Void lattice formation as a nonequilibrium phase transition
A. A. Semenov and C. H. Woo
*
Department of Electronic and Information Engineering, The Hong Kong Polytechnic University, Kowloon, Hong Kong SAR, China
共Received 3 November 2005; revised manuscript received 29 May 2006; published 20 July 2006
兲
The evolution of a void ensemble in the presence of one-dimensionally migrating self-interstitials is con-
sidered, consistently taking into account the nucleation of voids via the stochastic accumulation of vacancies.
Including the stochastic fluctuations of the fluxes of mobile defects caused by the random nature of diffusion
jumps and cascade initiation, the evolution of the void ensemble is treated using the Fokker-Planck equation
approach. A system instability signaling a nonequilibrium phase transition is found to occur when the mean
free path of the one-dimensionally moving self-interstitials becomes comparable with the average distance
between the voids at a sufficiently high void-number density. Due to the exponential dependence of the void
nucleation probability on the net vacancy flux, the nucleation of voids is much more favored at the void lattice
positions. Simultaneously, voids initially nucleated at positions where neighboring voids are nonaligned will
also shrink away. These two processes leave the aligned voids to form a regular lattice. The shrinkage of
nonaligned voids is not a usual thermodynamic effect, but is a kinetic effect caused entirely by the stochastic
fluctuations in point-defect fluxes received by the voids. It is shown that the shrinkage of the nonaligned voids,
and thus the formation of the void lattice, occurs only if the effective fraction of one-dimensional interstitials
is small, less than about 1%. The formation of the void lattice in this way can be accomplished at a void
swelling of below 1%, in agreement with experimental observation. The dominance of void nucleation at
void-lattice positions practically nullifies the effect of void coalescence induced by the one-dimensional self-
interstitial transport.
DOI: 10.1103/PhysRevB.74.024108 PACS number共s兲: 61.80.Az, 61.80.Hg, 61.72.Cc
I. INTRODUCTION
The formation of void lattices remains a controversial
subject without a generally accepted theory for more than
thirty years since its first observation.
1
Early theories
2–5
at-
tempt to model void ordering as an equilibrium phase tran-
sition via free energy considerations.
6–8
However, the evolu-
tion of a void ensemble under irradiation is a far-from-
equilibrium dynamical process in an open system, a fact that
is recognized by subsequent attempts. The intrinsic difficulty
of the next generation of models
9–12
is their orientation de-
generacy. These theories cannot account for the observed
geometric correlation between the void lattice and the host
lattice. Extending
13
one of these models
12
to include the ki-
netic effect of the intracascade production of glissile self-
interstitial clusters introduces the necessary geometrical cor-
relation with the host lattice. However, the theory requires
the vacancy clusters to be the dominating point-defect
sink,
12,14
which is inconsistent with experimental observa-
tions.
Another group of models focuses on the effect of intrinsic
anisotropic diffusion of self-interstitial atoms 共SIAs兲.
5,15–18
As a part of their investigation of the effects of anisotropic
diffusion on the point-defect kinetics
19
in irradiation damage,
Woo and Frank
5,18
explore a possibility suggested by
Foreman
15
that crowdions moving one-dimensionally along
the close-packed crystallographic directions may introduce a
“void shadowing” effect. They construct a quantitative ki-
netic void-growth model to study the role played by crowdi-
ons on the evolution of a system of growing voids. It was
found that the resulting system of rate equations does not
tend to an asymptotically stationary solution, as usual, but
bifurcates. Following sound mechanistic arguments, Woo
and Frank
5
hypothesized that the bifurcation can be identi-
fied with the instability associated with the nonequilibrium
disorder-order phase transition in the system of voids, i.e.,
the formation of a void lattice. This theory requires that the
mean free path , which the crowdions travel before convert-
ing to three-dimensionally migrating dumbbells, is compa-
rable with the average distance between the voids. In this
case, two voids aligned along the close-packed directions
share a flux of crowdions generated in the overlapping
crowdion-supply cylinders 共CSCs兲 of characteristic length .
Randomly distributed voids 共simply called random or non-
aligned voids in the rest of this paper兲, on the other hand,
receive a full flux of crowdions from the unshared CSCs.
The reduced interstitial flux received by the aligned voids
allows them to grow faster than the random ones. Woo and
Frank
5
intuitively postulate that the resulting competition
will force the nonaligned voids to shrink away, leaving the
aligned voids to form a regular lattice. Indeed, linear stability
analysis showed that the homogeneous void distribution be-
comes unstable when the crowdion mean free path is com-
parable with the average distance between voids. Emerging
from the instability, a periodic structure starts to develop,
with symmetry and orientation following that of the host
lattice.
5,18
Although the theory sounds plausible, a more de-
tail understanding of the evolution of the void ensemble be-
yond the bifurcation, and how it leads to the formation of the
void lattice, is still necessary to complete the theory.
Following a different approach, Evans used a simplified
Monte Carlo simulation, and demonstrated that void lattices
may form if interstitials can be assumed to migrate two-
dimensionally in the close-packed atomic planes.
16,17
How-
ever, such hypothesis lacks experimental support. Nor do
molecular dynamic simulations support the existence of the
interstitial configurations of SIAs whose migration jumps are
PHYSICAL REVIEW B 74, 024108 共2006兲
1098-0121/2006/74共2兲/024108共15兲 ©2006 The American Physical Society024108-1
completely restricted to one plane. More importantly, void
lattices always form in Evans’ model with little exceptions,
contrary to experimental observations.
Thus, there is a fairly general consensus that interstitials
moving one-dimensionally along the close-packed crystallo-
graphic directions are involved in the formation of void lat-
tices, at least they are responsible for the lattice structure and
orientation. This is also consistent with the results of molecu-
lar dynamic simulations,
20–23
which show that clusters of
crowdions can be directly produced in collision cascades and
that these clusters can indeed migrate one-dimensionally
similar to glissile dislocation loops. The fact that it is the
crowdion clusters, but not single crowdions, does not change
the essence of the physics underlying the theory of Woo and
Frank.
5,18
Void ordering due to crowdion clusters has also been stud-
ied using Monte-Carlo 共MC兲 simulations.
24
It is shown that
the crowdion clusters must change the direction of its Bur-
gers vector fairly frequently, so that the mean free path be-
tween consecutive changes is less than about two void-lattice
nearest-neighbor distances. Otherwise, the void growth char-
acteristics will deviate significantly from those observed
experimentally.
24
A more recent MC simulation
25
shows that,
even under rather extreme assumptions 共very high initial
void concentration ⬃10
25
m
−3
, and the number of mobile
interstitials is 20% higher than the number of mobile vacan-
cies兲, the mean free path of crowdion clusters has to be about
four void lattice nearest neighbor distances in order to pro-
duce even a rather poorly defined void lattice 共i.e., compared
with the experimental ones兲. A complementary MC study
26
demonstrated that stable void lattice formation by one-
dimensional 共1D兲 SIA diffusion mechanism is impeded by
the coalescence of neighboring aligned voids. It was also
found that void ordering is delayed until the void swelling
reaches a value of about 1.5%. Experimentally, however,
void lattices are already observable when the swelling is still
below 1%.
3,27–29
Based on these results, the causal relation-
ship between 1D self-interstitial transport and the close-to-
perfect void lattices observed experimentally is questioned.
26
There is, however, an obvious inconsistency in the meth-
odology used in the MC simulation.
26
While the inhomoge-
neous nature of the net vacancy flux in modeling the void
growth is fully taken into account, the nucleation of voids,
which is essentially the growth of smallest void embryos to
the macroscopic size, was assumed to be spatially homoge-
neous. In reality, the latter assumption was clearly invalid
because of the well-known high sensitivity of void nucle-
ation to the magnitude of the net vacancy flux. Indeed, the
importance of correct treatment of the void nucleation prob-
lem in the understanding of microstructure development has
been clearly demonstrated,
30,31
particularly when the micro-
structure is spatially heterogeneous.
The present paper considers an evolving void ensemble
under the mixed fluxes of three-dimensional 共3D兲 isotropi-
cally migrating vacancies and mono-self-interstitials, con-
taining a small portion of 1D migrating self-interstitials.
There is no artificial separation of the formulation into nucle-
ation and growth regimes. The nucleation of voids via sto-
chastic vacancy accumulation from the smallest embryos and
their growth to macroscopic sizes is modeled as one continu-
ous process using the Fokker-Planck equation. Void coales-
cence due to 1D SIA transport is shown to be unimportant to
void-lattice formation, thus removing the concern of the role
played by one-dimensional interstitial diffusion in the void-
ordering process in this regard.
Since in the majority of cases, such as molybdenum,
28
void lattice formation is observed at temperatures for which
vacancy emission from the voids is not important,
32
this con-
tribution is neglected in the present investigation. The paper
is organized as follows: In Sec. II, a stochastic model for the
kinetics of the nucleation and further evolution of a multi-
component void ensemble under irradiation is formulated,
and its properties are investigated. In Sec. III the theory de-
veloped in Sec. II is applied to study various characteristics
of void-lattice formation caused by 1D interstitial transport
due respectively to single crowdion and crowdion clusters,
via which the feasibility of the model is assessed. Possible
effects of void coalescence are investigated in Sec. IV. The
paper is concluded by the discussion of the obtained results.
II. GENERAL FORMULATION
A. Kinetic model
To properly account for void nucleation in the evolution
of the void ensemble, the system has to be considered be-
yond the mean-field approximation in full recognition of the
stochastic and nonlinear nature of the problem. In the correct
treatment, the number of vacancies in a void is a random 共not
deterministic兲 variable that evolves with time, i.e., a stochas-
tic process, which can only be appropriately described by a
time-dependent probability distribution. In the simplest ap-
proximation, within which the stochastic effects due to the
random nature of both the point-defect migratory jumps and
cascade initiation can be included, the kinetic equation for
the void evolution then takes the form of the Fokker-Planck
equation
33
f共n,t兲
t
=−
n
再
V共n兲 −
n
D共n兲
冎
f共n,t兲 + j
0
␦
共n − n
0
兲.
共1兲
Here f共n ,t兲 is the void-density distribution function in the
space of void sizes at time t. We measure the size of a void
by the number n of vacancies in the void. V共n兲 is the void
growth rate defined by the mean-field theory, and D共n兲 is the
diffusivity that governs the “diffusive spread” of the void
distribution function due to stochastic fluctuations. D共n兲 is
related to the average point-defect fluxes and cascade prop-
erties and, in the absence of vacancy emission, is given
by
33,34
D共n兲 =
3n
1/3
2a
2
共
D
C
+ D
i
C
i
兲
+
3GN
d
4ak
关1+共1−
i0
兲
2
兴n
2/3
⬅ d
s
n
1/3
+ d
c
n
2/3
, 共2兲
where D
j
and C
j
共j= i,
兲 are the diffusion coefficient and the
concentration of point defects, respectively, G is the effective
generation rate of point defects, N
d
is the average number of
point defects generated in a single cascade, k
2
is the total
A. A. SEMENOV AND C. H. WOO PHYSICAL REVIEW B 74, 024108 共2006兲
024108-2
sink strength for the three-dimensionally mobile point de-
fects,
i0
is the fraction of the interstitials produced directly
in collision cascades, which do not participate in the conven-
tional three-dimensional motion, a =共3⍀/4
兲
1/3
, and ⍀ is
the atomic volume. The superscripts s and c refer to the
contributions of the stochastic fluctuations due to the random
migratory jumps and random cascade initiation, respectively.
Taking into account that small vacancy clusters consisting
of two or three vacancies are mobile,
35
and a void shrinking
below the minimum size n
min
cannot be treated as a void
anymore, as well as that total void number density has to be
finite, we solve kinetic equation 共1兲 with the zero-boundary
conditions
f共n = n
min
,t兲 =0, f共n → ⬁ ,t兲 → 0. 共3兲
Since the smallest void embryos 共microvoids兲 with sizes n
−n
min
⬃1 may originate either directly from the collision
cascades
22,23
or via the agglomeration of single vacancies, a
contribution j
0
due to the homogeneous production of void
embryos is added to the right-hand side of the kinetic equa-
tion 共1兲. For simplicity, we also assume that the initial sizes
of void embryos are the same 共n
0
−n
min
⬃1兲, which is re-
flected by the
␦
function in Eq. 共1兲.
Since both 1D and 3D migrating point defects are present,
the mean-field void growth rate V共n兲 in Eq. 共1兲 is a sum of
two corresponding contributions
V共n兲 =
冏
dn
dt
冏
3D defects
+
冏
dn
dt
冏
1D defects
, 共4兲
where
冏
dn
dt
冏
3D defects
=
3n
1/3
a
2
共D
C
− D
i
C
i
兲. 共5兲
The corresponding mean-field balance equations for the
three-dimensionally moving vacancies and interstitials at
steady state can be written as
36
共1−
⬘
兲G − D
C
共
+ k
c
2
兲 =0, 共6a兲
共1−
i
兲G − D
i
C
i
共Z
+ k
c
2
兲 =0, 共6b兲
where
i
is the fraction of mobile interstitials that do not
participate in the conventional three-dimensional motion, k
c
2
is the void sink strength,
is the total dislocation density,
and Z is the reaction constant between dislocations and three-
dimensionally moving interstitials. As we shall see in the
following, the values of
i
and
i0
in Eq. 共2兲 are directly
proportional, but not necessarily equal, to each other.
The quantity
v
⬘
in Eq. 共6a兲 is to account for the balance of
free vacancies due to the production and dissolution of mi-
crovoids
⬘
= n
0
j
0
/G − n
min
冋
n
D共n兲
G
f共n,t兲
册
n=n
min
. 共7兲
It is shown in Appendix A that for the stationary solution of
Eq. 共1兲
⬘
=
共n
0
− n
min
兲j
0
G
⬎ 0. 共8兲
Solving Eqs. 共6兲, we obtain from 共5兲 that
冏
dn
dt
冏
3D defects
=
3n
1/3
G
a
2
共
k
c
2
+
兲
冉
i
+
共1−
i
兲共Z −1兲
共
k
c
2
+
兲
−
v
⬘
冊
.
共9兲
Since 共Z−1兲 is usually a small fraction, and in the case of the
void lattice formation the ratio of the sink strengths
/k
c
2
is
of the order of 10
−2
,
27,37–39
we may neglect the correspond-
ing term in Eq. 共9兲 temporarily. The effect of dislocation bias
on void lattice formation will be reconsidered in a later sec-
tion.
Considering that the cross section of a spherical void of
radius r =an
1/3
in a given close-packed direction is
r
2
/s
⬜
,
where s
⬜
is the corresponding area of projection of the
Wigner-Seitz cell, the contribution of the influx of 1D inter-
stitials to V共n兲 is given by
冏
dn
dt
冏
1D interstitials
⬀ −
3
i
G⍀
4as
⬜
n
2/3
. 共10兲
Note that the relation in 共10兲 implicitly assumes that the
mean free path of the 1D interstitials is much larger than the
average void radius. Otherwise, their reaction with the voids
would obey kinetics similar to that of a three-dimensionally
moving point defects.
40
The contribution to the total void growth rate from Eq.
共10兲 is negative and is proportional to the void surface area,
while the contribution from Eq. 共9兲 is positive and is propor-
tional only to the void radius. Thus, at some void size n
s
the
two contributions are equal and opposite, and void growth
saturates. Then, in terms of n
s
, the mean-field void growth
rate V共n兲 can be written as
V共n兲 =
3n
1/3
G
i
a
2
共k
c
2
+
兲
冉
1−
n
1/3
n
s
1/3
冊
⬅
n
1/3
冉
1−
n
1/3
n
s
1/3
冊
. 共11兲
Although in the following subsections
and n
s
will only be
used as probing parameters for assessing the feasibility of the
present theory, it is important that n
s
remains finite as
i
→ 0. In this regard, since
⬘
is positive definite, it does not
cause any divergence of n
s
, and can be omitted from Eq.
共11兲. Nevertheless, its contribution to the final result can be
easily taken into account, simply by substituting 共
i
−
⬘
兲 for
i
in the expression for
. As it will be shown in the Sec. III,
such substitution is not actually necessary.
B. Stationary state
In Appendix A, the stationary solution f共n兲 for n ⱖn
0
of
the kinetic equation 共1兲 subject to the boundary conditions
共3兲 is obtained
f共n兲 =
j
0
共n
0
− n
min
兲
D共n兲
exp
冋
冕
n
*
n
V共n
⬘
兲
D共n
⬘
兲
dn
⬘
册
, 共12兲
where n
min
⬍n
*
⬍n
0
.
VOID LATTICE FORMATION AS A NONEQUILIBRIUM PHYSICAL REVIEW B 74, 024108 共2006兲
024108-3
Using the expressions of V共n兲 and D共n兲 in Eqs. 共11兲 and
共2兲, respectively, the integral in Eq. 共12兲 can be straightfor-
wardly evaluated in terms of the mean-field saturation void
size n
s
冕
V共n兲/D共n兲dn ⬅ I共n/n
s
兲 = I
1
共n/n
s
兲 − I
2
共n/n
s
兲, 共13兲
where
I
1
共x兲 =
3
2
␥
c
冉
␥
s
␥
c
冊
2
冋
冉
␥
c
␥
s
x
1/3
−1
冊
2
+2ln
冉
␥
c
␥
s
x
1/3
+1
冊
册
,
共14兲
I
2
共x兲 =
1
␥
c
冋
冉
␥
s
␥
c
冊
3
冉
␥
c
␥
s
x
1/3
−1
冊
3
+
3
␥
s
2
␥
c
x
2/3
−3
冉
␥
s
␥
c
冊
3
⫻ln
冉
␥
c
␥
s
x
1/3
+1
冊
册
, 共15兲
where
␥
s
and
␥
c
are related to d
s
and d
c
in Eq. 共2兲 according
to
␥
s
=
d
s
n
s
⬵
1
i
n
s
, 共16兲
␥
c
=
d
c
n
s
2/3
=
akN
d
4
i
n
s
2/3
关1+共1−
i0
兲
2
兴. 共17兲
The quantities
␥
s
and
␥
c
account for the stochastic effects
due to the random jumps and random cascade initiation, re-
spectively. The void distribution function in Eq. 共11兲 can
then be rewritten as
f共n兲 =
j
0
共n
0
− n
min
兲
n
s
1/3
共n/n
s
兲
n
s
, 共18兲
where
共x兲 =
exp关I共x兲 − I共n
*
/n
s
兲兴
共
␥
s
x
1/3
+
␥
c
x
2/3
兲
⬵
exp关I共x兲 − I共0兲兴
共
␥
s
x
1/3
+
␥
c
x
2/3
兲
. 共19兲
In Fig. 1, we plot
共n / n
s
兲 as a function of n / n
s
for various
values of
␥
s
and
␥
c
. For small values of
␥
s
and
␥
c
, the sta-
tionary distribution function has a well defined maximum in
the vicinity of the mean-field void saturation size n
s
. How-
ever, the mean-field description of the void system behavior
becomes less and less accurate as the stochastic effects, as
measured by
␥
s
and
␥
c
, increase. Indeed, the peak of the
void-size distribution near the mean-field saturation radius
first broadens, and then disappears all together. The void size
predicted by the mean-field theory is no longer the most
probable. Instead, voids with the size of small void embryos
n
0
dominate the ensemble population, associated with a dras-
tic reduction of their mean lifetime. The disappearance of the
peak signals the instability of the stationary state and the
occurrence of a nonequilibrium phase transformation,
41
which is entirely the result of the stochastic noise in the
point-defect fluxes. Note that thermodynamic phase transi-
tions can also be described in a similar way, because it is
well-known that the stable phase corresponds to the free-
energy minimum, or, in other words, to the peak of the equi-
librium distribution.
A physical picture of this phase transition can be envi-
sioned as follows. In one dimension, the probability of the
eventual capture of a random walker by a trap is well known
to be unity. Due to the stochastic nature of point-defect
fluxes, the growth of a void proceeds such as a random-walk
process in the one-dimensional space of void sizes n, but
with a positive drift defined by the mean-field growth rate.
Since very small vacancy clusters are mobile, a small region
near the origin of the space of void sizes can be considered
as a “sink,” where a shrinking void disappears, i.e., ceases to
exist as a separate entity of the void ensemble. Within this
picture, every void has a finite mean lifetime before destruc-
tion by the stochastic-induced shrinkage, which depends on
its size as n
2
/D共n兲. When stochastic fluctuations dominates
the kinetics of void growth, most voids exist only for a short
period of time due to a large D共 n兲. As a result, few voids can
grow to an observable size, and the void-size distribution
function maximizes at the size of creation of the smallest
voids embryos 共Fig. 1兲. On the other hand, when the charac-
teristic time n / V共n兲 for the mean-field growth of a void of
size n is smaller than the time n
2
/D共n兲 for it to shrink out of
existence stochastically, a positive drift is strong enough to
drive many voids to grow to observable sizes. These voids
will have a long lifetime, and will rarely shrink away. Thus,
the disappearance of the peak at the size-distribution func-
tion marks a qualitative change in the characteristics of the
void ensemble—the fading of the long-life 共LL兲 stable com-
ponent, and the dominance of the short-life 共SL兲 unstable
component.
The LL/SL phase boundary is determined by values of
␥
s
and
␥
c
, for which the stationary size distribution 共18兲 is a
monotonic decreasing function for all void sizes n ⬎ n
0
.
From Eq. 共12兲, this condition is satisfied when
FIG. 1. Void size distribution
共n /n
s
兲 for different values of
parameters
␥
s
and
␥
c
. Here A is the numerically calculated area
under the corresponding curve.
A. A. SEMENOV AND C. H. WOO PHYSICAL REVIEW B 74, 024108 共2006兲
024108-4
max关V共n兲 − dD共n兲/dn兴
n⬎n
0
⬍ 0. 共20兲
In terms of
␥
s
and
␥
c
and the ratio x=n/ n
s
, this condition can
be rewritten as
␥
s
⬎ max关3x −3x
4/3
−2
␥
c
x
1/3
兴
x⬎0
. 共21兲
The value of x
max
⬎0, corresponding to the rhs of Eq. 共21兲,is
determined by a cubic equation for x
max
1/3
. Treating 2
␥
c
x
1/3
as
a perturbation, the positive root of this equation can be very
well approximated by the first iteration. As a result, the
LL/SL phase boundary can be expressed in terms of
␥
s
and
␥
c
by the following equation:
␥
s
=
冉
3
4
冊
4
冉
1−
2
5
␥
c
3
4
冊
3
冋
1−
2
5
␥
c
9共1−2
5
␥
c
/3
4
兲
2
册
. 共22兲
C. Multicomponent system
In the context of the present paper, if the ensemble con-
tains more than one subsystem of voids 共component兲, each
characterized by a different saturation size, it is crucial to
realize that the kinetic equation 共1兲 still applies separately to
each component if the total sink strength k
2
includes the total
sink strength of all voids in the ensemble. Thus, the interac-
tion among the various components is provided via the total
sink strength. This point is central to our present theory. To
facilitate comparison of the behaviors of the different com-
ponents with different saturation radii r
s
and r
s
⬘
we rewrite
the phase boundary equation 共22兲 in the form
␥
s
=

3
冉
3
4
冊
4
冉
1−
2
5
␥
c
3
4

2
冊
3
冋
1−
2
5
␥
c
9

2
共1−2
5
␥
c
/3
4

2
兲
2
册
,
共23兲
where

=r
s
⬘
/r
s
.InEq.共23兲 we take into account that, when
the saturation size changes from r
s
to r
s
⬘
, the values of
␥
s
and
␥
c
are modified by the factors

−3
and

−2
, respectively.
Thus, for a two-component void ensemble with

=1 and

=1.4, for example, the corresponding LL/SL phase bound-
aries from Eq. 共23兲 are shown in Fig. 2. From Eq. 共17兲,
␥
c
is
seen to be an increasing function of the total sink strength.
Then, at the beginning of the radiation when the total sink
strength is sufficiently small, both components evolve in the
long-life phase, i.e., nucleation and growth of most voids to
observable sizes prevail and the system is well described
within the mean-field theory. As the irradiation continues, the
increasing size and number density of the voids result in a
growing total sink strength. For a single-component system,
the total sink strength saturates. The presence of the higher-

component in a two-component system allows the total sink
strength to increase beyond the limit for the single-
component system, pushing the system through the

=1
boundary into a regime where the higher-

component still
remains in the long-life phase in which voids of observable
size are stable. The lower-

component, on the other hand, is
forced to go deeper into the short-life phase in which only a
very low density of observable voids can be sustained
against stochastic shrinkage. Through this kinetic process,
Darwinian selection takes place, resulting in the develop-
ment of the higher-

component and the suppression of the
lower-

component, leading to the eventual dominance of
the higher-

component.
For a one-subsystem void ensemble, stochastic shrinkage
of voids rarely occur because the growth of the void sink
strength is limited by the stochastic effect itself. However, if
a sufficiently high void number density is achieved, some-
how, extra voids of observable sizes are expected to shrink
away during further evolution. Such reduction of void num-
ber density from an initially overpopulated system of voids
of observable sizes is experimentally observable and theo-
retically demonstrated by both the analytical solution of
Fokker-Planck equation
42
and the numerical solution of the
time-dependent master equation.
32
Note that, in the absence
of any vacancy emission from voids, the void shrinkage is
entirely a stochastic effect.
32,42
During the formation of a void lattice, we have a situation
that is quite different from the case of a single-component
void ensemble. Indeed, the key point in the Woo-Frank
theory
5
of void-lattice formation is that, due to the overlap of
diffusion fields, there is a significant reduction in the one-
dimensional interstitial flux in the regions where neighboring
voids in several close-packed directions are sufficiently close
to each other.
5,18
In terms of the present framework, voids
nucleated in these reduced interstitial flux 共RIF兲 regions,
which we may call RIF voids, have larger saturation sizes in
comparison with the “random voids,” i.e., voids that do not
have close nearest neighbors in the close-packed directions.
Considering the void ensemble as a two-component system,
in which the RIF voids constitute the higher-

component,
and the random voids the lower-

component, the phase
transitionlike behavior of void ordering during irradiation
can be readily understood via the Darwinian selection pro-
cess described in the foregoing. In the following, we con-
FIG. 2. Phase diagram for the nonequilibrium phase transition in
the void ensemble induced by the stochastic fluctuations in point-
defect fluxes. The arrow indicates an increase in the value of
␥
c
with the growth of total sink strength k
2
. Dashed lines show the
parameter
␥
s
calculated with the corresponding values of
i
and r
s
.
VOID LATTICE FORMATION AS A NONEQUILIBRIUM PHYSICAL REVIEW B 74, 024108 共2006兲
024108-5
