Self-Diffusion in a Triple-Defect A-B Binary System: Monte Carlo simulation

In this comprehensive and detailed study, vacancy-mediated self-diffusion of A- and B-elements in 'triple-defect' B2-ordered ASB(1-S) binaries is simulated by means of a kinetic Monte Carlo (KMC) algorithm involving atomic jumps to nearest-neighbour (nn) and next-nearest-neighbour (nnn) vacancies. The systems are modelled with an Ising-type Hamiltonian with nn and nnn pair interactions complete with migration barriers dependent on local configurations. Self-diffusion is simulated at equilibrium and temperature-dependent vacancy concentrations are generated by means of a Semi Grand Canonical MC (SGCMC) code. The KMC simulations reproduced the phenomena observed experimentally in Ni-Al intermetallics being typical representatives of the 'triple-defect' binaries. In particular, they yielded the characteristic 'V'-shapes of the isothermal concentration dependencies of A- and B-atom diffusivities, as well as the strong enhancement of the B-atom diffusivity in B-rich systems. The atomistic origins of the phenomenon, as well as other features of the simulated self-diffusion such as temperature and composition dependences of tracer correlation factors and activation energies are analyzed in depth in terms of a number of nanoscopic parameters that are able to be tuned and monitored exclusively with atomistic simulations. The roles of equilibrium and kinetic factors in the generation of the observed features are clearly distinguished and elucidated.


Introduction
The tendency for triple-defect disordering (TDD) is clearly defined for the stoichiometric A-50 at%B system with the B2 superstructure ( Fig. 1) and means a substantial difference between the formation energies for A-and B-antisite defects which leads to preferential generation of the antisites with lower formation energy [1]. The tendency for TDD implies: (i) a large difference between the A-and B-antisite concentrations; (ii) a large difference between the concentrations of vacancies residing on αand β-sublattices (the 'home' sublattices of A and B atoms); (iii) an increase of vacancy concentration with decreasing degree of chemical long-range orderi.e. with an increasing concentration of antisite defects. In the extreme case of the exclusive generation of A-antisites, their concentration is equal to one half of the vacancy concentrationi.e. the vacancy concentration strongly increases with decreasing degree of chemical order. In nonstoichiometric binaries the tendency for TDDi.e. lower formation energy for A-antisites, means that while A-antisites compensate for the deficit of B atoms in A-rich systems, the Batoms in B-rich systems remain on the β-sublattice and the departure from stoichiometry is compensated by 'structural' α-vacancies. The process of TDD should be contrasted from the so called triple-defect mechanism of diffusion which means atomic migration via specifically correlated atomic jumps mediated by vacancy-pairs [2]. The topic of self or tracer diffusion in stoichiometric and non-stoichiometric A-B B2 intermetallics with the tendency for TDD has been widely investigated. While the number of theoretical and computational studies is fairly large, experimental works are relatively rare. The main reason for this lies in the major difficulties posed by experimental tracer-diffusion methods that require radioactive isotopes of the constituents. The most common TDD alloys have aluminum as the second component (NiAl, FeAl, CoAl, ...) which unfortunately lacks suitable radioactive isotopes. The direct tracer-diffusion experiments concerned, therefore the transition-metal componentsmostly Ni, whereas the tracer diffusion coefficient of Al has been mainly deduced (approximately) from interdiffusion experiments, the transition-metal self-diffusion coefficient and the thermodynamic factor.
In 1949, Smoluchowski and Burgess [3] measured the tracer diffusion coefficient of Co in NiAl. Co is known for its general tendency to substitute for Ni on the Ni sublattice. Radioactive Co was plated on the NiAl sample and the decrease of activity due to the penetration of cobalt into material was recorded at 1150 o C. The overall shape of the graph showing the self-diffusion coefficient of Co vs. concentration of Ni perfectly resembles one obtained in a more recent experiment by Frank et al. [4].
In 1971, Hancock and MacDonnell [5] measured the tracer diffusion coefficient of the Indirect estimation of the tracer diffusion coefficients in Ni-Al from interdiffusion experiments was most recently performed by Paul et al. [7] and Minamino et al. [8]. Fig. 2 shows the results obtained by Paul et al. which suggests that when traced in a logarithmic scale versus concentration both Ni and Al tracer diffusivities show again the 'V'-shape with minima around the stoichiometric composition Ni-50at.%Al. Another important feature is the intersection of the and isotherms at < 0.5.  [7] Predicting rapid growth of the tracer diffusivities of Ni and Al with an increase of Ni and Al content respectively, the results of Paul and Minamino are in good qualitative agreement with those of Frank et al. [4] and Hancock and MacDonnell [5]. In addition, the absolute values of the tracer diffusion coefficients calculated for Ni and In [6] are very close to the quantities directly measured in the vicinity of = 0.5. However, the symmetrical growth of the Ni diffusivity with decreasing (the 'V'-shape) is in clear contrast with the most reliable results of Frank et al. [4].
The above experimental results have been widely analyzed in terms of the activation energy of diffusion and possible mechanisms of atomic migration responsible for this energy. Krachler et al. [9] remarked on the impact of short-range chemical order in Ni-Al resulting in the curved shape of the Arrhenius plots of the measured tracer diffusivities [4]. Mishin et al. [10][11][12] performed extensive studies of the energetics of point defect complexes and migration barriers for various diffusion mechanisms in Ni-Al modelled with interatomic potentials determined within the embedded atom method (EAM). Soule De Bas and Farkas [13] further extended that research by considering complex sequences of 10 and 14 atomic jumps. More recently, Chen et al. [14] and Yu et al. [15] analyzed the atomic migration barriers in Ni-Al applying either angle-dependent interactions or new EAM potentials fitted to the experimental data. Marino and Carter proposed a more direct computational approach based solely on density functional theory (DFT) and in a series of works [16,17] evaluated not only the migration barriers and activation energies, but also the diffusion coefficients related to particular mechanisms proposed for Ni tracer diffusion in NiAl.
In 2011 Evteev et al. [18] published an interesting paper showing the results on Ni-and Al-self diffusion in NiAl simulated directly by means of Molecular Dynamics. The process was simulated in a layer limited by [110]-oriented free surfaces through which vacancies entered the system from outside and reached an equilibrium concentration. The simulations were performed at a temperature close to the melting point and yielded a Ni-diffusivity ca. 2.5 times higher than the Al diffusivity. An almost complete computational study of diffusion in NiAl was performed by Xu and Van der Ven [19][20][21] who combined ab initio energy calculations with configurational thermodynamics by means of the Cluster Expansion method. The developed model of Ni-Al covered the equilibrium vacancy thermodynamics with, however, a priori assumptions concerning their presence on particular sublattices in the B2 superstructure. Equilibrium thermodynamics of the system including vacancy concentrations was determined by means of the Semi Grand Canonical Monte Carlo (SGCMC) method assuming a zero value of the chemical potential of vacancies. Separately, migration barriers were calculated in the DFT formalism for different jump types and local configurations. Consequently, it was possible to use kinetic Monte Carlo (KMC) to simulate diffusion processes in a system with a well-defined defect concentration and the degree of chemical order. The final results concerning the isothermal concentration dependence of Ni-and Al-diffusivities were in qualitative agreement with the experimental study of Frank et al. [4] reproducing the growth of Ni-diffusivity with increasing Ni-concentration in Ni-rich binaries. In agreement with the experimental works of Paul and Minamino [7,8] the same behaviour of Al-diffusivity has been observed. Much less attention has been paid to the Al-rich systems. The results shown in the work concern only one composition ( ≈ 0.47) and may suggest that the diffusivity growth for both Ni and Al in the Al-rich region is much weaker than that reported experimentally [4][5][6][7]. Exploring the tracerdiffusion computations, Xu and Van der Ven evaluated the interdiffusion coefficient for Ni-Al which, however, decreased with growing Al content below = 0.5. Although this clear contradiction with experiment could be attributed to the polycrystalline character of samples analyzed in the works [7,8], the authors suggested that it rather resulted from incorrect assumptions concerning the equilibrium vacancy concentration.
The present work aims at the determination and detailed analysis of the impact of the tendency for TDDdefined at the beginning of this section, on self-diffusion of the components in B2ordering A-B binaries. Particular reference to Ni-Ale.g. by adapting in the model specific relationships between the atomic-jump migration energies yielded by ab-initio calculations concerning this system [21], follows from the fact that the related experimental results, to which the simulation findings might be compared, concern almost exclusively Ni-Al. The presented simulation study addresses, therefore, vacancy-mediated atomic migration processes in a B2 superstructure of a TDD system loosely resembling Ni-Al.
By applying a straight forward Ising-type model it is possible to clearly demonstrate the strict correlation between the equilibrium thermodynamics of the system (equilibrium configurations of atoms and vacancies) and the kinetics of self-diffusion. Systems were simulated that represented uniformly a wide range of compositions both in the A-rich and B-rich side of the AB stoichiometry. The approach provides a deep understanding of the diffusion phenomenon which is crucial for any effective development of material technologies.

General remarks
The methodology of the reported study covers two aspects:  The determination of the temperature and composition dependence of the equilibrium atomic and point-defect configurations in the system.  The determination of the temperature and composition dependence of self-diffusivities and tracer correlation factors of the system components, as well as their activation energies.
In both cases, Monte Carlo (MC) simulations were performed. Supercells were composed of 252525 unit cells of the B2 superstructure ( Fig. 1)i.e. containing = 31250 lattice sites belonging to equi-numerous α-and β-sublattices and populated with A-atoms, B-atoms and vacancies. 3D periodic boundary condition (PBC) were imposed upon the supercells.

Model for equilibrium configuration of the system:
Of interest is the equilibrium atomic configuration of a binary A-B system with vacancies.
 Pair-correlations (short-range order parameters) for atoms and vacancies: The present study was based on the Schapink model for the equilibrium configuration of a multicomponent system with vacancies [22], whose simple version was previously applied by one of the authors [23,24]. In this approach, a lattice gas A-B-V is treated as a regular ternary systemi.e. vacancies are treated strictly as an additional chemical component. The crucial property of the lattice gas (and also the condition for the applicability of the model) is that it shows a miscibility gap with a critical temperature below which it decomposes into two phases: one with ≪ 1 and another (unrealistic) one with ≈ 1. Then, the basic assumption of the model is that the lattice-gas phase with ≪ 1 being in equilibrium with the one with ≈ 1 is identified with the binary A-B crystal in equilibriumi.e. the crystal with an equilibrium atomic configuration and equilibrium vacancy concentration.

Search for phase equilibria in the A-B-V lattice gas
Following the idea of Binder et al. [25] equilibrium compositions and configurations of the A-B-V lattice gas were determined at fixed temperatures for arbitrary values of the chemical potentials (X=A,B,V). The procedure aimed at finding their values ( ) yielding two solutions: one with ≪ 1 and another with ≈ 1. Similar to our previous papers (see e.g. [26]) the lattice gas was examined using a standard algorithm of Semi Grand Canonical Monte Carlo (SGCMC) simulations where due to a fixed value of the system is parameterized by two independent relative chemical potentials defined in the present paper as: The SGCMC algorithm works in the following scheme: where ∆ → denotes the change of the system configurational energy due to the → replacement. The quantity denotes the Boltzmann constant. ∆ → is evaluated within a particular model of the system implemented with the simulations and depends on the current composition and configuration of the lattice gas. (iv) Return to step (i).
Two series of SGCMC runs were performed at each temperature: in series 1 the simulations started with a perfect B2-ordered supercell with whereas in the simulations of series 2 the supercell was initially empty ( = = 0; = 1).
The SGCMC simulations run at temperatures below yielded typical (Δ , Δ ) isotherms as shown in Fig. 3. The almost cliff-like discontinuity of the (Δ , Δ ) surface reflected the coexistence of the vacancy-rich and vacancy-poor phases. The effect showed well-marked hysteresis and thus, the exact values of Δ ( ) and Δ ( ) (the white line on the Δ − Δ plane) must be evaluated by means of specific procedures, see [26,27].

Model of the vacancy-mediated atomic migration
Vacancy-mediated self-diffusion of A-and B-atoms was simulated by means of the standard Residence-Time KMC algorithm [28] in samples whose atomic and vacancy configurations were formerly equilibrated by SGCMC runs. Conservation of these configurations was guaranteed by fulfilment of the detailed balance condition by the KMC algorithm. In the algorithm, which was extended for atomic jumps not only to the nn but also to nnn vacancies, the probability for an atom X (X =A,B) to jump from the initial i lattice site to a vacancy residing on j lattice site ( Fig. 4) is given by: where: Π 0 is a pre-exponential factor whose value depends on the jump-attempt frequency of the X-atom and thus is, in general, a function of temperature and the type of jumping atom. The KMC-time increment of: is assigned to each executed atomic jump.
where the value of + ( ) depends exclusively on the type X of jumping atom. As was mentioned in our previous work (see e.g. [29,30]) such parameterization of , → ( ) partially accounts for its dependence on a local configuration around the atom-vacancy pair.
While justification of the negligence of the temperature dependence of Π 0 (Eq. (1) was discussed earlier [30], almost equal values of the jump-attempt frequencies reported for Ni and Al-atoms (see e.g. [21,31]) make it reasonable to assume in the present work a constant value of Π 0 equal to unity both for A-and B-atoms.

Evaluation and analysis of diffusivities and correlation factors
The self-diffusion coefficients for X-atoms (X = A, B) were evaluated from the standard Einstein-Smoluchowski relationship (see e.g. [32]): where 〈 2 ( )〉 denotes the monitored mean-square-distance (MSD) travelled by X-atoms (X = A, B) within the MC-time t.
Analysis of the evaluated diffusivities in terms of the dynamics of atomic jumps to vacancies was done within the model of Bakker [33] now extended upon atomic jumps to both nn and nnn vacancies. Expression of 〈 2 ( )〉 in Eq. (13) in terms of elementary atomic jumps leads to 10 where 〈 ( ) ( )〉 and 〈 ( ) ( )〉 denotes the average numbers of nn and nnn jumps performed by an X-atom within the MC-time t; and denote the distances of the nn and nnn jumps, respectively; ( ) denotes the tracer correlation factor given by [34]: The problem is conveniently parameterized with average atomic-jump frequencies are directly determined by counting the particular X-atomic jumps executed within a fixed number of KMC steps and by dividing the number of these jumps by the related KMC time interval and the number of X-atoms present in the supercell. Within the microscopic model [33] they are expressed in terms of the atom-vacancy pair correlations ( ) (Eq. (5)) and the migration energies associated with the elementary atomic jumps (Eq. (8)): where: denotes the number of -sublattice sites being nn ( ≠ ) or nnn ( = ) of a -sublattice site; 〈( , → ( ) )( , )〉 denotes the average over the migration barriers , → ( ) associated with the jumps yielding → ( ) ( , ).
Due to the steady-state character of the simulated self-diffusioni.e. conservation of the average atomic configuration guaranteed by the KMC algorithm Eqs. (16) and (17) yield, therefore, a link between the system energetics ({〈( , → ( ) )〉}) and the configuration parameters ({ ( ) }). They also determine the steady-state atomic configuration of the system at a temperature .
Eq. (17) implies that Combination of Eqs. (14) and (18) yields: Eq. (19) makes it possible to demonstrate contributions of particular atomic jumps to the observed diffusion coefficients and thus to analyze in such terms the features of the effectively observed self-diffusion.

Hamiltonian
Applied was an Ising-type model of the B2-ordering binary A It should be noted that because of the varying composition of the system the SGCMC algorithm involves the total configurational energy (not only the energy of mixing) and therefore, separate evaluation of all the individual pair interaction parameters (not only of the 'ordering energies' Evaluation of the { (1) } and { (2) } parameters was based on the following criteria to be fulfilled by the modelled ternary A-B-V lattice gas: (i) Ternary miscibility gap with a non-zero critical temperature .
(ii) B2-ordering of the vacancy-poor lattice-gas phase at temperatures below the orderdisorder temperature − : − < . (iii) Tendency for TDD in the vacancy-poor phasei.e. preferential formation of Aantisite defects whose signature is a constant value of = ( ) ≈ 1/2 in the stoichiometric AB binary through a finite temperature range [35].
As fulfilment of the above criteria determines only the relationships between { (1) } and { (2) } assignment of particular values of the pair potentials required an arbitrary evaluation of − in the stoichiometric system with = . It should be firmly stressed that by no means did the latter affect meaningful results of the study, which in most cases are presented with relative (reduced) parameters.
The preliminary search for the proper values of { (1) } and { (2) } was done by scanning their space and analytically checking the above criteria within the Bragg-Williams approximation (see [35]). As a starting point, the values of { (1) } found in ref. [35] were used. A relationship (2) = −0.5 × (1) was chosen as an arbitrary assumption. Further adjustment was performed by checking the equilibrium atomic configurations generated by SGCMC simulations for the fulfilment of the criteria listed above. It must be emphasized that no calculations are known by the authors that accounted for interactions with vacancies and no strict reference to literature data was possible. 12 The final values of { (1) } and { (2) } used in the study are displayed in Table 1. Table 1. Values of nn ( (1) ) and nnn ( (2) ) pair interaction parameters used in the study.

Migration barriers (saddle-point energies)
For the sake of the studies of atomic migration the extended Ising model was completed with four parameters responsible for atomic migration:  Although the evaluation of the , + ( ) and , + ( ) parameters was achieved by respecting the above criteria, the choice of the particular values was arbitrary with a lower limit yielded by the obvious condition of E p,i→j (m) > 0. Table 3

General remarks
Because of the arbitrary evaluation of the energetic parameters of the simulated A-B system, presentation of the simulation results in terms of absolute quantities was generally avoided. The values of particular parameters were, therefore, normalized to selected characteristic values as listed in Table 4.

TDD tendency of the system
As briefly described in Section 1, the departure from stoichiometry of a B2 binary A-B system with a tendency for TDD is compensated by structural α-vacancies in A-poor binaries ( < 0.5) and by structural A-antisites in A-rich ones ( > 0.5). The S-dependence of the concentrations of the structural point-defects at → 0 (i.e. at the absence of thermally activated defects) follows from the balance ∑   The plots showed curvatures close to the order-disorder transition point − , but the linear segments make it possible to evaluate the activation energies ( ) for A-and B-atoms selfdiffusion (Fig.10). Both activation energies showed maximum values close to = 0.5. While ( ) < ( ) held for > 0.4, a strong decrease of ( ) with decreasing S caused that the relationship inverted at ≈ 0.4. The decrease of ( ) for > 0.5 means there is qualitative agreement between the reported simulation results and the corresponding experimental data on Ni-tracer diffusion in NiAl [4,5]. Fig. 11 presents the isotherms ( ) ( ) and ( ) ( ) corresponding to = 0.47 and = 0.78. The curves showed the characteristic asymmetric 'V'-shapes with minima located at = 0.43 and = 0.5, respectively (see the inset in Fig.11a). The 'V'-shape of ( ) ( ) was definitely more pronounced and clearly visible in a logarithmic scale (Fig. 11c,d). Besides, the curve increased much stronger with decreasing S than did ( ) ( ) with increasing S. Temperature dependences of the positions of the diffusivity minima and of the intersection of ( ) and ( ) are displayed in Figs.12a and 12b. While the minima of both ( ) and ( ) shifted towards lower values of S with increasing temperature (Fig. 12a), the location of = remained at ≈ 0.4 in the whole range of 0.47 < < 0.8 (Fig. 12b)which obviously resulted in ( ) = ( ) observed at the same value of (Fig. 10).

Elucidation of atomistic origins of the features of A-and B-tracer diffusivities
The analysis was based on the atomistic model of self-diffusion and the relationships given by Eqs.  (16)). Each one of the above parameters, as well as its composition-and temperature-dependence was independently evaluable by means of MC simulations. The linear parts of the Arrhenius plots of ( ) yielded effective activation energies ( ( ) ) traced in Fig. 14a against S. Figs. 14b and 14c show the S-dependence of two relationships between ( ( ) ) and the total activation energies ( ) for self-diffusion ( Fig. 10): the contribution of ( ( ) ) to ( ) (Fig. 14b) and the difference between both activation energies (Fig.14c). therefore, the part of ( ) stemming directly from the kinetics of atomic jumps to vacancies. The graphs in Fig. 14c show, in turn, that the contribution of the activation energy (

Tracer correlation factors
to the total activation energy ( ) for X-tracer diffusion never exceeded 30%.

Analysis of ( , ) in terms of atomic jump frequencies, atom-vacancy paircorrelations and average migration barriers.
The pure effect of → ( ) on the diffusivities is manifested by the values of ( , ) evaluated with Eq.   (iii) B-atom diffusion proceeds via all three kinds of nn and nnn B-atom jumps, however, while the nnn α↔α jumps and the much less frequent nn α↔β ones definitely dominate in the range of < 0.5, the nnn β↔β jumps are the most frequent in the range of ≥ 0.5. Therefore, while the strong increase of at < 0.5 is due to the strong increase of → ( ) , the minimum of ( ) at ≈ 0.5 actually results from the minimum of → ( ) . where the A-atom jumps to a nn vacancy generates no change of the system configurational energy.

Composition dependence of the tracer correlation factors
Low values of  (Fig. 16b) suggests that contribution of jumps with a higher migration energy increases and this means more cases of oscillations.

General remarks
The effect of composition and temperature on a steady-state vacancy-mediated tracer diffusion of components was simulated in a B2-ordering A-B binary showing the tendency for TDD. The system was modelled with Ising-type nn ( (1) ) and nnn ( (2) ) pair interactions between atoms and vacancies (X,Y = A,B,V) and with migration-barrier parameters , + ( ), , + ( ) controlling the heights of the migration barriers encountered by the jumping atoms. The migration-barrier parameters assigned to A-and B-atom jumps to nn and nnn vacancies were evaluated with reference to some ab initio calculations concerning Ni-Al [21] and thus, to that extent, the simulated system might be considered as resembling that real one. According to the applied parameterization (Fig. 4, Eq.(12)) the resulting migration barriers were partially dependent on local configurations of the jumping atoms.
Tracer diffusion running in bcc supercells with equilibrium vacancy concentration, as well as with equilibrium atomic and vacancy configuration was simulated by means of a rigid-lattice KMC algorithm. The equilibrium states of the supercells were generated by means of an SGCMC algorithm applied to the Schapink model [22] of phase equilibria in the ternary A-B-V bcc lattice gas. The main interest was focused on the effect of the tendency for TDD; in particular, on the origin of the 'V'-shapes of the diffusivity isotherms and of the strong enhancement of the B-atom diffusion in the B-rich binaries. Because the applied rigid-lattice approximation does not allow for the loss of the bcc structure, the stability (e.g. melting), or of definite rearrangements of atomic and vacancy configuration (e.g. the formation of phases mimicking Al3Ni2 or Al3Ni5 which neighbour β-NiAl in the Ni-Al system [36]) were beyond the performed MC simulations.

Effect of temperature and composition on the system tendency for TDD
According to the definition (see Section 1), the strength of the system tendency for TDD can be measured by the difference between the formation energies for A-and B-antisites. The analysis was performed by applying two alternative parameters measuring the antisite formation energies: (i) ( ( ) ) ( ( ( ) )) equal to differences between average potential energies of A-(B-) atoms on antisite and right positions; (ii) ( ( ) → ) ( ( ( ) → )) equal to average increments/decrements of the system configuration energy due to atomic jumps to nn vacancies residing on antisite positions (i.e. due to ( ) ↔ and ( ) ↔ ) exchanges). In view of Eq.  Positive values of ∆ ( ( ) , ( ) ) and ∆ ( ( ) ↔ , ( ) ↔ ) over all of the range of S indicate the maintenance of the tendency for TDD, whose strength reached, however, a maximum at = 0.5 and continuously decreased when departing from the stoichiometric composition of the system.
It is remarkable that while the values of ( ( ) ) and ( ( ) ) were positive over all of the explored range of the chemical composition of the system (which guaranteed stability of the B2 superstructure), the energy ( ( ) → ) was negative for > 0.4 indicating that Aatom-β-vacancy exchanges decreased the system configuration energy within this range of the chemical composition.
As was demonstrated in Fig. 8, the ground-state configurations of the A-B binaries showed features typical for a TDD system: the departure from the stoichiometric chemical composition ( = 0.5) was compensated exclusively by structural A-antisites at > 0.5 and predominantly by structural α-vacancies at < 0.5 (few B-antisites started to appear already at = 0.5). The effect of temperature on the curves corresponding to → 0 K (Fig. 8) is illustrated by Fig.18 which displays the same curves together with the analogous equilibrium point-defect concentration isotherms corresponding to = 0.47. Finally, it should be mentioned that the results of the present study corroborate with recent experimental findings obtained for self-diffusion in NiAl by means of X-ray Photon Correlation Spectroscopy [37]. The results suggest a large contribution of atomic jumps in [100] directions, obviously meaning the nnn ones. 33

Outlook for further investigations
By means of diverse Hamiltonians of Ni-Al based either on quasi-empirical potentials (e.g. EAM) or ab-initio calculations [21] the virtual B2→A2 'order-disorder' transition point in these binaries is estimated close to 6000 Kfar above the experimentally observed melting point (inaccessible of course within the rigid-lattice MC simulations). The reduced temperatures corresponding to most of the reported diffusion experiments performed on Ni-Al (usually at ≈ 1000 K) do not exceed the level of 0.2. As MC simulations performed at such low temperature are inefficient and yield large uncertainties of the evaluated parameters, the computer experiments are performed much higher in the reduced-temperature scale. In this way, reliable temperature dependences of the parameters of interest are determined. By extrapolating these dependences to the experimental conditions, not only qualitative, but also quantitative correspondence between the simulated and real properties of Ni-Al and other strongly ordered systems might be attained. Such an option seems especially attractive for KMC simulations implemented with ab-initio based Hamiltonianse.g. parameterized with Effective Cluster Interactions (ECI) evaluated within the Cluster Expansion (CE) formalism (as was done in ref. [21]).

Conclusions
 Vacancy-mediated tracer diffusion of the components of a B2-ordering binary systems A-B showing a tendency for TDD and loosely resembling the Ni-Al compounds was simulated in a wide range of concentration by means of a KMC algorithm. The process was run in crystals with equilibrium configurations and vacancy concentrations generated by a SGCMC algorithm. Features of the temperature and composition dependence of the component diffusivities were elucidated in terms of the frequencies of elementary atomic jumps to nn and nnn vacancies.  High B(Al)-atom diffusivity in B(Al)-rich binaries was found to be due to enhanced B(Al)-antisite migration via jumps to nnn α-vacancies. The diffusivity strongly increased with increasing B(Al)-atom concentration because of the strong increase of both α-vacancy and B(Al)-antisite concentrations caused by a gradual decay of the system tendency for TDD.  The isothermal concentration dependence of the B(Al) atom tracer diffusivity definitely showed a 'V'-shape with a minimum at the stoichiometric composition ( = 0.5). The atomistic origins of the shape, as well as of the position of the minimum were explained.  Although the 'V'-shape was observed also in the case of the isothermal concentration dependence of the A(Ni) atom tracer diffusion, its atomistic origin was different. The minimum was located at < 0.5away from the stoichiometric composition. This finding might suggest the reason for the discrepancies between the related experimental results.

Conflict of Interest
There is no conflict of interest to declare.