 Original article
 Open Access
An analysis of two classes of phase field models for void growth and coarsening in irradiated crystalline solids
 K. Ahmed^{1, 2} and
 A. ElAzab^{3}Email author
Received: 10 July 2017
Accepted: 18 December 2017
Published: 31 January 2018
Abstract
A formal asymptotic analysis of two classes of phase field models for void growth and coarsening in irradiated solids has been performed to assess their sharpinterface kinetics. It was found that the sharp interface limit of type B models, which include only point defect concentrations as order parameters governed by CahnHilliard equations, captures diffusioncontrolled kinetics. It was also found that a type B model reduces to a generalized onesided classical Stefan problem in the case of a high driving thermodynamic force associated with the void growth stage, while it reduces to a generalized onesided MullinsSekerka problem when the driving force is low in the case of void coarsening. The latter case corresponds to the famous rate theory description of void growth. Type C models, which include point defect concentrations and a nonconserved order parameter to distinguish between the void and solid phases and employ coupled CahnHilliard and AllenCahn equations, are shown to represent mixed diffusion and interfacial kinetics. In particular, the AllenCahn equation of model C reduces to an interfacial constitutive law representing the attachment and emission kinetics of point defects at the void surface. In the limit of a high driving force associated with the void growth stage, a type C model reduces to a generalized onesided Stefan problem with kinetic drag. In the limit of low driving forces characterizing the void coarsening stage, however, the model reduces to a generalized onesided MullinsSekerka problem with kinetic drag. The analysis presented here paves the way for constructing quantitative phase field models for the irradiationdriven nucleation and growth of voids in crystalline solids by matching these models to a recently developed sharp interface theory.
Keywords
 Phase field models
 Voids
 Irradiated solids
 Asymptotic analysis
Introduction
Irradiation drives complex microstructure evolution in solids by producing large nonequilibrium densities of point defects (Olander, 1976; Was, 2017; Brailsford & Bullough, 1972). Features such as dislocation loops and voids are often observed in irradiated solids. The presence of voids, the microstructure type of interest here, strongly affects the thermal and mechanical properties of irradiated materials. As such, many investigations were conducted to understand the void formation and growth processes (Olander, 1976; Was, 2017; Brailsford & Bullough, 1972; Krishan, 1982; Dubinko et al., 1989; ElAzab et al., 2014; Hochrainer & ElAzab, 2015; Millet & Tonks, 2011a). In analogy to the theory of sintering (Lifshitz & Slyozov, 1961; Rahaman, 2003), theoretical models of void formation and growth usually treat voids as a second phase of vacancies that nucleate in a metastable supersaturated solid matrix with the metastability sustained by the production and accumulation of point defects. These models thus consider the process as an example of nonequilibrium firstorder phase transition in driven systems. The nucleation models predict a nucleation barrier for the void formation that decreases as the supersaturation of vacancies (interstitials) increases (decreases) (Olander, 1976; Was, 2017; Katz & Wiedersich, 1971; Russell, 1971; Mayer & Brown, 1980). The void growth models can be classified into two types; the rate theory models (Olander, 1976; Was, 2017; Brailsford & Bullough, 1972; Krishan, 1982; Dubinko et al., 1989) and the fieldtheoretic, spatiotemporal models (ElAzab et al., 2014; Hochrainer & ElAzab, 2015; Millet & Tonks, 2011a). In the former, a test void in an effective homogeneous medium is considered to represent the average kinetics of the system. Moreover, the process of void evolution is treated as diffusioncontrolled, and the void growth equation is usually derived assuming quasistatic diffusion in the solid matrix (Olander, 1976; Was, 2017; Brailsford & Bullough, 1972; Krishan, 1982; Dubinko et al., 1989). Some rate theory models known as cluster dynamics models take into consideration the size distribution of the evolving void system (Was, 2017). These models will not be mentioned further in the current discussion. In field theoretic models, local balance equations and void growth are treated within the principles of irreversible thermodynamics and the system of voids is resolved in space and time (De Groot & Mazur, 1962).
The fieldtheoretic models can be further classified into two categories, sharp and diffuseinterface models. In a sharpinterface model developed recently by Hochrainer and ElAzab (2014; 2015), the void surface is treated as a singular surface across which jump conditions are applied. The diffuseinterface models, also called the phase field models, consider the void surface to be diffuse, i.e., has a finite width, in order to circumvent the numerical difficulties associated with solving the moving boundary problem of evolving voids (ElAzab et al., 2014; Hochrainer & ElAzab, 2015). However, the void surface is atomicallysharp and hence one must ensure that diffuseinterface models recover the sharpinterface description as the interface thickness vanishes. As of now, two different classes of phase field models for void growth have appeared in literature (Yu & Lu, 2005; Hu et al., 2009; Hu & Henager, 2009; Hu & Henager, 2010; Li et al., 2010; Semenov & Woo, 2012; Rokkam et al., 2009; Millett et al., 2009; Millett et al., 2011c; Millett et al., 2011b; Li et al., 2013; Xiao et al., 2013). Following the terminology of the fieldtheoretic approach set by physicists for modeling heterogeneous materials (Hohenberg & Halperin, 1977; Binder, 1987; Provatas & Elder, 2010), these models are conveniently classified as models of type B and type C. Such models, while intended to represent one and the same phenomenon, void nucleation and growth, do appear physically and mathematically different. Therefore, an assessment of these phase field models showing their relations to the sharpinterface physics and the classical rate theory models is desirable for making further progress.
The method of matched asymptotic expansions (Provatas & Elder, 2010; Pego, 1989; Dai & Du, 2012; Elder et al., 2001; Fife, 1992; Emmerich, 2008; Garcke et al., 2004; Ahmed et al., 2016; Caginalp, 1989) is used here to deduce the sharpinterface limits of type B and type C phase field models for void evolution in irradiated solids. The asymptotic analysis serves two purposes: it demonstrates the consistency of the phase field models in terms of their physics content, and, in the process, it helps to relate their parameters to the regular thermodynamic and kinetic parameters that appear in the sharpinterface models. This is critical for making the phase field models quantitative. The asymptotic analyses performed here can be considered as generalizations of those by Pego (1989), Dai and Du (2012), and Elder et al. (2001) to the case of driven multicomponent systems.
Our analysis concludes that phase field models of type B, which utilize the point defect concentrations as the only order parameters, can describe diffusioncontrolled kinetics. Moreover, in the low driving forces limit (coarsening stage) they are equivalent to the rate theory models. On the other hand, phase field models of type C, which couple CahnHilliard type equations (Cahn, 1961) for the local balance of point defects with an AllenCahn equation (Allen & Cahn, 1979) for the motion of the void surface, are able to take into account the attachment kinetics of point defects to the void surface. The attachment kinetics of the point defects to the void surface affects the overall void growth rate, as was shown in the numerical simulations of the sharpinterface model (ElAzab et al., 2014; Hochrainer & ElAzab, 2015). It is shown that the additional timedependent AllenCahn (GinzburgLandau) equation in model C acts as the interfacial constitutive law that ensures positive interfacial entropy production associated with the surface motion due to its reactions with point defects (ElAzab et al., 2014; Hochrainer & ElAzab, 2015). The connections between the coarsening limits of phase field models B and C and the classical meanfield LifshitzSlyozovWagner theories of Ostwald ripening are further addressed here (Lifshitz & Slyozov, 1961; Rahaman, 2003; Mullins & Sekerka, 1963; Niethammer, 2000; Dai et al., 2010). These limits are all important in fixing the phase field model parameters and understanding their results.
The theoretical models of void growth in irradiated solids are reviewed first. The asymptotic analysis of the different phase field models for void growth is then presented, followed by a summary and concluding remarks.
Models for void growth in irradiated solids
Rate theory description
In the above, c_{v} and c_{i} are the average (fractional) vacancy and interstitial concentrations in the solid, with the superposed dot referring to their time rates of change, P_{v} and P_{i} are the respective production terms, k_{vi} is a rate constant for vacancyinterstitial recombination, k_{vs} and k_{is} are the rate constants for defects reaction with sinks of average concentration c_{s}, R is the void radius, Ω is the atomic volume, γ is the surface energy, \( {c}_{\mathrm{v}}^{\mathrm{R}} \) and \( {c}_{\mathrm{i}}^{\mathrm{R}} \)are vacancy and interstitial concentrations in equilibrium with a void of radius R, \( {c}_{\mathrm{v}}^{\mathrm{eq}} \) and \( {c}_{\mathrm{i}}^{\mathrm{eq}} \)are the equilibrium vacancy and interstitial concentrations with a flat surface, and D_{v} and D_{i} are the diffusion coefficients of vacancies and interstitials, respectively and k and T are Boltzman constant and temperature. The rate theory has two major shortcomings. The first is the treatment of the solid as a homogeneous medium. Under irradiation, high gradients of point defect concentrations exist in the solid matrix, particularly near the surfaces of sinks such as free surfaces (or voids), grain boundaries, and dislocations. Describing such an effective homogeneous medium is complicated and never exact. The second shortcoming pertains to the assumption of diffusioncontrolled growth (Eq. (1.2)). The process of void growth is not necessarily diffusioncontrolled, and the point defect concentrations at the void surface usually deviate from their equilibrium values (Eq. (1.3)). In fact, as was recently shown by Hochrainer and ElAzab (2014; 2015), the overall growth kinetics depends on the interaction between the point defects and the void surface.
Sharpinterface description
Spatiotemporal models of void evolution (ElAzab et al., 2014; Hochrainer & ElAzab, 2015; Millet & Tonks, 2011a) treat voids as a second phase that nucleates in a supersaturated matrix (Lifshitz & Slyozov, 1961; Rahaman, 2003). The sharpinterface description of void growth may thus be considered a generalization of the classical sharpinterface models of particle growth from a supersaturated matrix, as in solidification and precipitation. However, the classical precipitation/solidification models are examples of phase transitions in nondriven systems, i.e., systems relaxing toward equilibrium from close thermodynamic states, while the void nucleation and growth is an example of nonequilibrium phase transition under an external driver, irradiation. It is the ongoing production of point defects that renders the solid matrix metastable all the time and causes the nucleation and growth of voids. The lack of sharpinterface models for void growth under irradiation motivated Hochrainer and ElAzab (2014; 2015) to introduce an elaborate one. Nevertheless, it is useful to first review the classical models of precipitation since they represent limiting cases of the generalized sharp and diffuseinterface models for void growth, before introducing the sharp interface model of voids reported in (ElAzab et al., 2014; Hochrainer & ElAzab, 2015).
In the above, c_{ k } is the fractional concentration of species k, satisfying the condition \( \sum \limits_k{c}_k=1 \), J_{ k } is the flux of the corresponding species, μ_{ k } is its chemical potential, M_{ k } is its mobility. f is the Helmholtz free energy density, κ is the interface curvature, and λ is the interface relaxation constant (λ^{−1} is the interface kinetic coefficient); it is also sometimes called the coefficient of kinetic drag (Niethammer, 2000; Dai et al., 2010). The notation \( {\left[\bullet \right]}_{\alpha}^{\beta } \) refers to the jump across the interface, Γ, of the quantity in brackets, and n is the unit normal to the interface, pointing from Ω_{−} to Ω_{+}, which, respectively refer to α and β phases. Eqs. (2.1) and (2.2) represent the mass balance and constitutive laws for each species in both phases. Eq. (2.3) is a statement of the continuity of the chemical potential at the interface. Eq. (2.4) and Eq. ((2.5) are the interfacial balance and constitutive laws, respectively. Eq. (2.4) is also known as the Stefan jump condition while Eq. ((2.5) is called the dynamical GibbsThompson equation (Provatas & Elder, 2010; Pego, 1989; Dai & Du, 2012; Elder et al., 2001; Fife, 1992; Emmerich, 2008; Garcke et al., 2004; Ahmed et al., 2016; Caginalp, 1989). The system of eqs. (2.1)(2.5) is of course to be supplemented with appropriate initial conditions and boundary conditions on the outer boundary ∂Ω. For a growing particle with fixed concentration, the dynamical system becomes onesided, i.e., diffusion takes place only in the metastable parent phase.
It is worth noting that the dynamical GibbsThompson relation (Eq. ((2.5)) reduces to the static (or equilibrium) GibbsThompson relation for a stationary interface. Moreover, for the case of a flat interface in a binary system, it recovers the common tangent (Maxwell) construction rule. While the dynamical GibbsThompson condition was first introduced in an ad hoc manner based on experimental data (Langer, 1980), it is now known that it arises as a consequence of the nonnegative entropy production expression of the second law of thermodynamics. Specifically, the dynamical GibbsThompson relation can be obtained by requiring the bulk and interfacial entropy production to be nonnegative independently and assuming a linear relationship between the normal interface velocity (the normal fluxes at the interface) and the corresponding driving thermodynamic forces (ElAzab et al., 2014; Hochrainer & ElAzab, 2015). For the case of infinitely fast interface kinetics (λ^{−1} → ∞, λ → 0), the velocity term vanishes and we get the classical GibbsThompson condition as the interfacial constitutive law indicating a diffusioncontrolled growth. In other words, the diffusioncontrolled models ignore the interface kinetics and assume zero interfacial entropy production. When the velocity term is considered, and hence the interface kinetics is taken into account, the dynamical GibbsThompson relation states that the chemical potentials and concentrations of each species deviate from their equilibrium thermodynamic values at the interface.
Classical models of particle growth by mass diffusion in a binary system. Only one species is tracked in the system^{a}
Stefan type models  MullinsSekerka models  

Classical (equilibrium) model  Model with kinetic drag  Classical (equilibrium) model  Model with kinetic drag 
\( {\displaystyle \begin{array}{l}\mathrm{for}\ x\in {\Omega}_{\pm}\\ {}{\partial}_tc=\nabla \cdot \mathbf{J}\\ {}\mathbf{J}=M\;\nabla \mu (c)\\ {}\\ {}\mathrm{for}\ x\in \Gamma \\ {}{\mu}^{\alpha }={\mu}^{\beta }=\mu \\ {}\mathrm{v}{\left[c\right]}_{\alpha}^{\beta }={\left[\mathbf{J}\right]}_{\alpha}^{\beta}\cdot \mathbf{n}\\ {}\mu =0\end{array}} \)  \( {\displaystyle \begin{array}{l}\mathrm{for}\ x\in {\Omega}_{\pm}\\ {}{\partial}_tc=\nabla \cdot \mathbf{J}\\ {}\mathbf{J}=M\;\nabla \mu (c)\\ {}\\ {}\mathrm{for}\ x\in \Gamma \\ {}{\mu}^{\alpha }={\mu}^{\beta }=\mu \\ {}\mathrm{v}{\left[c\right]}_{\alpha}^{\beta }={\left[\mathbf{J}\right]}_{\alpha}^{\beta}\cdot \mathbf{n}\\ {}\lambda \mathrm{v}=\gamma \kappa +{\left[f\right]}_{\alpha}^{\beta }\mu {\left[c\right]}_{\alpha}^{\beta}\end{array}} \)  \( {\displaystyle \begin{array}{l}\mathrm{for}\ x\in {\Omega}_{\pm}\\ {}\nabla \cdot \mathbf{J}=0\\ {}\mathbf{J}=M\;\nabla \mu (c)\\ {}\\ {}\mathrm{for}\ x\in \Gamma \\ {}{\mu}^{\alpha }={\mu}^{\beta }=\mu \\ {}\mathrm{v}{\left[c\right]}_{\alpha}^{\beta }={\left[\mathbf{J}\right]}_{\alpha}^{\beta}\cdot \mathbf{n}\\ {}\mu {\left[c\right]}_{\alpha}^{\beta }=\gamma \kappa \end{array}} \)  \( {\displaystyle \begin{array}{l}\mathrm{for}\ x\in {\Omega}_{\pm}\\ {}\nabla \cdot \mathbf{J}=0\\ {}\mathbf{J}=M\;\nabla \mu (c)\\ {}\\ {}\mathrm{for}\ x\in \Gamma \\ {}{\mu}^{\alpha }={\mu}^{\beta }=\mu \\ {}\mathrm{v}{\left[c\right]}_{\alpha}^{\beta }={\left[\mathbf{J}\right]}_{\alpha}^{\beta}\cdot \mathbf{n}\\ {}\mu\;{\left[c\right]}_{\alpha}^{\beta }=\gamma \kappa +\lambda \mathrm{v}\end{array}} \) 
f_{ a } is the free energy of an atom in the perfect (defectfree) crystal, f_{i} and f_{v} are the free energy of formation of interstitials and vacancies, respectively, k_{ B } is the Boltzmann constant, and T is the temperature.
Here, δ is the surface layer thickness over which the reaction takes place, which is on the order of a lattice parameter, c_{ α } is the limiting value of the point defect concentration at the surface, ν_{ α } is the attempt frequency, Δg_{ α } is the surface kinetic barrier, f is the limiting value of the free energy at the surface, and κ is the local surface curvature.
It is clear that the sharpinterface formulation, Eqs. (3)  (7), is a generalization of the classical particle growth models, Eqs. (2.1)(2.5), that is suitable for irradiationdriven void evolution. The canonical form of the surface boundary condition, Eq. (7), derived from transition state theory can be considered as a generalized interfacial constitutive law. By expanding the exponential term in Eq. (7) up to first order by assuming (μ_{ α } ± ΔE)/k_{ B }T < < 1 and using Eq. (6), one recovers the linear dynamical GibbsThompson relation.
Hochrainer and ElAzab conducted numerical simulations of their model for different scenarios of void growth and shrinkage (ElAzab et al., 2014; Hochrainer & ElAzab, 2015). They demonstrated that the point defect reaction with the void surface has a strong effect on the overall kinetics of the problem. In particular, they showed that, as the surface barrier increases, the rate of void growth diminishes and the deviation of the point defect concentrations from their equilibrium values increases. This is of course consistent with a kinetic type interfacial constitutive law (such as Eq. ((2.5) or Eq. (7)) that accounts for kinetic drag. However, the diffusioncontrolled models that assume local equilibrium at the surface cannot predict such effect. Therefore, the sharpinterface formulation of the problem of void growth overcomes the limitations of the rate theory, and hence provides a reference for constructing the corresponding diffuseinterface models of the problem.
Diffuseinterface description
Here we take the void phase to be in Ω_{−}, and consider a zero flux boundary condition on the outer boundary ∂Ω, which has a unit outward normal m. The results of the asymptotic analysis presented here also hold for Dirichlet or periodic boundary conditions. We further require C_{v}(x) and C_{i}(x) to be outside of the spinodal (unstable) regime. This means that C_{v}(x) and C_{i}(x) locally satisfy the condition that the Hessian matrix of the Helmholtz free energy density is positivedefinite at any point in the solid phase. Note that C_{v}(x) and C_{i}(x) have also to be compatible with Eq. (12.3).
Here, α = i, v and L is the GinzburgLandau (AllenCahn) mobility (Allen & Cahn, 1979), which is considered to be independent of all order parameters for simplification. Absorption of defects at sinks can be easily added to this description by assuming smeared sinks or discrete sinks. Since the kinetic eqs. (17.1) and (17.2) guarantee a reduction of the free energy along the evolution path, the free energy functional (Eq. (13)) is a Lyapunov functional and the two global minimizers (homogeneous phases) are Lyapunov stable.
In current phase field models of void growth (ElAzab et al., 2014; Rokkam et al., 2009; Millett et al., 2009; Millett et al., 2011c; Millett et al., 2011b), and for consistency with the corresponding sharpinterface description, the term P_{ α } and the parameters M_{ α } and R_{iv} are required to vanish in the void phase. This is usually accomplished by expressing them as functions of the order parameters such that the values of these functions are identically zero when the order parameters take on their equilibrium values that define the void phase. We will show here using the method of matched asymptotic expansions that this is indeed necessary for the phase field models to recover the desired sharpinterface limit.
Asymptotic limits of phase field models for void growth
Formal asymptotic analysis of phase field models for voids is crucial for proving thermodynamic and mathematical consistency of the models and for determining their parameters from sharpinterface counterpart. Several analyses of phase field models of type B and C for other phenomena exist in literature (Provatas & Elder, 2010; Pego, 1989; Dai & Du, 2012; Elder et al., 2001; Fife, 1992; Emmerich, 2008; Garcke et al., 2004; Ahmed et al., 2016; Caginalp, 1989). The ones relevant to our situation are those by Pego (1989), Dai and Du (2012) for the case of model B and Elder et al. (2001) for the case of model C. Pego (1989) has shown that the CahnHilliard equation with a constant mobility in a binary system recovers the classical Stefan problem at fast time scale (growth stage) and the classical MullinsSekerka problem at slow time scale (coarsening stage). Dai and Du (2012) have generalized the analysis to the case of highly dissimilar mobility in a twophase system, which is relevant to voids since the diffusive mobility is zero in the void phase. Our asymptotic analysis for phase field models B for void growth generalizes their work to driven ternary systems. For the case of model C, Elder et al. (2001) presented an asymptotic analysis for the low supersaturation case in a nondriven binary system.
We present here an elaborate formal asymptotic analysis of phase field models C for void growth for both cases of high and low driving forces in driven ternary systems. For the case of low driving force limits of models B and C, we discuss the connection between the coarsening limits and the classical diffusion or interfacecontrolled LifshitzSlyozovWagner theories of Ostwald ripening (Lifshitz & Slyozov, 1961; Rahaman, 2003; Mullins & Sekerka, 1963; Niethammer, 2000; Dai et al., 2010).
Asymptotic analysis of model B
Similar equations hold for the higher order terms. Therefore, one recovers the regular balance equations in the relevant phases. It is to be noted here that requiring P_{ α } = 0, M_{ α } = 0, and R_{iv} = 0 when c_{v} = 1 and c_{i} = 0, i.e., in the void phase, makes Eq. (20) trivial. Boundary conditions on the front necessary for solving these equations will be derived from the inner expansion.
In the above, ∇_{ s }, ∇_{ s }⋅ and \( {\nabla}_s^2 \) are the (d1) surface gradient, divergence and Laplacian operators, respectively.
The last condition states that the initial supersaturation in the solid matrix is of order ε. The other initial and boundary conditions are the same as in Eq. (12).
We will see from the inner expansion that the leading order terms on the front take on their equilibrium values, and hence the outer solution is in fact given by Eq. (37).
The front velocity is thus determined solely from the normal fluxes coming to the interface from the solid matrix. The analysis so far shows that, for the low driving force limit that is suitable for describing the coarsening stage, the phase field model B for void growth under irradiation reduces to a generalized onesided classical MullinsSekerka problem (Eqs. (38), (46), and (50)).
In model B the interface is always in local equilibrium since the chemical potentials and concentrations at the interface take on their equilibrium thermodynamic values. This means that the attachment kinetics of point defects to the void surface is ignored and the void growth process is diffusioncontrolled. In addition, the curvature correction term (GibbsThompson condition) appears only as a first order correction term (Eq. (46)) and hence the curvature effect is absent from the leading order terms. While this approximation could be tolerated for large voids (R > 100 nm), it certainly breaks down for small voids(R < 10 nm). As we will show later, model C can obviate such shortcomings of model B. Moreover, if one assumes that M_{i} = ε^{−1}M_{v}, the contribution of the interstitial to the normal velocity of the interface will be of order ε, and hence can be ignored as in the phase field models that consider vacancies only (Yu & Lu, 2005; Hu & Henager, 2009; Hu & Henager, 2010; Semenov & Woo, 2012; Xiao et al., 2013). This could be a reasonable approximation at low temperature where the interstitial are highly more mobile than vacancies (interstitial migration energy is usually much smaller than its vacancy counterpart in most solids), but it does not hold at high temperature.
This confirms that in the low driving force limit, model B recovers the rate theory description or, more precisely, the classical diffusioncontrolled particle growth (Olander, 1976; Was, 2017; Brailsford & Bullough, 1972; Krishan, 1982; Dubinko et al., 1989; Lifshitz & Slyozov, 1961; Rahaman, 2003; Mullins & Sekerka, 1963; Niethammer, 2000; Dai et al., 2010). Xiao et al. (2013) have recently shown using numerical simulations that phase field model B predictions agree well with the rate theory predictions.
It is wellknown that the classical MullinsSekerka problem exhibits kinetic scaling such that R(t) ∝ t^{1/3} (Lifshitz & Slyozov, 1961; Rahaman, 2003; Mullins & Sekerka, 1963; Niethammer, 2000; Dai et al., 2010). Lifshitz and Slyozov were the first to show that a collection of second phase particles follows such cubic growth kinetics (Lifshitz & Slyozov, 1961). The connection between the phase field model B (CahnHilliard equation) and the diffusioncontrolled LSW theory was discussed before by Bray (1994). However, only weakly driven systems can be considered to behave asymptotically as nondriven systems. In general, such kinetic scaling laws will not hold for driven systems since the supersaturation is not decaying with time as in nondriven systems.
Asymptotic analysis of model C
The reasoning behind this scaling is as follows. The first scaling ensures that the leading order terms of the nonconserved order parameters take on their equilibrium values in the bulk phases regardless of the supersaturation. The second guarantees that the curvature effect appear at leading order, in contrast to model B where it was a first order correction. The last relation can be explained if one defines a nondimensionalized Peclet number as, \( pe=\frac{a_v}{D_{\alpha }} \), where D_{ α } is the defect diffusivity and a is the lattice parameter. Hence, Peclet number represents the ratio between the velocity of points on the surface and the speed of diffusion in the bulk. Clearly, the reaction of the point defect with the surface is relevant when Peclet number is small; otherwise the process is diffusioncontrolled. In phase field model C, one can define Peclet number as \( pe=\frac{L{\varepsilon}^2}{D_{\alpha }} \). Now, if we assume pe = O(ε), we get L = O(ε^{−1}) as in Eq. (56.3). It is worth noting that if one assumes that pe = O(1) or higher and hence L = O(ε^{−2}) or higher, the asymptotic limits of model C reduce to their counterparts of model B.
Again, similar to the case of model B, we require the initial conditions in the solid matrix to be consistent with the fact that the solid matrix is metastable, i.e., to avoid the unstable (spinodal) region.
From Eq. (62.2), one has u^{0} = 0. Since away from the interface the gradients of the nonconserved order parameter should vanish (regardless of the supersaturation in the point defect concentrations), one must require \( {\partial}_{\eta }f\left({\eta}^0={\eta}^{\mathrm{M}},{c}_{\mathrm{v}}^0,{c}_{\mathrm{i}}^0\right)={\partial}_{\eta }f\left({\eta}^0={\eta}^{\mathrm{V}},{c}_{\mathrm{v}}^0,{c}_{\mathrm{i}}^0\right)=0 \), so that η^{1}(x) = 0, for x ∈ Ω_{±}. As usual, the boundary conditions on the front required to solve Eq. (62.1) will be imposed from the matching with the inner expansion.
In the above, \( {\gamma}_{\eta }=\underset{\infty }{\overset{+\infty }{\int }} h\varepsilon {\left({\partial}_z{\tilde{\eta}}^0\right)}^2 dz \) is the surface energy and δ = h ε^{2}/γ_{ η } is the diffuse interface width that is on the order of a surface layer thickness.
According to Eqs. (62), (70), (74), phase field model C for void growth under irradiation reduces to a generalized onesided Stefan problem with kinetic drag.
For the point defect concentrations, Eq. (81.1) indicates that the first order terms in the bulk phases are in steady state. We will deduce the boundary conditions on the front required for solving Eq. (81.1) from the inner expansion via the matching conditions as before.
Hence in the coarsening stage phase field model C reduces to a generalized version of the onesided MullinsSekerka problem with kinetic drag (Eqs. (81), (91), and (95)).
This is the first time where a scaling limit of model C is shown to recover the MullinsSekerka model with a kinetic drag. Note that this limit is different from the one obtained by Elder et al. (2001). In their asymptotic analysis, they obtained a timedependent diffusion equation for the first order term of the concentration field in the bulk phases, i.e., a modified Stefan problem for the first order term. This is however inconsistent with the assumption of low driving force where one expects the diffusion to be quasistatic. The reason that they did not arrive at the same limit obtained here is the fact that they did not study the interface motion at the slow time scale suitable for the low driving force limit. Our derivations of the asymptotic limits of model C for high and low driving forces are consistent with and can be viewed as generalizations of their model B counterparts.
However, a general nonlinear (canonical) form of the interfacial constitutive law as in Hochrainer and ElAzab model (Hochrainer & ElAzab, 2015) can only be obtained if one assumes a similar nonlinear form for the AllenCahn equation.
The difference between the coarsening limits of model B and C, e.g., the regular MullinsSekerka problem and the MullinsSekerka with kinetic drag problem is obvious. By comparing Eq. (98.2) and Eq. (54), we deduce that model C predicts lower coarsening rates for growing/shrinking voids than model B (since from Eq. (98.3), we always have \( {L}^{\mathrm{eff}}<{M}_{\mathrm{v}}^0 \)) . Also from Eq. (99) and Eq. (52.2), the vacancy chemical potential (and hence concentration) at the void surface obtained from model C is higher than the value predicted from model B (from the equilibrium GibbsThompson condition) for a growing void. The situation is reversed for a shrinking void; both limits give of course the same result for a stationary void. Moreover, according to Eq. (99), for the case of a growing void the vacancy chemical potential (and hence concentration) at the void surface increases as the AllenCahn mobility decreases (or equivalently, as the interface kinetic coefficient decreases or the surface barrier increases). All these predictions of model C are in good agreement with the recent simulation results obtained by Hochrainer and ElAzab from their sharpinterface model (ElAzab et al., 2014; Hochrainer & ElAzab, 2015).
Hence, as one should expect, for infinitely fast bulk diffusion kinetics, the vacancy chemical potential/concentration at the void surface is the same as in the bulk.
It is clear from Eq. (98.2) that a transition from interfacecontrolled kinetics to diffusioncontrolled kinetics may always take place. For a particular material, the thermodynamic and kinetic parameters are given and the only factor that determines the prevailing kinetics is the particle radius. For small particles \( \left(R<<{M}_{\mathrm{v}}^0/L\;\delta \right) \), interfacecontrolled kinetics dominates. For large particles \( \left(R>>{M}_{\mathrm{v}}^0/L\;\delta \right) \) diffusioncontrolled kinetics dominates. Hence, while a small particle grows, its kinetics changes from interface to diffusioncontrolled. Therefore, in the meanfield limit a collection of second phase particles in weakly or nondriven systems coarsen such that the average particle size follows parabolic growth (R(t) ∝ t^{1/2}) at short times and cubic growth (R(t) ∝ t^{1/3}) at long times. Such behavior was recently captured by the numerical simulations of Dai et al. (2010). Therefore, the meanfield limit of the MullinsSekerka model with kinetic drag gives rise to a generalized LifshitzSlyozovWagner (LSW) theory from which the diffusion and interfacecontrolled theories emerge as limiting cases. Moreover, Kockelkoren and Chaté reported similar trend from their numerical simulations of the late stage of coarsening in phase field model C (Kockelkoren & Chaté, 2002). Their results are then in agreement with our asymptotic analysis presented here. Nevertheless, as we mentioned before, such kinetic scaling laws are not expected to hold in driven systems.
Summary and conclusions
The diffuseinterface void growth models of type B and C were analyzed by deriving their sharpinterface limits. The limits obtained from the asymptotic analyses are summarized below followed by concluding remarks. A sample simulation is included in the Appendix.
Sharpinterface limits of phase field model B
High driving force limit (growth stage)

Bulk balance laws and constitutive laws:

Interfacial balance laws and constitutive laws:
The quantity \( {\left[{\mathbf{J}}_{\alpha}^0\cdot \mathbf{n}\right]}^{+} \)is the normal flux of the defect species at the solid side of the void surface.
Low driving force limit (coarsening stage)

Bulk balance laws and constitutive laws:

Interfacial balance laws and constitutive laws:
Again \( {\left[{\mathbf{J}}_{\alpha}^0\cdot \mathbf{n}\right]}^{+} \)is the normal flux of the defect species at the solid side of the void surface.
Sharpinterface limits of phase field model C
High driving force limit (growth stage)

Bulk balance laws and constitutive laws:

Interfacial balance laws and constitutive laws:
Low driving force limit (coarsening stage)

Bulk balance laws and constitutive laws:

Interfacial balance laws and constitutive laws:
According to the asymptotic analyses conducted here, a phase field model of type B recovers the diffusioncontrolled growth and coarsening. This model thus ignores the surface attachment kinetics. On the other hand, a phase field model of type C is able to account for the surface attachment kinetics via the extra AllenCahn equation, which ensures nonnegative interfacial entropy production associated with the interface motion. Therefore, for constructing models of voids/bubbles growth and coarsening, phase field models of type C must be used.
It is obvious that sharp and diffuseinterface models are able to relax all the assumptions of the rate theory. In particular, they account for the heterogeneity of the void microstructure and consider the effect of surface attachment of point defects to the void surface on the overall kinetics. One can, however, to some extent account for the latter in the rate theory models by replacing the diffusioncontrolled growth equation in the rate theory (Eq. (1.2)) by a general growth equation as Eq. (98.2). One also should replace the equilibrium GibbsThompson condition (Eq. 1.3), by the dynamical GibbsThompson condition which leads to Eq. (99). Nevertheless, this is only valid for the low driving force case where the assumption of quasistatic diffusion in the solid matrix is valid.
In passing, we recall here the recommendation by Hochrainer and ElAzab that a proper treatment of defect attachement to void surfaces, which is rigorously treated in their development of the sharp interface model for void evolution (Hochrainer & ElAzab, 2015), must be incorporated into phase field model construction to gurantee an accurate account of defect removal by reactions at the void surface. In earlier models of type C (ElAzab et al., 2014), such reactions were treated indirectly by amplifying the vacancyinterstitial recombination term, R_{iv}, in the diffuse interface region. The form of this term, however, does not impact the results presented here.
While the analysis presented here was tailored for void growth and coarsening, it can be easily generalized to particle growth and coarsening by mass or/and heat diffusion in multicomponent and multiphase driven systems. Moreover, thanks to the varitional formulation of phase field models, one can account for longrange interactions such as elastic or electrostatic in a systematic fashion, as was done several times in literature (Provatas & Elder, 2010). The kinetic trends based on the analysis presented here will not change. One will only have to work with a generalized chemical potential that account for all the driving forces of shortrange or longrange instead of the usual chemical potential used here. Therefore, our analysis represents a clear way of constructing diffuseinterface models and deducing their sharpinterface limits for investigating the growth and coarsening stages of nonequilibrium phase transitions in a general driven system.
Declarations
Acknowledgements
The authors wish to thank Srujan Rokkam and Thomas Hochrainer for useful discussions.
Funding
This material is based upon work supported as part of the Center for Materials Science of Nuclear Fuel, an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Sciences, Office of Basic Energy Sciences under award number FWP 1356, through subcontract number 00122223 at Purdue University.
Availability of data and materials
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Author’s contributions
Both authors of this paper have contributed equally to the research presented in this paper, as well as to the writing of the paper. Both authors read and approved the final manuscript.
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