Interfacial energy in phase-field emulation of void nucleation and growth

Journal of Nuclear Materials 411 (2011) 144–149

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Interfacial energy in phase-field emulation of void nucleation and growth A.A. Semenov a,b, C.H. Woo a,⇑ a b

Department of Electronic and Information Engineering, The Hong Kong Polytechnic University, Kowloon, Hong Kong SAR, China Institute for Nuclear Research, Russian Academy of Sciences, Moscow, Russia

a r t i c l e

i n f o

Article history: Received 9 November 2010 Accepted 24 January 2011 Available online 3 February 2011

a b s t r a c t Phase-field model of void nucleation and growth is analyzed. The Ginzburg-type gradient energy term conventionally used under the diffuse-interface assumption is found to make only partial contribution to the interfacial energy. A similar, or even larger, contribution from the bulk terms in the free energy functional also has to be included. Only then, the critical radius for void nucleation under classical nucleation theory can be correctly reproduced. Ó 2011 Elsevier B.V. All rights reserved.

1. Introduction An important feature of the development of irradiation-damage microstructure in metals is the formation of voids due to continuous production and agglomeration of single vacancies and vacancy clusters [1–3]. Physically, a void is an empty space encased by a metallic surface in the metal (often referred to as the void surface). It grows or shrinks according to whether the arrival of irradiationgenerated point defects results in the addition or elimination of atoms on the void surface. Thus, a net flux of vacancies to the void causes it to grow and a net flux of interstitials to the void causes it to shrink. In turn, the point-defect fluxes to the voids are driven by thermodynamic forces acting on the point defects as reacting solutes in a solid solution. Void-swelling models based on this physical understanding [3–7] have stood the test of decades of experiments. To consider the initial formation of the irradiation-damage microstructure, the supersaturated solution of point defects is envisaged to be thermally metastable. Under this condition, small thermally unstable new phase embryos (e.g., microscopic voids) are continuously formed and then re-dissolved after a short characteristic lifetime. They are able to grow beyond the critical size only via sufficiently large stochastic fluctuations of local concentration [8–12]. Beyond the critical size, the new phase nuclei are thermally stable and can grow directly from the supersaturated solution without the help of fluctuations. Physically, the fluctuations required for the nucleation are due to the combined effects of the random nature of point-defect migratory jumps, vacancy emission from voids, and cascade initiation [13]. A recent wave of renewed interest in the study of irradiation damage accumulation has also brought into spotlight research

⇑ Corresponding author. Tel.: +852 2766 6646. E-mail address: [email protected] (C.H. Woo). 0022-3115/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.jnucmat.2011.01.100

methodologies customarily used in other areas of solid-state physics. One of these is the phase-field model, in which the void surface is modeled by a continuous diffuse interface, allowing diffusioncontrolled kinetics of void nucleation and growth to be considered continuously throughout the entire vacancy concentration field. Voids are considered as specific states of this field, at locations where the local concentration of single vacancies approaches unity. This is in contrast to the notion that voids are empty spaces without vacancies. In this regard, we cannot have vacancies without atoms, by definition. The main attractiveness of phase-field modeling, however, is the ability to circumvent difficulties in the treatment of moving boundaries representing the evolving void surfaces as voids form, grow or shrink. This ability makes it potentially very useful in studies where spatial correlation, such as due to the overlap of diffusion fields among different components of the damage microstructure, is important. Current phase-field models of irradiation damage customarily describe vacancy kinetics phenomenologically via the Cahn–Hilliard equations [14–16], originally developed to highlight the phase separation process due to spinodal decomposition [17–22]. Arising from the instability of the homogeneous state of the solid solution, this mechanism occurs without having to overcome a finite activation barrier. That is, phase separation is dictated by the instability condition, and not by the magnitude of the stochastic fluctuations [21–23]. In contrast, classical nucleation is activated via fluctuations that drive the system over a thermodynamic barrier separating the metastable state from the stable state (the new phase) [8–12,21,22]. The probability of sufficiently large fluctuations required for this purpose is usually very small, and decreases exponentially with the height of the activation barrier. Thus, unlike spinodal decomposition, in which the phase separation takes place globally throughout the initially homogeneous solution [21], classical nucleation occurs stochastically only locally at discrete sites. The corresponding void nucleation probability is highly sensitive to temperature, impurity contents, dose rate

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(via the supersaturation) and irradiating particle (via the stochastic fluctuations), which is consistent with experimentally observed behavior of void-number densities [1–3,6,7,13]. The absence of an activation barrier for spinodal decomposition means a zero critical size for the new phase embryo. The behavior of the resulting void development must therefore differ fundamentally from that derived from the classical nucleation theory [3,6,7,13]. Indeed, as a mechanism for void nucleation, spinodal decomposition does not reflect the same characteristics of void formation as predicted by the classical nucleation theory. Thus, all existent phase-field models [14–16] require very high global concentrations of vacancies for voids to be nucleated. However, accumulation of vacancies to such high concentrations is not likely in view of the concurrent production and mutual annihilation during irradiation of vacancies and interstitials, as well as their annihilation at sinks like the void surfaces, dislocations, grain boundaries and external surfaces. To ensure a correct formulation based on the appropriate kinetics, a critical examination of the thermodynamics that drives void shrinkage and growth processes in a supersaturate vacancy solution is of fundamental importance. In the present paper, this issue will be considered in relation to the diffuse-interface assumption made in the existent phase-field methodology. For the sake of readability and the subsequent comparison with the phase-field approach we start with the classical thermodynamic theory of new phase nucleation in the sharp interface approach [11]. 2. Thermodynamics of void formation in the classical nucleation model A physical void as an empty space encased by a metallic surface in the metal can be modeled as a precipitate separated from the metal by a sharp interface across which the vacancy concentration is discontinuous, being identically zero on the void side and finite and variable on the metal side. Consideration of the vacancy solution is only needed on the metal side, where the solution is actually dilute. For the case of dilute ideal solution the thermodynamic potential U (Gibbs free energy) can be written in the following form [11]:

UðP; T; N; nÞ ¼ Nl0 ðP; TÞ þ nkB T½lnðn=NÞ  1 þ nEf ðP; TÞ:

ð1Þ

Here N is the number of solvent particles in solution, l0(P, T) the chemical potential of the pure solvent, n the number of solute particles, Ef(P, T) the chemical energy per solute particle, T the absolute temperature, P the pressure, and kB is the Boltzmann’s constant. The chemical potential l(P, T) of a particle in solution follows from Eq. (1) and is given by

lðP; TÞ ¼ @ U=@n ¼ kB T ln c þ Ef ðP; TÞ;

ð2Þ

where c = n/N is the concentration of the solution (c  1). At finite temperatures, the change in the thermodynamic potential DU due to the precipitation of m solute particles is given by [11]

DU ¼ mðl0 ðP; TÞ  lðP; TÞÞ þ rR;

ð3Þ

0

where l is the chemical potential of a particle in the precipitate, r the surface tension (i.e., surface energy density), R the area of the interface created between the precipitate and the solution, and rR is the interfacial energy. Equality of the chemical potential across the phase boundary at equilibrium allows l0 to be expressed in terms of the equilibrium concentration c1(P, T) of the saturate solution by [11]

l0 ðP; TÞ ¼ kB T ln c1 þ Ef ðP; TÞ:

ð4Þ

Table 1 Material parameters for copper and molybdenum. Parameter

Copper

Atomic volume, X Vacancy formation energy, Efv Surface tension coefficient, r Lattice constant, a0 Melting temperature, Tm

Molybdenum 29

1.0  10 1.2 eV

3

m

1.0  1.7 J/m2 0.36 nm 1356 K

1.34  1029 m3 3 eV 2.05 J/m2 0.31 nm 2885 K

From Eqs. (2)–(4) it follows that the solution is metastable, i.e. DU < 0, when c  c1 is sufficiently large (r > 0). If the precipitate is spherical with a radius R, then m = 4pR3/3X, where X is the particle volume in the precipitate, and Eq. (3) can be rewritten as

  dDU R : ¼ 8prR 1  Rcr dR

ð5Þ

Here Rcr is a critical radius given by

Rcr ¼ 2rX=½kB T lnðc=c1 Þ;

ð6Þ

Obviously, the growth of the precipitate is energetically favored, i.e., (d(DU)/dR < 0), if R is larger than Rcr, and shrinkage of the precipitate is favored, i.e., (d(DU)/dR > 0), if R < Rcr. Another interpretation of Eq. (6) is that it defines the concentration cs(R) of the solution, which is in unstable equilibrium (d(DU)/dR = 0) with the nucleus of radius R. Eq. (5) shows that, an energy barrier DU(Rcr) must be overcome for the formation of a stable nucleus in a supersaturate solution. This energy barrier is proportional to the interface energy at the critical radius Rcr, and can be obtained by integrating Eq. (5) from R = 0 to Rcr

UðRcr Þ ¼ 

8pR2cr r 4pR2cr r þ 4pR2cr r ¼ : 3 3

ð7Þ

In this sense, Rcr may be considered as a measure of the degree of metastability of the supersaturate solution (c > c1) in the classical nucleation model. The absence of point defects in the space above a planar free surface (i.e., outside the metal) means l0 = 0 in the void, so that the vacancy concentration of a saturate vacancy solution in the metal is given by cv1 = exp(Ef/kBT), according to Eq. (4). For the foregoing thermodynamic theory to be valid, the vacancy solution must be dilute, i.e., cv  1. Void swelling is predominantly observed at temperatures below 0.5Tm (where Tm is the melting temperature). For bcc molybdenum and fcc copper it is clear from Table 1 that the dilute solution requirement, namely, cv  1, can be easily met even for a highly-supersaturated vacancy solution (cv  cv1). Thus, in molybdenum Efv ffi 3 eV, and, consequently, at T = 0.5Tm cv1 ffi 3.3  1011 (Tm = 2885 K). This should be compared with the 9 orders of magnitude higher values needed for cv1 in recent phase-field models of void formation, namely, 0.006 [15] and 0.055 [14]. Moreover, it can be easily shown with experimental sink strengths that typical steady-state vacancy concentrations even under continuous production of vacancies by irradiation are very much below these assumed ‘‘equilibrium’’ values. Having to make assumptions of an initial homogeneous vacancy concentrations of c0v ¼ 0:01 [16], 0.05 [15] or even 0.32 [14] is a cause of concern. 3. Diffuse interface in the phase-field methodology As we have discussed in the foregoing, vacancy concentration is non-zero only in the metal, but is identically zero in the void. In contrast, vacancies in the phase-field model are assumed to be continuously distributed in all regions including voids, where the

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vacancy concentration may build up to unity. The voids and the metal matrix are assumed to be separated by a diffuse interface, and the associated interfacial energy is modeled with a Ginzburg gradient term Ur [17–22] in the free energy functional:

Ur ¼

Z

kB T j2

X

V

ðrcv Þ2 dx:

ð8Þ

Here the integral is over the total volume V, cv(x) is the local vacancy concentration, and j is the gradient energy coefficient defined so that Ur produces the correct interfacial energy. Indeed, the usefulness of the diffuse interface model depends very much on the correct definition of j. When the local vacancy concentration approaches unity in regions representing the voids, the dilute solution approximation in Eq. (1) is no longer applicable. In such cases the thermodynamic potential is conventionally interpolated by the following Cahn–Hilliard-type functional [17–22]:

kB T

UðP; T; N; nÞ ¼ Nl0 ðP; TÞ þ

X

Z V

½/ðcv ðxÞÞ þ j2 ðrcv Þ2 dx:

ð9Þ

Without the loss of generality we can express the bulk free energy density / as

/ðcÞ ¼ c lnðc=ecv 1 Þ  uðcÞ;

ð10Þ

where u expresses the deviation from the dilute solution limit. In practice, u is usually represented phenomenologically as a polynomial fitted to experimental observations. To ensure the correct dilute ideal solution limit, we require juðcÞ=c lnðc=ecv 1 Þj ! 0 as c ? 0. /(cv) is also required to have a minimum at cv ffi 1 (in addition to the minimum at cv = cv1) to ensure the existence of a stable ‘‘void’’ phase. To find the correct expression for the gradient energy coefficient j, let us consider the effect of a perturbation dcv(x) = cv(x)  c0v on an initially homogeneous solution with uniform concentration c0v . We assume that the perturbation dcv(x) is small (|dcv(x)|  c0v ) everywhere, except a small but finite volume dV  V, and satisfies the conservation condition, i.e.,

Z VdV

dcv ðxÞdx þ

Z

dcv ðxÞdx ¼ 0:

ð11Þ

dV

Then the free energy change due to the perturbation, according to Eq. (9), is given by

DU ¼ Uðcv ðxÞÞ  Uðc0v Þ Z kB T ½/ðcv ðxÞÞ  /ðc0v Þ þ j2 ðrcv Þ2 dx: ¼ X V

ð12Þ

Since the initially homogeneous vacancy solution is assumed to be sufficiently dilute, we have uðc0v Þ ffi 0. By defining /ð1Þ ¼ c0v , or, equivalently, u(1) = c0v  1  ln(cv1) ffi  ln(ecv1), we can rewrite Eq. (12) as

DU ¼

kB T

(Z

X þ j2

dV

Z V

rR ¼

Z VdV

dV

4. Applications to planar and spherical void-metal interfaces Under equilibrium conditions the chemical potential of vacancies l(cv(x)) has to be the same throughout the whole volume. Since far away from the interface vacancy solution is dilute, so that uðc0v Þ ffi 0, this requirement means that l(cv(x)) = kBTln(c0v /cv1). It follows from Eqs. (9) and (10) that

lðcv ðxÞÞ kB T

X dU d/  2j2 Dcv ðxÞ: ) lnðc0v =cv 1 Þ ¼ dcv kB T dcv ðxÞ

¼

From Eq. (11), the last integral in (13) is dV/V and can be neglected for sufficiently large V. Noting that dV/X is equal to the number of vacancies that can be fitted in dV in the sharp interface case, the interfacial energy can be obtained by equating DU in Eqs. (3) and (13):

ð15Þ

For a planar surface, Eq. (15) becomes one-dimensional and is exactly solvable. Thus, taking cv as the ‘‘spatial coordinate’’, and x the ‘‘time’’, Eq. (15) can be considered as the kinetic equation of a particle motion in the ‘‘potential’’

Uðcv Þ ¼ /ðcv Þ þ cv lnðc0v =cv 1 Þ ¼ cv lnðcv =c0v Þ þ cv þ uðcv Þ:

ð16Þ

The ‘‘trajectory of the motion’’ is cv(x). In our case, only ‘‘trajectories’’ in a finite region of ‘‘space’’ satisfying 0 6 cv 6 1 are physically meaningful. The law of ‘‘energy conservation’’ corresponding to the one-dimensional Eq. (15) has the form

j2

 2 dcv þ Uðcv Þ ¼ h; dx

ð17Þ

where h is the ‘‘time’’-independent integration constant. Since at equilibrium the concentration of the vacancy solution far from the planar surface must satisfy cv = c0v = cv1, putting cv = cv1 in Eqs. (16) and (17) then gives h = cv1. Substituting the solution of Eq. (17) with h = cv1 into Eq. (14) gives the free energy associated with the plane interface S:

rS ¼

kB TS

(Z

X

dx

Z

1

0

ð13Þ

ð14Þ

Although the first integral in Eq. (14) is performed over the volume dV, both terms in the integral are important only in the diffuse interface where cv(x) is significantly different from 1. Thus, its contribution is mainly from the diffuse interface region. Moreover, since c0v 6 cv(x) 6 1 in this region, both contributions from the first integral are positive. Thus, requiring the Ginzburg term Ur to account for the entire surface energy, as it is assumed in most phase-field models of void swelling, would therefore cause an over-estimation, which, as we shall see, can be very large for small voids. Eq. (14) provides us with the framework via which the gradient energy coefficient j can be determined to reproduce the properties of the classical nucleation model. However, the functional u that expresses the deviation of the bulk free energy density from the dilute solution limit has to be known. This will have to be considered for each specific case.

þ j2

)

ðdcv Þ2 dx : 2c0v

½ð1  cv ðxÞÞ lnðc0v =cv 1 Þ þ ð/ðcv ðxÞÞ  /ð1ÞÞdx  Z þ j2 ðrcv Þ2 dx :

X

V

½cv ðxÞ lnðc0v =cv 1 Þ þ ð/ðcv ðxÞÞ  /ð1ÞÞdx

ðrcv Þ2 dx þ

Z

kB T

¼

kB TSj

X

½ð1  cv ðxÞÞ lnðc0v =cv 1 Þ þ ð/ðcv ðxÞÞ  /ð1ÞÞdx



dcv dx

2

) dx

ðlnðc0v =cv 1 ÞI0 ðcv 1 Þ þ 2I1 ðcv 1 ÞÞ;

ð18Þ

where

I0 ðcv 1 Þ ¼ ¼

Z

1

cv 1

ð1  cv Þ ðcv 1 þ /ðcv ÞÞ1=2 1

ð1  u v ðc0

Þ=f ðc0

1=2

v ÞÞ

dcv Z

1

cv 1

ð1  cÞ ðf ðcÞÞ1=2

dc;

ð19Þ

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(a)

(b)

20

cv∞ = 10−14

18

1.0

16 0.5

14 n = 20

cv∞ = 10−12

12

φ /cv

φ

8

10

n=8

8

0.0

cv∞ = 10 −8

cv∞ = 10−14

6

cv∞ = 10−12

cv∞ = 10−10

-0.5 4 2 0 0.0

n=2

-1.0 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

-15

-14

10

10

Vacancy concentration

-13

-12

10

10

-11

10

-10

10

-9

10

Vacancy concentration

Fig. 1. (a) Thermodynamic potential /(cv) = cvln(cv/ecv1)  un(cv) as a function of vacancy concentration for various values of vacancy equilibrium concentration cv1 and polynomial exponent n. Potential for weak solutions, i.e., / = cvln(cv/cv1e), is shown by the dashed lines. Solid lines corresponds to n = 8, and dotted lines to n = 2 and 20. Arrows indicate the points, where the thermodynamic potential qualitatively deviates from a weak solution approximation. (b) The same as (a) for vacancy concentrations near the equilibrium values cv1.

I1 ðcv 1 Þ ¼

Z

1

cv 1

ðcv 1 þ /ðcv ÞÞ1=2 dcv

¼ ð1  uðc00v Þ=f ðc00v ÞÞ1=2

Z

1

ðf ðcÞÞ1=2 dc:

ð20Þ

cv 1

Here f(c) = cln(c/ecv1) + cv1, and c0v ; c00v are some values from the interval (cv1, 1) by the mean-value theorem. Although the first ‘‘bulk’’ term in Eq. (18) is zero for the planar boundary because c0v ¼ cv 1 , we retain this term to keep the form of the equation general for cases where c0v – cv 1 . Evaluation of integrals (19) and (20) requires explicit form of the functional u(cv). As mentioned, u(cv) is assumed analytical and thus expandable as a polynomial. The condition juðcÞ= c lnðc=ecv 1 Þj ! 0 as c ? 0, that ensures the correct dilute ideal solution limit, also requires that the polynomial should at least be of second order. Accordingly, we consider the class of expansions of the following form:

un ðcv Þ ¼ ½n lnðcv 1 Þ þ ðn þ 1Þð1  cv 1 Þcnv þ ½ðn  1Þ lnðcv 1 Þ þ nð1  cv 1 Þcnþ1 v ðn P 2Þ;

ð21Þ

all of which satisfy /(c = 1) = cv1, (d//dc)c=1 = [ln(c/cv1)  dun/ dc]c=1 = 0, and (d2//dc2)c=1 = 1 + n[(n  1)ln(1/cv1)  (n + 1)(1  cv1)] > 0. The thermodynamic potential /(c) = cln(c/ecv1)  un(c) as a function of vacancy concentration for various values of cv1 and polynomial exponent n is shown in Fig. 1. The double well nature of / with minima at the concentrations cv1 and 1 is obvious.

Table 2 Integrals I0 and I1 calculated with un for n = 0, 2 and 8. cv1

I0(u = 0)

I1(u = 0)

I0(u2)

I1(u2)

I0(u8)

I1(u8)

1016 1014 1012 1010 108

0.23 0.24 0.27 0.3 0.34

3.95 3.69 3.4 3.08 2.73

0.34 0.37 0.41 0.45 0.52

1.55 1.44 1.32 1.19 1.05

0.24 0.25 0.28 0.31 0.35

3.1 2.88 2.66 2.41 2.13

The arrows indicate the region where the thermodynamic potential starts to deviate from the dilute solution approximation, i.e., when u becomes important. Integrals I0 and I1 calculated with these functions, for n = 0, 2 and 8, are listed in Table 2. Note that the range of equilibrium vacancy concentrations cv1 between 1016 and 108 considered in Table 2 is sufficient to cover the range of temperatures T between 0.33 and 0.66Tm in the case of molybdenum and T between 0.28Tm and 0.56Tm in the case of copper. From Table 2, the integral I0 can be seen to be generally much smaller than 2I1. However, from the same table and Eq. (18) it can also be seen that the corresponding contribution remains substantial for vacancy supersaturations of many orders of magnitude and cannot be neglected in general, i.e., even in the absence of irradiation when only thermal annealing is considered. The gradient energy coefficient j from Eq. (18) is given by

j ¼ rX=ð2I1 ðcv 1 ÞkB TÞ

ð22Þ

It is obvious from Fig. 1a, Eqs. (20) and (22) that as the exponent n of the interpolation polynomial un(cv) increases, j decreases and the interface boundary sharpens. Using Tables 1 and 2, one can also obtain via Eq. (22) that a characteristic value of j is around one lattice constant a0, which is what could be expected for a free metal surface. The foregoing methodology can also be used to consider voids of spherical symmetry in the diffuse-interface approach. In this case, instead of (17) the ‘‘energy’’ conservation law can be written as

d dr

2

j



dcv dr

!

2 þ Uðcv Þ

¼

 2 4j2 dcv < 0: r dr

ð23Þ

The ‘‘potential’’ U(cv) given by Eq. (16) is shown in Fig. 2. For boundary conditions cv(r = 0) = 1 and cv(r ? 1) = c0v the law of ‘‘energy’’ conservation takes the form:

4j2

Z

1 0

 2 1 dcv dr ¼ Uð1Þ  Uðc0v Þ ¼ lnðc0v =cv 1 Þ: r dr

ð24Þ

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ically unstable with respect to small concentration fluctuations. In the following, we will examine this statement specifically with respect to the present formulation. The change in the free energy caused by infinitesimal inhomoP geneous fluctuations dcv ðxÞ ¼ k 0AðkÞeikx in the diffuse-interface approach can be derived from Eq. (9). Thus,

20

cv∞ = 10

−12

15

10

DU ¼

U

n=8 5

¼

n=2

kB T X 2 V jAðkÞj2 ð1=c0v  u00 ðc0v Þ þ 2j2 k Þ 2X k–0 kB T X 2 V jAðkÞj2 ð/00 ðc0v Þ þ 2j2 k Þ; 2X k–0

ð28Þ

0

-5

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

Vacancy concentration Fig. 2. ‘‘Potential’’ U(cv) at cv1 = 1012 as a function of vacancy concentration for various values of supersaturation c0v and polynomial exponent n. Solid lines correspond to c0v = 107, and dashed lines to c0v = 105. Arrows show the drop in the ‘‘potential energy’’ when the ‘‘motion’’ takes place from cv = 1 at r = 0 to cv ¼ c0v at r ? 1.

With a constant

4j2

Z 0

1

j ffi a0 , Eq. (24) simplifies to

 2 Z  2 1 dcv 4j2 1 dcv dr ffi dr r dr Req 0 dr Z 1 4j jjdcv =drjdcv ¼ lnðc0v =cv 1 Þ: ¼ Req c0v

ð25Þ

Here r = Req can be identified as the spatial position of the interface at equilibrium. For sufficiently large Req the last integral in (25) can be approximated by the corresponding integral I1(c0v ) for planar boundaries. Then, combining (18) and (25) we have

2rX ffi lnðc0v =cv 1 Þ: kB TReq

ð26Þ 40

This is identical to Eq. (6) for the radius of a spherical void in equilibrium with a vacancy solution with concentration cv ¼ c0v . A supersaturation higher than c0v will push the void to grow to a larger radius and a lower supersaturation will cause the void to shrink. Similar to the classical nucleation case in Section 2, Req is the critical radius that provides a measure of the metastability of the vacancy solution and the energy barrier for void nucleation. Eq. (26) is a direct result of a gradient energy coefficient j analytically obtained from Eqs. (14) and (18). It should also be noted that Eq. (26) results mainly from the second ‘‘bulk’’ term in Eq. (18), where the first ‘‘bulk’’ term is negligible when the radius R is much larger than the lattice constant. Indeed, substituting ln(c0v /cv1) from Eq. (26) into the first term gives

2I0

j R

rS  rS ðR  a0 Þ:

ð27Þ

The calculation of the interfacial energy for small values of void critical radius, which, according to (6) and (26), correspond to very high vacancy supersaturations, requires the numerical solution of Eq. (15). This work will be reported in a separate publication. 5. Spinodal decomposition We mentioned in the Introduction that phase separation due to spinodal decomposition occurs when the system is thermodynam-

30

cv∞ = 10−14 cv∞ = 10 −10

20 10

cv∞ = 10−8

0 -10

μ/kBT

-10 0.0

where u00 and /00 are second derivatives. The homogeneous vacancy solution is thermodynamically unstable (DU < 0) when /00 is sufficiently large and negative. Physically, a negative /00 , which corresponds to dl(cv)/dcv < 0, gives a chemical potential that pushes the vacancy diffusion to go against the concentration gradient, i.e., so-called ‘‘uphill diffusion’’ [21]. In such cases, any small local inhomogeneities will be growing with time, leading to instability and eventual phase separation. This is very different from the behavior of a system assumed in classical nucleation model, where dilute solutions of vacancies with uðc0v Þ ffi 0 are metastable, and, consequently, the growth of infinitesimal fluctuations, as well as spinodal decomposition, can not occur (/00 > 0). In Fig. 3, the chemical potential derived from Eq. (10) is shown as a function of vacancy concentration for various values of equilibrium concentration cv1 and polynomial exponent n. Accordingly, the chemical potential only starts to decrease with increase of vacancy concentration when cv reaches values of 102 even for the lowest possible exponent n = 2. Since such high vacancy concentrations are difficult to achieve in metals even at the melting temperature, it is very unlikely that the precipitation of vacancies into voids in the temperature range 0.3–0.5Tm can be physically realized through spinodal decomposition. Void formation in Refs. [14–16] is thus likely the result of spinodal decomposition, rather than that of a classical nucleation process. This can be seen from the extremely high concentrations

-20 -30 -40 -50 -60 -70 -80 -3 10

-2

10

-1

10

0

10

Vacancy concentration Fig. 3. Chemical potential l(cv)/kBT = d/(cv)/dcv as a function of vacancy concentration for various values of vacancy equilibrium concentration cv1 and polynomial exponent n. Chemical potential for dilute solutions, i.e., l (cv)/kBT = ln(cv/cv1), is shown by the dashed lines. Solid lines corresponds to n = 8, and dotted lines to n = 2.

A.A. Semenov, C.H. Woo / Journal of Nuclear Materials 411 (2011) 144–149

( 102) of vacancies which have to be used in these calculations. Unlike the case of classical nucleation, in which the detailed behavior of the stochastic fluctuations plays a fundamental and quantitative role, in the spinodal decomposition stochastic fluctuations only serve as arbitrary noise in the vacancy concentration field to trigger the initiation of phase separation process.

149

Acknowledgments This project is initiated and funded by Grants 532008 and 534409 from the Hong Kong Research Grant Commission, to which the authors are thankful. References

6. Conclusion Attempts to emulate the kinetics of void nucleation and growth has been made in the literature using phase-field models based on Cahn–Hilliard-type free energy functional, and assuming a diffuse void-metal interface. Results of such emulations are found to differ drastically from those obtained using classical nucleation theory and rate-theoretic approaches. Indeed, formation of voids in such cases takes place at unrealistically high vacancy supersaturations, globally, and via spinodal decomposition. In this paper, a diffuse-interface approach to the void formation is developed, which quantitatively reproduces the results of classical thermodynamics for the shrinkage and growth processes of voids. It is found that, besides the conventional Ginzburg-type gradient energy term in the Cahn–Hilliard functional, the energy of void-metal interface also contains contributions from the ‘‘bulk’’ terms, which is generally larger in magnitude. The corrected gradient energy coefficient depends not only on the surface energy density, but also on the vacancy supersaturation, and is particularly significant when the vacancy supersaturation is high, or when the void radius is small. With the correction, realistic critical radius of voids within classical theory of void nucleation in supersaturated, but still dilute and ideal, vacancy solution can be recovered within the phase-field theory framework.

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