Françoise Sullet-Nylander

Malin Roitman

Juan-Manuel Lopez- Muñoz

Sophie Marnette

L A Campagne

« Mademoiselle

L A Case

E N Trop! »

Béatrice Fracchiolla

Circulation ordinaire des discours sexistes et sens symbolique. La campagne " Mademoiselle, la case en trop ! "

Introduction

Les associations féministes dont l'objectif est de défendre les femmes connaissent un renouveau, plus particulièrement depuis l'affaire DSK (mai 2011). Spécialisée depuis longtemps dans la lutte contre les propos sexistes dans les différents médias, l'association des Chiennes de Garde i est depuis septembre 2011 à l'origine d'une campagne en faveur de la suppression de la case « Mademoiselle » et du remplacement de « nom de jeune fille » par nom de naissance dans les formulaires administratifs ii . Il s'agit de lutter contre l'existence d'une double appellation pour les femmes, alors que seul Monsieur existe pour les hommes (versus un potentiel 'Mondamoiseau') (voir aussi Waki : 2006). Pourtant, ce sujet a déjà été objet de débat et circulaires administratives (1974, 1983, 2006...) à maintes reprises iii . Pourquoi alors s'est-il trouvé à nouveau mis en avant? La question n'est plus de s'interroger sur la légitimité ou non d'une telle demande, mais plutôt de tenter de comprendre les raisons pour lesquelles il est nécessaire de la renouveler régulièrement depuis la Révolution Française iv . Notre intérêt pour cette question consistera surtout à tenter de comprendre les raisons pour lesquelles elle continue de rencontrer aujourd'hui des résistances dans ses applications, comme on a pu l'entendre par exemple dans les propos de Natacha Polony, qui reprochait aux féministes de ne s'intéresser qu'aux symboles v . Nous aborderons ici la problématique de la nomination, de l'adresse et de la dimension performative qu'elle implique dans la mesure où nommer fait exister (comme). On constate en effet dans le corpus traitant de ce débat que deux camps s'opposent sur l'interprétation du sujet : d'une part celles et ceux qui jugent la dimension symbolique et politique de ce sujet suffisamment importante en soi pour être défendue ; d'autre part celles et ceux qui ne la jugeant « que » symbolique l'estimeraient comme moins urgente vi . Elle nous conduit alors à poser d'autres questions relevant d'une problématique plus générale sur le rôle politique du sens symbolique. Les symboles sont-ils secondaires? Comment fonctionnent-ils comme relais de représentations? Un symbole est-il assimilable à un détail ou est-il justement là pour hyper signifier? Quel est le rôle des médias dans la resignification d'un symbole? Entre construction sexuée, culturelle et discursive, comment se négocie et se modalise ici en langue et en discours la parole de l'autre, mais aussi l'adresse à l'autre? Largement relayée dans la foulée par d'autres actions, associations (Osez le féminisme, MixCité...) et les médias, cette demande de suppression a été soutenue au niveau institutionnel par Roselyne Bachelot, à l'époque Ministre des solidarités. Elle a finalement connu une forme de dénouement administratif par la publication de la circulaire Fillon (à l'époque, Premier Ministre) en date du 21 février 2012, qui semble avoir clôt le débat. Nous souhaitons aujourd'hui revenir sur ce débat afin d'en comprendre les enjeux réels. Plus largement, nous nous interrogeons sur la performativité du discours politique dès lors qu'il s'agit de réformer des éléments de dénomination symbolique susceptibles d'influer sur l'égalité des sexes [START_REF] Boutet | Le Pouvoir des mots[END_REF]. La problématisation du discours rapporté concerne ainsi deux aspects qui seront traités conjointement : l'un, sociolinguistique, est lié au fait de cocher la case Madame ou Mademoiselle ; le second relève plus de l'analyse de discours et renvoie aux circulaires déjà publiées et déclarations déjà faites à ce propos.

La question des discours

Historique d'une distinction lourde de sens

Historiquement Mademoiselle est sémiotiquement construit par rapport à Madame, sur la base d'une identité sociale dépassée mais pourtant difficile à déraciner qui faisait que, lorsque l'on se mariait, on devenait honorable et respectable. Le statut de femme mariée -donc désirée/acquise par un homme -était positif, voire salvateur, versus un statut qui correspondait à son contraire : jeune écervelée, se transformant en « vieille fille » passé un certain âge , avec des connotations de « mal baisée » car célibataire. Or, s'il est vrai que le mariage a aujourd'hui beaucoup moins d'influence sur la vie sociale réelle des femmes, tout le réseau de représentations et de sens associés poursuit cette distinction entre le fait de devenir madame ou non.

Aussi, là où elle existe, la case « Mademoiselle » est aujourd'hui à cocher comme discours rapporté d'état civil. Le fait d'être dite Mademoiselle ou Madame la fait exister comme telle. Se dire et être dite « Mademoiselle » signifie alors accepter de se reconnaître une identité par défaut de femme non socialement incluse, car non mariée et en cela historiquement lié au statut de non mère. Or, cette dimension socialement négative associée au terme mademoiselle n'a pas lieu d'être dans une société où le mariage n'est plus le fait de la majorité et où d'autres modèles de vie en couple se sont développés (concubinage, pacs, couples de même sexe, famille monoparentales, recomposées...) vii . La formulation administrative est donc marquée comme obsolète au vu de l'évolution de la société. On est ainsi passé en réalité à une resignification sociale de mademoiselle et d'un « ce que je suis pour l'état civil » faisant référence à un statut juridique et social relativement désuet à « l'identité que je choisis comme mienne ».

De la case à cocher à la mise en discours -rapporté

On peut alors imaginer que la question de la case à cocher se poserait comme un vrai choix, expression d'une liberté individuelle et non plus contrainte administrative. Par translation, la respectabilité associée à chacun des termes devient également un choix qui correspond à « ce que l'on dit de moi » en fonction de « comment je veux être nommée » et non plus de « comment j'ai le droit d'être nommée ». La demande exprimée est celle d'une prise en compte de la parole comme lieu d'expression possible des libertés individuelles. Certaines femmes se montrent satisfaites de cette liberté dans les interactions quotidiennes tout en distinguant la manière dont on s'adresse à elles à l'oral, de leur propre discours rapporté d'elle-même écrit, et à condition que cela demeure un choix sans obligation. Les extraits du corpus d'échanges recueillis entre octobre et décembre 2011 sur différents forums et blogs en attestent : l'article : « Mademoiselle » le mot de trop ? du Nouvel Observateur, en ligne 27-09-11 à 16:54) On a ainsi l'impression que l'on est passé d'un niveau de sens imposé à un sens finalement choisi. En réalité, deux niveaux de sens sont toujours en concurrence autour de Mademoiselle. Le premier, à l'oral, renvoie au mode d'adresse et concerne le choix que quelqu'un d'autre fait de s'adresser à moi d'une certaine manière -comme le montre l'exemple ci-dessus, le « je » n'intervient pas du tout dans ce choix et peut juste l'accepter, le refuser, le critiquer, y répondre. Le second niveau de sens concerne l'écrit et la question de la case à cocher qui nous occupe. Ce niveau de sens-là peut apparaître aujourd'hui relever également du choix individuel, portant sur la manière dont je veux « être adressée ». En réalité il s'agit d'un non choix dès lors que des amendes sont attribuées et la case « Mademoiselle » devient alors une évidente discrimination à l'égard des femmes, comme le montre ce cas précis : Parallèlement, un autre versant de la revendication porte sur le caractère intrusif de Mademoiselle comme adresse et désignation induisant un jugement sur la vie privée. La manière dont les deux territoires privé et public peuvent de la sorte se trouver confondus, uniquement à travers ce terme, est ainsi explicité par les chiennes de garde : « il est bien plus poli d'appeler une femme «madame», et ainsi de ne pas porter de jugement sur sa vie privée. Contrairement à une idée reçue, ce n'est pas flatteur de laisser entendre à une femme qu'elle est "disponible", notamment dans un contexte professionnel ! » ix . On constate qu'à cette dimension intrusive, dénoncée également par la Ministre Roselyne Bachelot x font écho de nombreux témoignages comme celui de Chris02 :

(en réponse à @frmg10) : moi aussi je suis d'accord. Car ça me gêne, à mon âge, mère et grand-mère, de recevoir du courrier administratif avec la mention "mademoiselle" parce que je suis divorcée et que je n'ai pas de nom d'usage (conjoint) à fournir. » xi (réaction à l'article : « Mademoiselle » le mot de trop ? du Nouvel Observateur, en ligne 16/11/2011 15:01:55) et un peu plus loin « (en réponse à @iba972) : ça fait peut-être plaisir aux femmes de 35/40 ans (ou aux vieilles filles) mais ça ne fait pas plaisir aux mamies de 65 ans et plus qui ont eu droit dans le passé au "madame" et sont redevenues subitement "mademoiselle" (qui fait penser à jeune fille) parce qu'elles n'ont plus de mari » (idem, 18/11/2011 14:53:38) D'autres témoignages parallèles parfois humoristiques voire ironiques, accentuent l'absurdité d'une distinction aujourd'hui inconséquente : Au-delà de la simple « case à cocher », on voit ainsi comment la controverse s'alimente sur une confusion qui n'est pointée par personne et comment un discours pro Mademoiselle se construit face à un discours contra. Si les détracteurs de « Mademoiselle » posent la question de l'écrit et des formulaires d'État -et donc d'une action politique et linguistique volontaire, qui relève plus en réalité du juridique xii , ils n'ont pas cependant les moyens ni le souci de débattre sur les usages de l'oral -qui relèvent d'une sociolinguistique du quotidien. En revanche l'objectif visé est bien celui écrit de la case à cocher, au sens large. Il concerne aussi les formulaires des entreprises du privé, qui sont en réalité les plus nombreuses -d'où le kit de dénonciation proposé sur le site de la campagne, comprenant une lettre type à envoyer xiii .

L'identité sociale en construction

Le nom en questions

La question du discours rapporté est au coeur de la question du nom et de la dénomination. Jusqu'à très récemment, la femme n'avait pas de nom en propre mais passait du nom transmis par le père à celui donné par le mari xiv . Certaines femmes, lorsqu'elles divorcent, souhaitent retrouver leur « nom de jeune fille » (sic) ; d'autres veulent conserver leur nom de femme mariée, parce que c'est celui qu'elles « se sont choisi », et aussi celui « de leurs enfants », mais il faut pour cela que leur ex mari acceptent qu'elles le conservent. Mais dans le nom même, dit « de jeune fille », la femme était toujours infantilisée et rendue dépendante. Le corrélat problématique entre les deux objets de revendication se fait toujours autour du sens : l'existence d'un « nom de jeune fille » impliquait en effet qu'existe un nom de femme (voire de « vieille fille » quand elle cesse d'être jeune?). Cette dénomination de « nom de jeune fille » est également la trace d'un temps révolu, d'où la seconde revendication, qui est de remplacer cette expression par « nom de naissance » : en cela, l'exigence est que l'administration applique la loi du 6 fructidor de l'an II (1794) qui pose le principe selon lequel les femmes gardent leur nom de naissance toute leur vie (comme les hommes!) ; et que, si elles souhaitent porter le nom de leur époux après leur mariage, celui-ci devient alors leur « nom d'usage ». La terminologie fonctionnant comme réminiscence de l'époque pas si lointaine en France xv , où les femmes passaient de l'autorité de leur père à celle de leur mari a donc supprimé « patronyme », issue de pater, patris comme « nom d'époux ». Le nom de naissance permet en cela une égalité régulière entre femmes et hommes. Or, ce qui continue de gêner dans mademoiselle est précisément ce statut de dépendance à l'égard de l'un ou de l'autre, père ou mari, qui contribue toujours à la construction symbolique et sociale des femmes comme infériorisées et ne pouvant toujours pas pleinement jouir de leurs droits sinon via un statut marital. Comble du paradoxe, ce passage par le mariage n'ayant vraiment plus rien d'obligatoire, cette co-dépendance de sens contribuerait à les cantonner de surcroît encore plus définitivement (et à vie) à ce statut d'infériorité/dépendance via le mademoiselle.

Le choix du nom

Les femmes peuvent aujourd'hui choisir le nom qu'elles veulent porter lorsqu'elles se marient et en France, depuis le 1 er janvier 2005, les parents peuvent transmettre à leurs enfants le nom du père, de la mère ou bien les deux noms accolés, sans obligation légale de mettre un tiret. Le PACS, lui, ne change pas le statut juridique mais le statut fiscal. Il n'a aucune incidence directe sur le droit de la famille. Il n'a jamais entraîné de « perte du nom ». Il y a donc une forme d'absurdité à proposer à une femme pacsée, mère de famille, de cocher « mademoiselle ».

Le nom fait exister socialement parce qu'il fait exister tout court

Or, c'est par le « rapport » du nom seul que nous existons et, en particulier, la déclaration de naissance et son inscription au registre d'état civil (né + Nom ; né sous X) ... et ce, jusqu'à l'inscription a minima, de notre nom et prénom sur une tombe ou une urne, ou un registre de décès. D'un point de vue juridique, la transmission du nom, comme du patrimoine, se fait par désignation du nom de quelqu'un. En l'absence de testament, c'est la généalogie familiale, identifiable par la transmission des noms et de « qui est enfant de qui » qui décide de la transmission du patrimoine à l'un plutôt qu'un autre. Aussi, il me semble que l'objet véritable de ce débat est peut-être à chercher du côté de la filiation, de la transmission et de la distribution à parts égales entre les deux lignées matrilinéaire et patrilinéaire; ce qui va très certainement de pair avec les enjeux de pouvoir liés à la différenciation des identités sexuées.

Dans la perspective de la nomination, de l'adresse (Lecolle et al. : 2009) et de la dimension performative qu'elle implique (Austin : 1962), au sens où nommer fait exister comme, « Madame (Mademoiselle) » associé au nom participe également à ce faire exister socialement comme. Cette création sociale d'identité tend ainsi à une catégorisation, voire à une essentialisation en même temps qu'à une asymétrie interactionnelle et sociale : les femmes catégorisées « mademoiselle » ne sont pas « madame », et vice-et-versa [START_REF] Rosier | Petit traité de l'insulte[END_REF]. Or l'on sait que la catégorisation fonctionne comme l'un des moteurs de montée en tensions agressives ou verbales, dans la mesure où elle crée, par définition, de l'inégalité par exclusion réciproque et simultanée d'un terme par l'autre. C'est pourquoi cette double dénomination renvoie à une différence sociale et nominative de l'identité de genre et de sexe, renforçant la dimension inégalitaire qui existe déjà entre hommes et femmes xvi . "Discourses can be seen as the 'rules' and 'guidelines' which we produce and which are produced for us in order to construct ourselves as individuals and to interact with others." (Mills 2008:8). La question posée par la confrontation des propos tenus par les deux camps est celle de savoir comment se négocie, dans la langue et le discours, la construction sexuée, culturelle et discursive à travers la parole d'autrui et l'adresse à l'autre. Notre postulat de départ est fondé sur une réciprocité : à savoir, que l'activité langagière est influencée par la relation entre les sexes et la relation entre les sexes, par l'activité langagière [START_REF] Monnet | La répartition des tâches entre les femmes et les hommes dans le travail de la conversation[END_REF][START_REF] Théry | La distinction de sexe. Une nouvelle approche de l'égalité[END_REF][START_REF] Mills | Language and Sexism[END_REF][START_REF] Fracchiolla | Anthropologie de la communication : la question du féminin en français[END_REF][START_REF] Fracchiolla | Politeness as a strategy of attack in a gendered political debate -The Sarkozy-Royal debate », Women, Power and the Media[END_REF]. Ainsi, par exemple, on trouve stipulé dans les réponses ministérielles que les mères célibataires qui en font la demande doivent pouvoir être appelées MadameY. Ce discours porte et contribue à diffuser ainsi deux implicites sociaux. Le premier rattache la possibilité d'être nommée Mademoiselle au fait d'avoir eu ou non un rapport sexuel. Cet élément de discours est facilement démontré non pertinent si l'on considère les femmes ayant eu recours à des méthodes de PMA xvii , où l'acte de procréation n'est pas nécessairement associé à un acte sexuel. Le second implicite social, lié au premier, suppose une forme de gain d'honorabilité via la statut maternel, élément discriminant pour les femmes ne pouvant pas avoir d'enfants ou ne souhaitant pas en avoir xviii .

L'inscription du symbolique dans le réel

Notre hypothèse est que la dimension sociale de cette revendication va bien au-delà du "féminisme" et d'un débat de principe. Elle pose en réalité la conception que peut se faire une société de l'individu capable de décider de sa vie de façon active et responsable. Cela peut conduire à se poser la question, ainsi que l'ont fait les suédois en revendiquant un unique pronom neutre « hen », de la pertinence d'une dimension genrée dans l'interaction. Ainsi, la question posée (à nouveau) à travers ce débat est-elle peut-être celle de la binarité du système identitaire, de son sens éthique alors qu'il ne semble fonctionner que sur un mode de catégorisation emprisonnant : avec une injonction à se définir, à être ou ne pas être, à être l'un ou l'autre, en dehors de toute forme de neutralisation xix . Or, cette obligation sociale et biologique d'être déclaré homme/ou femme s'oppose à des individus chancelant au milieu de ces deux pôles, comme dans le cas de l'hermaphrodisme d'une part ou du transsexualisme d'autre part. On peut alors s'interroger sur les possibilités sociales et administratives qu'offrent ou offriraient une désexuation des formulaires au profit de la seule qualification d'être humain, en termes de liberté d'existence et d'être. Dans cet esprit, on pense au Québec, qui a redéfini les « droits de l'homme » en « droits de la personne ».

Conclusion

Le droit complet à l'autodétermination sexuelle pourrait affaiblir la binarité sexuelle. L'idée semble affleurer d'une suppression/disparition civile du genre (associée aux possibilités de la PMA) qui fasse disparaître la sexualité de l'horizon humain (Héritier : 2005[START_REF] Théry | La distinction de sexe. Une nouvelle approche de l'égalité[END_REF]. Or, comme nous l'avons évoqué, la question du sexe et de la sexualité, et de la dénomination de l'individu par rapport à un caractère sexué est indéniablement liée in fine à la question de la filiation toujours très polémique : c'est en effet à la naissance que le nom est déclaré. De ce nom transmis parce que donné découlent ensuite toute une série d'implications symboliques et patrimoniales qui dessinent les contours de la vie d'un individu, ce qu'illustre de manière humoristique le film La vie est un long fleuve tranquille qui repose entièrement sur l'échange de deux bébés à la maternité.

Nous avons ainsi cherché à mettre en évidence que si le sens réel apparent de cette campagne peut sembler discutable, son sens symbolique est, lui, indéniable. Certes, une circulaire signée par la Premier Ministre François Fillon, en date du 21 février 2012 a été rédigée. Cette circulaire qui s'adresse aux Ministres et Préfets, concerne la « suppression des termes « Mademoiselle », nom de jeune fille », « nom patronymique », « nom d'épouse » et « nom d'époux » des formulaires et correspondances des administrations ». Pourtant, la rédaction-même de cette circulaire laisse présager qu'elle ne sera pas la dernière. La circulaire se conclut ensuite par « Vous voudrez bien, en conséquence donner instruction aux services placés sous votre autorité d'éliminer autant que possible de leur formulaires et correspondances les termes... ». On applaudit d'abord le sens symbolique de cette circulaire. Mais si dans un second temps l'on s'attache à son sens réel, que ditelle exactement ? Elle ne dit pas « vous veillerez à ce que les personnes sous votre autorité n'utilisent plus ces formules », mais enjoint « d'éliminer autant que possible » ces formules, qui restent donc « possibles » par l'aveu même de l'absence de volonté exhaustive contenue dans cette formulation. Cela est renforcé un peu plus loin par « Les formulaires déjà édités pourront néanmoins être utilisé jusqu'à épuisement des stocks »... On peut se demander, naïvement, si cet élément de phrase tient vraiment de l'argument économique en temps de crise... comment connaître en effet la quantité des stocks ? Cette circulaire connaîtra vraisemblablement le même sort que les autres qui l'ont précédées -ce qui renforce indirectement mon propos de mettre en évidence la valeur du symbolique comme induisant le réel. Ce texte, qui apparaît comme une concession plus que comme une injonction, invite donc à se demander finalement : à quand la prochaine circulaire ? En tant que chercheure et membre d'une société civile et politiquement démocratique, il ne me semble pas qu'il y ait de sujets requestionnant la définition des représentations et des identités sociales qui soit « petite ». On le voit par exemple avec l'exemple bien connu de la Chine où la valeur et le sens symbolique donnée à la naissance d'un garçon conduit parfois jusqu'à l'infanticide des bébés filles ou au rapt d'enfants de sexe masculin. Le territoire symbolique est un vrai territoire. La puissance du symbolique sur le réel demeure indéniable. On remarquera que l'anglais a deux expressions qui permettent de distinguer entre childless et childfree. xix Cet idéal de neutralisation échouant à la base même en français dés lors que le neutre y est désigné par le masculin.

Bibliographie

  « rebekah : Et pour les gays et les lesbiennes travesti(e)s, on doit dire comment ? »(idem, 16/11/2011 16:07:35) Ou encore, avec une volonté humoristique marquée : « frmg10@DeBeuLioU : Et pour Dieu, faut il dire madame ou mademoiselle? Dieu est peut être du sexe féminin ? »(idem,16/11/2011 13:59:28) et, à propos de Roselyne Bachelot :« privateer : Pour l'égalité des sexes , elle va surement prendre les mesures, comme le disait Le Luron !!! » (idem, 16/11/2011 12:23:30) D'autres témoignages renvoient en réalité à l'interpellation orale et quotidienne, qui ne concerne pas directement la campagne dont il est question, laquelle vise les formulaires administratifs. Mais il est justement intéressant de voir que c'est cette confusion discursive entre les deux plans écrit et oral, administratif (rapporté) et interactionnel, qui nourrit la controverse, plus particulièrement autour de la liberté individuelle et de l'expression des valeurs : « Iba 972 : et pourtant que ça fait plaisir aux dames de 35-40 ans quand on les appellent comme ça, mariée ou pas. »(idem, 16/11/2011 15:42:51) Contextualisé, le sens interactionnel est tout sauf rigide. Il participe de la mise en place d'une relation, un peu comme l'utilisation du tutoiement ou du vouvoiement, du prénom, du surnom, ou du nom de famille pour s'adresser à quelqu'un. Il s'agit donc de deux plans tout à fait distincts, et les interventions montrent que le débat est mal compris et que les révoltes qu'il suscite viennent de cette confusion. Prétendre en effet vouloir éradiquer Mademoiselle de la langue française dans son ensemble et dans toutes ses utilisations reviendrait à appliquer une forme de totalitarisme, dont il n'a jamais été question ici(Klemperer : 1947[START_REF] Boutet | Le Pouvoir des mots[END_REF], Aubry & Turpin : 2012)

  Aubry, L. Turpin, B. (eds) (2012) : Viktor Klemperer.Repenser le langage totalitaire. Paris, CNRS éditions. Austin, J. (1962) : How To Do Things With Words, Oxford, Urmson. besoin de prévenir la personne avec laquelle on était pacsé. xiii http://www.madameoumadame.fr/kit-pratique xiv Depuis 2003, loi n°2003-516 du 18 juin 2003, les parents peuvent choisir de transmettre l'un ou l'autre de leur nom de famille, ou leurs deux noms accolés à leurs enfants. Les conjoints peuvent par ailleurs indifféremment, mari comme femme, choisir de prendre le nom de leur conjoint en nom d'usage -sans jamais perdre sur les actes d'état civil leur nom de naissance. xv La réforme du régime matrimonial de 1804 ne date que de 1965 : la femme peut enfin gérer ses biens, ouvrir un compte en banque, exercer une profession sans l'autorisation de son mari. Il faudra attendre 1970 pour que la mère devienne l'égale du père en matière d'autorité parentale. xvi On ne dit pas, en effet, Mondamoiseau... xvii Procréation médicalement assistée. xviii

« Pas mariée, j'aime bien qu'on m'appellent "mademoiselle" au marché, même si c'est un clin d' oeil. NB : dès que j'ai eu un enfant, j'ai coché -madame-comme une grande fille, je n'ai pas attendu qu'on m'y autorise. Mam'zelle falibade, posté le16-11-2011 à 16:22 (réaction à

« L'association Mix-Cité a été saisie par Madame D.O. (lettre jointe) qui a adressé le 12 mars 2007 à la HALDE une demande de soutien face à un cas flagrant de discrimination: sa demande de remplacement de la mention « mademoiselle » par la mention « madame » sur sa carte grise lui a été refusée, bien qu'elle l'ait présentée à l'occasion d'un changement de domicile. Madame D.O. s'est vu réclamer la somme de 145 euros pour le simple changement de la mention « mademoiselle », alors que le changement de domicile a été effectué gratuitement. Il est évident que ceci introduit une discrimination par rapports aux hommes qui n'auront jamais à formuler une demande équivalente » viii .


Camilo Cruz
email: camilo.cruz@ensam.eu

Lounès Illoul

Francisco Chinesta

Gilles Regnier
email: gilles.regnier@ensam.eu

G Régnier

Effects of a Bent Structure on the Linear Viscoelastic Response of Carbon Nanotube Diluted Suspensions

Keywords: Brownian dynamics, Semi-flexible filament, Linear viscoelasticity, Carbon nanotubes

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Today, without any doubt nanotechnology is a hot topic. Belonging to this nano-world, carbon nanotubes are being intensively studied because their potential applications are extremely varied and the expected properties really impressive. A carbon nanotube (CNT) is a cylindrical structure constituted essentially by rolledup graphite sheets with diameters going from 0.4 nm to several nanometres and lengths up to some millimetres (Pan et al. 1998). This particular structure confers to CNTs an extraordinary compromise between mechanical, thermal and electrical properties.

Particularly, single-walled carbon nanotubes (SWNTs) are considered as a perfect constituent material for obtaining very strong and ultra light electromechanical components based on nano-filaments [START_REF] Chico | Pure carbon nanoscale devices: nanotube heterojunctions[END_REF][START_REF] Mceuen | Single-wall carbon nanotubes[END_REF], carbon fibres [START_REF] Vigolo | Macroscopic fibers and ribbons of oriented carbon nanotubes[END_REF], and transparent films [START_REF] Wu | Transparent, conductive carbon nanotube films[END_REF]). Additionally, SWNTs have a great biomedical potential as an effective gene and drug delivery system through cell membranes [START_REF] Shim | Functionalization of carbon nanotubes for biocompatibility and biomolecular recognition[END_REF][START_REF] Kam | Nanotube molecular transporter: internalization of carbon nanotubeprotein conjugates into mammalian cells[END_REF].

Pristine SWNTs exhibit very strong tube-tube van der Waals forces that produce their natural agglomeration in bundles. In consequence, solubilisation of SWNTs as individual entities is difficult and remains the main obstacle to develop their industrial applications [START_REF] Ajayan | Nanotubes from carbon[END_REF]. To solve this problem, chemical modification of the tube side wall by covalent bonding has been the most extended attempt to enhance the dispersion of SWNTs [START_REF] Bahr | Highly functionalized carbon nanotubes using in situ generated diazonium compounds[END_REF].

A large range of the applications of SWNTs requires a transformation processing in liquid phase. For such reason, surface-treated SWNTs are usually suspended into a Newtonian fluid or a viscoelastic matrix (melted polymer). Control and optimization of those transforming processes, based on the flow of such suspensions, need a deep comprehension of their rheology [START_REF] Kharchenko | Flow-induced properties of nanotube-filled polymer materials[END_REF]. [START_REF] Davis | Phase behaviour and rheology of SWNTs in superacids[END_REF] reported the dispersion of SWNTs in superacids (102% sulphuric acid), where the sidewall SWNT protonation is the driving mechanism that counteracts the tube-tube van der Waals forces and promotes the dispersion of individual SWNTs. They found that at dilute regime (concentration <120 ppm vol) SWNTs act as simple Brownian rods and the suspension exhibits an expected shear thinning behaviour.

In 2006, Rahatekar et al. studied the rheology of multi-wall carbon nanotubes (MWNTs) within an epoxy resin. They verified a two decades enhancement of the low shear viscosity when incorporating carbon nanotubes into the epoxy. When suspension was progressively sheared the suspension evidenced a rheo-thinning behaviour, a phenomenon mainly explained by the reduction of aggregates and not by the nanotube alignment. On the contrary, Fan andAdvani (2005, 2007) revealed by transmission electron microscopy that MWNTs are aligned in the flow direction when their suspensions were sheared. They found also that the extent of such alignment is function of the shear rate. A more recent work showed that the shear-thinning behaviour of CNT suspensions can be explained considering both the aggregation and the alignment effects (Ma et al. 2008a).

In 2005, Xu et al. studied the morphology and rheology of carbon nanofiber suspensions. Carbon nanofibers (diameter of about 100-200 nm) are bigger than CNTs, but their structural similarities let to compare their dynamical behaviour. Xu et al. used several standard elastic and rigid dumbbell models to predict the rheological behaviour of aqueous carbon nanofiber suspensions. They conjectured that the elasticity in such suspensions comes mainly from the recoverable bending of the carbon nanofibers.

Recently, Ma et al. (2008b) presented a study about the extensional rheology of surface-treated SWNTs diluted within an epoxy matrix. They validated a filament thinning experiment to determine the extensional viscosity of CNT suspensions. It was observed that the filament relaxation process for the surface-treated SWNT samples (low concentration, ≤0.3% wt.) follows the typical Newtonian filament relaxation behaviour where filament diameter decreases linearly with time.

On the contrary, the extensional viscosity of the treated SWNT suspensions did not show the classical relationship between the zero-shear viscosity and the extensional viscosity for Newtonian liquids, giving evidence of an enhanced extensional viscosity, probably due to the high degree of alignment of treated SWNTs in the flow direction. This behaviour was modelled by a simple Fokker-Planck equation-based orientation model.

Given the high aspect ratio and tensile rigidity of carbon nanotubes the classical mesoscopic-macroscopic approaches for fibre suspensions [START_REF] Hinch | The effect of Brownian motion on the rheological properties of a suspension of nonspherical particles[END_REF][START_REF] Petrie | The rheology of fibre suspensions[END_REF][START_REF] Sepehr | Comparison of rheological properties of fiber suspensions with model predictions[END_REF][START_REF] Rajabian | Suspensions of semiflexible fibers in polymeric fluids: rheology and thermodynamics[END_REF]) can be employed to describe the rheological behaviour of CNT suspensions. However, diffusion coefficients usually depend on empirical relationships [START_REF] Folgar | Orientation behavior of fibers in concentrated suspensions[END_REF][START_REF] Larson | The structure and rheology of complex fluids[END_REF]) and lack of physical meaning. [START_REF] Ma | The rheology and modelling of chemically treated carbon nanotubes suspensions[END_REF] reported a generalized Fokker-Planck-based model to predict the rheology of surfacetreated SWNTs suspended within an epoxy matrix in the dilute to semi-dilute concentration regime. The classical shear-thinning behaviour when the suspension is subjected to a steady shear flow was explained in terms of the competition between two phenomena: SWNT orientation in the flow direction and SWNT random misalignment due to Brownian motion (thermal interaction with the solvent). The shear-thinning data were properly fitted by a Fokker-Plank orientation model tuning the rotary diffusion coefficient. On the other hand, when the surface-treated SWNT suspension was submitted to a small amplitude oscillatory deformation a non-negligible elastic response was identified in the tested frequencies. Figure 1 shows the viscoelastic contribution of functionalized SWNTs to the dynamic response of a 0.2 wt.% SWNT-epoxy resin suspension. A constant-slope evolution for the storage and loss modulus is observed in the tested frequency interval. This kind of rheological behaviour was found characteristic of the SWNT suspensions from the dilute to the semi-dilute concentration regime. [START_REF] Ma | The rheology and modelling of chemically treated carbon nanotubes suspensions[END_REF] fitted those linear viscoelastic data with the Fokker-Planck-based orientation model, but using an empirical relationship for the rotary diffusion coefficient that depends on the applied frequency. Even though there had been several hypotheses to explain this behaviour as, for example, existence of a weak CNT network [START_REF] Hough | Viscoelasticity of single wall carbon nanotube suspensions[END_REF]) and electrostatic interaction between surface-treated SWNTs, the physical origins of this mild elasticity in the linear viscoelasticity spectra are not clear yet.

SWNTs can contain naturally topological defects [START_REF] Lijima | Pentagons, heptagons and negative curvature in graphite microtubule growth[END_REF]. Several studies have showed the existence of those defects by using Fresnel projection microscopy [START_REF] Binh | Local analysis of the morphological properties of single-wall carbon nanotubes by Fresnel projection microscopy[END_REF], scanning tunnelling microscopy (STM; [START_REF] Ouyang | Atomically resolved single-walled carbon nanotube intramolecular junctions[END_REF]) and voltagecontrast scanning electron microscopy (VC-SEM; [START_REF] Vijayaraghavan | Imaging defects and junctions in single-walled carbon nanotubes by voltage-contrast scanning electron microscopy[END_REF]. Most common topological defects in SWNTs are due to the existence of a nonhexagonal carbon ring (e.g. pentagon-heptagon defect) in middle of the classical honeycomb-like lattice structure of a CNT. Molecular dynamics (MD) is largely employed to identify the structural consequences of this kind of defects. For example, it has been demonstrated that a bent structure is thermodynamically more stable than a straight one once a localized topological defect appears in the SWNT growth [START_REF] Xue | Comparison of the properties of bent and straight single-walled carbon nanotube intramolecular junctions[END_REF]). Additionally, it has been shown that a single pentagonheptagon pair can bend the SWNT tubular structure forming a localized junction [START_REF] Ajayan | Surface reconstruction and dimensional changes in single-walled carbon nanotubes[END_REF]) with angles varying from 0 • to 34 • [START_REF] Han | Observation and modelling of single-wall carbon nanotube junctions[END_REF][START_REF] Wako | Bending deformation of single-walled carbon nanotubes caused by 5-7 pair couple defect[END_REF] depending on the distance between the pentagon and heptagon pair defect [START_REF] Lambin | Structural properties of junctions between two carbon nanotubes[END_REF].

So, in order to contribute elucidating the physical phenomena behind the mild elasticity exhibited by the dynamics of individual SWNTs this paper investigates the influence of the naturally bent SWNT structure on the linear viscoelastic response of their dilute suspensions. To do so, we proposed an equivalent mechanical model that represents an individual SWNT as a semi-flexible filament (or a non-freely multi-beadrod model in a discretized version) with a natural bent configuration and we evaluated by Brownian dynamics the linear viscoelastic response of this physical model into a diluted suspension.

Modelling

Persistent length (L p ) considerations are necessary when modelling high aspect ratio systems that exhibit a very important resistance to tangential extension or compression [START_REF] Morse | Viscoelasticity of concentrated isotropic solutions of semiflexible polymers. 1. Model and stress tensor[END_REF][START_REF] Shankar | Theory of linear viscoelasticity of semiflexible rods in dilute solution[END_REF]. In absence of external forces (e.g. a flow field), persistent length is a characteristic length that relates the bending rigidity of the system to the thermal energy around it. In practice, this parameter determines a length scale over which the high aspect ratio system presents significant curvature due to thermal interactions with the surroundings. In consequence, a system of total length L where L/L p << 1 can be considered as a simple rigid rod and a system where L/L p >> 1 can be considered as a flexible filament. Traditionally, individual SWNTs have been modelled as Brownian rigid rods [START_REF] Davis | Phase behaviour and rheology of SWNTs in superacids[END_REF][START_REF] Duggal | Dynamics of individual singlewalled carbon nanotubes in water by real-time visualization[END_REF][START_REF] Mendes | Brownian dynamics simulations of single-wall carbon nanotube separation by type using electrophoresis[END_REF][START_REF] Ma | The rheology and modelling of chemically treated carbon nanotubes suspensions[END_REF].

Nevertheless, there are experimental results that lead to think that treated SWNTs can be considered as other than rigid-rod systems. Using a nearinfrared fluorescence technique, thermal SWNT-shape fluctuations has been observed into a diluted SWNT aqueous suspension [START_REF] Tsyboulski | Versatil visualisation on individual single-walled carbon nanotubes with near-infrared fluorescence microscopy[END_REF]. Additionally, [START_REF] Fakhri | Diameter-dependent bending dynamics of singlewalled carbon nanotubes in liquids[END_REF] demonstrated that the observed bending dynamics corresponds accurately with a semi-flexible filament model (L ∼ L p ), where L p scales with the d 3 (d, diameter of SWNT).

If we are considering now the action of an external flow field and the natural existence of a bent structure in SWNTs due to the presence of side-wall defects, the persistent length concept requires a more precise definition. In any case, those new considerations can only reduce the length scale of observable bending relative to the thermal classical persistent length and, hence, in this paper a semi-flexible filament is supposed to be a good equivalent micromechanical model describing the dynamics of an individual SWNT submitted to a homogeneous flow field.

In this context, we insist on considering that a SWNT contains naturally side-wall defects producing a deviation from the perfectly straight structure; i.e. the pristine SWNT has a slightly curved structure. This assumption is supported on two main facts:

• Even if there is a large discussion about the influence of sample preparation on the observed structure of CNTs, there are numerous images taken by SEM, STM, atomic force microscopy (AFM) and transmission electron microscopy that reveal clearly a natural bent structure in some CNT samples [START_REF] Shaffer | Dispersion and packing of carbon nanotubes[END_REF][START_REF] Clauss | Scanning tunneling microscopy of carbon nanotubes[END_REF][START_REF] Ouyang | Atomically resolved single-walled carbon nanotube intramolecular junctions[END_REF][START_REF] Kinloch | A rheological study of concentrated aqueous nanotube dispersions[END_REF][START_REF] Loos | Visualization of single-wall carbon nanotubes (SWNT) networks in conductive polystyrene nanocomposites by charge contrast imaging[END_REF][START_REF] Lamprecht | AMF imaging of function-alized double-walled carbon nanotubes[END_REF][START_REF] Vijayaraghavan | Imaging defects and junctions in single-walled carbon nanotubes by voltage-contrast scanning electron microscopy[END_REF]. For example, in Fig. 2 an image taken by AFM of surface-treated SWNTs protruding from a photocured acrylic surface is presented; the bent structure of the SWNTs is easily appreciable. • The natural existence of topological defects that plays an important role in the mechanical and thermo-electrical properties of CNTs [START_REF] Ruoff | Mechanical properties of carbon nanotubes: theoretical predictions and experimental measurements[END_REF]. A clear relationship between those CNT structural instabilities (pentagon/heptagon defect, for example) and the intrinsic curvature of CNT structures has been established [START_REF] Clinard | Electron microscopy, growth and defects of carbon nanotubes[END_REF][START_REF] Dimitrakopulos | The defect character of carbon nanotubes and nanoparticles[END_REF][START_REF] Ajayan | Surface reconstruction and dimensional changes in single-walled carbon nanotubes[END_REF][START_REF] Han | Observation and modelling of single-wall carbon nanotube junctions[END_REF][START_REF] Clauss | Scanning tunneling microscopy of carbon nanotubes[END_REF][START_REF] Lambin | Structural properties of junctions between two carbon nanotubes[END_REF][START_REF] Huang | Controlled growth and electrical characterization of bent single-walled carbon nanotubes[END_REF][START_REF] Wako | Bending deformation of single-walled carbon nanotubes caused by 5-7 pair couple defect[END_REF][START_REF] Xue | Comparison of the properties of bent and straight single-walled carbon nanotube intramolecular junctions[END_REF].

So, in order to test the dynamic mechanical response of this physical model, we decided to use an inertialess Langevin dynamics approach, also known as Brownian dynamics. The semi-flexible filament was discretized in a non-freely jointed multi-bead-rod model as observed in Fig. 3. This multi-bead-rod model is composed of n beads with positions r i , connected by n -1 rods of length a, where a approaches theoretically the persistence length under flow conditions of the system. The SWNT physical length is given by the total length of the multi-bead-rod model (n -1) a. Beads are considered the centres of hydrodynamic resistance, as usual for all the mechanical models in polymer kinetic theory [START_REF] Bird | Dynamics of polymer liquids[END_REF], and also mark the existence of a bent topological defect. On the other hand, rods are supposed infinitely rigid. We consider the existence of an internal bending potential that mimics the SWNT bending flexibility. Bending potential is given by the action of a hypothetical torsion spring between each pair of rods.

Concerning particularly the simulation of a dynamic mechanical test on a diluted suspension of SWNTs, several hypotheses are established:

• Dynamic mechanical solicitation occurs at constant temperature.

• Individual SWNTs are embedded in a Newtonian matrix of viscosity η s . • Suspension is supposed to be diluted, in consequence, no physical interaction between SWNTs is considered. • Applied flow field is considered homogeneous, i.e.

the rate-of-strain tensor is supposed constant within the suspension [START_REF] Bird | Dynamics of polymer liquids[END_REF]). • No external forces (electrostatic, magnetic or gravitational) play a role in the dynamics of the suspension. Possible electrostatic interaction between surface-treatment groups is neglected because it has been shown that those repulsion forces do not produce an increment of the storage modulus at high frequencies (Ma et al. 2008c).

Kinematic formulation

Keeping in mind all previous hypotheses, the kinematic description of the multi-bead-rod system is given by the next n stochastic equations (n force balances for each bead):

0 = F d i + F b i + F φ i - j n ij λ j for i = 1 . . . n and j = 1 . . . n -1 (1)
where F d i is the hydrodynamic drag force acting on bead i, F b i is the Brownian force acting on bead i, F φ i is the internal bending potential force acting on bead i and λ j is a constraint force associated to the rod connecting bead j to j + 1. Operator n ij is defined as follows:

n ij = u j δ i, j+1 -δ i, j (2)
in which u j = (r j+1r j )/a is a unit vector collinear to the rod connecting beads j and j + 1. In order to complete correctly the kinematic description of the multibead-rod chain in a Cartesian coordinates system is necessary to impose a set of n -1 constraints over the bead velocities:

ṙi • u i -ṙi+1 • u i = 0 for i = 1 . . . n -1 (3)
where ṙi is the instantaneous velocity of bead i. In summary, using the n equations in (1) plus the n -1 equations in (3) a linear system of equations describing the motion of the multi-bead-rod system can be obtained, where the unknown variables are n instantaneous bead velocities ṙi plus n -1 constraint forces λ j .

In what follows a formal mathematical description for each term in Eq. 1 is developed and some important conditions are emphasized given the time-integration algorithm implemented in this work:

• Hydrodynamic drag force F d i
This force describes the resistance experienced by the bead as it moves through the fluid. One of the simplest ways to express this force is given by the Stoke's law. This law considers that the hydrodynamic drag force is proportional to the difference between the fluid velocity and the averaged bead velocity. A more general expression takes into account a hydrodynamic interaction term that comes from the interactions inside the same system as it is moving. Mathematically, this force writes:

F d i = ζ [κ • r i ] -ṙi -ṙ * i (4)
where ζ is a second-order friction tensor, κ is the gradient of the homogeneous velocity field acting in all the fluid and ṙ * i is a correction velocity that accounts for the variation in the local homogeneous flow field due to the motion of the other beads in the same system. In this work, an isotropic second-order friction tensor is supposed, reducing this parameter to a scalar friction coefficient ζ (i.e. ζ = ζ I), and hydrodynamic interaction is neglected (ṙ * i = 0) due to the small strain imposed and the mild bending rigidity of SWNTs. Previous suppositions are also necessary when implementing a simple one-step integration scheme (Öttinger 1996), as is the case of the current work.

• Brownian force F b i

The Brownian forces account for the thermal effects of the solvent molecules over the SWNTs. Taking into account that the time scale of this thermal activity is quite small compared with the time scale of SWNT motion, Brownian forces are mathematically computed as a stochastic process. In unconstrained systems, the set of Brownian forces must to follow a normal distribution satisfying the next conditions:

F b i (t) = 0 (5) F b i (t) ⊗ F b i (t + t) ≈ 2ζ k B T t I (6)
where k B is the Boltzmann constant, T is the absolute temperature and t is the time step of the simulation. In Brownian dynamics, the Brownian force generation is a critical aspect, especially when the system contains constraints. A proof of this subtlety is the intense discussion about the simulations of coarse-grained models in the singular limit between rigid links and very stiff springs [START_REF] Fixman | Simulation of polymer dynamics. I. General theory[END_REF][START_REF] Fixman | Implicit algorithm for Brownian dynamics of polymers[END_REF][START_REF] Hinch | Brownian motion with stiff bonds and rigid constraints[END_REF][START_REF] Öttinger | Brownian dynamics of rigid polymer chains with hydrodynamic interactions[END_REF][START_REF] Öttinger | Stochastic processes in polymeric fluids[END_REF][START_REF] Grassia | Computer simulations of Brownian motion of complex systems[END_REF][START_REF] Grassia | Computer simulations of polymer chain relaxation via Brownian motion[END_REF].

In the present modelling rods are considered intrinsically inextensible and they do not intend to mimic the effect of a very stiff spring. In order to satisfy the fluctuation dissipation theorem for systems containing physical constraints a geometrical projection is done over the unconstrained Brownian forces obtained using Eqs. 5 and 6. Brownian forces must to satisfy the next condition:

0 = i F b i • n ij for j = 1 . . . n -1 (7)
A complete description of the algorithm used to satisfy Eq. 7 is given elsewhere [START_REF] Montesi | Brownian dynamics algorithm for bead-rod semiflexible chain with anisotropic friction[END_REF]. Projection of Brownian forces is absolutely necessary when an anisotropic friction tensor is considered or second or higher order integration schemes are employed.

• Internal bending potential force

F φ i
Based on the viscoelasticity theory of concentrated solutions of semi-flexible polymers [START_REF] Morse | Viscoelasticity of concentrated isotropic solutions of semiflexible polymers. 1. Model and stress tensor[END_REF], [START_REF] Pasquali | An efficient algorithm for metric correction forces in simulations of linear polymers with constrained bond lengths[END_REF] defined a discrete bending potential for a multi-bead-rod model perfectly straight in equilibrium:

φ = - K b a n-1 i=2 u i • u i-1 ( 8
)
where φ is the bending potential and K b is a bending rigidity constant. Using Eq. 8, a natural definition of the bending force for each bead can be obtained:

F φ k = - ∂φ ∂r k = K b a n-1 i=2 ∂(u i • u i-1 ) ∂r k (9)
Appendix A shows that the total bending force on bead k only depends on the local chain configuration and that this force can be interpreted as the sum of independent contributions coming from all the two-rod sub-sections containing bead k.

Considering now the main hypothesis mentioned in this paper, to take into account the naturally bent configuration of the multi-bead-rod model at equilibrium, the bending potential must to be redefined in the next way:

φ = - K b a n-1 i=2 Z i u i • u i-1 = - K b a n-1 i=2 u i • u i-1 (10)
where Z i is a linear operator that rotates vector u i of θ eq i (interior angle between u i and u i-1 at the equilibrium configuration) and u i is the rotated vector Z i u i . In an analogous way, the bending force expression in (9) takes the next form:

F φ k = - ∂φ ∂r k = K b a n-1 i=2 ∂ u i • u i-1 ∂r k (11)
Considering that the total bending force in Eq. 11 can also be interpreted as the sum of independent contributions coming from all the two-rod sub-sections containing bead k, as shown in Appendix A, the next algorithm is proposed to compute the total bending forces in coherence with the bent configuration:

1. Decomposition of the multi-bead-rod (n beads) system into n -2 sub-section systems of two con- secutive rods as shown in Fig. 4 (note that subsection p is composed by beads p, p + 1 and p + 2). 2. Calculation of the non-zero contributions to the bending forces over all beads composing each subsection p as follows:

• Bending force on bead p of sub-section p as:

F φ p, p = - K b a 2 I -u p ⊗ u p • u p+1 (12)
• Bending force on bead p + 2 of sub-section p as:

F φ p+2, p = Z -1 p+1 • K b a 2 I -u p+1 ⊗ u p+1 • u p (13)
Bending force on bead p + 2 of sub-section p has been rotated back (operator Z -1 p+1 ) in order to correct the direction of the force, in coherence with the actual bent configuration.

• As the bending potential in each sub-section is locally independent of the adjacent subsections, it must to satisfy mechanical equilibrium and hence bending force on bead p + 1 of sub-section p writes:

F φ p+1, p = -F φ p, p -F φ p+2, p (14)
3. Finally, the total bending force on bead k is calculated as follows:

F φ k = n-2 p=1 F φ k, p (15)
Once the instantaneous velocity ṙi of the system is known (by solving the system constituted by the 2n -1 equations defined in (1) and ( 3)), the system position is updated using a one-step algorithm that conserves the rod lengths throughout the time step. For example, the numerical time integration for a bead-rod chain confined in the plan xy is obtained as follows:

1. Calculation of angular velocities ω i = [0 0 ω i ] of rods by resolving adequately the over-defined system:

ṙi+1 -ṙi = ω i ⊗ au i-1 for i = 1 . . . n -1 (16)
2. Update of the position of bead 1 by:

r t+ t 1 = ṙt 1 • t (17)
3. Update of the director vectors of rods by:

θ t+ t i = θ t i + ω i • t for i = 1 . . . n -1 (18)
4. Update of the rest of bead positions by resolving consecutively from i = 2 to n:

r t+ t i = r t+ t i-1 + au t+ t i-1 (19)
Previous one-step algorithm is coherent with the explicit corrector-predictor scheme [START_REF] Liu | Flexible polymer chain dynamics and rheological properties in steady flows[END_REF][START_REF] Somasi | Brownian dynamics simulations of bead-rod and beadspring chains: numerical algorithms and coarse-graining issues[END_REF] because it ensures that the rotational velocity of each rod at the end of the time step is perpendicular to its orientation [START_REF] Doyle | Dynamic simulation of freely draining flexible polymers in steady linear flows[END_REF]).

Stress tensor calculus

The total shear stress tensor τ in a suspension is supposed to be the sum of the contribution coming from the solvent τ s and another one coming from the suspended entities τ p [START_REF] Bird | Dynamics of polymer liquids[END_REF]:

τ = τ s + τ p (20)
As defined in classical Newtonian fluid mechanics, τ s in Eq. 20 can be also written as follows:

τ s = η s γ (21)
where η s is the solvent viscosity and γ is the homogeneous rate-of-strain tensor. To model the shear dynamic response, we are particularly interested on the shear stress contribution coming from the SWNTs (represented as non-freely multi-bead-rod chains) τ p .

In order to quantify this contribution in the context of Brownian dynamics we used the Kramers-Kirkwood expression for the shear stress tensor:

τ p = c v R v ⊗ F d v ( 22
)
where c is the number of SWNTs per volume and R v = r vr c is the relative location of bead v to the centre of mass of the multi-bead-rod chain r c .

In order to correctly account for the Brownian stress contribution during a time step the shear stress tensor contribution coming from the SWNTs τ p is calculated using the next algorithm:

1. Calculation of the hydrodynamic drag forces F d i , the geometrical projected Brownian forces F b i and the internal bending potential forces F φ i at time t as described in the kinematic description section. 2. Update of the system position at time t + t. 3. Using the same geometrically projected Brownian forces obtained at time t, recalculation of the system velocity and hydrodynamic drag forces F d,+ t i at the updated configuration t + t. 4. Determination of the current shear stress tensor at time t by computing:

τ t p = c v R + t v ⊗ F d,+ t v (23)

Computational details

Previous algorithm was programmed on C++, compiled in Linux 64-bits and run on a cluster 6 x Quad-Core AMD Opteron™ 2376 (2300 MHz) processors.

Validation

In order to validate the Brownian dynamics algorithm we decided to simulate the dynamic response of the freely jointed three-bead model, also known as the trumbbell, trimer or (three-bead)-(two-rod) system. The dynamic response of bead-rod models for macromolecular solutions has been studied before by [START_REF] Hassager | Kinetic theory and rheology of bead-rod models for macromolecular solutions. II. Linear unsteady flow properties[END_REF]. His results are in coherence with the Fokker-Plank equation for a bead-rod chain in kinetic theory [START_REF] Kramers | Het gedrag van macromoleculen in een stroomende vloeistof[END_REF][START_REF] Bird | Dynamics of polymer liquids[END_REF], actually deviated from that one for bead-spring chains [START_REF] Gottlieb | A molecular dynamics calculation to confirm the incorrectness of the random-walk distribution for describing the Kramers freely jointed bead-rod chain[END_REF]. Using asymptotic expansions, Hassager found a numerical solution for the trimer model, i.e. a system composed of three beads with friction coefficient ζ connected by two freely jointed rigid massless rods of length a. To simulate the trimer system using our Brownian dynamics algorithm we used a bidimensional modelling and we neglected the internal bending potential by establishing the bending rigidity constant K b equal to 0. To compare the Hassager's results with our Brownian dynamics simulation, complex modulus is represented in a proper reduced frequency scale.

In Fig. 5 the Hassager's numerical solution is compared with results of the previously described Brownian dynamics simulation for the case of a trimer system. Hassager's complex modulus is presented in a reduced frequency scale ωτ 1 where τ 1 is the main relaxation time given by the numerical solution [START_REF] Hassager | Kinetic theory and rheology of bead-rod models for macromolecular solutions. II. Linear unsteady flow properties[END_REF]:

τ 1 = ζ a 2 5.4376k B T (24)
On the other hand, our Brownian dynamics simulation for the trimer system is plotted in a reduced frequency scale ωλ rod , where λ rod is the rotational diffusion time of a (three-bead)-(one-rod) system of equal length [START_REF] Bird | Dynamics of polymer liquids[END_REF]:

λ rod = ζ a 2 3k B T (25)
The Brownian dynamics algorithm yielded a complex modulus in good agreement with the Hassager's numerical solution for the trimer system. As bi-dimensional modelling validated the Hassager's results, it was not necessary to implement the three-dimensional modelling, which would be more expensive in computational time. Figure 6 presents the performance of the Brownian dynamics code in function of the population considered. The performance of the Brownian dynamics code is measured in terms of the mean of relative errors for 17 storage modulus values homogeneously distributed in an interval of reduced frequencies going

G simul. (ω R ) - log G Hassager (ω R ) log G Hassager (ω R )
from 10 -2 to 10 2 . Loss modulus signal is practically not sensitive to the Brownian dynamics population; mean relative error for the loss modulus signal is 2.9%. A population of 10 4 systems is selected for all the Brownian dynamics simulations because a good compromise between computational time and rheological prediction performance is guaranteed.

Results and discussion

The Brownian dynamics algorithm described in the modelling section was used to predict the dynamic response of a non-freely jointed multi-bead-rod system with a natural bent configuration, a physical model intended to emulate the mechanical response under flow conditions of a SWNT that contains bendable junctions.

In the first stage, we decided to analyze the simplest case, i.e. the (three-bead)-(two-rod) system. The freely jointed version of this model (i.e. with K b = 0) is presented in Fig. 7. In a general way, two different regions are clearly differentiated in the dynamic response of a freely jointed (three-bead)-(two-rod) system: a low frequency regime exhibiting a typical Maxwell terminal behaviour, where the thermal activity (or Brownian forces) drives the kinematic of the system; and a high frequency regime, where the domination of flow

θ max = 30 • ). BD parameters: n = 2, c = 1, a = 1, ζ = 1, k B T = 0.
1. λ rod is the rotational diffusion time of an equivalent multibead rigid-rod system of equal length forces over the diffusivity terms produces a plateau for the storage modulus and a limiting constant viscosity (G (ω)/ω) -η s larger than zero. The existence of a non-zero limiting viscosity is physically more coherent than the classical Rouse-Zimm responses, where the model contribution to the dynamic viscosity disappears at high frequencies.

If we consider now the action of the bending potential, i.e. a non-freely jointed (three-bead)-(two-rod) system, a large range of dynamic responses is deployed in function of the equilibrium configuration and the bending rigidity constant. To explain this fact, in Fig. 7 the dynamic responses of a freely jointed system, two non-freely jointed systems with natural straight configuration and a non-freely jointed system with a naturally bent configuration are compared. Rod misalignment (i.e. the natural bent SWNT configuration) is generated following the next formula:

θ 0 i = θ 0 i-1 -θ max + U(0, 1) • 2 θ max (26)
where θ 0 i is the equilibrium director vector of rod i, θ max is the maximal misalignment between two consecutive rods and U (0, 1) is a continuous uniform random distribution function defined between 0 and 1. In a general way, when the bending rigidity constant K b is equal or greater than the thermal coefficient k B T, the loss modulus is enhanced over the entire frequency range and the storage modulus is increased in the low frequency regime. Otherwise, if the bending rigidity constant is much lower than the thermal coefficient (K b << k B T), a dynamic response similar to that one of a freely jointed system is obtained, showing just one dominant relaxation time associated with the thermalto-flow-induced motion transition.

Additionally, in Fig. 7 the apparition of new relaxation processes when the bending rigidity constant is considerably greater than the thermal coefficient (i.e. K b >> k B T) is observed. For instance, the activation of a mild elasticity at intermediate frequencies is clearly noticed in the dynamic response of the non-freely jointed (three-bead)-(two-rod) system with bending rigidity constant K b = 10. When comparing the straight configuration with the bent one, even though the loss modulus responses are similar, an interesting difference in the storage modulus is appreciated. In fact, on the one hand the activation of the mild elasticity for the system with a natural bent configuration is observed at a lower characteristic frequency than the system with a straight configuration; on the other hand a more steeped mild elasticity is noticed for the bent system in comparison with the straight one. In front of this qualitative feature the appropriateness of considering a natural bent configuration in SWNTs is a priori justified when analyzing its dynamic response under flow.

In Fig. 8 the Brownian dynamics predictions of the complex modulus for a non-freely jointed (three-bead)-(two-rod) system with a natural bent configuration for different values of the bending rigidity constant are presented. In Fig. 8 the typical thermal-to-flow-induced relaxation process occurring around the rotational diffusion time λ rod (as in the equivalent straight systems) is easily identifiable. Additionally, an interesting second relaxation process (as showed before) activating a mild elasticity at intermediate frequencies is again appreciated for the natural bent systems. Brownian dynamics simulations have showed that the characteristic frequency of this relaxation process and the magnitude of the storage modulus enhancement are function of the bending rigidity constant. The activation frequency of this mild elasticity is increased as the bending rigidity constant is increased as well. This feature is physically coherent taking into account that a higher energy input (i.e. solicitation frequency) is required to activate the first bending mode of a stiffer system. On the other hand, it is also noticed that the storage modulus enhancement

c = 1, a = 1, ζ = 1, K b = 1, k B T = 0.1.
Storage modulus is scaled with the value of storage modulus at the unit reduced frequency. λ rod is the rotational diffusion time of an equivalent multi-bead rigid-rod system of equal length. Vertical pointed line represents the transition between the thermalinduced to the flow-induced regime Fig. 10 Predicted loss modulus of a non-freely jointed multibead-rod system with a natural bent configuration (maximal misalignment angle θ max = 30 • ) for different number of rods n. BD parameters:

c = 1, a = 1, ζ = 1, K b = 1, k B T = 0.1.
Loss modulus is scaled with the value of storage modulus at the unit reduced frequency. λ rod is the rotational diffusion time of an equivalent multi-bead rigid-rod system of equal length. Vertical pointed line represents the transition between the thermalinduced to the flow-induced regime and the frequency range associated to this new relaxation process also rise as the bending rigidity increases. This qualitative feature can be physically explained considering that the energy stocking capacity of a bending mode is directly proportional to the bending rigidity constant.

The reduced storage modulus predictions of a nonfreely jointed multi-bead-rod system with a natural bent configuration for different number of rods (n = 3, n = 5 and n = 10) are showed in Fig. 9. As in a Kramers chain, in the present modelling the number of rods reflects the flexibility of the bead-rod system, but in addition it captures the density of bent defects in the SWNT structure. In other words, this parameter is directly related to the SWNT persistent length under flow conditions that, at the same time, we suggest strongly depends on the presence of bent defects in the SWNT structure. For information, an example of the equilibrium internal angles θ 0 i-j = θ 0 iθ 0 j between consecutive rods used for the simulations in Figs. 9 and 10 is presented in Table 1. Differences in the dynamic response are only appreciable in the flow-induced regime, i.e. at frequencies greater than the characteristic relaxation frequency of an equivalent multi-bead rigid-rod system. In other words, different An enhancement of the elastic response when increasing the number of rods is observed and different slopes for the storage modulus in function of frequency are obtained (from 0.5 for n = 3 to 1.2 for n = 10 at reduced frequencies between 10 1 and 10 2 ). This mild elasticity can be explained by the coupled contribution of the tension forces required to maintain the inextensibility condition (as in the Kramers chain) and the forces coming from the torques activated by the bending of the system. On the other hand, in Fig. 10 the reduced loss modulus responses for the same systems deployed on Fig. 9 are presented. In the low frequency regime the typical thermal motion response is observed for the three systems, i.e. a loss modulus evolving proportional with frequency. As in the storage modulus case, differences between the loss modulus responses for the three multibead-rod systems (n = 3, n = 5 and n = 10) are only appreciable in the high frequency range. Differences are quite less important than those found for the storage modulus because just light variations in the slopes were encountered (from 0.7 for n = 3 to 0.8 for n = 10 at reduced frequencies between 10 1 and 10 2 ).

In order to properly compare the experiments on Fig. 1 with our simulations we decided to represent the experimental data in a reduced scale. To do so, an estimation of the equivalent rotational diffusion time for the actual surface-treated SWNT is needed. The rotational diffusion time for a multi-bead rigid-rod defined by [START_REF] Bird | Dynamics of polymer liquids[END_REF] writes as follows:

λ rod = ζ L 2 n(n + 1) 72(n -1)k B T ( 27
)
where the isotropic bead friction coefficient ζ is a simple discretization of the bulk friction coefficient ζ bulk = nζ . The bulk friction coefficient for a rigid rod system of length L and diameter d can be calculated following the next equation [START_REF] Mendes | Brownian dynamics simulations of single-wall carbon nanotube separation by type using electrophoresis[END_REF]:

ζ bulk = 2πη s L log[L/d] (28)
Rheological tests presented in Fig. 1 were effectuated at 25 • C using a suspension of HiPco SWNT (Nanocomposites, Houston, TX, USA) within an epoxy resin with a Newtonian viscosity of 10 Pa s. Length of HiPco SWNTs largely depends on factors such as whether the tubes have been functionalized and how long they have been sonicated for. From the literature, we found that HiPco SWNTs have average lengths ranging from 400 to 700 nm and diameters between 0.8 and 1.3 nm [START_REF] Parra-Vasquez | Simple length determination of single-walled carbon nanotubes by viscosity measurements in dilute suspensions[END_REF][START_REF] Mendes | Brownian dynamics simulations of single-wall carbon nanotube separation by type using electrophoresis[END_REF][START_REF] Fakhri | Diameter-dependent bending dynamics of singlewalled carbon nanotubes in liquids[END_REF]. Using mean values for the physical dimensions of HiPco SWNTs and employing Eqs. 27 and 28, we found equivalent rotational diffusion times for the SWNT going from 5.5 s for n = 10 to 9.0 s for n = 3. In Fig. 11 the scaled experimental data are compared with the dynamical response of a naturally bent non-freely jointed multi-bead-rod system of five rods. It appears from Fig. 11 that the mild elasticity measured experimentally is a phenomenon occurring in the intermediate frequency range and that the simulation results produce coherent magnitudes. Observe, for Fig. 11 Comparison in a proper reduced scale of the experimental SWNT dynamic response in diluted suspension (Fig. 1) with the BD predicted complex modulus of a naturally bent non-freely jointed multi-bead-rod system of five rods (Figs. 9 and 10) example, that the predicted ratio between G and G at intermediate frequencies is similar to that one encountered in experimental dynamical tests of SWNT diluted suspensions. A more precise model fitting will require identifying the relaxation time spectra of an individual SWNT and the dynamic response of the diluted suspension in a higher frequency range. We emphasize that the principal objective of this paper is to demonstrate that a natural bent SWNT structure (due to presence of synthesis defects) can play an important role in the dynamical response of a SWNT diluted suspension and constitutes, in consequence, a physical parameter to take into account when modelling the mechanical behaviour of SWNTs.

The physical model proposed in this paper could be applied for simulating non-linear regimes, but probably we should consider intra-and inter-tube interactions, that can be introduced via the hydrodynamic interaction hypothesis (as done in polymer kinetic theory) coupled with an adapted Brownian dynamics strategy. The analysis of this extended model constitutes a work in progress. The results reported in this paper consider small amplitude oscillatory flows where the maximum deformation was chosen within the interval in which the results were independent on the maximum deformation applied, in order to ensure the validity of the linearity hypothesis.

Conclusions

Isolated treated SWNTs have been modelled as semiflexible filaments with a naturally bent configuration considering the probable existence of side-wall defects on carbon nanotubes. Continuous semi-flexible filament model was discretized as a non-freely jointed multi-bead-rod system with a non-straight configuration at equilibrium in order to implement a Brownian dynamics simulation. In this context, a mechanically coherent re-definition of the discretized version of the bending potential for a natural straight semi-flexible filament is required in order to tackle the cases of non-straight configurations.

New relaxation processes in the dynamic mechanical response of a naturally bent system are observed from the simulation results. Those processes are characterised by the activation of a mild elasticity at intermediate frequencies (revealed as an enhancement of the storage modulus). Moreover, non-negligible differences are found with respect to the dynamic response of a naturally straight system. It is observed that the enhancement of the storage modulus and the range of frequencies associated with those new relaxation processes depend directly on the bending rigidity constant or, in other words, on the local structural stiffness of the SWNT junctions. Additionally, the dynamic response of such model is function of the persistent length under flow conditions (i.e. number of rods in the discretized version). An augmentation of the storage modulus slope in the intermediate frequency range is observed when increasing the number of rods. This work will be useful to give richer explanations to the physical origins of the impressive stiffening effects coming from carbon nanotubes on the rheological behaviour of their diluted, semi-diluted and even concentrated suspensions. Future work should be pointed towards the identification of physical parameters and the consideration of more complex bending potentials and poly-dispersed length distributions.

Acknowledgements Authors would like to thank Dr. Anson Ma (Rice University) and Prof. Malcolm Mackley (University of Cambridge) for fruitful discussions and for providing helpful experimental information (Fig. 1). We would like to acknowledge Dr. Anson Ma (Rice University), Dr. Loren Picco (Bristol University) and Prof. Mervyn Miles (Bristol University) for allowing us to use the AFM image (Fig. 2). We would like to thank also the GeM institute (Institute de Recherche en Génie Civil et Mécanique) at Nantes (France) for providing the computational resources.

Appendix A

Total bending force in a multi-bead-rod system writes:

F φ k = - ∂φ ∂r k = K b a n-1 i=2 ∂ (u i • u i-1 ) ∂r k (29)
Expanding Eq. 29 we have:

F φ k = K b a ⎛ ⎜ ⎜ ⎝ ∂(u 2 • u 1 ) ∂r k +...+ ∂(u k-1 • u k-2 ) ∂r k + ∂(u k • u k-1 ) ∂r k + ∂(u k+1 • u k ) ∂r k +...+ ∂(u n-1 • u n-2 ) ∂r k ⎞ ⎟ ⎟ ⎠ (30)
F φ k = K b a ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ (31)
Equation 31 can be evaluated using the identity:

∂ ∂r k u i = 1 a δ k,i+1 -δ k,i (I -u i ⊗ u i ) (32)
Observing identity (32) is clear that the derivative ∂ ∂r k u i takes a non-zero value only when i is k -1 or k. Hence, considering a bead k in the middle of an infinite chain, Eq. 31 becomes:

F φ k = K b a ∂u k-1 ∂r k u k-2 + K b a ∂u k ∂r k u k-1 + ∂u k-1 ∂r k u k + K b a ∂u k ∂r k u k+1 ( 33
)
Total bending force on bead k, as given in the previous equation, can be interpreted as the sum of three independent contributions. In order to explain the origin of those contributions consider a multi-bead-rod system containing n beads decomposed into n -2 independent sub-sections of two consecutive rods as showed in Fig. 4. Taking each of those sub-sections as an independent system, the expressions for the bending forces associated to each sub-section are given by Eq. 29.

For example, sub-section p (composed by beads p, p + 1 and p + 2) has the associated bending forces F From the previous equations it results clear that bending forces are in mechanical equilibrium. For this reason:

F φ p+1, p = -F φ p, p -F φ p+2, p (40)
Now, using the previous results about the decomposition of a multi-bead-rod system is easy to explain the origin of different contributions in Eq. 33 as follows:

F φ k = F φ k,k-2 + F φ k,k-1 + F φ k,k (41)
In other words, total bending force on bead k can be interpreted as the sum of all the independent bending forces coming from sub-sections containing bead k.

Fig. 1

 1 Fig. 1 Viscoelastic contribution of treated SWNTs to the dynamic response of a 0.2 wt.% SWNT-epoxy resin suspension. SWNT contribution to the storage and loss modulus are calculated as G CNT = G suspension -G epoxy and G CNT = G suspensionωη epoxy , respectively. Epoxy resin is supposed to be Newtonian (G epoxy ∼ 0 and η epoxy = 10 Pa s). Courtesy of Dr. Anson Ma (Rice University)

Fig. 2

 2 Fig. 2 AFM image of treated SWNTs protruding from a photocured acrylic surface (image area corresponds to a 5-μm-side square). Courtesy of Dr. Anson Ma (Rice University), Dr. Loren Picco (Bristol University) and Pr. Mervyn Miles (Bristol University)

Fig. 3

 3 Fig. 3 Non-freely jointed multi-bead-rod model composed of n beads and n -1 rods of length a. Bending potential between rods in the multi-bead-rod model is mimicked with a hypothetical torsion spring

Fig. 4

 4 Fig. 4 Multi-bead-rod system of n beads (in black) decomposed in n -2 sub-sections p of two consecutive rods (in gray)

Fig. 5

 5 Fig. 5 The dynamic modulus of a freely jointed (three-bead)-(two-rod) system in a proper reduced frequency scale: comparison between the Hassager's numerical solution and the current Brownian dynamics (BD) simulation. τ 1 is the main Hassager's relaxation time and λ rod is the rotational diffusion time of an equivalent multi-bead rigid-rod system of equal length. BD parameters: n = 2, c = 1, a = 1, ζ = 1, k B T = 0.1 and K b = 0

Fig. 6

 6 Fig. 6Mean relative error of the storage modulus (for a trimer system) and total computational time in function of the Brownian dynamics population. Mean relative errors are calculated as the average of 17 points distributed homogenously in a reduced frequency range going from 10 -2 to 10 2 . Relative error is calculated as follows: log G simul. (ω R ) -

Fig. 7

 7 Fig. 7 Predicted complex modulus responses of different (threebead)-(two-rod) systems: freely jointed, non-freely jointed with natural straight configuration (K b = 0.1 and K b = 10) and nonfreely-jointed with natural bent configuration (K b = 10 and θ max = 30 • ). BD parameters: n = 2, c = 1, a = 1, ζ = 1, k B T = 0.1. λ rod is the rotational diffusion time of an equivalent multibead rigid-rod system of equal length

Fig. 8

 8 Fig. 8 Predicted complex modulus responses of a non-freely jointed (three-bead)-(two-rod) system with a natural bent configuration (maximal misalignment angle θ max = 30 • ) for different values of the bending rigidity constant K b . Freely jointed system (K b = 0) is presented as reference. BD parameters: n = 2, c = 1, a = 1, ζ = 1, k B T = 0.1. λ rod is the rotational diffusion time of an equivalent multi-bead rigid-rod system of equal length

Fig. 9

 9 Fig. 9Predicted storage modulus of a non-freely jointed multibead-rod system with a natural bent configuration (maximal misalignment angle θ max = 30 • ) for different number of rods n. BD parameters are:c = 1, a = 1, ζ = 1, K b = 1, k B T = 0.1.Storage modulus is scaled with the value of storage modulus at the unit reduced frequency. λ rod is the rotational diffusion time of an equivalent multi-bead rigid-rod system of equal length. Vertical pointed line represents the transition between the thermalinduced to the flow-induced regime

  , where first sub-index refers to the bead and second sub-index refers to the sub-section. Mathematically, those bending forces write: to the three previous equations we have:F φ p, p = -K b a 2 Iu p ⊗ u p u p+1 (37) u p+1 ⊗ u p+1 u p (39)

Table 1

 1 Internal angles between consecutives rods θ 0 i-j = θ 0 i -θ 0 j of the multi-bead-rod systems with natural bent configurations presented in Figs. 9 and 10

		θ 0 2-1	θ 0 3-2	θ 0 4-3	θ 0 5-4	θ 0 6-5	θ 0 7-6	θ 0 8-7	θ 0 9-8	θ 0 10-9
	3 rods	18.9	24.3	-	-	-	-	-	-	-
	5 rods	-22.4	24.8	7.9	-24.1	-	-	-	-	-
	10 rods	-13.3	2.8	27.4	27.9	-20.5	28.2	27.4	-0.9	18.0
	relaxation processes at intermediate frequencies are
	activated in function of the number of rods, leading
	invariably to different curves of the storage modulus.


Pierre Gaillard
email: pierre.gaillard@u-bourgogne.fr

Mickaël Gastineau
email: gastineau@imcce.fr

The thirteenth Peregrine breather and its deformations depending on twenty four parameters solutions to the NLS equation

Keywords: 35Q55, 37K10 NLS equation, Peregrine breather, rogue waves

We go on with the study of the solutions to the focusing one dimensional nonlinear Schrödinger equation (NLS). We construct here the thirteen's Peregrine breather (P13 breather) with its twenty four real parameters, creating deformation solutions to the NLS equation. New families of quasi-rational solutions to the NLS equation in terms of explicit ratios of polynomials of degree 182 in x and t multiplied by an exponential depending on t are obtained. We present characteristic patterns of the modulus of these solutions in the (x; t) plane, in function of the different parameters.

Introduction

From the first quasi-rational solution to NLS equation constructed in 1983 by Peregrine [START_REF] Peregrine | Water waves, nonlinear Schrödinger equations and their solutions[END_REF], a lot of works have carried out. Akhmediev, Eleonskii and Kulagin obtained the first higher order analogue of the Peregrine breather [START_REF] Akhmediev | Generation of periodic trains of picosecond pulses in an optical fiber : exact solutions[END_REF][START_REF] Akhmediev | Exact first order solutions of the nonlinear Schrödinger equation[END_REF] in 1986 and construted also other families of higher order 3 and 4 were constructed in a series of articles [START_REF] Akhmediev | Rogue waves and rational solutions of nonlinear Schrödinger equation[END_REF][START_REF] Akhmediev | Rogue waves, rational solutions, the patterns of their zeros and integral relations[END_REF], using Darboux transformations. Since the years 2010, many works were published using various methods, in particular in 2013, it was found in [START_REF] Gaillard | Families of quasi-rational solutions of the NLS equation and multi-rogue waves[END_REF] solution in terms of wronskians and in [START_REF] Gaillard | Degenerate determinant representation of solution of the NLS equation, higher Peregrine breathers and multi-rogue waves[END_REF], solutions expressed in terms of determinants of order 2N depending on 2N -2 real parameters. A new representation has been found as a ratio of a determinant of order N + 1 by another one of order N by Ling and Zhao in [START_REF] Ling | Trajectory Characters of rogue waves[END_REF]. Very recently in 2014, another approach have been given in [START_REF] Gelash | Superregulier solitonic solutions : a novel scenario for the nonlinear stage of modulation instability[END_REF] using a dressing method where the solutions are expressed as the quotient of a determinant of order N + 1 by another one of order N .

We construct here new solutions to the focusing one dimensional nonlinear Schrödinger equation which appear as deformations of the (analogue) Peregrine breather of order 13 with 24 real parameters. These solutions are completely expressed as a ratio of two polynomials of degree 182 in x and t by an exponential depending on t. We cannot give the solutions in terms of polynomials of x and t in this text because of their lengths; only plots in the (x; t) plane of the modulus of the solutions to analyze the evolution of the solutions in function of the different parameters are presented in details here.

Solutions to NLS equation as a ratio of two determinants

We recall the following result which have been proved in [START_REF] Gaillard | Wronskian representation of solutions of the NLS equation and higher Peregrine breathers[END_REF][START_REF] Gaillard | Multi-parametric deformations of the Peregrine breather of order N solutions to the NLS equation and multi-rogue waves[END_REF] :

Theorem 2.1 The function v defined by v(x, t) = det((n jk) j,k∈[1,2N ] ) det((d jk) j,k∈[1,2N ] ) e (2it-iϕ)
is a quasi-rational solution to the NLS equation

iv t + v xx + 2|v| 2 v = 0,
where

n j1 = f j,1 (x, t, 0), n jk = ∂ 2k-2 f j,1 ∂ǫ 2k-2 (x, t, 0), n jN +1 = f j,N +1 (x, t, 0), n jN +k = ∂ 2k-2 f j,N +1 ∂ǫ 2k-2 (x, t, 0), d j1 = g j,1 (x, t, 0), d jk = ∂ 2k-2 g j,1 ∂ǫ 2k-2 (x, t, 0), d jN +1 = g j,N +1 (x, t, 0), d jN +k = ∂ 2k-2 g j,N +1 ∂ǫ 2k-2 (x, t, 0), 2 ≤ k ≤ N, 1 ≤ j ≤ 2N
The functions f and g are defined for 1 ≤ k ≤ N by :

f 4j+1,k = γ 4j-1 k sin A k , f 4j+2,k = γ 4j k cos A k , f 4j+3,k = -γ 4j+1 k sin A k , f 4j+4,k = -γ 4j+2 k cos A k , f 4j+1,N +k = γ 2N -4j-2 k cos A N +k , f 4j+2,N +k = -γ 2N -4j-3 k sin A N +k , f 4j+3,N +k = -γ 2N -4j-4 k cos A N +k , f 4j+4,k = γ 2N -4j-5 k sin A N +k , g 4j+1,k = γ 4j-1 k sin B k , g 4j+2,k = γ 4j k cos B k , g 4j+3,k = -γ 4j+1 k sin B k , g 4j+4,k = -γ 4j+2 k cos B k , g 4j+1,N +k = γ 2N -4j-2 k cos B N +k , g 4j+2,N +k = -γ 2N -4j-3 k sin B N +k , g 4j+3,N +k = -γ 2N -4j-4 k cos B N +k , g 4j+4,N +k = γ 2N -4j-5 k sin B N +k , (1)
The arguments A ν and B ν of these functions are given for 1 ≤ ν ≤ 2N by

A ν = κ ν x/2 + iδ ν t -ix 3,ν /2 -ie ν /2, B ν = κ ν x/2 + iδ ν t -ix 1,ν /2 -ie ν /2. The terms κ ν , δ ν , γ ν are defined by 1 ≤ ν ≤ 2N κ j = 2 1 -λ 2 j , δ j = κ j λ j , γ j = 1 -λ j 1 + λ j , κ N +j = κ j , δ N +j = -δ j , γ N +j = 1/γ j , 1 ≤ j ≤ N, (2)
where λ j are given for 1 ≤ j ≤ N by :

λ j = 1 -2j 2 ǫ 2 , λ N +j = -λ j . (3)
The terms x r,ν (r = 3, 1) are defined for 1 ≤ ν ≤ 2N by :

x r,ν = (r -1) ln γ ν -i γ ν + i . (4)
The parameters e ν are given by

e j = i N -1 k=1 a j ǫ 2k+1 j 2k+1 - N -1 k=1 b j ǫ 2k+1 j 2k+1 , e N +j = i N -1 k=1 a j ǫ 2k+1 j 2k+1 + N -1 k=1 b j ǫ 2k+1 j 2k+1 , 1 ≤ j ≤ N, (5)
3 Solutions of order 13 with twenty four parameters

We construct here deformations of the Peregrine breather P 13 of order 13 depending on 24 parameters. We cannot give the analytic expression of the solution to NLS equation of order 13 with twenty four parameters because of their lengths. For simplicity, we denote

d 3 := det((n jk) j,k∈[1,2N ] ), d 1 := det((d jk) j,k∈[1,2N ] ).
In this study, one reaches the current limits of capacity of calculation; all the explicit expressions were calculated for the parameters a i and b i for indices i from 2 to 12. The only expression which was not calculated is that corresponding to i equal to 1 : the calculation of the expression needing 32 core stopped at the end of 6 months. We construct figures to show deformations of the Peregrine breather of order 13. We get different types of symmetries in the plots in the (x, t) plane. We give some examples of this fact in the following discussion.

Peregrine breather of order 13

If we choose a i = b i = 0 for 1 ≤ i ≤ 10, we obtain the classical thirteenth Peregrine breather (here, the peak going until 27 has been truncated)

Variation of parameters

With other choices of parameters, we obtain all types of configurations : triangles and multiple concentric rings configurations with a maximum of 91 peaks. The configuration of the solutions of the NLS equation in the (x; t) plane being the same for a j or b j , we restrict ourself to present the solutions only in the case of the parameters a j , for 1 ≤ j ≤ 12. We present figures of deformations with only one parameter non equal to zero, all other parameters are fixed equal to zero. For more accuracy, one presents in the following only the views of top.

Conclusion

We have given here an explicit construction of multi-parametric solutions to the NLS equation of order 13 with 24 real parameters. We cannot give the explicit representation in terms of polynomials of degree 182 in x and t because of his length. The properties about the shape of the breather in the (x, t) coordinates, the maximum of amplitude equal to 2N + 1 and the degree of polynomials in x and t here equal to N (N + 1) are checked. We obtained different patterns in the (x; t) plane by different choices of these parameters. So we obtain a classification of the rogue waves at order 13.

We obtain two types of patterns : the triangles and the concentric rings for the same index i for a i or b i non equals to 0.

In the cases a 1 = 0 or b 1 = 0 we obtain triangles with a maximum of 91 peaks; for a 2 = 0 or b 2 = 0 , we have rings with 5 or 10 peaks. For a 3 = 0 or b 3 = 0, we obtain rings with 7 or 14 peaks with in the center P 2 . For a 4 = 0 or b 4 = 0, we have 5 rings with 9, 18, 18, 19, 18, peaks with in the center the Peregrine This conjecture was also given for this time by the authors of this paper in 2015.

It would be relevant to prove the preceding conjectures.

Figure 1 :

 1 Figure 1: Solution to NLS, N=13, all parameters equal to 0, Peregrine breather P 13 .

Figure 2 :

 2 Figure 2: Solution to NLS, N=13, a 1 = 10 3 : triangle with 91 peaks, sight from top.

Figure 3 :

 3 Figure 3: Solution to NLS, N=13, a 2 = 10 4 : 11 rings with 5, 10, 10, 10, 10, 10, 10, 5, 5, 10, 5 peaks and one peak in the center, sight from top.

Figure 4 :

 4 Figure 4: Solution to NLS, N=13, a 3 = 10 6 : 10 rings with 7, 14, 7, 7, 14, 14, 7, 7, 7, 7, peaks, sight from top.

Figure 5 :

 5 Figure 5: Solution to NLS, N=13, a 4 = 10 8 : 5 rings with 9, 18, 18, 18, 18, peaks, in the center P 4 , sight from top.

Figure 6 :

 6 Figure 6: Solution to NLS, N=13, a 5 = 10 9 : 7 rings of 11, 11, 22, 11, 11, 11, 11 peaks one peak in he center P 2 , sight from top.

Figure 7 :

 7 Figure 7: Solution to NLS, N=13, a 6 = 10 12 : 7 rings with 13 peaks without peak in the center, sight of top.

Figure 8 :

 8 Figure 8: Solution to NLS, N=13, a 7 = 10 14 : 6 rings with 15 peaks and in the center one peak, sight from top.

Figure 9 :

 9 Figure 9: Solution to NLS, N=13, a 8 = 10 15 : 5 rings with 17 peaks and in the center the Peregrine breather of order 3, sight from top.

Figure 10 :

 10 Figure 10: Solution to NLS, N=13, a 9 = 10 19 : 4 rings with 19 peaks and in the center the Peregrine breather of order 5, sight from top.

Figure 11 :

 11 Figure 11: Solution to NLS, N=13, a 10 = 10 20 : three rings with 21 peaks and in the center the Peregrine breather of order 7, sight from top.

Figure 12 :

 12 Figure 12: Solution to NLS, N=13, a 11 = 10 20 : two rings with 23 peaks and in the center the Peregrine breather of order 9, sight from top.

Figure 13 :

 13 Figure 13: Solution to NLS, N=13, a 12 = 10 20 : one ring with 25 peaks and in the center the Peregrine breather of order 11, sight from top.

P 4 . 2 ]

 42 For a 5 = 0 or b 5 = 0 , we have 7 rings of11, 11, 22, 11, 11, 11, 11 peaks with in the center P 3 . For a 6 = 0 or b 6 = 0 , we have 7 rings with 13 peaks on each of them without peak in the center. For a 7 = 0 or b 7 = 0 , we have 6 rings with 15 peaks on each of them and in the center the Peregrine breather of order 1 (one peak). For a 8 = 0 or b 8 = 0, we have 5 rings with 17 peaks on each of them and in the center the Peregrine breather of order 3. For a 9 = 0 or b 9 = 0, we have 4 rings with 19 peaks and in the center the Peregrine breather of order 5. For a 10 = 0 or b 10 = 0, we have 3 rings with 21 peaks and in the center the Peregrine breather of order 7. For a 11 = 0 or b 11 = 0, we have only two rings with 23 peaks and in the center the Peregrine breather of order 9. At least, for a 12 = 0 or b 12 = 0, we have only one ring with 25 peaks and in the center the Peregrine breather of order 11. We verify in this study the following conjectures : at order N , for a 1 = 0 or b 1 = 0, the modulus of the solution to the NLS equation presents the configuration of a triangle with N (N + 1)/2 peaks; this has been formulated already by different authors, in particular Akhmediev and Matveev. at order N , in the case1 ≤ i ≤ [ N2], for a N -i = 0 or b N -i = 0, the modulus of the solution to he NLS equation presents i concentric rings with 2N -2i + 1 peaks and in the center the P N -2i breather; this conjecture was given for this time by the authors of this paper in 2015.at order N , in the case [ N < i ≤ N -2, for a N -i = 0 or b N -i = 0,the modulus of the solution to the NLS equation presents n k rings of k(2N -2i + 1) peaks, for 1 ≤ k ≤ r, rn r (2N -2i + 1) < N (N + 1)/2) with in the center the P N -2i breather, verifying r n k =1n k k(2N -2i + 1) = 2iN -2i 2 + i.

Table 1 :

 1 On the other hand, one could generate the corresponding images in this case i equal to 1 without calculating the determinants explicitly. Calculations are the more long as indices i are close to 1. Precisely for a 12 , calculations last approximately 1 hour on 20 cores; for a 2 , it is approximately about 3 months on 32 cores.The number of terms of the polynomials of the numerator d 3 and denominator d 1 of the solutions are shown in the table below (Table1) when other a i and b i are set to 0. Number of terms for the polynomials d 3 and d 1 of the solutions to the NLS equation in the case N = 13.

	a i	d 3	d 1
	2 1051741 531675
	3	559480 282841
	4	348403 176084
	5	245676 124198
	6	183131	92581
	7	143053	72324
	8	115599	58448
	9	94974	48014
	10	76243	38529
	11	55015	27787
	12	30003	15154