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Defect and Diffusion Forum Vols. 323-325 (2012) pp 95-100
Online available since 2012/Apr/12 at www.scientific.net
© (2012) Trans Tech Publications, Switzerland
doi:10.4028/www.scientific.net/DDF.323-325.95
Atomistic simulation of clustering and annihilation of point defects in
Molybdenum
Alexey Yanilkin,1,a Zeke Insepov,2 Genri Norman,1 Jeff Rest,2
and Vladimir Stegailov1
1
Joint Institute for High Temperatures, Moscow, Russia
2
Argonne National Laboratory, Argonne, Illinois, USA
a
aleyanilkin@gmail.com
Keywords: Interstitial, vacancy, cluster, recombination, 1D diffusion
Abstract. Evolution of a molybdenum system containing self-interstitials and vacancies was
studied by molecular dynamics simulation using a new molybdenum interatomic potential. The
potential was parameterized by using formation and migration energies of the defects. Clustering
and annihilation of the defects were investigated in terms of the defect concentration changes during
the calculation. The rate constants were evaluated and compared with the diffusion coefficients.
Also investigated was the influence of one-dimensional diffusion on kinetics, as well as the effects
of temperature and defect concentrations on the reaction rates.
Introduction
Evolution of self-interstitial atoms (SIAs) and vacancies is the first stage of the defected
structure relaxation after generation of a radiation cascade. This stage plays an important role in the
nucleation processes of dislocations and voids. Modern kinetic theory of the radiation damage
describes the kinetics of these processes in terms of the rate constant and interaction radius [1].
Results for the SIA and vacancy interaction were obtained from recovery experiments of
elements irradiated with electrons. Specifically, in [2], the recovery of Mo irradiated at 4.2K was
investigated, and the following activation energies were obtained: 0.066 and 0.076 eV of the
recombination at Stage 1 of recovery. The trapping radius of vacancy was estimated to be 0.7 Å.
These values correspond to a recombination process before free migration of SIA started, which
was shown to be noticeable at temperatures above 40K for Mo [3,4]. Researchers also have studied
[5, 6] the recovery of copper irradiated at different temperatures (50–200K). The temperature
dependence of the radius of correlated recombination was rv=13.7T-0.32a, where a is the lattice
constant.
Experimental results based on the analysis of the time dependence of resistivity were obtained
based on the kinetics of the defects production and probability of correlated recombination [7]. The
probability was based on the assumption that SIA formation occurs at a distance rp and
recombination of these defects at radius rv and is rv/ rp. A theoretical estimation of rv was made in
[8], and the temperature dependence of the recombination radius rv ~ T-n was obtained for a simple
spherical-power attractive potential, where n is the power. The authors assumed that the
recombination radius was equal to the distance at which the interaction in the saddle point was an
order of magnitude of the temperature. The values of recombination radii were obtained in [9] by
using an expression for angular dipole interaction of point defects. These results show a decreasing
radius with increasing temperature and are in a good agreement with experimental results [6]; the
value of the radius is close to 2a.
The recombination volume was calculated by atomistic simulation with a many-body interatomic
potential [10]. The recombination volumes for SIA crowdion configurations around vacancy in
tungsten and vanadium were 42 and 84 times the atomic volume (Ω).
The cited experiments were obtained for 3D diffusion. For 1D diffusion, the theory for diffusioncontrolled reactions predicts a third-order rate of recombination [11]. This prediction agrees with
kinetic Monte Carlo (KMC) studies of absorption of pure 1D, 1D/3D, and 3D gliding interstitial
clusters by voids [12].
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Diffusion in Materials - DIMAT 2011
For the work presented here, static and dynamic simulations were carried out for interactions of
SIA with SIA and with vacancy in Mo, for 1D diffusion of SIAs. The interactions between atoms
were described by an EAM potential [13]. This potential was parameterized by an ab initio forcematching method, with a large set of configurations of defects, and therefore was capable of
reproducing a correct potential energy map and a hierarchy of the formation and migration energies
of the defects. The calculations are carried out with the LAMMPS code [14].
The first part of paper is devoted to molecular static (MS) simulations of SIA and vacancy
interaction. The result of such simulation is an estimate of a reaction cross-section and a reaction
volume. The kinetics of evolution of an ensemble of defects is presented in the second section.
Comparison of the static and dynamic simulations with another theory is discussed in the third
section.
Static simulation
MS model. The MS model represents the interaction of an SIA with a vacancy and another SIA
placed at some distance from the former. In the initial state, the SIA was formed by adding a Mo
atom in a crowdion or a vacancy position by deleting one atom in the center of the calculation cell
and by relaxing the obtained configuration by minimization of the potential energy. Then the SIA
was placed near the first defect by adding a Mo atom at a crowdion position and by optimizing the
structure further. The results obtained for the clustering, annihilation, and conservation were used as
the initial positions of the point defects. In the case of clustering and annihilation, the initial
positions of the point defects were considered unstable.
Results. Figure 1a shows the unstable SIA positions around a vacancy, indicated as gray circles.
The total number of unstable positions is 200, which corresponds to the reaction volume 200ΩSIA,
where ΩSIA is the SIA volume a3/8. The effective radius is 2.8a.
Fig. 1. (a) Unstable positions of SIA around the vacancy. (b) (11 1 ) cross-section of the unstable
positions; the black dots are the diffusion paths along [11 1 ].
The molecular dynamics (MD) simulations of SIA demonstrate the 1D character of SIA
diffusion at low temperatures [11]. Hence, we should be interested in the effective cross-section of
the vacancy, which can differ from the reaction volume. The 1D diffusion of SIA occurs along the
four symmetrical directions [111], [1 1 1], [11 1 ], [1 1 1 ]. The unstable positions lie along the
directions of 1D diffusion, which form the cross section. Figure 1b shows the (11-1) cross-section
of the SIA unstable positions around the vacancy. The number of diffusion paths that pass through
the cross section along the [11 1 ] direction is 13. Each path corresponds to a2/5. The total crosssection is 2.6 a2, which is two times smaller than from the number of unstable positions.
The interaction of SIA and SIA in the crowdion structures can be divided into two cases based
on the cylindrical symmetry of the displacements around the crowdion. The first case is the parallel
configuration of crowdions. Figure 2a shows the unstable position of SIAs around the [111]
crowdion in the (111) cross-section. The number of the diffusion paths along [111] is 6, so the cross
Defect and Diffusion Forum Vols. 323-325
97
section is 1.2a2. The second case is the intersecting configuration of crowdions. Figure 2b shows the
unstable position of the SIAs around the [111] crowdion in the (11 1 ) cross-section. The number of
the diffusion paths along [11 1 ] is 20, so the cross section is 4a2. The average cross-section is 3.5a2.
One can conclude that the cross section of the SIA and SIA interaction is larger than that of the SIA
and vacancy interaction, so the rate of clustering should be larger than annihilation at the same
conditions.
The total number of unstable positions is 242. The total volume is 242 ΩSIA.
Fig. 2. Unstable positions of SIAs around the crowdion (open circle in the center of cell). (a) The
(111) cross-section of the unstable positions, where the black dots are the diffusion paths along
[111]. (b) The cross-section of the unstable positions, where the black dots are the diffusion paths
along [11 1 ].
Dynamic simulation
MD model. We used the MD model to study the evolution of the point defects ensemble. The
point defects in an ideal bcc Mo crystalline structure were introduced into the system at the
beginning of the simulation. The damage rate during the calculations was set to zero.
Two types of the initial microstructures were considered: crystalline containing only SIAs in the
form of crowdions and a system with both crowdions and vacancies. Such an initial setup enables
us to study the separate kinetics of cluster formation and recombination. A typical size of simulation
cell for the simulations of SIA and SIA reactions was 100x100x100 lattice units (31x31x31 nm).
The simulated system contained about 2 million atoms. The initial numbers of SIAs were 300, 400,
and 600. For evaluating the vacancy and SIA recombination rates, a 100x100x100 system with 200
SIAs and 800, 1000, 1500, and 2000 vacancies was used, and a 150x150x150 system with 200 SIAs
and 3000, 4000, 5000, and 6000 vacancies was applied. During the calculation the vacancies were
immobile. Throughout, 3D periodic boundary conditions were used.
The initial space distributions of defects were uniform for both defect types. The crowdions have
four equivalent directions in the basic crystal cell: [111], [ 1 11], [1 1 1], and [11 1 ]. It was assumed
that the equivalent orientations were distributed uniformly.
Kinetics of SIA clustering. During the first 10 ps the system reaches the quasi-equilibrium state
at constant temperature and pressure. Then SIA clustering occurs. In the system with only
interstitials we have the simple kinetic equation
1 − 1 = α ii t ,
ci
ci 0
where ci0 and ci are initial and instantaneous SIA concentrations and αii is rate constant of SIA
clustering. We can represent our results with the coordinates 1/ci – 1/ci0 vs t in order to obtain the
rate constant. Figure 3a shows the evolution defects obtained by MD. We consider three initial
concentrations. Figure 3b plots the dependencies for two temperatures: 300K and 1000K,
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Diffusion in Materials - DIMAT 2011
approximated by a linear equation with the slope αii. At both temperatures the slope does not
depend on the initial concentration. We conclude that the reaction is second order at both
temperatures.
Fig. 3. (a) Dependence of the SIA number on time for different initial concentrations ci0:
1 –1.5*10-4, 2 – 2*10-4, 3 – 3*10-4. T=300K. (b) Dependence of rate constant αii on the initial
concentration for two temperatures T: 1 - 300K, 2 - 1000K. The system size is 2 million atoms.
Kinetics of SIA and vacancy annihilation. We use a system where the concentration of
vacancies is much larger than the concentration of interstitials. With a first-order approximation we
can consider the cv ≈ cv(t = 0) = cv0, where cv0 and cv are the initial and instantaneous concentrations
of vacancies, respectively. The time dependence of SIA is
ci = ci 0 exp(− α iv cv 0t )
Fig. 4. (a) Dependence of the SIA number on time for different initial concentrations cv0: 1 –3*10-4,
2 – 6.8*10-4, 3 – 10*10-4; T=300K; systems size is 6 million atoms. (b) Dependence of rate constant
αiv on the initial concentration; T=1000K; system size is 2 million atoms.
We can represent the results with the scales log (ci/ci0) vs t in order to obtain the rate constant.
Figure 4a shows the evolution defects obtained by MD. We consider three initial concentrations. In
the logarithmic scale the dependencies are approximated by a linear equation with the slope αiv*cv0.
At both temperatures the value αiv does not depend on the initial concentration. We conclude that
the reaction is second order at both temperatures.
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Discussion
We calculated the rate constants of clustering αii and recombination αiv at different temperatures
(Fig. 5a). Two types of points correspond to two different analyses for each reaction. In the case of
SIA and SIA interaction there are the decreasing rate of SIA and the production rate of clustering.
In the case of vacancy and SIA interaction there are the decreasing rates of SIA and vacancies.
Fig. 5. (a) Dependence of the rate constants αii (1,2), αiv (3,4) on the temperature. (b) Schematic
model of SIA and vacancy annihilation. The square is the vacancy. Open circles are the SIA stable
positions. Black circles are the unstable SIA positions.
Although the diffusion at low temperature is 1D, there is no confirmation of the influence on the
defect interaction. The kinetics is described by an equation of second order. This conclusion differs
from that presented in [11, 12], in which the reaction is third order. The main difference is the
postulating of the unsaturated absorbers in the theory and KMC. An alternative approach, referred
to as the “jump method,” results in a second-order reaction in the case of 1D diffusion [15]. The
main proposal of that approach is the uniformly smearing of the defect concentrations throughout
all regions up to the edge of the reaction volume surrounding sinks. Such an approach is applicable
to vacancy-interstitial recombination: because the vacancy is annihilated on absorption of a
migrating interstitial, no gradient can arise. Based on this approach and the cross section obtained
by static simulation, the recombination coefficient can be expressed as αvi = Avivd, where Avi is the
number of absorbers and vd is the hop frequency. The hop frequency can be expressed through the
diffusion coefficient D111 = 3/2vd a2; it follows that αvi = 1.6*1013 s-1. The rate constant of clustering
is αii = 4*1013 s-1. The relation between constants is similar to that obtained by MD, but an
overestimation of the reaction coefficient is inherent in this approach.
The “jump method” can be used to describe the dependence of the reaction cross-section on
temperature. The diffusion coefficient has a weak dependence on temperature, so the effective
cross-section should increase with increasing temperature, a conclusion that contradicts the results
of the dependence of the reaction volume presented in [6,8]. In those works 3D diffusion is
considered. In the case of 1D diffusion, the simple model results in increasing dependence (Fig.5b).
The SIAs diffuse along some direction around the vacancy. MD simulation indicates that the SIAs
should change diffusion direction to annihilate with the vacancy. Thus, the probability of the
trapping of SIAs by the vacancy is p=vrot/(vrot+ vd), where vrot is the rotation frequency. The
activation energy of rotation is about 0.3 eV [13]. The interaction with the vacancy decreases the
activation energy, but the vrot << vd. Therefore the probability is controlled by temperature
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dependence of vrot. It results in an increasing of the cross-section with increasing temperature. To
describe the effect much more precisely, one should sum the trapping probabilities over all
positions around the vacancy.
Summary
We carried out MD simulations in order to investigate the clustering and annihilation of SIAs and
vacancies, which play a crucial role in the beginning of radiation damage. The influence of 1D
diffusion on kinetics was considered by simulations at different temperatures. Although the
diffusion of SIAs is strongly one dimensional at the temperatures considered, the recombination of
SIAs and vacancies is described as a second-order reaction as well as at high temperatures. We
drew the same conclusion for the clustering of SIAs and SIAs.
The results contradict current theory about diffusion reactions and KMC. They can be described
in part, however, by the “jump method.”
A simple model is presented to describe the increase of the reaction cross-section with increasing
temperature. The model is based on the trapping probability of 1D diffusing SIAs around the
vacancy.
Acknowledgement
This work was supported by the U.S. Dept. of Energy under Contract DE-AC02-06CH11357,
president grant MK-3174.2011.8 and RFBR grant 09-08-01116-a.
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Diffusion in Materials - DIMAT 2011
10.4028/www.scientific.net/DDF.323-325
Atomistic Simulation of Clustering and Annihilation of Point Defects in Molybdenum
10.4028/www.scientific.net/DDF.323-325.95