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3696 WANG and CHEN: SOLUTE SEGREGATION the B atom is at r, and the A atom is at r+d. Then the rate of change of the probability that the site r occupied by an A atom is given by dP A …r†=dt ˆ S d S fxg P BA fX g…r, r ‡ d, fxg† R BA …fX g† S d S fxg P AB fX g …1† …r, r ‡ d, fxg†R AB …fX g† Fig. 1. The b.c.c. lattice is divided into two sublattices: a and b. The numbers 1±20 indicate the ®rst and second neighbors of the pair AB or BA. composition), A and B may have a preference to occupy either the a- or b-sublattice. For instance, for composition c ˆ 0:5 at low temperatures, there is a higher probability of ®nding A atoms on the a- sublattice and B atoms on the b-sublattice, or vice versa, resulting in the so-called B2-ordered phase. At high temperatures, due to the mixing entropy e€ect, A (or B) atoms have the same probability of occupying a- and b-sublattices, forming a disordered phase. In the microscopic master equation method, the structural state of an alloy at a given temperature and composition is described by a set of multiparticle distribution functions or cluster probabilities [11]. These multiparticle distribution functions satisfy the normalization conditions [12]; for example, summing the single-site distribution function P a (r) [occupation probability of a-type atom (A or B) on site r] over all lattice sites gives S r P a …r† ˆN a where N a is the total number of a-type atoms in the alloy. Away from equilibrium all of the multiparticle distribution functions are at nonequilibrium and change with time as the atomic di€usion takes place on the lattice. For simplicity, we assumed a direct exchange mechanism for atomic di€usion and isothermal environment. Let us consider a pair of interchange sites at r and a nearest-neighbor site, r+d, and a set {x} of nearby sites on which atoms interact with the two interchanging atoms. If we have an A atom at r, a B atom at r+d, and a set of atoms {X} at{x}, we denote by R AB ({X}) the rate at which the AB pair interchanges under the in¯uence of the set of neighboring atoms {X} on{x} up to the second neighbors as shown by the numbers 1±20 in Fig. 1. Similarly, R BA ({X}) is the rate at which the BA pair will interchange under the same environment when where the ®rst and second terms on the right-hand side are the average rates at which, on site r, A atoms are appearing and disappearing, respectively; S d denotes the summation over all the nearestneighbor sites, d, ofr; and P AB fX g…r, r ‡ d, fxg† is the probability of ®nding an A atom on r, a B atom on r+d, and the set {X} on the neighboring sites {x} simultaneously. A similar equation can be written for dP B (r)/dt since P A …r†‡P B …r† ˆ1. The reaction rate constant in equation (1) is calculated according to [11±13] R AB …fX g† ˆ u expf…U 0 ‡ 1=2DE †=k B T g …2† where U 0 is the average activation energy for AB exchange, u is the vibrational frequency associated with the AB exchange, DE is the energy di€erence before and after an atom exchange and dependent on the local atomic arrangements, T is the absolute temperature and k B is Boltzmann's constant. Since u exp…U 0 =k B T † occurs in all con®gurations, it can be combined with the time, t, in the kinetic equations to give a dimensionless time, t* t ˆ tu exp…U 0 =k B T †: …3† Therefore, the time unit of t in our simulation is 1=u exp…U 0 =k B T † and the dimensionless time step is 0.01. In order to carry out the sum on the right-hand side of equation (1), we need the joint probability distribution, P AB fX g…r, r ‡ d, fxg†. In this paper, we use the simplest point approximation by assuming statistical independence among occupation probabilities. In this approximation P AB fX g…r, r ‡ d, fxg† ˆ P A …r†P B …r ‡ d†P X1 …x1†P X2 …x2† ... P Xn …xn† …4† where x1, ... , xn are the individual sites in the set of neighboring sites around the pair, r and r+d, and X1, ... , Xn are the types of atoms occupying those sites. The summation on the right-hand side of equation (1) is over all possible arrangements of A and B atoms on the sites which in¯uence the atomic interchange at r and r+d. For the b.c.c. binary alloy with a second-neighbor interaction model (Fig. 1), there are a total of 2 20 possible terms in the summation. For each of the 2 20 di€erent con- ®gurations, the rate constants R AB and R BA have to
e calculated. The number of con®gurations will increase dramatically if a longer range interaction model is employed. However, in the point approximation, it is computationally much more ecient to replace the sum over products of individual singlesite probabilities using the product of sums over A and B [12, 13], i.e. S fxg P AB fxg…r, r ‡ d, fxg†R AB …fX g† ˆ P A …r†P B …r ‡ d† P A …1† WANG and CHEN: SOLUTE SEGREGATION 3697 exp‰ …V 1 AB V1 AA †=2k BT Š‡P B …1† exp‰ …V 1 BB V1 BA †=2k BT Š ... fP A …20†exp‰ …V 1 AB V1 AA †=2k BT Š ‡ P B …20†exp‰…V 1 BB V1 BA †=2k BT Šg where P A (k) and P B (k), k ˆ 1, ... , 20, are the values of single-site distribution functions P A and P B at site k around the pair r and r+d, and the V n AA is de®ned as the nth neighbor interaction between A and A atoms and the same for A±B and B±B. To solve the kinetic equations, we apply the simple Euler technique P A …r, t ‡ Dt† ˆP A …r, t†‡dP A …r†=dt Dt: …6† …5† In the traditional Bragg±Williams mean-®eld approximation the phase diagram for the b.c.c. binary alloy can be calculated [14]. The bond energies are chosen to be V 1 AA ˆ 1:0, V1 AB ˆ 0:0, V1 BB ˆ 0:0, and V 2 AA ˆ0:86, V2 AB ˆ 0:0, V2 BB ˆ 0:0. Therefore, the e€ective interaction energies are given by w 1 ˆ V 1 AA ‡ V1 BB 2V1 AB ˆ 1:0 w 2 ˆ V 2 AA ‡ V2 BB 2V2 AB ˆ0:86 …7† where w 1 and w 2 are the ®rst- and second-neighbor e€ective interchange energies and the interactions beyond the second coordination shell are neglected. The free energy per lattice site can then be obtained for the B2 phase of the b.c.c. structure as follows: F ˆ 1=2fc 2 V 0 ‡…Z† 2 V 1 ‡ k B T‰…c ‡ Z†ln…c ‡ Z† ‡…1 c Z†ln…1 c Z†‡…c Z†ln…c Z† ‡…1 c ‡ Z†ln…1 c ‡ Z†Šg …8† where c is the mole fraction of component A, Z is the order parameter, V 0 ˆ 7w 1 ‡ 6w 2 and V 1 ˆ7w 1 ‡ 7w 2 . The conventional common-tangent construction for the F vs c curves at di€erent temperatures determines the equilibrium compositions of the disordered D phase and the ordered B2 phase and allow one to draw the solubility lines. Therefore, the phase boundary and spinodal curves for the point Fig. 2. The calculated phase diagram for b.c.c. alloy in a point approximation with interactions up to second neighbors. approximation are calculated using equation (8) and are presented in Fig. 2. There are three regions in the phase diagram: B2-ordered phase (B2), disordered phase (D) and two-phase coexistence (B2+D), and the solid lines are the phase boundaries of the low temperature two-phase ®eld, dotdashed line is the ordering transition line of the second kind extended into the D+B2 ®eld, thin line is the stable ordering transition line of the second kind, dotted line is the conditional spinodal and letters X represent the alloy compositions and temperatures chosen for the computer simulation. Reduced temperature T ˆ k B T=jV 1 j is used in the phase diagram representation. This phase diagram is topologically very similar to the upper part of the Fe±Al diagram describing the two-phase ®eld, the disordered+ordered B2 phase [15]. In order to characterize the B2-ordered phase, we de®ned the long-range order parameter Z in terms of single-site occupation probabilities using Z…r† ˆ‰P a A …r†Pb A …r†Š=2, where a and b label the two sublattices. The corresponding local composition is de®ned as c…r† ˆ‰P a A …r†‡Pb A …r†Š=2. The local solute segregation is measured by de®ning s…r† ˆc…r†c 0 , where c 0 is the average composition of the system. 3. RESULTS AND DISCUSSION 3.1. Initial con®guration of APB We employed a simulation supercell 64 64 2 conventional b.c.c. unit cells with two lattice sites per unit cell (Fig. 1). Periodic boundary conditions are applied along all three directions. We considered a cylindrical antiphase domain with a radius R ˆ 30 (unit is lattice constant). The initial condition is generated as follows: inside the cylinder, P a A …r† and Pb A …r† are, respectively assigned high and low values corresponding to the equilibrium B2- ordered single phase at the temperature and composition of interest, and outside, P a A …r† and Pb A …r† are assigned low and high values, respectively.
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