JERK, an event-based Kinetic Monte Carlo ... - Erwan ADAM

Mar 1, 2005 - continuous coordinates in the simulation cell, their nature (e.g. the number of ... from the combination of the preceding objects like gas-or-solute ю defects [1, 2] ... This requires a prior knowledge of the interacting laws between.
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Philosophical Magazine, Vol. 85, Nos. 4–7, 01 February–01 March 2005, 549–558

JERK, an event-based Kinetic Monte Carlo model to predict microstructure evolution of materials under irradiation J. DALLA TORRE*y, J.-L. BOCQUETy, N. V. DOANy}, E. ADAMz and A. BARBUy yCommissariat a` l’E´nergie Atomique, CEA Saclay, DEN-SRMP, 91191 Gif-sur-Yvette Cedex, France zCommissariat a` l’E´nergie Atomique, CEA Saclay, DEN-DM2S-SFME-LFLS, 91191 Gif-sur-Yvette Cedex, France }Retired

JERK is a Kinetic Monte Carlo model which aims at describing the evolution of materials under irradiation over large time and length scales. The evolution is calculated by collecting only those events which actually modify the objects making up the microstructure (defect clusters, various complexes, dislocations), by sampling their probability of occurrence, deciding in view of the chosen delays whether the events will take place or not within a given time interval. The details of the atomic transport are ignored and the jumps of mobile species are bunched into trajectories, which comply with continuous diffusion laws. After some tuning, the increase in efficiency over a Kinetic Monte Carlo procedure which processes each jump event after another should be noticeable.

1. Introduction to the JERK model In this model the different defects are considered as objects, i.e. the detailed atomic configuration is not treated but, instead, defects are characterized by their continuous coordinates in the simulation cell, their nature (e.g. the number of self-interstitials, vacancy, impurity atoms that they contain), their shape (e.g. cluster or loop), their mobility and eventually their dissociation rate if necessary. Some examples of the objects that have been included in JERK are surfaces, dislocations, self-interstitials, vacancies, clusters, gases and solute atoms; complexes stemming from the combination of the preceding objects like gas-or-solute þ defects [1, 2]. Obviously, when these objects are mobile, they diffuse and interact together, i.e. annihilate on an anti-defect (e.g. a self-interstitial on a vacancy cluster) or agglomerate to other defects. When immobile, they can also simply react, by emitting a mobile defect. These are examples of possible events. In our model, the migration of a mobile object is not an event in itself: it is only when this diffusion step ends *Corresponding author. Email: [email protected] Philosophical Magazine ISSN 1478–6435 print/ISSN 1478–6443 online # 2005 Taylor & Francis Ltd http://www.tandf.co.uk/journals DOI: 10.1080/02678370412331320134

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up with a final reaction which changes one of the objects. The migration and subsequent reaction are then treated as a single event, which is performed in a single Monte Carlo step. This requires a prior knowledge of the interacting laws between the defects, which are a function of space coordinates and time. A detailed description of the model can be found in [2]. In the present paper we present new interaction laws for migration and reaction of mobile objects, together with a general description of the model. These interaction laws constitute the core of the predictive capabilities of the model and their validation is presented in section 3. We focus in this presentation on a system consisting of a solid under irradiation with various geometries (infinite medium, thin foil) which contains only self-interstitials and vacancies. We start with the simulation procedure.

2. Description of the model 2.1. Simulation procedure The irradiation simulation proceeds by selecting one of the events: injection of primary damage resulting from neutron, ion or electron irradiation, defect emission from a surface or from clusters and defect interaction via diffusion. The total simulated time is divided into small t intervals that are large compared to the time scale of events but small compared to the total simulated time. The role (and the choice) of t is discussed below. Within a given t, the simulation proceeds as follows: 1. Primary damage defects are introduced in the simulation cell. This event is repeated with a rate that corresponds to the desired dose rate and follows a Poisson distribution. This production competes with the other events. Once these defects are created, the delay of all possible events between objects are computed by using a Monte Carlo scheme to sample the probability distribution of their occurrence. 2. The next event is selected by looking for the shortest delay,  s, in the list. 3. The event is executed by, if necessary, deleting the defects that have interacted, creating the new defects, deleting the events associated with the deleted defects and computing the delays associated with the newly created defects if any. 4. The actual time, ta, is updated (increased by  s), the remaining delays are reduced by  s and steps 2 through 4 are repeated until no other event is possible before the end of t. At the end of each time step, t, the defects which did not take part in any reaction are moved, according to continuous diffusion laws, to a new position that corresponds to their migration since the date of their last displacement or reaction. The chosen displacements are checked in order to comply with the particular boundary conditions (for instance, the interstitial defect must not cross the thin foil surfaces, since this event was not selected as an absorption event in the event list, etc.).

2.2. Interaction between spherical defects Two defects may meet if at least one of them is mobile with a diffusion coefficient D. If both are mobile, the slowest defect is considered as immobile while the second

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diffuses with a diffusion coefficient D ¼ D1 þ D2. According to the distance, d, separating defects different cases may be met: 1. d < r ¼ r1 þ r2 þ dr where r1 and r2 are the radii of defects 1 and 2, and dr is a reaction distance between the defects. The reaction is immediate and the associated pffiffiffiffiffiffiffi delay  ¼ 0. 2. d > r þ 4 Dtr where tr is the remaining time before the end of t i.e. tr ¼ t  ta . Defects are too far apart and the interaction is not considered. 3. Within these two cases the probability distribution P(d,t) that two defects at distance d meet during t is obtained considering that the two defects are alone in an infinite medium. The classical solution of this diffusion problem is obtained after transposing it into spherical symmetry. A unit source is placed along an infinitely thin spherical shell of radius d: an inner surface at radius r < d with zero concentration plays the role of a non saturable trap. The time dependence of the impinging flux onto the trap is then evaluated and reads ([3], p. 247):   r dr Pðd,tÞ ¼ erfc pffiffiffiffiffiffi d 2 Dt

ð1Þ

where erfc is the complementary error function. The delay of interaction is therefore obtained by sampling P(d,t), i.e. by inverting equation (1): ¼

ðd  rÞ2 4D½erfc1 ðZd=rÞ2

ð2Þ

where erfc–1 is the inverse complementary error function and Z a random number uniformly distributed over [0,1] The above delay is calculated as if the two meeting defects were alone in an infinite volume, which is at variance with the actual situation where encounters with all the other competing partners all along the trajectory should be taken into account. We assume, and check below, that superposing such binary interactions remains, however, quantitatively correct. The success of this approximation rests on the fact that the shortest delays, which correspond to the closest partners, are processed first. These events are the very ones for which the binary collision approximation is most fitted: no other competing partner can be met on the way with a noticeable probability (otherwise the probability of meeting with this competitor would have been larger and would have corresponded to a shorter delay). As a consequence the kinetics is weakly influenced by the calculation of farthest interactions which are calculated with an accuracy decreasing with distance. 2.3. Periodic boundary conditions In a simulation cell of finite dimensions, periodic boundary conditions duplicate the main cell into an infinity of image cells. The mobile defect may also interact with the images of the defect sink created by the periodic boundary conditions. The trapping by an array of traps has been treated long ago [4]: it is known that the long-time behaviour of the trapping probability reduces to an absorption characterized by a unique relaxation time. The trapping effect of the background can consequently by

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expressed by a simple exponential decay. The practical problem to be tackled with is then to decide how and when to switch from the short-time behaviour to the longtime one. In a previous version of the code [1, 2], the crossover point was placed between the main cell and its images: the encounter probability was evaluated with the above expression for the main cell and the contributions of all images were merged into a single background term. This method turned out to be insufficiently accurate and yielded results departing too much from the exact solutions for the benchmark cases detailed hereafter. This is the reason why this question was tentatively solved in two steps: pffiffiffiffiffiffiffi . By considering only those cells i at a distance di < 4 Dtr as competing with the defect sink in the main cell at distance d with a noticeable probability (same expression as above with di in place of d). The corresponding trapping delays are calculated and added to the event list; the shortest delay is selected and this selection procedure is considered as being representative of the interaction with the defect sink and its images (approximation named ‘all images’). . By considering the same cells as above but partitioning them into concentric shells and attributing to each of the N image defect sinks in a given shell the same distance from the mobile defect in order to spare computation time (approximation named ‘simplified images’). In this frame, if we sample N probability distributions they will appear as equally partitioned in the [0,1] interval. By sampling P(d,t) in the [0,1/N] interval, we consider that the delay obtained is representative of the interaction of the mobile defect with the N defect sinks in a given shell.

2.4. Interaction of defects with surfaces Free surfaces are highly dimensioned traps and must be handled in a specific way. The classical solution of the diffusion problem in a thin foil is given by two mathematically equivalent expressions, using a series of gaussians on one hand and exponentially decaying functions of time on the other hand ([3], p. 59). The first formulation is well suited for short times, whereas the second one is better for long times. From a practical point of view, such series cannot be used easily for inversion: only the leading term in each series will be retained for the purpose of tractability. After a proper integration, it yields finally the probability distribution for hitting a surface under the form:   d pffiffiffiffiffiffi Pðd,tÞ ¼ erfc for short times ð2 DtÞ !   4 p2 Dt pd Pðd,tÞ ¼ 1  exp  2 sin for long times p e e where d is the distance to the nearest surface and e the foil thickness. The shorttime approximation corresponds to the only contribution of the nearest surface. It is easily checked numerically that the exact solution departs from the shorttime approximation in the very same region as it merges with the approximate solution for long times. As a consequence, the short-time approximation is chosen first but, after a convenient check, is replaced by the long term approxi-

Probability of hitting a surface P(d,t)

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d/e = 0.4

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Nearest surface approximation: erfc [d/(2 sqrt (Dt))] Thin foil surfaces: exact result (whole series)

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(b) Figure 1. Probability of reaching a surface of the thin foil for a walker starting at a distance d from one surface (e stands for the foil thickness): (a) d/e ¼ 0.4; (b) d/e ¼ 0.1 (zoomed around the point of closest approach of the two approximations).

mation beyond their intersection (figure 1a). Unfortunately, the two distributions do not intersect for small d/e values. In this case the envelope probability distribution presents a discontinuity at the point corresponding to the distance of closest approach of the two probability distributions (figure 1b). At this crossing point, a probability may be associated either with the short- or long-time approximation: choosing one or the other introduces a relative difference on the computed value on the x-axis within 5% in most cases. 2.5. Choice of typical time interval t Analysing the behaviour of the code shows that most of the time consumption is spent in the calculation of delays and in the frequent modifications to be brought to the event list after the processing of the event having the shortest delay. The time

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interval t, which was up to now introduced as a free and arbitrary parameter, becomes in fact a matter of compromise. Choosing a very small time interval of the order of a jump delay would come back to the atomic Monte-Carlo procedure, which is too slow. Choosing a very large time interval (for instance, the final date at which the system has to be macroscopically observed) should in principle be possible; however, there would be two practical drawbacks: . The larger the time interval, the larger the number of events to be processed and delays to be calculated, the larger the number of event list modifications to be performed. . For long time intervals, the sampling of the probability distributions given above will be probed in their long time tails and the associated events will have a noticeable weight in the final result. It is unfortunate that part of the delay distributions is badly known. Indeed they were evaluated in the frame of a binary collision approximation which becomes less and less accurate as time elapses, since, in the absence of any exact solution for the complex diffusion problem as a whole, we do not have at hand the proper distribution law for delays which take due account of the interference effects between reaction partners and of the shadowing effect between closely located sinks or nonlocal sinks like dislocations [5]. Up to now, only expensive and numerical treatments using the Laplace transform have been proposed, which cannot be included simply into the present procedure [6]. This is the reason why, in the present version of the code, an intermediate value of the time interval is chosen and its optimum is tuned empirically by trial and error to get a minimum computing time, as a function of the volume of the cell and the temperature, both of these ingredients influencing the number of events to be processed. In the following benchmark tests, the time interval was set to 5  104 s, whereas for the long runs presented in the companion paper, it is set to 6  103 s. This choice is apparently conveniently tuned for the higher temperatures, since the speed up with respect to Object-based Kinetic Monte-Carlo amounted to a factor of 30 (16 h against 24 days at 300 C). Comparison with Object-based Kinetic Monte-Carlo and Cluster Dynamics are detailed in the companion paper [7]. 2.1.5. Emission event. The emission of a defect from any source (cluster, surface, cavity, etc.) is introduced through a characteristic time. Its temperature dependence is dictated by the binding energy of the emitted defect with the source (the larger this energy, the less frequent the emission event). In a rateequations formalism like Cluster Dynamics, the emission rates are introduced through coefficients including this binding energy and the mobility of the emitted particle D, in such a way as they balance exactly absorption and emission events (local equilibrium assumption). In this frame, the emission is a transition from one class of cluster to another without any reference to their location. A Kinetic Monte-Carlo simulation like the present one has to face the question of where to put the emitted defect. In the present version of the model, two procedures are available to determine the defect location: . Once the emission event has been ffi selected with a delay  e, the defect is put pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi at a distance Dðt  ta  e Þ from the emitting cluster in a random

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direction, in order to keep the probability of an eventual return to (and a re-absorption by) the emitter to a sufficiently low level during the remaining time interval from the emission date to the end of the time step. Indeed, putting the defect closer and allowing the recombination would cancel out the emission, thus contradicting the fact that the emission event was chosen to be effective over the time interval t. . The second procedure mimics the Cluster Dynamics procedure in which the emission of defects from clusters increases uniformly the average defect concentration in the simulation cell. When transposed to the Monte Carlo procedure, it means that the space coordinates of the emitted defect are chosen at random in the simulation cell.

3. Model validation 3.1. Electron irradiation of a model of -Fe with recombination The check of the approximations of interactions between spherical defects was performed in the following way. Choosing a model material mimicking -Fe, we simulated an electron irradiation with a dose rate G ¼ 9  10 4 dpa s1 at various temperatures ranging from 300 to 900 K in a cell with periodic boundary conditions. The interstitial and vacancy diffusivities for this material are expressed by D ¼ Do expðEm =kT Þ with  ¼ i,v and the input parameters set to Doi ¼ 3  104 cm2 s1 , Emi ¼ 0:28 eV, Dov ¼ 5 cm2 s1 , Emv ¼ 1:36 eV. We let the interstitial and vacancy concentrations reach their stationary values thanks to the only elimination process at work, namely their mutual recombination. The latter is uniquely determined by the recombination radius rrec ¼ 0:547 nm. The stationary concentrations were then compared to thepffiffiffiffiffiffiffiffiffi exact ffi solution of the diffusion problem which in this case reads Ci ¼ Cv ¼ G=R where the recombination constant R ¼ 4prrec ðDi þ Dv Þ=O, and O is the atomic volume. The relative departure of the volume averaged vacancy concentration in the simulations from the exact solution are displayed in figure 2a and amounts at most to 5% at all temperatures. Note that all the images (first approximation in section 2.3) were included in these calculations. 3.2. Electron irradiation of a thin foil of a model of -Fe with recombination The check of the approximations related to the introduction of surfaces was performed by simulating the electron irradiation (same dose rate G) of a thin foil (e ¼ 172.2 nm) made of the same model material as above, at various temperatures ranging from 600 to 900 K. The defects annihilate through mutual recombination and elimination at the free surfaces. After the steady-state is established, the volume-averaged vacancy concentration is numerically evaluated. It is then compared to the numerical solution of the diffusion equation in a thin foil under the same conditions (diffusion equations were solved using a standard ODE solver package Lsoda [8]). The departure of the simulated results from the exact solution is displayed in figure 2b and does not exceed 15%. We used both methods to include periodic images of the defects (see section 2.3). The first method turned out to be much more computer demanding, because of the very large number of different distances and delays to be evaluated at each time step.

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Figure 2. Relative departure of the volume-averaged vacancy concentration from exact results: (a) irradiated bulk with recombination and periodic boundary conditions in all directions (all images); (b) irradiated thin foil with recombination and periodic boundary conditions parallel to the surfaces (all images and simplified images).

The second method proved to be more efficient since the computer time increases only as the number of shells increase. Furthermore, it was not less accurate than the first and consequently was retained for all subsequent calculations presented in the companion paper [7]. 3.3. Time- and space-dependent vacancy concentrations in a thin foil of a model of -Fe with recombination The second check of the approximations related to the introduction of surfaces was performed by simulating the electron irradiation (G ¼ 1.5  104 dpa s1) of a thin foil (e ¼ 287 nm) made of a model material mimicking -Fe (parameters, slightly different from the above ones, are given in the companion paper [7]). Diffusion equations were solved using a standard ODE solver package Lsoda [8] in order to obtain the time- and space-dependent defect concentrations. In these calculations only recombination of self-interstitials with vacancy and elimination of monomers at the free surfaces were allowed (simulations allowing vacancy and self-interstitial clustering are presented in the companion paper [7]). Figure 3 shows the vacancy concentration profile obtained by solving the diffusion equations. The high vacancy concentration near the surface results from fast diffu-

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Figure 3. Vacancy concentration profile after 120 s irradiation (G ¼ 1.5  104 dpa s1) at 200 C versus depth into the thin foil.

sion of self-interstitials to the surface that causes reduced self-interstitial–vacancy recombination in this zone. JERK simulations are in excellent agreement with these results at intermediate and long times but deteriorates slightly at very short times. Indeed, the recombination probability of a self-interstitial atom with a vacancy near a surface is a three-body problem (self-interstitial, vacancy and surface) approximated through binary interactions: the probability that the self-interstitial meets the vacancy should ideally account for all diffusion pathways that the surface intersects. The larger the extended defect, the larger the number of pathways intersected and the worse the binary approximation: this situation prevails at smaller times, when concentrations are low and surface effects dominate over recombination. 4. Conclusions Event-based Kinetic Monte-Carlo can be a very useful tool to explore macroscopic time scales and provides precious spatial information about the evolving microstructure with a reasonable amount of computation. The basic laws can be easily checked, and improved if necessary. The binary collision approximation appears sufficient in the explored cases and yields quantitative results with a good accuracy. The model results are compared to other methods in a companion paper. References [1] J.M. Lanore, Radiat. Eff. 22 153 (1974). [2] F. Maurice and N.V. Doan, Simulations de l’e´volution de population de de´fauts dans un cristal par la me´thode de Monte Carlo (Technical Report CEA-R-5101, 1981).

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[3] H.S. Carslaw and J.C. Jaeger, Conduction of Heat in Solids (Oxford Science Publications, 1959). [4] F.S. Ham, J. Phys. Chem. Solids 6 335 (1958). [5] V.A. Borodin, Physica A 211 279 (1994). [6] M. Fixman, Phys. Rev. B 15 5741 (1977). [7] A. Barbu, C. Becquart, J.L. Bocquet, J. Dalla Torre and C. Domain, this conference. [8] A.C. Hindmarsh, in ODEPACK, A Systematized Collection of Ode Solvers, in Scientific Computing, edited by R.S. Stepleman et al. (North-Holland, Amsterdam, 1983), p. 55.