Article A sharp interface model for void growth in ...

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Philosophical Magazine Vol. 00, No. 00, 00 Month 2013, 1–35

Article A sharp interface model for void growth in irradiated materials Thomas Hochrainera and Anter El-Azabb∗ a

Bremen Institute of Mechanical Engineering, University Bremen, 28359 Bremen, Germany b School of Nuclear Engineering and School of Materials Engineering, Purdue University, West Lafayette, IN 47907, USA (February 19, 2015) A thermodynamic formalism for the interaction of point defects with free surfaces in single component solids has been developed and applied to the problem of void growth by absorption of point defects in irradiated metals. This formalism consists of two parts, a detailed description of the dynamics of defects within the non-equilibrium thermodynamic frame, and the application of the second law of thermodynamics to provide closure relations for all kinetic equations. Enforcing the principle of non-negative entropy production showed that the description of the problem of void evolution under irradiation must include a relationship between the normal fluxes of defects into the void surface and the driving thermodynamic forces for the void surface motion; these thermodynamic forces are identified for both vacancies and interstitials and the relationships between these forces and the normal point defect fluxes are established using the concepts of transition state theory. The latter theory implies that the defect accommodation into the surface is a thermally activated process. Numerical examples are given to illustrate void growth dynamics in this new formalism and to investigate the effect of the surface energy barriers on void growth. Consequences for phase field models of void growth are discussed. Keywords: Defect thermodynamics; defect kinetics; radiation effects; void growth

1.

Introduction

Particle irradiation leads to microstructure evolution in crystalline solids [1, 2]. In this regard, voids are particularly important because they are related to the dimensional stability of irradiated materials [3, 4]. Under irradiation, large densities of vacancies and interstitials are often produced. The biased removal of interstitials by sinks such as dislocations results in vacancy supersaturation leading to nucleation and growth of voids. The classical void evolution models in irradiated metals include nucleation models [5–7], master-equation based models [8, 9], rate theory models for growth [10–12] and void lattice models [13–16]. Nucleation models assume that vacancy clusters form as thermodynamic flucuations but grow stably ∗ Corresponding

author. Email: [email protected]

ISSN: 1478-6435 print/ISSN 1478-6443 online c 2013 Taylor & Francis

DOI: 10.1080/14786435.20xx.xxxxxx http://www.informaworld.com

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once a critical size is reached, a process that is characterized by a nucleation barrier [17]. The master equation approach is based on a description of elementary transitions leading to size changes within the cluster population, and is cast in the form of rate equations for the cluster density at all sizes. In the continuum limit, these equations are cast in the form of a Fokker-Planck equation for an appropriate distribution function [8, 9]. In the rate theory approach, a test void embedded in an effective medium grows by a net absorption of vacancies from a background defect concentration that is fixed by point defect production, recombination and absorption at sinks. In a single component material, sinks include dislocations, voids, and grain boundaries [18, 19]. The rate theory model assumes that the defect concentrations at the void surface take on their thermal equilibrium values, and that the void growth rate is controlled by defect diffusion to the void surface from within a control volume determined by the void density. Within the rate theoretical approach, the effect of reactions of defects with surfaces has only been considered with regard to the question whether void growth is diffusion limited or surface (reaction) limited [20, 21]. The void lattice models focus on the factors that drive self organization of voids in patterns [13–16]. Some ordering mechanisms have been proposed, including spinodal-like behavior of the defect system, elastic interactions of voids and reaction-diffusion aspects of point defects [22]. Void evolution models based on the phase field approach started to appear recently [23–30]; these models are motivated by the need to predict both nucleation and growth of voids simultaneously and to resolve the defect and void dynamics in both space and time. These benefits come at the cost of having to deal with a number of difficulties. For example, while the phase field approach is based on the notion of diffuse interfaces, the void-matrix interface in crystalline solids is not diffuse. As such, it must be guaranteed that the phase field models are consistent with their sharp-interface counterparts. Also, phase field models involve several parameters describing the energetics and kinetics of moving fronts, which have to be related to the physics at the interfaces in order to produce quantitative results. In the phase field literature, these parameters are usually derived by asymptotic or numerical matching of the phase field predictions to the sharp-interface model results. The successful development of phase field models for void growth thus re-

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quires a parallel development of the sharp interface formalism of the same problem. We develop a sharp interface formulation of the void growth problem under irradiation by investigating the problem of defect-surface interactions based on the principles of non-equilibrium thermodynamics. The exchange of defects between the matrix and the void surface is found to contribute to entropy production and is thus connected with a thermodynamic driving force. This connection is fixed using the transition state theory, and, as a result, an energy barrier for defect accommodation into the surface is defined. A similar sharp interface model is available in [31], which deals with charged vacancies in dielectric and piezoelectric materials. However, this model does not consider interstitials and remains rather simplistic in the constitutive model of the defect reactions at the surface. The capabilities of the model developed here is demonstrated by solving a set of sample problems of void growth under initially uniform defect supersaturation conditions and under irradiation. The significance of the sharp interface model as a required step toward the development of accurate phase field models for void evolution is discussed.

2. 2.1.

Mass and energy conservation in connection with surface motion Mass and defect conservation

Consider a single component crystalline solid containing vacancies and self interstitials. The defect state of this crystalline solid is assumed to have only isolated point defects (monomers). If present, small defect clusters could affect the balance of point defects in the bulk as well as void growth. As voids will shrink or grow by absorbing or losing atoms individually or in clusters, the formalisms presented here, which considers only point defects for simplicity, may be generalized in a systematic fashion to include defect clusters. By neglecting furthermore elastic distortions due to point defects we work on a fixed lattice with a given number of lattice sites N = 1/Ωa per unit volume, obtained as the inverse of the atomic volume Ωa . Moreover, we assume that the self interstitials appear as so called dumbbell (or split) interstitials where two atoms share a regular lattice position. This assumption is consistent with the state of knowledge about such defects in face-centered and body-centred cubic crystals. These considerations mean that each lattice site

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is either occupied by a regular atom, a dumbbell interstitial or a vacancy. Consequently, the site fractions of regular atoms, ca , interstitials, ci , and vacancies, cv , satisfy: ca + ci + cv = 1. As atom diffusion may only be mediated by the diffusion of either vacancies or interstitials, the consideration of regular atoms is redundant. In the sequel we will thus describe the system solely by the fractional concentrations of interstitials, ci , and vacancies, cv . As elastic effects are ignored in the following for simplicity, we only deal with conservation of mass and energy and discard momentum conservation (stress equilibrium). With regard to mass conservation we note that due to the assumption of dumbbell interstitials the number density of atoms per unit volume is given by ρ = N (ca + 2ci ) = N (1 + ci − cv ). In the absence of elastic deformation, changes in density within the crystal may only be due to changes in defect concentrations. The conservation of atoms (mass conservation), therefore, translates to conservation laws for defects. However, interstitials and vacancies may be produced from displacement cascades due to irradiation and may annihilate each other when recombining to a regular atom (see schematic in Figure 1). Mass conservation requires that interstitials and vacancies need to be produced in equal amounts (if we ignore the so called source bias) and also removed in equal amounts by mutual recombination. The defect concentrations in the bulk change due to defect fluxes, Ji and Jv , defect reactions (recombination), Ri and Rv , and irradiation acting as a (random)

Recombination Field variables Diffusion

V Void

Figure 1.: Schematic of a material region with a void ensemble under irradiation.

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defect source, Qi and Qv . That is,

c˙i = −∇·Ji − Ri + Qi ,

(1)

c˙v = −∇·Jv − Rv + Qv ,

(2)

where the recombination rates are equal and given by the so called velocity of reaction: Ri = Rv = dξ/dt. In the above, removal of defects at dispersed sinks is ignored as it will not affect the results we seek. For the production terms, by R R contrast, we only require them to be balanced globally, i.e., Qi dV = Qv dV . The reason is that vacancies and interstitials do not necessarily have to be produced in equal amount at each point if the spatial resolution is smaller than the cascade size. For example, displacement cascades may be comprised of a vacancy rich core surrounded by an interstitial rich shell [32, 33]. Note that the fluxes appearing in (1) and (2) are fluxes of fractional concentrations and not fluxes of densities and, therefore, they have the dimension of a velocity. They are related to the (number) density fluxes Jρi and Jρv by the atomic volume Ωa = 1/N ; that is

Ji = Ωa Jρi and Jv = Ωa Jρv .

(3)

While conservation of mass in the bulk is ensured by requiring the reactions to remove equal amounts of vacancies and interstitials and the sources to be globally balanced, conservation of mass at a surface determines the normal velocity v of this surface. Whether a void surface or an outer material boundary, a free surface may act as sink or source for both interstitials and vacancies and, in turn, moves according to the amount of defects absorbed or emitted—the crystal grows by consuming interstitials or emitting vacancies from the surface and shrinks by the opposite process. Furthermore, we assume that atoms may diffuse as described ˆ ρ along the surface, the divergence of which yields an by a surface atom flux J according build up or removal of material. Atom conservation requires that the atoms contained in the added or removed volume, which is proportional to the instantaneous atom density times the normal velocity, matches the (normal) flux

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of atoms across the surface, Jρ · n, plus the (surface) divergence of the surface atom ˆ ρ . Hence, the normal velocity is obtained as flux, J

v=

ˆρ ˆρ ˆ J ˆ J JJρ · nK − ∇· JJρi · n − Jρv · nK − ∇· = . JρK N J1 + ci − cv K

(4)

ˆ is the surface divergence operator, n is the outer normal to the In the above, ∇· crystal and the double brackets J · K denote the jump of the argument across the surface. Because we focus on the boundary to a void and thus empty space, and aside from setting cv = 1 therein, all quantities vanish in the void and we will typically leave out the brackets in the following. By switching entirely to a description in terms of site fractions, that is working with Ji and Jv and introducing ˆ = Ωa J ˆ ρ , the velocity can then be rewritten as an accordingly scaled surface flux J

v=

ˆ ˆ J Ji · n − Jv · n − ∇· . 1 + ci − cv

(5)

We remark that the surface does not need to be a source or sink of equal strength for vacancies and interstitials and that the velocity of the surface hence accounts for the imbalance between the respective removal or production rates at the surface. We conclude this section by taking a look at the defect conservation at the surface. It is important to note that while the defect fluxes appear to correspond to the rates of loss or production of defects at the surface, the motion of the surface also requires to build in or remove defects, because the crystal growth or shrinkage has to be in accordance with the defect content of the adjacent bulk. The rates of loss (or gain) of defect species at the surface, σi and σv , are thus obtained as the difference between this accretive creation or removal of defects due to the motion of the surface and the flux of defects to or from the surface: 1 (vci − Ji · n) δ 1 σv = (vcv − Jv · n) . δ σi =

(6) (7)

Here, δ denotes the distance the surface is advanced by an atom layer, which is on

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the order of Ωa 1/3 . Taking the velocity from atom conservation (5), we obtain the following expressions for the rate of interstitial and vacancy removal or production

σi =

ˆ ˆ J (cv − 1)Ji · n − ci Jv · n − ci ∇· δ (1 + ci − cv )

(8)

σv =

ˆ ˆ J cv Ji · n − (1 + ci )Jv · n − cv ∇· . δ (1 + ci − cv )

(9)

These rates (and not the normal fluxes) determine the absolute change of the amount of defects in the system and will appear in the next section within the energy balance at the surface. Again, the difference between the defect removal rates, σi and σv , and the normal fluxes of defects into the surface arises because the newly created or removed material layers at the surface contain defects.

2.2.

Energy conservation

We assume the internal energy to be made up of a bulk internal energy density e and a surface internal energy density eˆ. Both quantities are taken as related to the number of (bulk) lattice sites per unit volume, giving the bulk internal energy density e the dimension energy per lattice site and the surface internal energy eˆ the dimension energy times length per lattice site. In the bulk we consider the evolution of the internal energy density to follow the form

e˙ = −∇·Je − ∆H

dξ + Qe , dt

(10)

where ∆H denotes the heat of reaction of recombination between interstitials and vacancies and Qe is the energy production due to irradiation. Similarly, we consider the evolution of the surface internal energy density to be given by a surface flux and reaction terms associated with the defect reaction rates with the surface as

ˆ eˆ − ∆H ˆ J ˆ i σi − ∆H ˆ v σv . eˆ˙ = −∇·

(11)

ˆ i and ∆H ˆ v denote the heat of reaction of interstitials and vacancies with the ∆H surface, respectively. We do not consider irradiation events directly in the surface,

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which is why there appears no source term for the internal energy of the surface. We now turn to the conservation equation at the surface. The bulk reactions and sources do not affect the energy balance across the surface so long as the source terms Qi and Qv do not overlap with the surface. We assume the latter and, consequently, neglect the reaction and source terms in the following derivation. In this case, energy conservation requires that for any given control volume Ω (compare Figure 2), possibly intersecting the free surface S in Σ = Ω ∩ S, changes of the overall internal energy content may only be due to (i) fluxes of bulk density through the boundary of the control volume ∂Ω, (ii) the heat produced by surface reactions, (iii) fluxes of surface energy density through the boundary line ∂Σ = ∂Ω ∩ S of the intersection of Ω with the surface S, or (iv) through changes in the surface area of the intersecting surface Σ due to the motion of S. Consequently, the following conservation equation must hold for the control volume [34] (We formulate the conservation laws based on integrals of per site quantities. Note that a formulation in terms of per volume (or per surface area) quantities can be obtained from the used integral expressions through a multiplication with the constant lattice site density N ): d dt

Z e dV + Ω



Z eˆ dA

Z =−

Je · n dA +

Σ

∂Ω

Z 

ˆ i σi − ∆H ˆ v σv −∆H



dA

Σ

  nS · n∂Ω ˆ eˆ · n ˆ − vˆ J e dL. knS × n∂Ω k ∂Σ

Z −

(12)

  S





Void

Figure 2.: Schematic of a control volume Ω (light blue) partly intersecting a void with surface S (red). The part of the void surface contained in the control volume is the intersecting surface Σ. The intersection line ∂Σ of ∂Ω and the void surface S is depicted in black. (Color version online)

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ˆ denotes the normal to the Here, v denotes the normal velocity of the surface S, n intersection line ∂Σ within the surface plane and n∂Ω and nS denote the normals to ∂Ω and S, respectively. The local normal n∂Ω to ∂Ω along the intersecting line is chosen such that nS × n∂Ω = λt is a positive multiple (λ > 0) of the oriented unit tangent t along ∂Σ, as determined from the orientation of Σ. The last term in the line integral in (12) results from the change in surface area of Σ = Ω ∩ S due to the motion of the intersecting line ∂Σ within S (compare Appendix A). Without loss of generality we will in the sequel take the control volume Ω such that the normal to S is tangential to ∂Ω along the intersecting line, i.e. such that the normal n∂Ω is orthogonal to nS and the term vanishes. This assumption is made without ever mentioning it in nearly all literature concerned with conservation laws at moving singular surfaces. Because this term is so seldom found in the literature, we give a derivation of it in Appendix A. The left hand side of (12) is given by d dt

Z



Z e dV +

Ω

eˆ dA

Z =

Σ

e˙ dV + Ω

Z 

 ve + eˆ˙ + vˆ eκ dA,

(13)

Σ

where κ denotes the mean curvature of the surface. Because within the bulk e˙ = −∇ · Je , we find from Gauss theorem that equation (12) is equivalent to Z −

Je · n dA + ∂Ω

Z 

 ve − Je · n + eˆ˙ + vˆ eκ dA =

(14)

Σ

Z −

Je · n dA + ∂Ω

Z 

ˆ i σi − ∆H ˆ v σv −∆H

Σ



Z dA −

ˆ eˆ · n ˆ dL. J

∂Σ

We rewrite this equation using Gauss theorem on the surface and the evolution equation of the surface internal energy density (11) to determine the energy flux normal to the surface Σ:

Je · n = v (e + eˆκ) .

(15)

This result can be understood by considering the build up of material, when v is positive. This build up must be accompanied by an outward flux of internal energy as the energy content of the new material needs to come from the crystal.

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If the crystal growth moreover leads to a change in surface area due to a non-zero curvature, κ, the according surface energy content has to be exchanged too. Mass and energy conservation determine the form of the evolution equations both in the bulk and at the surface. The nature of the fluxes in the bulk and along the surface, however, is not yet specified; these fluxes are driven by generalized forces defined in the next section from a postulated entropy functional and the requirement of non-negative entropy production.

3.

Thermodynamic forces driving bulk and surfaces fluxes

In this section we give a brief derivation of requirements for thermodynamically consistent constitutive equations as obtained from the condition of non-negative entropy production. Mathematical details of the derivations are provided in a general form in Appendix B. Similar to the internal energy, we assume the entropy S of the system to have contributions due to a bulk entropy density s and a surface entropy density sˆ. The bulk entropy density is taken to be dependent on the defect concentrations ci , cv and the bulk internal energy e, while the surface entropy density depends on the surface internal energy density eˆ only. We note that the concentration changes due to the velocity of reaction dξ/dt in the evolution Equations of the defect concentrations (1) and (2) contribute to entropy production. By contrast, the heat of reaction −∆H dξ/dt in Equations (10) and (11) acts as a source term for the internal energy and therefore contributes to entropy generation (thus not being subject of the constraint of non-negative entropy production). We define the overall source ¯ e = −∆H dξ/dt + Qe for the internal energy and obtain term Q

¯ e. e˙ = −∇·Je + Q

(16)

The same applies for the heat of reaction at the surface and we write

ˆ eˆ + Q ˆ J ˆ eˆ, eˆ˙ = −∇·

(17)

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ˆ eˆ = −∆H ˆ i σi − ∆H ˆ v σv . with Q Referring again to Figure 2, non-negative entropy production requires the overall change in entropy in the control volume Ω to be larger or equal to the change due to entropy fluxes across the boundaries ∂Ω and ∂Σ and entropy generation within the control volume itself. In mathematical terms, this statement reads d dt

Z



Z sˆ dA

s (e, ci , cv ) dV + Σ

Ω

Z

Z

Js · n dA

sg dV −

≥

(18)

∂Ω

Ω

Z

Z sˆg dA −

+ Σ

ˆ sˆ · n ˆ dL, J

∂Σ

where sg and sˆg denote the entropy generation in bulk and surface, respectively, ˆ sˆ denote the bulk and surface entropy fluxes, respectively. The fluxes and Js and J and generation terms are given by (compare (B14)–(B19)) ∂s ∂s ∂s Ji + Jv + Je ∂ci ∂cv ∂e sˆ ˆ sˆ = ∂ˆ J Jeˆ ∂ˆ e ∂s ∂s ∂s ¯ sg = Qi + Qv + Q e ∂ci ∂cv ∂e ∂ˆ sˆ Qeˆ. sˆg = ∂ˆ e

Js =

(19) (20) (21) (22)

By applying Equation (B20) to the right hand side nad subtracting the flux and generation terms from both sides of Equation (18) the entropy inequality becomes  Z  ∂s ∂s ∂s dξ ∂s ∂s dξ ∇ · Je + ∇ · Ji − +∇ · Jv − dV (23) ∂e ∂ci ∂ci dt ∂cv ∂cv dt Ω  Z  ∂s ∂s ∂s ∂ˆ s ˆ eˆ + vκˆ ˆ J + − Je · n − Ji · n − Jv · n + vs + ∇ s dA ≥ 0, ∂e ∂ci ∂cv ∂ˆ e Σ where we note that all fluxes vanish in the void, which is why the jump terms reduce to the fluxes from the bulk side only. Due to energy conservation at the surface we have Je · n = v(e + eˆκ), see Equation (15). The entropy inequality may

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then be rewritten as  Z  ∂s ∂s dξ ∂s dξ ∂s ∂s ∇ · Je + ∇ · Ji − · Jv − dV (24) +∇ ∂e ∂ci ∂ci dt ∂cv ∂cv dt Ω    Z  ∂s ∂s ∂ˆ sˆ ∂s ∂ˆ s ˆ − Ji · n − + Jv · n + ∇ Jeˆ + v s − e + κ sˆ − eˆ dA ≥ 0. ∂ci ∂cv ∂ˆ e ∂e ∂ˆ e Σ As the control volume may be chosen arbitrarily, the inequality must hold pointwise and, herefore, non-negative entropy production requires in the bulk   ∂s ∂s ∂s dξ ∂s ∂s · Ji + ∇ · Jv + − − ≥ 0. ∇ · Je + ∇ ∂e ∂ci ∂cv ∂ci ∂cv dt

(25)

Similarly, we find using the normal velocity obtained from mass conservation (equation (5)) and upon sorting by flux terms, that at each point on the surface must hold ∂s ∂ˆ s s − e ∂e + κ sˆ − eˆ ∂ˆ ∂s e + − ∂ci 1 + ci − cv

ˆ +∇


!

∂ˆ s ∂s + κ sˆ − eˆ ∂ˆ s − e ∂e e 1 + ci − cv

Ji · n −
! ˆ+∇ ˆ ·J

∂s ∂ˆ s s − e ∂e + κ sˆ − eˆ ∂ˆ ∂s e + ∂cv 1 + ci − cv

∂ˆ s ˆ · Jeˆ ≥ 0 ∂ˆ e


! Jv · n

(26)

Because all summands in the bulk part (25) and surface part (26) of the entropy inequality are independent, each must be non-negative by itself. We recall the following thermodynamic relations which are assumed to be valid for the surface quantities in full analogy to the bulk (see e.g. [35]): ∂s 1 ∂ˆ s 1 ∂s µi ∂s µv = , = , s−e/T = −f /T , sˆ− eˆ/T = −fˆ/T , = − and =− , ∂e T ∂ˆ e T ∂ci T ∂cv T where f denotes the free energy per lattice site and fˆ = γ/N = Ωa γ denotes the surface free energy density per lattice site, while µi and µv denote the chemical potentials of interstitials and vacancies, respectively. Inserting this into (25) and (26) and multiplying all terms by T we find the following requirements for

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thermodynamic consistency in the bulk: ∇T · Je ≥ 0 T   ∇T − ∇µi − µi · Ji ≥ 0 T   ∇T · Jv ≥ 0 − ∇µv − µv T −

(µi + µv )

dξ ≥ 0. dt

(27) (28) (29) (30)

On the surface we find accordingly   f + Ωa κγ µi − Ji · n ≥ 0 1 + ci − cv   f + Ωa κγ µv + Jv · n ≥ 0 1 + ci − cv   f + Ωa κγ ˆ≥0 ˆ −∇ ·J 1 + ci − cv −

ˆ ∇T ˆ eˆ ≥ 0. ·J T

(31) (32) (33) (34)

The last two sets of inequalities define the thermodynamic forces that drive the defect and energy fluxes both in the bulk and on the void surface and the vacancyinterstitial reaction. These inequalities further serve as restrictions on the relationships between these forces and fluxes and defect reactions, which asserts the requirement that the evolution of the system may not spontaneously reduce entropy. These restrictions also prescribe a sign for reactions (in the bulk and at the surface) and a sign of a scalar product for vector fluxes.

4. 4.1.

Constitutive modeling Bulk free energy and equilibrium concentrations

The current formalism is concerned with a non-equilibrium situation involving point defect diffusion and evolution of internal void surfaces. A constitutive closure of the equations describing the associated boundary value problem is required, which consists of three parts: (a) the relationship between local free energy den-

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sity and the local state variables, and (b) the constitutive rules connecting fluxes to thermodynamic forces, including those connecting the normal defect fluxes at the surface to the respective thermodynamic forces. The local free energy density and the defect chemical potentials are developed here assuming that defects are introduced in a (locally uniform) system of given volume and temperature. For such a system, the Helmholtz energy is an appropriate thermodynamic potential. By including vacancies and interstitials concurrently, we generlize the standard formalism described, for example, in [36]. The free energy of a system of n lattice sites, consisting of nv vacancies, ni interstitials, and na = n − nv − ni atoms is given by

F (n, nv , ni , T ) = fa na + fv nv + fi ni − kB T [n ln n − na ln na − nv ln nv − ni ln ni ] ,

(35)

where fa is the free energy of an atom in the reference crystal and fv and fi are free energies of formation of a vacancy and an interstitial, respectively. The above yields the local free energy per lattice site, f (cv , ci , T ), as

f (cv , ci , T ) = fa ca + fv cv + fi ci + kB T [cv ln cv + ci ln ci + ca ln ca ] ,

(36)

where cv + ci + ca = 1. The chemical potentials for vacancies and interstitials are defined by the change in free energy F with respect to the number of the respective species keeping temperature, the number of other species, and the number of sites fixed; where the latter constraint is tantamount to fixing volume. This yields   ∂f cv ∂F = = fv − fa + kB T ln µv := ∂nv ∂cv 1 − cv − ci   ∂F ∂f ci µi := = = fi − fa + kB T ln . ∂ni ∂ci 1 − cv − ci

(37) (38)

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By requiring the chemical potentials to vanish (µv = 0, µi = 0) the well known equilibrium concentrations (cf. [1, 2]) are obtained through

ceq v (T ) =

exp [− (fv − fa ) /kB T ] 1 + exp [− (fv − fa ) /kB T ] + exp [− (fi − fa ) /kB T ]

= exp (−fv /kB T ) ceq i (T ) =

(39)

exp [− (fi − fa ) /kB T ] 1 + exp [− (fv − fa ) /kB T ] + exp [− (fi − fa ) /kB T ]

= exp (−fi /kB T ) ,

(40)

eq eq eq as follows from ceq a (T ) = exp (−fa /kB T ) and cv (T ) + ci (T ) + ca (T ) = 1.

As accepted in defect thermodynamics we will in the sequel neglegt the entropy of formation and approximate the free energies of formation of the defects by the according energies of formation Evf and Eif , i.e., we employ fv ≈ Evf and fi ≈ Eif .

4.2.

Constitutive relations for bulk and surface fluxes

By assuming the deviation from equilibrium to be small one usually assumes a linear relationship between thermodynamic forces and fluxes. We adopt this for bulk and surface fluxes and assume ∇T T   ∇T Ji = −Mi ∇µi − µi T   ∇T Jv = −Mv ∇µv − µv T Je = −Me

ˆ ˆ eˆ = −M ˆ eˆ ∇T J T   f + Ωa κγ ˆ ˆ ˆ J = −M∇ , 1 + ci − cv

(41) (42) (43) (44) (45)

ˆ eˆ and M. ˆ In the following, with positive definite mobility matrices Me , Mi , Mv , M the mobility matrices will always be taken as isotropic, i.e. as a multiple of the unit tensor I in the bulk and of the metric tensor ˆI on the surface. We assume the coefficients for the fluxes of bulk and surface energy as related to bulk and surface

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ˆ and the defect mobilities to be related to the thermal conductivities K and K, scalar diffusivities (Di , Dv for interstitials and vacancies, respectively, and Ds for surface atoms) through

Me = KT I

(46)

Mi =

Di ci (1 − ci − cv ) I kB T

(47)

Mv =

Dv cv (1 − ci − cv ) I kB T

(48)

ˆ eˆ = KT ˆ ˆI M ˆ = Ds (1 − ci − cv ) ˆI. M kB T

(49) (50)

Reactions in the bulk and boundary conditions for the fluxes may principally also be modeled in a linear fashion. However, the canonical form for the rate of a chemical reaction (e.g. [37]) suggests for the bulk reactions that dξ = ν sinh (− (µi + µv ) /kB T ) dt

(51)

where ν is an attempt frequency. A linearization of this expression yields the more common form containing a rate coefficient k and the product of the concentrations
of the involved substances, i.e., dξ/dt = k ci − ceq (cv − ceq v ). i

4.3.

Constitutive relations for surface reactions

The surface reactions of vacancies and interstitials are modeled in the spirit of transition state theory. The earlier derived restriction on thermodynamic consistency helps define the energy state of defects when they reach the surface. It is instructive to first recall what the surface reactions mean in the current case: interstitials and vacancies lose their defect identity once they reach the surface. Interstitials become regular surface atoms while a vacancy reaching the surface is tantamount to a surface atom filling a vacant site close to the surface. Likewise, the process of defects ’entering’ the bulk from the surface is in fact a creation of defects. In the case of interstitial creation, a surface atom jumps to an interstitial site, while

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a bulk atom jumps to a surface position to create a vacancy. In the following we will not distinguish different kinds of surface atoms or consider surface defects to determine the available reaction sites at the surface. Instead we assume ’empty’ surface sites and the surface atoms to be always available if needed for a reaction. This corresponds to assuming an atomically rough surface. Generically we write the reaction of a defect species α ∈ {i, v} with the surface as a forward reaction rate from bulk to surface

  Rαbs = cα ναbs exp −∆gαbs /kB T .

(52)

Similarly, the rate of a backward reaction from surface to bulk (defect emission) is expressed in the form

  Rαsb = ca ναsb exp −∆gαsb /kB T .

(53)

Here ναbs and ναsb denote attempt frequencies, cα is the limiting value of the concentrations of defects at the surface, ca = 1 − ci − cv is the limiting value of the site fraction of regular atoms at the surface and ∆gαbs and ∆gαsb denote the kinetic barriers which have to be overcome by the reactions in either direction. We again note that due to the assumption of dumbbell interstitials surface atoms jumping into the bulk will form an interstitial only when they jump to a position currently occupied by a regular atom. Similarly, the creation of a vacancy at the surface requires a regular bulk atom to jump to the surface. Therefore, the ’atom density’ ca determines the available reaction sites for the backward direction for either defect. The net reaction rate is given by the difference between forward and backward rates

    RαS = Rαbs − Rαsb = cα ναbs exp −∆g bs /kB T − ca ναbs exp −∆g sb /kB T .

(54)

We assume the barrier from bulk to surface to be given as ∆gαbs = Eαm + ∆gαS , where Eαm denotes the migration barrier of species α in the bulk and ∆gαS denotes an additional barrier for the final jump to the surface. For the backward direction

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we assume that it additionally contains the net free energy of formation Eαf − fa and a possible further deviation ∆Eα due to the surface geometry and the adjacent defect state, i.e.: ∆gαsb = Eαf −fa +∆Eα +∆gαbs . With these definitions and assuming ναbs = ναsb = να we rewrite the net reaction as   h   i RαS = cα να exp −∆gαbs /kB T − ca να exp − Eαf − fa + ∆Eα + ∆gαbs /kB T  n h  = cα να exp −∆gαbs /kB T 1 − exp − Eαf − fa   cα +kB T ln + ∆Eα /kB T 1 − ci − cv   = cα να exp −∆gαbs /kB T {1 − exp [− (µα + ∆Eα ) /kB T ]} . (55)

The logarithmic term in the second line is obtained from the identity ca /cα = exp [ln (ca /cα )] = exp [− ln (cα /ca )]. It was noted earlier that the fluxes corresponding to the site fractions considered here have the dimension of a velocity. Consequently, these fluxes across the surface Jα · n derive from the reaction rates by multiplication with the distance δ the surface moves per atom as Jα · n = δRαS , where we remind that δ ∼ Ωa 1/3 . Because the reaction rate RαS will have the same sign as µα + ∆Eα we obtain by comparison with the requirements from thermodynamic consistency (31) and (32) that f + Ωa κγ 1 + ci − cv f + Ωa κγ ∆Ev = 1 + ci − cv ∆Ei = −

and

(56) (57)

are thermodynamic consistent choices for the influence of geometry and defect state on the energetics of the backward reactions at the surface. We summarize that we model the flux boundary conditions from the surface reactions of interstitials and vacancies through     f + Ωa κγ Ji · n = δci νi exp 1 − exp − µi − /kB T (58) 1 + ci − cv       f + Ωa κγ bs Jv · n = δcv νv exp −∆gv /kB T 1 − exp − µv + /kB T . (59) 1 + ci − cv 

−∆gibs /kB T



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In the current section we derived thermodynamically consistent coupled evolution equations for interstitial and vacancy concentrations and heat flow in a single component material which are coupled to a moving boundary. The motion of the surface is governed by reaction equations of defects with a free surface. Although the evolution equations are developed in the general framework of irreversible thermodynamics including variable temperature, the numerical implementation so far focused on the constant temperature case. This means that we do not consider heat fluxes and assume the heat of reactions both in the bulk and at the surface to be negligible.

5. 5.1.

Results and discussion Numerical implementation

The equations for the constant temperature case are discretized by a finite volume method. Finite volume methods provide greater flexibility for the underlying spatial discretization than finite difference methods in that non-uniform and nonrectangular meshes can be used (see e.g. [38]). This is relevant for solving a system with moving surfaces, where a non-uniform motion of the surface, i.e. of surface nodes, leads to irregular meshes. Moreover, finite volume approaches are suited for discretizing curved surfaces and for dealing with flux boundary conditions arising from reactions (58) and (59). There are two finite volume schemes used in the current work. The first one is a one-dimensional implementation of spherically symmetric diffusion around a three-dimensional void. The second scheme is developed for quadrilateral two-dimensional meshes. The schemes exclusively use unknowns at cell centers (barycenters), not at cell vertices. The surface velocity is calculated from the fluxes across the surface, (58) and (59), and atom conservation (5). Time integration is done by a forward Euler scheme.

5.2.

Material parameters

All calculations have been performed with material parameters for copper, taken from the literature [39, 40]. The values are given in Table 1. The diffusivities are

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Table 1.: Material parameters of Copper used in the simulations. Eif 2.2

Evf 1.0

Eim 0.24

Evm 0.8

Es 0.9

eV

Di0 200

Dv0 2.5×103

Ds0 2.6×104

nm2 ns−1

obtained as

Di = Di0 exp (−Eim /kB T )

(60)

Dv = Dv0 exp (−Evm /kB T )

(61)

Ds = Ds0 exp (−Es /kB T )

(62)

The attempt frequency for the surface jumps ν0 is assumed as ν0 = 72300 ns−1 . ˚3 which yields the jump distance The atomic volume is approximated as Ωa = 12 A ˚ Bulk recombination was not considered in the examples shown below. δ = 2.3 A.

5.3.

Equilibrium at the surface

As a first check for the current theory we take a look at the equilibrium concentrations of defects at the surface, which in the current case derive from the requirement of detailed balance, i.e., by requiring the net fluxes across surface (58) and (59) to vanish. These fluxes vanish if and only if f + Ωa κγ =0 1 + ci − cv f + Ωa κγ µv + = 0. 1 + ci − cv µi −

(63) (64)

Because these expressions explicitly contain the free energy, they depend on the reference level of the free energy. However, as is easily checked, the free energy as defined in Eq. (36) vanishes for the equilibrium concentrations at every temperature. For a flat surface (zero curvature κ), the requirements of detailed balance (63) and (64) thus recovers the equilibrium concentrations in the bulk, which correspond to µi = µv = 0. Due to the concentrations appearing in the denominator of the surface term there are no analytical solutions for equations (63) and (64) in the case of non-vanishing curvature κ. However, numerical solutions indicate that the

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curvature dependent equilibrium concentrations coincide with the classical result (see e.g. [2]), given by

    f cclass (k) = exp − E − Ω κγ /k T a B i,eq i     f cclass (k) = exp − E + Ω κγ /k T . a B v,eq v

(65) (66)

for all practical matters. A sample result, the vacancy equilibrium concentrations at void surfaces of different radii as obtained from (65) and from numerical solutions done with Mathematica

R

of detailed balance (63) and (64) are compared in Figure

3 at a temperature of 1000 K. We note that the equilibrium concentrations are independent of the height of surface barriers ∆gibs and ∆gvbs which only affect the kinetics at the surface.

5.4.

Void growth

As a first test on the dynamics we look at the growth or shrinkage of a twodimensional void of 30 nm radius in a matrix of different supersaturation of vacancies and interstitials. In these simulations, the barrier for defects to jump to the surface ∆gαbs was set to the migration barrier Eαm . In Figure 4 the growth curves of different combinations of supersaturations of vacancies and interstitials are shown,

1.75e-04

Current work Classical

1.50e-04 1.25e-04 cveq

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1.00e-04 7.50e-05 5.00e-05 2.50e-05 0.00e+00 1

2

3

4

5

6

7

8

9 10

Radius in nm Figure 3.: Near surface vacancy equilibrium concentration as function of void radius.

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again at a temperature of 1000 K. Supersaturation refers to the equilibrium con−12 , ceq = 9.03 × 10−6 . In centration, see (39), given at 1000K by ceq v i = 8.00 × 10

the case of no supersaturation, no motion of the surface is visible, although there is a small shrinkage rate, because the void emits vacancies in order to establish the higher curvature dependent equilibrium concentration at the surface. In all supersaturated cases, the growth or shrinkage shows an initial steady state regime controlled by diffusion. The lower migration barrier of the interstitials makes them the faster species. Therefore, the shrinkage is faster for high interstitial supersaturation (Sv = 1, Si = 108 ) than the growth in a similar vacancy concentration (Sv = 100, Si = 1). The interstitials also dominate in the case where both species are present in approximately the same amount (Sv = 100, Si = 108 ). For the high interstitial supersaturation cases the influence of the void curvature displays in the shrinkage developing faster than linear. This superlinear shrinkage is due to the following feedback loop: the surface curvature decreases the equilibrium concentration of interstitials at the surface (compare (65)) which accelerates the diffusion of interstitials to the surface and thus the shrinkage rate, which in turn increases the surface curvature. Unique to the current approach is the possibility to consider barriers for the final step to the surface which differ from the migration barriers in the bulk. Figure 5 shows how the motion of a void surface in a vacancy super-

Void radius in nm

February 19, 2015

òòòò òòòò ò ò ò òòò òòòò ò ò æ ò æ æ æ à æ æ æ æ ô ì 30 à à à à à à à æ æ æ æ æ æ æ æ æ æ æ æ æ àààààà ô ì àààààà ô à ì ô ìô S = 1; S = 1 æ v i ìô 25 7 à Sv = 1; Si = 10 ìô 8 ô ì Sv = 1; Si = 10 ì ô ò Sv = 100; Si = 1 ì 20 ô S = 100; Si = 108 ì ôô v

15

ì

0

10

æ

ô æ

20 Time in ms

30

40

Figure 4.: Void radius evolution for a void of initial radius 30 nm surrounded by different defect supersaturations.

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saturation of Si = 100 and equilibrium interstitial concentration is varying with the final barriers taken as fractions or multiples of the migration barriers, both for interstitials and vacancies. For barriers lower than the migration barrier the motion of the surface is slightly sped up, while assuming higher barriers slows down the void growth considerably. In Figure 6 the time evolution of the surface velocity for the cases of ∆gαbs = Eαm and ∆gαbs = 0.25Eαm are compared to a calculation where the concentration at the surface was fixed at equilibrium concentrations, as usually assumed in the literature (see e.g. [1]). The comparison shows, as expected, that the assumption of equilibrium concentrations at the surface corresponds to assuming a very low or vanishing barrier for the final jump of the defects to the surface. Figure 7 shows the time evolution of the concentration of vacancies at the surface (that is in the last finite volume element) and the according motion of the surface for the different barriers at the surface. For comparison we also show the equilibrium concentration. The initial position of the void surface is displayed at the initial radius of 30 nm. In all cases the void grows monotonically (moving to the right) and the symbols indicate the surface position and concentration after the first time step and subsequently at time intervals of 50 ns. The right most points correspond to a simulation time of 400 ns. For the low barriers from bulk to surface, the æ à æ à ì æ à bs m æ à ì æ 33.0 à Dg = 0.5 E àìì æ à bs m ì Dg = 1.0 E æ àìì æ àì 32.5 bs m æ àì ò Dg = 1.5 E æ àì àì 32.0 ô Dgbs = 2.0 Em æ æ àì æ àì 31.5 æ àì æ àì æ àì ì 31.0 æ à ì ò æ à ì òòò æ à òòò 30.5 ò ì ò òò æ à ì òòò òòò ò ò òôôôôôôôôôôôôôôôôôôô æ à òô ô 30.0 ì bs m æ Dg = 0.25 E

33.5

Void radius in nm

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0

10

20 Time in ms

30

40

Figure 5.: Influence of extra barrier for surface reaction on the motion of a void surface (of initial radius 30 nm) in a vacancy supersaturation of Sv = 100 and equilibrium interstitial concentration.

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2.4e-04

Eq. c at surface ∆gbs = 0.25 Em ∆gbs = 1.0 Em

Velocity in nm/ns

2.0e-04 1.6e-04 1.2e-04 8.0e-05 4.0e-05 0

0.5

1

1.5

2

2.5

3

3.5

4

Time in µs Figure 6.: Comparison of the surface velocity in a vacancy supersaturation of Sv = 100 obtained with two different extra barriers for surface reactions with a velocity obtained when assuming equilibrium concentrations of defects at the surface. surface concentration of vacancies reaches almost immediately a value very close to the equilibrium value. However, when the barrier from bulk to surface is higher, the convergence rate to the equilibrium concentration is slowed down considerably. We now turn to a two-dimensional simulation of a void in a matrix under irradiation. The irradiation is introduced randomly in space and time. A core-shell structure of a cascade is mimicked by creating a high vacancy concentration at a randomly chosen volume element, while an equal amount of interstitials is dis-

0.001

cveq ∆gbsbs= 0.25 Em ∆gbs = 0.5 Em ∆gbs = 1.0 Em ∆gbs = 1.5 Em ∆g = 2.0 Em

0.0008 0.0006 cv

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0.0004 0.0002 0 30

30.1

30.2

30.3

30.4

Radius in nm Figure 7.: Influence of extra barrier for surface reaction on the time evolution of the surface concentration of vacancies starting from constant vacancy supersaturation of Sv = 100 and equilibrium interstitial concentrations.

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tributed to the surrounding eight elements. The probability p for an element Vi to be hit by a cascade is taken proportional to the elements volume, i.e. |Vi | p(Vi ) ∝ P . j |Vj |

(67)

The number of vacancies introduced is obtained from a further random number as a concentration value, which is in case adjusted to not exceed a concentration of 1 in the volume element. Because we introduce defects by changing the concentration, the amount of defects introduced currently depends on the volume of the chosen volume element. While the amount can be easily adjusted to be independent of the volume element by scaling the concentration with the inverse of the volume element, this introduces very high gradients in regions with small elements which caused numerical problems. The current simulation of a void with radius r = 30 nm applies an extremely high dpa rate in order to reach a reasonable number of cascades within the simulation time of 20 µs. Figure 8 shows snapshots of the interstitial (a) and vacancy (b) concentration fields after 4 µs. Recent cascades are visible through the peaks in the concentrations. The concentration of the faster diffusing interstitials is lower than the vacancy concentration because (i ) the cascades spikes are lower and diminish quicker and (ii ) because the interstitials reach the void surface faster.

(a)

(b)

Figure 8.: Snapshot of interstitial (a) and vacancy (b) concentrations under random irradiation around a void after 4 µs.

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6.

VoidGrowthPhilMag˙RG Taylor & Francis and I.T. Consultant

Concluding remarks

A thermodynamically consistent model of reactions of vacancies and interstitials with free surfaces was developed based on mass conservation and the entropy inequality for general non-isothermal situations. The surface reactions were found to depend on the chemical potential and free energy state of the defect species close to the surface and on the surface curvature. Transition-state theory based expressions for the defect reaction rates at the surface which account for these dependencies were also derived. For the case of voids, the resulting model equations were solved with a finite volume method that is able to track the surface motion. Numerical solution showed that the current formalism recovers the well-known equilibrium concentrations at curved surfaces. It also showed that the growth rates of voids depend on the defect-surface reaction barriers. The central motivation for the current work is the development of a physically sound sharp interface model serving as the asymptotic limit of phase field models used for the simulation of void nucleation and growth. The current sharp interface model for point defect driven motion contains several features which are so far not rigorously reflected in all phase field models used in this regard. There are two key elements which are typically not or not well incorporated in phase field simulations of surface motion due to defect fluxes. These are (a) the extra defect-surface reaction kinetics at the surface, and (b) the non-conservation of point defects due to reactions with the surface. As for the defect-surface reaction kinetics, because they are largely unknown and not easily determined from experiment or from atomistic simulations, the current work essentially creates awareness of this issue. But qualitative steps in this direction have been proposed in phase field modeling, for instance in [29], where, besides the bulk recombination rate parameter, an extra surface recombination rate parameter of vacancies and interstitials is introduced. For the case of a simple material treated in this paper the surface kinetics may not be too important, but they will play an essential role in more complex materials like oxides where surfaces may be of markedly different structure than the bulk. Whether a higher or lower recombination rate at the surface can be used to faithfully reflect the extra kinetics at the barrier must be checked by matching the

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phase field models to the sharp interface description, e.g., by the method of asymptotic matching. The non-conservation of point defects is also of direct relevance to the pertinent phase field models. Simple phase field models used for void structure evolution are pure vacancy models [23, 27] which do not consider interstitial concentrations and treat voids as a pure vacancy phase. Within these model assumptions it is justified to treat vacancies as a conserved species at the surface. Phase field models including interstitials [28, 29] also work with a pure vacancy phase for voids, where the energetics is such that the equilibrium interstitial concentration in the void vanishes. The extra recombination rate parameter at the surface used in [29] is a qualitative way of incorporating the non-conservation of interstitials at the surface, which indeed is a reaction with vacancies outside the crystal. But in the sharp interface model this recombination rate is essentially coupled to the surface motion. If there are no vacancies in the crystal, the recombination of interstitials with external vacancies would actually be the only means of surface motion. Because Millett and co-workers [29] use a non-conserved order parameter to describe the surface motion, one would expect a more direct (two-way) coupling between the surface recombination and the evolution of the non-conserved order parameter. Such a coupling is found, for example, in phase field models of epitaxial growth (see e.g. [41] and citations therein), where the diffusion of ad-atoms on a surface and their attachment to surface steps is largely analogous to the interstitial diffusion and their attachment to free surfaces. Such phase field models of epitaxial growth can be asymptotically matched to corresponding sharp interface models, but the additional vacancy field in the current case prevents one from simply adopting the structure for the void problem. In [28] non-conserved order parameter is not used and the void is solely described by a high vacancy concentration. In that case, recombination at the surface directly reduces the region of high vacancy concentration and thus shrinks the void. However, in the latter case there are no extra kinetics applied at the surface, e.g. by distinguishing recombination in bulk and surface. The current model marks an important step towards a physically based theory of void growth in defected crystals. We deliberately chose to work with a manageable level of generality in order to emphasize the important contributions to the surface

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reactions. However, the model is extendable to include more complex behavior as eigenstrain, in case we must consider elastic effects, or electrical charge associated with point defects in ionic materials. The model can also be expanded to include point defect cluster densities in the bulk but that would not have affected the findings in this paper, which is the fact that the motion of voids surfaces is a result of reactions with point defects that are governed by energetic quantities different from those governing the kinetics of diffusion. Such a surface role also revives interest in an old hypothesis: that the void growth can be surface (reaction) controlled [20, 21] and implies in general that surface reactions are a controlling factor in the dynamics of voids in irradiated materials. Most importantly, in this context, the current model defines precisely the underlying thermodynamic force and the activation barriers. We finally note that, with a suitable nucleation criterion, the void evolution framework presented here can be used to investigate the full phenomenon of void nucleation and growth in irradiated materials.

Appendix A. Derivation of the change in surface area

The change in surface area of the intersection Σ = Ω ∩ S of a moving surface S (red) with a control volume Ω is locally given by the normal velocity vS of the intersecting line ∂Ω ∩ S within the surface S. In the following we show that this normal velocity within the surface is related to the normal velocity v of the surface itself by

vS = v

nS · n∂Ω , knS × n∂Ω k

(A1)

which in turn explains the appearance of this term in Equation (12). Here nS denotes the normal vector to S and n∂Ω a normal vector to ∂Ω, the direction of which is chosen such that the positively oriented tangent vector t to the intersecting line ∂Ω ∩ S is given by

t=

nS × n∂Ω . knS × n∂Ω k

(A2)

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We introduce two further local unit vectors (see Figure A1): (i ) The (bi-)normal b∂Ω to the intersecting line within the surface ∂Ω is given by

b∂Ω = n∂Ω × t

(A3)

and (ii ) the normal bS to this line within the moving surface S is

bS = nS × t

(A4)

Note that besides the tangent vector t itself, all vectors defined above lie in a plane orthogonal to t. The derivation of the velocities can therefore be done in a planar geometry as depicted in Figure A1. Equation (A1) can now easily be read from Figure A1 by applying basic trigonometry. Obviously, the angle θ between bS and b∂Ω is the same as the angle between nS and n∂Ω . Basic trigonometry yields vS = v cot θ. Because we have cos θ = nS · n∂Ω and sin θ = knS × n∂Ω k this yields Equation (A1).

∂Ω

S(t)

S(t+ Δt)

n∂Ω θ v nS θ b ∂Ω bS vS v∂Ω Σ(t+ Δt) Ω Σ(t) t

Figure A1.: Schematic of the change in surface area of the intersection Σ = Ω ∩ S of a moving surface S (red) with a control volume Ω.

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Appendix B. Generic conservation laws B.1.

Generic evolution law

Suppose the body M under consideration is composed of two parts M1 (bulk) and M2 (voids) with empty intersection (M1 ∩ M2 = ∅) and a (not necessarily simply connected) surface S separating M1 and M2 . That is, M = M1 ∪ M2 ∪ S and ¯1 ∩ M ¯ 2 , where the superscribed bar denotes the set theoretic closure. Let Ω S=M be a control volume which allows to apply Gauss’ integration theorem. We define Ω1 = Ω ∩ M1 , Ω2 = Ω ∩ M2 and Σ = Ω ∩ S = ∂Ω1 ∩ ∂Ω2 . The orientation of S and likewise of Σ is chosen such that the normal points from M1 to M2 , i.e. from the bulk into the voids. The generic evolution law for the integral of some quantity composed of a volume density ω which is possibly discontinuous across S and a surface density ω ˆ on S reads [34] d dt

Z

Z ω dV +

Ω

Σ

 Z Z   ω ˆ dA = ω˙ dV + vJωK + ω ˆ˙ + v ω ˆ κ dA, Ω

(B1)

Σ

where v denotes the normal velocity and κ the mean curvature of the surface, while double brackets denote the jump across the surface JωK = ω 1 − ω 2 . This expression is true regardless of whether the quantity is conserved or not. Conservation laws prescribe this to be equal to some flux and source expressions (see e.g. Equation (13) in the current manuscript). In the case of the entropy inequality the above expression is required to be not smaller than the contribution from entropy fluxes and generation, thus prescribing entropy production to be non negative.

B.2.

Conservation of derived quantities

Assume now that the volume and surface densities ω and ω ˆ are described solely as functions of other evolving volume densities qα and surface densities qˆβ , respectively. Let α ∈ {1, . . . , n} and β ∈ {1, . . . , n ˆ }. Then the evolution equation (B1)

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reads d dt

Z



Z ω ˆ dA

ω dV +

Z ω˙ dV +

=

 vJωK + ω ˆ˙ + v ω ˆ κ dA

(B2)

Σ

Ω

Σ

Ω

Z 

Z X n ∂ω = q˙α dV (B3) ∂q α Ω α=1   Z n ˆ X ∂ω ˆ ˙ + vJωK + qˆβ + v ω ˆ κ dA. (B4) ∂ q ˆ β Σ β=1

If we assume that the elementary quantities obey conservation laws, q˙α = −∇·Jα +Qα − Rα , in M1 and M2 we obtain Z X Z X n n ∂ω ∂ω q˙α dV = (−∇·Jα + Qα − Rα ) dV (B5) ∂q ∂q α α Ω α=1 Ω α=1    Z X n  ∂ω ∂ω ∂ω −∇· = Jα + ∇ · Jα + (Qα − Rα ) dV, ∂qα ∂qα ∂qα Ω α=1

where the product rule of the divergence operator (∇·(f X) = ∇f · X + f ∇·X) was applied. The divergence term can be transformed by Gauss’ integration theorem as Z X n Ω α=1

 −∇·

∂ω Jα ∂qα



n X

Z =

 −∇·

Ω1 α=1

∂ω Jα ∂qα



Z +

n X

 −∇·

Ω2 α=1

∂ω Jα ∂qα

 (B6)

Z n n X X ∂ω ∂ω Jα · n1 dA − Jα · n2 dA (B7) =− ∂qα ∂qα ∂Ω2 ∂Ω1 Z

α=1

Z =−

n X

∂Ω1 \Σ α=1

Z −

n X

∂Ω2 \Σ α=1

Z =−

n X

∂Ω α=1

α=1

Z X n ∂ω ∂ω Jα · n1 dA − Jα · n1 dA (B8) ∂qα ∂q α Σ α=1

Z X n ∂ω ∂ω Jα · n2 dA − Jα · (−n1 ) dA ∂qα ∂q α Σ α=1

Z X n ∂ω ∂ω Jα · n dA − JJα · nK dA, ∂qα ∂qα Σ

(B9)

α=1

where we intermediately introduced the normals n1 and n2 to Ω1 and Ω2 , respectively, and used that at Σ holds n2 = −n1 . With this result at hand we can rewrite

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Equation (B5) as Z X Z X Z X n n n ∂ω ∂ω ∂ω q˙α dV = − Jα · n dA − JJα · nK dA (B10) ∂Ω α=1 ∂qα Σ α=1 ∂qα Ω α=1 ∂qα  Z X n  ∂ω ∂ω + ∇ · Jα + (Qα − Rα ) dV. (B11) ∂qα ∂qα Ω α=1

A similar reasoning applies for the surface terms in Equation (B4) if we assume ˆβ + Q ˆ J ˆβ − R ˆ β , which the surface quantities to obey conservation laws, qˆ˙β = −∇· gives us Z X Z X n ˆ n ˆ ∂ω ˆ ˙ ∂ω ˆˆ ˆ dL qˆβ dA = − Jβ · n ∂ qˆβ ∂ qˆβ Σ ∂Σ β=1

(B12)

β=1

+

Z X n ˆ  Σ β=1

 ∂ω ˆ ˆ ∂ω ˆ ˆ ˆ ˆ dA, ∇ · Jβ + Qβ − Rβ ∂ qˆβ ∂ qˆβ

by means of the product rule for the divergence and Gauss’ integration theorem on the surface. Summing up, the evolution equation for a derived quantity on a domain containing a singular surface is d dt

Z

Z ω dV +

Ω

Σ

Z X n n X ∂ω ∂ω Jα · n dA − JJα · nK dA ∂Ω α=1 ∂qα Σ α=1 ∂qα  Z X n  ∂ω ∂ω + ∇ · Jα + (Qα − Rα ) dV ∂qα ∂qα Ω

 Z ω ˆ dA = −

α=1

n ˆ X ∂ω ˆˆ ˆ dL Jβ · n ∂ qˆβ ∂Σ

Z −

β=1

+

Z X n ˆ  Σ β=1

 ∂ω ˆ ˆ ∂ω ˆ ˆ ˆ ˆ dA ∇ · Jβ + Qβ − Rβ ∂ qˆβ ∂ qˆβ

Z +

v (ω + ω ˆ κ) dA Σ

(B13)

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The flux of ω in the bulk may naturally be defined through n X ∂ω Jω = Jα . ∂qα

(B14)

α=1

In the context of entropy we would split the volumetric term ωs = ωp + ωq into a production and a generation term by

ωp =

n  X α=1

∂ω ∂ω ∇ · Jα − Rα ∂qα ∂qα

 (B15)

n X ∂ω ωg = Qα . ∂qα

(B16)

α=1

We may define in full analogy flux, production and generation terms on the surface by

ˆ ωˆ = J

n ˆ X ∂ω ˆˆ Jβ ∂ qˆβ

(B17)

β=1

ω ˆp =

n ˆ  X β=1

ω ˆg =

ˆ ˆ ∂ω ˆ ˆ ˆ ∂ω ∇ · Jβ − Rβ ∂ qˆβ ∂ qˆβ

n ˆ X ∂ω ˆ ˆ Qβ , ∂ qˆβ

 (B18)

(B19)

β=1

where now ω ˆs = ω ˆp + ω ˆg. Accordingly, we may split the integral expressions into external flux terms (across

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REFERENCES

∂Ω or ∂Σ), bulk and surface production terms and generation terms as

d dt

Z



Z ω ˆ dA

ω dV + Ω

Z X n n ˆ X ∂ω ∂ω ˆˆ ˆ dL Jα · n dA − Jβ · n ∂qα ∂ qˆβ ∂Ω ∂Σ

Z =−

Σ

α=1

β=1

 Z X n  ∂ω ∂ω ∇ · Jα + Qα dV + ∂qα ∂qα Ω α=1

+

Z X n ˆ  Σ β=1

 ∂ω ˆ ˆ ∂ω ˆ ˆ ˆ ∇ Qβ dA · Jβ + ∂ qˆβ ∂ qˆβ

# Z "X n ∂ω + JJα · nK + v (ω + ω ˆ κ) dA ∂qα Σ α=1

Z X Z X n n ˆ ∂ω ∂ω ˆ ˆ − Rα dV − Rβ dA. ∂qα ∂ qˆβ Ω Σ α=1

(B20)

β=1

We note that the jump of the flux across Σ and the terms from changes in volume and surface area are internal production terms with respect to the control volume Ω.

Acknowledgments

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 Science, Office of Basic Energy Sciences under award number FWP 1356, through subcontract number 00122223 at Purdue University.

Notice

This is an Authors Original Manuscript of an article published by Taylor Francis Group in the Philosophical Magazine on 16/02/2015, available online: http://www.tandfonline.com/doi/full/10.1080/14786435.2015.1009516

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