Atomistic modeling of long-term evolution of twist boundaries under ...

PHYSICAL REVIEW B 86, 214109 (2012)

Atomistic modeling of long-term evolution of twist boundaries under vacancy supersaturation Enrique Mart´ınez* and Alfredo Caro Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA (Received 6 June 2012; published 20 December 2012) Vacancy accumulation in 4◦ {110} bcc Fe and 2◦ {111} fcc Cu twist boundaries (TBs) has been studied. These interfaces are characterized by different sets of screw dislocations: two sets of a20 111 and one set of a20 100 in Fe and three sets of a60 112 in Cu. We observe that vacancies agglomerate preferentially at the misfit dislocation intersections (MDIs), where their formation energy is lower. In bcc the dislocation structure remains stable, but in fcc the interface rearranges itself increasing the stacking fault area. To perform this study a kinetic Monte Carlo algorithm coupled with the molecular dynamics code LAMMPS has been developed. Atomic positions are relaxed at every step after an event takes place to account for long-range strain fields. The events considered in this work are vacancy migration hops. The rates are calculated via harmonic transition state theory with the energy at the saddle point obtained either by a linear approximation considering the relaxed energy of the initial and final configurations or the nudged-elastic band method depending on the vacancy position in the sample. Vacancy diffusivities at both interfaces have also been calculated. For the {110} TB in Fe the diffusivity is of the Fe = 2.60 × 10−13 m2 /s) while at the {111} TB in Cu, diffusivities are two same order of magnitude as in bulk (DTB Cu = 2.06 × 10−12 m2 /s). The correlation factors at both interfaces orders of magnitude larger than in bulk (DTB Cu Fe = 1.61 × 10−4 and fTB = 3.34 × 10−4 ), highlighting the importance of trapping sites at are extremely low (fTB these interfaces. DOI: 10.1103/PhysRevB.86.214109

PACS number(s): 61.80.Az, 61.72.Cc, 61.72.J−, 61.82.Bg

I. INTRODUCTION

The study of the long-term microstructural evolution of heterogeneous systems under irradiation is one of the main modeling goals for the radiation damage community. The importance of studying heterogeneous systems comes from the fact that interfaces are well-known sinks for point defects created under irradiation.1–4 Many deleterious effects of irradiation on material properties, like void swelling and radiationinduced hardening and embrittlement, can be traced back to the formation of these defects. The response to irradiation is characterized by the atomic structure at the interfaces, which in turn prescribes also the mechanical properties, diffusivities, and susceptibilities to embrittlement.5–8 The main goal for materials science is to be able to tune the microstructure in such a way that defects annihilate themselves, such that the system self-heals and withstands higher irradiation doses. Nanocrystalline materials and nanoscale foams show promise as radiation-tolerant structures due to their large sink densities.9–11 Twist boundaries (TBs) in particular provide a playground for models and experiments to test these ideas. The difficulty in modeling irradiation effects in heterogeneous systems comes from the fact that the irradiation process involves physics from very different time and length scales. The starting ballistic process is extremely fast, around a fraction of a picosecond, and drives the system out of equilibrium. On the other hand, thermally activated diffusion tries to restore the system equilibrium but its time scale is much slower, usually beyond the millisecond. The ballistic phase can be modeled atomistically using molecular dynamics (MD) techniques whereas the diffusional phase is beyond the current MD capabilities. Accelerated MD algorithms12–14 extend the physical time that can be studied using MD approaches and provide good insight into complex mechanisms but their application is still limited in the system size as well as the simulated time. 1098-0121/2012/86(21)/214109(11)

The kinetic Monte Carlo (KMC)15–20 algorithm provides a methodology to overcome the MD time limitation. It relies on the a priori knowledge of the probabilities per unit time of different events to happen. An event is defined depending on the problem under consideration. For the diffusion problem an event might be the hop of an atom into a vacancy site or in the context of chemical reactions it might be the occurrence of one specific reaction. In the radiation damage field, the algorithm has been extensively used (see, e.g., Ref. 21 and references therein). In most cases the system is treated as homogeneous, without considering explicitly the presence of dislocations or grain boundaries. Some approximations have been done, both in the context of lattice KMC (LKMC)22 in which a rigid atomic lattice is used taking into account a precalculated segregation energy in the calculation of the rates, and following an object KMC (OKMC) methodology,23–25 in which the atoms are not considered in the calculation and defects are assumed to be objects that can migrate and react among themselves. On the other hand, new algorithms have been developed which are capable of calculating the event catalog on the fly, making no assumptions about what these events might be. The adaptive KMC (AKMC),26,27 k-ART,28 or the self-evolving atomistic KMC (SEAK-MC)29 are some examples. These algorithms would be able to accurately describe the evolution of a complex system under irradiation but, in general, are limited in size and physical time. In this study we develop an approximate algorithm which explicitly takes into account the atomistic heterogeneities (dislocations, grain boundaries, and eventually alloying effects) allowing us to study accumulation of damage in the presence of these heterogeneities. Moreover, the algorithm allows for the microstructure to evolve during the calculation, which, in turn, will modify the response of the system to irradiation. In this paper we study the accumulation of vacancies in a {110} 4◦ TB in bcc Fe and in a {111} 2◦ TB in fcc Cu,

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as a first step towards the full treatment of microstructure evolution under irradiation. The methodology is described in Sec. II, in which the KMC algorithm is sketched along with the calculation of the rates and the coupling with the MD code LAMMPS. Section III shows the results. We have calculated the interaction energy of a single vacancy with the pristine interfaces to study the thermodynamic driving force that the vacancy is likely to be subjected to, we have analyzed the migration path of the vacancy towards the interfaces, and we have performed KMC calculations at different vacancy insertion rates to observe the microstructural changes as vacancies accumulate at these interfaces. In Sec. IV we analyze the results and give an estimate of the error. Finally, in Sec. V the main conclusions of this study are highlighted. II. METHODOLOGY

where ξ is a random number uniformly distributed in the  interval [0,1) and R = N μ μ is the aggregate of all the rates. The probability of event μ to be performed in the interval (t + τ,t + τ + dτ ) is given by the conditional probability μ . (3) R It can be easily sampled by drawing another random number ζ uniformly distributed in the unit interval and taking μ to be the integer for which P2 (μ|τ ) =

μ−1  ν

ν < ζ R 

μ 

ν .

(4)

ν

Once the event is performed, the catalog of rates for every possible event has to be updated. Then, the sampling procedure is repeated until the stop condition is reached.

A. Kinetic Monte Carlo algorithm

The KMC algorithm gives one realization of the master equation provided that all the probabilities per unit time are known for the events available in the problem under consideration. Gillespie18,19 showed how, knowing the probabilities, the master equation is constructed and how, by defining the reaction probability density function P (τ,μ), we can obtain a transition path in phase space that complies with the master equation. This probability is defined such that P (τ,μ)dτ is the probability at time t that the next event V will occur in the differential time interval (t + τ,t + τ + dτ ) and will be the event μ. In general, the algorithm solves the right dynamics provided that the rates (considered equivalent to probabilities per unit time in this case) are given accurately. One of the main challenging problems to be solved in the context of stochastic processes is the accurate calculation of the probabilities per unit time for each event while maintaining the high efficiency of the KMC algorithm, such that the simulated times are long enough to be able to extract meaningful physics from the calculations. In this work, we have developed a hybrid KMC algorithm coupled to the molecular dynamics (MD) code LAMMPS.30 The events that we consider in our calculations are vacancyatom exchanges. Provided that the harmonic transition state theory holds, as it is most likely the case for vacancy diffusion in crystalline systems31,32 and the temperatures considered, the probability per unit time for an atom to jump into the vacancy is given by the expression33 3N i   sp   k ωk μ = i→j = 3N−1 exp − Ej − Ei /kB T , (1) sp ωk k where ωki are the normal-mode frequencies in the initial sp configuration (i), ωk are the normal-mode frequencies when the system is at the saddle point (sp), Ei is the initial energy sp at configuration i, and Ej is the energy of the system at the saddle point escaping towards configuration j . kB is the Boltzmann constant and T is the temperature. Following the direct method,18 once we have calculated the probabilities for every event we can sample the time increment (τ ) from an exponential distribution simply by τ=

−lnξ , R

(2)

B. Transition rate calculations

The accurate calculation of the reaction paths is of paramount importance in theoretical chemistry and condensed-matter physics due to the fact that the knowledge of the mechanism of escaping from configuration i to configuration j and its transition barrier opens the door to the study of the long-term kinetic evolution of the system. A great amount of work has been done in the development of methods to extract such information, both knowing initial and final states34 (and references therein) and without the knowledge of the final state35 (and references therein). All these methodologies are computationally quite demanding. The recently developed Adaptive KMC (AKMC) method (same as k-ART or SEAKMC)26,36 calculates on the fly the transition paths to accurately predict the catalog of events and their rates. Once the catalog of event rates is calculated, the KMC methodology is followed to advance the system in time. Because the saddle-point searches are computationally expensive, the calculations are restricted to rather small systems which implies that the number of events that might take place is also small and likewise for the simulated times. It is worth noting that these calculations are performed in an off-lattice framework where the atoms are free to move following the forces acting on them. On the other hand, traditional lattice KMC (LKMC)37–40 has relied on different approaches to calculate the saddle-point energies,41 from considering that the energy at the saddle point depends neither on the atomic species that jumps nor on the composition around the saddle point to the model where the energy at the saddle point depends on both the exchanging species and the local composition. These approaches have been successfully applied to the calculation of diffusion coefficients as well as precipitation kinetics.42–44 Mason et al.45,46 used the expression given by Flynn,47 derived from dynamical theory, which in essence is a linear approach taking into account the energies of the initial and final configurations, to develop an off-lattice KMC methodology. In a previous work, we followed the same framework to study vacancy accumulation in a 2◦ {100} TB in fcc Cu.48 Basically the same approach is used in Refs. 49 and 50 to calculate the precipitation kinetics in the Fe-Cr system, although in a rigid lattice framework. If the initial and final configurations are known, a more accurate approach to calculate the energy at the saddle point would be to

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use the nudged-elastic band method (NEB), which follows the minimum-energy path to obtain the value of the saddle-point energy.51,52 This method is more computationally demanding, but it gives better results near system heterogeneities where the linear approximation (LA) might fail. Comparisons between these two methods have been shown by Bocquet53 for a diffusion problem in Au-Ni alloys. Our approach uses both methodologies, i.e., the linear approximation and the nudged-elastic band, extending the work presented in Ref. 48. Our system is divided into subdomains. Depending on the atomic configuration of those subdomains, known a priori, the rates are calculated using either the linear approximation for configurations that do not deviate much from bulk, or NEB in the regions where heterogeneities are present. Following this methodology we maintain the efficiency of the linear approximation as much as possible and we calculate the migration barriers accurately where the linear approximation is expected to fail (see Secs. III B and III E). We have performed NEB calculations to define clearly in what parts of the sample we need to use NEB and where the LA is accurate enough. Once these values are obtained, the migration barrier is calculated depending on the position of the vacancy. The whole universe of processors is divided into different worlds. In the case of Fe we have used 80 processors with 8 worlds of 10 processors each, while for Cu we used 120 processors and 12 worlds with 10 processors each. On average, a KMC step takes on the order of 1 s. If the vacancy is in a region where LA is to be used, each world calculates the final energy of a trial configuration (in which the vacancy has exchanged with a first-nearest-neighbor atom) whereas if the vacancy is in a NEB region, each world performs a NEB calculation with each processor taking care of one replica.54 This approach allows us to handle large systems with many defects and we are able to run it for long enough time to extract meaningful statistics out of the calculations. Figure 1 shows the subdomain decomposition and the parallel scheme used in the simulations.

FIG. 1. (a) Sketch showing the region definition and the method used to calculate the migration rates. (b) Parallel scheme used in the calculations. If the vacancy is in a NEB region a set of processors will be used to calculate the saddle-point energy. If the vacancy is in a LA region the set of processors will be used to minimize the local domain to find the final energy of the exchange.

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The migration energy in the LA is defined as follows: sp

sp

Ej = Ej − Ei = E0α +

Ef − Ei , 2

(5)

where E0α is a value for the barrier in bulk that depends on the atomic species that performs the hop, Ei is the energy of the initial configuration, and Ef is the energy of the final configuration. The values for E0α are taken from NEB calculations in bulk samples with the same interatomic potentials used along the study, namely, Ackland 200455 for Fe and Mishin 200156 for Cu. In our case, we have used 0.64 eV for bcc Fe and 0.68 eV for fcc Cu. The attempt frequencies used in this work have been taken as constants with the same value as the Debye frequencies for each element, νFe = 8.75 × 1012 s−1 for Fe and νCu = 6.56 × 1012 s−1 for Cu. C. Hybrid MD-KMC algorithm

Two instances of LAMMPS are defined in the code. The global instance relaxes the whole sample after an event is performed, whereas the local instance is used to calculate the migration barriers. The whole universe of processors is used in the global relaxation. The local instance is used to calculate the rates. The processor universe is subdivided in different worlds. Each world handles the calculation of the rate for a different vacancy-atom exchange. The initialization step comprises the relaxation of the whole system and the calculation of the whole catalog of possible events that are accessible to the current configuration. To calculate the rates, a cutoff defined by the ˚ is used to subdivide the user, in this case we used 8.0 A, domain in small cells. The cell containing the vacancy and its 26 neighboring cells are used to calculate the rates. The atoms at the boundary of this configuration are fixed. If the vacancy is inside a NEB region then a NEB calculation is performed. On the other hand, if the vacancy is considered to be in a close-to-bulk region, each world in parallel calculates the initial and final energies for the different possible atomvacancy exchanges. The typical number of atoms used in this minimization process is around 1000 but it obviously changes along the simulation. The minimization style can be set by the user among the styles available in LAMMPS. We have used conjugate gradient for the calculations in bulk regions and the fire optimizer57 when NEB is performed. Once we have the whole catalog of rates we invoke the KMC algorithm to perform one event. After the event is performed the full system is relaxed via the global LAMMPS instance and the rate catalog is updated. To update the rates we check if the vacancy that has moved has vacancies nearby within a tunable ˚ after convergency tests. All radius that we have set to 7.0 A the vacancies inside the sphere defined by this radius and the center of the hopping vacancy are updated, i.e., their rates are calculated and the frequency line is updated accordingly. III. RESULTS A. Interaction energy of a vacancy with a (110) 4◦ twist boundary in bcc Fe

The sample has been created by a symmetric rotation along the (110) direction of two perfect crystals by 2◦ in

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FIG. 2. (Color online) (a) Screw dislocation network characterizing the 4◦ {110} TB in Fe as given by the dislocation extraction algorithm (DXA) (Ref. 58). Line directions are also shown. (b) Atomic structure showing non-bcc atoms according to the common neighbor analysis (Ref. 59) (white atoms represent the dislocation cores).

clock and anticlockwise directions, respectively, followed by relaxation to 0 pressure using a conjugate gradient algorithm. It contains 256 320 atoms, with dimensions 114.235 × 80.823 × ¯ and [011] directions. ˚ 3 oriented in the [100], [011], 323.126 A Periodic boundary conditions are applied to every direction. After relaxation we observe that the {110} TB in bcc Fe is characterized by a hexagonal network of dislocations composed by two sets of a20 111 screw dislocations and one set of a0 100 screw dislocations as shown in Fig. 2. This configuration has been reported experimentally by Kinsman et al.60 The misfit dislocation intersection (MDI) and the extended misfit dislocation intersection (eMDI), like the segments with Burgers vector a0 100, are thought to be preferential sites for heterogeneous nucleation of voids. To analyze their behavior as sinks we have studied first the interaction energy between the TB and one vacancy. This interaction energy is defined as EVint−TB = E[V − TB] + E[X] − {E[TB] + E[V ]}

FIG. 4. (Color online) Vacancy migration energies in the 111 direction towards the interface as given by NEB and LA calculations. RC is a normalized reaction coordinate in that direction. The bottom configurations show the initial position of the vacancy in light grey and its final position in black at different distances from the interface (white atoms).

driving force for vacancies to segregate to these sites. We also see that the core of the misfit dislocations with Burgers vector a0 111 presents attractive values, with vacancy formation 2 energies lower than in bulk. We see in Fig. 3(b) that there is a basin of attraction around the eMDI and along the a0 111 dislocations. Vacancies falling to a dislocation core 2 will try to reach the eMDI. Depending on their migration barriers, the time to reach it will be different. Moreover, if more vacancies arrive at the dislocation before the single vacancy reaches the eMDI, the mobility of the complex might change and vacancies will accumulate at the dislocation cores as well.

(6)

where E[V − TB] is the energy of the system with the vacancy and the TB, E[X] is the energy of the pure X system (Fe in this case), E[TB] is the energy of the system with the TB, and E[V ] is the energy of the X system with one vacancy. The interaction energy values are shown in Fig. 3. We notice that the minimum of the interaction energy takes place at the eMDI and its value is approximately −1.15 eV. These values imply an attraction between these structural features and vacancies, and therefore, thermodynamically, there is a

FIG. 3. (Color online) Twist boundary-vacancy interaction energy in a 4◦ {110} twist boundary in Fe. (a) Interaction energy map around a MDI. (b) Interaction energy projection along the x direction.

B. Vacancy migration barriers in bcc Fe

Nudged-elastic band calculations have been carried out to determine the range in which the LA approximation can be used in order to obtain the migration barriers. Figure 4 shows the results from these calculations. One of the atoms with the lowest vacancy formation energy at the MDI is replaced by a vacancy and the energy landscape along the minimum energy path in and out of the plane 111 direction is calculated using NEB. We observe that the main distortion from the bulk migration landscape introduced by the interface is fairly short range. Beyond two {110} planes from the interface [configurations (i) and (ii) in Fig. 4] the LA is able to capture the small drift in the landscape. This short-range behavior is indeed expected since the elastic field associated to the twist boundary decays exponentially in the direction perpendicular to the interface plane. The double hump shown in the figure for configurations beyond this second layer is an artifact of the potential and is not considered in the calculation of the rates (see Ref. 61 for details). To calculate these rates the maximum and the minimum values are obtained and used as the migration barriers. Beyond the fourth layer the landscape is indistinguishable from the one in bulk. It is worth noticing that the difference in energy between a stable position in bulk

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(RC = 5, where RC stands for reaction coordinate) and the stable position right at the MDI (RC = 0) is about 1.2 eV, very close to the maximum value shown in Fig. 3. On the other hand, closer to the interface [configurations (iii) and (iv) in Fig. 4], the landscape deviates strongly from the one in bulk and the LA is not accurate enough. In this case NEB is used to obtain the migration barrier, and consequently the migration rates, in the KMC simulations. The energy barrier calculated for the vacancy to migrate from the third layer to the second one [(iii) in Fig. 4] is 0.68 eV while the inverse path gives a barrier of 1.04 eV. For the last migration layer [(iv) in Fig. 4] the obtained value was 0.19 eV while the inverse path has a barrier of 1.00 eV. The path chosen is thought to be one of the most distorted ones in the sample and therefore representative to set the different regions in which the LA or NEB are to be used. As shown in Fig. 4, NEB has been used when the vacancy is two {110} planes far from the interface or closer while LA calculations are performed everywhere else.

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C. Vacancy accumulation in a {110} 4◦ twist boundary in bcc Fe

The TB-V interaction energy calculated in Sec. III A provides evidence of the thermodynamic driving force for vacancies to accumulate at the eMDI. In a vacancy supersaturated environment, the interface atomic structure changes as vacancies accumulate at the interface. To study how this structure actually evolves we have performed KMC calculations for several dose ˚ 3 s, 10−3 V/A ˚ 3 s, and 10−2 V/A ˚ 3 s, equivalent rates, 10−4 V/A −3 −2 to 1.16×10 dpa/s, 1.16×10 dpa/s, and 1.16×10−1 dpa/s, respectively.62 Vacancies are inserted at random locations in the sample with a probability proportional to the insertion rates. The simulations are run for 105 Monte Carlo steps at 500 K. The algorithm captures the microstructure evolution of the dislocation network as it accommodates the excess volume induced by vacancy segregation (see Fig. 5). The black circles in the figure represent vacancies while the white circles are non-bcc atoms as given by a common neighbor analysis. We observe that vacancies agglomerate at the interface. In particular, as the thermodynamic driving force suggests, they tend to accumulate at the eMDI. For this dose rate range, although large, most of the vacancies have enough time to diffuse to the eMDI. When they arrive at these sites, either they delocalize after the relaxation step or get trapped in a deep well. A vacancy is considered delocalized when an ˚ of the stored position of the vacancy. atom is closer than 1 A If this happens, the vacancy will be disregarded for the rest of the calculation, setting its rates to 0. The interface then restructures itself to absorb the free volume, but the dislocation structure remains unchanged for the total doses studied here, i.e., the same dislocation segments can be found in the sample. This means that the accumulation of vacancies does not lead to the formation of a void at the dislocation cores for the doses explored. Instead, the atoms re-accommodate themselves, lowering the density. For the simulation with a ˚ 3 s we have reached a physical time dose rate of 10−4 V/A −1 ˚ 3 s dose rate, the time of 1.086 × 10 s, for the 10−3 V/A −2 ˚3 s attained has been 1.842 × 10 s, and for the 10−2 V/A −3 dose rate we obtain 1.662 × 10 s. In Fig. 6(a) the time step evolution along the simulations is shown. We observe that ˚ 3 s there are both high barriers for a dose rate of 10−4 V/A

˚ 3 s, 10−3 V/A ˚ 3 s, FIG. 5. Twist boundary structure for 10−4 V/A −2 3 ˚ s insertion rates. Vacancies are shown in black while and 10 V/A non-bcc atoms are in white. Panels (a)–(c) show the normal view of the middle interface. Panels (d)–(f) show the view perpendicular to the y axis.

that increase the time step and low barriers that decrease it. There seem to be a correlation between both, happening close in time. The high migration barrier events take place because they are the only ones that may happen in the system, i.e., there is only one mobile vacancy in the sample and it is trapped in a deep minimum. These events are improbable because their rates are low and they might be an artifact of the algorithm. If some other mechanism involving complex arrangements of atoms has lower activation energy, it will not be captured by the present methodology although it would be more probable than the vacancy hop, and the physical time will be biased. Increasing the dose rate makes more events with higher rates available, and therefore the large time steps ˚ 3 s no large are less probable. When the dose rate is 10−2 V/A time step is observed. The small time steps are related to small migration barriers and are observed for all dose rates explored. Some more comments on this time deviation will be given in Sec. IV. We have calculated the vacancy diffusion coefficients in bulk and at the interface. From our calculations we have measured the average time that a vacancy takes to reach the interface. Following the Einstein equation for the diffusivity (D = VFe R 2  ) we obtain a value for Dbulk = 2.74 × 10−13 m2 /s, where 6t ˚ is the mean distance that vacancies migrate in R = 79.71 A bulk and t = 3.86 × 10−5 s is the average time for vacancies to reach the interface. The diffusivity in bulk can be also calVFe 2 culated using the expression Dbulk = aFe νFe exp(−E/kT ), ˚ is the Fe lattice parameter, νFe = 8.75 × where aFe = 2.866 A 1012 s−1 is the vacancy attempt frequency, E = 0.64 eV is the vacancy migration barrier, k is the Boltzmann constant, and T = 500 K is the temperature. This last equation gives

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80 70 60

FIG. 6. (a) Kinetic Monte Carlo time step evolution for the 105 steps run in the Fe {110} TB depending on the dose rate. (b) Vacancy trajectories at the interface for a dose rate of 1.17×10−2 dpa/s.

y

50 40 30 20 10 0 0

20

40

60

80

100

x

(a)

(b)

VFe a diffusivity Dbulk = 2.54 × 10−13 m2 /s in agreement with the calculated value. We have also calculated the average distance that the vacancy diffuses at the interface and the average time that it takes before it is trapped at an eMDI. ˚ on the The average distance has a value of R = 11.71 A, order of half the distance between eMDIs, while the time t = 2.64 × 10−6 s. As it is shown in Fig. 6(b), the vacancy trajectories are mostly one-dimensional along the dislocation cores. Using the Einstein equation in one dimension (1D), 2 (D = R2t  ), we obtain a value for the vacancy diffusivity VFe = 2.60 × 10−13 m2 /s, similar to the at the interface of DTB diffusivity in bulk. The average migration barrier given by the NEB calculations has been ETB  = 0.31 eV. These two last values imply that there are trapping sites where the vacancy spends time without contributing to the mean-square displacement, and therefore, to the diffusivity. For a purely random VFe walk the value for the diffusivity in 1D would be D1D = 2 −9 2 3aFe νFe exp(−ETB /kT ) = 1.62 × 10 m /s which implies that the correlation factor at the interface has a value of DTB f =D = 4.61 × 10−4 . This extremely low value for the 1D vacancy correlation factor gives an idea of the strength of the trapping sites.

D. Interaction energy of a vacancy with a (111) 2◦ twist boundary in fcc Cu

A {111} TB in fcc Cu is characterized by three sets of screw dislocations. These dislocations are Shockley partials formed from the splitting of a20 110 screw dislocations on the interface {111} plane. Because the stacking fault energy in Cu is relatively low, the Shockley partials react among themselves to create the final microstructure (see Fig. 7).63 a0 112 2

(a)

These Shockley partials act as boundaries for stacking faults, which results in the interface patterning observed in the figure, with coherent patches and stacking fault regions separated by the partial dislocations. The sample was prepared rotating symmetrically 1◦ the upper-side crystal in the clockwise direction, and the lower-side crystal in the anticlockwise direction. Similar to the calculations in Sec. III A, the sample was relaxed at 0 pressure using a conjugate gradient algorithm. ˚3 The dimensions of the system are 153.44 × 88.59 × 219.16 A ¯ [12 ¯ 1], ¯ and [111] with 252 210 atoms and oriented in the [101], directions. Periodic boundary conditions are again applied in every direction. The most distorted atoms are located in two consecutive planes which form the interface. As it was done above, we have calculated the interaction energy between the TB and a vacancy to obtain the driving forces acting on the vacancy to be trapped at the interface. The interaction energy is defined as in Sec. III A and the results are shown in Fig. 8. The calculations are done in a prismatic region around one MDI, containing four planes in the z direction. We observe that the interaction is the most attractive right at the MDI, with some negative values (attractive) at the core of the Shockley partial dislocations. At the MDI we obtain a value of −0.4 eV for the interaction energy. We also notice that there are some sites at which the interaction is repulsive. From these data we may conclude that the vacancies will tend to agglomerate at the MDI, where their formation energy is lower. To check this hypothesis we have performed KMC calculations in vacancy supersaturated environments, at the same dose rates ˚ 3 s, 10−3 V/A ˚ 3 s, and as in Sec. III C, namely, 10−4 V/A −2 3 ˚ 10 V/A s. The results are discussed in Sec. III F.

(b)

FIG. 7. (Color online) (a) Dislocation network characterizing the 2◦ {111} twist boundary in Cu as given by the dislocation extraction algorithm (DXA) (Ref. 58). Line directions are also shown. (b) Atomic structure showing non-fcc atoms according to the common neighbor analysis is shown (Ref. 59) (white atoms have unknown structure).

(a)

(b)

FIG. 8. (Color online) Interaction energy of a 2◦ {111} twist boundary in Cu with a single vacancy. (a) Interaction energy map around a MDI. (b) Interaction energy projection along the x direction.

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FIG. 9. (Color online) Vacancy migration energies in the 110 direction towards the interface as given by NEB and LA calculations. RC is a normalized reaction coordinate in that direction. The bottom configurations show the initial position of the vacancy in light grey, and its final position in black at different distances from the interface (white atoms).

E. Vacancy migration barriers in fcc Cu

The same methodology as in Sec. III B has been followed to analyze the energy landscape of the migrating vacancy in {111} planes parallel to the interface. One atom at the MDI with the lowest vacancy formation energy has been identified as the final vacancy position. Nudged-elastic band calculations have been performed to obtain the vacancy migration barriers along a 110 direction out of the interface planes. The migration path followed by the vacancy is shown in the bottom configurations of Fig. 9. The same figure shows the results of the NEB calculations and their comparison with the LA approximation. The results are qualitatively very similar to those observed for the {110} interface in Fe, with the same short-range effect as in the previous case. NEB has to be used up to the second layer from one of the interface planes. In this region, the morphology of the landscape is strongly distorted compared to bulk. The height of the barrier is also different. The direct barrier (from reaction coordinate 2 to 1) is 0.62 eV and the inverse (from reaction coordinate 1 to 2) is 0.63 eV. The last jump for the vacancy to reach the interface [(iv) in Fig. 9] has a migration barrier of 0.20 eV while the barrier for the vacancy to escape from the MDI has a value of 0.60 eV. In the next layer [(ii) in Fig. 9] the morphology is much more similar to the one in bulk. Although the width of the barrier is remarkably lower, the height is very similar. The LA approximation slightly overestimates the barrier (0.66 eV vs 0.68 eV). From the fourth to the third layer [(i) in Fig. 9] both NEB and LA give very similar results as they do in bulk (0.68 eV). We observe again that the energy difference between the equilibrium position at RC = 5 and when the vacancy is at one of the interface planes (RC = 0) is about 0.4 eV, close to the value given in Fig. 8 in the previous section. Similarly to the Fe case, we have used NEB to calculate the rates in the KMC simulations for the vacancy migration when the defect is within two layers from one of the interface planes. The total number of planes

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(f)

˚ 3 s, 10−3 V/A ˚ 3 s, FIG. 10. Twist boundary structure for 10−4 V/A −2 3 ˚ and 10 V/A s insertion rates. Vacancies are shown in black while non-fcc atoms are in white. Panels (a)–(c) show the normal view of the middle interface. Panels (d)–(f) show the view perpendicular to the y axis.

where NEB is used was 6. In the rest of the sample the LA approximation was used to calculate the rates. F. Vacancy accumulation in a {111} 2◦ twist boundary in fcc Cu

Results from the KMC calculations are shown in Fig. 10. Vacancies are shown in black while the non-fcc atoms (as given by a common neighbor analysis) are displayed in white. We observe that at the end of the calculation, after 105 KMC steps, most of the vacancies have reached the interface for the three dose rates investigated. We notice also that most of these vacancies accumulate at the MDIs. The structure of the interface changes at these sites, with new dislocation arrangements created as a result of the vacancy agglomeration. This results in an increase in the stacking fault area of the interface, which in turn modifies the thermodynamic driving force for the vacancies to be absorbed at the interface. This will lead to a modification of the sink efficiency of this TB as damage accumulates. Again, once vacancies reach the MDIs they either delocalize or stay trapped at deep wells. No void formation is observed; the free volume is absorbed by the interface generating more stacking fault area, as mentioned above. Even though the insertion rates are quite large, most of the vacancies have time to diffuse to those sites, lowering the overall energy of the sample. The physical times attained in the simulations were 2.84 × 10−1 s, 2.53 × 10−2 s, and ˚ 3 s, 10−3 V/A ˚ 3 s, and 3.44 × 10−3 s, for dose rates 10−4 V/A −2 3 −3 ˚ s, respectively, equivalent to 1.17×10 dpa/s, 10 V/A 1.17×10−2 dpa/s, and 1.17×10−1 dpa/s. The time step along the simulations [see Fig. 11(a)] follows the same trend as for the Fe TB interface, with some large and small values

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80 70 60 50 40 30 20 10 0

FIG. 11. (Color online) (a) Kinetic Monte Carlo time step evolution for the 105 steps run in the Cu {111} TB depending on the dose rate. (b) Vacancy trajectories at the interface for a dose rate of 1.17×10−2 dpa/s. 0

20

40

60

80 100 120 140

x (b)

IV. DISCUSSION

The approach developed in this study only considers vacancy hops as possible events in the dynamic evolution of the system. In general, complex events, involving different mechanisms, will be present in the real dynamics, and therefore, our approach might be biased. How important these other mechanisms are for the values of the observables studied is a difficult question to answer. To try to shed some light on it and check the accuracy of the KMC calculations we have performed MD simulations at the same temperature (500 K) for 1 ns on the final samples as given by the KMC for the different dose rates. After running the dynamics, the sample was quenched and the atomic positions compared to the KMC sample. This was done in order to characterize the low activation energy processes that may take place during the dynamic evolution of the system and that the algorithm does not capture. We observe that there are some atomic ˚ mostly at the eMDI for the Fe displacements larger than 0.5 A, sample and at the MDI for Cu. Figure 12 shows the frequency ˚ depending of displacements larger than 0.5, 1.0, 1.5, and 2.0 A on the vacancy insertion rate. These displacements correspond to events that the algorithm has not captured. We observe that the number of events is directly proportional to the dose rate. The larger the dose rate the larger the events captured by MD. This suggests that the KMC algorithm is more accurate at low dose rates. At the same time, these results confirm that the final microstructure is close to a relaxed state and that the number of events missed by the algorithm is not dramatically large. It is worth noticing that the number of events larger than ˚ is higher for the Fe sample than for the Cu sample. In 0.5 A general, the dependence of the number of displacements with

600 500 400

3

70

3

60

φ=10

−3

V/Å s

φ=10

−2

V/Å s

300 200

φ=10−4 V/Å3 s

(b)

φ=10−3 V/Å3 s φ=10

−2

3

V/Å s

50 40 30

100 0

80

φ=10−4 V/Å3 s

(a)

Counts

indicating large and small migration barriers, respectively. These time steps above and below the average occur when the vacancy gets close to a MDI, where the atomic deformation is large. We also notice that the number of large time steps decreases with the dose rate, for the same reason as explained for the Fe TB. Because the only event that we consider is a vacancy exchange, it might again be possible that lower activation energy mechanisms are being neglected, which in turn would modify the physical time as well as the actual microstructure. We have followed here the same methodology as in Sec. III C to calculate the vacancy diffusivities in bulk and at the interface. We have used the Einstein equation to obtain the diffusion coefficient since we know the mean-square distance that the vacancy travels before reaching the interface and we can calculate the average time that it takes to be trapped at the boundary. The average distance that the vacancy ˚ and the time that on migrates in bulk is R = 54.25 A average takes to reach the TB was t = 5.58 × 10−5 s which gives a diffusivity of D = 8.79 × 10−14 m2 /s. Comparing to the expression for an entropy driven diffusion in which VCu 2 Dbulk = aCu νCu exp(−E/kT ) = 9.51 × 10−14 m2 /s we see that, again, the values agree. In this last expression a0Cu = ˚ is the Cu lattice parameter, νCu = 6.56 × 1012 s−1 3.615 A is the vacancy attempt frequency, E = 0.68 eV is the vacancy migration barrier, k is the Boltzmann constant, and T = 500 K is the temperature. At the interface, the vacancy ˚ on average, which is about has migrated R = 17.76 A half the distance between MDIs. It has taken t = 7.65 × 10−7 s to be trapped at a MDI. Figure 11(b) shows the vacancy trajectories at the interface for a insertion rate of 1.17×10−2 dpa/s. We observe that, again, the migration is mostly one-dimensional along the dislocation cores. Therefore, the one-dimensional diffusion coefficient at the interface VCu results in DTB = 2.06 × 10−12 m2 /s, which is two orders of magnitude larger than the diffusivity in bulk. The average migration barrier at the interface was ETB  = 0.26 eV. With this migration barrier, the diffusivity of a pure random VCu 2 walk would be D1D = 3aCu νCu exp(−ETB /kT ) = 6.16 × −9 2 10 m /s. Therefore, the correlation factor at this interface, neglecting the effect of the accumulated vacancies, would be DTB = 3.34 × 10−4 . Again, the correlation factor is low f =D 1D which highlights the importance of the trapping sites in this interfaces.

Counts

(a)

20 10 > 0.5

> 1.0

> 1.5

> 2.0

Displacement (Å)

> 0.5

> 1.0

> 1.5

> 2.0

Displacement (Å)

FIG. 12. (Color online) Number of displacement events larger ˚ (a) For the Fe (110) sample. (b) For the than 0.5, 1.0, 1.5, and 2.0 A. Cu (111) sample.

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ATOMISTIC MODELING OF LONG-TERM EVOLUTION OF . . . 1100 1000 900 800

KMC Fe MD Fe KMC Cu MD Cu

700 600 500 400 300 200

10−4

10−3 V/Å3 s

10−2

FIG. 13. Yield point for shearing simulations at 109 s−1 for the 4 {110} bcc Fe sample and the 2◦ {111} fcc Cu sample at different vacancy insertion rates. KMC in the legend refers to samples as given by KMC after 105 Monte Carlo steps while MD refers to the annealed KMC final samples for 1 ns using MD. ◦

the dose rate is lower in the case of Cu while the number of large displacements is slightly larger in Cu. On the other hand, these MD results do not provide conclusive information about the real path followed by the system, in part due to the fact, as it was said in Sec. I, that MD is not able to reach the time scales involved in this study. Moreover, there might be some other processes, with slightly higher barriers, that are not captured by either the KMC or the MD approaches. To capture all these mechanisms, more accurate methods, like accelerated MD or AKMC, might be used, although the ultimate validation of these KMC predictions should come from the experiments themselves. The presence or not of these more complex processes modifies also the physical time attained by the simulations. As it was mentioned in Secs. III C and III F, some long time steps are observed at low insertion rates. These long time steps take place when the vacancy is trapped in a deep well at the MDIs. It might be the case that lower activation energy events compete with the vacancy hops in the real system, and because their rate may be larger, their occurrence probability will be larger too. The existence of these processes would modify the rate catalog, and therefore, the time probability distribution will be different. How far is the physical time obtained just considering vacancy hops from the real time is again difficult to know. It is unlikely that AKMC models will become efficient enough to be able to accumulate the dose considered here, but they seem to be the only methodology capable of giving a more accurate answer to this kind of problem. To analyze the possible implications of the discrepancy between the KMC final sample and MD-relaxed sample on an observable of interest we have studied the stress-strain curves of both configurations given by a shear simulation at a shear rate of 109 s−1 . In Fig. 13 the different yield points are reported. For the Fe sample, we observe that at low dose ˚ 3 s) the yield point remains basically the same. rates (10−4 V/A On the other hand, for higher dose rates the values differ by about 30%, with the annealed samples having a lower yield point. The higher the dose rate the higher the difference. This result suggests that the events shown in Fig. 12 for these

PHYSICAL REVIEW B 86, 214109 (2012)

dose rates actually are important and modify substantially the mechanical properties of the system. Regarding the Cu sample, we observe no appreciable difference in the yield point at any dose rate. This seems to be correlated with the lower number of displacements that we obtain for these samples after annealing. In this latter case, the displacement events reported in Fig. 12 do not modify the mechanical response of the heterogeneous system. Again, this analysis does not give a conclusive answer of whether or not the algorithm follows the right dynamics. It just gives some insight on the importance of events other than vacancy hops that the methodology does not capture. We find that the reliability of the method is dose rate and material dependent. Further studies on different configurations and materials will be carried out in order to assess properly this dependence. From the NEB calculations shown in Secs. III B and III E we observe that the interface effect on the vacancy migration barriers is short ranged. Beyond the fourth plane parallel to the interface, the migration barrier is indistinguishable from bulk. In the next two layers there is an observable drift which can be captured by the linear approximation. In the two planes closer to the interface the barrier is completely distorted and NEB calculations have to be performed to obtain accurate values for the barriers. This result was expected from elasticity theory since the elastic field of these dislocation networks decays exponentially in the direction perpendicular to the interface plane. These NEB results characterize the system in a set of regions in which different approaches are used to calculate the migration barriers, i.e., LA or NEB might be used. Vacancy diffusivities have been calculated in bulk and at the interface for both Fe and Cu systems. In bulk regions, diffusivities in both materials are consistent with the continuum description of atomic transport. At the interface, vacancies are observed to migrate in 1D along the dislocation cores. In Fe, the diffusivity at the interface is of the same order of magnitude as in bulk while in Cu it is two orders of magnitude faster. We have calculated the correlation factors at the interface showing a Cu Fe = 1.61 × 10−4 and fTB = 3.34 × 10−4 ), very low value (fTB which highlights the importance of trapping sites at the boundaries. One of the main goals of this KMC tool is to provide a constitutive law for the interface sink efficiency evolution. The sink efficiency is a function of temperature, dose, dose rate, interface character, etc. The idea is that this methodology will help in the development of a function for the sink efficiency that can be used in continuum models to extend the time and length scales of the systems to be studied, taking into account the change of the microstructure as defects accumulate at the interface. The presence of self-interstitial atoms created by irradiation will most certainly alter the dislocation rearrangement and therefore the evolution of the sink efficiency will be different. Treating self-interstitials in an off-lattice tool retaining a high efficiency is a challenge. However, the self-interstitial diffusivity is usually much larger than the one for vacancies and for the dose rates explored in this study we could, as a first approximation, decouple the migration of each defect and study them individually. This will be the focus of a future work in which we will study the evolution of the sink efficiency in the presence of both vacancies and interstitials.

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PHYSICAL REVIEW B 86, 214109 (2012)

V. CONCLUSIONS

We have studied the long-term microstructural evolution of heterogeneous systems in vacancy supersaturated environments. As test cases we have investigated the vacancy migration and accumulation in a 4◦ {110} bcc Fe and a 2◦ {111} fcc Cu twist boundaries. These interfaces are characterized by different sets of screw dislocations. In the case of Fe, two sets of a20 111 and one set of a20 100 are present while in Cu three sets of Shockley partials ( a60 112) characterize the interface. We have calculated the interaction energy of a vacancy with the described boundaries and observed that it is the most attractive at the MDIs ( a20 100 segments in Fe). We have performed NEB calculations following the vacancy path towards the interface. In both Fe and Cu we observe that the effect of the interface on the migration barriers is short ranged. To study vacancy migration and accumulation we have developed a methodology based on the KMC algorithm which accounts for the displacement fields in an approximate manner. With the proposed algorithm we have studied vacancy diffusivities in bulk and at the interfaces. We observe a onedimensional vacancy migration along the dislocation cores. The interface diffusivity in Fe is of the same order of magnitude as in bulk while in Cu it is two orders of magnitude faster. We have also analyzed agglomeration at system heterogeneities, while the atomic structure evolves on the fly as damage accumulates. As a matter of fact, we observe a different dislocation structure in the Cu samples as vacancies aggregate at the interface while in Fe the dislocation structure remains stable. We have studied the effect of the vacancy

*

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insertion rate on the system evolution in large systems and up to doses as of yet unachieved by more sophisticated approaches while maintaining the atomistic description of the problem. We have given an approximate quantification of the error introduced by just considering vacancy hops through shearing experiments and observed that for low dose rates the final configuration obtained by KMC gives the same results as the same sample after MD annealing, giving confidence in the validity of the present methodology. On the other hand, for large insertion rates in the Fe sample, a 30% discrepancy in the yield strength is observed, suggesting a less realistic final microstructure as given by KMC. More systematic comparisons with AKMC are needed in order to clearly identify the limitations of the presented method. This development opens the door to the study of problems like segregation in heterogeneous systems in which the vacancy rates depend strongly on the microstructure.

ACKNOWLEDGMENTS

Work was performed with support from the Center for Materials at Irradiation and Mechanical Extremes, an Energy Frontier Research Center funded by the US Department of Energy (Award Number 2008LANL1026) at Los Alamos National Laboratory. The authors want to thank L. Vernon, C. Brandl, B. P. Uberuaga, and A. F. Voter for useful discussions. A.C. also acknowledges support from the Laboratory Directed Research and Development Program.

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