S Fréour

O Gloaguen

M François

R Guillén

S Freour

D Gloaguen

M Fran

R Guillen

Influence d'une microstructure biphasée sur les CERet l'analyse des contraintes par DRX

Les Constantes Élastiques Radiocristallographiques (CER) dépendent de l'état de biphasage d'un système, c'est-à-dire de la nature et de la proportion de chacune des phases présentes dans le matériau. Or, les CER jouent un rôle primordial dans les méthodes de détermination des contraintes pseudomacroscopiques par Diffraction des Rayons X (ORX). Des modèles de calcul micromécaniques ont été développés, afin d'effectuer le calcul des CER au sein de divers matériaux biphasés. Ceci à rendu possible l'analyse quantitative des effets exercés par le biphasage sur le résultat d'une analyse de contrainte par ORX.

INTRODUCTION

La détermination des contraintes résiduelles à l'aide de la diffraction des rayons X est devenue une technique classique en sciences des matériaux. Cette méthode résulte de l'application des lois de la mécanique des milieux continus à la jauge de déformation que constitue la distance interréticulaire d'une famille de plans {hkl} en position de diffraction. Seuls les états de déformations sont directement accessibles au travers de cette méthode. Cependant, l'équation dite " en sin 2 w , fournit une relation entre les états de déformations et de contraintes pseudomacroscopiques existant au sein de la phase qui participe au phénomène de diffraction. Cette relation en sin 2 w fait intervenir des coefficients de proportionnalité entre déformations et contraintes [START_REF] Noyan | Residual stress measurement by diffraction and interpretation[END_REF]. Ces coefficients, notés 1/2 S 2 et S 1 sont communément appelés Constantes d'Élasticité Radiocristallographiques ou CER.

Si on néglige les effets de texture, la valeur des CER dépend du choix de la famille de plans diffractants, de la nature de chaque phase du matériau, par l'intermédiaire de leurs propriétés mécaniques élastiques, et de la fraction volumique des phases en présence. Les méthodes permettant d'accé• der aux valeurs des CER dans les matériaux monophasées sont bien connues et relèvent soit de méthodes expérimentales, soit de modèles numériques de transition d'échelle [START_REF] Voigt | Lehrbuch der Kristallphysik[END_REF][START_REF] Voigt | [END_REF](4)(5)(6)(7)(8).

Un certain nombre de précautions sont nécessaires lors de l'étude d'un matériau biphasé. Connaissant les fractions volumiques et le comportement mécanique du monocristal de chacune des phases, il est possible de calculer les CER nécessaires à l'expérimentateur. En effet, à notre connaissance, aucune démarche expériment,Jie n'a été proposée à ce jour pour la détermination des CER au sein de ces matériaux. Cependant, on suppose généralement que les CER d'une phase donnée, prise dans un matériau biphasé, sont identiques aux CER de la phase pure. Cette hypothèse conduit tout naturellement à introduire les CER des composés monophasés dans la relation" en sin2w "• afin de déterminer les contraintes pseudomacroscopiques existant au sein d'un échantillon biphasé (9)(10)[START_REF] Ceretti | Colloque de métallurgie[END_REF]. Cette méthode néglige implicitement l'influence de la seconde phase sur le comportement élastique du volume diffractant du matériau. t.:objectif de cette étude consiste précisément à évaluer de façon quantitative l'influence du biphasage sur les valeurs des CER et, par voie de conséquence, l'impact du biphasage sur les contraintes mesurées par ORX.

Experimenters require proportionality constants, called X-Ray Elasticity Constants (XEC), in order to determine the residual stress state of a material from X-Ray strain measurements.

Apart from effects due to crystalfites morphology or orientation (crystalline texture), XEC depend on several parameters such as the diffracting planes chosen, the nature of each phase through their elastic mechanical properties, and the volume fraction of the different phases constituting the multiphase polycrystal. Methods leading to XEC values determination are well known in the case of single-phase materials. Experimental and scale transition simulation schemes lead to concordant numerical results in this particular case. The study of two-phases materials requires more sophisticated means. To our knowledge, no procedure has been yet proposed in order to determine the XEC of two-phases materials. Knowing the volume fraction and the mechanical behaviour of the single crystal of each phase, one needs to calculate the XEC necessary for his experiments. In fact, it is usually assumed that XEC remain the same in two-phases materials as in single-phase ones. This hypothesis yields straightforwardly to introduce the classical, available in the literature, values of singlephase XEC in the "sin2'Jf" relation, in order to process the experimental pseudo macro strain data obtained through measures in two-phases samples, e. g. (9)(10)[START_REF] Ceretti | Colloque de métallurgie[END_REF]. This methods implicitly neglects the effects of the second phase on the mechanical elastic behaviour of the diffracting volume. The aim of this work consists in the evaluation of the two-phases effect on the XEC values and its consequences on the stresses determined by XRD analysis in cubic crystallographic structures.

A one site self consistent calculation scheme has been developed in order to reproduce the diffracting volume behaviour of two-phases samples. This model is based on the treatment of the elastic interactions of spherical anisotropic crystallites embedded in an infinitely extended elastically isotropic continuum (matrix), in accordance with the formalism described by Kroner, Eshelby and Morris (5, 6, 7, 8, 15}. This model has been used to simulate the cases of several two-phases material : duplex steels, aluminium -[3 silicon carbide Metal Matrix Composites (MMC), aluminium -diamond carbon composites, copper -Feu. alloys. The different systems have been especially chosen in order to separate the effects of the following parameters on XEC values: single crystal anisotropy and the ratio of the elastic moduli of the two considered phases.

Simulations results (cf. tables 11-V) show that the XEC of a given phase generally vary with the volume fraction of the phase in a given system, and with the nature of the second phase. In physical terms, this discrepancy between XEC of a phase, calculated in a single-phase material and in a two-phases sample can be explained by the elastic interactions existing between the polycrystalfine matrix and any crystallite. In fact, the grains of a diffracting volume often have a and [3 grains as first neighbours simultaneously, as shown on figure 1. The presence of these first neighbours [3 grains generates interactions which affect a diffracting volume elastic mechanical behaviour.

In the cases of the aluminium -[3 SiC MMC and Feu.copper alloys, XEC calculated are strongly dependent on each phase volume fraction. This property, characteristic to such two-phases polycrystals, results from the strong mechanical elastic heterogeneities existing between the phases. On the opposite, if the macroscopic behaviour of the two phases is similar, such as in the case of duplex steels, the XEC do not vary significantly with the volume fraction of each phase.

Aluminium -diamond like carbon systems, illustrates another property. In fact, diamond XEC remains almost constant, in spite of the large mechanical discrepancies existing between the two phases. This property has been attributed to diamond single crystal isotropy. In consequence, a phase with elastically isotropic single crystal keeps its XEC constant, whatever the two-

phases state of the polycrystal in which it is embedded.

On the opposite, results obtained on copper underline that multiphases effects on XEC arise in the case of strongly anisotropic diffracting phases.

As mentioned before, the usual assumption consists in introducing single phase XEC values in order to process numerically the strains measured in two-phases polycrystals. The results given by our simulations demonstrate that this approach leads to an approximation of the polycrystal/ine sample mechanical effective behaviour. In consequence, this assumption generates a systematic error o on the determined stresses.

As part of a residual stresses analysis through XRD, the relative error on stresses values due to experimental and data processes general/y remains between 5 and 15% for the considered phases. Figure 2 shows that the systematic relative error ô resulting from the usual assumption is often lower than 5 % but may reach 20 % in particularly unfavourable cases.

INFLUENCE DE L:ÉTAT DE BIPHASAGE D'UN SYSTÈME SUR SES CER -MODÈLE AUTOCOHÉRENT ÉLASTIQUE À 1 SITE

Un modèle autocohérent à un site a été développé afin de simuler le comportement du volume diffractant de matériaux biphasés (a+~). Ce modèle est fondé sur le traitement de l'interaction élastique de cristaux sphériques anisotropes plongés dans une matrice infinie, homogène et élastiquement isotrope, en accord avec les formalismes décrits par However, it is necessary to take into account effects due to two-phases interactions in order to minimize the deviation between the residual stresses determined through XRD experiments and true stresses in the sample.

Krôner, Eshelby et Morris (5,6,8,[START_REF] Morris | [END_REF]. Ce modèle a permis de déterminer les valeurs de CER dans un grand nombre de biphasés aux caractéristiques différentes. On a ainsi traité le cas des aciers austéno-ferritiques, de même que celui de composites à matrice métallique constitués par exemple d'aluminium et de carbure de silicium renees de comportement mécanique élastique des deux phases sont à l'origine de la faible dépendance de leur GER vis-à-vis de la composition de l'acier. De façon générale, pour un biphasé (a+~). on peut démontrer que les GER de la phase a s'écartent d'autant plus de la valeur obtenue pour la phase a pure, lorsque l'état de biphasage du système augmente, c'est-à-dire lorsque la fraction volumique de la phase a s'éloigne de 1, et lorsque la phase ~ est caractérisée par un comportement mécanique élastique sensiblement différent de celui de la phase a.

Les résultats obtenus pour le biphasé aluminium -diamant suivent cette règle, du moins pour ce qui concerne les GER de l'aluminium. Le comportement du diamant semble par contre s'écarter de ce modèle. Ses GER restent en effet constantes quelle que soit la composition du biphasé, et ce en dépit des considérables différences de comportement élastique existant entre l'aluminium et le diamant. Cependant, on peut montrer que ce phénomène est l'expression d'une propriété propre au diamant que l'on ne retrouve dans aucune des autres phases étudiées (aluminium, cuivre, ferrite, austénite et SiC). ~expression de S a été donnée par (3). On peut montrer que le coefficient ARx d'anisotropie aux rayons X de la phase a est égal à 1 lorsque l'équation [4] est vérifiée: [4] Ceci est caractéristique d'un monocristal isotrope, ce qui est le cas du diamant (A= 1 ,01). Dans ce cas, fe coefficientS tend vers O. Il s'ensuit que les GER d'une telle phase sont presque constantes, pour une fraction volumique donnée. Mais, on remarque également que l'effet du biphasage s'annule pour une valeur r ::: r 0 proche de 0,2. En ce point, on a évidemment égalité entre les GER du monophasé et du biphasé, pour la phase a.

Or, l'expression [START_REF] Noyan | Residual stress measurement by diffraction and interpretation[END_REF] de la GER peut encore s'écrire : Dans le cas d'une phase ayant un monocristal presque isotrope du point de vue élastique, cette expression se simpfi• fie pour donner : [6] avec e -7 O. Les GER d'une telle phase sont donc, en première approximation, indépendantes de l'état du biphasage du matériau. Ceci explique le comportement particulier du diamant, mais on peut s'attendre à observer un phénomène identique pour le tungstène (A = 0,98) et, dans une moindre mesure, pour le TiC (A == 0,90) et le molybdène (A = 0,91 ).

_!_gz (r)=!s2 (ro )+[3ro -3r]s 2 2 [5]

CONSÉQUENCES DES EFFETS DU BIPHASAGE SUR LA MÉTHODE DE DÉTERMINATION DES CONTRAINTES PAR ORX

~hypothèse usuelle conduit à introduire dans fa relation en " sin 2 ljl , les valeurs de GER des phases pures, pour exploiter les résultats de mesures effectuées dans des biphasés. Les résultats donnés précédemment montrent que cette approche ne donne qu'une approximation du comportement mécanique effectif de l'échantillon analysé. En conséquence, l'usage de cette hypothèse génère une erreur systématique ô sur la détermination des contraintes.

Pour une famille de plans diffractants {hkl} donnée, dans le cas d'un matériau contenant une fraction volumique f de la phase analysée, cette erreur systématique peut être exprimée comme suit :

.!.s 2 (r,r)-.!.s 2 (r,r=t)

ô=~--------~2 ________ _ [7] ô (f = O) correspond au cas limite de l'inclusion d'Eshelby, qui fixe la valeur maximale de l'écart à laquelle on peut s'attendre. . ,c::. ~ , . . , c::. "' c::. ' :le::. roc::. ' \c::. q, C: : . OJc: : . c; : , C: : .

1:>• 1:>• 1:>• 1:>• 1:>• 1:>• ~:>• 1:>• 1:>• 1:>• ,.

3*r

Erreur systématique, sur la contrainte mesurée dans c; : , c: : . ,c::. ~ , . . , c::. ~ ' :le::. roc::. ' \c::. q,c::. Ojc::. <: : >c: : . <::><::> ,c::. ~ , . . , c::. "' c::. ' :le::. ro<: : > '\<::> q, C: : . OJc: : . c; : , C: : .

<::)• 1:>• <::>• ~:>• 1:>• 1:>• 1:>• 1:>• <::)• <::)• , , 3

CONCLUSION

La valeur des CER joue un rôle très important dans les méthodes de détermination des contraintes résiduelles par la diffraction des rayons X. Un modèle autocohérent élastique fondé sur le formalisme de Krôner et Eshelby a été développé afin de simuler le comportement mécanique du volume diffractant du matériau. Ceci a permis de mettre en évidence l'influence de l'état de biphasage sur les valeurs de CER. On a pu montrer que cette influence pouvait être considérée comme une source d'erreur supplémentaire du point de vue de la détermination des contraintes. La prise en compte des effets dus au biphasage participe donc à la démarche qui tend à identifier les autres sources d'incertitudes affectant les résultats des contraintes déterminées par les techniques de ORX. bibliographie

Figure 1 a

 1 Figure 1 a) Volume diffractant uniquement connexe avec des grains de la même phase (égalité des CER du monophasé et du biphasé) ; b) volume diffractant de a en interaction avec des grains de !) premiers voisins.

Figure 1 -

 1 Figure 1 -a) Diffracting vofume constituted by a grains on/y connected to other a grains (case of equality of the XEC in the single phase and two phases material); b) diffracting volume constituted of a grains interacting with a and !) first neighbours grains.

!

  .:application de cette relation aux biphasés ferrite • cuivre, aluminium -SiC et austéniteferrite a conduit aux tracés des courbes reproduites sur la figure 2. Les incertitudes expérimentales sur des contraintes résiduelles déterminées par la DRX sont généralement comprises entre 5 et 15 %. !.:erreur systématique o due à l'emploi de l'hypothèse •Erreur systématique, sur la contrainte mesurée dans ô l'Aluminium, pour un biphasé Al-SiC 0,04 ; ----------------. f = 0,25 • f = 0,00 0,00 -l--,..--.--..---.--.:::~..c::::~.....-......--l ~ ~ ~ ~ ~ ~ ~ ~ * ~ ~ ~:>• ~:>• 1:>• ~:>• 1:>• 1:>• ~:>• 1:>• 1:>• 1:>• ,. 3*r Erreur systématique. sur la contrainte mesurée dans la ferrite, pour un biphasé austéniteferrite 0 , 0 2 . --------------, +f=0,75 •f=0,50 ... f = 0,25 • f = 0,00 0,01 o.oo -l-.....--.,...-~::::s~~~::JI c; : , C: :

  0,75 • f = 0,50 ... f = 0,25 • f = 0,00 0,00 -f---.---,---,.....:_,.....:::::::::$jl~~:::::~

  Erreur systématique, sur la contrainte mesurée dans le 0 SiC, pour un biphasé Al-SiC 0,20 . . , ---------------.... + f = 0,75 • f = 0,50 l>tématique, sur la contrainte mesurée dans l'austénite, p<!ur un biphasé austénite-ferrite 0,02 . . . . ----------------. . + f = 0,75 • f = 0,50 ... f= 0,25 0,01 0,00 t-...,.-...,.--,.--.,.-~~~~:::::::~

1 :

 1 >• 1:>• 1:>• <::)• <::)• <::)• ~:>• <::)• <::)• 1:>• ,, 3*r Erreur systématique, sur la contrainte mesurée dans ô le cuivre, pour un biphasé cuivreferrite 0,08 -r-----------------. 0,75 • f = 0,50 ... f = 0,25 0,00 -l--.---.----r-.,..---,--.....:;::!!!lr.4J;:e::::;:.......,....-ll c;:,<::> ,c::. "v<::> , . . , c::. ~ ' :le::. ro<: : > ' \c::. q, c::. ~ c;:,<::>

Figure 2 -

 2 Figure 2 -Tracé de l'erreur systématique ô due au biphasage sur la contrainte mesurée dans différentes phases de matériaux biphasés. f indique la fraction volumique de la phase analysée.

Figure 2 -

 2 Figure 2 -Calculated relative systematic error ô due to multiphase effects, on the stresses determined in severa/ cubic structure phases of two-phases polycrystals. t denotes the volume traction of the analysed phase.

TABLE 1 -

 1 ~• Les différents systèmes ont été spécialement choisis afin d'identifier l'influence des paramètres suivants sur la valeur des CER : l'anisotropie élastique du monocristal et le rapport des modules élastiques des deux phases en présence. TABLEAU 1-Constantes d'élasticité et coefficient d'anisotropie des monocristaux[START_REF] Noyan | Residual stress measurement by diffraction and interpretation[END_REF]10,[START_REF] Hosford | The mechanics of crystals and textured polycrystals[END_REF] 14). Single crystals elastic constants and anisotropy ratio[START_REF] Noyan | Residual stress measurement by diffraction and interpretation[END_REF]10,[START_REF] Hosford | The mechanics of crystals and textured polycrystals[END_REF] 14).

		Austénite	Ferrite	Aluminium	SiC~	C-diamant
	c11 [TPa]	0,1975	0,2374	0,1082	0,3523	1,0200
	c12 [TPa]	0,1245	0,1347	0,0613	0,1404	0,2500
	c44 [TPa]	0,1220	0,1164	0,0285	0,2329	0,3900
	A	3,34	2,27	1,23	2,20	1,01

TABLEAU Il -CER déterminées dans un acier austéno-ferritique, en fonction des proportions respectives de chacune des phases.

TABLE Il -

 Il XEC simulated in duplex steels, as a function of the proportion of each phase.

							Cuivre
							0,1684
							0,1214
							0,0755
							3,20
	Fraction volumique de la phase diffractant	100%	75%	50%	25%	0%
	Austénite	1/2 S 2 (h00) [1Q•6fMPa]	8,77	8,74	8,70	8,66	8,61
		S (10-6 /MPa]	3,68	3,63	3,57	3,52	3,45
	Ferrite	1/2 S 2 (hOO) [1Q•6fMPa]	7,31	7,33	7,36	7,38	7,40
		s [1 o-6 /MPa]	2,29	2,33	2,36	2,40	2,43
		TABLEAU Ill-CER déterminées dans un CMM aluminium-carbure de silicium~ en fonction
		des proportions respectives de chacune des phases.

TABLE Ill -

 Ill XEC simulated in aluminum -~ silicon carbide MMC, as a function of the proportion of each phase.

	Fraction volumique de la phase diffractant	100%	75%	""	1	0%
	Aluminium	1/2 S 2 (h00) [10-6/MPa]	19,99	19,77	19,54	19,37	19,26
		S [1Q•6fMPa]	1,71	1,35	0,98	0,68	0,50
			3,63	3,80	4,00	4,18	4,32
		S [1Q•6fMPa]	1,15	1,39	1,66	1,91	2,08

SiC~ 1/2 S 2 (h00) [1Q-6/MPa]

  TABLEAU IV -GER déterminées dans un biphasé ferrite -cuivre, en fonction des proportions respectives de chacune des phases.

TABLE IV -

 IV XEC simulated in Fe a copper two-phases alloy, as a function of the proportion of each phase.

	Fraction volumique de la phase diffractant	100%	75%	50%	25%	0%
	Ferrite	1/2 S 2 (h00) (10-6 /MPa]	7,31	7,41	7,53	7,65	7,77
		S [1Q•6fMPa]	2,29	2,44	2,61	2,77	2,95
	Cuivre	1/2 S 2 (h00) [10•6fMPa]	13,71	13,43	13,17	12,92	12,70
		S [1Q•6fMPa]	5,46	5,05	4,66	4,30	3,96
		TABLEAU V GER déterminées dans un biphasé aluminium -carbone diamant, en fonction
		des proportions respectives de chacune des phases.

TABLE V -

 V XEC simulated in the case of aluminum -diamond carbon two-phases alloy, as a function of the proportion of each phase. Pour que les GER de la phase a soient identiques dans la phase pure et dans le biphasé, il faudrait que le volume diffractant de a soit uniquement connexe avec des grains de phase a, comme sur la figure ta. Cependant, dans la réalité, les grains du volume diffractant de a peuvent avoir pour premiers voisins des grains de p, comme indiqué sur la figure tb. La présence de ces

	Grains de phase a pa1iicipant à la diffraction
	a)	b)
	Fraction volumique de la phase diffractant	100%	75%	50%	25%	0%
	Ferrite	1/2 S 2 (h00) [10•6fMPa]	19,99	19,70	19,38	19,18	19,10
		S [10•6fMPa]	1,71	1,23	0,70	0,37	0,23
	Carbone diamant	1/2 S 2 (h00) [1Q•6fMPa]	1,29	1,29	1,30	1,30	1,30
		S [1Q•6fMPa]	0,01	0,01	0,01	0,01	0,01
	Le comportement élastique des matrices biphasées a été	EXISTENCE D'INTERACTIONS
	simulé grâce à un calcul préliminaire sur 4 000 cristallites.	ENTRE LES PHASES
	Les constantes élastiques micromécaniques employées
	pour modéliser les monocristaux de chacune des phases
	impliquées sont données dans le tableau 1. Le comporte-
	ment du volume diffractant a ensuite été modélisé, pour cha-
	cune des phases et pour différentes compositions du maté-
	riau (a+ p): 0% p, 25% p, 50% p, 75% p, et 100% P (ce
	qui correspond au cas limite de l'inclusion d'Eshelby). À titre
	d'exemple, on a rassemblé certains résultats obtenus pour
	les GER dans les tableaux Il à V.
	On rappelle que dans le cas d'une phase cubique, les GER
	peuvent toujours être exprimées de la façon suivante [15] :
	où le paramètre d'orientation r s'écrit :	grains p modifie en toute logique le comportement élastique du volume diffractant de a.
		[2]	Dans le cas du CMM aluminium -SiC p, les GER calculées
		[3]

avec ARx le coefficient d'anisotropie aux rayons X de la phase.

Ces résultats montrent que le modèle permet de rendre compte des effets exercés par l'état de biphasage d'un système sur les GER d'une phase donnée. Du point de vue physique, l'inégalité entre les GER d'une phase a pure et les GER de la phase a appartenant à un matériau biphasé a+ p s'explique par l'existence d'interactions élastiques entre le volume diffractant de la phase a et l'ensemble de la phase p. Ce phénomène peut être décrit par les figures ta et tb. dépendent fortement de la fraction volumique de chacune des phases. On observe un résultat identique dans le biphasé ferrite -cuivre. Cette propriété s'explique par les fortes hétérogénéités de comportement mécaniques existant, d'une part, entre l'aluminium et le carbure de silicium, et, d'autre part, entre la ferrite et le cuivre. Inversement, dans le cas de l'acier austéno-ferritique, les faibles diffé-

  *r usuelle reste la plupart du temps largement inférieure à 5 %. Cependant, cette approche peut conduire à des écarts importants entre les états de contraintes déterminés et la contrainte réelle existant dans le matériau, en particulier lorsque l'état de biphasage est très prononcé (cf cas de Al+ SiC).


Stéphanie Challita

Fawaz Paraiso

Philippe Merle

A Study of Virtual Machine Placement Optimization in Data Centers

Keywords: VM placement, Data center, Multi-objective optimization, Energy-aware, Traffic-aware, Cloud computing

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

INTRODUCTION

Cloud computing is a technology that enables computing resources, software, or data to deliver ondemand services over the Internet. These resources have become cheaper, more powerful and more ubiquitously available than ever before [START_REF] Mell | The NIST Definition of Cloud Computing[END_REF]. The cloud computing stack consists of three types of cloud service models: Infrastructure, Platform and Software, which are built upon the Hardware layer. Since provisioning Virtual Machines (VMs) is fundamental to provide infrastructure services, one can say that virtualization is the key concept of cloud computing. According to Mike Adams, director of product marketing at VMware1 , "Virtualization software makes it possible to run multiple operating systems and multiple applications on the same server at the same time. It enables businesses to reduce IT costs while increasing the efficiency, utilization and flexibility of their existing computer hardware. " [START_REF] Angeles | Virtualization vs Cloud Computing: What's the Difference? BusinessNewsDaily[END_REF] However, VMs need to be adequately placed to fulfill performance goals, to optimize network flows, and to reduce CPU, storage and energy costs. VM placement optimization processes may be traffic-aware, energy-aware, application-aware, network topology-aware, data-aware, or a combination of these.

In recent years, the problem of allocating VMs to suitable Physical Machines (PMs) has been studied for efficiency and quality purposes [START_REF] Xu | Multi-Objective Virtual Machine Placement in Virtualized Data Center Environments[END_REF], [START_REF] Feller | Energy-Aware Ant Colony Based Workload Placement in Clouds[END_REF], [START_REF] Fang | VMPlanner: Optimizing Virtual Machine Placement and Traffic Flow Routing to Reduce Network Power Costs in Cloud Data Centers[END_REF], [START_REF] Gao | A Multi-Objective Ant Colony System Algorithm for Virtual Machine Placement in Cloud Computing[END_REF], [START_REF] Vu | A Traffic and Power-Aware Algorithm for Virtual Machine Placement in Cloud Data Center[END_REF]. On the provider side, these solutions map VMs to PMs to optimize server efficiency, allowing some servers to hibernate or shut down depending on load conditions. On the consumer side, these solutions maximize Quality of Service (QoS) and Quality of Experience (QoE). In addition to this, such solutions lead to better utilization of resources and less frequent overload situations, leading to cost savings.

Although VMs have shown considerable opportunities to the IT industry, their placement brings many challenges that need to be carefully addressed. In this paper, we present a survey of various approaches studying VM placement, highlighting their key concepts, as well as the state-of-the-art implementations. The remainder of this paper is organized as follows.

In Section 2, we motivate the need of VM placement optimization. Section 3 presents the state-of-the-art that details the four approaches addressing energy, cost and data flow issues in data centers. Section 4 discusses potential comparison metrics between presented solutions. In Section 5, we discuss related surveys that allow us to situate our work. Finally, Section 6 concludes and outlines future work.

As stated before, the motivation behind VM placement optimization can be traffic-aware, energy-aware, application-aware, network topology-aware, dataaware, and multi-objective. In our paper, we are interested in investigating solutions that reduce energy consumption [START_REF] Ajiro | Improving Packing Algorithms for Server Consolidation[END_REF], [START_REF] Verma | pMapper: Power and Migration Cost Aware Application Placement in Virtualized Systems[END_REF], [START_REF] Chaisiri | Optimal Virtual Machine Placement across Multiple Cloud Providers[END_REF], resource costs [START_REF] Xu | Multi-Objective Virtual Machine Placement in Virtualized Data Center Environments[END_REF], [START_REF] Fajjari | A New Virtual Network Static Embedding Strategy within the Clouds Private Backbone Network[END_REF], and properly manage data flow [START_REF] Kanagavelu | Virtual Machine Placement with Two-path Traffic Routing for Reduced Congestion in Data Center Networks[END_REF], [START_REF] Aral | Network-Aware Embedding of Virtual Machine Clusters onto Federated Cloud Infrastructure[END_REF] in data center architectures. Each of these objectives is detailed below.

Energy Management

Enhancing energy efficiency in data centers can be resolved by applying a suitable VM placement algorithm that minimizes the cost of powering at the hardware level. Moreover, turning off unused machines, on the basis of server consolidation2 and energyaware job scheduling, can also constitute a solution for the energy problem. In this context, "Green Data Centers" [START_REF] Basmadjian | Green Data Centers. Large-scale Distributed Systems and Energy Efficiency: A Holistic View[END_REF] are nowadays a must to fight against huge power consumption and bills caused by inappropriate virtualization.

Resource Usage Optimization

In order to maintain the application performance, isolation and security, each VM requires a certain amount of resources, such as CPU, memory and links bandwidth, etc. In order to minimize their cost, these resources should be made available to applications only as needed and not allocated statically based on the peak workload demand [START_REF] Kusic | Power and Performance Management of Virtualized Computing Environments via Lookahead Control[END_REF]. This is known as the "elasticity of the cloud" [START_REF] Herbst | Elasticity in Cloud Computing: What It Is, and What It Is Not[END_REF].

Traffic Engineering

Measuring and optimizing data center traffic is important to maintain the efficiency of applications. A data center, which hosts thousands of devices like servers, switches and routers, needs an accurate planning of the network architecture. One can distinguish several architectures [START_REF] Wang | Rethinking the Data Center Networking: Architecture, Network Protocols, and Resource Sharing[END_REF] such as Fat-tree [START_REF] Al-Fares | A Scalable, Commodity Data Center Network Architecture[END_REF], VL2 [START_REF] Greenberg | Vl2: A Scalable and Flexible Data Center Network[END_REF] and BCube [START_REF] Guo | BCube: A High Performance, Server-centric Network Architecture for Modular Data Centers[END_REF].

APPROACHES

In this section, we review existing methods related to the optimization of VM placement, embedded in the cloud computing domain. Placement algorithms, that collect and study information from deployed applications, can be either static or dynamic. Static algorithms, which mainly do offline calculations, take as an input the information that is formerly collected. After a primary placement, a relocation may not be considered for several months. Static allocation normally indicates initial VM placements that will be actively running in subsequent phases of the system administration. As for dynamic VM placement, it is implemented on shorter timescales, preferably shorter than periods of significant variability of the resource demand [START_REF] Shankar | Virtual Machine Placement in Computing Clouds[END_REF], [START_REF] Abdelsamea | Virtual Machine Consolidation Challenges: A Review[END_REF]. It does online VM placement, including VM migrations. One very important difference between static and dynamic VM placement algorithms is the fact that dynamic solutions consider potential VM live migrations and therefore require larger amount of resources than the static solutions, which can badly affect the performance of the hosted applications. Since the static solutions are primitive and outdated [START_REF] Usmani | A Survey of Virtual Machine Placement Techniques in a Cloud Data Center[END_REF], we are interested to study the dynamic VM placement optimization solutions. As shown in Figure 1, our classification is based on four main approaches: 1) Constraint Programming, 2) Bin Packing, 3) Stochastic Integer Programming, and 4) Genetic Algorithm. We detail in the following the algorthims that were implemented through these approaches.

Constraint Programming

Foremost Constraint Programming is used where we have already completely collected the input information, so we start the cost function calculations after we are aware of the VM requirements. Since this approach can always consider additional constraints, it can always be expandable. However, in cases where we have several constraints to take into consideration, this approach may take too much time to find the most suitable VM placement. Therefore, the main challenge consists in finding the optimal solution before any modification in terms of the constraint parameters. Although this approach formally expresses the demands of VMs in terms of constraints, it has been modestly adopted. In 2009, [START_REF] Hermenier | Entropy: A Consolidation Manager for Clusters[END_REF] introduce the Entropy resource manager, which is a constraint programming-based solution allowing for a dynamic server consolidation. It tackles both problems of finding the available servers, and migrating the VMs to these servers. This solution is only applied in homogeneous data centers. In 2010, [START_REF] Van | Performance and Power Management for Cloud Infrastructures[END_REF] propose an interactive resource management framework that ensures both dynamic VM provisioning and placement, by treating them as two constraint satisfaction problems.

Bin Packing

Bin Packing is used for dynamic VM placement when all servers have the same amount of resources (CPU, memory, storage, etc.), but the requirements of VMs are variable in time. In order to minimize the number of PMs, this approach may host two dependent VMs on one PM. The Bin Packing problem is an NP-hard3 problem that can be solved using i) Greedy heuristics like First Fit Decreasing (FFD) algorithm [START_REF] Ajiro | Improving Packing Algorithms for Server Consolidation[END_REF], [START_REF] Tang | Dynamic Forecast Scheduling Algorithm for Virtual Machine Placement in Cloud Computing Environment[END_REF], [START_REF] Verma | pMapper: Power and Migration Cost Aware Application Placement in Virtualized Systems[END_REF], Best Fit (BF) algorithm [START_REF] Jiankang | Energy-Performance Tradeoffs in IaaS Cloud with Virtual Machine Scheduling[END_REF], Least Loaded (LL) algorithm [START_REF] Ajiro | Improving Packing Algorithms for Server Consolidation[END_REF], etc., ii) Ant Colony Optimization (ACO) heuristics [START_REF] Dorigo | Ant Colony System: A Cooperative Learning Approach to the Traveling Salesman Problem[END_REF], [START_REF] Dorigo | Ant System: Optimization by a Colony of Cooperating Agents[END_REF], [START_REF] Stützle | Max-Min Ant System[END_REF], iii) Subgraph isomorphism algorithms [START_REF] Ullmann | An Algorithm for Subgraph Isomorphism[END_REF].

Greedy heuristics

FFD algorithm is a greedy heuristic. It places each VM into "the first bin in which it will fit". This approach is more efficient when first sorting the list of elements into a decreasing order. For example, Ajiro et al. [START_REF] Ajiro | Improving Packing Algorithms for Server Consolidation[END_REF] present an Improved FFD + LL heuristic. Both FFD and LL heuristics pack VMs with certain resources into destination servers. However, other resources may interfere and increase the number of destination servers. The proposed algorithm aims to address this issue and reduce the number of destination servers.

The pMapper system [START_REF] Verma | pMapper: Power and Migration Cost Aware Application Placement in Virtualized Systems[END_REF] packs the VMs in a small number of physical servers in order to minimize the migration costs, but always under a fixed constraint of performance. The pMapper system is a subset of the FFD heuristic and it aims for a compromise between performance and cost.

In 2013, VMPlanner [START_REF] Fang | VMPlanner: Optimizing Virtual Machine Placement and Traffic Flow Routing to Reduce Network Power Costs in Cloud Data Centers[END_REF]) is proposed as another greedy Bin Packing algorithm for reducing power costs of network elements in data centers. VMPlanner takes advantage of the flexibility provided by dynamic VM migration and programmable flow-based routing, available in modern data centers, to optimize network power consumption while satisfying network traffic demands. In order to evaluate VMPlanner, the authors implement it in a simulated environment running real data center traffic workloads, with a VL2 architecture [START_REF] Greenberg | Vl2: A Scalable and Flexible Data Center Network[END_REF]. [START_REF] Kanagavelu | Virtual Machine Placement with Two-path Traffic Routing for Reduced Congestion in Data Center Networks[END_REF] develop a fast heuristic algorithm called Greedy Virtual Machine Placement with Two Path Routing (GVMTPR) that is based on some constraints. Its goal is to reduce the maximum load on any link, by reducing the network traffic load, followed by the number of used servers and the server resource utilization, while guaranteeing the specified protection grade requirements. It uses a greedy approach to assign the hypervisor for each VM, in order to satisfy the computing and memory resource requirements. This will be done due to the VM management and routing decisions. In addition, instead of a single-path routing, it chooses a two-path routing to ensure load balancing on the two least-congested paths. Traffic splitting helps to reduce congestion, which is largely influenced by the VM placement, and ensures traffic protection (minimum bandwidth) in the event of failures.

Likewise, in 2014, Tang et al. [START_REF] Tang | Dynamic Forecast Scheduling Algorithm for Virtual Machine Placement in Cloud Computing Environment[END_REF]) present VM-DFS, an algorithm based on a dynamic forecast scheduling method and that is a sort of Bin Packing problem which can also be solved by FFD. VM-DFS analyzes the historical memory consumption of each VM, to select the most appropriate PM to place the running VM according to the future consumption prediction. The purpose of this method is to reduce the number of active servers, such that every VM can conform to the required memory consump-tion SLA 4 .

In [START_REF] Jiankang | Energy-Performance Tradeoffs in IaaS Cloud with Virtual Machine Scheduling[END_REF], the authors propose BF-HC and VM-Mig, two algorithms based on the BF heuristic. The difference from FFD heuristic is that the VMs are placed in the bin that can hold them with minimum empty space. The objective of this two-stage heuristic algorithm is to minimize the energy consumption and to optimize the traffic management, by respectively reducing the number of activated PMs and network elements and maximizing the link utilization. The evalutaion is performed by simulation experiments in Amazon EC2 data center using Fat-tree topology [START_REF] Al-Fares | A Scalable, Commodity Data Center Network Architecture[END_REF].

ACO heuristics

Broadly speaking, ACO is a probabilistic technique for solving computational problems, which can be reduced to finding good paths through graphs. The collective behaviour of social insects is an inspiration source for researchers. When searching for food, ants tend to choose paths marked by strong pheromone concentrations. As soon as an ant finds a food source, it studies the quantity and the quality of the food and takes some of it back to the nest. During the return trip, the quantity of pheromones that an ant leaves on the ground may depend on the quantity and quality of the food. The pheromone trails will guide other ants to the food source and enables them to find the shortest paths between their nest and food sources [START_REF] Blum | Ant colony optimization: Introduction and recent trends[END_REF]. Briefly, ants mark the best solutions and take into account the previous markings to optimize their search [START_REF] Yu | Rethinking Virtual Network Embedding: Substrate Support for Path Splitting and Migration[END_REF]. This behaviour, aiming for the shortest paths, can be used for the VM migration optimization between PMs. Some extensions of ACO algorithms are presented in the literature such as Ant System (AS) [START_REF] Dorigo | Ant System: Optimization by a Colony of Cooperating Agents[END_REF], Ant Colony System (ACS) [START_REF] Dorigo | Ant Colony System: A Cooperative Learning Approach to the Traveling Salesman Problem[END_REF] and MAX-MIN Ant System (MMAS) [START_REF] Stützle | Max-Min Ant System[END_REF]. Let us note that AS is the progenitor of all the research efforts with Ant algorithms. It is widely used to solve the Traveling Salesman Problem (TSP5 ) [START_REF] Hoffman | Traveling Salesman Problem[END_REF].

MMAS [START_REF] Stützle | Max-Min Ant System[END_REF] is a variant of ACO that aims to exploit the best solutions found during the search in order to reach an optimal solution. MMAS updates the pheromone according to the best solution found from the beginning of the process until the current iteration. Feller et al. [START_REF] Feller | Energy-Aware Ant Colony Based Workload Placement in Clouds[END_REF] propose Single-objective ACO (SACO), a MMAS metaheuristic-based algorithm that is focused on reducing the number of physical servers, which are needed to handle the workload. Moreover, Fajjari et al. [START_REF] Fajjari | A New Virtual Network Static Embedding Strategy within the Clouds Private Backbone Network[END_REF] propose a new, efficient and scalable, strategy named VNE-AC. It is based on a MMAS metaheuristic too. Its main aim is to reduce the amount of allocated resources for each virtual network request. This will help to minimize the reject rate and maximize the cloud provider6 's revenue. Based on extensive simulations, this proposal makes the VM placement more effective and achieves better resource utilization even when assuming a high arrival rate of VM requests. Its fundamental stages are: i) formation of solution components, ii) localization of potential candidates, iii) stochastic selection of the candidate, iv) selection of the best solution, and v) updating the pheromone trail. Moreover, VNE-AC goes beyond related strategies investigated in existing literature, such as VNE-Greedy [START_REF] Yu | Rethinking Virtual Network Embedding: Substrate Support for Path Splitting and Migration[END_REF], VNE-Cluster [START_REF] Zhu | Algorithms for Assigning Substrate Network Resources to Virtual Network Components[END_REF], VNE-Subdividing [START_REF] Zhu | Algorithms for Assigning Substrate Network Resources to Virtual Network Components[END_REF], and VNE-Least [START_REF] Zhu | Algorithms for Assigning Substrate Network Resources to Virtual Network Components[END_REF].

Most of the solutions that attempt to optimize the placement of VMs are shifted towards processing only one criterion. However, similar to any other problem, it is usually useful to simultaneously consider several criteria. For this, a new research direction aims to realize multi-objective optimized allocation of VMs. Therefore, Gao et al. [START_REF] Gao | A Multi-Objective Ant Colony System Algorithm for Virtual Machine Placement in Cloud Computing[END_REF] propose a solution that tackles at a time the problem of resource waste and energy utilization. Knowing that ACS is built upon the previous AS, a modified version of ACS algorithm is proposed and designed to properly deal with the potential large solution space for large-scale data centers. Its name is VMPACS. A performance benchmark is carried out between the proposed Ant algorithm and other algorithms, such as Multi-objective Grouping Genetic Algorithm (MGGA7 ) [START_REF] Xu | Multi-Objective Virtual Machine Placement in Virtualized Data Center Environments[END_REF], SACO8 algorithm [START_REF] Feller | Energy-Aware Ant Colony Based Workload Placement in Clouds[END_REF] and the Improved FFD + LL algorithm [START_REF] Ajiro | Improving Packing Algorithms for Server Consolidation[END_REF]. To summarize the results, we provide in Table 1 the energy utilization and resource consumption comparison between the algorithms under consideration.

As depicted in Table 1, it can be noticed that VM-PACS goes beyond MGGA. Indeed, the choice of Energy Utilization Resource Consumption MGGA [START_REF] Xu | Multi-Objective Virtual Machine Placement in Virtualized Data Center Environments[END_REF] Improved FFD + LL [START_REF] Ajiro | Improving Packing Algorithms for Server Consolidation[END_REF] VMPACS [START_REF] Gao | A Multi-Objective Ant Colony System Algorithm for Virtual Machine Placement in Cloud Computing[END_REF] SACO [START_REF] Feller | Energy-Aware Ant Colony Based Workload Placement in Clouds[END_REF] placement given by VMPACS consists of: i) the data being collected at the moment and ii) the traces kept by a non-optimal solution. VMPACS also updates, in a continuous way, the pheromones, which allows a much more relevant VM placement. Evenly, VM-PACS goes beyond Improved FFD + LL heuristic and SACO. These results are mainly related to the number of servers employed and to the resource utilization of each algorithm.

Subgraph isomorphism algorithms

The subgraph isomorphism problem is a NP-hard computational task in which two graphs are given as input, and one must determine whether the first graph contains a subgraph that is isomorphic to the second graph. Recently, several algorithms have used subgraph isomorphism to formulate the problem of VM placement, i.e., to model data center topologies and VM clusters. [START_REF] Zong | Cloud Service Placement via Subgraph Matching[END_REF] proposes Gradin, a new graph index framework that accelerates subgraph matching on dynamic graphs of numerical labels. Gradin efficiently calculates frequent index updates and eliminates unpromising matches to minimize the cost. Gradin's performance is evaluated over BCube topology [START_REF] Guo | BCube: A High Performance, Server-centric Network Architecture for Modular Data Centers[END_REF], on query processing, index update and scalability. Results show that Gradin outperforms competitive approaches such as VF2 [START_REF] Cordella | A (sub) Graph Isomorphism Algorithm for Matching Large Graphs[END_REF] up to 10 times. [START_REF] Aral | Network-Aware Embedding of Virtual Machine Clusters onto Federated Cloud Infrastructure[END_REF] provides Topology Based Mapping (TBM) algorithm that uses a subgraph search to locate federated clouds with a topology isomorphic to the VM cluster.

Additionally, it presents RalloCloud framework for modeling and simulating distributed VM allocation. Using RalloCloud for TBM evaluation, results indicated that TBM outperforms greedy heuristics in latency, throughput, cost and acceptance rate.

Stochastic Integer Programming

Stochastic Integer Programming is introduced for situations where future demands and prices of resources are unknown, but their expected distributions are either known or can be computed. This is the best tech-nique to be used in the case where we have two or more uncertain parameters on which the cost depends. It is helpful in estimating the variation in demands and costs. Thereby, frequent recomputations are not needed, but if there is an error in the estimation, users might end up paying more. For example, the authors in [START_REF] Chaisiri | Optimal Virtual Machine Placement across Multiple Cloud Providers[END_REF] define Optimal Virtual Machine Placement (OVMP), an algorithm for optimally placing the VMs on the suitable PMs. The goal is to minimize the cost of deploying and running VMs in a cloud provider environment, where future demands and prices are not necessarly stable. This objective is reached by reducing the number of used nodes. The algorithm they define is particularly based on linear and quadratic programming. In [START_REF] Speitkamp | A Mathematical Programming Approach for Server Consolidation Problems in Virtualized Data Centers[END_REF], the authors translate the server consolidation problem into a linear programming formulation. Their approach, Branch & Bound (B & B), aims at limiting the number of VMs in each PM. As a result, it is certain that VMs are not hosted in a single physical server. Some design constraints were added in order to achieve this approach, very useful to ensure fault-tolerancy in case of outages.

Genetic Algorithm

Finally, Genetic Algorithm is particularly convenient in cases where we need to operate on groups and for problems where objective functions9 dynamically change. It considers additional constraints while optimizing the cost function, so it solves the VM interference problem encountered in the Bin Packing approach, but it requires more computing time and higher computing resources as compared to Bin Packing. Mi et al. [START_REF] Mi | Online Self-Reconfiguration with Performance Guarantee for Energy-Efficient Large-Scale Cloud Computing Data Centers[END_REF] propose a genetic algorithm-based approach, namely GABA, to adaptively self-reconfigure the VMs in cloud data centers consisting of heterogeneous nodes. GABA can efficiently decide the optimal physical placement of VMs according to application requirements and dynamic environmental conditions, that may vary over time. Moreover in the same year, the VM placement problem is formulated in [START_REF] Xu | Multi-Objective Virtual Machine Placement in Virtualized Data Center Environments[END_REF] as a multi-objective optimization problem of simultaneously minimizing total resource wastage, power consumption and thermal dissipation costs. MGGA with fuzzy multi-objective evaluation is investigated for efficiently searching the large solution space and conveniently combining possibly conflicting objectives.

DISCUSSION

A proper VM placement is essential for easier failure recovery, better geographical coverage and vendor lock-in avoidance in the cloud computing. The existing VM placement techniques consider various approaches, system assumptions, features of data centers like the network topology, as well as different evaluation approaches. Moreover, VM placement and migration is a broad research area with various optimization and objectives. Some of the techniques are only focused on a single objective, such as [START_REF] Ajiro | Improving Packing Algorithms for Server Consolidation[END_REF], [START_REF] Chaisiri | Optimal Virtual Machine Placement across Multiple Cloud Providers[END_REF] and [START_REF] Feller | Energy-Aware Ant Colony Based Workload Placement in Clouds[END_REF]. Other techniques try to incorporate multiple objectives while making VM placement decisions, such as [START_REF] Xu | Multi-Objective Virtual Machine Placement in Virtualized Data Center Environments[END_REF] and [START_REF] Gao | A Multi-Objective Ant Colony System Algorithm for Virtual Machine Placement in Cloud Computing[END_REF]. We can state that Bin Packing is the most employed approach. It always generates a good solution in a correct amount of time.

It is crucial to choose a VM placement technique that suits the needs of both the cloud user and cloud provider. However, due to the presence of several parameters, comparative analysis in a uniform fashion of such techniques becomes quite tricky. In fact, each of the VM placement algorithm works well under certain specific conditions/objectives. Thereby, in order to compare the efficiency of the previous algorithms, we propose that future empirical studies will be based on the three following metrics to measure and evaluate the algorithms performance. Firstly, one should take into account the energy amount consumed by data center resources, due to the application workloads. The second metric to be considered is the SLA violation percentage, which expresses the level by which performance requirements defined between the resource provider and consumers are violated. The SLA violation can happen when VMs sharing the same PM need a CPU performance that cannot be provided because of energy-aware resource management and consolidation. The provider pays a penalty to the client in case of SLA violation. The third metric is the number of VM migrations during the adaptation of the VM placement [START_REF] Beloglazov | Energy-Aware Resource Allocation Heuristics for Efficient Management of Data Centers for Cloud Computing[END_REF]. VM migrations consume time, energy and network bandwidth. Thus, it is important to minimize the number of VM migrations.

RELATED SURVEYS

Three recent surveys also discuss the state-of-the-art of VM placement techniques in data centers. [START_REF] Pires | A Virtual Machine Placement Taxonomy[END_REF] covers several axes such as objective functions (i.e., energy consumption minimization, network traffic minimization, economical costs optimization, and performance maximization), solution techniques (i.e., algorithms and meta-heuristics), cloud architectures (i.e., single cloud, multi-clouds, and federated clouds), and experiment types (i.e. simulation and applications). However, [START_REF] Pires | A Virtual Machine Placement Taxonomy[END_REF] remains mainly statistics-oriented. Our work provides though better details especially regarding the solution techniques. It is also focused on energyaware and traffic-aware solutions.

Also, we consider [START_REF] Jennings | Resource Management in Clouds: Survey and Research Challenges[END_REF]) as a decent and comprehensive survey. It details the different kinds of cloud resources and discusses approaches based on eight subdomains: global scheduling, local scheduling, demand profiling, utilization estimation, pricing, application scaling, workload management, cloud management, and measurement studies. Being focused on resource management for cloud environments in a general way, [START_REF] Jennings | Resource Management in Clouds: Survey and Research Challenges[END_REF] misses quite a number of approaches in VM placement field. The authors refer to the latter as "global scheduling" subdomain. In contrast, our taxonomy focuses on VM placement problem and covers all the approaches that have been investigated in this field.

Finally, while [START_REF] Usmani | A Survey of Virtual Machine Placement Techniques in a Cloud Data Center[END_REF] offers a similar classification to ours, its motivation is mainly focused on server consolidation. Our survey addresses the topic of VM placement with a broader motivation and takes into account the different types of system architecture. Moreover, we explain the optimal case for the use of each of the four discussed approaches. We certainly fill a void in the literature by providing this up-to-date survey, which is useful for the cloud computing domain.

CONCLUSION AND FUTURE DIRECTIONS

In this paper, we survey the state-of-the-art of VM placement optimization, covering its essential approaches and key existing solutions that reduce network power cost and prevent congestion of data flow.

As the cost and the performance of cloud data centers become a practical concern and attract attention of both service provider and consumer, our survey provides a better comprehension of the existing VM placement algorithms that deal with power cost and handle traffic in data centers. However, many future directions and perspectives have not been explored yet and can be contemplated for the future. For example, data security remains unresolved in cloud infrastructures. Since the migration of VMs require a secure connection between both source and target servers, we pave the way for further work to address this issue and we mark the urge of defining reliable protocols for establishing and managing a protected communication. Moreover, in order to optimally benefit from the features of the four approaches, designing an hybrid solution combining several approaches represents a future challenge. Finally, nowadays there are several resource managers that are mostly doing the placement of VMs, like vMotion [START_REF] Murphy | Enabling Long Distance Live Migration with F5 and VMware vMotion[END_REF], the commercial product of VMware and OpenStack [START_REF] Sefraoui | Open-Stack: Toward an Open-Source Solution for Cloud Computing[END_REF], the open-source cloud manager. Other resource managers like Kubernetes [START_REF] Brewer | Kubernetes and the path to cloud native[END_REF], Swarm [START_REF] Thapatsuwan | Modifying Particle Swarm Optimisation and Genetic Algorithm for Solving Multiple Container Packing Problems[END_REF], and Mesos [START_REF] Hindman | Mesos: A Platform for Fine-Grained Resource Sharing in the Data Center[END_REF], to cite a few, are responsible for the placement of containers. Therefore, it will be interesting to conduct an exhaustive study of the existing static and dynamic resource managers, and to map between them and the placement algorithm(s) they use.

Figure 1 :

 1 Figure 1: VM placement approaches.

Table 1 :

 1 Comparison of VMPACS and other algorithms.

VMware is an innovator in virtualization and cloud infrastructure, http://www.vmware.com/

Server consolidation is an approach to the efficient usage of physical resources in order to reduce the total number of servers that an organization requires. It involves gathering several VMs into a single physical server.

A problem is NP-hard if the algorithm for solving it can be translated into one for solving any NP-problem (nondeterministic polynomial time) problem. NP-hard therefore means "at least as hard as any NP-problem," but it can eventually be harder!

A Service-Level Agreement (SLA) is a contract between a network service provider and a customer that specifies, in measurable terms, what services the network service provider will

furnish.5 Given a list of cities and the distances between each pair of cities, TSP looks for the shortest possible route that visits each city exactly once and returns to the origin city.

Such as Google, Microsoft and Amazon.

Further information about MGGA are provided in Section 3.4

8 SACO is a modified MMAS algorithm for VM placement.

An objective function is a function to maximize or minimize.

ACKNOWLEDGEMENTS

This work is supported by the OCCIware 10 research and development project funded by French Programme d'Investissements d'Avenir (PIA). Likewise, this work is partially funded by Nord-Pas de Calais Regional Council.


Thomas Voisin

Arnaud Erriguible

David Ballenghien

David Mateos

André Kunegel

François Cansell

Cyril Aymonier

Solubility of inorganic salts in sub-and supercritical hydrothermal environment: Application to SCWO processes

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Liquid water, as we know it in the normal conditions of temperature and pressure, is a strong polar solvent. This property designates water as a very good solvent for inorganic salts such as alkali and metallic salts or complexes, and a very bad one for organic compounds (oils, petroleum derivatives, polymers. . . ). But in particular conditions of high temperature and pressure (374°C, 22.1 MPa), water becomes supercritical, meaning a single phase is formed between liquid and vapor (c.f. Figure 1), resulting in intermediate properties (c.f. Figure 2).

One of these particularities is that the water polarity collapses, leading to an overturn of the solubility abilities. More precisely, the polarity of a compound is mainly quantied with its dielectric constant ε which decreases quickly with the temperature. This leads to the supercritical water polarity to be as low as non-polar solvents (like hexane. . . ) so that water can now easily dissolve organic compounds, but with the drawback of inorganic precipitation. This is the foreground for the Supercritical Water Oxidation process (SCWO). Once the organic components of the waste have been dissolved, liquid oxygen is introduced in order to oxidize most of the compounds into basic inert molecules. The drawback of this phenomenon is that most of the inorganic compounds become nonsoluble in the media, leading to a massive precipitation into solids which can induce plug formation inside the * cyril.aymonier@icmcb.cnrs.fr The aim of this review is to sum up the experimental methods that have been developed in order to measure salt solubility in sub-and supercritical water as well as the theoretical modelling proposed for predictive calculation. After a short presentation of the SCWO processes and technologies, the notion of salt types will be introduced in a second part regarding water-salt equilibria.

Consecutively, the third part concerns experimental setups for salt solubility measurement, with a historical review of the research in the eld and a summary of all the solubility data available in sub-and supercritical water.

Then, a part concerning salt mixtures and the inuence on solubility will follow. And the last part will present the dierent theoretical and semi-empirical models for predictive calculation of inorganic salt solubility in suband supercritical conditions. The principle of the process is to take advantage of the properties of supercritical water, which is both a good solvent of organic molecules. In order to assure a complete oxidation, liquid oxygen, air, or hydrogen peroxide is also injected in the process. The working conditions are usually a pressure of 25 MPa and temperatures from 300°C to up to 600°C.

One of the advantages of the SCWO process is that it does not produce fumes or ashes, but mainly pure gases (CO 2 , N 2 ) and liquids (H 2 O, acids/basis), depending on the composition of the waste. Furthermore the amount of NOx produced is negligible. As oxidation reactions are exothermic, it enables an autothermal process (meaning the heat released from oxidation is enough for keeping the process at the right temperature). For quite high COD (> 100 g.L-1) energy can be gained during the operation. Another main interest comes from the fact that SCWO process can easily reach 99.99% of waste degradation eciency in very low residence time (few minutes) [START_REF] Loppinet-Serani | Chapter 7 -Hydrolysis in Near-and Supercritical Water for Biomass Conversion and Material Recycling A2 -Anikeev, Vladimir[END_REF]. Figure 3 A large number of papers has been published about SCWO processes, dealing with reactor designs, scale control or waste treatment eciency. A 2001 study [START_REF] Kritzer | An assessment of supercritical water oxidation (SCWO): existing problems, possible solu-tions and new reactor concepts[END_REF] sums up most of the existing technologies for SCWO, but also the dierent remaining issues regarding corrosion, plugging and eciency. A few years later, two reviews [START_REF] Hodes | Salt precipitation and scale control in supercritical water oxidationPart A: fundamentals and research[END_REF][START_REF] Marrone | Salt precipitation and scale control in supercritical water oxidationpart B: commercial/full-scale applications[END_REF] focused on the salt precipitation phenomenon in SCWO.

Salt precipitation is a recurring subject in SCWO processes and lots of works have been done on the engineering of systems and reactors to try to counteract this drawback. One example is the MODAR reactor (c.f. Figure 4 (c)), also called reverse ow reactor, and is described by several works [START_REF] Xu | A novel concept reactor design for preventing salt deposition in supercritical water[END_REF][START_REF] Killilea | Supercritical water oxidation: Microgravity solids separation[END_REF][START_REF] Cohen | Hydrothermal oxidation of Navy excess hazardous materials[END_REF] and patents [START_REF] Hong | United States Patent: 4822497 -Method for solids separation in a wet oxidation type process[END_REF][START_REF] Huang | United States Patent: 5100560 -Apparatus and method for supercritical water oxidation[END_REF]. Reverse ow reactor principle is to work with a supercritical zone (upper area of the reactor) where oxidation and waste degradation occurs, and a subcritical zone (lower area)

where solid salt can be re-dissolved and be taken out as a concentrated brine. Some other SCWO reactors have been designed in order to reduce the corrosion and precipitation issues by keeping the reactor's wall cooler than the supercritical zone with an injection of compressed air or cold water (c.f. Figure 4 (a)). Examples are the wallcooled hydrothermal burner [START_REF] Roche | Design rules for the wallcooled hydrothermal burner (WHB)[END_REF], the transpiring-wall reactors [2022] or the cold wall reactor [START_REF] Cocero | Cool wall reactor for supercritical water oxidation[END_REF]. One company who is currently running a SCWO process at industrial scale is INNOVEOX in Europe, and is using a multiinjection tubular reactor, consisting in injecting pressurized oxygen at dierent stages of the process (c.f. Figure

(b)).

At a smaller scale, some other prototypes have been imagined for SCWO applications. One has been developed, using centrifugal forces to split the two phases based on the dierences in weight and density between solid salts and supercritical water. Usually called vortex or cyclone reactors, several variety of reactors have been developed following this process, such as the centrifuge reactor [START_REF] Reid | United States Patent: 5425883 -Method for separation and removal of impurities from liquids[END_REF], the hydrocyclone [START_REF] Grith | The rst commercial supercritical water oxidation sludge processing plant[END_REF] or rotational spin reactor [START_REF] Titmas | United States Patent: 4594164 -Method and apparatus for conducting chemical reactions at supercritical conditions[END_REF]. Another interesting design is the sonochemical reactor [START_REF] Aymonier | Ultrasound for Hydrothermal Treatments of Aqueous Wastes: Solution for Overcoming Salt Precipitation and Corrosion[END_REF] (c.f. Figure 4 (d)), combining an ultrasound probe to activate oxidation reactions at lower temperatures to reduce the precipitation and corrosion issues.

Regarding the oxidative process eciency, the reviews written by BRUNNER [START_REF] Brunner | Near critical and supercritical water. Part I. Hydrolytic and hydrothermal processes[END_REF][START_REF] Brunner | Near and supercritical water. Part II: Oxidative processes[END_REF][START_REF] Brunner | Supercritical process technology related to energy and future directions An introduction[END_REF] , in 2009 are focused either on the destruction of organic biomass (lignin, cellulose. . . ) or on the corrosion, depending on the solvent used, and the dierent available materials meeting the requirements to prevent it. One year later, a more specic work was done [3032] on the salt precipitation problem, analysing dierent salt behaviors, with binary mixtures as well but few solubility measurements. A recent work on the salt issue [START_REF] Xu | Salt deposition problems in supercritical water oxidation[END_REF] is focused on the deposition of sticky salts (Na 2 CO 3 , Na 2 SO 4 . . . ) and the means to avoid it.

With respect to the commercially full scale process, a recent review [START_REF] Vadillo | Problems in Supercritical Water Oxidation Process and Proposed Solutions[END_REF] proposes an overview of most of the existing SCWO plants, with the description of the type of waste treated, the capacity and the type of reactors used. This work links up with a previous thesis done on the subject [START_REF] Aymonier | Traitement hydrothermal de déchets industriels spéciaux. Données pour le dimensionnement d'installations industrielles et concepts innovants de réacteurs sonochimique et électrochimique[END_REF] in 2000. In this thesis, many dierent chemical wastes and compounds have been treated and analyzed in regard to the oxidative eciency as well as a sum up of the dierent type of process, reactors and materials was done. Dierences between the diagrams are characterized by the immiscibility or miscibility domains and the conguration of the critical lines between the two critical points of the pure compounds. A seventh type has then been discovered [37] with the Lennard-Jones equation of state.

Following this work, VALYASHKO [3840] used some of these diagrams which enable the description of solid-uid interactions, such as salt-water systems. However, due to the large dierence in critical temperature between inorganic salts and water (in comparison with two uids for example), unusual phase equilibrium can occur when reaching the critical domain. In order to better build these diagrams, some specic assumptions to the water-salt systems have been added (taken from [START_REF] Valyashko | Fluid phase diagrams of ternary systems with one volatile component and immiscibility in two of the constituent binary mixturesPresented at the International Bunsen Discussion Meeting of the Deutsche Bunsen-Gesellschaft für Physikalische Chemie[END_REF]): Using a continuous topological transformation approach, the results reveal two main behaviors for salts (type I and type II) , which are split into many sub-diagrams.

Operating small changes on thermodynamical parameters allow to process a continuous transformation from one diagram to another [START_REF] Valyashko | Fluid phase diagrams of ternary systems with one volatile component and immiscibility in two of the constituent binary mixturesPresented at the International Bunsen Discussion Meeting of the Deutsche Bunsen-Gesellschaft für Physikalische Chemie[END_REF][START_REF] Valja²ko | Hydrothermal experimental data: [experimental data on aqueous phase equilibria and solution properties at elevated temperatures and pressures[END_REF].

These thermodynamic binary phase diagrams nicely classify most of the salts according to their behavior in supercritical water. Following this classication, type I salts present a continuous solubility curve at supercritical temperature which does not cross the critical curve, whereas type II salts present an intersection between the solubility curve and the critical curve, leading to two critical endpoints in this domain (c.f. Figure 5). These two general diagrams can then be complicated with immiscibility domains occurring in some cases.

MARSHALL [START_REF] Marshall | Water and its solutions at high temperature and high pressure[END_REF] made a dierence between type I and type II salts saying that the rst group generally has a higher solubility in supercritical water than the others, and classied most of inorganic salts according to this criteria (c.f table I). In addition, VALYASHKO [START_REF] Valyashko | Phase equilibria in water-salt systems: some problems of solubility at elevated temperature and pressure, in: High temperature and high pressure electrochemistry in aqueous solutions[END_REF] classied them according to their melting temperatures (c.f. table I). Type I salts have a melting temperature between 800°C and 1000°C whereas type II salts have their melting temperature above 700-800°C. What is interesting is that the two classications are not in opposition with each other, but complementary. It is also important to notice that according to this classication, for a given cation (anion), the salts solubility in supercritical water will increase with the size of the associated anion (cation).

Table I: MARSHALL and VALYASHKO classications

for type I and type II salts.

As appearing in the dierent classications, sodium chloride (NaCl) is a very good illustration of a type I salt, with a high solubility in water, even at sub-critical or supercritical conditions (compare to other salts), and the appearance of gas-liquid like equilibrium. The binary diagram at high pressure and high temperature for NaCl is well dened (c.f. Figure 6) and has been checked with several experimental data [START_REF] Armellini | Experimental methods for studying salt nucleation and growth from supercritical water[END_REF]. The lower limit of the diphasic zone has also been directly observed, but its upper limit remains theoritically xed to a certain temperature from which a unique supercritical phase remains and salt precipitates.

A good example of a type II salt is sodium sulfate (Na 2 SO 4 ), which presents an intersection of the solu- bility curve with the critical curve, leading to a simple precipitation behavior at subcritical conditions, without 3-phase equilibrium (see Figure 6). Some recent experiments have been performed on the subject, towards improvement of gasication processes [3032]. The set up is a Modar like reactor, with a subcritical part at the bottom and a supercritical zone on the top. The aim of this set up is to study the dierent salt behavior, depending on their type, by looking at their ability to be recovered from the bottom part of the reactor. Experiments were performed with binary mixtures of water-salt systems, with type I or type II salts. The rst trend observed is that type I salts (K 2 CO 3 , K 3 PO 4 , K 2 HPO 4 , KH 2 PO 4 , NaNO 3 , KNO 3 and Ca(NO 3 ) 2 ) precipitate in supercritical water and can be recovered as a brine solution, whereas type II salts (Na 2 CO 3 , Na 2 SO 4 , K 2 SO 4 and Na 3 PO 4 ) precipitate but directly stick to the reactor walls and rapidly plug the process, disabling any recovery as a brine. But even if the behavior follows the same trend for a salt type, dierences still exist due to the solubility variations. This is seen in the nitrate compounds, the solubility of NaNO 3 is higher than the one of KNO 3 , itself higher than Ca(NO 3 ) 2 (meaning Ca(NO 3 ) 2 is easily recovered than NaNO 3 because it precipitates more eas- ily). As a similar result, KH 2 PO 4 is easily recovered in comparison with KNO 3 . These results seem to be on the opposite trend than the one predicted by MARSHALL [START_REF] Marshall | Water and its solutions at high temperature and high pressure[END_REF] and VALYASHKO [START_REF] Valyashko | Phase equilibria in water-salt systems: some problems of solubility at elevated temperature and pressure, in: High temperature and high pressure electrochemistry in aqueous solutions[END_REF] classications (solubility increases, for a given anion, with the size of the cation and vice versa).

In the end, this classication in types of salt cannot give all the information regarding solubility at supercritical conditions but it can be used as a trend for the general behavior. This solubility is strongly dependent on the dielectric constant of water which is dropping down when reaching the supercritical domain. Thus, inorganics precipitate, leading to the well-known problem of reactor plugging.

The initial studies for salt solubility at high temperatures and pressures were rst done for geological purpose [START_REF] Benedict | Properties of Saturated Aqueous Solutions of Potassium Chloride at Temperatures above 250 • C[END_REF][START_REF] Keevil | Vapor Pressures of Aqueous Solutions at High Temperatures1[END_REF] in the 40s. These rst analyses were performed with vapor pressure measurements at dierent temperatures, up to 650°C. Various salted solutions of potassium based salts [START_REF] Benedict | Properties of Saturated Aqueous Solutions of Potassium Chloride at Temperatures above 250 • C[END_REF] and sodium based salts [START_REF] Keevil | Vapor Pressures of Aqueous Solutions at High Temperatures1[END_REF] were investigated.

Then, several authors have investigated the phase equilibrium of the NaCl-H 2 O system at high temperatures and pressures, including the supercritical domain, with vapor pressure measurement [5155]. Following the keen interest in the phase equilibria for the binary system NaCl-H 2 O, a comparison of all the data of the previous studies was carried out [START_REF] Bischo | The system NaCl-H2O: Relations of vapor-liquid nar the critical temperature of water and of vapor-liquid-halite from 300 • C to 500 • C[END_REF], looking at the trends and disagreements between the results. Experiments were also performed to complete the lack of data near the critical temperature of water and re-determine some vaporliquid-halite phases where disagreement existed. In order to better understand the salt deposition (mainly sodium based salts) occurring in turbines and steam machines, which induces fatigue cracking, wearing and mechanical failures, the solubility of sodium chloride in steam at high temperature (450-500°C) and pressure (5-10 MPa) was mainly studied [START_REF] Galobardes | In Situ preconcentration for the determination of sodium ion in dry steam[END_REF][START_REF] Galobardes | Solubility of sodium chloride in dry steam[END_REF]. The water steam was analysed in a continuous process with a ame spectroscopy detector at the end, in order to detect the sodium concentration according to time. Among all the previous cited papers, it was the only one to provide details about experimental setup and procedure. Other continuous processes can be found in previous works [5961] where solubility measurements were made using chloride titrations (colorimetric or volumetric analysis). Alternative methods for NaCl concentration measurements used the titration of Na + ions by ion-chromatography, or even 22 Na radioactive tracers in batch system [START_REF] Alekhin | Thermodynamic parameters and solubility of NaCl in water vapor at 300-500 • C up to 300 bar[END_REF][START_REF] Vakulenko | Solubility and thermodynamic properties of alkali chlorides in steam[END_REF], where this method is used as an in situ analysis.

In addition to all researches on sodium chloride solutions in sub-or supercritical water, some experiments have also been performed on sodium sulfate solutions, in the same range of temperature, pressure, and with an equivalent geological purpose. Studies on Na 2 SO 4 sol- ubility in steam have also been performed [START_REF] Styrikovich | Solubility of salts in high pressure steam[END_REF][START_REF] Morey | The Solubility of Some Minerals in Superheated Steam at High Pressures[END_REF][START_REF] Martynova | Solubility of inorganic compounds in subcritical and supercritical water[END_REF].

Salt concentration measurements were mostly done us-ing a 35 S radioactive tracer in continuous systems.

In 1993, the rst paper about experimental results

for the solubility of salts in sub-and supercritical water, towards the SCWO application, was published by ARMELLINI and TESTER [START_REF] Armellini | Solubility of sodium chloride and sulfate in sub and supercritical water vapor from 450-550 • C and 100-250 bar[END_REF] . This work summarizes most of the previous results on sodium chloride and sodium sulfate solubility in steam, comparing the measurement methods and editing the temperature and pressure domains screened by the past researches. A detailed description of the experimental setup and method is given, as well as a tting of the data with a Gibbs-Helmholtz model which is compared with some other theoretical solubility models.

In opposition to the past experimental methods, which usually consisted of heating a brine solution and then analysing the salt concentration remaining inside, this new setup (c.f. Figure 7 (a)) proposes a dierent approach: a pure water solution is introduced, pressurized and heated up, then the water passes through a tube lled with salt crystals in order to dissolve them and saturate the water to the maximum concentration. Then the solution is analysed using ICP measurements. The interest in using such a method is that it prevents the pipes from plugging, but it can only be used with low saturation concentration value due to the limited amount of salt inside the pipe.

This kind of process has also been used by other authors [START_REF] Leusbrock | Quantitative approaches for the description of solubilities of inorganic compounds in near-critical and supercritical water[END_REF][START_REF] Higashi | Measurement and correlation for solubilities of alkali metal chlorides in wa-ter vapor at high temperature and pressure[END_REF] for several inorganic salts. In opposition, some use the reverse process, consisting in heating a brine solution in order to precipitate the salts. Thus, the remaining solution is at the saturation concentration for the chosen conditions of pressure and temperature. This method is particularly useful when analysing salts with a low melting temperature, as with nitrate salts [START_REF] Dell'orco | The solubility of 1-1 nitrate electrolytes in supercritical water[END_REF] (c.f. Figure 7 (c)). Following this, LEUSBROCK et al. [START_REF] Leusbrock | Solubility of 1:1 Alkali Nitrates and Chlorides in Near-Critical and Supercritical Water[END_REF] (2009-2010) used this alternative method for dierent salts, without limitation to the low melting temperature ones (c.f. Figure 7 (b)).

It is interesting to notice that one set up [START_REF] Khan | Solubility of Na2SO4, Na2CO3 and their mixture in supercritical water[END_REF] combines the precipitation and the dissolution of the salts (Na 2 SO 4 and Na 2 CO 3 ) to analyse the solubility in super- critical water (c.f. Figure 7 (d)). In fact, sodium sulfate and sodium carbonate salts are commonly considered as sticky salts, which precipitate easily on reactor walls.

Using a ow conductivity cell at the end of their setup, the salt concentration is known continually according to time. A brine solution is passed through the system, pressurized and heated up. The conductivity will decrease while the salt precipitate in the reactor; when the steady state is reached (conductivity drops to a plateau), pure water is passed through, dissolving the salt inside.

At this point, the pure water is saturated to a concentration similar to the one of the exiting brine before, thus conductivity remains the same until there is not enough salt to saturate the water anymore. This method enables the comparison of solubility values between solubilisation and precipitation methods.

All the solubility data regarding inorganic salt in sub- 382 -397 24 [START_REF] Khan | Solubility of Na2SO4, Na2CO3 and their mixture in supercritical water[END_REF] Table II: Presentation of all the solubility data available in the literature for inorganic salt in sub-and supercritical water.

ganic compound precipitates [START_REF] Hurst | Optical ow cell and apparatus for solubility, salt deposition and Raman spectroscopic studies in aqueous solutions near the water critical point[END_REF] or to develop an in situ turbidity measurement set up [START_REF] Dell'orco | The solubility of 1-1 nitrate electrolytes in supercritical water[END_REF] in order to detect the nucleation point of precipitates and the dierent phases for inorganic salt mixtures.

As a summary, dealing with quantitative salt solubility measurements in sub-and supercritical conditions often lead in using two dierent methods. One consisting of the precipitation of a brine, while the other saturates pure water through a column lled with salt crystals. But even if these methods have been used several times by dierent research groups, solubility data remains quite scarce for a lot of inorganic salts. Moreover, these studies only concern binary mixture with water, without inuence of any other species present in the solution. This leads us to the next topic, concerning salt mixtures.

Part IV

Solubility with salt mixtures

Despite the fact that some of the common salt's behavior under sub-and supercritical conditions are quite known, very few studies investigated the behavior of a mixture of two salts, and the eect that one can have on the solubility. One of the rst studies on mixtures [START_REF] Dipippo | Ternary phase equilibria for the sodium chloridesodium sulfatewater system at 200 and 250 bar up to 400 C[END_REF] managed to reconstruct a part of the ternary phase diagram of the H 2 O NaCl Na 2 SO 4 mixture under supercritical con- ditions, up to 550°C and 25 MPa (c.f. Figure 9). For the ternary gures, they used previous data [START_REF] Armellini | Experimental methods for studying salt nucleation and growth from supercritical water[END_REF][START_REF] Armellini | Solubility of sodium chloride and sulfate in sub and supercritical water vapor from 450-550 • C and 100-250 bar[END_REF] to build the ternary phase diagram with a continuous topological method. These results correlate with VALYASHKO's prediction [START_REF] Valyashko | Fluid phase diagrams of ternary systems with one volatile component and immiscibility in two of the constituent binary mixturesPresented at the International Bunsen Discussion Meeting of the Deutsche Bunsen-Gesellschaft für Physikalische Chemie[END_REF][START_REF] Valyashko | Phase equilibria of water-salt sytems at high temperatures and pressures[END_REF] that the presence of a type I salt with a type II would increase the type II solubility with temperature.

Most of the experimental work regarding salt mix- mixture is obtained, meaning that the formation of a type I salt in equilibrium with a type II salt is favored. Some exchanges between common ions of dierent salts can also be seen with NaNO 3 /K 2 CO 3 (type I type II) where at the end, Na 2 CO 3 is mainly recovered. Regarding the mixtures of two types II salts (which cannot recombined into a type I salt), the behavior remains the same as a binary solution: precipitation and plugging occurs. As a conclusion of all the works about the behavior of salt mixtures, the authors declare that it is not possible to predict the separation performance of a given salt mixture just by knowing the separation performance of the single salt solution [START_REF] Schubert | Continuous salt precipitation and separation from supercritical water. Part 3: Interesting eects in processing type 2 salt mixtures[END_REF].

In the meantime, KRUSE et al. [START_REF] Kruse | Brines in supercritical biomass gasication: 1. Salt extraction by salts and the inuence on glucose conversion[END_REF] published their results on another study on the improvement of supercritical gasication and the ways to catch poisonous salts for metal catalysts. Beginning with a liquid or solid brine inside a semi-continuous reactor (c.f. Figure 10), it is shown that some ions can be trapped inside, depending on the brine used. For example, a potassium brine (KHCO 3 ) is deposited inside before running the exper- with either Na 2 SO 4 or K 2 SO 4 , dierent eects occur de- pending on the molar ratio (salting in and salting out eects) between the salts. Surprisingly, a complicated behavior between K 2 CO 3 and Na 2 CO 3 does not allow the quantication of any salting eect, and no precipication is observed for phosphate salts (which could be in opposition with the solubility values for phosphate salts from the litterature [START_REF] Leusbrock | The solubilities of phosphate and sulfate salts in supercritical water[END_REF], but the temperatures were actually not high enough to precipitate). Besides the fact that the work is not performed at constant pressure, it brings interesting information concerning salt mixtures.

The results enable to establish a trend in the ions solubility (comparing the precipitation temperature) in the following decreasing order : K + > Na + > Mg 2+ for cations and HPO 4 2-> SO 4 2-> CO 3 2for anions. This solubility scale means that for example, Na 2 CO 3 salt will be less soluble than Na 2 SO 4 which will be less soluble than K 2 SO 4 . This solubility scale is in opposition with the trend from MARSHALL and VALYASHKO (Table I). The possible explanation given by the authors comes from the consideration of the ionic radius. A small ion with a single charge, like Na + , will lead to a small ionic radius, thus a strong electrostatic interaction with other ions leading to clustering and ion association prior to precipitation. Whereas a bigger ion like K + with a larger radius will have a weaker electrostatic interaction and will be less inuenced by the drop in dielectric constant with temperature. Regarding Mg 2+ , the atom may be bigger, but as the ionic charge is also bigger, the resulting ionic radius is smaller than K + or Na + and it is very to one application. Still, several types of predictive models have been developed for pure salt solubility in supercritical water conditions, and the experimental approach using DSC analysis is quite interesting and brings new hypothesis for salt classication depending on their solubility value and ionic radius. In addition, the inuence of the oxidative compound or the degradation products like CO 2 , on the salt solubility is quite unknown although its presence in the SCWO process is non-negligible. This lack of data is quite surprising, as some studies show that the presence of dissolved salt in water inuence the solubility of dissolved gas [106117], it is very likely that dissolved oxidative compounds inuence salt solubility as well.

For future research work, it is probably important to better understand the inuence of the presence of other species on the solubility of inorganic salts (salting-in and salting-out eects) under near-and supercritical conditions with the aim of understanding the precipitation phenomena in SCWO processes. But it requires rst some basic data for pure salts, as it is very probable that complex behaviors like diphasic domains (type I salts), miscibility limits or critical end-points will strongly inuence the interactions with other salts or compounds.

Figure 1 :

 1 Figure 1: Pressure-temperature phase diagram of pure water (adapted from [1])

Figure 2 :

 2 Figure 2: Changes in the dielectric constant of water according to the temperature, at 25 MPa. Comparison with common solvent values at room pressure and temperature [2].

Figure 3 :

 3 Figure 3: (a) Scheme of the main products obtained with SCWO processes. (b) General scheme of a SCWO process

Figure 4 :

 4 Figure 4: Some of the reactors specically designed for SCWO in order to avoid plugs with precipitation (adapted from [3]).

1 .

 1 The melting temperature of the pure nonvolatile component (salt) is higher than the critical temperature of the volatile component 2. No solid-phase transition (polymorphism, solidsolution...) or azeotropy in liquid-gas equilibria is considered 3. Liquid immiscibility is terminated by the critical region at high pressures and cannot be represented by more than two separated immiscibility regions of dierent types 4. All geometric elements of phase diagrams, their reactions and shapes (but not the combinations of these elements) can be illustrated by existing experimental examples.

Figure 5 :

 5 Figure 5: P-T-x diagrams for a type I salt (a) and a type II salt (b). For a type I salt, the binary critical curve is continuous and distinct from the saturation curves, whereas for the type II the binary critical curve is interrupted by the saturation curves, leading to critical end points P and Q. Abbreviations: A, volatile compound ; B, nonvolatile compound ; TP, triple point ; CP, critical point ; E V , eutectic vapor coordinates ; E L , eutectic liquid coordinates ;P, A-rich S-L-V critical end point ; Q, B-rich S-L-V critical end point (from[START_REF] Hodes | Salt precipitation and scale control in supercritical water oxidationPart A: fundamentals and research[END_REF]).

Figure 6 :

 6 Figure 6: Binary diagrams at 25 MPa for water-NaCl (left) and water-Na 2 SO 4 (right). L=liquid phase;V=vapor phase; S=solid phase (from[START_REF] Hodes | Salt precipitation and scale control in supercritical water oxidationPart A: fundamentals and research[END_REF])

Figure 7 :Figure 8 :

 78 Figure 7: Schemes of dierent experimental set up for salt solubility measurement. (a) Set ups for the salt bed method used by ARMELLINI & TESTER [47] ; (b) Structure for salt bed or precipitation method used by LEUSBROCK et al. [48] ; (c) Precipitation apparatus used by Dell'ORCO et al. for low melting temperature salts [49]; (d) SCWO plant for salt deposition & salt bed method used by KHAN & ROGAK[50].

Figure 9 :

 9 Figure 9: Example of the H 2 O -NaCl -Na 2 SO 4 ternary diagram at 400°C, 25 MPa, from DiPIPPO et al.

  iment. A brine solution containing sodium ions is then injected through the apparatus. The analysis performed on the exiting solution shows that the sodium concentration is lower than the initial concentration, and that potassium ions are mainly present at the outlet. Several other similar experiments have been done following the same procedure, with dierent kinds of brine or solid bed inside the reactor. These results tend to show the same conclusions as the previous papers [3032] which is the ability of particular brines to trap some ions by playing on the solubility dierence between two salts, and the mechanism by which certain salts are favored.Earlier in the decade, a study was performed on the solubility of Na 2 SO 4 and Na 2 CO 3 under supercritical conditions[START_REF] Khan | Solubility of Na2SO4, Na2CO3 and their mixture in supercritical water[END_REF], but also the individual solubility of both salts. According to this work, solubility remains the same above supercritical conditions, but is slightly reduced for Na 2 SO 4 at near critical conditions, because of the pres- ence of Na 2 CO 3 in the solution. This inuence on the solubility of one salt on the other would be due to the fact that they have a common ion. This leads to an excess in the concentration which would favor the precipitation of the less soluble salt (meaning Na 2 SO 4 ), according to the common ion eect. Common ion eect is a direct implication of Le Chatelier's principle, which implies that a chemical equilibrium is favored in the direction of the products if the reactant concentrations are in excess. In the case of the salt mixture, the presence of Na + ions coming from Na 2 CO 3 increases the sodium concentra- tion for the precipitation of Na 2 SO 4 , leading to a small increase in the products side, thus a decrease in sodium sulfate solubility.This eect from one salt on another, commonly called salting in eect if increasing the solubility and salting out when decreasing it, has been studied more recently[START_REF] Reimer | Inuence of anions and cations on the phase behavior of ternary salt solutions studied by high pressure dierential scanning calorimetry[END_REF]. Using a sealed inconel crucible and Differential Scanning Calorimetry (DSC), dierent ternary salt mixtures were studied by detecting the precipitation phenomenon, immiscibility limit and supercritical homogenization. This allows them to highlight salting in or salting out eects, looking at the changes in the precipitation temperatures. The study was focused on three type I salts (K 2 CO 3 , Na 2 HPO 4 and K 2 HPO 4 ) and three type II (Na 2 SO 4 , K 2 SO 4 and MgSO 4 ) and small salting in eects are observed for Na 2 SO 4 /K 2 SO 4 and Na 2 SO 4 /Na 2 CO 3 mixtures. Regarding MgSO 4 mixtures

Figure 10 :

 10 Figure 10: Scheme of the semi-continuous reactor used to trap poisonous ions, for gasication applications (adapted from [80]).

Figure 11 :

 11 Figure 11: (a) Comparison of the dierent models with experimental data for NaCl from MASOODIYEH et al.

[ 95 ]

 95 ; (b) Extension in the validity domain for the SAA model.The limitation of Pitzer EoS on aqueous electrolyte solutions is that it requires a large number of characteristic parameters to be able to predict the solubility behavior for inorganic salts. Moreover, Pitzer equations are more convenient to be used for phase diagram determination rather than solubility curves. In order to dispose of a predictive model for solubility under high temperature and pressure, HELGESON et al.[START_REF] Helgeson | Theoretical prediction of the thermodynamic behavior of aqueous electrolytes at high pressures and temperatures: IV. Calculation of activity coecients, osmotic coecients, and apparent molal and standard and relative partial molal properties to 600 • C and 5kb[END_REF] (1981) published a signicant paper for their new model, called HKF for Helgeson-Kirkham-Flowers. This model uses lots of dierent thermodynamic parameters, semi-empirical interaction functions, and some variables from the BORN theory for electrolyte interactions. Several other papers have been published later [97100] to improve the model, which will then take the name of R-HKF model (R stands for Revisited). Beyond its good abilities to predict solubility, the R-HKF model requires some specic thermodynamic parameters which can be dicult to determine (like the molar volume of the salt-solvent mixture according to the temperature and pressure), especially for uncommon salts. In order to try to dodge this problem, SUE et al. [101] (2002) created a simplied version of the R-HKF model, considering the temperature-density dependence of the semi-empirical part of the model, instead of the temperature-pressure dependence. This model is known as the SAA model (Sue-Adschiri-Arai) and is easier to handle. Recently, MASOODIYEH et al. [95] (2014) published a paper to summarize these dierent correlation models (R-HKF, SAA and Chrastil) and compare them with the literature data available to determine their precision (c.f. Figure 11 (a)). Without surprise, the R-HKF model provides the best ts with experimental data, but the SAA model is most of the time very close as well, with very few dierences with R-HKF. However, the R-HKF model requires at least 7 parameters for each compound involved in the considered chemical reaction. In the meantime of simplifying the R-HKF model into the SAA model, SUE et al. also extended the validity domain to the near-and supercritical region (c.f. Figure 11 (b)), using more recent experimental data of NaCl and KCl [102105]. This improvement allows the SAA and R-HKF models to predict quite nicely the solubility of some common salts under supercritical conditions. Even though these predictive models well t experimental data, their use remains quite complex, due to the number of parameter required, which are most of the time determined from experimental data. And as soon as experimental data are available, it will become more convenient to adopt a Chrastil like model, which is the most common method employed in the literature. Conclusion Research on SCWO processes for waste treatment has expended a lot during the last 20 years, and many technological gaps have been overpassed. Despite these im-provements, the precipitation of inorganic salt remains an important issue for most of the SCWO applications. Thanks to the broad development and accessibility of corrosion-resistant alloys and sapphire windows technologies, new investigations on the salt precipitation phenomena under supercritical conditions become possible. Since, lots of solubility data, phase diagrams and modelling have been established in sub-and supercritical conditions for the most common salt like NaCl or Na 2 SO 4 , but much less data exist for other salts. As a consequence, incoherencies appear between experimental results and predictive behaviors, such as the predicted increase in solubility of a given cation (anion) with the size of the anion (cation), which has been invalidated with the more recent salt mixture experiments. Moreover, the behavior of a single salt in water remains a very simplistic example compared to the complexity of a real waste treated in SCWO. Regarding the solubility of salt mixtures, as cited before, it is for now not possible to predict the behavior of a given salt mixture just by knowing the behavior of the two separated salts. As the salt-water solubility data under supercritical conditions are still scarce in the literature, only very few papers are published regarding the study of the mixture, due to the complexity to analyse a solubility mixture without knowing the single solubility of each salt. Despite the great investment put into the classication of salt in order to highlight a trend for each type of salt, the behavior of ternary mixtures remains very dicult to predict and is most of the time specic