David Simeone
email: david.simeone@cea.fr

Philippe Garcia
email: philippe.garcia@cea.fr

Laurence Luneville
email: laurence.luneville@cea.fr

Radiation-Induced Patterning at the Nanometric Scale: A Phase Field Approach

Keywords: radiation-induced patterning, phase field approach, nano-structures

The phase field approach was developed in the last 20 years to handle radiation damage in materials. This approach bridges the gap between atomistic simulations extensively used to model first step of radiation damage at short time and continuum approach at large time. The main advantage of such an approach lies in its ability to compute not only the microstructure at the nanometric scale but also to calculate generalized susceptibilities such as elastic constants under irradiation. After a brief description of the rate theory, used to model the microstructure induced by irradiation, we briefly discuss the foundation of the phase field method, highlighting not only its advantages, but also its limitations in comparison with the rate theory. We conclude this presentation by proposing future orientations for computing the microstructure in irradiated materials.

Introduction

Modern materials science has enormously progressed from a descriptive endeavor to more physically sounded foundations. The ultimate goal of such a fundamental tilt is to design material with tunable properties. The use of new experimental techniques such as particle irradiation made possible to escape the tyranny of the phase diagram. Overcoming this restriction, it becomes possible to observe the formation of patterns at the nanometric scale. It is now clear that such patterns affect the properties of materials and thus need to be understood and controlled.

On the other hand, materials under irradiation may be considered as toy models to study systems maintained far from equilibrium. Varying the control parameters (temperature, particles flux), experimentalists can then test different approaches for modeling instabilities and pattern formation far from equilibrium.

The slowing down of energetic particles (electrons, ions and neutron) in crystalline solids were studied in detail over the past 60 years [START_REF] Sigmund | Particle Radiation and Radiation Effects[END_REF]. Even if mechanisms occurring during the slowing down of particles are complex [START_REF] Simeone | Continuum approaches for modeling radiation-induced self-organization in materials: From the rate theory to the phase field approach[END_REF], it is nowadays admitted that a part of the energy dissipated by incident particles during their slowing down generates a distribution of point and extended defects in materials. This distribution is far from being uniformly distributed along the material but localized in spatial areas, named displacement cascades [START_REF] Simeone | Disrupted coarsening in complex Cahn-Hilliard dynamics[END_REF]. The atomic fraction of defects in these areas is largely superior to those expected from the thermal equilibrium. Such an over-concentration of defects is responsible for the formation of unexpected microstructures over few tens of nanometers: voids, bubbles and nano-precipitates. Moreover, self-organized nano-structures may emerge under some specific conditions of irradiation flux and temperature. Many examples of such patterns have been observed in metals, alloys, ceramics and amorphous materials [START_REF] Simeone | Disrupted coarsening in complex Cahn-Hilliard dynamics[END_REF][START_REF] Enrique | Self organized Cu-Ag nanocomposites synthesized by intermediate temperature ion-beam mixing[END_REF][START_REF] Ghoniem | Instabilities and Self Organization in Materials[END_REF][START_REF] Baldinozzi | Evidence of extended defects in pure zirconia irradiated by swift heavy ions[END_REF][START_REF] Beauford | In situ evolution of He bubbles in SiC under irradiation[END_REF].

The conditions associated with the formation of these radiation-induced nano-patterns, as well as mechanisms associated with their formation is still subject of debate. In this paper, we present new achievements to answer these questions applying the Phase Field formalism. The first part of this work is devoted to the rate theory approach and its extension applied to discuss the formation of nano-patterns. A brief description of this technique, its successes and limitations is discussed. The second part of this work describes in detail the foundation of the Phase Field method, its application to radiation damage and its limitation. Some prospects for future works is proposed as a conclusion.

The Rate Theory

The formation of steady state patterns in driven chemical systems has long been a puzzling problem, both theoretically and experimentally. In 1952, Turing [START_REF] Turing | The Chemical Basis of Morphogenesis[END_REF] proposed that the interplay between diffusion and reacting species could lead to the formation of spatial patterns due to non linear interactions. The interest of this original approach is to describe the dynamics of the system under investigation by sets of ordinary or partial differential equations. Conditions and mechanisms associated with the formation of chemically driven systems can thus be associated with the stability of these dynamical equations. Patterns result from Hopf or Turing bifurcations [START_REF] Simeone | Continuum approaches for modeling radiation-induced self-organization in materials: From the rate theory to the phase field approach[END_REF][START_REF] Ghoniem | Instabilities and Self Organization in Materials[END_REF]. Extension of this approach to radiation damage in metals was developed in the early 70 [START_REF] Brailsford | The rate theory of swelling due to void growth in irradiated metals[END_REF][START_REF] Bullough | The temperature dependence of irradiation creep[END_REF].

Foundation of the Rate Theory

The aim of the rate theory was first developed to study the radiation-induced microstructure at large time such as for instance the radiation-induced swelling of iron steel or nuclear fuels. Two major assumptions are applied to build up a rate theory:

• Defects produced during the slowing down of incident particles in the medium are produced over a time scale τ C , i.e., the characteristic time of a displacement cascade (10 -9 s), much smaller than the time scale τ D associated to the growth or shrink of extended defects (voids, dislocations) which is controlled by mass transport (10 -6 s). Such an assumption makes a distinction between point defects (vacancy, interstitials) and extended defects (voids, dislocations, self interstitial tetrahedron ) produced in displacement cascades.

•

The flux of incident particle is large enough to neglect the correlation between subcascades and the the correlation between produced defects. The production rate of point defects can then be considered as uniform in the irradiated area.

Under these two assumptions, a master equation may be formulated for modeling the dynamics of extended defects. Such a formulation of the problem can be obtained performing a coarse-graining over a space scale larger than the sub cascade size (few angstroms). The master equation holds over a characteristic time scale larger than the displacement time scale of a cascade formation [START_REF] Sigmund | Theoretical aspects of atomic mixing by ion beams[END_REF]. By analogy with the Doring and Becker formalism of nucleation [START_REF] Ghoniem | Instabilities and Self Organization in Materials[END_REF], the size of each extended defect, i.e., void cavities for instance, grows and shrinks by a process of accretion and emission, controlled by the fluctuation of point defects in its neighborhood [START_REF] Karlin | A First Course in Stochastic Processes[END_REF]. Thus, an appropriate framework for describing such events is the theory of stochastic processes [START_REF] Nicolis | Self Organization in Nonequilibrium Systems: From Dissipative Structure to Order through Fluctuations[END_REF][START_REF] Van Kampen | Stochastic Processes in Physics and Chemistry[END_REF] for which the size x of extended defects appears to be a random variable defined by a probability P(x, t) in the continuum approximation:

P(x, t) = dx dtW(x + δx → x, t)P(x + δx, t) -W(x → x + δx, t)P(x, t) (1)
where δx refers to the indexation of different states and W(x → x + δx, t) is the transition rate from state x + δx to state x. This description is extensively used to discuss a large class of phenomena such as chemical reactions [START_REF] Nicolis | Self Organization in Nonequilibrium Systems: From Dissipative Structure to Order through Fluctuations[END_REF] or in nucleation process under thermal equilibrium as pointed out by Becker and Doring (for a summary of the theory of nucleation under thermal equilibrium, see the book of Ghoniem [START_REF] Ghoniem | Instabilities and Self Organization in Materials[END_REF]).

To make things more concrete, we can determine the transition rate produced under ballistic mixing. Applying the seminal approach of Sigmund [START_REF] Simeone | Continuum approaches for modeling radiation-induced self-organization in materials: From the rate theory to the phase field approach[END_REF][START_REF] Sigmund | Theoretical aspects of atomic mixing by ion beams[END_REF], it reduces to:

W(x + δx → x, t) = p(x, δx, t) = p(x, δx)δ(t) (2)
Since the life time of a displacement cascade is of about 10 -9 seconds [START_REF] Sigmund | Particle Radiation and Radiation Effects[END_REF], W(x + δx → x, t) can be considered as a instantaneous function of time, i.e., its time dependence reduces to a Dirac. Under irradiation, the characteristic relocation length of a recoil depend on its kinetic energy and then varies. The function p(x, δx, t) can thus be approximated by a probability density function p R (x) for high energy displacement cascades, i.e., displacement cascades produced for high energetic particles (E > 10 keV.) The term R is a characteristic relocation distance which can be understood as the mean free path for the atomic relocation. Its second moment x 2 p R (x)dx is the ballistic diffusion coefficient and can be computed from binary collision approximation [START_REF] Sigmund | Theoretical aspects of atomic mixing by ion beams[END_REF] or extracted from Molecular Dynamics simulations.

It must be kept in mind that the kinetic energy of ion beams, used for simulating radiation damage produced in nuclear plants [START_REF] Baldinozzi | Evidence of extended defects in pure zirconia irradiated by swift heavy ions[END_REF], are of about 10 keV/amu, well below the Coulombian barrier, and they do not induce any fission nor disintegration in the materials. In nuclear plants, neutrons will not only generate damage, but also they will modify the chemical composition of the material [START_REF] Luneville | Impact of nuclear transmutations on the primary damage production: The example of Ni based steels[END_REF]. These chemical modifications are not taken into account in this paper.

The Rate Theory Approach

It has been clearly shown [START_REF] Nicolis | Self Organization in Nonequilibrium Systems: From Dissipative Structure to Order through Fluctuations[END_REF] that if the process W(x, t) modeling the effect of sub cascades on the microstructure of materials under consideration is a Poisson distribution, i.e describes a rare event, resolution of the Master Equation can be simplified. For a Poisson distribution, the absorption and emission rates depend no more on P(n, t) but only on < x > the first moment of the random variable x, i.e., the mean value of the distribution P(x, t). The dynamics associated to the growth of the cluster is no more given by a set of coupled differential equations but reduces to a simple differential equation describing the dynamics of the averaged cluster size < xP(x, t) >. In other words, it is possible to compute from the Master Equation, the first moment of the distribution according to:

< x >=< f (x) >≈ f (< x >) (3)
This simplification procedure is equivalent to the one used in reaction rate theory for chemical reactions [START_REF] Nicolis | Self Organization in Nonequilibrium Systems: From Dissipative Structure to Order through Fluctuations[END_REF]. The evolution of cluster size is now given by a deterministic equation in which all fluctuations, i.e., higher moments of P(x, t), are neglected in the function f . The Master Equation may then be understood as a mean field approximation.

Assuming the production rate of point defects are uniformly distributed, dynamics of point defects, i.e., vacancies(v) and interstitials (i), can be modeled by the following quasi-equilibrium "chemical" reactions [START_REF] Sizmann | The effect of radiation upon diffusion in metals[END_REF]:

dC i dt = K i -K iv C i C v -K vi C i C s dC v dt = K v -K iv C i C v -K vi C v C s (4)
where "s" refers to the different sinks (free surface, grain boundaries, dislocations loops, cavities) associated with the micro-structure. K i(v) = ν i(v) σ d φ where φ is the flux of impinging particles, σ d is a displacement cross section and is an estimation of the displacement cascades formation averaged over all crystalline direction, ν i(v) an efficiency factor varying from few percents to few tens of percent depending on the temperature and the displacement cascade density. The vacancy-interstitial recombination term K iv = 4πr 2 iv

(D i +D v ) Ω
where r iv is the radius capture in which a vacancy and an interstitial recombines and D i(v) is the thermal diffusion of the point defect in the medium and Ω the atomic volume within the effective medium theory framework [START_REF] Brailsford | The rate theory of swelling due to void growth in irradiated metals[END_REF]. The term K i(v)s can also be computed. In the assumption of the equivalent medium [START_REF] Brailsford | The rate theory of swelling due to void growth in irradiated metals[END_REF], it reduces to

k 2 i(v)s D * (c i(v) -c eq i(v) (T) where k -1 i(v)s
characterized the mean free path associated with the diffusion of a point defect to a sink and has the dimension of a length and c eq (T) is the equilibrium concentration of point defect in the medium (for a summary of coefficients used in Equation ( 4), see the article of Bullough et al. in [START_REF] Bullough | The temperature dependence of irradiation creep[END_REF]). Similar equations can be extended to study the growth rate of radiation-induced precipitates or dislocations. This is achieved by adding evolution laws [START_REF] Ghoniem | Instabilities and Self Organization in Materials[END_REF] leading to a set of differential equations for the radius of precipitates R or the density ρ of dislocations. Within the framework of the rate theory [START_REF] Bullough | The temperature dependence of irradiation creep[END_REF][START_REF] Sizmann | The effect of radiation upon diffusion in metals[END_REF], the late time evolution of the microstructure is given by the resolution of such a set of differential equations for which the temperature, the flux, the diffusion coefficients or the k -1 i(v)s terms, usually called the sink's strengths, are external parameters, given by different models. In the theory of dynamic systems [START_REF] Berge | L'ordre dans le Chaos[END_REF], these parameters ate called as control parameters and they control the possible bifurcation of the dynamical systems to different steady states named attractors. Solving the set of (partial) differential equations defining the dynamical system is often hopeless and need lots of computational resources. Knowing the list of possibles attract, i.e., steady states, allows the prediction of all possible radiation-induced microstructures able to emerge at late time. As the kinetic energy of incident particles (ions, neutrons, electrons) is dissipated into the formation of displacement cascades under irradiation, materials under irradiation belong to the universality class of dissipative systems. The energy dissipation leads to the formation of microstructure, unpredictable from the analysis of the thermodynamic phase diagram. The number and the nature (attractive, repulsive, etc.) of these attractors and the dependence of the long-term behavior of the system to its initial conditions and control parameters accessible to the experimentalists can then be achieved by an inspection of the set of (partial) differential equations [START_REF] Simeone | Continuum approaches for modeling radiation-induced self-organization in materials: From the rate theory to the phase field approach[END_REF][START_REF] Somorjai | Physical Chemistry[END_REF]. For instance, the qualitative evolution of the system can be obtained from the simple computation of the phase portrait as displayed in Figure 1. The phase portraits of the dynamical system, displayed in Figure 1, for instance, describe the evolution of a small number of variables and are appropriate for systems with small spatial extension (the mean free path k -1 is smaller than the distance between sinks). For larger systems, transport processes and/or non local (elastic) interactions can take place:

dX dt = G(X, µ) + D∇ 2 X ( 5
)
where X is a vector describing the variables of the problem (c i , c v , r ho, R). This equation can be considered as the archetype of Turing reaction-diffusion equations. Introducing in the last term of Equation ( 4) the divergence of fluxes J i = ∑ j D ij ∇X, such an equation can describe the evolution of the microstructure defined by X in large systems with appropriate boundary conditions.

Limitations of the Rate Theory Approach

Despite that rate theory is extensively used to describe the evolution of the microstructure of materials under irradiation, it suffers from two distinct limitations. Since atomic clustering in materials under irradiation is driven by the concentration of point defects (vacancies, interstitials), the complete distribution of clusters (di-mere, tri-mere, etc.) is thus given by a coupled hierarchy of discrete equations. This results in the cluster distribution not only being determined by the average size of clusters < xP(x, t) >, but also by all the different moments of P(x, t). only in a pure birth process, i.e., W(x + δx, x) = cte and other W term are null, P(x, t) is given by a Poisson distribution ( λ i = λ; µ i = 0) and the variance of the process is equal to its mean value. No theorem ensures that in radiation damage, W terms follow a Poisson distribution. If it is not the case, fluctuations (second and higher moments of P(x, t)) need to be explicitly taken into account even in the averaged equation. A hierarchy of equations describing the evolution of the different moments of P(x, t) must be computed. In these equations, the moments of P(x, t) are related to those of P(x + 1, t).

To obtain solutions, we thus need to perform some kind of truncation. Realistic truncation, if it exists, is obviously related to the system size and the correlation length over which two parts of the system "feel" each others. This leads to a drastic increasing of the set of differential equations and obviously the number of control parameters, restricting the predictive power of the rate theory.

Even if the rate theory offers the fascinating possibility to determine different steady states for the microstructure depending values of control parameters, underlying physically sounded mechanisms responsible for the evolution of the microstructure remains unclear. No symmetry argument can be applied to select some of all possible microstructures and up to now model parameters may be force fitted to experimental data. The large amount of fitted parameters for reaction constants limits the predictive nature of such an approach.

Moreover, the continuum theory of elasticity needs to be introduced in the rate theory to discuss diffusion-controlled kinetics due to interactions with applied fields.

The Phase Field Approach

The main difficulties of the rate theory is due to the fact that many parameters are needed to efficiency model the large time microstructure such as the strength of sinks k * for instance. These parameters evolve with the microstructure via elastic (or electric for ceramics) interactions leading to a feed back effect difficult to understand.

On the other hand, the spatial distribution of sub cascades responsible for the anisotropy of the point defects production terms are difficult to take into account in such a formalism. Kinetic Monte Carlo methods were then developed handling the spatial anisotropy of the defects formation. However, the accuracy of these simulations is directly related to the knowledge of main defects produced under irradiation and their associated diffusion paths. Up to now, no clear formalism such as for instance the detailed balance in thermal equilibrium exists under irradiation, assessing the validity of the choice of the set of defects and their diffusion paths.

During the last decade, the phase field approaches initially developed by Landau [START_REF] Landau | Theoretical Physics[END_REF] for describing phase transition in thermal equilibrium, has been successfully applied for understanding and modeling radiation-induced nano-patterns. This approach seems to reproduce kinetic Monte Carlo simulations and offers a guide to understand key parameters responsible for the nano-structures induced by irradiation. On the other hand, this approach provides the unique opportunity to compute generalized susceptibilities (elastic constants, dielectric constants, etc.) associated with radiation-induced nano-structures. We briefly describe the application of the phase field to radiation-induced patterning.

Foundation of the Phase Field

As it is the case for the rate theory, the phase field may be understood as a mean field theory. To understand this, we can assume that an atom in a disordered solid occupies a vertex of a geometric lattice, assumed cubic for simplicity. At the atomic scale, all sites of the underlying lattice are occupied by species (atoms or vacancy). Calling σ i the occupation number at the i -th site and d the dimension of the lattice defined by a characteristic length a, we can define "d-blocks". Every "d-block" will have a volume la d . We can define the averaged occupation σ(r) by a coarse-graining procedure:

σ(r) = 1 N l ∑ i σ i = ( a la ) d ∑ i σ i (6)
N l can be understood as the number of degrees of freedom. Obviously, different coarsegraining procedures can be used instead of a simple summation. However, this definition seems reasonable as long as N l is large. After performing an averaging, σ l (r) does not fluctuate much on microscopic scale but varies smoothly in space.

Of course, in general we need to specify l in order to determine σ l (r), but the coarse graining procedure we are applying will be useful only if the final results are independent of l at least in the spatial scales considered. Under equilibrium, we now must express the partition function in terms of H e f f (σ l (r)), where H e f f can be considered as an effective Hamiltonian [START_REF] Bellac | Equilibrium and Non Equilibrium Statistical Thermodynamics[END_REF]. Since we now have a system made up of "blocks" this effective Hamiltonian will be composed of two parts: a bulk component relative to the single blocks and an interface component relative to the interactions between the blocks.

When we suppose that every block of volume la d is separated from the rest of the system, σ l (r) inside every one of them is uniform only if the dimension of the blocks is much smaller than the correlation length, i.e., the characteristic length of variation of σ l (r) labeled ξ. When la << ξ, a non equilibrium free energy density f (σ(r)) may be introduced according to Landau [START_REF] Ginzburg | On the Theory of Superconductivity[END_REF][START_REF] Tolédano | Reconstructive Phase Transitions: In Crystals and Quasicrystals[END_REF]. This free energy density, f (r), describes the evolution of a uniform system made of non interacting blocks. The coarse graining procedure insures that σ l (r) results from a mean field theory.

To go a step beyond, we now must take into account the fact that adjacent blocks do interact. Since, as we have stated, σ l (r) does not vary much on microscopic scales, the interactions between the blocks must be such that strong variations of magnetization between neighboring blocks are energetically unfavorable. If we call δ l a vector of magnitude la that points from one block to a neighboring one, the most simple analytic expression that we can guess for such a term can be an harmonic one:

H int e f f = ∑ r ∑ δ l κ 2 (σ l (r) -σ l (r + δ l )) 2 (7)
where the factor 1/2 multiplying κ has been inserted for convenience. The term κ defines the stiffness of the material under consideration, i.e., the ability of the material to create interfaces between blocks. This term known as the Ginzburg term [START_REF] Landau | Theoretical Physics[END_REF][START_REF] Tolédano | Reconstructive Phase Transitions: In Crystals and Quasicrystals[END_REF] can also be considered as a first approximation of a general interaction between the blocks, namely as the first terms of a Taylor expansion of the real interaction energy. Since the linear dimension of the blocks la is much smaller than the characteristic length L of the system, we can treat r as a continuous variable and thus substitute the sum over r with an integral:

∑ r → 1 (la) d dr (8)
Finally, we obtain the following expression for the Hamiltonian:

H e f f = 1 (la) d [ f (σ l (r) + κ 2(la) d-2 |∇(σ l (r)| 2 ]dr d (9)
Application of the saddle point approximation allows computing the partition function Z(β) and allows identifying F = -k B Tlog(Z) to the stationary value σ(r) of the functional H e f f . The term |∇(σ l (r)| 2 insures that the Ginzburg-Landau free energy F may take into account inhomogeneities of the order parameter σ(r) [START_REF] Ginzburg | On the Theory of Superconductivity[END_REF][START_REF] Tolédano | Reconstructive Phase Transitions: In Crystals and Quasicrystals[END_REF]. The term κ = κ 2(la) d-2 is the cost associated with the formation of an interface. The Landau free energy describes the evolution of the system not only at equilibrium but also far from equilibrium. It must be kept in mind that the non equilibrium Landau free energy exhibits some minima corresponding to those associated with phase diagrams expected from equilibrium. As the free energy is not restricted to the equilibrium free energy, such an approach allows computing a kinetic path for the transition. As it is the case for the rate theory, application of the PF approach to radiation damage may allow computing radiation-induced microstructures.

The kinetics of the system is now given by a partial differential equation:

∂σ(x, t) ∂t = ∇ p L(σ(x -y, t))∇ p δF(T, σ(y, t) δσ(y, t) (10)
L(σ(x, t) is the Onsager coefficient associated with the kinetic path [START_REF] Khatchaturyan | Theory of Structural Transformation in Solids[END_REF]. This kinetic coefficient defines the relaxation time scale. From the definition of the order parameter (atomic position, atomic fraction, polarization, strain field, etc.) and the non equilibrium free energy, more generally thermodynamic potential, it is possible from Equation [START_REF] Bullough | The temperature dependence of irradiation creep[END_REF] to infer some features of the system: list of all possible low symmetry phases for a given symmetry of the "parent" phase, degeneracy of the "low symmetry" phases (number of variants) and susceptibilities (latent heat, heat capacity, etc.). In Equation [START_REF] Bullough | The temperature dependence of irradiation creep[END_REF],

δF(T,σ(x,t) δσ(y,t)
has a clear meaning. It is the chemical potential of the system. Kinetics occurring during a lower time scale cannot be handle by such relaxation equations. For non conservative OP [START_REF] Ginzburg | On the Theory of Superconductivity[END_REF], p is null. It is equal to 1 for conservative OP [START_REF] Cahn | On Spinodal Decomposition[END_REF]. Neglecting the dependence of Onsager coefficient with the OP, classical PF equations are recovered [START_REF] Hohenberg | Theory of dynamic critical phenomena[END_REF]:

∂σ(x, t) ∂t = -L δF(T, σ(x, t) δσ(x, t) (11) ∂σ(x, t) ∂t = M∆ δF(T, σ(x, t) δσ(x, t) (12)
The first equation holds for non conservative OP associated with ordering whereas the second equation holds for conservative OP associated with phase separation. L can be understood as a damping coefficient controlling the time relaxation of the OP to its equilibrium value. Usually, the term M is the mobility and is related to the diffusion coefficient of diffusing species. Introducing the H -1 inner product defined by

< f |g > H -1 =< (-∆) -1 f |g > L 2 for functions of L 2 (R d ) (d
is the dimension of the embedded space), the two equations are formally identical. Both dynamics are then gradient flows of the same Lyapunov functional, L[σ],associated with the H -1 or the L 2 metrics. Computing the inner product in the Sobolev space H -1 is equivalent to solve the Poisson equation ∆w =f associated with boundary conditions. Equation ( 13) may be derived from the application of the principle of minimal energy dissipation postulated by Onsager [START_REF] Onsager | Reciprocal Relations in Irreversible Processes I[END_REF]. This principle is based on the assumption that the system under consideration is close to an equilibrium condition, so that locally thermodynamic equilibrium can be assumed. As a consequence the concepts of equilibrium thermodynamics can be extended to these non-equilibrium conditions based on the linearization of different fluxes as a function of the generalized thermodynamic forces.

Assuming kinetic coefficients L and M do not depend on the order parameter, Equation ( 13) is a gradient systems and F(T, σ(r, t) is the Lyapunov of the system [START_REF] Nicolis | Self Organization in Nonequilibrium Systems: From Dissipative Structure to Order through Fluctuations[END_REF] and dF dt < 0. This implies that dynamics evolves to different minima of F(T, σ(r, t). As it is the case for the rate theory, different minima of F(T, σ(r, t) define a set of different attractors of the system. These attractors are separated by separatrices. These separatrices themselves make up an invariant set for the gradient flow dynamics. These gradient systems are dissipative systems since dF dt < 0 even when M and L are assumed independent of the order parameter.

These partial differential equations can be solve as soon as boundary and initial conditions are defined:

•

The initial condition is usually given by an uniform distribution σ of the order parameter in the integration volume V. Since σ is an attractor of these equations, a small random variable db is added to the uniform value of the order parameter. The mean value of this random variable db is null and its variance of the order of k B T [START_REF] Bray | Scaling and multi scaling in the ordereing kinetics of a conserved order paameter[END_REF].

•

For close systems, Neumann conditions must be applied to the the order parameter and the driving force, defined by the chemical potential

∂L[σ]
∂σ(x,t) . This implies ∇σ ad ∇ 3 σ are null over δV for the conservative Cahn-Hilliard equation. Obviously for the non conservative equation, only the second condition holds.

•

Neglecting edge effects, periodic conditions are usually applied for computing microstructures within the framework of the phase field theory. This implies that the order parameter σ(x, t) can be expanded in an infinite cosine set. Numerically, a finite set of discrete cosine transform (dct-I) is used to solve these equations for close systems [START_REF] Strang | The Discrete Cosine Transform[END_REF].

Application to Radiation Damage

Applying the PF approach for computing radiation-induced pattern implies to take into account the effect of radiation damage. Radiation damage is handled in two distinct ways:

•

Point defects produced during the collision step are assumed to only modify the mobility of species or more precisely the kinetic Onsager coefficients. Their concentration is supposed to evolve rapidly with time and reach a steady state. It may result from the resolution of RT equations. The interest of such an approach lies in the fact that this steady state may evolve with the nano-structure dictated par the solutions of the Phase Field approach. The nano-structure is no more freeze and defined at the beginning of the simulation such as in the RT but can evolve with time. Under this approach, the atomic fraction of point defects is always small and does not modify the value of the thermodynamic potential describing the formation of the nano-structure under equilibrium. For instance, the value of the stiffness coefficient trigging the formation of heterogeneous structures does not depend on the atomic fraction of point defects.

•

The motion of atoms under irradiation is dictated by an external dynamics resulting only from the ballistic mixing occurring inside sub-cascades. This ballistic mixing is due to a spatial averaging of atomic collisions occurring inside the sub-cascade. The flux of ejected particles is not proportional to the chemical potential of different species in the material as it is the case in thermal diffusion but depends on the atomic fraction of diffusing species. The key feature of the ballistic mixing lies on the fact that it exhibits a spatial dependence. As clearly pointed out by Sigmund [START_REF] Sigmund | Theoretical aspects of atomic mixing by ion beams[END_REF], it depends on the nature and the energy of incident particles and can be modeled by a simple diffusion equation. It can be analytically computed [START_REF] Sigmund | Particle Radiation and Radiation Effects[END_REF] or extracted from MD simulations [START_REF] Bukonte | Comparison of molecular dynamics and binary collision approximation simulations for atom displacements analysis[END_REF].

As a first approximation, the dynamics induced by the ballistic mixing may be modeled by the following equation:

∂σ ∂t = Γ(E, φ) (p R (y) -δ(y))σ(x -y)dy (13)
The term Γ(E, φ) defines the strength of the the mixing, i.e., the ability of irradiation to exchange atoms at a given position irrespective to their chemical nature. It is a function of the energy of primary knock-on atoms E and the flux φ of incident particles. p R x) is a probability density function modeling the ability to eject atoms at a distance x from its initial position in the sub-cascade. Generally speaking, this probability density function may be modeled by a Levy's function [START_REF] Simeone | Concentration profile distortion under ion beam mixing: An example of Levy flight[END_REF]. The term R can be understood as the mean free path associated to the ballistic mixing. It is a complex function of the energy and the mass of incident particles and atoms forming the material. For incident particles with kinetic energies below a few MeV, R is of the same order of magnitude than the unit cell parameter [START_REF] Bukonte | Comparison of molecular dynamics and binary collision approximation simulations for atom displacements analysis[END_REF].

Application of the phase field theory to radiation-induced microstructure implies that R must be larger than the coarse graining characteristic length la. Fixing the coarse-graining length to R implies that the PF approach is not able discussing modifications of interfaces below R. The PF approach then only provides a description of the microstructure over length scale larger than few nanometers forbidding the description of radiation-induced dislocations loops for instance.

Within the framework of the phase field theory, the radiation-induced microstructure results from a competition between thermal and ballistic effects. The first one is associated with the thermal evolution of the system and modeled by a PF equation and the second one results from a ballistic mixing generated inside sub-cascades:

∂σ(x, t) ∂t = M(T, φ)∆ ∂F(σ(x, t)) ∂σ(x, t) + Γ(E, φ) (p R (y) -δ(y))σ(x -y)dy (14)
where σ(x, t) is the order parameter proportional to the local atomic fraction of the species B in the AB alloys. This order parameter is by definition null at high temperature. The mobility M(T, φ) enhanced by the steady state concentration of point defects produced under irradiation appears to be the unique control parameter. Equation ( 14) illustrates the power of the phase field approach. Even if the late time evolution is modeled by a partial differential equation, as it is the case in the rate theory, only a unique control parameter physically sounded M(T, φ) is needed. Moreover, Equation ( 14) allows computing a Lyapunov functional for the system

L(t) = F[σ(x, t)] + Γ(E,φ) 2M(σ(x,t) G[σ(x, t)] where G(t) = 1 2 σ(y, t)g R (x -y)σ(x, t)dxdy and ∆g R = p R (σ(x)) -δ(x).
The function g R (x) quantifies the "averaged" spatial extension of the ballistic mixing induced by all sub cascades in the medium. As p R (x) displays an exponential-like tail associated with rare events, i.e., the ejection of atoms far from the center of the sub cascade [START_REF] Simeone | Concentration profile distortion under ion beam mixing: An example of Levy flight[END_REF], g R (x is also a decreasing function of x as expected. Assuming the ballistic mixing occurring in sub-cascades is isotropic and p R (x) = p R (x) =∝ exp(-x R ), g R (x) is isotropic and can be explicitly computed. A similar Lyapunov functional can be computed for non conservative OP [START_REF] Wilkes | Phase stability under irradiation-A rev iew of theory and experiment[END_REF].

The key point of this analysis is that dL[σ(r,t)] dt < 0. This inequality insures that the system is dissipative as expected. The energy of incident energetic particles is dissipated in many collisions during the slowing down of these particles in the medium and generate nano-patterns. The existence of such a Lyapunov implies that no chaotic nor oscillatory steady state exists [START_REF] Cross | Pattern Formation and Dynamics in Non Equilibrium Systems[END_REF] and its minimization allows drawing all possible steady states pro-duced under irradiation. This Lyapunov functional L[σ] can be understood as an "effective" free energy including irradiation damage. However, recent numerical simulations [START_REF] Luneville | Predicting Nonequilibrium Patterns beyond Thermodynamic Concepts: Application to radiation-induced Microstructures[END_REF] highlight the fact that radiation-induced patterns in non miscible binary alloys depend on the average composition of the alloy as displayed on Figure 2. This result, in agreement with kinetic Monte Carlo simulations [START_REF] Krasnochtchekov | Phase separation and dynamic patterning in Cu 1-x Co x films under ion irradiation[END_REF], highlights fundamental differences between systems under equilibrium and far from equilibrium. Such a dependence of radiation-induced steady states with the average composition forbids to describe the evolution of the microstructure in irradiated alloys applying the notion of phase diagram [START_REF] Enrique | Self organized Cu-Ag nanocomposites synthesized by intermediate temperature ion-beam mixing[END_REF][START_REF] Krasnochtchekov | Phase separation and dynamic patterning in Cu 1-x Co x films under ion irradiation[END_REF][START_REF] Martin | Phase stability under irradiation: Ballistic effects[END_REF][START_REF] Enrique | Nonequilibrium fluctuations, effective temperature, and effective interactions 10 driven by irradiation of alloys[END_REF], as it is the case under thermal equilibrium. The existence of an "effective" free energy is not sufficient to apply concepts of the thermal equilibrium to systems far from equilibrium. In fact, this functional is not homogeneous under irradiation but varies in space. Ground states of this functional are not homogeneous forbidding to establish any phase diagram under irradiation.

This example shows that the phase field offers the possibility to list all possible radiation-induced steady states tracking minima of the Lyapunov. A Lyapunov functional may also be computed from the rate theory formalism even if its computation is much more complicated and many parameters need to be known. Both theories then give the same information on the final microstructure. The main advantage of the phase field approach is its ability to propose a mechanism for the formation of radiation-induced patterns.

The self-organization of systems leading to the formation of patterns results from the invariance of the correlation function versus the space scale [START_REF] Nicolis | Self Organization in Nonequilibrium Systems: From Dissipative Structure to Order through Fluctuations[END_REF][START_REF] Hohenberg | Theory of dynamic critical phenomena[END_REF][START_REF] Cross | Pattern Formation and Dynamics in Non Equilibrium Systems[END_REF]. Computing the structure factor S(k, t) =< | σ(k, t)| 2 > (where < . > is a radial average and f is the spatial Fourier transform of f (r)) is the usual way to track the spatial invariance. Figure 3 illustrates this point showing structures factors (right panel) associated with a random (top graphs) and a self-organized (bottom graphs) distribution of Cu-rich precipitates in AgCu [START_REF] Simeone | Radiation-induced micro-structures as ground states of a Swift-Hohenberg Energy functional[END_REF].

Within the phase field theory, S(k, t) ∝ ( ∂L [σ] ∂[σ] ) -1 is directly related with the quadratic part of the Lyapunov via L The term ∆(T, φ) is the ratio between the a-thermal ballistic coefficient Γ(E, φ) and the radiation-enhanced mobility M(φ, T) and appears to be the unique control parameter of the problem:

[σ] = D(k)| σ(k, t)| 2 dk with D(k) = [-1 + k 2 + ∆(T, φ)g R (k)].

•

Low values of ∆(T, φ) implies that ballistic effects induced by irradiation can be neglected and that the system behaves as it does under thermal equilibrium. S(k, t) displays a maximum at k 0 ∝ t -1/3 as expected for a spinodal decomposition [START_REF] Khatchaturyan | Theory of Structural Transformation in Solids[END_REF]. • High values of ∆(T, φ) implies that ballistic effects dominate the micro-structure of the material under irradiation. As thermal effects are negligible, the large time microstructure results from a random distribution of Cu atoms in an Ag matrix. S(k, t) displays a maximum at k 0 = 0. • When ∆(T, φ) ≈ 1, a competition between thermal and ballistic effects occurs. At large time, S(k, t) displays a maximum at a non null k 0 value independent of time [START_REF] Luneville | Patterning in systems driven by non local external forces[END_REF]. The pattern shows the existence of this non null k 0 value of about few nm -1 . This implies that a nano-patterning occurs under irradiation for given values of temperature T and flux φ.

A large time, D(k) has a minimum k 0 and can be expanded in the neighborhood of this minimum:

D(k) = D(k 0 ) + d 2 D(k 0 ) 2dk 2 (k -k 0 ) 2 ≈ D(k 0 ) + d 2 D(k 0 ) 8k 2 0 dk 2 (k 2 -k 2 0 ) 2 (15)
Since k 0 is a minimum of D(k),

d 2 D(k 0 ) dk 2
is positive and the first order expansion of D(k) leads to the conservative form of the Swift-Hohenberg equation [START_REF] Cross | Pattern Formation and Dynamics in Non Equilibrium Systems[END_REF]. This equation is extensively applied to discuss liquid/crystal transition in unary systems [START_REF] Elder | Modelling elasticity in crystal growth[END_REF] It was successfully used to model dynamics of microstructural defects [START_REF] Rodriguez | Capturing Dislocation Half-Loop Formation and Dynamics in Epitaxial Growth Atomistically at Diffusive Time Scales[END_REF], diffusive atomistic dynamics of edge dislocations [START_REF] Berry | Diffusive atomistic dynamics of edge dislocations in two dimensions[END_REF][START_REF] Berry | Defect stability in PFC models: Stacking faults and partial dislocations[END_REF], plasticity and micro-mechanics of emergent patterns in plastic flows [START_REF] Biswas | Neutron small-angle scattering study of unmixing in f eCR alloys[END_REF] out of irradiation.

This example illustrates how the Phase Field can help to rationalize the appearance of radiation-induced nano-patterns in irradiated materials. Therefore, more information can be deduced from the phase field model. It may explain the formation of nano-patterns. In the example of the AgCu alloy, the minimum of D(k), occurring for ∆(T, φ) ≈ 1, is isotropic. However, ground states of L[σ(r) describing radiation-induced microstructures are non isotropic symmetry adapted functions for satisfying boundary conditions. The mechanism associated with the formation of nano-patterns can be clarify. It results from the symmetry breaking of L[σ(r) by ground states. This symmetry breaking generates an instability such as the Eckaus instability [START_REF] Ghoniem | Instabilities and Self Organization in Materials[END_REF]. This instability propagates via Goldstone modes [START_REF] Goldstone | Field theories with superconductor solutions[END_REF], i.e., without dissipation of energy [START_REF] Simeone | Continuum approaches for modeling radiation-induced self-organization in materials: From the rate theory to the phase field approach[END_REF].

The existence of nano-structures displayed in Figure 2 generalizes the notion of topological defects defined for equilibrium structures and are generally the rule beyond pattern forming instabilities, as it is the case in equilibrium phase transitions [START_REF] Hohenberg | Theory of dynamic critical phenomena[END_REF][START_REF] Cross | Pattern Formation and Dynamics in Non Equilibrium Systems[END_REF]. In graph c of Figure 2, disclination and geometrical dislocation are clearly visible.

The phase field approach points out that defects result from the existence of a set of unstable modes. Although each individual spatio-temporal patterns break transnational and rotational symmetries, different patterns grow in different part of the system implying the formation of various defects connecting these different steady states.

The formation of energy-less nano-structures via the propagation of Goldstone modes impacts the properties of materials. In a more precise way, the appearance of radiationinduced nano-structures may modify the response of the materials to external perturbation such as constraints or temperature gradients. By its ability to compute generalized susceptibilities [START_REF] Tolédano | Reconstructive Phase Transitions: In Crystals and Quasicrystals[END_REF], the phase field method then offers the unique opportunity to compute for instance variations of elastic constant induced by the formation of B-rich precipitates under irradiation [START_REF] Simeone | Symmetry Breaking Resulting from Long-Range Interactions in Out of Equilibrium Systems: Elastic Properties of Irradiated AgCu[END_REF]. As irradiation stops the coarsening of B-rich precipitates in non miscible alloys [START_REF] Luneville | Patterning in systems driven by non local external forces[END_REF], the characteristic size of nano-structured B-rich precipitates is about few nanometers in agreement with experimental results [START_REF] Simeone | Symmetry Breaking Resulting from Long-Range Interactions in Out of Equilibrium Systems: Elastic Properties of Irradiated AgCu[END_REF][START_REF] Chee | Dynamic self-organization in Cu alloys under ion irradiation[END_REF] in AgCu and are always in coherence with the underlying atomic lattice. Applying the formalism developed by Katchaturyan [START_REF] Katchaturyan | Elastic strain energy in inhomogeneous solids[END_REF] for computing elastic constants, variation of elastic constants resulting from the radiation-induced microstructure has been computed. Figure 4 displays the comparison between Young moduli computed from the application of the Katchaturyan's methods [START_REF] Simeone | Symmetry Breaking Resulting from Long-Range Interactions in Out of Equilibrium Systems: Elastic Properties of Irradiated AgCu[END_REF] and extracted from nano-indentation [49] experiments performed in Ag 0.42 Cu 0.58 thin films irradiated at different temperatures (for all experiments, the Poisson ratio is equal to 0.24 in agreement with numerical simulations). The experimental increasing of about 10% of the Young modulus with the temperature (navy triangle on Figure 4) can only be modeled taking into account the precipitate-precipitate dipoles elastic interactions (blue stars on Figure 4) highlighting the impact of infinite range on the elastic properties of irradiated alloys. Neglecting this interaction leads to a decreasing of the Young modulus with the temperature (red line on Figure 4) not experimentally observed.

Limitations of the Phase Field Approach

Despite that the phase field method seems to reproduce nano patterns observed in some irradiated alloys, its application to radiation damage is still questionable:

•

The theoretical foundations of phase field models lies in the ability to perform a coarsegraining procedure over a length scale la small in comparison with the characteristic length of the system L but large enough to neglect fluctuations of the order parameter proportional to (la) d/2 under equilibrium. Under irradiation, specific mechanisms of energy deposition may impose larger fluctuations over this length scale and then may be taken into account to define the coarse graining length scale. Whatever the choice of the coarse graining length, this approach is inaccurate to compute for instance the formation of radiation-induced dislocation nor evolution of the short range order.

•

The control parameter ∆(φ, T) is the ratio between a-thermal ballistic diffusion term and the radiation-enhanced mobility M(φ, T). This last term is directly related with the concentration of point defects produced under irradiation. This concentration depends on the late time microstructure, resulting from the calculation of the microstructure using the phase field approach, via the numbers and the density of sinks. This feedback effect must be studied in detail.

•

Since the dynamics of vacancies and interstitials created under irradiation cannot be described by any free energy functional, the phase field approach is inefficient for computing the evolution of point defects under irradiation.

Many authors apply the phase field approach for modeling the evolution of point defects under irradiation [START_REF] Zhang | Symmetry breaking during defect self-organization under irradiation[END_REF][START_REF] Li | Applications of the Phase Field methods in predicting microstructure and properties evolution of irradiated nuclear materials[END_REF]. They combine a Landau-Ginzburg functional to describe the spatial evolution of the concentration of point defects and a rate theory approach. However, the use of a Landau-Ginzburg term for describing agglomeration of vacancies is doubtful. Recent publications [START_REF] Li | Compositional patterning in irradiated alloys: Effective potential and effective interfacial energy[END_REF] confirm the analogy between the Swift-Hohenberg equation extensively used for describing out of equilibrium structures and the PF approach of ballistic mixing modelled by Equation ( 14) firstly pointed by Simeone et al. [START_REF] Simeone | Radiation-induced micro-structures as ground states of a Swift-Hohenberg Energy functional[END_REF]. However, they pointed out that this analogy may not be satisfied in all the patterning regime. Recent works [START_REF] Luneville | Predicting Nonequilibrium Patterns beyond Thermodynamic Concepts: Application to radiation-induced Microstructures[END_REF] confirm that patterns similar to those derived from the SH model can be observed in the pattern domain and that approach extends also to the phase separation regime induced by low irradiation fluxes.

Conclusions and Perspective

Studies of the behavior of materials maintained far from equilibrium at length scale larger that what we can handle by direct atomic simulations and smaller than what allows macroscopic continuum averaging displays difficulties. The fair agreement between simulation and experimental results demonstrates that the PF approaches is then a powerful tool able to simulate not only microstructure but also properties of irradiated materials. Application of this method allows predicting the unexpected patterns observed in non miscible alloys such as AgCu [START_REF] Chee | Dynamic self-organization in Cu alloys under ion irradiation[END_REF] under irradiation at large time. This method demonstrated that the notion of phase diagram cannot be applied under irradiation. By its ability to compute generalized susceptibilities, it also offers the unique opportunity to understand and explain modifications of properties of materials under irradiation.

The main interest of this mean field approach in comparison with the rate theory formalism, extensively used to describe the evolution of the microstructure in irradiated materials, lies in the small physically sounded number of control parameters in comparison with those used in the rate theory. The strength of this approach is also its weakness. Since the phase field approach is based on a coarse graining procedure and a minimization of an "effective" free energy density, it fails to model evolution of radiation induced point defects.

The existence of kinetic Onsager's coefficients, i.e., L and M, under irradiation as well as their dependence with the defect concentrations need to be clarified. In particular their dependence to the the order parameter will dictate the dynamics of the system. Since these coefficients can be considered as independent of the order parameter, the system is is gradient flow system and at equilibrium.

On the other hand, both rate theory and Phase field approach assume that reaction paths (meta-sable states and minima of the energy landscape) are independent of the environmental noise which does not reduced to k B T under irradiation. This point needs to be understood in detail [START_REF] Freidlin | Random Perturbations of Dynamical Systems[END_REF].

The combination of rate theory and phase field should give an accurate description of the microstructure of material under irradiation. To be achieved, efficient methods need to be developed to plug these two approaches. Combination of rate theory and PF could achieve both the computation of a realistic population of point defects and nano-patterns at large time scale. Extension of this method to irradiation by swift heavy ions which mimic the impact of fission products and to oxidation processes under irradiation, is still an open field.

Figure 1 .

 1 Figure 1. Example of phase portrait of solute atoms in precipitates (x) and vacancies (n) calculated under irradiation. Reprinted with permission from Ref. [19]. Copyright 1977 Elsevier. At large time, the system can reach two distinct attractors: a stable node (0,0) and a the saddle critical point (x*,n*). The existence of these attractor partition the abstract x-n plane in four distinct quadrants. The arrows displays the velocities of the solute atoms and vacancies in each quadrant (I to IV).

Figure 2 .

 2 Figure 2. Simulation of 3D nano-patterns of Ag cCu 1-c irradiated with 1 MeV Kr ions at 330 K with a flux of φ = 10 11 cm -2 s -1 . Different periodic microstructures emerge depending upon the average Ag concentration c ((a): 1D stripes (S); (c): cylinders forming hexagonal patterns (C); (e): 3D body centered cubic structures ( c = 0.65) (B); (g) solid solution ( c = 0.75) (SS)). Micro-structures associated with the co-existence of such domains are also plotted ((b): S+C; (d): C+B; (f): B+SS). The color stick defines the composition of radiation-induced nano-patterns. The blue and red colors refer to a null concentration of Ag and a null concentration of Cu respectively.

Figure 3 .

 3 Figure 3. Radiation-induced patterns simulated via the phase field approach of Ag 0.38 Cu 0.62 . The initial solid solution (graph a) leads to a labyrinthine structure (graph c) after irradiation with 1 MeV Kr ions at φ = 6 × 10 12 cm -2 s -1 , and T = 440 K. Associated Power Spectral Density |S(k, t)| 2 in the Fourier space are also plotted (graphs b,d). In these simulations, A rich (black) (B rich (white)) domains are associated with positive (negative) σ values (the 3D domain size is equal to 200 in reduced units). The PSD is radially symmetric and peaks sharply around k 0 (intense white ring in graph d) as expected for a patterning whereas it is flat as expected for a random distribution of atoms (graph b). On the 2D cut of the 3D pattern (graph c), extended defects such as disclination and geometrical dislocation lines are clearly visible.

Figure 4 .

 4 Figure 4. Comparison between experimental Young moduli (navy triangles) extracted from nanoindentation experiments performed on Ag 0.42 Cu 0.58 thin films and computed from application of the Phase Field approach including elastic terms (blue stars) versus the temperature. The red line is displays Young moduli resulting from a simple lever rule neglecting the formation of nano-structures under irradiation.

Acknowledgments: We acknowledge C. Deranlot (CNRS/Thales) for providing AgCu thin films, A. Millard (CEA/DES/DM2S) for helpful advices on mechanics, C.O. Bacri (CNRS/IN2P3/SCALP) for having performed different irradiations of AgCu samples and S. Urvoy (CEA/DES/DMN) for having measured the Young moduli on irradiated AgCu samples.

Funding: This research received no external funding.

Institutional Review Board Statement: Not applicable.

Informed Consent Statement: Not applicable.

Data Availability Statement: Not applicable.

Author Contributions: Conceptualization, L.L., P.G. and D.S.; methodology, D.S. and L.L.; software, L.L. and D.S. All authors have read and agreed to the published version of the manuscript.

Conflicts of Interest:

The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.


domain: info spi

text:
Julie Farkouh
email: julie.farkouh@etd.univ-evry.fr

Audrey Ahoukeng Donwoung
email: audrey.ahoukeng-donwoung@etd.univ-evry.fr

Nazim Agoulmine
email: nazim.agoulmine@univ-evry.fr

Intelligent Orchestration of Containerized Applications in a Cloud Infrastructure (Short Paper)

Introduction

Resource contention and performance interference between co-located applications can be a serious problem in a multi-tenant infrastructure. This problem is difficult to address by existing orchestrators and may cause applications performance downgrade, extra maintenance costs, as well as service-level agreement (SLA) violations. Therefore, a refined and robust container orchestration system is the key factor in controlling overall resource utilization, energy efficiency, and application performance. However the highly dynamic nature of modern applications, diverse features of cloud workloads and environments to take into account considerably raise the complexity of orchestration mechanisms [START_REF] Zhong | Machine learning-based orchestration of containers: A taxonomy and future directions[END_REF]. One particular orchestration problem is the deployment of containers in appropriate hosts with the right required resources as resources play a major role in the containerized applications performance. This problem is not fully taken into account by modern orchestrator since the type of application and the deployment mode are not known to the orchestrators and therefore it is not able to determine the appropriate required resources.

Deploying Applications or Websites that are in production mode is an important concern for many companies willing to externalize their Applications and services in the Cloud Computing. Since allocated resources play a major role in their performance and the end-users satisfactions, determining the most appropriate level of resources is of paramount importance. Today mainstream deployment is using containers and orchestrator such as Kubernetes. The concerns is the difficulty to identify the required level of resources to allocate and maintain to these applications during their life cycle.

To overcome this problem, we propose to artificial intelligence and more precisely Decision Tree to predict the applications types and modes to take into account this information during orchestration. We adopt in this work a use case where the deployed applications are containerized Web Applications (CWA) and containerized DataBase Applications (CDA) which could be in test mode or production mode. The orchestrator that receives this information can deploy therefore deploy them in the most appropriate nodes with the most appropriate resources (CPU/Memory/Network) and achieve better resource allocation decision and performances.

Some related works on this issues are presented here. In [START_REF] Ishak Harichane | A proposal of kubernetes scheduler using machine-learning on cpu/gpu cluster[END_REF], Ishak et al. proposed an intelligent scheduler in Kubernetes environment using ML based on the execution prediction of the applications and the nature of the processed data in order to better select the appropriate device, CPU or GPU. In [START_REF] Santos | Towards network-aware resource provisioning in kubernetes for fog computing applications[END_REF] , Jose et al. proposed Network-Aware scheduler for container-based applications in Smart City deployments that added the parameters of network quality to the calculation score of the Kubernetes scheduler in order to take it into account when selecting the appropriate deployement node.

Kubernetes which is one of the most popular orchestrating tools for containers. It is an open source scheduling platform that automates Linux container operations: Distribution, Replication, Load-balancing, Availability, Higher-level interfaces to composition features and Docker which is an open source containerization platform -with standardized executable components that combines application source code with the operating system(OS), libraries and dependencies that are required to run that code in any environment.

Kubernetes is doing a great job in scheduling, deploying and provisioning containers based on the user-requested RAM and CPU for their containers in proportion to the free RAM and CPU available in the infrastructure. Unfortunately, those are not only the resources that should be taken into account when deploying containers as other resources may play an important role in the application performance based on what it really needs such as Network resources and Disk Input and Output.

Eventually, these addition requirements should be taken into account during orchestration and appropriate related resources should be computed based on the type of application container service and the context in which they are deployed. If applications are deployed in production mode, more attention to all resources needs to be addressed to cope with the potentially large amount of service requests. Whereas deploying an application in a development mode for a single user developer is not that important as there is no online demands and as a result, high performance requirements are not necessary. Unfortunately, Kubernetes Scheduler that it is responsible for scheduling and deploying the containers into the nodes does not take into account these parameters (e.g. Network Quality , I/O ..) and the mode in which applications are deployed (development, stage, production). As a result, the deployment solution could be inappropriate resulting in bad performances.

Proposed Solution

As a solution, we introduce a model which uses ML to automatically predict the type of containerized applications to execute and their corresponding mode for deployment. Once predicted, the solution triggers the appropriate deployment of the applications in the appropriate nodes of the containerized infrastructure using an orchestrator (in our case Kubernetes).

In our model, we have addressed two types of applications: Containerised Web-based Applications (CWA) and Containerised Database Applications (CDA) in production mode. The rational behind the selection of these two applications is that there are the most commonly deployed industry grade applications.

Therefore, the ML Model will predict the Container Type and Mode and based on these predicted values, the model will assign the container to the appropriate computing node. This means that if the ML Model predicts that the container is for a Web application Service in a production mode, it will automatically assign and deploy it into one of the nodes with the highest network speed interface and if the predicted value is database container in production mode, then it will assign it to one of the nodes with the highest available RAM capacity.

During the prediction phase, the model reads the desired containers configurations and extracts the features that correspond to their type and deployment mode. These features are then introduced in a previously trained ML Model that uses classification learning to classify the containers based on their type and mode.

Regarding the deployment phase, the proposed solution first searches for the nodes with the highest network interface speed from all the available nodes in its cluster to host the Webapplication containers (since in production mode, this feature is one of the most important). Similarly, it searches for the node with the highest available RAM capacity to host to the Database Server containers.

• Container Image and Port that the container will use -these two features are the most important indicators to predict the application type and the ones commonly used. The used images are the ones available in the docker hub -official website of the dockers images -and widely known for web services like Apache, Nginx or IIS and the relevant ports would be also the generally the most used one: 80, 443, 8080 . For Data Base Service, the name of the images that are the most used are Mysql, Oracle or Postgress Sql and corresponding ports are usually 3306, 1521.

• Name of the Container, as in IT, we should always use as a best practices in naming process, names that indicate clearly the service to help maintenance such as, logging, debugging, auditing, etc. For the web application, it could vary between the following names "Websites", "Front ends", "Webs". Similarly, for the Database application, names could be "Database", "MyDataBase", "DB" in repeated way like for web services.

RAM we assume in our work the RAM would start from 256/512 M bytes of RAM for the development mode to up to 4 G bytes of RAM for applications in the production mode..

These are the features we took into account to train our ML Model to classify them into two types of outputs (Web or database) and two predicted environment mode outputs ( Production or Development ).

Implementation and Experimentation

For our Deployment, we first trained our model in Matlab Classification App -Analysis Trees on simple Dataset of the Web and Database services properties previously introduced based on containers' image, Port, Name indicating reserved RAM. We then exported our Model into Matlab Production Server in order to use it for production. The provided interface is a Rest API. Next, we used Kubernetes configured into one master node and two Worker nodes with different RAM Capacities and Network Interface Speed. Worker Node1 = 8 Gg RAM , 10,000 Mb/s Traffic Speed Worker Node2 = 10 Gg RAM, 1000 Mb/s Traffic Speed After training the ML Mode, the sequence of performed tasks are the following: 1. Our Model first finds the node wirh the highest speed interface which is worker-node 1 and label it "prodweb" as for Webservices in Production.

2. It also finds the node with the highest RAM Capacity Node which is worker-node 2 and labels it "proddb" for DataBase Production 3. Then it reads the configuration file of the pod configuration containing the container specifications ( YAML File ) and extracts the features of the containers and send them via a HTTP Request API to our trained ML Model, The predicted values of the ML Model would be one of three cases:

• Containerised Web-based Applications (CWA) in Production Mode.

• Containerised Database Applications (CDA) in Production Mode.

• CWA or CDA in Development Mode or Other application types in Production / Development Mode but we didn't follow-up thi czse in the deployment case we did not handle the deployment, just the first two first.

4. The pod configuration file is then updated to launch the the deployment in the corresponding node.

We tested the accuracy of our ML Model. Since our data was simple and the output is only composed of three possible values, the model achieved 93% accuracy.

We tested also the robustness of the prediction with general values in the dataset. For example, we changed the names to some unrelated strings (that do not contain the pattern ) to the application while keeping the other values as it in the datase (e.g. the Ram to a lower/higher values than from the ones in our dataset , Changing two features like the Name and Image. Despite these changes the ML was still able to predict correctly the type and mode, thanks to the Decision Tree (DT) approach.

Limitations and Future Works

In our work we presented a simple Model which is able to predict the containerized application type and mode based on features contained in the configuration file. Based on this prediction, the scheduler aims to deploy these applications in the most appropriate nodes in the cluster. The prediction worked well with the type and mode, but we aim to extend it to address other type of services like Domain Name Service ( DNS ), Mail Services, Virtual Private Networks services (VPN), etc. However, this work has several limitations we plan to overcome. Our scheduler state is considered in an initial status with no consideration for other potentially running containers competing for the same computing and networking resources. Indeed, the solution identifies the nodes with the highest network interface and RAM capacity without considering the current bandwidth or RAM utilization. A possible approach would be to combining our model with the native Kubernetes scheduler score tool that it used in node selection i.e. using the result as factor to computing the node selection in a pod, similarly to the work presented in Jose et al. [START_REF] Santos | Towards network-aware resource provisioning in kubernetes for fog computing applications[END_REF]. we envision also to expand our dataset to predict several more containerized applications types and have more real accuracy results for better orchestration mechanisms based on the container application type.

Conclusion

In this work, we presented a ML based scheduler which relays on "Type" and "Mode" Awareness of containerized applications for better resources allocation and node selections decisions. With this solution, it is possible to detect the nature of applications: Web Containerized Applications and Data Base Containerized Applications and their production mode (test, production). This information is used to improve the allocation of resources (Network/RAM). We used two popular open-source projects that are Docker and Kubernetes, to validate our model which helps to identify additional needed resources beyond RAM and CPU (that are used in the original Kubernetes scheduler). We implemented a poc of our model a separate scheduler. Our future works is to integrated our model with Kubernetes scheduler and allow is to be aware of more information to perform a better scheduling and deployment .


Randane Boulahia

Fahmi Zaïri
email: fahmi.zairi@polytech-lille.fr

Yuri Voznyak

Jean-Michel Gloaguen

Repeated equal-channel angular extrusion of polypropylene: Processing routes and back-pressure influence

Keywords: High plastic deformation, Equal-channel angular extrusion, Multi-pass, Microstructure, Mechanical properties

Multi-pass equal channel angular extrusion (ECAE) process of polypropylene was investigated at room temperature using a 90° die-angle tooling outfitted with a back-pressure.

The process was repeated until eight times using two different extrusion orientation routes: route A where the sample orientation is kept constant between each extrusion and route C where the sample is rotated by 180°. The crystalline textures were examined by WAXS measurements and the changes on the mechanical response were observed by full-field strain measurements during tensile tests. The use of a back-pressure turns the deformation behavior, along the flow direction of the ECAE products, from highly heterogeneous (characterized by a peculiar periodic shear-banding behavior) to roughly homogeneous and leads to completely different plastic processes in the crystalline phase. The textural differences subsequent to the multi-pass processing and the use of a back-pressure are the source of significant differences in tensile response along the flow direction. The observations show that the polypropylene processed via route C reaches more easily a saturation state than that obtained via route A with a strong beneficial influence of a back-pressure.

Introduction

Equal-channel angular extrusion (ECAE) is a high-plastic-deformation technique consisting to extrude a specimen through two equal cross-section channels intersecting at a certain angle to form a single internal channel. The material is sheared across a thin layer when it crosses the angular zone of the channel. In ECAE products, substantial microstructure changes are thus induced by high degrees of simple shear plastic deformation. The geometrical constraints assigned to the channel keep constant the initial cross-section of the extruded specimen that can be thus extruded repetitively to accumulate severe plastic strains. The orientation of the specimen between successive extrusions (e.g. processing routes) may be changed provoking various textures and microstructures. The ECAE-induced deformation behavior depends on intrinsic material features [START_REF] Zaïri | Influence of the initial yield strain magnitude on the materials flow in equal-channel angular extrusion process[END_REF] and various processing parameters such as number of extrusion sequences, processing route, friction, temperature, back-pressure (BP) and die geometry [START_REF] Aour | A computational study of die geometry and processing conditions effects on equal channel angular extrusion of a polymer[END_REF][START_REF] Aour | Finite element analysis of plastic strain distribution in multipass ECAE process of high density polyethylene[END_REF].

These parameters impact in turn the properties of the ECAE products. The application of the process on polymers has become the subject of considerable interest as reported in recent overviews of the state-of-the-art [START_REF] Beloshenko | Equalchannel angular extrusion of polymers[END_REF][START_REF] Beloshenko | Severe plastic deformation of polymers[END_REF].

In this paper, the ECAE process is applied to polypropylene. Although the large deformation ability of this semicrystalline polymer is a well-known feature [START_REF] Favier | Microstructure transformation and stress-strain behavior of isotactic polypropylene under large plastic deformation[END_REF][START_REF] Codolini | Mechanical characterization of thin injectionmoulded polypropylene specimens under large in-plane shear deformations[END_REF], the limit of possible accumulated plastic strains upon multi-pass ECAE remains largely undefined [START_REF] Campbell | Equal channel angular extrusion of polyalkenes[END_REF][START_REF] Phillips | Simple shear deformation of polypropylene via the equal channel angular extrusion process[END_REF][START_REF] Kitagawa | ECAP process in polypropylene[END_REF][START_REF] Zaïri | Steady plastic flow of a polymer during equal channel angular extrusion process: experiments and numerical modeling[END_REF][START_REF] Boulahia | Deformation behaviour and mechanical properties of polypropylene processed by equal channel angular extrusion: Effects of back-pressure and extrusion velocity[END_REF][START_REF] Aour | Experimental and numerical study of ECAE deformation of polyolefins[END_REF][START_REF] Beloshenko | Equal-channel multiangular extrusion of semicrystalline polymers[END_REF][START_REF] Wang | Effects of processing route on morphology and mechanical behavior of polypropylene in equal channel angular extrusion[END_REF][START_REF] Wang | Microstructural evolution and its effect on mechanical properties of isotactic polypropylene by shear plastic deformation at elevated temperatures[END_REF].

Furthermore, the peculiar behavior of periodic shear-banding in polypropylene reported by Boulahia et al. [START_REF] Boulahia | Deformation behaviour and mechanical properties of polypropylene processed by equal channel angular extrusion: Effects of back-pressure and extrusion velocity[END_REF] in one ECAE pass was never investigated by repetitions of the process.

The goal of this paper is to examine in the case of polypropylene the relationships between key processing parameters (number of extrusion sequences, processing route and use of a BP), load/deformation behavior of the sample upon the solid-state extrusion process and microstructure/mechanical properties of the extruded sample.

Material and methods

Material and specimens

The studied material is a semicrystalline polypropylene supplied by Goodfellow in the form of 10 mm thick compression-molded plates. The number-average molar weight Mn and the weight-average molar weight Mw, given by the supplier, are 25kg/mol and 180kg/mol, respectively. Thermal properties and crystal content were determined by means of differential scanning calorimetry experiments. The glass transition temperature Tg and the melting temperature Tm are -20°C and 170°C, respectively. The crystal content is about 55%.

Prismatic specimens of 75 × 10 × 10 mm were cut from the as-received polypropylene plates.

They were then surfaced on the cutting facets and polished. Although material isotropy is expected, the specimens were cut along the same direction.

ECAE experiments

A stainless steel ECAE die adapted on an electromechanical Instron-5800 testing machine was used to extrude the specimens through a L configuration channel. A picture of the experimental set-up is shown in Figure 1. The polypropylene piece is placed into the entrance channel and then pressed into the exit channel under the action of a ram that moves vertically thanks to the displacement of the cross-head of the testing machine. To avoid the formation of an open corner gap at the intersection area, i.e. shearing zone, between the die wall and the piece bottom surface, leading to a bending mechanism [START_REF] Zaïri | Steady plastic flow of a polymer during equal channel angular extrusion process: experiments and numerical modeling[END_REF], a BP may be applied in opposition to the flow direction. The polypropylene pieces were extruded until eight times under two extrusion orientation scenarios illustrated in Figure 1: route A (0° rotation between extrusions) and route C (180° rotation between extrusions). All ECAE experiments were conducted at an extrusion speed of 45 mm/min and at room temperature. The two channels were lubricated using silicone grease before each extrusion.

WAXS measurements

Wide-angle X-ray scattering (WAXS) patterns were recorded at room temperature with the Xray beam along the flow and transverse directions of the ECAE-processed pieces. The slices were prepared by careful diamond sawing. WAXS experiments were performed using the Nifiltered Cu-Kα radiation from a 2kW Panalytical sealed tube operated on a INEL 2000 generator at 40kV and 20mA. The 2D-WAXS patterns recorded on a digital CCD camera from Photonic Science Ltd were corrected for background scattering.

Tensile tests and full-field strain measurements

Dumbbell-shaped specimens with dimension 12 × 4 × 2 mm were punched out from the ECAE-processed pieces as illustrated in Figure 2, the tensile axe being the ECAE flow direction. The tensile tests were carried out at room temperature and at a nominal strain rate of 10 -3 s -1 using a displacement-control function. Load and displacement versus time were recorded during the tests and converted to average stress-strain curves. The digital image correlation method was used to determine the full-field displacements during the tensile tests.

Randomized gray levels distributions were obtained using India ink sprayed with an airbrush at the specimen surface. The latter was illuminated by a strong white light beam. Images were registered by means of a CCD camera Imager E-lite 2M (LaVision GmbH) at regular intervals with resolution of 290 pixels/mm and size of 1024 × 1024 pixels. The full-field displacement data were extracted using DaVis Software 8.0 (StrainMaster, LaVision GmbH) and then converted into true strains by means of a homemade visual basic program.

Results and discussion

Extrusion behavior

The extrusion load is plotted as a function of the ram displacement in Figure 3 for the different ECAE processing parameters including number of extrusions, processing route and use of a BP. The figure gives an overview of the different types of plastic flow displayed by the polypropylene upon ECAE process. When no BP is applied, periodic load oscillations associated with plastic flow instabilities are observed. This peculiar behavior was firstly reported by Boulahia et al. [START_REF] Boulahia | Deformation behaviour and mechanical properties of polypropylene processed by equal channel angular extrusion: Effects of back-pressure and extrusion velocity[END_REF] in one ECAE pass. It would be the result of a "stick-slip" process due to an unstable balance between the intrinsic material yield strength and the friction force between the die wall and the specimen surface. As a consequence, a periodic shear-banding behavior of the ECAE-processed piece occurs manifested by alternating sheared/unsheared bands tilted at roughly 45° from the flow direction along the sample length. The peculiar behavior is accompanied by more or less pronounced periodic waves along the top surface of the ECAE-processed piece. The unsheared bands remained opaque in transmitted light whereas the sheared bands appeared translucent due to the reduction of the light scattering resulting from the destruction of the spherulitic structure [START_REF] Phillips | Simple shear deformation of polypropylene via the equal channel angular extrusion process[END_REF][START_REF] Boulahia | Deformation behaviour and mechanical properties of polypropylene processed by equal channel angular extrusion: Effects of back-pressure and extrusion velocity[END_REF]. Since the processing conditions are the same for routes A and C, the extrusion behavior is identical until the beginning of the second extrusion. The final deformation along the flow direction of the sample after the second and subsequent extrusions is strongly different when comparing both routes. This is evidenced by significant differences in load response. In route A, homothetic responses are obtained since it involves no rotation of the ECAE-processed piece between successive extrusions. The responses using route C show alternate increase and decrease in load after the maximum since the second and subsequent shearing of the ECAE-processed piece follow effective rotations of 180°. The shift of the maximum load corresponds to a material densification resulting in a sample length reduction. It worth noticing that due to severe deformation at room temperature, the multi-extruded specimen can undergo surface damage but only conditions in which surface damage does not appear are presented in this study. The application of a BP leads to an increase in load magnitude due to both the pressure-sensitivity of the yield strength and the friction. The BP has a significant effect on the filling status of the extruded specimen and its deformation behavior upon ECAE. It significantly contributes to the disappearance of the alternating bands (and the periodic waves along the top surface of the specimen) as well as the load oscillations and consequently promotes the homogeneity of the plastic deformation along the ECAE-processed piece.

Whereas no saturation is observed in route A, the application of a BP in route C results in a stationary state of the load after the second extrusion, since the evolution observed beyond that is almost negligible.

X-ray scattering

Figure 4 provides the 2D-patterns after one ECAE pass. The results evidence a quite perfect crystalline isotropy along the flow direction whereas a strong crystallographic texturing of the (hk0) planes is observed along the transverse direction. The result for a shear-banding behavior, appearing when no BP is applied, is provided in Figure 4a for three 2D-patterns recorded from three successive translucent-opaque-translucent bands. A quite perfect crystalline isotropy remains in the middle unsheared band whereas a strong diagonal texturing appears in the two adjacent sheared bands. The latter corresponds to a preferred orientation of the (110), ( 040) and (130) planes. As shown in Figure 4b, the application of a BP results in a more uniform texturing along the extruded specimen and changes the plastic processes in the crystalline phase. They are characterized by scattering reinforcements on quadrant positions of the inner reflections. This reveals a double crystalline texturing of the (hk0) planes (with a tilted orientation of the chain axis with respect to the ECAE shear plane) that would be due to the formation of a double population of oriented crystallites [START_REF] Boulahia | Deformation behaviour and mechanical properties of polypropylene processed by equal channel angular extrusion: Effects of back-pressure and extrusion velocity[END_REF].

Figures 5 and6 presents the 2D-patterns in the sheared bands after four and eight passes, respectively. The specimen orientation during successive passes plays a significant role in the crystallographic texturing due to the change in the ECAE shear plane orientation (Figure 1). A tilting of the strong diagonal texturing is observed with the consecutive shearing in route A.

The application of a BP changes the plastic processes in the crystalline phase due to a tilting of the active crystallographic shear plane due to the pressure sensitivity. The route C leads to quite similar 2D-patterns with quadrant (hk0) reflections symmetric with regard to the ECAE shear plane. The observed saturation state at the macro-scale induced by the application of a BP in route C is confirmed by these local observations.

Tensile response

Figure 7 shows the stress-strain curves of the polypropylene specimens machined from the ECAE-processed pieces. The ECAE process induces profound modifications of the polypropylene response with a significant effect of the processing parameters. The textural modifications subsequent to the first ECAE pass lead to both lower yield strength and strainsoftening after yielding as compared to the as-received material. In route A, there is a significant increase of the yield strength between the second and the eight passes, but the differences observed on the yield strength are attenuated beyond the second pass when applying a BP. In route C, the number of passes has a slight effect on the stress magnitude but the specimens present superimposed stress-strain behaviors when applying a BP. The latter results highlight the fact that the extruded material polypropylene approaches its saturation state after the second passes; as observed also in the load response. It appears that the multipass processing has a significant effect on the strain-hardening ability and the strain at break. The strain at break increases as a function of extrusion number in route A and decreases in route C. When a BP is used for extrusion, the strain at break does not change significantly in multi-pass processing in route C whereas it decreases in route A.

Illustrative examples of the strain localization along the gage length of the polypropylene specimens are presented in Figure 8 for a given cross-head displacement. The pattern of strain localization is completely different following the processing parameters. The periodic shearbanding occurring without BP results in multiple strain localizations (Figures 8a andc). The sheared bands with their crystallographic texturing are indeed highly favorable to plastic shear during the tensile testing along the flow direction. Beyond the yield point, the increase in stress allows to gradually deform the unsheared bands. As expected, the local strain is quite homogeneous when a BP is applied (Figures 8b andd).

Conclusions

Multi-pass ECAE experiments were conducted on polypropylene using a 90° die. The ECAEinduced deformation behavior, the resulting structural evolution and the resulting mechanical properties were investigated as a function of processing parameters. Several conclusions may be drawn from the present work:

• Without BP, the deformation behavior of the ECAE-processed piece is manifested by peculiar periodic shear-banding for routes A and C. The application of a BP triggers homogeneous plastic deformation by preventing instabilities and leads to completely different crystalline textures.

• The tensile responses of polypropylene processed via route C reach easily a saturation state contrary to those obtained via route A. No significant modifications in terms of strain-hardening ability and strain at break are found in route C when a BP is used for extrusion while the strain at break decreases without applying a BP.

Figure 1 .

 1 Figure 1. Solid state extrusion equipment placed in the testing machine. The drawing shows the processing routes A and C.

Figure 2 .Figure 3 .Figure 4 .Figure 5 .Figure 6 .Figure 7 .Figure 8 .

 2345678 Figure 2. Preparation of a tensile specimen from an ECAE-processed piece.