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Comprehensive nuclear materials 1 15 phase field methods Comprehensive nuclear materials 1 15 phase field methods Comprehensive nuclear materials 1 15 phase field methods Comprehensive nuclear materials 1 15 phase field methods Comprehensive nuclear materials 1 15 phase field methods Comprehensive nuclear materials 1 15 phase field methods
1.15 Phase Field Methods P Bellon University of Illinois at Urbana-Champaign, Urbana, IL, USA ß 2012 Elsevier Ltd All rights reserved 1.15.1 Introduction 411 1.15.2 1.15.3 1.15.4 1.15.4.1 1.15.4.2 1.15.4.2.1 1.15.4.2.2 1.15.4.2.3 1.15.4.2.4 1.15.5 References General Principles and Applications of PF Modeling Quantitative PF Modeling PF Modeling Applied to Materials Under Irradiation Challenges Specific to Alloys Under Irradiation Examples of PF Modeling Applied to Alloys Under Irradiation Effects of ballistic mixing on phase-separating alloy systems Coupled evolution of composition and chemical order under irradiation Irradiation-induced formation of void lattices Irradiation-induced segregation on defect clusters Conclusions and Perspectives 412 418 420 420 421 421 423 426 427 428 430 Abbreviations 1D CVM KMC ME PF PFM SIA One-dimensional Cluster variation method Kinetic Monte Carlo Master equation Phase field Phase field model Self-interstitial atom 1.15.1 Introduction Electronic and atomistic processes often dictate the pathways of phase transformations and microstructural evolution in solid materials For quantitative modeling of these transformations and evolution, it is thus effective, and sometime necessary, to rely on methods using some representation of atoms and of their dynamics, as for instance in molecular dynamics simulations (see Chapter 1.09, Molecular Dynamics) and atomistic Monte Carlo simulations (see Chapter 1.14, Kinetic Monte Carlo Simulations of Irradiation Effects) While these atomistic methods can now simulate quite accurately the evolution of specific alloy systems, these simulations are nevertheless limited to small length scales, from a few to 100 nm Molecular dynamics is furthermore limited to small time scales, typically in the nanosecond range, although in some cases, new developments have made it possible to obtain atomistic simulations at much longer times (see Chapter 1.14, Kinetic Monte Carlo Simulations of Irradiation Effects) An alternative modeling approach is to replace the many microscopic degrees of freedom of the system of interest by the few mesoscopic variables that are sufficient to provide a realistic description This approach has been widely used in many disciplines, and well-known examples are the Fourier and Fick equations, which describe the diffusive transport of heat and chemical species, respectively This approach is also commonly used in modeling the evolution of point defects, in particular, during irradiation (see Chapter 1.13, Radiation Damage Theory and Sizmann1) The work of Cahn and Hilliard2– and Landau and Lifshitz (see for instance Tole´dano and Toledano6) provided a way to include the contributions of interfaces to chemical evolution, thus making it possible to model heterogeneous and multiphase materials Kinetic models based on these descriptions are broadly referred to as phase field (PF) methods, since the microstructure of a material is fully characterized by a few mesoscopic field variables such as concentration, magnetization, chemical order, or temperature One key assumption of this approach is that the variables chosen to describe the state of the system vary smoothly across any interface or, in other words, that interfaces are diffuse This assumption finds a natural justification in the theory of critical phenomena, since the 411 412 Phase Field Methods interface thickness diverges at the critical temperature.7 Diffuse interface models offer some advantages over sharp interface models,8 in particular, for the modeling of complex microstructures Furthermore, the PF approach can be extended to include macroscopic variables other than the local composition, making it possible to describe chemical order–disorder transitions, solid–liquid reactions, displacive transformations, and more recently dislocation glide PF methods and applications have been recently reviewed by Chen,9 Emmerich,10 and Singer-Loginova and Singer.11 This chapter focuses on solid–solid phase transformations, with a particular emphasis on transformations and microstructural evolution relevant to irradiated materials While conventional PF modeling lacks atomic resolution, the main interest in this technique comes from the fact that it can provide the evolution of large systems, exceeding the micrometer scale, over very long time scales, from seconds to centuries Recent developments have led to the introduction of PF models (PFMs) that possess atomic resolution,12–26 the so-called PF crystal models This model, which can be seen as a density functional theory for atoms, appears very promising, although at this time it is not clear whether it can reproduce correctly the discrete nature of pointdefect jumps from one lattice site to a neighboring lattice site The PF crystal model is not covered in this chapter, so the interested reader should consult the above references This chapter is organized as follows Section 1.15.2 introduces the key concepts and steps employed in conventional, that is, phenomenological PF modeling, and provides some illustrative examples Section 1.15.3 focuses on important recent developments toward quantitative PF modeling, whereby evolution equations are rigorously derived by coarse-graining a microscopic model This approach provides a full treatment of fluctuations and thus makes it possible to study fluctuation-controlled reactions, such as nucleation of a second phase The capability of PFMs to reach large time and length scales makes them an attractive tool for simulating the evolution of materials relevant to nuclear applications, in particular, for alloys subjected to irradiation Applying PF modeling to these nonequilibrium materials, however, raises new challenges, as is discussed in Section 1.15.4.1 Some selected results of PF modeling applied to irradiated materials are presented in Section 1.15.4.2 Finally, conclusions and perspectives are given in Section 1.15.5 1.15.2 General Principles and Applications of PF Modeling The first step in PF modeling lies in choosing and defining the fields of interest These continuous variables are functions of space and time, and they are in most cases scalar fields, such as temperature, or the concentration of some chemical species of interest In systems with solid–liquid interfaces, a phenomenological field variable is introduced in such a way that it varies continuously from to as one goes from a fully solid to a fully liquid phase Multidimensional fields can be used as well, for instance, to describe the local composition of a multicomponent alloy, the local degree of chemical order, or the local crystallographic orientation of grains These multidimensional fields may transform like vectors under symmetry operations, thus leading to a vectorial representation of the system and tensorial expressions for mobilities (as will be discussed later), but there are cases for which the multidimensional fields cannot be reduced to vectors.27 In all cases, an averaging procedure is necessary to define continuous field variables for systems that are intrinsically discrete at the atomic scale Various averages can be used, including (1) a spatial average over representative volume elements, which will correspond to the cells used for evolving the PF variables; (2) a spatial and temporal average; or (3) a spatial and ensemble average The spatial averaging method is used most often, although in many cases the exact conditions of the averaging procedure are not defined Section 1.15.3 will cover a model where this coarse-graining is performed explicitly and rigorously The last two averaging procedures are rarely explicitly invoked, although one of their advantages is that a smaller volume can be used for the spatial average, thanks to the additional averaging performed either in time or in the configuration space of a system ensemble Turning now to the kinetic equations used to describe the evolution of these field variables, an important distinction is whether the field variable is conserved or nonconserved For the sake of simplicity, the following discussion focuses on alloy systems Let us consider two simple examples, one where the field variable is the local composition, C(r,t), in a binary A–B alloy system, and a second example, also for a binary alloy system, but this time with a fixed composition and where chemical ordering takes place The degree of chemical order is described by the field S(r,t) For the sake of convenience, one Phase Field Methods may normalize that field such that S(r,t) ¼ corresponds to a fully disordered state and S(r,t) ẳ ặ1 to a fully ordered state The first field variable C(r,t) is globally conserved – assuming here that the system of interest is not exchanging matter with its environment This imposes the constraint that the time evolution of the field variable at r is balanced by the divergence of the flux of species exchanged between the representative volume centered on r and the remainder of the system: @Cr;t ị ẳ rJ r;t ị ½1 @t One then makes use of linear response theory in the context of thermodynamics of irreversible processes28 to linearly relate the flux J(r,t) to the driving force responsible for this flux Here, this driving force is the gradient of the chemical potential mr ; t ị ẳ dF =dCr ; t Þ, where F is the free energy of the system for a compositional field given by C(r,t) The resulting evolution equation is thus @Cðr;t Þ dF ẳ r Mr ẵ2 @t dCr;t ị where M is a mobility coefficient In contrast, for the nonconserved order parameter S(r,t), its evolution is directly related to the free energy change as S(r,t) varies, so that by making use of linear response theory again @Sr;t ị dF ẳ L @t dSr;t ị ẵ3 where L is the mobility coefficient for the nonconserved field S(r,t) Two important consequences of eqns [2] and [3] are worth noticing First, although all extrema of the system free energy (i.e., minima, maxima, saddle points) are stationary states, often in practice only the minima can be obtained at steady state due to numerical errors Second, the stationary state reached from some initial state may not correspond to the absolute minimum of the free energy In order to overcome this problem, noise can be added to transform these deterministic equations into stochastic (Langevin) equations, as will be discussed in Section 1.15.3 Following the work of Cahn and coworkers2–4 and Landau and Ginzburg,6 the free energy F is decomposed into a homogeneous contribution and an heterogeneous contribution Treating the inhomogeneity contribution as a perturbation of a homogeneous state, one finds that, in the limit of small amplitude and long wavelength for this perturbation, the lowest order correction to the homogeneous free 413 energy is proportional to the square of the gradient of the field variable For instance, returning to the simple example of an alloy described by the concentration field C(r,t), the total free energy can be written as ẵ4 F fCr ; t ịg ẳ dV ẵf Cị ỵ krCị2 V where f (C) is the free-energy density of a homogeneous alloy for the composition C, and k the gradient energy coefficient, which is positive for an alloy system with a positive heat of mixing A similar expression can be used in the case of a nonconserved order parameter, for example, S(r,t), or more generally, in the case of an alloy described by nC conserved order parameters and nS nonconserved order parameters ð F ¼ dV f ðC1 ; C2 CnC ; S1 ; S2 SnS Þ V ỵ nC X pẳ1 kp rCp ị2 ỵ nS X q ij ri Sq rj Sq ẵ5 qẳ1 An implicit summation over the indices i and j is assumed in the last term of eqn [5] The number of nonzero and independent gradient energy coefficients q ij for the nonconserved order parameters is dictated by the symmetry of the ordered phase Specific examples, for instance for the L12 ordered structure, can be found in Braun et al.27 and Wang et al.29 The free energy can also be augmented to include other contributions, in particular those coming from elastic fields using the elasticity theory of multiphase coherent solids pioneered by Khachaturyan,30 in the homogeneous modulus case approximation This makes it possible to take into account the effect of coherent strains imposed by phase transformations or by a second phase, for example, a substrate onto which a thin film is deposited.31 Two important interfacial quantities, the excess interface free energy and the interface width, can be derived from eqn [5] for a system at equilibrium We follow here the derivation given by Cahn and Hilliard.2 Considering the case of a binary alloy where two phases may form, referred to as a and b, and with respective B atom concentrations Ca and Cb, the existence of an interface between these two phases results in an excess free energy s ð s ẳ dV ẵf Cị ỵ krCị2 CmeB CịmeA ẵ6 V where meA and meB are the chemical potential of A and B species when the two phases a and b coexist at 414 Phase Field Methods equilibrium At equilibrium, this excess free energy is minimum A homogeneous free energy Df referenced to the equilibrium mixture of a- and b-phases is introduced as Df Cị ẳ f Cị ẵCmeB ỵ CịmeA ẳ CẵmB Cị meB ỵ CịẵmA Cị À meA ½7 (Note that the ‘D’ symbol in Df in eqn [7] does not refer to a Laplacian.) The variational derivative of this excess energy with respect to the concentration field is given by ds @Df @k ½8 ¼ À 2kDC À ðrCÞ2 dC @C @C At equilibrium, the excess free energy s is minimum, and the concentration field must be such that ds=dC ¼ Thus, @Df @k @ ẳ 2kDC ỵ rCị2 ẳ krCị2 ị ẵ9 @C @C @C Equation [9] must hold locally for any value of the concentration field along the equilibrium profile joining the a- and b-phases, and this can only be satisfied if krCị2 ẳ Df ẵ10 for all values of C(r) It is interesting to note that eqn [10] means that the equilibrium concentration profile is such that, at any point on this profile, the homogeneous and inhomogeneous contributions to the total free energy are equal The interfacial excess free energy is thus given by ẵ11 s ẳ dV Df ẳ dV krCị2 V V This last integral over the spatial coordinates can be rewritten as an integral over the concentration field Assuming a one-dimensional (1D) system for simplicity, Cðb Cðb dC krC ¼ s¼2 Ca dC pffiffiffiffiffiffiffiffiffi kDf ½12 Ca In order to proceed further, it is necessary to assume a functional shape for the concentration profile or for the homogeneous free energy Df Expanding the free energy near the critical point Tc yields a symmetric double-well potential for the homogeneous free energy,2 which we write here as 2C À 2C Df ẳ Dfmax 1ỵ ½13 Cab Cab with Cab ¼ Cb À Ca ¼ À 2Ca ¼ 2Cb À Using eqn [10], the equilibrium concentration profile can now be obtained: rffiffiffiffiffiffiffiffiffiffiffi ! Cab Dfmax x ỵ ẵ14 Cxị ẳ k Cab Integration along this equilibrium profile from eqn [11] yields the interfacial energy pffiffiffiffiffiffiffiffiffiffiffiffiffi s ¼ Cab kDfmax ½15 Furthermore, the width of the equilibrium profile we, which is defined as the length scale entering the argument of the hyperbolic tangent function in eqn [14], is given by r Cab k ẵ16 we ẳ Dfmax In this conventional approach to PFMs, Dfmax and k are phenomenological coefficients Equations [15] and [16] play an important role in assigning values to these coefficients for a specific alloy system The excess interfacial energy may be known experimentally or it may be calculated separately, for instance by ab initio calculations.32 If the interfacial width we is also known, one can obtain Dfmax and k from an inverse solution of eqns [15] and [16] (note that Cab is given by the equilibrium phase diagram) Even if we is not known, values for Dfmax and k can be chosen to yield a prescribed value for s In all cases, it is important to recognize that any microstructural feature that develops during the simulations is expressed in units of we At elevated temperatures, as T ! Tc , s vanishes2 while we goes to infinity, and therefore, at high enough temperatures, interfaces are diffuse, thus meeting this essential requirement underlying the PF method The PF eqns [2] and [3] are usually solved numerically on a uniform mesh with an explicit time integration, using periodic boundary conditions, when surface effects are not of interest When the free energy contains an elastic energy contribution, it is quite advantageous to use semi-implicit Fourierspectral algorithms (see Chen9 and Feng et al.33 for details) Variable meshing can also be employed, in particular to better resolve interfaces when they tend to be sharp, for instance at low temperatures A few examples selected from the literature serve to illustrate the capacity of PFMs to successfully reproduce a wide range of phenomena In particular, Khachaturyan30 and his collaborators34–38 proposed a microelasticity theory of multiphase coherent solids, Phase Field Methods which has been widely used to include a strain energy in the overall free energy A method for systems with strong elastic heterogeneity has been proposed by Hu and Chen,39 which includes higher order terms that are usually neglected in Khachaturyan’s approach Figure illustrates the anisotropic morphology of Al2Cu precipitates growing in an Al-rich matrix.32 Bulk-free energies were calculated using a mixedspace cluster expansion technique, with input from first-principle calculations for about 40 different ordered structures with full atomic relaxations Interfacial energies were calculated at T ¼ K from first-principle calculations as well, using configurations where the Al-rich solid solution and the tetragonal y0 -Al2Cu coexist For the elastic strain energy calculations, the elastic constants of y0 -Al2Cu were calculated ab initio An important feature of this system is that both elastic and interfacial energies are strongly anisotropic, and the PF approach makes it possible to include these anisotropies Furthermore, when the high-aspect-ratio y0 -phase forms, its growth kinetics will be anisotropic as well, which can be included in a phenomenological way by introducing a dependence of the mobility on the orientation of the precipitate–matrix interface Figure illustrates that these three anisotropies, interfacial, elastic, and kinetic, are required to reproduce the morphology of y0 precipitates Isotropic Figure illustrates another effect of coherency stress on microstructural evolution, this time for an A1–L10 order–disorder transition in a Co–Pt alloy.40 The tetragonal distortion accompanying the ordering reaction leads to the formation of self-organized tweed patterns of coexisting (cubic) A1- and (tetragonal) L10-phases As seen from Figure 2, the agreement between experimental and simulated microstructures is remarkable Ni and Khachaturyan proposed recently that, in order to minimize elastic energy during transformations involving symmetry changes and lattice strain, a pseudospinodal decomposition is likely to take place, leading to 3D chessboard patterns.41 PF modeling has also been used extensively to study martensitic transformations,34–38,42–44 phase transformations in ferroelectrics45–57 (see also the recent review by Chen58 on that topic), transformations in thin films,47,59–65 grain growth and recrystallization,66–81 and microstructural evolution in the presence of cracks or voids.82–84 A recent extension of PFMs has been the inclusion of dislocations in the models,85–87 by taking advantage of the equivalence between dislocation loops and coherent misfitting platelet inclusions.88 This approach has been applied, for instance, to study the interaction between moving dislocations and solute atoms,89 or to study the influence of dislocation arrays on spinodal decomposition in thin films.61 Rodney et al.87 have pointed out, Interface only (b) (a) Int + strain Strain only (c) Int + strain + kinetics Experiment 50 nm 50 nm (d) 415 (e) (f) Figure Phase field simulations of y0 -phase precipitation in Al–Si–Cu alloys at 450 C, illustrating that strain, interfacial, and kinetic anisotropies are required to reproduce experimental morphologies Reprinted with permission from Vaithyanathan, V.; Wolverton, C.; Chen, L Q Phys Rev Lett 2002, 88(12), 1255031–1255034 Copyright by the American Physical Society 416 Phase Field Methods (a) (b) (c) Time (d) (e) (f) Figure Comparison between transmission electron microscopy experimental observations (a–c) and phase field modeling (d–f) of formation of chessboard pattern in Co39.5Pt60.5 cooled from 1023 K to (a) 963 K, (b) 923 K, and (c) 873 K The scale bar corresponds to 30 nm Reproduced from Le Bouar, Y.; Loiseau, A.; Khachaturyan, A G Acta Mater 1998, 46(8), 2777–2788 however, that the artificially wide dislocation cores required by the above approach lead to weak shortrange interactions These authors have introduced a different PFM for dislocations, which allows for narrow dislocation cores As an illustration of that model, Figure shows the development of dislocation loops and their interaction with hard precipitates in a 3D g/g0 single crystal It is interesting to note that dislocation loop initially expands by gliding in the soft g channels, until the local stresses are large enough for the dislocation to shear the hard g0 -phase The above presentation of the PF equations leaves certain questions open First, the maximum homogeneous free energy difference Dfmax involves the free energy of the unstable state separating the two minima at Ca and Cb It is thus been questioned90 whether this quantity can be rigorously defined from thermodynamic principles If one employs mean field techniques such as the cluster variation method (CVM)91–95 to derive the homogeneous free energy of an alloy, Dfmax is in fact very sensitive to the approximation used, and generally decreases as the size of the largest cluster used in the CVM increases.96 Kikuchi97–99 has argued that, in order to resolve this paradox, Df should not be considered as the free energy of any homogeneous state, but that it should be understood as the local contribution to the free energy of the system along the equilibrium composition profile The second set of questions relates to the gradient energy coefficients k and in eqn [5] In many applications of PF modeling, these coefficients are taken as phenomenological constants that can be adjusted at will, as long as the microstructures are scaled in units of k1/2 or 1/2 Such an approach, however, is problematic for many reasons First, when one scalar field variable is employed, for instance C(r,t), a regular solution model,2,100 or equivalently a Bragg–Williams approximation,101 establishes that the gradient energy coefficient is not arbitrary but that it is directly proportional to the interaction energy between atoms, that is, to the heat of mixing of the alloy Furthermore, in the most general case, k should in fact be composition and temperature dependent Starting from an atomistic model, rigorous calculations of k are possible by monitoring the intensity of composition fluctuations as a function of their wave vector, and using the fluctuation–dissipation theorem.100 In the case of a simple Ising-like binary alloy, it is observed that k varies as Cð1 À CÞ, where C is the local composition of the alloy.100 Furthermore, when more than one field variable is employed, care should be taken to consider all possible contributions of field heterogeneities to the free energy of the system, as the different fields may be coupled Symmetry considerations are important to identify the nonvanishing terms, but it Phase Field Methods (a) (b) the interfacial anisotropy can be fitted to experiments or to atomistic simulations Let us now return to the mobility coefficients M and L introduced in eqns [2] and [3] For the sake of simplicity, many PF calculations are performed while assigning an arbitrary constant value to these coefficients An improvement can be made by relating the mobility to a diffusion coefficient In the case of M, for instance, in order to make eqn [1] consistent with Fick’s second law for an ideal binary alloy system, one should choose M¼ (c) (d) (e) (f) Figure Phase field modeling of the evolution of a dislocation loop (red line) in a g (dark phase)/g0 (white phase) under applied stress Reproduced from Rodney, D.; Le Bouar, Y.; Finel, A Acta Mater 2003, 51(1), 17–30 may remain challenging to assign values to these nonvanishing terms that are consistent with the thermodynamics of the alloy considered Another important point is that interfacial energies are in general anisotropic In order to obtain realistic morphological evolution, it is often important, and sometimes even absolutely necessary, to include this anisotropy, for example, in the modeling of dendritic solidification The symmetry of the mesh chosen for numerically solving the PF equations introduces interfacial anisotropy but in an unphysical and uncontrolled way One possible approach to introduce interfacial anisotropy is to let k vary with the local orientation of the interface with respect to crystallographic directions.11,32 Another approach is to rely on symmetry constraints27,30 to determine the number of independent coefficients in a general expression of the inhomogeneity term, see eqn [5] In both approaches, the different coefficients entering 417 C1 Cị ~ D kB T ẵ17 where C is the average solute concentration and D~ the interdiffusion coefficient In both cases, the simulated times are expressed in arbitrary units of MÀ1 or LÀ1, thus precluding a direct connection with experimental kinetics This problem is also directly related to the lack of absolute physical length scales in these simulations Moreover, using a 1D Bragg–Williams model composed of atomic planes, Martin101 showed that M is not a constant but is in fact a function of the local composition along the equilibrium profile A complete connection between atomistic dynamics and M will be made in Section 1.15.3 Similar to the discussion on coupling between various fields for the gradient energy terms, kinetic coupling is also expected in general The kinetic couplings between composition (a conserved order parameter) and chemical ordering (a nonconserved order parameter) are revealed by including sublattices into Martin’s 1D model and deriving the macroscopic evolution of the fields from the microscopic dynamics In that case, atoms jump between adjacent planes.102,103 As a result, instead of the mere superposition of eqns [2] and [3], the kinetic evolution of coupled concentration and chemical order in a binary alloy is given by @Cr;t ị dF ẳr M1 r @t dCr;t ị dF ỵ r M2 r dSðr;t Þ @Sðr;t Þ @F dF ẳ L1 ỵ r L2 r @t @Sr;t ị dSr;t ị dF ỵ r L3 r dCr;t Þ ½18 where L1 is a mobility coefficient, and L2, L3, M1, and M2 are second-rank mobility tensors, since they 418 Phase Field Methods relate diffusional fluxes (vectors) to chemical potential gradients (vectors) In the case of cubic crystalline phases, second-rank tensors reduce to scalars, but in many ordering reactions, noncubic phases form, thus leading to anisotropic mobility Vaks and coworkers104 have also derived PFMs for simultaneous ordering and decomposition starting from microscopic models These works, however, illustrate the fact that it would be quite difficult, especially for multidimensional field variables, to assign correct values to the kinetic coefficients for a given alloy system by relying solely on a phenomenological approach 1.15.3 Quantitative PF Modeling The PF equations introduced in Section 1.15.2, that is, eqns [2] and [3], are phenomenological, and one particular consequence is that they lack an absolute length scale All scales observed in PF simulations are expressed in units of the interfacial width we of the appropriate field variable As discussed in the previous section, for the case of one scalar conserved order parameter, this width we and the excess interfacial free energy s are directly related to the gradient energy coefficient k and the energy barrier between the two stable compositions Dfmax (see eqns [15] and [16]) Beyond the difficulty of parameterizing k and Dfmax to accurately reflect the properties of a given alloy system, the phenomenological nature of these coefficients creates additional problems In particular, as the number of mesh points used in a simulation increases, the interfacial width, expressed in units of mesh point spacing, remains constant if no other parameter is changed Increasing the number of mesh points thus increases the physical volume that is simulated but does not increase the spatial resolution of the simulations If the intent is to increase the spatial resolution, one would have to increase k so that the equilibrium interface is spread over more mesh points Equilibrium interfacial widths in alloy systems typically range from a few nanometers at high temperatures to a few angstroms at low temperatures In the latter case, if the interface is spread over several mesh points, it implies that the volume assigned to each mesh point may not even contain one atom This raises fundamental questions about the physical meaning of the continuous field variables, and practical questions about the merits of PF modeling over atomistic simulations Another important problem related to the lack of absolute length scale in conventional PF modeling concerns the treatment of fluctuations Fluctuations arise owing to the discrete nature of the microscopic (atomistic) models underlying PFMs Furthermore, fluctuations are necessary for a microstructure to escape a metastable state and evolve toward its global equilibrium state, such as during nucleation Fluctuations, or numerical noise, will also determine the initial kinetic path of a system prepared in an unstable state The standard approach for adding fluctuations to the PF kinetic equations is to transform them into Langevin equations, and then to use the fluctuation– dissipation theorem to determine the structure and amplitude of these fluctuations For instance, in the case of one conserved order parameter, the Cahn–Hilliard diffusion equation, that is, eqn [2], is transformed into the Langevin equation: @Cr;t ị dF ẳ r Mr ỵ xr;t Þ ½19 @t dCðr;t Þ where xðr;t Þ is a thermal noise term The structure of the noise term can be derived using fluctuation dissipation105,106: hxr;t ịi ẳ hxr;t Þxðr0 ;t Þi ¼ À2kB TMr2 dðr À r0 Þdðt À t Þ ½20 where the brackets h i indicate statistical averaging over an ensemble of equivalent systems However, eqn [20] does not include a dependence of the noise amplitude with the cell size, which is not physical Even if this dependence is added a posteriori, it is observed practically that this noise amplitude gives rise to unphysical evolution, as reported by Dobretsov et al.107 While these authors have proposed an empirical solution to this problem by filtering out the short-length-scale noise in the calculation of the chemical potentials, a physically sound treatment of fluctuations requires a derivation of the PF equations starting from a discrete description Recently, Bronchart et al.100 have clearly demonstrated how to rigorously derive the PF equations from a microscopic model through a series of controlled approximations We outline here the main steps of this derivation The interested reader is referred to Bronchart et al.100 for the full derivation These authors consider the case of a binary alloy system in which atoms migrate by exchanging their position with atoms that are first nearest neighbors on a simple cubic lattice A microscopic configuration is defined by the ensemble of occupation variables, or Phase Field Methods spin values, for all lattice sites, C ¼ fsi g, where si ẳ ặ1 when the site i is occupied by an A or a B atom, respectively The evolution of the probability distribution of the microscopic states is given by the following microscopic Master Equation (ME): à X @PðCÞ ẳ W C ! Cij ịPCị @t i; j ỵ X i; j W Cij ! CịPCij ị ẵ21 where the * symbol in the summation indicates that it is restricted to microscopic states that are connected to C through one exchange of the i and j nearest neighbor atoms, resulting in the configuration Cij The next step is to coarse-grain the atomic lattice into cells, each cell containing Nd lattice sites It is then assumed that local equilibrium within the cells is achieved much faster than evolution across cells The composition of the cell n, cn, is given by the average occupation of its lattice site by B atoms, and thus cn ¼ 0; 1=Nd ; ; Nd =Nd A mesoscopic configuration is fully defined on this coarse-grained system ~ by C ¼ fcn g A chemical potential can be defined within each cell and, if this chemical potential varies smoothly from cell to cell, the microscopic ME, eqn [21], can be coarse-grained into a mesoscopic ME: 2 à ~ X @PðCÞ a ba ~ ~ ~ ~ y lmn Cịexp ẳ mm Cị mn Cịị PCị @t d 2d n;m ỵ gain term ẵ22 where a is the lattice parameter and d the number of lattice planes per cell (i.e., Nd ¼ (d/a)3), y is the ~ attempt frequency of atom exchanges, lmn ðCÞ is a mobility function that is directly related to the microscopic jump frequency, b ¼ ðkB T ÞÀ1 , and ~ mn ðCÞ is the chemical potential in cell n The * symbol over the summation sign indicates that the summation over m is only performed over cells that are adjacent to the cell n; the first term on the righthand side of eqn [22] represents a loss term, and there is a similar gain term, which is not detailed The mesoscopic ME eqn [22] can be expanded to the second order using 1/Nd as the small parameter for the expansion The resulting Fokker–Planck equation is then transformed into a Langevin equation for the evolution of the composition in each cell n: ðnÞ @cn a y X ~ ~ ~ lnm C ịẵmm C ị mm C ị ỵ zn t ị ½23 ¼ @t d kB T m 419 where the noise term zn ðt Þ is a Gaussian noise with first and second moments given by hzn ðt Þi ¼ hzn ðt Þzn ðt Þi ¼ a2 X ~ lnp ðC Þdðt À t Þ Nd d p hzn ðt Þzm ðt Þi ¼ À2 a2 ~ lnm ðC Þdðt À t Þ Nd d ðnÞ ½24 While the structure of eqns [23] and [24] is quite similar to that of the phenomenological eqns [19] and [20], there are several key differences in these two descriptions First, thermodynamic quantities such as the homogeneous free-energy density and the gradient energy coefficient are now cell-size dependent These quantities can be evaluated separately using standard Monte Carlo techniques.100 Second, the mobility coefficients, and thus the correlations in the Langevin noise, are functions of the local concentration, as well as of the cell size Bronchart et al.100 applied their model to the study of nucleation and growth in a cubic A1ÀcBc system for various cell sizes, d ¼ 6a, d ¼ 8a, and d ¼ 10a The supersaturation is chosen to be small so that the critical nucleus size is large enough to be resolved by these cell sizes As seen in Figure 4, for a given supersaturation, the evolution of the volume fraction of precipitates is independent of the cell size and in very good agreement with fully atomistic kinetic Monte Carlo (KMC) simulations (not shown in Figure 4) The above results are important because they show that it is possible to derive and use PF equations that retain an absolute length scale defined at the atomistic level The point will be shown to be very important for alloys under irradiation On the other hand, the work by Bronchart et al.100 clearly highlights the difficulty in using quantitative PF modeling when the physical length scales of the alloy under study are small, as for instance in the case of precipitation with large supersaturation, which results in a small critical nucleus size, or in the case of precipitation growth and coarsening at relatively moderate temperature, which results in a small interfacial width In these cases, one would have to reduce the cell size down to a few atoms, thus degrading the validity of the microscopically based PF equations since they are derived by relying on an expansion with respect to the parameter 1/Nd Phase Field Methods Volume fraction 420 0.04 C = 0.160, d = 8a C = 0.165, d = 8a C = 0.170, d = 8a 0.02 C = 0.170, d = 6a C = 0.170, d = 10a 0.00 ´ 106 ´ 106 −1 Time (unit: q ) Figure Evolution of the volume fraction of precipitates with time for a three-dimensional binary alloy A1ÀcBc using the microscopically derived phase field eqns [23] and [24] Parameters a and d are the lattice parameter and the number of lattice planes in h100i directions For a given concentration, C ¼ 0.17, the precipitation kinetics is equally well resolved with three different cell sizes Reprinted with permission from Bronchart, Q.; Le Bouar, Y.; Finel, A Phys Rev Lett 2008, 100(1) Copyright by the American Physical Society 1.15.4 PF Modeling Applied to Materials Under Irradiation 1.15.4.1 Challenges Specific to Alloys Under Irradiation The PFMs discussed so far are broadly applied to materials as they relax toward some equilibrium state In particular, the kinetics of evolution is given by the product of a mobility by a linearized driving force, see for instance eqns [2] and [3] In the context of the thermodynamics of irreversible processes,28 the mobility matrix is the matrix of Onsager coefficients Irradiation can, however, drive and stabilize a material system into a nonequilibrium state,108 owing to ballistic mixing and permanent defect fluxes, and so it may appear questionable at first whether linearized relaxation kinetics is applicable A sufficient condition, however, is that these different fields undergo linear relaxation locally, and this condition is often met even under irradiation A complicating factor arises from the presence of ballistic mixing, which adds a second dynamics to the system on top of the thermally activated diffusion of atoms and point defects A superposition of linearized relaxations for these two dynamics is valid as long as they are sufficiently decoupled in time and space, so that in any single location, the system will evolve according to one dynamic at a time KMC simulations indicate that, for dilute alloys, this decoupling is valid except for a small range of kinetic parameters where events from different dynamics interfere with one another.109 A second issue is that PFMs, traditionally, not include explicitly point defects Vacancies and interstitials are, however, essential to the evolution of irradiated materials, and it is thus necessary to include them as additional field variables The situation is more problematic with point-defect clusters, which often play a key role in the annihilation of free point defects Since the size of these clusters cover a wide range of values, it would be quite difficult to add a new field variable for each size, for example, for vacancy clusters of size (divacancies), size (trivacancies), size 4, etc Moreover, under irradiation conditions leading to the direct production of defect clusters by displacement cascades, additional length scales are required to describe the distribution of defect cluster sizes and of atomic relocation distances These new length scales are not physically related to the width of a chemical interface at equilibrium, we, and therefore, they cannot be safely rescaled by we This analysis clearly suggests that one needs to rely on a PFM where the atomic scale has been retained This is, for instance, the case in the quantitative PFM reviewed in Section 1.15.3 Another possible approach is to use a mixed continuous–discrete description, as illustrated below in Section 1.15.4.2.4 We note that information on defect cluster sizes and relocation distances should be seen as part of the noise imposed by the external forcing, here the irradiation, on the evolution of the field variables The difficulty is thus to develop a model that can correctly integrate this external noise It is Phase Field Methods well documented that, for nonlinear dissipative systems, the external noise can play a determinant role and, for large enough noise amplitude, may trigger nonequilibrium phase transformations.110–113 One last and important challenge in the development of PFMs for alloys under irradiation is the fact that in nearly all traditional models the mobility matrix is oversimplified, for instance Mirr ẳ C1 CịD~irr =kB T , which is a simple extension to eqn [17] where D~ has been replaced by D~irr to take into account radiation-enhanced diffusion In the common case of multidimensional fields, for instance for multicomponent alloys, or for alloys with conserved and nonconserved field variables, the mobility matrix is generally taken as a diagonal matrix, thus eliminating any possible kinetic coupling between these different field variables As discussed at the end of Section 1.15.2, this approximation raises concerns because it misses the fact that these kinetic coefficients are related since they originate from the same microscopic mechanisms This is, in particular, the case for the coupled evolution of point defects and chemical species in multicomponent alloys This coupling is of particular relevance to the case of irradiated alloys since irradiation can dramatically alter segregation and precipitation reactions owing to the influence of local chemical environments on point-defect jump frequencies While new analytical models have been developed recently using mean field approximations to obtain expressions for correlation factors in concentrated alloys,114–117 work remains to be done to integrate these results into PFMs 1.15.4.2 Examples of PF Modeling Applied to Alloys Under Irradiation 1.15.4.2.1 Effects of ballistic mixing on phase-separating alloy systems Consider the simple case where the external forcing produces forced exchanges between atoms (such relocations are found in displacement cascades), and let us assume for now that these relocations are ballistic (i.e., random) and take place one at a time For this case, one can use a 1D PFM to follow the evolution of the composition profile C(x) during irradiation.118 This evolution is the sum of a thermally activated term, for which the classical Cahn diffusion model can be used, and a ballistic term: dCxị dF ẳ Mirr r2 Gb CðxÞ À wR ðx À x ÞCðx Þdx ½25 dt dC 421 where Mirr is the thermal atomic mobility, here accelerated by the irradiation, F the free energy of the system, Gb the jump frequency of the atomic relocations forced by the nuclear collisions, and wR is the normalized distribution of relocation distances, characterized by a decay length R Since most of these atomic relocations take place between nearest neighbor atoms, in a first approximation one may assume that R is small compared to the cell size In this case, the second ballistic term in eqn [25] reduces to a diffusive term: dCxị dF ẳ Mirr r2 À Gb a r2 C dt dC ½26 In this case, the model thus reduces to the one initially introduced by Martin,119 and the steady state reached under irradiation is the equilibrium state that the same alloy would have reached at an effective irr temperature Teff ¼ T ỵ Gb =Girr th ị, where Gth is an average atomic jump frequency, enhanced by the point-defect supersaturation created by irradiation In particular, in the case of an alloy with preexisting precipitates, depending upon the irradiation flux and the irradiation temperature, this criterion predicts that the precipitates should either dissolve or continuously coarsen with time Some relocation distances, however, extend beyond the first nearest neighbor distances,120,121 and it is interesting to consider the case where the characteristic distance R exceeds the cell size An analytical model by Enrique and Bellon118 revealed that, when R exceeds a critical value Rc, irradiation can lead to the dynamic stabilization of patterns To illustrate this point, one performs a linear stability analysis of this model in Fourier space, assuming here that the ballistic jump distances are distributed exponentially The amplification factor w(q) of the Fourier coefficient for the wave vector q is given by oqị=M ẳ @ f =@C ịq 2kq4 gR2 q =1 ỵ R2 q ị ẵ27 where f (C) is the free-energy density of a homogeneous alloy of composition C, k the gradient energy coefficient, and g ¼ Gb =M is a reduced ballistic jump frequency The analysis is here restricted to compositions and temperatures such that, in the absence of irradiation, spinodal decomposition takes place, that is, @ 2f/@C < The various possible dispersion curves are plotted in Figure Unlike in the case of short R, it is now possible to find irradiation intensities g such that the 422 Phase Field Methods Thermal w(q) Total qmin 128 q Irradiation Figure Sketch of the dispersion curve given by the linear stability analysis eqn [27], in the case when the ballistic relocation distance R is large The total dispersion curve is decomposed into its thermal and irradiation components Wave vectors below qmin are stable against decomposition ballistic term in eqn [27] is greater than j@ f =@C j at small q, but smaller than that at large q In such cases, the amplification factor is first negative for small q values, but it becomes positive when q exceeds some critical value qmin, while for larger q, the amplification factor is negative again Therefore, decomposition is still expected to take place, but only for wave vectors larger than qmin, that is, for wavelengths smaller than 2p/qmin It can thus be anticipated that coarsening will saturate, since at large length scales, the alloy remains stable with respect to decomposition Enomoto and Sawa122 have investigated this model using a 2D PFM based on eqn [25] The interest here is that the PFM, unlike the above linear stability analysis, includes both linear and nonlinear contributions to the evolution of composition inhomogeneity and also permits following the morphology of the decomposition Using this model, Enomoto and Sawa have confirmed the existence of the patterning regime, see Figure 6, and showed that this patterning can take place in the whole composition range The PFM approach allows for a direct determination of the patterning length scale as a function of the irradiation conditions, as illustrated in Figure Similar results have also been obtained using a variational analysis of eqn [25], leading to the dynamical phase diagram displayed in Figure As seen in this diagram, when the characteristic length for the forced relocation is smaller than the critical value Rc, the system never develops patterns at steady state Above Rc, patterning takes place when the irradiation 64 Figure Irradiation-induced compositional patterning in a binary alloy with (a) CB ¼ 50% and (b) CB ¼ 35%, using a two-dimensional phase field model based on eqn [25] with 1282 cells Reproduced from Enomoto, Y.; Sawa, M Surf Sci 2002, 514(1–3), 68–73 conditions are chosen so as to result in an appropriate g value Another result obtained from the KMC simulations is that the steady state reached by an alloy is independent of its initial state Experimental tests performed on a series of dilute Cu–M alloys, with M ¼ Ag, Co, Fe, have confirmed some of the key predictions of the above simulations and analytical modeling In particular, irradiation conditions that result in atomic relocation distances exceeding a few angstroms lead to the dynamical stabilization of precipitates at intermediate irradiation temperature.123 These results provide also a compelling rationalization of the puzzling results reported by Nelson et al.124 on the refinement of g0 precipitates in Ni–Al alloys under 100-keV Ni irradiation at 550 C The origin of the above irradiation-induced compositional patterning lies in the finite range of the atomic mixing forced by nuclear collisions.125 Enrique and Bellon126,127 have shown that the effect of this finite-range dynamics can be formally recast as effective finite-range repulsive interaction between like particles It is interesting to note that PF Phase Field Methods Patterning 423 g2 g1 R/Ö(C/A) I(k,t)/(S02L04) t/t0 = 500 2.5 Solid solution (Rc,gc) Macroscopic phase separation 0.1 0 (a) kL0 10 101 L0 -1/3 g /(A2/C) Figure Analytical dynamical phase diagram yielding the most stable steady state in a phase-separating A50B50 alloy as a function of the forced relocation distance R and the relative ballistic jump frequency g Insets are (111) sections of three-dimensional kinetic Monte Carlo (KMC) results; the lateral size of the KMC inset is 17 nm Reprinted with permission from Enrique, R A.; Bellon, P Phys Rev Lett 2000, 84(13) Copyright by the American Physical Society (a) (b) (c) (d) (e) (f) 100 10-1 10 (b) 102 t/t0 103 Figure (a) Spherically averaged and rescaled structure factor for a low ballistic mixing frequency g ¼ 0.0005 (○) and a higher ballistic mixing frequency g ¼ 0.05 (●) (b) Evolution of the first moment of this structure factor as a function of rescaled time, demonstrating that the alloy undergoes continuous coarsening with the low mixing rate, but stabilizes at the finite length scale (irradiation-induced patterning) for the higher mixing rate Reproduced from Enomoto, Y.; Sawa, M Surf Sci 2002, 514(1–3), 68–73 simulations of alloys with Coulomb interactions also predict a patterning of the microstructure.128 The parallel with the treatment of finite-range mixing is in fact quite strong, since a screened Coulomb repulsion is described by a decaying exponential, as also assumed for the probability of finite-range ballistic exchanges in deriving eqn [27] The contribution of the Coulomb repulsion to the linear stability analysis is thus proportional to 1/(q2 ỵ qD2), where qD is the Figure Phase field modeling of the evolution of ordered precipitates in the presence of electrostatic (repulsive) interactions, with increasing time from A to F The average precipitate size reaches a finite value at equilibrium Reprinted with permission from Chen, L Q.; Khachaturyan, A G Phys Rev Lett 1993, 70(10), 1477–1480 Copyright by the American Physical Society screening wavelength For reasons similar to the ones discussed in the case of finite-range mixing, it is then anticipated that these interactions will suppress coarsening This is confirmed by PF simulations, as illustrated in Figure 1.15.4.2.2 Coupled evolution of composition and chemical order under irradiation Many engineering alloys contain ordered phases or precipitates to optimize their properties, in particular mechanical properties It is thus important to 424 Phase Field Methods investigate how these optimized microstructures evolve under irradiation It is anticipated that, under appropriate conditions, ballistic mixing can lead to the dissolution of precipitates, and to the disordering of chemically ordered phases.129 Matsumura et al.130 used a 1D PF approach on a model binary alloy system to specifically investigate what evolution irradiation may produce In that model, the composition field is represented by the globally conserved order parameter X(r), while the degree of order is represented by the nonconserved order parameter S(r) X is chosen to vary from to ỵ1 for pure A and pure B composition, respectively, and S takes a value ranging from to for fully disordered and fully ordered phases, respectively The free energy functional of the system is written as F ẵfX rị; Srị; T g H ðt Þ > 2> > > > rX ị f X ; S; T ị ỵ ð> = < dr ¼ > > > > K t ị > > : rSị ; ỵ ½28 where f (X, S, T ) is the mean field free energy of a homogeneous alloy, and H and K are positive constants of the interfacial energy coefficients in the presence of varying field X and S, respectively The homogeneous free-energy density is given by a Landau expansion f ðX ; S; T Þ 2 2 Þ À bðT Þfx ðT Þ À X gS X x m aT ị4 ẳ f0 ỵ ỵ bT ị2 x T ị2 S ½29 where f0 is the mean field free energy of the disordered phase with composition xm, and a, b, x12 are positive constants depending on temperature The equilibrium phase diagram for this model system is given in Figure 10 Notice, in particular, that at low temperature and for compositions sufficiently far from the equiatomic composition, an ordered phase coexists with a disordered phase The kinetic evolution of these two fields is governed by @X ¼D0mix fr2 X @t ỵ LT ; fịr dF fX ; S; T gị m dX ẵ30 and @S dF fX ; S; T gị ẳ efS MT ; fị @t dS ½31 where f is the atomic displacement rate, m is the chemical potential, D0mix and e are positive coefficients characterizing the efficiency of mixing and disordering 1.2 Disorder Temperature, T/Tc 1.0 0.8 x0(T) Order 0.6 x2(T) x2(T) 0.4 Order + disorder 0.2 x1(T) Order + disorder x1(T) x0(T) 0.0 -1.0 x0(T) -0.5 0.0 Composition, X 0.5 1.0 Figure 10 Equilibrium phase diagram for model alloy system given by eqns [28] and [29] The dotted lines correspond to the metastable extrapolation of the order–disorder transition into the miscibility gap Reprinted with permission from Matsumura, S.; Muller, S.; Abromeit, C Phys Rev B 1996, 54(9), 6184–6193 Copyright by the American Physical Society Phase Field Methods by irradiation, and L and M are the mobility coefficients for the conserved and nonconserved fields Note that here there is no kinetic coupling between these two fields There is, however, thermodynamic coupling through the expression chosen for the homogeneous free-energy density, eqn [29] Although point defects are not explicitly used as PF variables, the dependency of the mobility coefficients M and L with temperature, irradiation flux, and sink density (cS) is obtained from a rate theory model for the vacancy concentration under 425 irradiation in a homogeneous alloy.1 The steady-state phase diagrams for two irradiation flux values are given in Figure 11 At the higher flux, the phase diagram is composed of homogeneous disordered and ordered phases only At low enough temperature, the ballistic mixing and disordering dominate the evolution of the alloy, leading to the destabilization of the ordered phase at and near the stoichiometric composition X ¼ 0, and to the disappearance of the two-phase coexistence domains for off-stoichiometric compositions 1.2 CS = 10−5 Disorder f = 10−5f c Temperature, T/Tc 1.0 0.8 x1irr x1irr Order 0.6 0.4 x2 x0irr (T,f ) O+D 0.2 x1 0.0 -1.0 -0.5 Disorder O+D x1 0.0 Composition, X (a) 0.5 1.0 1.2 CS = 10−5 Disorder f = 10−4f c Temperature, T/Tc 1.0 0.8 Order x1irr x1irr 0.6 0.4 x0irr (T,f ) x2 x2 0.2 x1 0.0 -1.0 (b) -0.5 Disorder 0.0 Composition, X x1 0.5 1.0 Figure 11 Steady-state phase diagrams under irradiation for (a) a low irradiation flux and (b) an irradiation flux 10 times larger The two-phase field is barely present in (a), and is no longer stable in (b) Reprinted with permission from Matsumura, S.; Muller, S.; Abromeit, C Phys Rev B 1996, 54(9), 6184–6193 Copyright by the American Physical Society 426 Phase Field Methods The model has also been used to study the dissolution of ordered precipitates under irradiation In agreement with prior lattice-based mean field kinetic simulations,131 it is found that two different dissolution paths are possible, depending upon the composition and irradiation parameters Ordered precipitates may either disorder first and then slowly dissolve or they may dissolve progressively while retaining a finite degree of chemical order until their complete dissolution These two kinetic paths have indeed been observed experimentally in Nimonic PE16 alloys irradiated with 300-keV Ni ions.132,133 1.15.4.2.3 Irradiation-induced formation of void lattices The formation of voids in irradiated solids results from the clustering of vacancies, which can be assisted by vacancy clusters produced directly in displacement cascades and by the presence of gas atoms Vacancy supersaturation under irradiation may locally reach a level large enough to trigger clustering owing to the biased elimination of interstitials on sinks, especially since interstitial atoms and small interstitial clusters usually migrate much faster than vacancies Evans134 discovered in 1971 that under irradiation voids may self-organize into a mesoscopic lattice The symmetry of the void lattice is identical to that of the underlying crystal, but with a void lattice parameter about two orders of magnitude larger than the crystalline lattice parameter (see also the reviews by Jager and Trinkaus135 and Ghoniem et al.136 for irradiation-induced patterning reactions) It has been suggested that the formation of the void lattice results from the 1D migration of self-interstitial atoms (SIAs) and of clusters of SIAs, although elastic interactions between voids could also contribute to self-organization.137 This 1D migration of SIAs would stabilize the formation of voids along directions of the SIAs migration by a shadowing effect.138–140 The model proposed by Woo141 indicates, in particular, that the mean free path of SIAs needs to exceed a critical value for a void lattice to be stable Atomistic KMC simulations have been performed142 to evaluate the dynamics of void formation, shrinkage, and organization during irradiation Due to the large difference in mobility of vacancies and interstitials, the slow evolution of the microstructure, and the large range of length scales, assumptions had to be used, in particular, regarding the void position and size Recently, Hu and Henager143 have approached the problem of void lattice formation in a pure metal using a PFM Their model relies on the traditional approach presented in Section 1.15.2 for the evolution of the vacancy field, but it makes use of continuumtime random-walk kinetics for modeling the fast transport of interstitials 2D simulations indicate that irradiation can stabilize a void lattice if the ratio of SIA to vacancy diffusion coefficients is large enough (see Figure 12) and if the defect production rate is not too large (see Figure 13) It would be interesting to extend this first model to include interstitial clusters The model lacks an absolute length scale, for the reasons discussed in Section 1.15.2, and thus nucleation of new voids is treated in a deterministic and phenomenological manner based on the local vacancy concentration It would clearly be beneficial to use a quantitative PFM of the type presented in Section 1.15.3 to treat void nucleation This would also then make it possible to directly compare the void size stabilized by irradiation with experimental observations We note also that Rokkam et al.144 recently introduced a simple PFM for void nucleation and coarsening in a pure element subjected to irradiation-induced vacancy production In addition to the local vacancy concentration, these authors introduced a nonconserved order parameter to model the matrix–void interface, similar to the nonconserved order parameter t * = 16 000 t * = 16 000 t * = 16 000 (a) (b) (c) t * = 16 000 (d) Figure 12 Phase field simulations of void distributions for a low generation rate of vacancies and self-interstitial atoms (SIAs), g_ V ¼ g_ SIA ¼ 10À5 , for different diffusivity ratios between SIA and vacancies, DSIA =DV , (a) 10, (b) 102, (c) 103, and (d) 104 Reproduced from Hu, S.; Henager, C H., Jr J Nucl Mater 2009, 394, 155–159 Phase Field Methods t * = 16 000 t * = 16 000 t * = 16 000 t * = 16 000 (a) (b) (c) (d) 427 Figure 13 Phase field simulations of void distributions for a high diffusivity ratio between self-interstitial atoms (SIAs) and vacancies, DSIA =DV ¼ 104 , and various generation rates of vacancies and SIAs, g_ V ¼ g_ SIA , (a)  10À3, (b)  10À4, (c) 10À4, and (d) 10À5 Reproduced from Hu, S.; Henager, C H., Jr J Nucl Mater 2009, 394, 155–159 used for solid–liquid interfaces It is shown144,145 that this model reproduces many known phenomena, such as nucleation, growth, coarsening of voids, as well as the formation of denuded zones near sinks such as free surfaces and grain boundaries This phenomenological model is currently limited by the absence of interstitial atoms in the description It may also suffer from the fact that the void–matrix interfaces are intrinsically treated as diffuse, whereas real void–matrix interfaces are essentially atomically sharp This problem is further discussed in the following section 1.15.4.2.4 Irradiation-induced segregation on defect clusters In order to circumvent the problems raised in the previous paragraph and in Section 1.15.4.1 for the inclusion of defect clusters in a PFM, Badillo et al.146 have recently proposed a mixed approach that combines discrete and continuum treatments of the defect clusters, so that each cluster is treated as a separate entity Point-defect cluster size is treated as a discrete quantity for cluster production, whereas the long-term fate of clusters is controlled by a continuum-based flux of free point defects New field variables are thus introduced to describe the size of these clusters: p p p Nc;A p Nc;V p Nc;B ¼ d ; Cc;A ¼ d ; Cc;B ẳ d ẵ32 N N N p where Nc;V is the number of vacancies in the vacancy cluster in the cell p, and Nd is the number of substitup p tional lattice sites per cell; Nc;A and Nc;B are the numbers of A and B interstitial atoms, respectively, forming the interstitial cluster in the cell p Each cell contains at most one cluster The production of point defects by irradiation takes place at a rate dictated by the irradiation flux p Cc;V f in dpa sÀ1 and by the simulated volume In the case of irradiation conditions leading to the intracascade clustering of point defects, the total number of point defects created in a displacement cascade, the fraction of those defects that are clustered, and the size and spatial distribution of these clusters are used as input data The production of Frenkel pairs is treated in the same way as defect clusters, except that the variables affected are the free vacancy and interstitial concentrations This treatment of defect and defect cluster production makes it possible to compare irradiation conditions with identical total defect production rates, that is, identical dpa sÀ1 values, but with varied fractions of intracascade defect clustering and varied spatial distribution of these clusters Furthermore, it is also very well suited for system-specific modeling, since all the above information can be directly and accurately obtained from molecular dynamic simulations sampling the primary recoil spectrum.121 In particular, one can build a library of such displacement cascades, so that the PFM will inherit the stochastic character in space and time of the production of defect clusters by displacement cascades The continuous flux of free point defects to the clusters results in the growth or shrinkage of these clusters, which translates into the continuous p p evolution of the cluster field variables Cc;V , Cc;A , and p Cc;B When any cluster field variable drops below 1=N d , this cluster is assumed to have dissolved, and the remaining one point defect is transferred to the corresponding free point-defect variable of that cell For the sake of simplicity, defect clusters are treated as immobile, but the approach can be extended to include mobility, in particular for small interstitial clusters Further details are available in Badillo et al.146 The potential of the above approach is illustrated by considering a 2D A8B92 alloy with a zero heat of 428 Phase Field Methods mixing, so that at equilibrium, it always forms a random solid solution The production of interstitials is, however, biased so that only A interstitial atoms are created This could, for instance, simulate an alloy where there is a rapid conversion of B interstitial atoms into A interstitial atoms via an interstitialcy mechanism The preferential transport of A interstitial atoms to defect clusters should lead to an enrichment of A species around defect clusters, since these clusters act as defect sinks The effect of the primary recoil spectrum on this irradiation-induced segregation is studied by comparing two cases: the first one where a small fraction of cascades, 1/Ncas ¼  10À4, produces defect clusters, and the second one where that fraction is 100 times higher, 1/Ncas ¼  10À2 In both cases, however, the displacement rate per atom per second is the same, here 10À7 dpa sÀ1 Figure 14 shows instantaneous concentration maps of the A solute atoms for 1/Ncas ¼  10À4 In this case, the PFM uses 64  64 cells, each containing  lattice sites Segregation of A species is clearly observed at a few locations, typically 2–5 This number is close to the average number of defect clusters The sharp peaks with high levels of segregation correspond to segregation of existing defect clusters, either interstitial or vacancy ones This is confirmed by visualizing the defect and defect cluster fields, see Figure 15 The broader segregation profiles in Figure 14 are the remnants of sharp segregation profiles after their corresponding clusters shrank CA 1.0 400 0.5 300 0.0 200 100 200 100 300 400 Figure 14 The concentration field of A atom (CA ) for an A8B92 alloy with zero heat of mixing where all interstitials are created as A atoms The two-dimensional model system contains 448  448 lattice sites, decomposed into 64  64 cells for defining the phase field variables, each cell containing  lattice sites Irradiation displacement rate is 10À7 dpa sÀ1; the cascade frequency rate is 1/Ncas ¼  10À4, and the irradiation dose is dpa Reproduced from Badillo, A.; Bellon, P.; Averback, R S to be submitted and disappeared As a result, the nonequilibrium segregation that build up on those clusters is being washed out by vacancy diffusion, as expected for an A–B alloy system with zero heat of mixing In the case where defect sinks have a finite lifetime, as in the present case, one should thus expect a dynamical formation and elimination of segregated regions In the case of much higher defect cluster production rate, 1/Ncas ¼  10À2, a very different microstructure is stabilized by irradiation, as illustrated in Figure 16 Now a high density of clusters is present, typically 40 interstitial clusters and 20 vacancy clusters, as seen in Figure 17(a) and 17(b), and the segregation measured on these clusters is reduced by about one order of magnitude compared to the previous case These results are reminiscent of the experimental findings reported by Barbu and Ardell147 and Barbu and Martin,148 showing that, with irradiation conditions producing displacement cascades (e.g., 500-keV Ni ion irradiation), the domain of irradiation-induced segregation and precipitation in undersaturated Ni–Si solid solutions is significantly reduced compared to the case where irradiation produces only individual point defects (e.g., 1-MeV electron irradiation) 1.15.5 Conclusions and Perspectives Thanks to fundamental advances, coupled with the development of efficient algorithms and fast computers, the PF technique has become a very powerful and versatile tool for simulating phase transformations and microstructural evolution in materials, as illustrated in this chapter This technique provides simulation tools that are complementary to atomistic models, such as molecular dynamics and lattice Monte Carlo simulations, and to larger scale approaches, such as finite element models With some modifications, it can also be employed for materials subjected to irradiation In the case of materials subjected to irradiation, specific issues need to be addressed to fully realize the potential of PF modeling First, a proper description of point defects and atom transport requires mobility matrices (or tensors) that capture the kinetic coupling between these different species In particular, the models reviewed in this chapter not account for the correlated motion of point defects and atoms, thus leading to unphysical correlation factors in the mobility coefficients These correlation effects, however, play an essential role in phenomena 0.0004 0.0003 0.0002 0.0001 0.0000 400 0.00020 0.00015 0.00010 0.00005 300 200 400 300 100 200 100 200 300 0.08 0.06 0.04 0.02 0.00 400 300 200 100 400 400 300 200 100 (d) 300 0.05 0.04 0.03 0.02 0.01 0.00 100 200 400 (b) CClus Int CClus Vac 400 (c) 100 200 100 300 (a) 429 Ci CV Phase Field Methods 200 100 300 400 ~V, (b) free interstitials Cp þ Cp Figure 15 Defect concentration fields corresponding to Figure 14: (a) free vacancies C int A int B (A and B atoms), (c) clustered vacancies Cc;V , and (d) clustered interstitials Cc;A ỵ Cc;B (A and B atoms) Reproduced from Badillo, A.; Bellon, P.; Averback, R S to be submitted CA 1.0 0.5 400 300 0.0 200 100 100 200 300 400 Figure 16 Concentration field of A atom CA for the same A8B92 alloy as in Figure 14, except at a higher cascade frequency, 1/Ncas ¼  10À2 Notice the significant reduction in segregation on defect clusters compared to Figure 14 Reproduced from Badillo, A.; Bellon, P.; Averback, R S to be submitted such as irradiation-induced segregation and precipitation and are thus required in PFMs aiming for system-specific predictive power Second, it remains challenging to include in a PFM all the elements of the microstructure relevant to evolution under irradiation, namely point-defect clusters, dislocations, grain boundaries, and surfaces, although it has been shown here that models handling adequately a subset of these microstructural elements are now becoming available Third, the numerical integration of the evolution equations is more challenging than for conventional PFMs in the sense that the continuous defect production, as well as the large difference in vacancy and interstitial mobility, usually prevents the use of long integration time steps, even in coarse microstructures Finally, materials under irradiation constitute nonequilibrium systems that are quite sensitive to the amplitude and the structure of fluctuations, in particular the fluctuations resulting from Phase Field Methods 0.08 0.06 0.04 0.02 0.00 400 300 200 100 CClus Int CClus Vac 430 100 200 400 300 200 100 200 300 (a) 0.05 0.04 0.03 0.02 0.01 0.00 100 300 400 (b) 400 Figure 17 Defect cluster concentration fields corresponding to Figure 16: (a) clustered vacancies, Cc;V and (b) clustered interstitials Cc;A ỵ Cc;B (A and B atoms) Reproduced from Badillo, A.; Bellon, P.; Averback, R S to be submitted point defect and point-defect cluster production, and from ballistic mixing of species A self-consistent and tractable PFM that would include both thermal and irradiation-induced fluctuations is still missing Such a model would be very beneficial for the study of microstructural evolution under irradiation, especially that involving the nucleation of new phases, defect clusters, or gas bubbles see Chapter 1.13, Radiation Damage Theory; Chapter 1.14, Kinetic Monte Carlo Simulations of Irradiation Effects; and Chapter 1.09, Molecular Dynamics Acknowledgments The author gratefully acknowledges stimulating discussions with Robert Averback, Arnoldo Badillo, Yan Le Bouar, Alphonse Finel, and Maylise Nastar The author also thanks Robert Averback for his critical reading of the manuscript The support from the US DoE-BES under Grant DEFG02-05ER46217 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