL are no more the minima, but saddles. This can be understood by the fact that at high-temperature,

Theoretical considerations

Saddles are unstable equilibrium points (∇E P = 0), characterized by an arbitrary number K of negative eigenvalues of the Hessian of the potential energy [START_REF] Cavagna | Fragile vs. strong liquids: a saddles-rules scenario[END_REF]. If we assume working with N particles in a 3-dimensional space, then the index K can take any values between K = 0 and K = 3N . We can define:

• the index density of negative direction denoted k and defined as k = K/3N ;

• the potential energy density u = U/N where U is the average potential energy

E P .
An observable useful to consider, is the average index density k of saddles located at potential energy density u, which is assumed to be a increasing function of u and this was indeed proven initially by Broderix et al. [START_REF] Broderix | Energy landscape of lennard-jones liquid: Statistics of stationary points[END_REF] and Angelani et al. [START_REF] Angelani | Saddles in the energy landscape probed by supercooled liquids[END_REF] who published independent studies in which they observed that for LJ liquids k(u) increases monotonously with u.

The notion of potential energy barrier can be introduced for saddles and is defined as the energy ∆U required to pass from a saddle of index K to an adjacent higher saddle of index K + 1. We immediately see that, in the case where K = 0, we recover the standard definition of an energy barrier between a minima and a simple saddle. If k(u) is known, the average energy ∆U can be estimated [START_REF] Cavagna | Fragile vs. strong liquids: a saddles-rules scenario[END_REF]:

∆U ≈ 1 3
du dk (5.30) This last approximation means that potential barriers between saddles change as the slope of the function k(u).

As Goldstein did with minima, Cavagna proposed a description of the diffusion in terms of motion of the state point among the neighborhood of saddles [START_REF] Cavagna | Fragile vs. strong liquids: a saddles-rules scenario[END_REF]. A relevant question is now how to define the neighborhood of a saddle point ? Stationary points are known to be zeros of function W = |∇U | 2 [START_REF] Weber | Local order and structural transitions in amorphous metal-metalloid alloys[END_REF][START_REF] Mciver | Structure of transition states in organic reactions. general theory and an application to the cyclobutene-butadiene isomerization using a semiempirical molecular orbital method[END_REF]. Therefore saddle points can be seen as solutions of W = 0 for which the density index k > 0. Similarly to the approach of Stillinger and Weber, a quench of W can therefore be performed and results depends on temperature:

• if T < T x , U is small enough to ensure that on average k = 0, thus a quench is likely to lead to a minimum of U ;

• if T > T x , a quench leads to a saddle of U .

However it is important to stress that contrary to the approach Stillinger and Weber, the method described previously does not partition the whole phase space into basins and saddles as there exists local minima of W which are > 0 [START_REF] Cavagna | Fragile vs. strong liquids: a saddles-rules scenario[END_REF]. However in 2003, Wales and Doye proposed a theoretical and a computational scheme that allows to partition the PEL into basins whatever the stationary point [START_REF] Wales | Stationary points and dynamics in high-dimensional systems[END_REF]. These different approaches developed for computing saddles are discussed in the following subsection.

Cavagna identified two main mechanisms denoted A and B, that could lead to the motion of the state point among saddles:

(A) The first mechanism is driven by activation as it implies the crossing of an potentialenergy barriers. The system close to a saddle of index K, takes an uphill stable direction to reach a saddle of index K + 1 and then take an downhill direction to end in a saddle with the same potential energy as the initial one. As in the case of inherent structures, overcoming an energy barrier implies structural rearrangement in the system involves a finite number of particles. The probability to have a transition is given by

P A ∼ exp[-∆U/k B T ]
where ∆U is the energy barrier estimated above.

(B) The second mechanism is driven by entropy. Saddles have generally nonzero K, the system can use one of the unstable downhill directions to reach a lower potential energy level with an excess of kinetic energy that could be used to take an uphill direction in order to reach finally a saddle with the same potential energy as the initial one. At very low temperature when only minima are visited this mechanism becomes inefficient by the system as there is no more negative direction.

Efficiency of both mechanisms decreases upon cooling and for a given T we can assume that the diffusion is run by the most efficient one. To have a better insight into the mechanism responsible for the slowing-down of the dynamics it is interesting to compare both mechanisms. To do so it is useful to start by looking at mechanism B and to determine the temperature T B at which the mechanism would be frozen.

Therefore we introduce the threshold energy u th defined as the energy below which minima prevails and consequently there is no more negative directions:

k(u th ) = 0 (5.31)
From u th we can find T B . First we notice that u th corresponds to the energy density of the threshold minima without taking into account the vibrational contributions [START_REF] Cavagna | Fragile vs. strong liquids: a saddles-rules scenario[END_REF].

Then assuming an harmonic approximation for the vibration, the total potential energy density of a system vibrating around an IS is given by:

u(T ) = e IS + 3 2 k B T (5.32)
Therefore if we consider the peculiar case where e IS = u th , the temperature T B is defined as:

u th = u(T B ) - 3 2 k B T B (5.33)
Thus below T B we are sure that mechanism B is totally inefficient. On the opposite mechanism A could always apply as it requires only thermal activation. However depending on the size of energy barriers at ∆U (u th ), the slowing-down of the dynamic would not be impacted the same way. This observation allows for a classification of liquids in two distinct classes. For this class of liquids, the relaxation time is expected to exhibit a sharp increase which is related to the slowing-down of mechanism B. This observation corresponds to the definition of fragile glass-formers.

We only deal with the two extreme cases but we can assume that cases where ∆U (u th ) ≈ These theoretical considerations on saddles can be advantageously tested by means of numerical simulations. In the beginning of the 2000s, some studies focusing on saddle analysis were published [START_REF] Cavagna | Fragile vs. strong liquids: a saddles-rules scenario[END_REF][START_REF] Broderix | Energy landscape of lennard-jones liquid: Statistics of stationary points[END_REF][START_REF] Angelani | Saddles in the energy landscape probed by supercooled liquids[END_REF][START_REF] Doye | Saddle points and dynamics of lennard-jones cluster, solids dans supercooled liquids[END_REF]. The principal interest was to find T B as it is assumed to correspond to the mode coupling temperature T c for fragile glass-formers.

Evaluations of energy barriers for T > T x have been proposed [START_REF] Angelani | General features of the energy landscape in lennard-jones-like model liquids[END_REF][START_REF] Coslovich | Understanding fragility in supercooled lennard-jones mixtures. ii. potential energy surface[END_REF]. Some attempts were also made to correlate unstable directions and dynamical heterogeneities [START_REF] Coslovich | Are there localized saddles behind the heterogeneous dynamics of supercooled liquids?[END_REF].

All these works face the difficulty to compute efficiently true saddle points. In the following we discuss two computational approaches commonly used to get saddle points in the case of supercooled liquids. We will use these two methods to perform a saddle analysis of the bidisperse Voronoi liquid.

5.2.2

Computing saddles: Saddles vs. Quasisaddles

Minimizing W : the problem of Quasisaddles

As suggested above a way to extract saddles is to minimize the function

W = |∇U | 2 .
This minimization was originally proposed by Weber and Stillinger [START_REF] Weber | Local order and structural transitions in amorphous metal-metalloid alloys[END_REF] as saddles correspond minima of W . However Doyes and Wales have shown that minimizing W leads most of time to local minima of W and actually only a few numbers of absolute minima are actually found [START_REF] Angelani | General features of the energy landscape in lennard-jones-like model liquids[END_REF][START_REF] Doye | Saddle points and dynamics of lennard-jones cluster, solids dans supercooled liquids[END_REF][START_REF] Angelani | Quasisaddles as relevant points of the potential energy surface in the dynamics of supercooled liquids[END_REF][START_REF] Doye | Comment on "quasisaddles as relevant points of the potential energy surface in the dynamics of supercooled liquids[END_REF]. These local minima of W correspond to inflection points of the potential energy U and therefore they are not stationary points of U and cannot be considered as true saddles [START_REF] Angelani | Quasisaddles as relevant points of the potential energy surface in the dynamics of supercooled liquids[END_REF].

Angelani et al. proposed to refer to these local minima as quasisaddles (QS) [START_REF] Angelani | Quasisaddles as relevant points of the potential energy surface in the dynamics of supercooled liquids[END_REF]. They make the assumption that if the Hessian has a small numbers zero eigenvalues 1 , apart the one associated to translations, then QS point is very close from the true saddles and therefore it is possible to identify QS with saddles. Some evidences point toward the validity of their hypothesis as they showed numerically that for LJ glass-formers the index density of negative directions expressed as a function of the temperature is the same of QS 1. It was actually shown by Doyes and Wales [START_REF] Doye | Comment on "quasisaddles as relevant points of the potential energy surface in the dynamics of supercooled liquids[END_REF] that the QS investigated by Angelani et al. in [START_REF] Angelani | Quasisaddles as relevant points of the potential energy surface in the dynamics of supercooled liquids[END_REF] have only one zero eigenvalue which is not associated with translations. and true saddles. Therefore it appears that QS and saddles convey the same information that can be used to understand the slowing down of the dynamics in supercooled liquids [START_REF] Angelani | General features of the energy landscape in lennard-jones-like model liquids[END_REF][START_REF] Angelani | Reply to "comment on "quasisaddles as relevant points of the potential energy surface in the dynamics of supercooled liquids[END_REF]. This observation leads to numerous study where the the minimization of W = |∇U | 2 was the criterion selected to determine "saddles" [START_REF] Angelani | General features of the energy landscape in lennard-jones-like model liquids[END_REF][START_REF] Broderix | Energy landscape of lennard-jones liquid: Statistics of stationary points[END_REF][START_REF] Angelani | Saddles in the energy landscape probed by supercooled liquids[END_REF][START_REF] Coslovich | Are there localized saddles behind the heterogeneous dynamics of supercooled liquids?[END_REF][START_REF] Angelani | Quasisaddles as relevant points of the potential energy surface in the dynamics of supercooled liquids[END_REF].

Any minimization algorithm could be used to compute saddles but in practice, the L-BFGS algorithm (where L-BFGS stands for Limited-memory Broyden-Fletcher-Goldfarb-Shanno algorithm) is generally used [START_REF] Liu | On the limited memory bfgs method for large scale optimization[END_REF].

Finding true saddles by mean of the Eigenvector-Following method

Doye and Wales pointed to the problem of finding non-stationary points while investigation the minimization of W [START_REF] Doye | Saddle points and dynamics of lennard-jones cluster, solids dans supercooled liquids[END_REF]. To get rid of this problem, they propose an alternative minimization scheme based on the Eigenvector-Following (EF) method developed in 1981 by Cerjan and Miller to locate transition state [START_REF] Cerjan | On finding transition states[END_REF]. The EF method is based on two central ideas 1. The potential energy surface can be approximate by a quadratic form for each point of the phase state.

2. A saddle of index K can be viewed as a maximum in K directions and like a minimum in the (3N-K) other directions. Thus to find the saddle, one must maximize the energy in K uphill directions and minimize it in the other downhill directions. Doye and Wales proposed to embed the EF scheme into a Newton-Raphson method in order to converge to stationary-points of index K of the Hessian [START_REF] Wales | Stationary points and dynamics in high-dimensional systems[END_REF][START_REF] Doye | Saddle points and dynamics of lennard-jones cluster, solids dans supercooled liquids[END_REF] Technical descriptions of the Eigenvector-Following method can be found in [START_REF] Cerjan | On finding transition states[END_REF] (for its original formulation) and in [START_REF] Wales | Energy Landscapes, chapter Exploring the landscape[END_REF] (for details on EF embedded in Newton-Raphson).

The algorithm proposed by Wales and Doye contains 3 main steps: 1) a minimization using LBFGS 2) a transition step search using an Newton-Raphson-EF method and 3) a minimization of W using LBFGS [START_REF] Wales | Stationary points and dynamics in high-dimensional systems[END_REF].

Later in 2006, Grigera proposed an alternative to the scheme of Doye and Wales [START_REF] Grigera | Geometrical properties of the potential energy of the soft-sphere binary mixture[END_REF] where the Newton-Rapshon-EF method is directly used to converge to a saddle point from a thermalized state. This alternative scheme gave good results and appears easier to implement. Thus we selected this method to compute saddles in the bidisperse Voronoi liquid.

To work properly the EF method requires the computation of eigenvalues and eigenvectors of the Hessian at each iteration. The Newton-Raphson step ∆x is evaluated in the base that diagonalize the Hesssian [START_REF] Grigera | Geometrical properties of the potential energy of the soft-sphere binary mixture[END_REF]:

∆x µ = S µ 2g µ |h µ |(1 + 4g 2 µ /h 2 µ ) (5.34)
where h µ are the eigenvalues of the Hessian and g µ are the components of the gradient in the diagonal base. For the directions where h µ = 0, ∆x µ is set to zero.

S µ = ±1 defined the sign of the Newton-Raphson step. When S µ = 1, the energy increases along the uphill direction µ, whereas when S µ = -1, the energy decreases along the downhill direction µ. Thus by choosing S µ = sgn(h µ ) the algorithm should converge to a saddle of index K. However the convergence may be not reached when dealing with high-temperature [START_REF] Wales | Stationary points and dynamics in high-dimensional systems[END_REF]. A solution to this problem is to perform a certain number N ns of steps by leaving K "free" and then to fix K to its final value until reaching the stationary point. Wales and Doye have chosen N ns = 20 [START_REF] Wales | Stationary points and dynamics in high-dimensional systems[END_REF] and so did Grigera in [START_REF] Grigera | Geometrical properties of the potential energy of the soft-sphere binary mixture[END_REF]. We started also by fixing N ns = 20, however we encountered problems of convergence and in our case very small values of N ns are required to observe a convergence. In practice, N ns = 1, meaning that we fixed K directly from the theamalized configuration.

To remain in the region where a quadratic approximation of the potential energy is valid, a set of trust radii {δ µ } is considered. The step ∆x µ is rescaled in order to satisfy ∀µ, |∆x µ | ≤ δ µ . Initially δ µ = 0.001 and at each step these values are increased (or decreased) by a factor 1.2 if the quantity r = (h e -h µ )/h µ is less (or larger) than 1. h e is an estimation of the eigenvalue at current step. Wales in [START_REF] Wales | Energy Landscapes, chapter Exploring the landscape[END_REF] and Grigera in [START_REF] Grigera | Geometrical properties of the potential energy of the soft-sphere binary mixture[END_REF] have estimated that:

h e = g µ (n) -g µ (n -1) δx µ (n -1) (5.35)
where n is the current step. However with this approximation of h e , we were unable to make the algorithm converge for the bidisperse Voronoi liquid. We thus proposed an improvement in the estimation of h e .

In order to estimate h e , we evaluate the evolution of the gradient of the potential energy between two points of space phase which are very close. We denote {e j } the set of eigenvectors of the Hessian.

In X the gradient of the potential energy U (X) is expressed in the base that diagonalizes the Hessian H as ∇U (X) = j g j e j . Thus in the peculiar direction j the component of the gradient g j is: Now we consider a small evolution of the point in phase space X + dX. At first order the jth component of the gradient is now given by:

g j = ∇U (
g ′ j = ∇U (X) • e j + ∇U (X) • δe j + H(X)dX • e j e 2 j
(5.37)

We immediately see that the first term in the sum in numerator corresponds to the gradient evaluated in X, therefore the difference between the jth component of gradient evaluated in two points is:

δg j = g ′ j -g j = ∇U (X) • δe j + H(X)dX • e j e 2 j
(5.38)

By using the relation H(X)e j = h j e j , previous equation becomes:

δg j = h j e j • dX + ∇U (X) • δe j e 2 j
(5.39)

Now we select the direction of X parallel to e j , we can write that dX • e j = ∆x j . It is finally possible to estimate the eigenvalue h e = h j :

h j = δg j ∆x j - ∇U (X) • δe j ∆x j e 2 j ( 5.40)
In equation 5.40 we recognize that the first term of the R.H.S corresponds to the estimation 5.35 of the eigenvalue. Interestingly our result presents an additional term to the formulation proposed by Grigera et al. [START_REF] Grigera | Geometrical properties of the potential energy of the soft-sphere binary mixture[END_REF]. Thus we observe that in our case the estimation at first order of h e implies a corrective term related to the force field in X which is absolutely required for the convergence of the algorithm.

Finally it is important to mention that W = |∇U | 2 is monitored at each iteration and the convergence of the EF-Newton-Raphson procedure is assumed when W ∼ 10 -16

which corresponds to limiting precision of the calculation.

We detailed the two processes that could be used to obtain QS and saddles. In the following, we compare the two methods to test the assumption that QS and saddles are conveying the same kind of information is true for a system with a potential that differs from the LJ one. Then by means of a saddle analysis we determine the mode coupling temperature T c that we can compare to the value of T c found by ideal MCT analysis.

Finally we evaluate the energy barrier between saddles to see first to what extent we can interpret the increase of relaxation time for T > T c by means of motion of the system in a PEL dominated by saddles.

Numerical results

General observations on saddles and quasisaddles

Using both methods described above we compute QS and saddles for systems with N = 125, 512 and 729 particles. These choices are motivated by the facts that:

(i) For very small systems N ≤ 125, the LBFGS algorithm is efficient. The final value of W ∼ 10 -10 which corresponds to a zero value relatively to the numerical precision. Thus there are higher chances to reach true stationary points.

(ii) The EF method requires the computation of the Hessian (a 3N × 3N matrix) at each iteration. Therefore we must favor small systems to perform calculation in a reasonable time and with a limited amount of memory.

(iii) We could have limited our analysis to N = 125, however we are interested by the N -dependence of the quantities measured.

For each system of N particles we perform the calculation for temperatures T ∈ [0.86, 2.00] and for each saddle or quasisaddle computed, we collect the index density k and the energy density u.

In Figure 5.15 is represented the evolution of the density index k as a function of the potential energy density u for a system of N = 125 particles. In this case saddles have been computed by mean of the EF method. [START_REF] Broderix | Energy landscape of lennard-jones liquid: Statistics of stationary points[END_REF]. Indeed we observe that k is an increasing function of u. Moreover as suggested by Broderix and Cavagna [START_REF] Cavagna | Fragile vs. strong liquids: a saddles-rules scenario[END_REF][START_REF] Broderix | Energy landscape of lennard-jones liquid: Statistics of stationary points[END_REF] the curve of k(u) seems to be an underlying geometric feature of the PEL which can be seen as temperatureindependent. As a matter of fact, sampling saddles as different temperatures enable to reveal different parts of k(u) as different regions of the PEL are explored [START_REF] Broderix | Energy landscape of lennard-jones liquid: Statistics of stationary points[END_REF].

To support this conjecture, Broderix proposed to perform two kind of averages on the scatter plot 5.15. They were labeled as geometric and parametric averages [START_REF] Broderix | Energy landscape of lennard-jones liquid: Statistics of stationary points[END_REF] In both cases we observe the good collapse between the two ways of averaging meaning that the sampling of saddles is correctly done and it allows for a fair exploration of the geometric curve k(u) [START_REF] Broderix | Energy landscape of lennard-jones liquid: Statistics of stationary points[END_REF]. We compare now the geometric averages of true saddles and QS for each system size.

Results are represented in Figure 5.17. First we notice a relatively good agreement between the two methods used to compute the saddle points at the exception of k 0.07

where the minimization of W seems to lead to QS with a density index higher than the corresponding index for true saddle points.

We also observe that apart from the smallest system, the EF method leads to a sampling of saddles points with large values of (k, u) and the region where k ≤ 0.02 is not populated for N = 512 and N = 729. This is more visible in the lower part of Figure 5.18 where we have represented the geometric average for the three system sizes. We notice the good collapse between the three systems meaning that the curve k(u) is independent of the number of particles as expected by its definition. (This observation is also valid in the case of QS represented in the upper part in Figure 5.18). This lack of information for k < 0.02 can be explained by the way we used the EF method. We recall that we fix the index of negative directions K at the beginning the Newton-Raphson procedure meaning that K is very close (or equal) to the number of negative directions that could be found with an instantaneous normal mode analysis (INM).

We noticed that if we leave K varying during a certain number of iterations before fixing its value as it was suggested by Doye and Wales [START_REF] Wales | Stationary points and dynamics in high-dimensional systems[END_REF] or Grigera [START_REF] Grigera | Geometrical properties of the potential energy of the soft-sphere binary mixture[END_REF], K stabilized around a lower value than the initial one. However in most of cases the convergence is not reached and W = |∇U | 2 gets stuck around 10 -2 .

We tried another approach to populate the region k < 0.02. Assuming that sampling saddles from low temperatures corresponds to exploring the lower part of the curve k(u)

for T ≤ 1.00, we fixed K to values that allow for k ∈]0; 0.2]. Again the convergence was not observed.

These unsuccessful attempts reveal that the difficulties encountered to find true saddles when the number of particles is increased. Even by fixing K from the beginning the convergence is not ensured and for instance the success rate is ≈ 96% for N = 125, it decreases to ≈ 70% for N = 512 and it falls to ≈ 40% for N = 729. In Figure 5.17 we observe finally that for N = 125, k seems to be a linear function of u whereas for N = 512 or 729 there is an apparent change in the slope when k ≤ 0.2.

For the reasons mentioned just above, this change is only visible when considering QS.

As explained by Cavagna [START_REF] Cavagna | Fragile vs. strong liquids: a saddles-rules scenario[END_REF] the derivative of k(u) is directly related to the energy barriers which exist between adjacent saddles. Thus we can assume that this change of slope is the signature of differences that occur at the level of the energy barriers. More precisely the apparent change of slope is likely to be associated to the transition from the Arrhenian to the super-Arrhenian regime. As a matter of fact, by using the parametric average we can determine the range of temperature for which the change in the slope of u(k) occurs.

In Figure 5.20, we have represented for N = 729 both parametric and geometric averages of QS. We observe that the slope of u(k) changes for a temperature 1.00 < T < 1.10

which is cooler (but close) than the crossover temperature T * = 1.25 for which we have identified the transition from the Arrhenian to the super-Arrhenian behavior in Chapter 4. This observation suggests that the slowing down of the relaxation time and its departure from the Arrhenian behavior might be interpreted in terms of a change in the potential energy barriers between saddles. Of course, one must be careful by doing such an interpretation as the activation barriers related to the Arrhenian regime are free energy barriers. However if we assume that the entropic contribution could be neglected, the evolution of the potential energy barriers might give a reasonable insight into the underlying phenomenon responsible of the change from Arrhenian to super-Arrhenian behavior. To check this assumption we compute for saddles and QS the energy barrier ∆U b using equation 5.30 proposed by in Cavagna [START_REF] Cavagna | Fragile vs. strong liquids: a saddles-rules scenario[END_REF].

Energy barriers

For saddles and QS we determine the energy barrier for the three system sizes. To do so, we followed the procedure described by Coslovich and Pastore in [START_REF] Coslovich | Understanding fragility in supercooled lennard-jones mixtures. ii. potential energy surface[END_REF]. We performed a linear regression on the point cloud (u, k) associated to each temperature as shown in In practice we computed also the slope du/dk by performing a linear regression on (k, u).

In the case where the statistics would be sufficient, the slope dk/du would be equal to (du/dk) -1 . However in our case, we do not have enough statistics and we note some disparities between the two measurements. Therefore we made an average with both estimations and the energy barrier is now given by: We observe that:

∆U b = 1 6 du dk + dk du
• For high temperature T ≥ 1.20 ((T -T c )/T c ≥ 0.5), the height of the energy barriers remains constant ∆U b = E h . We notice however that the value of the energy barrier E h is higher for saddles, indeed, E h = 9.8 for saddles whereas E h = 7.3 for QS. These values reflect the disparities observed in Figure 5.17 at high temperatures.

• For temperature T < 1.20 ((T -T c )/T c < 0.5), the height of the energy barriers increases while the system is cooled toward T c . This observation is in agreement with the presence below T * of temperature-dependent energy barriers whose height increases when the temperature is decreased. We notice however the limited increase of the energy barriers for the true saddle points. This is related to the fact that even at low temperature we were not able to access the regime k ≤ 0.2 where significant changes in the curve of k(u) are expected.

Coslovich and Pastore perfomed a saddle analysis on LJ glass-formers [START_REF] Coslovich | Understanding fragility in supercooled lennard-jones mixtures. ii. potential energy surface[END_REF] and observed the same phenomenology described above for ∆U b . They also showed that the more fragile the mixture the higher ∆U b [START_REF] Coslovich | Understanding fragility in supercooled lennard-jones mixtures. ii. potential energy surface[END_REF]. To see whether this effect is also visible in For the moment we focused only on the evolution of the density index k as a function of density energy u to describe phenomena that occurs at the saddles level in the PEL.

Angelani et al. proposed an alternative view of the problem by considering the elevation in potential energy with respect to the corresponding IS [START_REF] Angelani | Saddles in the energy landscape probed by supercooled liquids[END_REF]. They investigated the quantity u -e IS as a function of k for several LJ simple liquids and glass-formers and found that they all collapse on the same master curve [START_REF] Angelani | General features of the energy landscape in lennard-jones-like model liquids[END_REF]. This suggests that these different systems share common features regarding the PEL.

In order to see if this observation is more general and concerns also liquids whose potential is different from the LJ potential, we computed the difference u -e IS for the bidisperse Voronoi liquid. Results are shown in Figure 5.23 where we have represented u -e IS scaled by k B T c as a function of k for the bidisperse Voronoi liquid and for four different LJ liquids [START_REF] Angelani | General features of the energy landscape in lennard-jones-like model liquids[END_REF]:

• The modified monatomic LJ (MLJ) at ρ = 1.0, the potential is given by V M LJ (r) = 4ǫ[(σ/r) 12 -(σ/r) 6 ] + δV where δV is a small many-body term used to inhibits crystallization, δV = α q θ(S(q) -S 0 )[S(q) -S 0 ] 2 with α = 0.8 and S 0 = 10 and the sum is made over all q such as q max -∆ < |q| < q max + ∆, q max = 7.2ρ 1/3 .

• The modified monoatomic soft spheres (MSS) at ρ = 1.0, the potential is given by V M SS (r) = 4ǫ(σ/r) 12 + δV where δV is defined just above.

• The Kob-Andersen (KA) mixture described in Chapter 4.

• A modified version of KA (KA2) where σ AA and σ BB are exchanged. We observed that the bidisperse Voronoi liquid seems to collapse on the same master curve as the LJ systems, meaning that all these systems share universal common features.

Moreover Angelani et al. defined the elementary elevation ∆E, which corresponds to the energy to pass from a saddle of order n to a saddle of order n + 1 [START_REF] Angelani | General features of the energy landscape in lennard-jones-like model liquids[END_REF]:

∆E = 1 3 d(u -e IS ) dk (5.42)
Apart from the presence of the IS energy, e IS , this formulation is similar to the definition of ∆U (see equation 5.30) provided by Cavagna [START_REF] Cavagna | Fragile vs. strong liquids: a saddles-rules scenario[END_REF].

The value of ∆E is easily found by doing a linear fit on the data in Figure 5 [START_REF] Angelani | General features of the energy landscape in lennard-jones-like model liquids[END_REF]. More important this phenomenon is not related to a specific class of potential, i.e LJ potential. Indeed, it was already shown that the Morse potential satisfied this observation [START_REF] Angelani | General features of the energy landscape in lennard-jones-like model liquids[END_REF]. Now we notice it is also the case for the bidisperse Voronoi liquid which is a many-body potential.

Evaluation of the mode-coupling temperature T c

In his letter [START_REF] Cavagna | Fragile vs. strong liquids: a saddles-rules scenario[END_REF], Cavagna explained that the mode-coupling temperature T c is often associated to the transition from a non-activated to an activated dynamics. Regarding the mechanisms he proposed to explain diffusion, in the case of fragile glass-formers T c could be associated to the temperature T B below which mechanism B becomes inefficient as there is no more negative directions at saddle points. Consequently, we can test numerically this assumption by considering equation 5.43 which links T c to the threshold energy E th below which there is no more negative direction.

E th = E P (T c ) N - 3 2 k B T c (5.43)
E th is obtained by fitting linearly the part of k(u) below the crossover regime that we identified in Figure 5.20. As a consequence, we have to performed these fits on quasisaddles for which the regime of small values of k(u) is accessible. Example of such a fit is shown in Figure 5.24 for a system of N = 729 particles and values of E th for the other system sizes can be found in These values of T c are shown in Table 5.3.

We notice that for the smallest system, the value found for T c with a saddle analysis This observation highlights that below T c ≈ T M CT c , Goldstein's scenario applies and the dynamics in the system is activated. When the temperature is increased, the dynamics becomes non-activated and this crossover is related to the presence of saddle points whose number of negative directions increase with temperature. The analysis in terms of saddles shows that, as suggested by Angelani et al. [132? ], QS convey important information about dynamics that occur in real space. The remaining problem for saddles analysis is that, contrary to IS, there is no unique partitioning of the space and therefore using different methods to compute saddles would lead to different saddle points.

In the following chapter, we come back to the real space to investigate the relation between the slowing down of the dynamics and the presence of locally favored structures. In Chapter 3, we have investigated the behavior of classical microscopic observables such as pair correlation function or its Fourier transform, the static structure factor. The relatively weak sensitivity of these observables upon cooling might lead to the wrong conclusion that structure plays a minor role in the dynamical arrest at glass transition.

However one just has to remind that static structure factor is taken as input in MCT calculations to immediately realize that, even if the static density fluctuations are weakly varying when temperature decreases, these variations are sufficient to capture the slowing down of the dynamics.

Angell also referred to the differences in the structure of liquids when he proposed an explanation to the fragile/strong behavior [START_REF] Angell | Perspectives on the glass transition[END_REF][START_REF] Angell | Relaxation in liquids, polymers and plastic crystals -strong/fragile parrterns and problems[END_REF]. Indeed, strong liquids such as SiO 2 do not present important changes in their local order due to the strong directional bondings. On the opposite, the local environment of fragile liquids changes upon cooling.

The super-Arrhenian behavior is assumed to be related to the growth of domains composed of an underlying structure that minimizes locally the free energy, but which does not manage to tile the Euclidian space [START_REF] Royall | The role of local structure in dynamical arrest[END_REF][START_REF] Shintani | Frustration on the way to crystallization in glass[END_REF] leading to the theoretical concept of geometrical frustration [START_REF] Tarjus | The viscous slowing down of supercooled liquids as a temperature-controlled super-arrhenius activated process: a description in terms of frustration-limited domains[END_REF][START_REF] Tarjus | The frustration-based approach of supercooled liquids and the glass transition: a review and critical assessement[END_REF].

The idea that some non-tiling local structures could minimize the local free energy dated back to the 50's when Frank showed that the ground state of 13 identical particles subjected to the Lennard-Jones potential, was the icosahedron [START_REF] Royall | The role of local structure in dynamical arrest[END_REF][START_REF] Frank | Supercooling of liquids[END_REF]. More recent studies identified the ground state of isolated clusters of different potentials and investigated how the spatial extension of these clusters could have a direct impact on the dynamics of glass formers. [START_REF] Malins | Identification of long-lived clusters and their link to slow dynamics in a model glass former[END_REF][START_REF] Doye | The favored cluster structures of model glass formers[END_REF].

The link between the presence of favored structures and slowing down of the dynamics have been observed in several studies. In 2002, Dzugutov et al. investigated a system of monoatomic particles where the potential was specially designed to favor icosahedral structures. He noticed that the dynamics of particles inside icosahedra was slower than the other ones [START_REF] Dzugutov | Decoupling of diffusion from structural relaxation and spatial heterogeneity in a supercooled simple liquid[END_REF]. In 2007 Coslovich and Pastore investigated the locally favored structures of numerous LJ binary mixtures by means of Voronoi tessellations and found that the icosahedron is the favored structure for the Wahnström model, whereas it is the bicapped square antiprism for the Kob-Andersen mixture. [START_REF] Coslovich | Understanding fragility in supercooled lennard-jones mixtures. i. locally preferred structures[END_REF]. These results were thereafter confirmed by Malins et al. who investigated the locally favored structures using the Topological Cluster Classification (TCC) [START_REF] Malins | Identification of long-lived clusters and their link to slow dynamics in a model glass former[END_REF][START_REF] Malins | Lifetimes and lengthscales of structural motifs in a model glassformer[END_REF].

The first section is devoted to the theory of geometric frustration which predicts a certain formulation of the free energy barriers in the super-Arrhenian regime. We will test to what extent this prediction applies for the bidisperse Voronoi liquid. Then in a second section, we will focus on finding the locally favored structures for the bidisperse Voronoi liquid. To that purpose, we will use the same method as Coslovich and Pastore [START_REF] Coslovich | Understanding fragility in supercooled lennard-jones mixtures. i. locally preferred structures[END_REF].

The aim is to see if 1) a peculiar structure emerges upon cooling, 2) if it's the case, how the dynamics is impacted by its presence. In the final section we will present some results that tend to show that at low temperature a quasicrystal forms characterized by the presence of Frank-Kasper phases.

Geometric frustration

The theory of geometric frustration is based on three propositions that have already been well-verified for spherically symmetric particles but still remain postulates for molecular liquids, mixtures and polymers [START_REF] Tarjus | The frustration-based approach of supercooled liquids and the glass transition: a review and critical assessement[END_REF]. We summarize here the ideas that can be found [START_REF] Tarjus | The viscous slowing down of supercooled liquids as a temperature-controlled super-arrhenius activated process: a description in terms of frustration-limited domains[END_REF][START_REF] Tarjus | The frustration-based approach of supercooled liquids and the glass transition: a review and critical assessement[END_REF][START_REF] Viot | A heterogeneous picture of alpha relaxation for fragile supercooled liquids[END_REF].

(i) A liquid is characterized by a locally favored structure (LFS) that minimizes the local free energy of a small number of particles.

(ii) The LFS cannot tile the Euclidian space. The incompatibility between the local order and the global space filling is at the origin of frustration. The spatial extension of LFS generates superextensive strain as the space cannot be tiled with the

LFS.

(iii) There exists a reference system in which the effect of the frustration can be turned off. For instance, it has been shown that in curved space a complete tilling of the space with icosahedra is possible [START_REF] Nelson | Defects and Geometry in Condensed Matter Physics[END_REF]. In this reference system, there is a critical point reached when T = T * F L for which a transition toward an ordered phase [START_REF] Viot | A heterogeneous picture of alpha relaxation for fragile supercooled liquids[END_REF]. The theory supposes the existence of a critical point when there is no frustration. However this critical point would vanish as soon as a very small amount of frustration is introduced. Due to the presence of frustration, long-range order is forbidden. Therefore below T * F L , the temperature at which would occur the transition when no frustration is present, the liquid breaks up into different domains composed of LFS whose size and future growth is limited by frustration [START_REF] Berthier | Dynamical Heterogeneities in Glasses, Collloids and Granular Media[END_REF][START_REF] Tarjus | The viscous slowing down of supercooled liquids as a temperature-controlled super-arrhenius activated process: a description in terms of frustration-limited domains[END_REF].

In the frustration-limited domain picture, T * F L is the temperature at which the behavior of the liquid is impacted by the presence of domains. A scaling description of the slowing-down of the dynamics is proposed and it predicts a super-arrhenian behavior of the relaxation time below T * F L :

τ = τ 0 exp E A (T ) k B T (6.1)
where the free energy barrier E A (T ) is given by:

E A (T ) =          E ∞ T > T * F L E ∞ + BT * F L 1 - T T * F L ψ T < T * F L (6.2)
B is a parameter measuring the departure from the Arrhenian behavior, so in other words it measures the fragility of the system [START_REF] Tarjus | The viscous slowing down of supercooled liquids as a temperature-controlled super-arrhenius activated process: a description in terms of frustration-limited domains[END_REF]. It can be shown that B increases as the frustration decreases [START_REF] Tarjus | The frustration-based approach of supercooled liquids and the glass transition: a review and critical assessement[END_REF].

The exponent ψ has a universal character as it has been shown experimentally that for a wide range of liquids, ψ = 8/3 [START_REF] Kivelson | Fitting viscosity: Distinguishing the temperature dependences predicted by various models of supercooled liquids[END_REF] and then it was analytically demonstrated that the value of the exponent is 8/3 [START_REF] Tarjus | The viscous slowing down of supercooled liquids as a temperature-controlled super-arrhenius activated process: a description in terms of frustration-limited domains[END_REF]. However for certain systems it can weakly deviate from 8/3 and it has been shown experimentally that it could vary between 7/3 to 3 [START_REF] Tarjus | The frustration-based approach of supercooled liquids and the glass transition: a review and critical assessement[END_REF].

To test the predictions of the frustration-limited theory we try to fit the data of relaxation time of the bidisperse Voronoi liquid by using equations 6.1 and 6.2. The result of the fits are represented in Figure 6.1. We noticed first that equations 6.1 and 6.2 allow a quite good description of the data.

We briefly discuss the results for the fitting parameters (B, T * F L and ψ): • The temperature at which the crossover to the domains-dominated dynamics takes place is T * F L = 1.15 and is close from T * = 1.25, the crossover temperature to the super-Arrhenian regime.

• In our case we found that the parameter B ∼ 128. It is a relatively small values in comparison to what has been observed for experimental systems [START_REF] Kivelson | Frustrationlimited clusters in liquids[END_REF]. This means that our system has a relatively high degree of frustration.

• Finally in our case we found that ψ = 8/3 which is in agreement with most of results found in litterature [START_REF] Tarjus | The frustration-based approach of supercooled liquids and the glass transition: a review and critical assessement[END_REF][START_REF] Viot | A heterogeneous picture of alpha relaxation for fragile supercooled liquids[END_REF][START_REF] Kivelson | Frustrationlimited clusters in liquids[END_REF].

All these observations suggest the presence of an underlying locally preferred structure that we need now to identify. In the next section we use a Voronoi analysis to determine the most frequent polyhedra present in the bidisperse Voronoi liquid.

Locally favored structures (LFS) 6.2.1 Computing LFS

In 2007, Coslovich and Pastore investigated the link between fragility and the emergence of locally favored structures for several LJ glass formers by means of Voronoi tessellations [START_REF] Coslovich | Understanding fragility in supercooled lennard-jones mixtures. i. locally preferred structures[END_REF]. We naturally chose to proceed in the same way as the Voronoi analysis is particularly straightforward for our model. However, contrary to Coslovich and Pastore, who proceeded to a classical Voronoi analysis, all the LFS analysis will be performed within the specific framework of Voronoi-Laguerre tessellation since the bidisperse Voronoi liquid is directly built from the geometrical properties of this tessellation.

In Chapter 1 we have seen that to each particle is associated a Voronoi polyhedron and that this polyhedron is composed of faces which are polygons. To identify the LFS, we compute the distribution of faces, i.e we look for each polyhedron to the number of faces associated to a specific polygon composed of k vertices. Thus in practice each polyhedron can be described by a succession of integers (n 3 , n 4 , n 5 , n 6 , • • • ) where n k is the number of faces composed of k vertices [START_REF] Coslovich | Understanding fragility in supercooled lennard-jones mixtures. i. locally preferred structures[END_REF]. In other words, it corresponds to the number of times a specific polygon (△, , , , • • • ) appears in the Voronoi polyhedron.

The smallest polygons that can be observed are triangles and so we start our classification with triangles. For each temperature T ∈ [0.83; 2.00] we investigated the LFS of 1000 independent thermalized configurations and IS configurations. By independent we mean that configurations are space out by at least τ α .

We started by investigating the LFS on all particles and as Coslovich and Pastore, we observed that the small B particles are more likely to be at the center of a welldefined geometry whereas no peculiar structure seems to be associated to large particles.

Therefore in what follow we focused only on small particles and in Table 6.1 is presented the three most frequent structures at T = 2.00 and T = 0.83. We notice that:

• At high temperature, in thermalized configurations the fraction of LFS is relatively weak. The most frequent structure represents less than 5% of all the structures associated to B particles. However when looking at IS configurations, we observe that favored structures are already substantial, more than 16% of B particles are involved in a distorted icosahedron ((0, 2, 8, 2) signature).

• At low temperature, icosahedron ((0, 0, 12) signature) is the most frequent structure that appears in both thermalized and IS configurations. Distorted icosahedra represent still a non-negligible fraction of the LFS. (0, 0, 12) (0, 2, 8, 2) (0, 3, 6, 3) (0, 3, 6, 4) The evolution of the LFS over the whole range of temperatures is shown in Figure 6.3

where we observe that:

• When the temperature is decreased, there is a huge increase of the number of icosahedra for both thermalized and IS configurations.

• The number of distorted icosahedra in thermalized configurations is increasing slowly upon cooling whereas it remains barely constant in IS configurations.

• The LFS (0, 3, 6, 4) and (0, 3, 6, 3) slightly increase in thermalized configurations and decrease in IS configurations.

The increase of icosahedra is certainly the most striking result. We also notice in Figure 6.3 that this increase is much more pronounced when T c /T 0.7 which corresponds to T 1.15. Interestingly this temperature corresponds to the temperature T * F L below which the formation of domains of LFS is expected.

The temperature is also very closed from the crossover temperature T * = 1.25. Therefore this suggests a possible correlation between the strong increase of icosahedra and the presence of a marked super-Arrhenian regime. Increase of icosahedra have already been observed for fragile glass formers such as binary alloys [START_REF] Shimono | Icosahedral symmetry, fragility and stability of supercooled liquid state of metallic glasses[END_REF] or the Wahnström model [START_REF] Coslovich | Understanding fragility in supercooled lennard-jones mixtures. i. locally preferred structures[END_REF][START_REF] Malins | Identification of long-lived clusters and their link to slow dynamics in a model glass former[END_REF]. Knowing that, we computed the LFS of the Wahnström model and the AMLJ-0.80 mixture and we indeed found an increase of the number of icosahedra. In a second step, we compared results obtained for the bidisperse Voronoi liquid with those of LJ glass formers. In Figure 6.4 we have represented for thermalized and IS configurations, the evolution with temperature of the fraction of icosahedra as a function of the scaled temperature T r /T . We observe that the increase of the fraction of icosahedra is qualitatively the same for the three glass formers. All these systems have barely no icosahedra at height temperature when they are thermalized and only a small fraction of icosahedra exists in the IS configurations. As the temperature decreases, the fraction of icosahedra increases.

Moreover, for both thermalized and IS configurations when the temperature T < T * , the increase is much more pronounced which correlates with the entrance into the super-Arrhenian regime. When considering only the IS configurations, we notice that the bidisperse Voronoi liquid exhibits a steeper increase of the fraction of icosahedra than the two other LJ glass formers. Moreover Coslovich and Pastore have determined, for the Wahnström model, that the other most frequent Voronoi polyhedron associated to small particles, is (0, 2, 8, 2) [START_REF] Coslovich | Understanding fragility in supercooled lennard-jones mixtures. i. locally preferred structures[END_REF].

Our analysis reveals that the same observations also hold for AMLJ-0.80. Therefore there must be some common underlying features between all these glass formers. be verified for the KA model (which is non-additive) as it shares the same LFS as non additive metallic glasses [START_REF] Crowther | The nature of geometric frustration in the kob-andersen mixture[END_REF]. We also notice in this study that WAHN and AMLJ-0.80, which are both additive, share the same LFS.

Although the definition of σ AB might give intuition about the possible LFS, it is not sufficient to predict the LFS. Indeed Coslovich and Pastore performed isobaric simulations on various additive mixture of LJ by changing only the value of σ BB . They found that when σ BB is decreased, the fraction of icosahedra is also decreasing and for σ BB < 0.7, icosahedra are not found among the most frequent structure [START_REF] Coslovich | Understanding fragility in supercooled lennard-jones mixtures. i. locally preferred structures[END_REF]. Thus it seems that one needs to consider also the ratio σ BB /σ AA .

Moreover in the case of the bidisperse Voronoi liquid, we cannot speak about additive or non-additive interactions, as a term like σ AB does not exist, only the definition of η = R 2 A -R 2 B /v 1/3 makes sense. However we have demonstrated in Chapter 3 that for the pair correlation function, the position of the first peak r * AB associated to A-B interaction is always equal to 0.5(r * AA + r * BB ), which is a characteristic of additive mixture at low temperature. Thus we could assume that the bidisperse Voronoi liquid possesses some characteristics of additive interactions and if we consider that the size ratio R B /R A = 0.83 is the same as for the Wahnström model, that might explain why our system shares similarities with WAHN and AMLJ-0.80 regarding the LFS.

In Figure 6.5 is represented the evolution of the fraction of icosahedra with temperature for different compositions A : B of the bidisperse Voronoi liquid. We observe that the larger the number of A particles, the weaker is the increase of icosahedra upon cooling for both thermalized and IS configurations. We even notice that for the 80 : 20 mixture, the number of icosahedra remains constant for IS configurations. Moreover we observe for the IS configurations that the fraction of icosahedra at T r /T = 0.4 is decreasing when the number of large particles A is increased. For each configuration we computed the average numbers n A of large A particles and n B of small B particles that composed the icosahedra. Results are shown in Table 6 For mixtures 70 : 30 and 80 : 20 the average numbers of neighbors A and B is similar at high and low temperature as it was observe for additive mixtures and Wanström model in [START_REF] Coslovich | Understanding fragility in supercooled lennard-jones mixtures. i. locally preferred structures[END_REF]. For 50 : 50 and 60 : 40 we observe slight variations and we notice that for the equimolar mixture, there are more large neighbors of type A at T = 0.83. For this low temperature, this change in the neighborhood, even if small, could be an indicator of the ease to generate Frank-Kasper network. This point will be more detailed in the last section of this chapter. We also identify the LFS for each composition for thermalized configuration as it is shown in Table 6.3. We notice that when the number of large particles is increased the icosahedron is no more the favorite structure and it is little by little replaced by other structures such as (0,2,8,1) or the distorted icosahedron, which seems to be the LFS common to all the mixture composition with a fraction always greater than 10%. This change of LFS with composition is likely to be related to the fragility of each mixtures. In Chapter 4 we have seen that when the number of large particles is increased, the bidisperse Voronoi liquid is less and less fragile. Thus we see that the more icosahedra, the more fragile the mixture. On the opposite when the mixture is strong other LFS seem to emerge. However their fraction does not exceed 14% of small particles meaning that domains of LFS should not be abundant, if any. Composition also influences the LFS of usual LJ glass formers and it is interesting to notice that Coslovich and Pastore also found for the AMLJ systems a reduction of the fraction of LFS when the parameter σ BB is decreased or equivalently when systems become less fragile [START_REF] Coslovich | Understanding fragility in supercooled lennard-jones mixtures. i. locally preferred structures[END_REF].

n A n B n A n B n A n B n A n B n A n B n A n B n A n B n A n B 6 6 7
Moreover by using TCC, Crowther et al. have also observed modifications in the LFS of the Kob Andersen model when changing the composition and observed that the fraction of bicapped square antiprism increases when the number of small particles is increased [START_REF] Crowther | The nature of geometric frustration in the kob-andersen mixture[END_REF].

Additionally Crowther et al. have shown that bicapped square antiprisms are associated with an important slowing down of the dynamics. Some similar observations occur when the LFS is composed of icosahedron [START_REF] Dzugutov | Decoupling of diffusion from structural relaxation and spatial heterogeneity in a supercooled simple liquid[END_REF]. These lead us to investigate the effect of icosahedra on the dynamics of the bidisperse Voronoi liquid, but prior to this we want to characterize the spatial extension of this icosahedral order.

Presence of domains

As suggested by the frustration-limited theory, below T * LF S , the preferred structures tend to spatially extend upon cooling and as a result the formation of clusters have been reported [START_REF] Coslovich | Understanding fragility in supercooled lennard-jones mixtures. i. locally preferred structures[END_REF][START_REF] Dzugutov | Decoupling of diffusion from structural relaxation and spatial heterogeneity in a supercooled simple liquid[END_REF][START_REF] Doye | The favored cluster structures of model glass formers[END_REF][START_REF] Malins | Lifetimes and lengthscales of structural motifs in a model glassformer[END_REF]. To test whether this observation also applies for the bidisperse Voronoi liquid we looked for the presence of clusters of icosahedra for T < T * LF S .

To determine the existence of such clusters, we proceeded in the following way:

1. We determine the particles i located at the center of icosahedra. To do so we focus on particles whose Voronoi polyhedron is (0,0,12).

2. Then for each i, we identify the 12 neighbors j that form the icosahedron.

3. Before defining the algorithm for the clusters search, we need to define how two icosahedra i and j can be connected. We highlight the fact that we only consider the problem from a geometrical point of view and we do not search for clusters that minimize locally the potential energy as it was for instance done by Doyes et.

al [START_REF] Doye | The favored cluster structures of model glass formers[END_REF]. As a result 4 connections can be considered. An illustration of these four connections is provided in Figure 6.7.

(a) The minimal connection that can link two icosahedra consists on one common nearest neighbor shared by i and j.

(b) Two icosahedra can be linked through a common edge. This implies that i and j have 2 common nearest neighbors.

(c) It is possible that two icosahedra share a common facet, in this case 3 common neighboring particles of i and j are involved in the process.

(d) Finally j can be one of the nearest neighbors of i and in this case they both share 5 common neighbors.

We consider that the minimal connection in our cluster-search procedure and therefore two icosahedra are considered as belonging to the same cluster if they share at least one common neighbor.

Chapter 6 Locally favored structures and crystallization

185 Start level ℓ {ico} ℓ card({ico} ℓ ) = N ℓ i ∈ {ico} ℓ i ≤ N ℓ ? i already seen? i = i + 1 j ∈ {neigh(i)} card({neigh(i)}) = N b j ≤ N b (i) ? j already seen? j = j + 1 (i, j) common neighbor ≥ 1? level = newlevel {ico} ℓ = {ico}N ℓ N ℓ = card({ico}N ℓ ) newlevel = newlevel + 1 j ∈ {ico}N ℓ
all particles ico. already seen?

.6: Algorithm for the clusters search. To find a cluster we consider a particle at the center of an icosahedron (level ℓ = 0) and we identify the neighbors which are also at the center of icosahedra (level ℓ = 1). Then for each of these specific neighbors, their own neighbors are also analyzed to determine those inside icosahedra. We repeat this procedure until the particle analyzed has no unseen icosahedra in its neighborhood (level=newlevel). To avoid the possibility of considering two times or more the same particles we label as "seen" each particle analyzed. Now that we have a tool to identify clusters, we can measure the probability P (n i ) of having clusters composed of n i icosahedra. This probability is represented in Figure 6.9 (upper graph) for temperatures T ∈ [0.83, 1.00]. We observe that:

• For the highest temperature P (n i ) is monotonously decreasing as n i increased.

However we underline the fact that clusters of more than 50 icosahedra can already be observed at T = 1.00.

• When the temperature is decreased, the behavior of P (n i ) is no longer monotonous.

A decrease is observed and then for n i > n c , a critical size, P (n i ) is increasing and a maximum is reached. We notice that this phenomenon is enhanced upon cooling.

• For T ≤ 0.85, the presence of large clusters is significative in the system as it is suggested by the growth of maximum of P (n i ) for n i ∼ 90. In Figure 6.8 is shown a snapshot of a configuration where 96 icosahedra were counted and 90 of them have formed a cluster. • As expected, the formation of large clusters is favored by the possibility of sharing one common neighbor. We notice also that sharing a face is also increasing P (n i )

for relatively large n i .

• Surprisingly sharing an edge, i.e N nn = 2, does not enable to form large clusters.

This might be due to steric effects between neighboring particles of both icosahedra.

• Finally we observe that when N nn = 5, P (n i ) continuously decreases but there is a non-zero probability to find a large cluster where the center of icosahedron i + 1

is located in the closest neighborhood of icosahedron i.

The probability P (n i ) have shown the existence of large clusters composed of icosahedra.

One can wonder to what extent these clusters are representative. In other words, if one selects a particle arbitrarily, what is probability that this latter is located at the center of an icosahedron included in a cluster of size n i ?

To answer this question we define the probability Q(n i ) that for a configuration sampled from thermal equilibrium, a particle chosen at random is located at the center of an icosahedron which is included in a cluster of size n i .

For a configuration with a relatively large number of particles, if N c is the total number of clusters, N c P (n i ) measures the number of clusters of size n i , then N c P (n i )n i counts the number of particles located at the center of icosahedra in a cluster of size n i .

Therefore the probability Q(n i ) is given by Q(n i ) = N c P (n i )n i /N c ∝ n i P (n i ).

As a result we compute n i P (n i ) which conveys the relevant informations. It is shown in the lower graph in Figure 6.10. We observe that when the temperature is lowered, there are more chances that a particle chosen at random is located at the center of an icosahedron in a large cluster. This is suggested by the pronounced maximum exhibited by n i P (n i ). Therefore the bidisperse Voronoi liquid has a clear tendency to form large clusters of icosahedra upon cooling. Figures 6.8, 6.9 and 6.10 suggest a possible percolation of icosahedra in the system at low temperatures where the size of the percolating cluster would be given by the value of n i when P (n i ) is maximal (n i ≈ 95). An immediate perspective to this work would be to determine the percolation threshold and to see how it evolves with temperature.

Moreover Figure 6.8 raises the question of the fractal character of such clusters and one could try to characterize it.

As predicted by the frustration-limited theory, the formation of domains has been observed for the bidisperse Voronoi liquid, we are now interested in the impact of these domains on the dynamics. .10: Evolution of n i P (n i ) with n i for different temperatures. n i P (n i ) is proportional to the probability that a particle chosen at random is in cluster of size n i for a configuration sampled from thermal equilibrium (see text for details).

Effect on the dynamics

The LFS are generally associated to slow domains as shown by Dzugutov et al. [START_REF] Dzugutov | Decoupling of diffusion from structural relaxation and spatial heterogeneity in a supercooled simple liquid[END_REF],

Coslovich and Pastore [START_REF] Coslovich | Understanding fragility in supercooled lennard-jones mixtures. i. locally preferred structures[END_REF] in the case of icosahedra, and then by Crowther et al. [START_REF] Crowther | The nature of geometric frustration in the kob-andersen mixture[END_REF] for the bicapped square antprism. It is interesting to see to what extent clusters of icosahedra found above are likely to have an influence on the dynamics of the bidisperse Voronoi liquid.

Previously we focused only on the particles located at the center of icosahedra. We want now to differentiate all the N particles contained in the bidispserse Voronoi liquid and to do so we use the same scheme as in [START_REF] Coslovich | Understanding fragility in supercooled lennard-jones mixtures. i. locally preferred structures[END_REF]. We define thus three sub-ensembles:

• the ico-ensemble which corresponds to the particles located at the center of icosahedra;

• the neigh-ensemble which deals with nearest neighboring particles belonging to the icosahedra but which are not located at the center. Particles are considered as neighbors if they share a face in the Voronoi tessellation.

• the other-ensemble which consists of particles that are neither at the center of an icosahedron nor in the LFS.

Therefore using equation 4. The idea is then to compare the relaxation times of particles involved in LFS, meaning τ ico and τ neigh , to the relaxation time τ other which concerns particles which do not belong to LFS. Results of ratios τ ico /τ other and τ neigh /τ other are presented respectively in Figures 6.11 and 6.12. In Figure 6.13 we have represented the ratio τ other /τ α to see how τ other behaves with respect to the structural relaxation time. In both Figures 6.11 and 6.12 we observe that for T c /T ≤ 0.72 (T ≥ 1.10), the ratios τ ico /τ other and τ neigh /τ other are constant and ≈ 1.5, meaning that dynamics is already sensitive to LFS at high temperature. When T c /T > 0.72, both ratios are increasing and the ratio τ ico /τ other becomes more important than τ neigh /τ other . Therefore we can conclude that 1) the relaxation of particles involved in LFS is indeed slower than other particles, 2) particles at the center of icosahedra relax even more slowly than their neighbors. In Figure 6.13 we observe that particles which do not belong to LFS formed by icosahedra relax faster with respect to the structural relaxation time. This implies that the structural relaxation τ α is partly dominated by the relaxation of underlying LFS.

In Chapter 4, we explained that the breakdown of the Stokes-Einstein relation is related to the presence of dynamical heterogeneities. We can now explain the violation of SE relation by the presence of LFS which, as in Figure 4.25, tends to separate the liquid in regions where the relaxation of particles is more or less slowly.

The influence of LFS on the dynamics is more and more pronounced when the temperature is decreased. At sufficiently low temperatures, we have seen that the system exhibits a strong tendency to form large clusters of icosahedra. In what follows we will see that, for the lowest temperatures, the system tends to form a quasicrystal which seems to be associated with the formation of Frank-Kasper phases.

6.3 "Crystallization"

Decrease of the potential energy

When we investigated the low temperature T ∈ [0.83, 0.84], we faced difficulties to obtain equilibrated trajectories of the supercooled liquid. For T = 0.83 and T = 0.84 we generated 10 trajectories of duration t = 6 • 10 5 . To appreciate the length of the trajectory, we recall that the relaxation time at these temperature is τ α ≈ 2000.

Among these trajectories only a few ones were representative of the supercooled state and most of them tend to observe a decrease of the potential energy which cannot be assimilated to simple fluctuations of E P . In Figure 6.14 is represented the typical behavior of potential energy per particles E P /N as a function of the time t when such phenomenon occurs. We selected two specific trajectories where both the decrease of E P /N and its stabilization are visible. For both temperatures, we clearly see that the system passes from an high energy level to a lower one. The transition seems to be rather smooth, there is no brutal change in the value of E P and it takes place on a period of time which is relatively long compared to the total length of the trajectory. After a time t ≥ 5 • 10 6 the system seems to stabilize around an average value of the potential energy. Same kind of phenomena have been observed by Pedersen et al. when they investigated the Wahnström model at low temperatures [START_REF] Pedersen | Crystallization of the wahnström binary lennard-jones liquid[END_REF]. They have shown that for the Wahnström model, the decrease of the potential energy is associated with the formation of a quasicrystal, and Frank-Kasper phases (defined below) were also found.

Owing to the similarities that exist between the bidisperse Voronoi liquid and the Wahnström model, one can think that quasicrystals are also forming in the bidisperse Voronoi liquid and we therefore investigate the nature of this quasicrystal and the possible presence of such Frank-Kasper phases. We also highlight that for the rest of this section, for T = 0.84 and T = 0.83, we consider that equilibrated crystalline phase corresponds to portion of the total trajectory where the energy is stabilized, which occurs when t ≥ 5 • 10 6 .

The structure factor : another proof of crystallization

The structure factor is a useful tool when looking for the characteristics of a crystal, as the position of peaks may inform on the underlying structure. We compute the partial structure factors S ab (k) for crystalline configurations at T = 0.84 and T = 0.83. The results are represented in Figure 6.15.

For each partial structure factor we observe that they are more structured that their supercooled counterparts shown in Chapter 3 in Figure 3.12. We note the presence of a succession of narrow peaks which are characteristic of a crystalline structure.

A Frank-Kasper phase ?

Regarding the Wahnström binary mixture, Pedersen et al. [START_REF] Pedersen | Long-time simulations of viscous liquids from strong correlations to crystallization[END_REF] have notice that the LFS around small particles are the same in the liquid and the crystalline phase. However they observe a change in the LFS of large particles. Therefore for the bidisperse Voronoi liquid we investigate, in the crystalline phase, the LFS for small and large particles at T = 0.84 and T = 0.83. The results are shown in Table 6.4.

• The LFS of the small particles remain unchanged, the icosahedron and the distorted icosahedron are still the two preferred structures. However their fraction has spectacularly increased. More than the half of small particles are now involved in icosahedra and one can easily imagine that the clusters they form are much larger.

In Figure 6. [START_REF] Hanakata | Interfacial mobility scale determines the scale of collective motion and relaxation rate in polymer films[END_REF] we have represented such a cluster composed of 260 particles at the center of icosahedra and we notice that the cluster has percolated.

• The neighborhood of large particles is different in the crystalline phase with respect to the liquid phase. We observe clearly the presence of preferred structures with fraction > 10%. We therefore have qualitatively the same kind of results than the Frank and Kasper [START_REF] Frank | Complex alloy structures regarded as sphere packings. i. definitions and basic principles[END_REF], are tetrahedrally close-packed periodic structures [START_REF] Sadoc | Quasiperiodic frank-kasper phases derived from the square-triangle dodecagonal tiling[END_REF]. In these structures, the atoms occupy sites which are characterized by the value Z of the coordination number (or number of nearest neighbors).

Frank and Kasper showed that Z = 12, 14, 15 or 16 [START_REF] Frank | Complex alloy structures regarded as sphere packings. i. definitions and basic principles[END_REF] and furthermore they investigated the Voronoi polyhedron associated to particles characterized by Z, and they demonstrated that faces of the polyhedra must be pentagons or hexagons. In other words faces have 5 or 6 vertices [START_REF] Frank | Complex alloy structures regarded as sphere packings. i. definitions and basic principles[END_REF]. As a consequence, the particles with Z = 12 are icosahedra which have only faces with 5 vertices. For the other particles with higher values of Z, the number of faces with 6 vertices is given by Z -12. The different types of polyhedra are represented in Figure 6.17.

Frank and Kasper considered in a second step, the global structures formed by the connection between the different polyhedra. They named minor the sites where Z = 12

and major the sites where Z ≥ 14. An atom in a minor site has no neighbor with which it has 6 neighbors in common. On the contrary an atom A in a major site has Z -12

B neighbors with which it shares 6 common neighbors. Frank and Kasper defined as major ligand the line between A and B. There are 3 possible topologies for connected the major ligands as it is represented in Figure 6.17 (black lines). Therefore it is possible to consider an extended structure composed of all the connected networks of major ligands that is called the major skeleton of the structure.

One important point is that the network of major ligands is infinite as there is no case where Z = 13 (or equivalently there is no polyhedron with only one face with 6 vertices).

Frank and Kasper rejected the possibility that a network of major ligands close on itself and they therefore concluded that layering of the major skeleton must occurs [START_REF] Frank | Complex alloy structures regarded as sphere packings. i. definitions and basic principles[END_REF].

The previous description of the Frank-Kasper phase informs us that if such structures are present in the bidisperse Voronoi liquid, then one must observe a huge number of icosahedra and a smaller fraction of structures such as Z = 14, 15 or 16 [START_REF] Pedersen | Geometry of slow structural fluctuations in a supercooled binary alloy[END_REF]. This implies the presence of the following Voronoi polyhedra (0,0,12,2), (0,0,12,3) or (0,0,12,4) [START_REF] Bennett | Local atomic environments of hard magnets, metallic glasses and icosahedral phases[END_REF].

When comparing the signatures of this polyhedra with the ones found in Table 6.4, we immediately see that the polyhedron (0,0,12,4) is the favored structure around A particles for T = 0.84, and (0,0,12,3) or (0,0,12,4) are both the LFS of A at T = 0.83.

Therefore, there are strong evidences in favor of the existence of Frank-Kasper structures in the crystalline phase of the bidisperse Voronoi liquid. It is also interesting to notice that as polyhedra (0,0,12,3) are only present at T = 0.83, Frank-Kasper structures are likely to be different at T = 0.83 and T = 0.84. This is in agreement with the observations done for the partial structure factors.

At this stage of the work, the crystalline structures have not been identified and further investigations are required to identify the underlying Frank-Kasper phases.

Conclusion & Perspectives

During this thesis, we introduced a brand-new class of liquids, the Voronoi liquids, where the interaction among particles are directly related to the geometrical properties of Voronoi tessellations. Therefore the interactions are intrinsically manybody. In its simplest form, the monodisperse Voronoi liquid is unable to avoid crystallization.

Therefore, as our aim was to probe the neighborhood of glass transition, we proposed a bidisperse extension where the geometrical properties are now those of the Voronoi-Laguerre tessellations. We investigated theoretically and through molecular dynamics We firstly focused on the behavior of thermodynamic observables upon cooling. Contrary to thermodynamic potentials which were not really sensitive to the cooling, the excess heat capacity at constant volume C e v , shows a marked increase when the temperature is decreased below the temperature T * , which marks the crossover from the Arrhenian to the super-Arrhenian regime. Considering the underlying PEL, we attributed the behavior of C e v to the huge decrease of the average value of the IS energy for T < T * . Then the computation of the excess pressure revealed that this quantity is strongly negative. Moreover it was possible to determine a pair decomposition of the manybody force field for which we observed that two neighboring particles attract each other. This means that the bidisperse Voronoi liquid is always under tension but the nature of the forces prevents cavitation to occur, leading thus to a stable fluid always under tension.

We investigated then the microscopic structural observables such as the pair correlation function and the structure factor. We found for both observables that the bidisperse Voronoi liquid is compliant with usual phenomenology of glass transition, i.e structural observables are weakly sensitive to cooling. Moreover, we were able to derive analytically the position of the first peak of partial pair correlation function and structure factor.

An analysis of the angle distribution revealed a tendency to form icosahedra that was confirmed during the study of the locally favored structures. Indeed, we found that upon cooling, an increasing fraction of small particles is located at the center of icosahedra which spatially extend into large clusters that seem to percolate at very low temperature.

The characterization of this percolation has not been done yet and represents a future work that would give a better understanding of the behavior of these clusters. Moreover we notice that the presence of clusters of icosahedra impacts directly the dynamics as the relaxation time inside LFS is slower than the relaxation time of particles which are not involved in LFS. This clearly shows the heterogeneous nature of the dynamics at low temperatures which was also revealed by the breakdown of the Stokes-Einstein relation.

A particular interest was given to analyze of dynamical observables. We noticed that upon cooling, both relaxation and coefficient diffusion strongly deviate from the Arrhenian regime. We quantified this deviation by measuring the fragility for relaxation time and diffusion coefficient and then by comparing the fragility of the bidisperse Voronoi liquid with the ones of the LJ glass formers. We found that the fragility of the Voronoi liquid is changing in function of the scale observed. As a matter of fact, when investigating microscopic scales, thus when the structural relaxation time is considered, the system has a marked fragility. However, when probing the hydrodynamic limit, thus when looking at the diffusion coefficient, the fragility appears less marked. This difference was not observed for LJ glass formers. This led us to propose an alternative to Angell-plot and to search for intrinsic microscopic time and energy scales that would be common to different systems.

We proposed to define the microscopic intrinsic time-scale as the inverse of the Einstein's frequency Ω -1 0 , as this latter is the fastest physical time in liquids. The definition of a proper energy scale E µ was much more tricky and we proposed three possible candidates: the microscopic work, the fluctuations of the potential energy and the infinite-frequency shear modulus. The representations in terms of τ α Ω 0 as a function of E µ /(k B T ) brought, on the first hand, physical informations on the increase of the relaxation time. On the other hand they emphasized the difference between classes of system as clear separations between the bidisperse Voronoi liquid and the LJ glass formers were observed. A remaining major challenge would be to find the right couple (Ω -1 0 , E µ ) that would enable to reproduce the T -dependence of the energy barriers in such a way that the relaxation time could be written τ α = Ω -1 0 exp(E µ /(k B T )). Then the Holy Grail would be to find a representation that would take into account the diversity of microscopic interactions and offer a universal interpretation of the behavior of relaxation time.

Furthermore our investigations of the collective relaxation time on different length-scales

Conclusion & Perspectives 200 revealed that, in the mesoscopic range for k ≈ 4.90, the relaxation does not couple to the structural relaxation time τ α . To the best of our knowledge, this behavior is specific to the bidisperse Voronoi liquid and we did the hypothesis that this length scale is associated to the presence of a transition state, that we still have to identify.

We also investigated the dynamics by testing the predictions of the ideal MCT. We found that for wavevectors k ≥ k * , the ideal MCT managed to describe the relaxation time relatively well, whereas in the mesoscopic range, we noticed that the value of the von Schweidler exponent does not remain constant, which suggests a breakdown of the theory. Understanding the origin of this violation would be of great interest to capture the specific phenomenology of the mesoscopic range.

To get more insight into the physical origin of the glass transition, we investigated the PEL of the bidisperse Voronoi liquid and of the 3 LJ glass formers. The study of the underlying inherent structures revealed that their distribution is well-described by the Gaussian approximation. However the width σ 2 of the distribution is 10 times greater for the Voronoi liquid than for the LJ systems. We correlated this observation to the huge increase C e v at the neighborhood of the glass transition. Moreover a particular interest was given to the determination of the configurational entropy S c . In the harmonic approximation, we can distinguish between the usual configurational entropy defined as S c = S liq -S sol , and the effective configurational entropy Ŝc which is linked to S c via the spectrum of the Hessian of the potential energy. Harmonic approximation implies that Ŝc should be given by ln(P (e IS |T )) + e IS /T for T < T * . By comparing the results of ln(P (e IS |T )) + e IS /T with an analytical expression of Ŝc we find an agreement only at very low temperature. However an estimation of the anharmonic contribution to the potential energy showed that there are non-negligible even for T T * . We therefore took them into account into the analytical expression Ŝc . We observed that in the limit of weak anharmonicity, the results were even worst than without taking them into account. However when we considered stronger anharmonicities, the agreement between Ŝc and ln(P (e IS |T )) + e IS /T appeared better. We therefore concluded that anharmonicity played a significant role in the bidisperse Voronoi liquid even for low temperature. Their precise origin have still to be determined as it will allow a better understanding of the thermodynamics of inherent structures.

From the knowledge of the configurational entropy S c , we tested the theory of Adam-Gibbs for the relaxation time and at very low temperature, we found a good agreement between the theory and our numerical results. Considering the limit where k → 0, we were able to determine the threshold energy below which there is no more negative directions and thus to give an estimation of the mode coupling temperature T c . We found that by increasing the system size, the value of T c determined by this method tends to the value found with ideal MCT analysis. Unfortunately the EV method allowed only an exploration high k-high u regime (associated with the Arrhenian regime) and we didn't succeed to explore the low klow u regime to validate the results found for quasisaddles. However understanding why quasisaddles, which are not proper stationary points, seem to convey correctly information about the dynamics in the real space, remains an open question.

Finally we observed that, at very low temperature, the equimolar bidisperse Voronoi liquid crystallizes. The precise crystalline structure has not been identified yet but we found strong evidences in favor of the formation of Frank-Kasper phases.

During this work, we didn't pay to much attention to the dynamical heterogeneities despite strong evidences of their presence in the bidisperse Voronoi liquid. In the past two decades, they deserved much attentions as it has been shown that the four-point susceptibility χ 4 (t) informs on the spatial extension of these heterogeneities [START_REF] Berthier | Dynamical Heterogeneities in Glasses, Collloids and Granular Media[END_REF]. Moreover in the hydrodynamic limit, the dynamical correlation length ξ 4 (t) can be extracted from the four-point structure factor S 4 (q, k, t) which exhibits a power-law behavior [START_REF] Flenner | Large and long-range dynamic correlations in supercooled fluids revealed via four-point correlation functions[END_REF]. It could be interesting to compute these quantities for the bidisperse Voronoi liquid to obtain precise informations about the dynamical heterogeneities. However this will be technically difficult. For the monodisperse Voronoi liquid, it has been shown that the hydrodynamics regime developed only for very small values of k [START_REF] Ruscher | Anomalous sound attenuation in voronoi liquid[END_REF] much more smaller than for the usual LJ systems. The same behavior is also likely to be present for the bidisperse model and therefore having access to ξ 4 (t) would require to work, at constant density, with systems with more than N = 2 • 10 6 resulting on very long simulation time.

Due to the difficulties encountered to express the stress tensor, we didn't have time to look at the behavior of the shear-stress in the system. It would be interesting to see how the bidisperse Voronoi liquid behaves under shear as we assumed in [START_REF] Ruscher | Anomalous sound attenuation in voronoi liquid[END_REF], that it is relatively easy to shear the system. Monitoring the evolution of the shear relaxation upon cooling could bring further information about what happens close to glass transition.

Moreover as proposed in [START_REF] Pinney | Structure in sheared supercooled liquids: Dynamical rearrangements of an effective system of icosahedra[END_REF] for the Wahnström model, it could be interesting to see Bibliography

Figure 1 :

 1 Figure 1: Représentation 2d d'une tessellation de Voronoï-Laguerre pour un mélange bidisperse.

Figure 2 :Figure 3 :

 23 Figure 2: Evolution de la chaleur spécifique d'excès à volume constant par particule C e v /N en fonction de la température pour le liquide de Voronoï bidisperse ainsi que pour 3 liquides de Lennard-Jones.

83 Figure 4 :Figure 5 :

 8345 Figure 4: Haut : Evolution avec la température de la fonction de corrélation de paire partielle g AA (r). Bas : Evolution avec la température du facteur de structure statique partiel S AA (k).

  F s (k, t) et du MSD nous a permis d'extraire respectivement le temps de relaxation structurale τ α et le coefficient de diffusion D. La représentation de ces grandeurs à la Figure 6 montre qu'à haute température, le liquide de Voronoï bidisperse présente un caractère Arrhénien et que lorsque la température est abaissée, en deçà d'une température T * , dite température de crossover, τ α et D manifestent un comportement super-Arrhénien. Ces observations sont valables quelque soit la composition (A : B).

  Nous avons ensuite comparé nos résultats avec ceux des 3 mélanges LJ. Cette comparaison est loin d'être triviale car les unités du liquide de Voronoï diffèrent des unités Lennard-Jones. Il a donc été essentiel de déterminer des échelles de longueur, de temps et d'énergie afin de pouvoir adimensionner nos observables. Une première possibilité consiste à representer un graphe d'Angell où la quantité de transport est, elle aussi, adimensionnée. Pour se faire nous avons adimensionné τ α et D respectivement par le temps de relaxation τ ∞ et le coefficient de diffusion D ∞ que nous avons extrait à l'amorce du régime Arrhénien. Cette représentation permet de comparer l'évolution des grandeurs de transports entre deux points de référence, l'un pris à haute température et l'autre à basse température. En effet, comme mentionné précédemment, la température dans un graphe d'Angell est adimensionnée par T G . Pour les systèmes étudiés, T G est largement inférieure à la température la plus basse accessible numériquement. Nous avons donc utilisé une temperature de référence T r jouant le rôle d'une T G numérique, définie de telle sorte que τ α (T r ) = 4 • 10 4 . De façon similaire, pour le coefficient de diffusion, nous avons défini T D r telle queD(T D r ) = 10 5 . Les résultas présentés sur la Figure 7 montrent que "l'ordre" dans le graphe d'Angell n'est pas respecté pour le liquide de Voronoï bidisperse. Ainsi lorsque T → T r le liquide est relativement fragile lorsqu'on sonde l'environnement microscopique, c'est à dire τ α . En revanche, lorsque la limite hydrodynamique est considérée, i.e lorsqu'on regarde D, cette fragilité est beaucoup moins prononcée.

Figure 6 : 4 .

 64 Figure 6: Evolution du temps de relaxation τ α (haut) et du coefficient de diffusion D (bas) en fonction de la température et pour différentes compositions du mélange binaire. T r est définie telle que τ α (T r ) = 4 • 10 4 .

Figure 7 : 80 Figure 8 :

 7808 Figure 7: Comparaison de l'évolution de τ α /τ ∞ (haut) et de D/D ∞ (bas) en fonction de la température adimensionée T r /T . τ ∞ et D ∞ sont des constantes extraites du fit Arrhénien à haute température. T r et T D r sont des températures de référence arbitrairement choisies comme les températures pour lesquelles τ α = 4 • 10 4 et D(T D r ) = 10 5 .

Figure 9 :

 9 Figure 9: Evolution pour différentes températures de la relaxation individuelle (haut) et de la relaxation collective (bas) pour différentes valeurs du vecteur d'onde k adimensionné par sa valeur k * prise au premier pic de S(k).

  xiii

Figure 10 :

 10 Figure 10: Test pour le facteur de structure cohérent (haut) et pour le facteur de structure intermédiaire (bas) du fit de von Schweidler avec contraintes pour différents vecteurs d'onde q.

Figure 11 :

 11 Figure 11: Evolution de l'énergie moyenne des IS e IS /N en fonction de l'inverse de la température 1/T pour le liquide de Voronoï (+), le mélange KA ( ) , le modèle de Wahnström ( ) et AMLJ-0.80 (△).

)Figure 12 :

 12 Figure 12: Evolution du temps de relaxation en fonction de 1/(T S c ).

Figure 13

 13 Figure 13 haut). La seconde méthode permet d'évaluer l'écart énergétique u -e IS entre un point-selle ayant K directions négatives et la structure inhérente la plus proche. Nous avons comparé nos résultats avec ceux de différents systèmes LJ et nous avons ainsi pu remarquer que tous les résultats se trouvaient sur une courbe maitresse (voir Figure 13 bas). Ceci suggère qu'il existe une phénoménologie commune entre tous ces systèmes.

Cette tendance à former

  des icosaèdres n'est pas spécifique au liquide de Voronoï. Elle est par exemple partagée avec le modèle de Wahnström ou AMLJ-0.80, comme on peut le voir sur la figure 15. Il semble donc que le liquide de Voronoï partage des spécificités avec WAHN ou AMLJ-0.80, ce qui permet d'observer ces similitudes structurales.

Figure 14 :

 14 Figure 14: Représentation de la fraction des principales LFS en fonction de T c /T où T c = 0.7989 est la température de couplage de modes. Les fractions ont été calculées pour des configurations thermalisées (haut) et également pour des configurations correspondant à des IS (bas).

Figure 15 :

 15 Figure 15: Evolution de la fraction d'icosaèdres en fonction de T r /T pour le liquide de Voronoï bidisperse et deux liquides LJ. Les fractions ont été calculées pour des configurations thermalisées (haut) et également pour des configurations correspondant à des IS (bas).

Figure 16 :

 16 Figure16: Evolution de la probabilité P (n i ) de trouver un cluster composé de n i icosaèdres.

Figure 17 et

 17 Figure 17 et cette transition s'accompagne par un accroissement très brutal du nombre d'icosaèdres. De plus, nous avons noté que les LFS associés aux grandes particules A changeaient radicalement par rapport à la phase liquide. Ces nouveaux LFS suggèrent la présence d'une phase de Frank-Kasper que nous n'avons pas encore identifiée pour le moment.

Chapter 1 A brand new class of liquids Contents 1 . 1

 111 Geometrical properties of Voronoi tessellations . . . . . . . 1.1.1 Classical Voronoi tessellations . . . . . . . . . . . . . . . . . . . 1.1.2 Voronoi-Laguerre tessellations . . . . . . . . . . . . . . . . . . . 1.2 Energy and force field : general expressions . . . . . . . . . 1.3 The monodisperse Voronoi liquid . . . . . . . . . . . . . . . . 1.3.1 Scaling relation for the potential energy and thermodynamics . 1.3.2 Microscopic observables: similarities with usual simple liquids 1.3.3 Anomalous sound attenuation . . . . . . . . . . . . . . . . . . . 1.3.4 Metastability and crystalline phase . . . . . . . . . . . . . . . . 1.4 The polydisperse Voronoi liquid . . . . . . . . . . . . . . . . . 1.4.1 The bidisperse Voronoi liquid . . . . . . . . . . . . . . . . . . . 1.4.2 Molecular dynamics (MD) simulations . . . . . . . . . . . . . . 1.1 Geometrical properties of Voronoi tessellations 1.1.1 Classical Voronoi tessellations We consider N point particles contained into a cubic box of volume V = L d where L is the length of the box and d is the dimension of the space. In this work we will focus on dimension d = 2 and d = 3. Voronoi tessellations are a mathematical way of partitioning the space into Voronoi cells. Given the set of points

Figure 1 . 1 :

 11 Figure 1.1: Representation of 2-dimensional Voronoi tessellation.The polygon associated to the particle j is built from the nearest neighbors of j. Each segment composing the polygon corresponds to the median plane between j and one of its nearest neighbors. The intersection of these median planes defines the polygon.

Figure 1 . 2 :

 12 Figure 1.2: Geometric representation of the power of M with respect to the circle of center M J and radius R j .

Figure 1 . 3 :

 13 Figure 1.3: Localization of the hyperplane between a small particle (black circle) and a large particle whose size is increased (coloured).

C 3 and C 2

 2 are contained into the Voronoi cell of C 1 .

Figure 1 . 4 :

 14 Figure 1.4: Geometric representation of Voronoi-Laguerre tessellation. Figure is taken from [1].

Figure 1 . 5 :

 15 Figure 1.5: Representation of 2-dimensional Voronoi-Laguerre tessellation.

Figure 1 . 6 :

 16 Figure 1.6: Representation of the vector r in 2 dimensions.

  liquid, where the N point particles playing an identical role are contained into a box of volume V = L d at density v = V /N . By choosing the function f (r) = r 2 in equation 1.8, the Hamiltonian becomes

  8000 point particles contained into a cubic box of length L = 20 such as v = 1. The constant γ is arbitrarily chosen equal to 1000. Due to the dimensionality of γ, γv d+2 d has the dimension of an energy and this ensures that the characteristic energy of interaction as E v = 0.001γv d+2 d .

3 Figure 1 . 7 :

 317 Figure 1.7: Left: Evolution of the scaled excess free energy per particle F e /(N γv 5/3 ) as a function of the scaling variable x. Right: Evolution of the scaled excess chemical potential as a function of x (blue line). The red crosses represent the value of x ln(g(0)) for several values of x.

Figure 1 . 8 :

 18 Figure 1.8: For different temperatures are represented the pair correlation function g(r) (left) and the structure factor S(k) (right).

Dγ - 1 / 2 v - 7 / 6 Figure 1 . 9 :

 127619 Figure 1.9: (Left) Evolution of scaled mean square displacement for different temperature. From bottom to top : T = 1.05, 1.50, 1.75, 2.00, 4.00, 6.00, 8.00, 15.00. The circles corresponds to the Enskog theory for the highest temperature. (Right) Logarithm of the scaled diffusion coefficient as a function of 1/x. The dashed line corresponds to an Arrhenian fit. Insert: Scaled diffusion coefficient as a function of the scaling variable x.The dashed line underlines the asymptotic behavior where D ∼ T 5/2 .

Figure 1 . 10 :

 110 Figure 1.10: Schema of wavelength as a function of frequency shows three types of regimes observed in liquids. Wavelength and frequency have been rescaled respectively by the mean free path ℓ mf p and the collision time τ c .

Figure 1 . 11 :

 111 Figure 1.11: Test of scaling kS(k, ω)/S(k) as a function of ω/k for T /E V = 1.50, N = 65536 particles and k ∈ [0.3, 1.0]. The same data are represented for the scaling k 2 S(k, ω)/S(k) vs. [ωω max ]/k 2

Figure 1 . 12 :

 112 Figure 1.12: Representation of the potential energy per particule in the metastability zone for temperatures 1.05 ≤ T ≤ 1.85. Grey and orange curves correspond to cooling and heating process at rates respectively dT /dt = 2.5 • 10 -4 and dT /dt = 2.5 • 10 -5 . The black and blue crosses are mean values of E P /N evaluated on several isotherms with respectively N = 8000 and N = 8192.

Figure 1 . 13 :

 113 Figure 1.13: Representation of the pair correlation function for three temperatures taken in the metastable zone. The straight blue lines correspond to isotherms in the liquid phase with N = 8000 particles whereas the dashed red lines are data from isotherms in the crystalline phase for N = 8192 particles.

Figure 1 . 14 :

 114 Figure 1.14: Representation of the MSD for three temperatures taken in the metastable zone. The straight blue lines correspond to isotherms in the liquid phase with N = 8000 particles whereas the dashed red lines are data from isotherms in the crystalline phase for N = 8192 particles.

Figure 1 . 15 :

 115 Figure 1.15: Left: Truncated octahedron corresponding to the BCC Voronoi cell. Right: Rhombic dodecahedron corresponding to the FCC Voronoi cell.

Figure 1 . 16 :

 116 Figure 1.16: Upper part left and right : Cooling processes between T = 2.00 and T = 0.1 for different values of η ∈ [0.20, 0.60]. For clarity high values of η are presented on separated graphs (Right). Lower part : Cooling processes between T = 4.00 and T = 0.1 for η = 0.70.

Figure 1 . 17 :

 117 Figure 1.17: Top: Pair correlation function for different values of η at T = 2.00. Inset: Pair correlation function for η = 0.70 and T = 4.00. Bottom: For η = 0.70, distribution of the volume of the Voronoi cell. Inset Distribution of volumes v B for very small values of v. The power law behavior is emphasized by the black straight line.

Figure 2 . 1 :

 21 Figure 2.1: Evolution of the potential energy per particle E P /N with temperature T for several cooling rates k ∈ [5 • 10 -6 , 2 • 10 -4 ]. The inset corresponds to a zoom on the glass part.

Figure 2 . 2 :

 22 Figure 2.2: Illustration of the method used for finding T G .

Figure 2 . 3 :

 23 Figure 2.3: Left: Representation of the demixing case. The dashed grey line represents the interface between the two phases of the mixture. Right: Perfect mixing of the bidisperse Voronoi liquid.

Figure 2 . 4 :

 24 Figure 2.4: Schematic η-T diagram illustrating the path (blue line) chosen to reach the state represented by the red point.

Figure 2 . 5 :

 25 Figure 2.5: Upper graph: Representation of E (a)

Figure 2 . 6 :

 26 Figure 2.6: Excess free energy as a function of temperature T .

Figure 2 . 7 :

 27 Figure 2.7: Excess entropy as a function of temperature T .

. 23 ) 2 Nk B T 2 Figure 2 . 8 :

 232228 Figure 2.8: Excess heat capacity as a function of temperature T .

Figure 2 . 9 :

 29 Figure 2.9: Representation of the two components of the total bidisperse potential energy: in blue the monodisperse part and in orange the component relative to polydispersity.

Figure 2 . 10 :

 210 Figure 2.10: Schematic representation of what would happen for the bidisperse Voronoi liquid in case of cavitation.

  (i) The Clausius expression for the Virial σ e zz (k = 0) = i F i,z r i,z is only valid for a system without P.B.C.(ii) In the case of P.B.C Louwerse and Baerends have shown that an explicit dependence of the box size appears in the formulation of the stress tensor[START_REF] Louwerse | Calculation of pressure in case of periodic boundary conditions[END_REF]

Chapter 2 .Figure 2 . 11 :

 2211 Figure 2.11: Excess pressure P e as a function of temperature. The solid blue line corresponds to the computation directly made via equation 2.12 and dark blue crosses corresponds to the computation from the Virial theorem.

3 . 1 55 3. 1 . 2 55 3. 1 . 3 61 3. 2 69 3. 2 . 1 69 3. 2 . 2

 315512551361269216922 Pair correlation function . . . . . . . . . . . . . . . . . . . . . 53 3.1.1 Evolution with temperature . . . . . . . . . . . . . . . . . . . . Cosine distribution . . . . . . . . . . . . . . . . . . . . . . . . . Relative position of the first peaks of g ab (r) . . . . . . . . . . . Static structure factor . . . . . . . . . . . . . . . . . . . . . . Evolution with temperature . . . . . . . . . . . . . . . . . . . . Relative position of the first peaks of S ab (k) . . . . . . . . . . . 71

Chapter 3 T 83 Figure 3 . 1 :

 38331 Figure 3.1: Partial pair correlation function g ab (r) evaluated for several temperature T ∈ [0.83, 2.00] and for N = 1000. From top to bottom g AA , g AB and g BB . Each curve has been shifted of 0.1 upward for clarity.

5 )Chapter 3 1 - 1 dFigure 3 . 2 :

 531132 Figure 3.2: Illustration in 2d of the triplet of particles (i,j,k) included in the sphere of radius R c . The angle θ is formed by bonds IJ and IK.

Figure 3 . 3 :

 33 Figure 3.3: Representation of the cosine distribution P jik (R c , cos(θ)) when (Top) the central particle is of type A and R c,A = 1.5 and when (Bottom) the central particle is of type B and R c,B = 1.25. Both distributions have been computed at T = 0.85 for N = 1000.

  [START_REF] Heyes | Chemical Modelling Applications and Theory, volume 2, chapter Simulation of the Liquid State[END_REF] shows particles in an icosahedral environment and the right upper schema is a zoom over the upper part of the icosahedron. Each edge is of length a and the length of the central particles to each of the other particles is given by the circumradius ≈ 0.95a. We look for the angle formed by the bond between top and central particles and the bond between the central particle and any particles belonging to the pentagon. The lower right illustration is the typical triangle formed by the particles of interest and we search for the value of ϕ which is given by ϕ ≈ 2 arcsin(1/(2•0.95) ≈ 63 • .

Figure 3 . 4 :

 34 Figure 3.4: Left: Representation of particles in an icosahedral configuration. Right: Typical distances and angle in the icosahedron.

Figure 3 . 5 :

 35 Figure 3.5: If θ 1 ≈ 60, Left: Representation of the possible ways to find angles ∼ 2θ 1 . Particles colored in red or yellow are examples where the angle formed with the central particle ∼ 2θ 1 . Right: Representation when ∼ 3θ 1 , the green particles are representative of the typical angle.

Figure 3 . 6 :

 36 Figure 3.6: Representation of the three types of configurations that can be found when the central particle i is small.

85 Figure 3 . 7 :

 8537 Figure 3.7: Evolution of P AAA (R c , cos(θ)) (Upper graph) and P BBB (R c , cos(θ)) (Lower graph) upon cooling.

3. 1 . 3

 13 Relative position of the first peaks of g ab (r)3.1.3.1 Splitting of g AA (r) and g BB (r)The position r * of the first peak of the pair correlation function informs on the average position of nearest neighbors. If we distinguish the different species by focusing on partial pair correlation functions, it is possible to determine r * ab and consequently the relative position of the first peaks for the two species. In Figure3.8 are represented the three partial pair correlation functions.

Figure 3 . 8 :

 38 Figure 3.8: Representation of the relative position of the three partial pair correlation functions for T = 0.85 and N = 8000.

Figure 3 . 9 :

 39 Figure 3.9: Explanation of the equilibration of pressure inside a Voronoi cell.

Figure 3 . 10 :

 310 Figure 3.10: Representations of the monodisperse pair correlation function g 0 (k) in blue and δ(rr ij ) (v i -v) 0 in dashed red line for T = 2.00 and N = 8000.

Figure 3 . 11 :

 311 Figure 3.11: Representation of the relative positions of the two first peaks of the three partial pair correlation functions for T = 2.00, N = 1000 and for different compositions A : B of the mixture.

T 83 Figure 3 . 12 :

 83312 Figure 3.12: Partial pair structure factors S ab (r) evaluated for several temperatures T ∈ [0.83, 2.00] and for N = 1000. From top to bottom S AA , S AB and S BB . Each curve has been shifted of 0.1 upward for clarity.

Figure 3 . 13 :

 313 Figure 3.13: Representation of the total structure factors and of the different partial pair structure factor for T = 0.85 and N = 1000.

jFigure 3 . 14 :

 314 Figure 3.14: Representation of the monodisperse structure factor S(k) in blue solid line and j e ik•rij (v i -v) 0 in dashed red line for T = 2.00 and N = 8000.

Figure 3 . 15 : 3

 3153 Figure 3.15: Representation of the relative positions of the first peaks of the three partial structure factors for T = 0.85, N = 1000 and for different compositions A : B of the mixture. The curve of S AB has been shifted on purpose to better compare the position of the different peaks.

  Two-step relaxation processes . . . . . . . . . . . . . . . . . . 4.1.1 Intermediate scattering function . . . . . . . . . . . . . . . . . 4.1.2 Mean-square displacement . . . . . . . . . . . . . . . . . . . . . 4.2 Fragility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Influence of the mixture composition . . . . . . . . . . . . . . . 4.2.2 Comparing models: Lennard-Jones glass formers . . . . . . . . 4.2.3 Comparing models : the problem of units . . . . . . . . . . . . 4.2.4 Other nondimenzionalization of microscopic time and energy scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Stokes-Einstein relation . . . . . . . . . . . . . . . . . . . . . . 4.4 Individual and collective relaxations . . . . . . . . . . . . . . 4.5 Ideal Mode Coupling Analysis . . . . . . . . . . . . . . . . . . 4.5.1 Theoretical predictions of MCT . . . . . . . . . . . . . . . . . . 4.5.2 Time-temperature superposition principle . . . . . . . . . . . . 4.5.3 Factorization theorem . . . . . . . . . . . . . . . . . . . . . . . 4.5.4 von Schweidler fit . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.5 Determining T c . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 4 . 1 :

 41 Figure 4.1: Illustration of the decay of a correlation function φ(t) at high temperature (dashed line) and at low temperature (solid line). Figure taken from [69].

4 )Chapter 4 observables 82 In Figure 4 . 2

 448242 Given a specific wavevector k, the incoherent intermediate scattering function measures the correlation between the position of a tagged particle at time t = 0 and its position at time t. Incoherent and coherent intermediate scattering function complement one another in the description of the liquid. Whereas the incoherent scattering function describes the dynamics of individual particles, the coherent scattering function informs on collective behaviors of particles [75]. The incoherent intermediate scattering function can be computed for any values of the wavevector k. However in Chapter 3 we have seen that the static correlations of density measured by the partial structure factors S ab (k) are maximal when k ∼ k * ab (where k * ab is the position of the first peak of S ab (k)). As a consequence it corresponds to the region where the spatial fluctuations of density are maximal. It is thus interesting to see how these fluctuations are going to evolve with time. Dynamical are represented for both species A and B the incoherent intermediate scattering functions computed for k = k * aa .

Figure 4 . 2 :Figure 4 . 3 :

 4243 Figure 4.2: From top to bottom are respectively represented the coherent intermediate scattering functions, the intermediate scattering functions for A particles and the intermediate scattering function for B particles. They are all computed for different temperatures at k = k * or k = k * aa .

Figure 4 . 4 :

 44 Figure 4.4: Representation of the MSD of A particles for several temperatures.

Figure 4 .

 4 Figure 4.5 where the diffusion D A and D B of A and B species respectively are plotted as a function of temperature.

Figure 4 . 5 :

 45 Figure 4.5: Diffusion coefficient computed at different temperatures for A and B particles.

Such a rescaling of

  the temperature by introducing the glass transition temperature was firstly proposed by Oldekop in 1957 for comparing the evolution of the viscosity of different oxide and fluoride glass-formers[START_REF] Ngai | Relaxation and Diffusion in Complex Systems[END_REF]77]. In 1972, Laughlin and Uhlmann reintroduced the T G scaled temperature and compared different classes of species ranging from oxide to organic glass-formers. As Oldekop, they noticed that different systems belonging to the same class of materials have qualitatively the same evolution on the log(η) vs. 1/T plot[START_REF] Laughlin | Viscous flow in simple organic liquids[END_REF]. Furthermore they also observed without giving any explanation, that different classes of systems have different curvatures in the log(η) vs. 1/T plot.In the 80s, Angell pushed further the work of Laughlin and Uhlmann by including other species to the log(η) vs. T plot, as for instance polymeric materials. Moreover Angell proposed and interpretation of the pattern observed. To this purpose he classified the behavior of the materials regarding their fragility index m[START_REF] Böhmer | Correlations of the nonexponentiality and state dependence of mechanical with bond connectivity in ge-as-se supercooled liquids[END_REF] which is a measure of the slope of η when T = T G :

  species do not conform to these observations as for instance methylcyclohexane which presents an intermediate behavior in the log(η) vs. T G /T plot also known as Angell plot (see Figure4.6). Nowadays the term fragility has a bit lost its original meaning and refers more to the way transport quantities such as viscosity or diffusion coefficient are evolving with temperature[START_REF] Berthier | Theoretical perspective on the glass transition and amorphous materials[END_REF].

Figure 4 . 6 :

 46 Figure 4.6: Angell-plot of the viscosity of several species as a function of T G /T . (Reprinted with permission from [2], copyright of Elsevier, 1988.)

  Numerically we could think of defining a temperature of reference denoted T r which would correspond to a numerical glass transition. This idea was first suggested by Bordat et al.[START_REF] Bordat | Does the interaction potential determine both the fragility of a liquid and the vibrational properties of its glassy state?[END_REF] and then used by Coslovich and Pastore[START_REF] Coslovich | Understanding fragility in supercooled lennard-jones mixtures. i. locally preferred structures[END_REF]. In both works T r is arbitrarily defined as the temperature at which τ α = 4 • 10 4 and its determination would require an extrapolation of the relaxation time to τ α = 4 • 10 4 . Results for the structural relaxation times are shown in Figure 4.7 and for the diffusion coefficient in Figure 4.8. The parameter extracted from the fit for the relaxation time and diffusion coefficient are presented respectively in

•

  The higher temperature regime where the Arrhenian regime occurs is only weakly sensitive to the composition of the mixture. Indeed, for both relaxation time and diffusion coefficient, τ ∞ and D ∞ are almost constant whatever the nature of the mixture. Moreover it is also the case for the activation energy E ∞ which is in average E ∞ ∼ 4.89 for the relaxation time and E ∞ ∼ 5.21 for the diffusion coefficient.

Figure 4 . 7 :

 47 Figure 4.7: Angell-plot of the evolution of the relaxation time τ α as a function of T r /T for several compositions of the binary mixture.

Figure 4 . 8 : 4

 484 Figure 4.8: Angell-plot of the evolution of the diffusion coefficient D as a function of T r /T for several compositions of the binary mixture.

Figure 4 . 9 :

 49 Figure 4.9: Angell-plot of the evolution of the relaxation time τ α as a function of T r /T .

Figure 4 . 10 :

 410 Figure 4.10: Angell-plot of the evolution of the diffusion coefficient D as a function of T r /T .

Figure 4 . 11 : 4 Figure 4 . 12 :

 4114412 Figure 4.11: Angell-plot of the evolution of the scaled relaxation time as a function of T r /T for the bidisperse Voronoi liquid and the 3 LJ glass-formers.

Table 4 . 6 :

 46 Reference temperature T D r extracted from the diffusion coefficient when D(T = T r ) = 10 -5 . When comparing Figure 4.12 and Figure 4.11 we observe that:

4. 2 . 4 Chapter 4 80 Figure 4 . 13 :

 24480413 Figure 4.13: Evolution of the scaled relaxation time τ / τ ∞ as a function of the scaled temperature E ∞ /(k B T ).

Figure 4 . 14 :

 414 Figure 4.14: Evolution of the inverse of the Einstein frequency rescaled by the intrinsic time units as a function of temperature for the bidisperse Voronoi liquid (blue +) and for LJ systems (orange) KA ( ), WAHN ( ), AMLJ-0.80(△) .

Figure 4 .

 4 Figure 4.14 is a typical illustration of the difference of the intrinsic time units calculated

80 Figure 4 . 15 :

 80415 Figure 4.15: Evolution of the inverse of the Einstein frequency as a function of temperature for the bidisperse Voronoi liquid and for the 3 LJ glass formers. Notice that the choice of γ = 1000 put the typical values of Ω -1 0 close to that of LJ systems.

Figure 4 . 17 :

 417 Figure 4.17: Evolution of the typical energy E µ = F 2 (0) v 1/3 with temperature for the Voronoi liquid and the 3 LJ glass formers.

80 Figure 4 . 18 :Chapter 4

 804184 Figure 4.18: Representation of the relaxation time rescaled by the Einstein frequency as function of the inverse of the temperature rescaled by E µ /k B .

Figure 4 . 19 :

 419 Figure 4.19: Representation of the heat capacity at constant volume as a function of the scaled temperature for the bidisperse Voronoi liquid and the LJ glass-formers.

Figure 4 . 20 :

 420 Figure 4.20: Scaled relaxation time as a function of the scaled temperature C e V /(N k B ).

Figure 4 .

 4 21 is shown the results of this fit for the bidisperse Voronoi liquid and the Wahnström model.

Figure 4 . 21 :

 421 Figure 4.21: Scaled relaxation time as a function of the scaled temperature C e V /(N k B ). The solid lines correspond to the exponential fit.

Figure 4 .

 4 22 we have represented the dimensionless quantity G ∞ /(ρT ) as a function of T /T * . Rescaling T by the crossover temperature is motivated by the fact that the energy barriers below T * are known to be temperature-dependent. As in the shoving model the energy barriers ∝ G ∞ (T ), it is interesting to see how this quantity evolves above and below T * .

  as consequence a linear behavior is expected for all the systems in the semi-log representation in Figure4.23.

Figure 4 . 22 :

 422 Figure 4.22: Evolution of the scaled instantaneous shear modulus as a function of the scaled temperature.

Figure 4 . 23 :

 423 Figure 4.23: Evolution of the scaled relaxation time as a function of G ∞ v/T .

4 •

 4 2 /D and ηR 3 /T lead to the same timescale up to constant 1/6πR[START_REF] Berthier | Theoretical perspective on the glass transition and amorphous materials[END_REF].The Stokes-Einstein relation requires at first sight the knowledge of the viscosity η that we didn't compute. However we have access to the structural relaxation time which is linked to the viscosity via η = G ∞ τ α where G ∞ is the instantaneous shear modulus defined above. The evolution of G ∞ as a function of temperature has been computed in the previous section, seeFigure 4.22. In comparison to the relaxation time which Chapter Dynamical observables 112 increases of 3-4 orders of magnitude upon cooling, G ∞ observes a very slow variation in such a way that it can be considered constant and thus η ∼ τ α . Therefore a good way to check the validity of the Stokes-Einstein relation is to compute the ratio Dτ α /T and to see to what extent it remains constant over the whole range of temperature investigated. In Figure 4.24 we have represented the Stokes-Einstein relation rescaled by G ∞ v 1/3 as a function of T /T * At high temperature for T > T * the presence of a plateau suggests that the Stokes-Einstein relation is valid and therefore Dτ α /T ∼ C te .

Figure 4 . 24 :

 424 Figure 4.24: Semi-log representation of the scaled Stokes-Einstein relation as a function of the scaled temperature T /T * . Note that the values at high temperature are very small 10 -4 .

Figure 4 . 25 :

 425 Figure 4.25: Sketch of the different domains in the liquid.

It is also interesting

  to investigate collective and individual relaxation times evaluated for other wavevectors. To do so we compute respectively the coherent and incoherent intermediate scattering function for several k ∈ [2π/L; 14.0]. The relaxation times τ are still defined as the value of the time when F (k, t = τ )/S(k) = 0.1 (or F s (k, t = τ ) = 0.1).InFigures 4.26

and 4 .Figure 4 . 26 :

 4426 Figure 4.26: Scaled individual relaxation time as a function of the scaled wavevector computed for different temperatures. The black line represents a relaxation ∼ k 2 .

Figure 4 . 27 :

 427 Figure 4.27: Scaled collective relaxation time as a function of the scaled wavevector computed for different temperatures. The black line represents a relaxation ∼ k 2 .

Chapter 4 observables 115 •

 4115 Dynamical Whereas the individual relaxation decreases in a monotonic way as k is increased, the collective relaxation exhibits extrema and among them a very deep minima for wavevector k ∼ 4.90. When looking in Figure 4.28 at the coherent structure factor F (k = 4.90, t) computed for k = 4.90 we notice the presence of very fast decay followed by numerous oscillations reminiscent of the relaxation of the hydrodynamic sound modes at low k.

Figure 4 . 28 :

 428 Figure 4.28: coherent intermediate scattering function computed at k = 4.90 for T = 0.85.

  and 4.30. The collective relaxation is relatively similar, apart from the huge decrease in the mesoscopic range. This suggests that for k ∼ 4.90 the collective relaxation does not couple to the structural relaxation. A possible explanation of this phenomenon would be that the associated length 2π/4.90 be exclusively associated to a transition state. This way, the relaxational dynamics of that wavelength would not couple to the slow dynamics induced by structural relaxation.

Figure 4 . 29 :

 429 Figure 4.29: Comparison of the individual relaxation times obtained for the bidisperse Voronoi liquid at T = 0.88 and the polydisperse system of quasi-hard spheres of Weysser et al. [3] at ϕ = 0.585.

Figure 4 . 30 :

 430 Figure 4.30: Comparison of the collective relaxation times obtained for the bidisperse Voronoi liquid at T = 0.88 and the polydisperse system of quasi-hard spheres of Weysser et al. [3] at ϕ = 0.585.

Chapter 4

 4 Dynamical observables120 4.5.1.

  be determined. The aim of the following work is thus to find γ by using the predictions of the MCT mentioned above and in particular extracting the value the exponent b from the von Schweidler equation 4.33. By knowing its value, equations 4.31 and 4.30 enable the determination of γ. However a special care should be taken if one would obtain a reasonable and coherent value of b.

Figure 4 . 31 :

 431 Figure 4.31: Test of TTSP for temperatures T ∈ [0.84, 0.88]. The dash-dot grey line corresponds to an estimation of the lower bound of the non-ergodicity parameter f c q .

Figure 4 . 32 :

 432 Figure 4.32: Test of the factorization theorem. R q (t 1 = 6.0) = 1.0 and R q (t 2 = 16.0) = 0 by definition.

Figure 4 . 33 :

 433 Figure 4.33: Coherent intermediate scattering function computed for q = 6.85 at T = 0.84. The dot grey lines correspond to the borders used to define the fit interval.

Figure 4 . 34 :

 434 Figure 4.34: Coherent intermediate scattering function computed for q = 6.85 at T = 0.84 (blue +). The black solid line corresponds to the von Schweidler fit where the fitting parameters are the average values defined in Table 4.8.

Chapter 4

 4 Dynamical observables 127 improved by shifting the left border to a larger time as it is suggested by the dashed black line in Figure 4.35.

Figure 4 .

 4 Figure 4.35: Test of the von Schweidler fit on φ q (t) for several wavevectors q. The data points are represented by crosses (+). The dot grey line marks the value of the crossover time t co = 0.37. The solid lines represent the von Schweidler fits where the left border of the time interval is given by the crossover time. The dashed black line is the von Schweidler fit for q = 9.35 but this time the left border has been shifted to larger time t lef t = 1.5. The blue dashed line is von Schweidler fit for q = 3.00 where the left border has been shifted to t lef t = 10. The right border is by t right = 100 apart from q = 3.00 where t right = 1500.

  Schweidler fit for the incoherent intermediate scattering function of A and B particles and results are respectively shown in Figures 4.36

  and 4.37. As for the coherent intermediate scattering functions, the values of q are chosen to be extrema of S aa (k).

Figure 4 . 38 :

 438 Figure 4.38: Representation of τ -1/γ α as a function of the temperature. The linear fit gives the value of T c when it intercepts the T -axis.

1

 1 Inherent structures . . . . . . . . . . . . . . . . . . . . . . . . 133 5.1.1 Theoretical framework . . . . . . . . . . . . . . . . . . . . . . . 134 5.1.2 Computing IS and Gaussian landscape . . . . . . . . . . . . . . 137 5.1.3 Configurational entropy . . . . . . . . . . . . . . . . . . . . . . 140 5.1.4 Adam-Gibbs theory . . . . . . . . . . . . . . . . . . . . . . . . 150 5.2 Saddles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 5.2.1 Theoretical considerations . . . . . . . . . . . . . . . . . . . . . 153 5.2.2 Computing saddles: Saddles vs. Quasisaddles . . . . . . . . . . 156 5.2.3 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . 160

Figure 5 . 1 :

 51 Figure 5.1: Sketch of the typical quenching of a thermalized configuration into its basin of attraction [4].

. 13 )

 13 If we consider the R.H.S of equation 5.[START_REF] Angell | Relaxation in liquids, polymers and plastic crystals -strong/fragile parrterns and problems[END_REF], we immediately see that if the factorization approximation holds, only the configurational entropy would depend on e IS . Therefore Sciortino et. al highlighted that for different T , a shift of a temperature dependent quantity would lead to the collapse of the curves ln[P (e IS |T )δe IS ] + βe IS onto the same master curve which corresponds to the configurational entropy apart from an unknown constant.

Figure 5 . 2 :

 52 Figure 5.2: Evaluation of ( e IS /N -e IS,∞ /N )/T * as a function of the scaled temperature T * /T . Where we recall that T * is the temperature at which the system enters into the super-Arrhenian regime. T * = 1.25 for Voronoi, T * = 1.00 for KA and WAHN and T * = 0.60 for AMLJ-0.80.

Figure 5 . 3 :

 53 Figure 5.3: Evolution of e IS /N as function of 1/T for the bidisperse Voronoi liquid (+), KA ( ), WAHN ( ) and AMLJ-0.80 (△). The dark straight line corresponds to a linear fit performed on lowest temperatures.

Figure 5 . 4 :

 54 Figure 5.4: Evolution of the entropies per particle S liq and S sol as a function of the temperature.

Figure 5 . 5 :

 55 Figure 5.5: Configurational entropy S c per particles as a function of temperature T .

  E anh to see to what extent our observation is specific to the bidisperse Voronoi liquid. In the lower part of Figure 5.6, E anh is shown for the three LJ glass formers. Whereas a polynomial c 4 T 4 + c 3 T 3 + c 2 T 2 describes well the data for the Kob-Andersen model and the Wahnstöm mixture, a polynomial of the form c 6 T 6 + c 5 T 5 + c 4 T 4 is required to fit properly the data of AMLJ-0.

80 Figure 5 . 6 :Figure 5 . 7 :

 805657 Figure 5.6: Energy per particle of the anharmonicity as a function of temperature for the bidisperse Voronoi liquid (upper graph) and for the 3 LJ glass formers (lower graph).

••Chapter 5

 5 8 is represented P (e IS |T ) as a function of e IS /N , we notice that the distribution of e IS reflects what has already been observed in Figure 5.2: At high temperature, the distribution of e IS is weakly influenced by temperature at which the IS are extracted. All distributions are centered around the same average value e IS ≈ 113.95. As the temperature decreases, e IS of lower energy could be extracted, the distributions of e IS are influenced by temperature and a shift towards smaller values of e IS is observed upon cooling. Potential Energy Landscape 146 It is possible to give an analytical expression of the probability P (e IS |T ) if we assume

85 Figure 5 . 8 : 85 Figure 5 . 9 :

 85588559 Figure 5.8: Representation of the probability P (e IS |T ) of having an IS of energy e IS at a given temperature T as a function of the energy of IS per particle. The dot lines corresponds to the gaussian fit using equation 5.22.

5 Figure 5 . 10 :

 5510 Figure 5.10: Effective configurational entropy as a function of the inherent structure energy (colored circles). Notice that the collapse is not working for T ≥ 1.20. The black dotted line corresponds to a polynomial fit (-4.30 • 10 -4 e 2 IS + 98.9e IS -5.6 • 10 6 ) done on e IS ∈ [1.136 • 10 5 , 1.138 • 10 5 ].

[ 1 .

 1 1365 • 10 5 , 1.1375 • 10 5 ] which correspond to very low temperatures.

Figure 5 . 11 :

 511 Figure 5.11: Comparison between Ŝc (e IS ) obtained from the polynomial fit of order 2 performed in Figure 5.10 and Ŝc (e IS ) obtained from equation 5.23.

Figure 5 . 12 :

 512 Figure 5.12: Comparison between Ŝc (e IS ) obtained from the polynomial fit of order 2 performed in Figure 5.10 and Ŝc (e IS ) obtained from equation 5.24.

Figure 5 . 13 :

 513 Figure 5.13: Comparison between Ŝc (e IS ) obtained from the polynomial fit of order 2 performed in Figure 5.10 and Ŝc (e IS ) obtained from equation 5.27.

)Figure 5 . 14 :

 514 Figure 5.14: Evolution of the structural relaxation time τ α (extracted from F s (k, t)) as a function of the 1/(T S c ). The data points are symbolized by (+) and the solid line represents the exponential fit performed on the lowest temperatures.

(

  I) ∆U (u th ) ≪ k B T B : potential energy barriers are small compared to thermal energy, as a result when mechanism B freezes, mechanism A is relatively efficient. The decrease of the temperature below T B leads to a slowing down of mechanism A until reaching Goldstein's temperature T x where ∆U ∼ k B T x . In the case where ∆U does not depend a lot on u, the energy barriers remain relatively constant and it is legitimately to say that the relaxation time follows an Arrhenian law. This description is thus in agreement with the definition of a strong liquid.(II) ∆U (u th ) ≫ k B T B : potential energy barriers are large compared to thermal energy and consequently mechanism A is already slow when mechanism B becomes inefficient. We can assume that for T > T B , mechanism B is more advantageous than mechanism A as the system do not need to overcome energy barriers. Thus when T B is reached, the only way to diffuse is by activation and consequently T x = T B .

k

  B T B would lead to a description of relaxation time compatible with intermediate fragility. This qualitative reasoning on relaxation mechanisms related to saddles enable to have a better physical insight into what could lead to strong/fragile behaviors.

X) • e j e 2 j( 5 . 36 )Chapter 5

 25365 Potential Energy Landscape 159

86 Figure 5 . 15 :

 86515 Figure 5.15: Representation for a system of N = 125 particles of the index density of negative direction k as function of the potential energy density u for "true" saddle points computed with the EF method at several temperature T .

  . The geometric average enables to exclude the temperature dependence by averaging all the values of k associated to the same energy density u whereas the parametric average implies averaging over k and u for each temperature. These two kinds of averages are shown in Figures 5.16 (for true saddles) and 5.20 (for QS).

Figure 5 . 16 :

 516 Figure 5.16: Comparison of the two possible ways of averaging the scatter plot 5.15. The parametric averages ( ) has been done for temperatures T ∈ [2.00, 1.75, 1.50, 1.25, 1.20, 1.15, 1.00, 0.95, 0.92, 0.91, 0.89] (from top to bottom).

Figure 5 . 17 :

 517 Figure 5.17: Representation for the 3 systems size of the geometric average of k as a function of the energy density u. The blue triangles correspond to results obtained by minimizing W (Quasisaddles) and the red circles are results of the EF method (Saddles).

125 Figure 5 . 18 :

 125518 Figure 5.18: Representation of the geometric average of k as a function of the energy density u for the 3 systems size. The upper graph is for QS whereas the lower graph is for true saddles.

Figure 5 . 19 :

 519 Figure 5.19: Representation of both geometric (△) and parametric ( ) averages computed from QS for a system of N = 729 particles.

Figure 5 . 20 .Figure 5 . 20 :

 520520 Figure 5.20. The slope of the linear regression gives dk/du and thus thanks to equation 5.30 the energy barrier ∆U b can be estimated.

- 1 ( 5 . 41 )

 1541 Results are shown in Figure 5.21 where we have represented the evolution of ∆U b as a function of the scaled temperature (T -T c )/T c .

Figure 5 . 21 :

 521 Figure 5.21: Estimation of the potential energy barriers ∆U b as a function of the rescaled temperature (T -T c )/T c . Energy barriers are determined by using equation 5.41 for QS (upper graph) and for saddles (lower graph). The dashed black line corresponds in each case to the value of the constant energy E h found for high-temperature regime: E h = 7.3 for QS and E h = 9.8 for saddles.

Figure 5 . 22 :

 522 Figure 5.22: Evaluation of the energy barrier ∆U b for systems of N = 1000 particles with different compositions.

Figure 5 . 23 :

 523 Figure 5.23: Representation of the elevation in potential energy rescaled by the energy at T c as a function of the density index of negative direction. Data of LJ systems are taken from [5].

Figure 5 . 24 :

 524 Figure 5.24: For N=729, representation of the geometric average (+). The linear solid line corresponds to the fit performed for k ≤ 0.015.

(

  performed on QS) is overestimated in comparison to the value of T M CT c found with ideal MCT analysis. However when the number of particles is increased, the values of T c → T M CT c but remain slightly larger. This in agreement with what have been shown by Broderix et al. who determined from a saddle analysis the mode-coupling temperature for the Kob-Andersen model [124]. They found T c ≈ 0.44 which is a bit greater than T M CT c = 0.435 the value determined by ideal MCT [76].

Figure 5 . 25 :

 525 Figure 5.25: Potential energy per particle without vibration term as a function of temperature (+). The dot red line correspond to a fit performed on all data points of E P /N -3/2k B T and we extrapolate the fit to values of T lower than the lowest isotherm. The horizontal solid lines are the values measured for E th for the different system size. The vertical dashed line corresponds to the value of T c found with the ideal MCT analysis.

1

 1 Geometric frustration . . . . . . . . . . . . . . . . . . . . . . . 173 6.2 Locally favored structures (LFS) . . . . . . . . . . . . . . . . 175 6.2.1 Computing LFS . . . . . . . . . . . . . . . . . . . . . . . . . . 175 6.2.2 Presence of domains . . . . . . . . . . . . . . . . . . . . . . . . 184 6.2.3 Effect on the dynamics . . . . . . . . . . . . . . . . . . . . . . . 189 6.3 "Crystallization" . . . . . . . . . . . . . . . . . . . . . . . . . . 192 6.3.1 Decrease of the potential energy . . . . . . . . . . . . . . . . . 192 6.3.2 The structure factor : another proof of crystallization . . . . . 193 6.3.3 A Frank-Kasper phase ? . . . . . . . . . . . . . . . . . . . . . . 194

Figure 6 . 1 :

 61 Figure 6.1: Fit of the relaxation time τ α with equations 6.1 and 6.2. The pink solid line corresponds to the fit at high temperature when T > T * F L whereas the blue solid line corresponds to the one at low temperature when T < T * F L .

Figure 6 . 2 :

 62 Figure 6.2: Left: icosahedron. Right: distorted icosahedron. (Reproduced with permissions from [6], copyright The American Association for the Advancement of Science 2013.)

Figure 6 . 3 :

 63 Figure 6.3: Evolution of the fraction of the LFS with temperature for thermalized configuration (upper graph) and IS (lower graph). T c = 0.7989.

Chapter 6 80 Figure 6 . 4 :

 68064 Figure 6.4: Evolution of the fraction of icosahedra as a function of T r /T for the bidisperse Voronoi liquid and two LJ glass formers WAHN and AMLJ-0.80. The upper graph shows the evolution for thermalized configuration whereas the lower graph focuses on IS configurations.

T = 2 .

 2 00 T = 0.83 T = 2.00 T = 0.76 T = 2.00 T = 0.65 T = 2.00 T = 0.65

Figure 6 . 5 :

 65 Figure 6.5: Evolution of the fraction of icosahedra as a function of T r /T for different compositions of the bidisperse Voronoi liquid. The upper graph shows the evolution for thermalized configuration whereas the lower graph focuses on IS configurations.

4 .

 4 Figure 6.6.

Figure 6 . 7 :

 67 Figure 6.7: Representation of the different type of connections that can be found between two neighboring icosahedra: a) only one particle is common to the two icosahedra, b) in this case, an edge is shared and two neighboring particles are involved, c) the two neighboring icosahedra share a facet meaning that 3 neighbors are concerned and d) the two icosahedra interpenetrate. This implies that 5 neighbors are shared.

Figure 6 .

 6 Figure 6.8 suggests the ability of icosahedra to arrange in the different directions of the space. One can assume that connections involving 1,2 or 3 neighbors should make easier changes in any direction of space, whereas 5 neighbors connections may favor the formation of linear clusters. In order to test this hypothesis, we have computed the probability P (n i ) by considering the following rule: "two icosahedra belong to the same cluster if they share only N nn nearest neighbors." We recall that, initially, we require that these icosahedra share at least one nearest neighbor (N nn ≥ 1). Results of such process for the lowest temperature T = 0.83 is shown in the lower part in Figure6.9.

Figure 6 . 8 := 5 Figure 6 . 9 :

 68569 Figure 6.8: Snapshot of typical clusters of n i = 90 icosahedra at T = 0.83. 96 icosahedra are represented in total.

Figure 6

 6 Figure 6.10: Evolution of n i P (n i ) with n i for different temperatures. n i P (n i ) is proportional to the probability that a particle chosen at random is in cluster of size n i for a configuration sampled from thermal equilibrium (see text for details).

Figure 6 . 11 :Figure 6 . 12 :Figure 6 . 13 :

 611612613 Figure 6.11: Evolution of the ratio τ ico /τ other as a function of the scaled temperature T c /T . The ratio τ ico /τ other corresponds to the relaxation time of particles inside icosahedra with respect to particles that do not belong to icosahedra.

83 Figure 6 . 14 :

 83614 Figure 6.14: Representation of the decrease of the potential energy per particles during time observed for T = 0.84 (upper graph) and T = 0.83 (lower graph).

Surprisingly, the partial 6 Figure 6 . 15 :

 6615 Figure 6.15: Partial structure factors obtained when the bidisperse Voronoi liquid is in the "crystalline" phase at T = 0.84 (upper graph) and T = 0.83 (lower graph).

3 Table 6 . 4 :

 364 ones obtained by Pedersen et al. for the Wahnström binary mixture. 0, 12, 4) 17.8 (0, 0, 12) 54.8 (0, 0, 12, 4) 17.6 (0, 0, 12) 59.7 (0, 1, 10, 4) 10.6 (0, 2, 8, 2) 10.7 (0, 0, 12, 3) 15.0 (0, 2, 8, 2) 11.Table representing the two most frequent Voronoi polyhedra around large A particles and small B particles for T = 0.84 and T = 0.83.

Figure 6 . 16 :

 616 Figure 6.16: Snapshot of typical clusters that can be found when the system is in the "crystalline" phase at T = 0.83. The cluster is composed of n i = 260 particles.

Figure 6 . 17 :

 617 Figure 6.17: Upper graph: Representation of the four coordination polyhedra of Frank-Kasper. The white spheres are the minor sites whereas the black spheres correspond to the major sites. Lower graph: The associated Voronoi polyhedra. The black lines that start from the center of the Voronoi polyhedra and go through hexagonal faces show how major ligands can be connected and form for these local structures the major skeleton. (Reproduced with permissions from [7], copyright Springer 2017.)

  simulations the properties of the bidisperse Voronoi liquid for which we mainly focused on an equimolar binary mixture. During cooling experiments, we observed that the bidisperse Voronoi liquid was able to avoid crystallization and therefore we used it to probe the neighborhood of glass transition. To see to what extent the properties of the bidisperse Voronoi liquid are unusual, we compared our results with the ones found for three usual Lennard-Jones glass formers: the Kob-Andersen model, the Wahnström mixture and an additive mixture of Lennard-Jones particles where σ BB = 0.80.

Furthermore we pursued our 2 p

 2 analysis of the PEL by focussing on saddle points for which we underlined the difficulty to determine them numerically, especially when large systems are considered. We compared two different numerical methods: the minimization of ∇E 2 p and the Eigenvector-Following (EF) method. Whereas the minimization of ∇E Conclusion & Perspectives 201 allows only for the determination of quasisaddles, the EF method enables to find true saddle points. The investigation of the index of negative directions k as a function of the energy density of the saddle u revealed the presence of an underlying geometric feature of the PEL. Exploration of k vs. u with the different methods shows that the crossover from Arrhenian to super-Arrhenian regime is captured in the quasisaddles behavior.

  evolves under shear. One could also imagine working in the amorphous phase too.

  we find back the monodisperse case. A too large value of η, typically,

	η 0.6, would lead to cases where a fraction of particles could be outside their own cells
	or worse could have empty cells (zero volume). To avoid these unphysical possibilities
	we work with intermediate values of η which should ensure to avoid crystallization and
	systems where particles do not generate their own cells. Definition of size ratio x and
	equation 1.42 enable to define values of R

A and R B :

Table 2 .

 2

	T G	0.6983	0.6950	0.6866	0.6845	0.6716	0.6633

1: Estimation of the glass transition temperature T G for several cooling rates.

Table 3 . 1 :

 31 Values in degree of the angle θ taken for the different maxima observed from right to left in Figure3.3.

	.1.

Table 3 . 2 :

 32 Values of the position of the first and second peaks of the partial pair correlation functions for different compositions A : B of the mixture.

Table 3 . 3 :

 33 .3 for different compositions of the binary mixture. Values of the position of the first peak of structure factor.

	Mixture A : B	k *	k * AA	k * BB	k * AB	0.5(k * AA + k * BB )
	50 : 50	6.85 6.75 7.15 6.95	6.95
	60 : 40	6.85 6.75 7.15 6.95	6.95
	70 : 30	6.85 6.85 7.25 7.05	7.05
	80 : 20	6.95 6.85 7.25 7.05	7.05

Table 4

 4

	τ ∞	E ∞	T *	K	T 0	T r
	50 : 50 0.076 4.924 1.25 0.610 0.691 0.794
	60 : 40 0.077 4.892 1.20 0.474 0.612 0.726
	70 : 30 0.077 4.882 1.15 0.259 0.498 0.659
	80 : 20 0.074 4.885 1.10 0.195 0.443 0.629

.1 and Table

4

.2. It is important to stress that, even if in practice the crossover temperature T * is a fitting parameter, we have fixed its value in the case of the diffusion coefficient and this constraint does not alter the quality of the fit as it can be observed in Figure

4

.8.

Table 4 . 1 :

 41 Parameters obtained when fitting the structural relaxation time with equation 4.8. T r is found when the relaxation time τ α (T = T r ) = 4 • 10 -4 .

	D ∞	E ∞	K	T 0
	50 : 50 0.984 5.291 0.692 0.6634
	60 : 40 0.969 5.223 0.500 0.580
	70 : 30 0.944 5.168 0.324 0.487
	80 : 20 0.942 5.159 0.320 0.472

Table 4 . 2 :

 42 Parameters obtained when fitting the diffusion coefficient with equation 4.10. For this fit the value of T * corresponds to one determined when fitting τ α .

Table 4

 4

	.3.

Table 4 . 3 :

 43 Parameters of the Lennard-Jones potential of 3 glass-formers. Density, masses of particles and concentration number are also presented.

Table 4 . 4 :

 44 Fitting parameters extracted from equation 4.8 of the relaxation time for the three LJ systems. The reference temperature T r has been chosen when τ α

		184 3.183 1.00 0.268 0.325 0.436
	WAHN	0.140 3.201 1.00 0.400 0.428 0.524
	AMLJ-0.80 0.260 1.636 0.60 0.626 0.280 0.322

Table 4 . 5 :

 45 Fitting parameters extracted from equation 4.10 of the diffusion coefficient for the three LJ systems.

		255 3.140 0.90 0.3852 0.3381
	WAHN	0.319 3.227 0.90 0.534	0.429
	AMLJ-0.80 0.179 1.644 0.50 0.617	0.247

  80. Values of the fitting parameters A and B are presented in Table 4.7.

	Systems	Voro	KA	WAHN	AMLJ-0.80
	Fit interval C e v /(N k B ) [1.40, 1.70] [1.10, 1.28] [1.10, 1.26] [1.06, 1.23]
	Fit interval T	[1.00, 2.00] [0.90, 1.50] [1.00, 2.00] [0.60, 1.50]
	A	10.5	10.8	7.7	6.5
	B	11.5	8.9	5.2	3.6
	B/A	1.10	1.21	1.48	1.81

Table 4 . 7 :

 47 Fitting parameters found by using equation 4.16 for the fit of ln(Ω 0 τ α ) as a function of C e v /(N k B ). The fit intervals are given in term of C e v /(N k B ) and with the corresponding T -interval.

Table 4 . 8 :

 48 Table of the average and the standard deviation of the fitting parameters obtained from the fits performed on the coherent intermediate scattering function. The fit was done using the von Schweidler equation 4.33.

	8371	1404.1	0.5235	0.35512	1.0094
	standard deviation 0.0016668 7.3768 0.0016061 0.00053202 0.0043676

Table 4 . 9 :

 49 Evolution of the von Schweidler exponent b in the mesoscopic q range. b was computed from φ s q (t).

	Chapter 4 Dynamical observables

Table 4 . 10 :

 410 Table of the final results of the MCT exponents and mode-coupling temperature T c .

	b	λ	a	γ	T c
	0.5235 0.7711 0.2943 2.6541 0.7989

we can compute λ which has been defined in equation 4.31. Once λ is determined, it is straightforward to find the exponent a. Finally we have all the required ingredients to determine the exponent γ defined in equation 4.30. The value of the exponents can be found in Table

4

.10.

  .1. Whereas the three LJ systems have approximately the same value of σ 2 we observe that this value is 10 times greater in the case of the bidisperse Voronoi liquid.

	Voro	KA WAHN AMLJ-0.80
	σ 2 /N 1.474 0.127	0.189	0.139

Table 5 . 1 :

 51 Value of σ 2 found from a fit with equation 5.[START_REF] Dasgupta | Is there a growing correlation length near the glass transition?[END_REF]

	114
	113.8
	113.6
	0.5	0.6	0.7	0.8	0.9	1	1.1	1.2	1.3
	-7.5
	-7.6
	-7.7
	0	0.5		1		1.5	2		2.5
	-7
	-7.1
	-7.2
	0.2	0.4	0.6	0.8	1	1.2	1.4	1.6	1.8
	-6.7
	-6.8
	-6.9
	0	0.5	1	1.5	2		2.5	3	3.5

  [START_REF] Kivelson | Fitting viscosity: Distinguishing the temperature dependences predicted by various models of supercooled liquids[END_REF]. It appears thus that AMLJ-0.80 behaves similarly than the bidisperse Voronoi liquid. It is not straightforward to explain this phenomenon and for the moment an explanation is still lacking. Values of coefficients c i can be found in Table5.2.

		Voro	KA	WAHN AMLJ-0.80
	c 2	N.A	-0.4912 -1.028	N.A
	c 3	N.A	1.122	2.271	N.A
	c 4 -0.1362 -0.5858 -1.196	-1.706
	c 5 0.6351	N.A	N.A	11.56
	c 6 -0.3276	N.A	N.A	-13.38

Table 5 . 2 :

 52 Table of the coefficients c i involved in the polynomial expressions for anharmonicity. N.A stands for Non Applicable.

  .23 before rescaling by k B T c . Angelani et al. recorded the values of ∆E for the 4 LJ systems and then computed ∆E * = ∆E/k B T c . They found that for all models ∆E ≈ 10k B T c . The collapse of the bidisperse Voronoi liquid on the master curve implies that, in our case, the relation ∆E ≈ 10k B T c is also valid. As mentioned by Angelani et al., it appears that the landscape of different systems share common features and that only one parameter ∆E describes the organization of saddles above the IS

Table 5

 5

			.3.
	N	125	512	729	1000
	E th 113.7358 113.6619 113.6401 113.6438
	T c	0.8320	0.8088	0.8029	0.8038

Table 5 . 3 :

 53 Table of the threshold energy E th and mode-coupling temperature T c computed from QS.

	113.6 113.8 114 114.2 114.4 114.6

Table 6 . 1 :

 61 Tablerepresentingthe three most frequent Voronoi polyhedra around small particles for T = 2.00 and T = 0.83. For both temperatures polyhedra have been identified for the thermalized and the IS configurations. The last line of the table shows structures that are not the most representative at T but they are mentioned for purpose of comparison.

		0.2
		0.15	(0, 0, 12) (0, 2, 8, 2) (0, 3, 6, 3)	Thermalized config.
	Fraction	0.1	(0, 3, 6, 4)
		0.05
		0
		0.4	0.6	0.8	1
				T c /T
		0.3
				IS config.
		0.25
	Fraction	0.15 0.2
		0.1
		0.05
		0.4	0.6	0.8	1
				T c /T

Table 6 . 2 :

 62 Evolution with the composition of the average numbers n A and n B of A and B particles composing the icosahedra at T = 2.00 and T = T min .

	5	5	7	6	6	6	6	6	6	6	6	6	6

Table 6 . 3 :

 63 Table of the most frequent LFS for different compositions of the bidisperse Voronoi liquid at T = 2.00 and T = T min the lowest simulated temperature for each composition. The LFS were determined for thermalized state.

	Chapter 6 Locally favored structures and crystallization	183
	A60B40
	T = 2.00	T = 0.76
	Signature % Signature	%
	(0,3,6,3) 3.8 (0,2,8,2) 14.2
	(0,3,6,4) 3.6	(0,0,12)	12.5
	(0,2,8,2) 3.2 (0,2,8,1)	9.1
	A70B30
	T = 2.00	T = 0.65
	Signature % Signature	%
	(0,3,6,3) 3.8 (0,2,8,2) 15.3
	(0,2,8,2) 3.2 (0,2,8,1) 12.8
	(0,3,6,4) 3.0	(0,0,12)	9.3
	A80B20
	T = 2.00	T = 0.65
	Signature % Signature	%
	(0,3,6,3) 3.6 (0,2,8,1) 13.7
	(0,4,4,3) 3.5 (0,2,8,2) 12.2
	(0,2,8,1) 3.1 (0,2,8,4) 1 10.0

  3, it is now possible to compute the intermediate scattering function for each of these sub-ensembles, i.e F ico ico , τ neigh and τ other can be extracted with the criterion F sub s (k, t = τ sub ) = 0.1 where the subscript sub stands for "sub-ensemble".

	Chapter 6 Locally favored structures and crystallization	190

s (k, t), F neigh s (k, t), F other s (k, t)

[START_REF] Coslovich | Understanding fragility in supercooled lennard-jones mixtures. i. locally preferred structures[END_REF]

. The relaxation times τ

This LFS is found around large A particles. Its percentage was computed with respect to large particles.

Acknowledgements

Résumé

Comprendre l'origine microscopique du ralentissement de la dynamique au voisinage de la transition vitreuse reste l'un des problèmes fondamentaux de la physique de la matière condensée. Au cours de ce travail, nous introduisons un nouveau modèle de liquide, appelé liquide de Voronoï, et dont les interactions sont directement reliées aux propriétés géométriques des tessellations de Voronoï. Pour cette classe de liquides, les interactions sont à plusieurs corps et agissent de telle sorte que le système est toujours sous tension tout en restant stable.

Le but de ce travail est d'étudier un mélange binaire du liquide de Voronoï et de voir de quelles façons ces interactions exotiques affectent le scénario habituel de la transition vitreuse.

Tout au long de ce travail, nous caractérisons le liquide de Voronoï bidisperse théoriquement et par le biais des simulations numériques. Nous proposons également des comparaisons avec des liquides de Lennard-Jones surfondus bien décrit dans la littérature. -----------------------------------------------------------------

Mots clés : transition vitreuse, tessellations de Voronoï, simulation de dynamique moléculaire, ralentissement de la dynamique, fragilité, théorie du couplage de modes, paysage d'énergie potentiel, structure localement favorisée

Summary

Understanding the origin of the important slowing down of the dynamics near glass transition is still one of the remaining fundamental problems of condensed matter physics. During this work we introduced a brand-new model of liquids named Voronoi liquid, whose interactions are directly related to the geometrical properties of Voronoi tessellations. For these class of liquids interactions are intrinsically manybody and act in such a way that the liquid is always under tension but remains stable.

The aim of this work is to use a binary mixture of the Voronoi liquid to see to what extend these exotic interactions may affect the classical scenario of glass transition.

Throughout this work we characterize theoretically and by mean of numerical simulation the bidisperse Voronoi liquid. Comparisons with well-known Lennard-Jones glass formers are systematically performed.