Cluster Variation Method in the Computational Materials Science

Pergamon Calphad, Vol. 26, No. 1, pp. 33-54, 2002

© 2002 Published by Elsevier Science Ltd 0364-5916/02/$ - see front matter

PII: S 0 3 6 4 - 5 9 1 6 ( 0 2 ) 0 0 0 2 3 - 8

Cluster Variation Method in the Computational Materials Science

R. Kikuchi a and K. Masuda-Jindo b a Materials Science and Mineral Engineering,

University of California, Berkeley, CA 94720-1760 b Department of Materials Science and Engineering,

Tokyo Institute of Technology, Nagatsuta 4259, Midori-ku, Yokohama 226-8503, Japan

(Received September 18, 2001)

Abstract. Cluster Variation Method (CVM) has been very successful in the computations of alloy phase diagrams as well as in many problems of the materials science related to the phase transitions. Originally, CVM was developed in the framework of the so-called rigid lattice approximation, but it has recently been extended to include continuous atomic displacements due to thermal lattice vibration and local atomic distortion due to size mismatch of the constituent atoms. In the present study, we focus our attention on the latter continuous displacement treatment of CVM. The continuous displacement (CD) formulation of the CVM is applied to study the phase stability of the binary alloys. The basic idea is to treat an atom which is displaced by r from its reference lattice point as a species designated by r. The effects of continuous atomic displacement on the thermodynamic quantities and phase transitions of binary alloys are investigated in detail. We also discuss the extension of the CD treatment of CVM to the calculations of solid-liquid and gas liquid phases transitions. © 2002 Published by Elsevier Science Ltd. Keywords: Cluster variation method, continuous displacement treatment, pair approximation, grand potential, phase separation, order-disorder transition

I. Introduction

The Cluster Variation Method (CVM) has been developed, improved and applied for more than half century. The key concept of the entropy formula was introduced in 1949 by Kikuchi and published in 1951 [51Kik]. The entropy and the free energy of a binary alloy system were written in terms of long- and short- range order parameters by Takagi [41Tak]. He derived the equilibrium state by minimizing the free energy with respect to these variables, and showed that the equations were identical to those of Bethe [35Bet]. When the CVM was introduced, the immediate purpose was to find a technique to go beyond the approximation of Bethe and Takagi. The derivation of the entropy expression in the first report of the CVM [51Kik] made use of the requirement of translational symmetry as the lattice was constructed point by point. It was later found that the method was closely related to the superposition approximation ofKirkwood [35Kir].

The CVM provides us analytical expressions of the internal energy, configurational entropy and free energy of the system in terms of the cluster probability variables. The basic cluster is usually taken in such a way that it contains the maximum interaction range one wants to take into account. The equilibrium state is determined by a variational principle, i.e., by a variation of these cluster probabilities until a maximum of the thermodynamic potential is obtained. The simple computational technique, natural iteration method, is often used to get the free energy minimum equilibrium states.

The CVM has been successfully applied, for example, to investigate temperature-composition phase diagrams of alloys for complex Hamiltonians [94Fon], which include pair and many-body interactions. In these CVM calculations, the entropy expression was formulated for permutations of atoms among rigid lattice points [94Kik]. This type of approach will be referred to as the conventional CVM in the present paper. In real alloys, however, there are problems which need more than permutations of atoms among rigid lattice points. These

33

34 R. KIKUCHI AND K. MASUDA-JINDO

problems include local lattice distortion caused by the atomic size differences [95Hor,88Moh,89Ter], thermal lattice vibration and the effect of elastic interactions on phase diagrams [73Lar]. Responding to the need of removing the restriction of the rigid lattice, the continuous displacement (CD) formulation has been introduced recently [92Kik,97Kik,94Fin]. Since the CD formulation is an extension of the conventional CVM, we present the basis of the CD by comparing it with the conventional CVM treatment.

2. Principle of Conventional CVM

Within the conventional CVM, the free energy of the given system is composed of the internal energy Uo and configurational entropies, which will be evaluated by the so-called cluster approximation. We consider the system as a large cluster composed of N sites and generalize the entropy expression S¢,K [94K_ik]

S~d(=~pGp +~a~qGp,q"~ P q "'"-'~-Gct K (l)

to the partial entropy SN. This expression of SN contains correction terms corresponding to all possible points, pairs, triplets, etc., of

lattice points in the entire system. The CVM simplification is done in a manner that all G: for clusters "larger" than C~K are neglected. In the remaining expression, all clusters and subclusters of CtK are grouped into orbits: If each cluster type is given an orbit index r, then the entropy may be written as

K

S u -- N~_~m,G,, (2) r=l

where m,.is the number of clusters of type r per lattice site. The correction terms are derived from eq.(1), which may be rewritten, as the sum over orbits

r-I

S u = N ~ m r _ j G r _ j , (3) j=l

in which the integers mrn are the number of subclusters of type n in the cluster of type r This linear triangular system can be solved recursively for the Gr terms, which are then inserted into expression eq.(2). Then, the entropy per lattice site can be expressed in terms of the partial entropies Sr, in the CtK approximation, as

Su = ks~-~ y~-'~pr(Or)In p,(Or), (4) m r=l ~,

where Kikuchi-Barker coefficients Yr are given by the successive formulas ?K = --mK, (5-a)

x YK-I = - m x - 1 - mx-{YK" (5-b)

In general, K

7r = - m r - ~-'~ m~ y: r = l ..... K, (6) j= r+l

have been derived initially and independently by Barker [53Bar] and by Hijmans and de Boer [55Hij,56Hij]. A very compact and elegant treatment and derivation of the Barker formulas has also been given by Inden and Pitsch [9lind] and more recently by Finel [17], Thus, the configurational entropy of alloy systems can be obtained quite efficiently and accurately by a "cluster" approximation.

For instance, for phase separating fcc binary alloys, tetrahedron cluster approximation gives a sufficient accuracy in deriving the transition temperature kBT/2iV]J=0.3545 compared to the "best known" value of 0.81638 [94Kik]. It also gives ordering temperature of L10 phase kBT/Vl=l.8933, which can favorably be compared with the best known value of 1.741~1.751. The values of all other correlation functions can be found with the help of a set of irreversible transformations on basic clusters which equilibrate the cluster distributions with respect to the given distribution of their basic cluster. The free energy of Ising model can be described as a function of the very small set of basic variables.

The extensions of CVM to non-cubic materials have been done by a number of authors, e.g., for hcp materials [82Gra,84Kan,92Bic,93Ast,94Ono,00Shi], silicates [97Vin,98Vin], oxides [84Bur,88Kik] and topologically complex phases [90Fin,95Slu]. The dynamic aspects associated with phase transitions, e.g., pre-transition phenomena can also be studied by extending the conventional CVM so as to include time dependent path variables [00Moh]. Although the phase stability is a 'static' property of a given phase, phase

CLUSTER VARIATION METHOD IN THE COMPUTATIONAL MATERIALS SCIENCE 35

transition involves more 'dynamic' phenomena originating from atomic diffusion and lattice dynamics. Recently, the pre-transition phenomena (pseudo critical slowing down phenomena) of Ll0 ordered phase have been studied by Mohri [00Moh] within the CVM and path probability method (PPM). The extension of conventional CVM to the time dependent PPM has been developed by Kikuchi [66Kik,74Kik].

3. Continuous Displacement Treatment of CVM

We now describe the essence of the CD treatment of CVM [92Kik,97Kik,94Fin]. The CD treatment of CVM differs essentially from the treatment of the conventional CVM in the sense that the atomic displacements are allowed in the vicinity of the reference lattice sites. We start with a reference lattice, to each lattice point of which we assign an atom. An atom assigned to a lattice point is actually located at r away from the lattice point.

The atomic displacements from reference sites i and j are written as r~ and r'j. The vector from the i atom at P to the j atom at Q is a + r'j - ri, where a denotes the lattice spacing vector. For the pair variables, we distinguish three kinds of pair distribution functions gij, and use three potentials g~i:

gij(ri, r'j) = ~0(I a + r'j - ri[). (7) For two-dimensional (2D) lattice, we use the polar coordinates system. For convenience, the polar

coordinates are written as, p(m) =- rnb; m = 1,2 ..... 5, (8-a)

=- n~m;n = 1,2,3 ..... 8m. (8-b) O(m,n) By this discretization in the polar coordinates, one can allow an atom to be displaced to one of the (=1 +8+ 16+24+32+40+...) discrete points around each reference lattice point.

Then, we define the functionJ(r)dr as the probability of finding an atom at r in the volume element dr. We call fir) the probability distribution function (PDF) for a point. In principle, the displacement r can take continuous values. For a pair of atoms displaced to r and r ' from nearest-neighboring lattice points, we define a PDF, g(r,r') for the pair. The probability distribution functions can also be defined for larger clusters. In the following subsections, we will formulate the CD treatments of CVM for single component systems and then for binary (fcc and bcc) alloys. We do not treat here the applications to crystal surfaces [98Mas] and lattice defects [96Mas].

3.1 Single Component System

We apply the CD treatment of CVM to single component systems (pure metals) and as an example firstly consider two dimensional (2D) hexagonal lattice. We take a trimer as the basic cluster in the 2D hexagonal lattice as shown in Fig. 1. We introduce the triplet probability function h(rl, r2, r3) for the trimer (tri-angular shaped atomic cluster) in addition to the pair probability function g2(rl, r2). The Helmholtz free energy of the system can be given in terms ofh(rb r2, r3) as,

F j3 Idrl j" dr 2 Idr3 [t:(r.r2) + ~(r2.rs) + ~(rs. rl)] q~_-- = N,T

x h(rm,r2,r,)- 3 ~dr, fdr21n[g2(r,,r2) l

+2 ~dr I ~dr 2 ~dr3 ln[h(r,,rvr3)]+ Sdrln[f(r)]

+ J'dr, .br +2 Idr~ ffdr 2 ~dr3[~](r,,r2)-n(rvr,)]h(r,,rvr3) +2 Idr, f d r 2 [dr3[rl(rvr,)-q(r3,r2)]h(r,,r2,r,) +2 far, j'dr= j'dr,[rl(r3,r,)-rl(r,,r,)]h<,r:,r3) +2 Idr, I d r 2 Idr3[ot(r,)-ot(Rr,)]h(r,.rvr.) + 2 ~er, Idr 2 Iar,[cx(r2)-ct(R~)]h(r,,r2,r3) + 2 f dr~ I dr2 Idr3[°t(r3)-ot(Rr3)]h(rl,r2,r3),

(9)


where In[X] -= X lnX - X. (10)

In the above eq.(9), g2(rl, r2) denotes the pair probability function with atomic displacements ri and r2 at atomic sites 1 and 2, respectively, and ~, the Lagrange multiplier for the normalization ofh(rt, r2, r3) probability function. ~(ri, rj,) and tx(ri) are the Lagrange multipliers for the lattice symmetry constraints and R denotes the rotation operator of angle ~/3. The trimer probability function h(rb r2, r3) is obtained by minimizing the free energy • of the system with respect to h(rh r2, r3) as

[3~, 1 h(rl, r2,r 3 ) = exp[-~-] ex'p[-~ [c(rl, r2) + e(r2,r3) + c(r3,rl)]] exp(-(Tl(r 1, r2) - vl(-r2,-rl))

- ( q ( r 2 ~ r 3 ) - T ~ ( - r 3 ~ - r 2 ) ) - ( ~ t ( r 3 ~ r z ) - ~ ( - r 1 ~ - r 3 ) ) ) x e x p ( - ( c x ( r ~ ) - ~ ( R r 1 ) ) - ( a ( r 2 ) - ~ ( ~ r 2 ) ) (11)

(oft. r r ~V/2 - (ct(r 3 ) -~ [Rr "~)) '~" i, 2, 3,J .

3~ (f(rl,r2,r3))l/6 Here, g(rb rz, r3) and f(rb r2, r3) functions are defined by

g(r~,h,r3) =_g2(rl ,r2)g2(r2,r3)g2(r3,rl) , (12) and f(rl,r2,r3) =- f ( r ~ ) f ( h ) f ( r 3 ) . (13)

The similar formulations can also be done for other single-component materials with different crystal structures, e.g., the pair approximation for 2D-square, fcc and bec crystals, and tetrahedron cluster approximation for fcc crystal. The CD treatment of CVM for the three dimensional (3D) single component systems can be formulated quite similarly to the case of 2D systems. For treating 3D systems, the atomic displacements from the reference lattice sites are taken to finite numbers of the order of ~500 sites as shown in Fig.2. Accordingly, the volume integrals are replaced by the discrete summation over the cubic meshes [34]. Crystal symmetries are taken into account in the constraint terms as in eq. (9). For instance, the rotational symmetries, R2x, R2y and R2z are taken into account for fcc lattice as shown in Fig.3a. The CD-CVM formulations for cubic alloys will be given in the subsequent subsections.

In Fig.4, we present the thermal lattice expansions a(T) both for 2D and 3D crystals, calculated by the CD treatments of CVM. The thermal lattice expansions are given by normalized quantities a(T)/a(0), as divided by those of the absolute zero temperature, and 12-6 type of Lennard-Jones potentials are used. The interatomic interactions are taken into account for ZI nearest-neighbor atom pairs, Zl being 4, 6 and 12 for 2D-square, 2D-hexagonal and face centered cubic crystals, respectively. In general, the normalized thermal lattice expansion decreases with increasing the number of nearest-neighbors Z1 using the same interatomic potentials. It is also noticeable that the thermal lattice expansion becomes smaller when the better CVM approximation is used [99Kik]. For instance, for fcc crystals, the quasi-chemical tetrahedron cluster approximation gives much smaller thermal expansion than those calculated by the CVM pair approximation, as shown in Fig.4a.

The more realistic calculations of thermal expansions of Cu and Au metals are performed by using the electronic many body potentials [93Cle]. The calculated thermal lattice expansions of Cu and Au metals are compared in Fig.4b and 4c, respectively. The thermal expansions calculated by pair and quasi-chemical tetrahedron cluster approximations are presented by symbols C) and I-l, respectively. In the figures 4b and 4c, they are also compared with the corresponding experimental results (solid curves) [72Tou] as well as with the molecular dynamics simulations (dashed lines) [90Hol]. The pair approximation of the CD treatment of CVM tends to overestimate to s.ome extent the thermal lattice expansion of fcc metals as in the MD simulations, but considerable improvements can be made by using the quasi-chemical tetrabedron cluster approximation.

3.2 FCC Binary Alloys

For treating 3D alloys, the atomic displacements from the reference lattice sites are also taken to finite numbers of the order o f -500 sites as shown in Table 1 and Fig.2. In this subsection, we apply the CD treatment of CVM to binary fcc alloys. First, we calculate the phase separating curves of fcc binary alloys by using the pair approximation of CVM. In order to assess the validity of the pair approximation, we also use the quasi-chemical tetrahedron cluster approximation [49Li], and compare the both results.


For treating fec binary alloys, the point distribution and pair probability functions are denoted byflr) and g(r~, r2), respectively. When it is needed to identify the two end points of a pair, we use 1 and r, and the point functions for the two end points are given as

fil (r0 and fir (r2) with i, j=l, 2. (14) On the other hand, the pair probability functions are written as

gij (rl, r2); rl is for 1, and rz for r. The reduction relations of the pair probability functions can be written as

f, (r,)= J'dr_, E gi~ (r, , r 2 ) i = l , 2, (15) J

fj, (r2)= ~ dr, Z g,j (r, , r 2 ) j = l , 2. (16) i

We denote the point probability of site i on the left end by x, and that of site j on the right end by xjr. They are written in terms of the point and pair probability functions as

x,,= J'dr, fia(r,)= Idr, j'dr2Zg0(r,,r2) . (17) J

The normalization equations for the probability functions are given by

1= Z xij = Idr, E f ,~ , )= ~dr_. x f,,(r.,)= j'dr, Idr,_ E g 0 (r,,rz) . (18) i i j ij

In deriving the probability functions, one must take into account the crystal symmetries of the given lattice. The fcc lattice has the crystal symmetries of 4-fold rotations, with respect to the [001 ] crystal axes as shown in Fig.3a. Clearly, for the 4-fold rotation symmetries, i.e., R4x, R4y and R4~ on the x, y and z axis, we have the following relations:

f,(r0=fil(R4xrl); fir(r2)=fi~(Raxr2) f i l ( r l ) = f i l ( R 4 y r l ) ; f i r ( r 2 ) = f i r ( R 4 y r 2 )

fil(rl)=f,(R4~rl ); fir(r2)=fi~(R4zr2). (19) In terms of the components, we have the following transformations:

R4~(x,y,z)=(x,-z,y); R4xt (x,y,z)=(x,z,-y), R4y(x,y,z)=(z,y,-x); R4yq (x,y,z)=(-z,y,x). R4~(x,y,z)=(-y,x,z); R4,q (x,y,z)=(y,-x,z). (20)

Then we see that the relations hold for R4x, R4y and R4~ as R4,R4, (x, Y,Z)= R4,R4y (x,Y,Z)= R4yR4~(x, Y, z)= (z, x.Y). (21)

Therefore, the three rotation operations R4~, R4y and R4, can be taken as independent, and one may use R4~ and R4~ operations as independent.

We now introduce the constraint terms C,, and C,~, in terms of the Lagrange multipliers ct C~il =- ~ drlaxil (rl){fil ( r l ) - f il (R4xrl )}

= ~ drl {Ctxil (rl) - atxil (R4x rl )}fi/(rl ) =- J drl Axil (rl)fi/(rl ), (22) where A~, (r,)-- et ~,, ( r l ) - c%, (R,~ r, ). (23)

The entire constraint terms are Cax + Caz for which

Coa" =- Y" Coxil + ~ Cajl = I drl I dr2 ~. {Ax//(rl )+ Axjr (r2 )~q (rl ,r2 ). (24) i j (j

and

Ca: -~ I drl I dr2Z {A=il(rl )+ A:jr (r2 )}gij(rl,'2 ) . (25) ij

Since there are no sublattices in the phase-separating system, symmetries among A's are the same and gi ven as A,,(r)= A,i,(R4, r), A,i,(r)= A,i.(R4:r). (26)

Thus, when one calculates A,ij(r) and Azil(r), and then A's for r are derived from them. For the energy expressions, we use the 12-6 type of Lennard-Jones potentials, otherwise stated, and we

choose

e(r::o 4 >0 ,e7)


Here, we use rl ~o as the unit of length and write it simply as r o --- r~to , and choose e as the unit of the energy. Then, the energy of an fcc alloy with N reference lattice points is written as

E = 6j-dr, j-dr, E eij(r, r,)gij(r, r, _). (28) N ~j

On the other hand, the entropy of the system in the pair approximation is given, in a similar manner as that of 2D system, as

- + Sdr I {L[fpa(rl)]+ Lkna(rl)]}+ [.dr 2 {L[fpb(r2)]+ Lknb(r2)]}) DN kN - 6 S dr 1Jdr 2 {Ltg pp (r,.r2 )]+ Ltg.. (rt, r2)]+ Llg.p (~, ,~2 )JL[g..(rt,r2 )]} (29)

When we work with a general alloy, we need the chemical potential and hence the chemical potential terms C. in the grand potential ~ The latter for a system of N reference lattice points is given by

C~, _ y, xi,g,, (30) N i

where x~ is the fraction of species "i" on the point "a". Therefore. when there are no vacancies in the system, one can write

lal = - tx , ~a2 = g, (31)

which (30) in terms of the pair probabili ty functions g(rj , rD as

c . _ J'dr, j'dr:Zg,g,j(r,, r2). (32) N ~j

which can be symmetrized as

Clt - ~ Idrlldr2Z{~i + pj}go.(rl,r2) . (33) N 0

Then, the grand potential h ° of the system is given using the four terms mentioned above as

n E S "-'~c~C"+6~-~+C~'. q~=_ =13 NkT N kN 13 N

(34)

Therefore, the final expression of the grand potential ~ i s expressed by adding the normalization condition as 1

W = 6,17 Idq I dr'- Z gii(r, r2)gii(rpr2)-~ fl I dr~ I ar,- Z {.i + ~i}go (r~,ra) s , ,., o kN

121 { Id'];L[i~it(rl)]+ Idrz~,iLki,(r2)])+6Idri Idr2~ijL[go(rl,r2)]

, ] ÷,LTA 1- fdrl Idr,~.s go(rl,r,) (35) L

+6 fdrl'Idr_Z {A,.i,(rl)+ A.ri,(rz)}go.(rl,rz)

+6 j<t, I,l,-_, Z G,(r,)+ 6,, (r,)}g,,(r., #

When we minimize the grand potential W with respect to gij(ri,r2) function, we obtain

taxi 71n{fil(rl)f J"(r2)}+61n{gij(rl'r2)}- l fl{ui + I'tJ} (36) z~s~i (r~, ~2 ) =- 6~0 (n,.2)- - + . , - , o ; / +

makes the chemical potential IX increase as B-atom concentration increases. Using eq.(31), we rewrite

Then. it is s traightfo~'ard to derive the basic equations of the pair distribution functions as


gij(rl,r2)=exp[fl-~--~ - . /jr 1 ,£7c (r , / + f l2 / -~ lk i i+ lA j ~'" }l{fil(rl)fjr(r2)~l/12 (37)

x exp{Axl(rl)+ Axjr(r2)+ Azil(rl)+ Azjr(r2)}i, j=l.2. When these equations are satisfied and the free energy is a minimum, the Lagrangean constant for normalization

in (35) is L=~/kT. The extension of the CD treatment of CVM to the quasi-chemical tetrahedron cluster approximation

[49Li] is straightforward. For this treatment, we take into account the symmetry of fcc lattice as shown in fig.3b. We introduce the tetrahedron variables w in addition to the pair probability variables. Then, the energy of a system with N lattice point is given by

E = N J dr o J dfi J d 5 J dr 5 %,25(ro;ri;r2;rs) w0~25(ro;r,;r2;rs) , (38)

where e012dr0;rl;rz;rs) is the energy per tetrahedron, and is written as a sum of pairwise energies when many-body interaction can be neglected

eo,2s (ro;r~ ;r2;r5 ) = %, (ro;q) + eo2(ro;r2)+%5 (ro;r5) +e,z(q;r2) +e2s(r2;rs )+es,(rs;rO. (39) In the present case, each pair belongs to one tetrahedron only, and thus we do not need the factor 1/2.

On the other hand, the entropy of the system can be written in terms of the point and tetrahedron variables fi(ri) and wo125(ro; rl; r2; rs) as

S S ° 3 ;droL[fo(ro)]+ ;drlL[fi(rl)]+ ~dr2L[f2(r2)]+ J'drsLr.fs(rs)]) kN - kN + ~ ( (40)

- ~dr o Sdrl ~dr2 ~dr5 L[wol2s(ro;rl;r2;rs)]. Then the free energy is given by

F

NkT

-7(3 j'droL[fo(ro)] + j'drlL[fl(rl) ] + ~dr2L[f2(r2)]+ Jdr, L[f,(r,)]

+ ~d'o ;drl ;dr2 ;drsL[wol:s(ro;rl;r2;,5)] (41)

- ; d r o ~dr I S d5 ;dr,(A:(ro)+ Az(rl)+ A(r2)+ A=(rs))woi2,(ro;rl;r2;rs)

- Sdr o ~drl ~dr2 ~drs(A,(ro)+ A~(ri)+ A,(r2)+ Ax(rs))Wol2s(ro;rl;r2;rs)

+flA(1- ; d r o ; d r l Sdr 2 ~drswoi2s(ro;ri;r2;rs)).

When one minimizes the free energy ~ of the system with respect to the tetrahedron variable, one obtains

aO _ 13eo,:5 (ro;r,;r2;r~) _ 3 ln(/o(ro)f (r,)~ (rDA (r5)) 8wo,25 ( ro;rl ;r2 ; rs )

+ ln(wo 2s (r0;r5 5;r0) - (Ax(ro) + A,. (r~) + A x (r2) + A, (r 5 )) - (A:(ro) + Az(r~) + A~(r 2) + A=(r 5 )) - ~ .

=0.

(42)


One can thus get the tetrahedron probability variable wm25(ro;rl ;r2;rs) as 3

w0j25 (ro; r~; r 2; r 5) = exp(flA - flSolzs (ro; r~; r 2; r s ))(fo (ro)f (r~)fz (r2)f5 (r5)) 4

x exp(A~ (r0) + A~(fi) + A,(r2) + Ax(rs) ) (43)

x exp(A, (r o) + A, (r t) + A, (r 2) + A z (r 5)).

When the equilibrium state is solved, the Lagrangean constant for normalization 3. gives the free energy F per single atom as in the pair approximation

X =F/N. (44) Before going to the continuous displacement calculations, in order to examine the accuracy and to see

the effects of the new CD treatment, we briefly discuss the phase separation of binary fcc alloys calculated by the conventional CVM. The three kinds of approximation schemes of CVM, i.e., pair, quasi-chemical tetrahedron and CVM tetrahedron approximations are used. The entropy expressions for the three cases are obtained by taking into account the characteristic cluster variables as [6]

S~i r/(ksN ) = 1 l~-~L(x,) - 6~]L(y 0) + 5, (45) i /j

Sq-ch,r/(ksN) = 3~L(x , ) - ~ L ( W e , . ) + 2, (46) i ijmn

ScvM ,r/(ksN) = 6~-~L(Y0) - 5~-~L(xi)- 2ZL(W0,,,) - 1, (47) iJ" i ijmn

where W/j,,, is the tetrahedron variable. For the phase-separating case, we take the interaction parameters as

4g? -= 2g, (1,2)- ~i(l, 1)- ai(2,2) > 0, (48) where the individual parameters are defined by

O, 0 =-,? + 149)

~i(1,2)= ~i(2,1) = + 4 - ~ . Further, when we choose e12=0, the phase diagram becomes symmetric, em is the overall cohesion and is irrelevant when there are no vacancies. The unit of energy e is defined as

2e~2 - { % + e22}- = 46>0. (50) Using the nearest-neighbor approximation the congruent temperatures are estimated to be

10.97 Pair,

kT~/g~ =110.8 Quasi-chemical TTR, (51)

L10.025 CVM TTR, for pair, quasi-chemical tetrahedron and CVM tetrahedron approximations, respectively. One can thus see that the critical point is around kTc/e=10 when one uses the conventional CVM. One can see in the above eq. (51) that pair approximation gives reasonable result compared to those obtained by two other schemes, especially for the dilute alloy region.

The continuous CVM can now be applied to the fee alloys within the pair and quasi-chemical tetrahedron cluster approximations. We have performed the numerical computations using the nearest-neighbor approximation in the fcc binary alloys. For the binary system, we use the chemical potential to control the composition and for convenience, we choose

~h=-/.t and la2=[.t, so that B atoms increase as ~t increases. We write the chemical potential terms in the grand potential as

, ) f±s (,)t - ~Pl +gzp2 = - I-tl drlf rl + I.tj dr ' i f 'j . (52) \ i=l j= l

In order to get general understanding of the continuous atomic displacements, we use the Lennard-Jones potential for the interatomic potentials. Specifically, we choose the potential parameter values symmetrically as


el 10=e220=3.0, el20 = 1.0, (53-a) r~ t 0----r120--r220 = 1.0. (53-b)

This choice makes the unit of energy e=l.0 and the phase separating curves in Figs. 5 and 6 are directly to be compared with those values of eq. (51). It is noted that the kT/s axis scale of Fig. 5 goes only to 7 while that of eq. (51) to 11, and one notices that the introduction of continuous atomic displacement reduces Te to about 60%. This is understood when we compare the definition of c in the conventional Ising model and that of e in (53) used in Fig. 5. In the continuous displacement formulation, the effective interaction potential e* corresponding to the conventional Ising model varies as the temperature increases, and may be defined as

4~* = 2e12(< rt2 > ) - {%(< rll >)+ e22 (< r22 >)} , (54)

where <rij> is the thermal average of the i-j interatomic distance at the temperature T of interest. As temperature increases, rij moves from the energy minimum position rij0 and the energy e~j(rij) increases. With the above energy parameters, we deal with two energy curves: one for en(rH)=e22(r22) and the other for e12(r12). The former is lower and one can verify the relation:

e~2( r u) - e~( rH ) = 3Abs[e,~( r) ], (55)

which decreases from 2.0 toward zero as one moves from the energy minimum to the right. Therefore, the value of e* is less than the energy unit e. Since we expect k T c / ¢ ~ kT[ / ~ ' , we can transform as

kT~*_kT~* e* kT~x ~* kT~ - x - - * ~ < (56)

E C* ~ ~ S S

The next feature which can be seen in Fig. 5 is that of our cases of Table 1 (with b/a=O.075 for n=5) which share the similar size of the peripheral sphere size lead to practically the same phase-separation diagram, although the accuracy of integration increases with n. In Fig. 5, the n=5 case is plotted for b/a=O.075. However, a spot check shows that n=5 with b/a=O.08 agrees very close with the n=4, b/a=O.lO results. Based on these findings, we perform the phase-separation diagram studies for the size different atoms using the simplest case n=2 of Table 1. The upper two curves in Fig. 5 are the phase separating curves calculated by using the 8-4 and 10-5 type of Lennard-Jones potentials and n=2 of Table 1. One can see that the transition temperatures of phase separation are considerably higher than those of the 12-6 type of Lennard-Jones potentials, but still much lower than those of the conventional CVM. The upward shifts of the phase separating curves when using 8-4 or 10-5 L.-J. type of potentials result from the longer range nature of the interatomic potential, and the smaller decrease in the segregation energy due to the thermal lattice expansions compared to those of 12-6 type of L.-J. potential.

In Fig. 6, we present the phase separating curves of fee binary alloys for input alloy parameters, (a), (b), (e), and (f) presented in Table 2. For parameters of (e), the energy minimum position is the same as that of (a). The boundary temperature is lower for (e) than that of (a), because the effective interaction energy e* is smaller than (e). The slight asymmetry in (e) is due to the difference in the occupation probabilities caused by the difference in the energy curves of e22(r) and ell(r). The curve (d) is higher than (e) by the same effect as that mashes (b) and (c) in Fig. 6 higher than (a). The triangular symbols in Fig. 6 represent the congruent temperatures calculated by using the quasi-chemical tetrahedron cluster approximation. One can see in Fig. 6 that the congruent temperatures become slightly, by a few per cents, lower by applying the quasi-chemical tetrahedron cluster approximation.

In case of the parameter set (0 both the energy level and the energy bottom positions for B are changed by a larger amount than (d). The short center curve is the average of the left and right branches and show how we locate x(B)¢ at T~. In contrast to the study of critical phenomena of the Ising model type, we use the different scheme to determine T¢ and x(B)¢, and plot the calculated values of {x(B)L-x(B)R} 2 and {aL-aR} 2 as a function of temperature. Since we can assume that x(B) and lattice constant curves are parabolic near T¢, we expect the squared quantities change linearly near Tc and vanish at To. This extrapolation scheme is quite accurate for the present calculations.

The above mentioned phase separating curves of fcc binary alloys can be also calculated by using the quasi-chemical tetrahedron cluster approximation, without enormously increasing computational times. In applying the quasi-chemical tetrahedron approximation, the Lagrange multipliers ct's must be modified so as to meet the constraint relations of the 2-fold rotations R2~ and R2~. It is noted that tetrahedron approximation in the quasi-chemical treatment treats only a half of tetrahedron in the fce lattice as shown in Fig.3b. However, the


general trends in the phase separating curves obtained by quasi-chemical cluster approximation remains unchanged compared to those of the pair approximation.

3.3 BCC Alloys

The CD treatment of CVM is now applied to bee binary alloys. For bee binary alloys, we discuss the order-disorder phase transition of B2 structure alloys. Usually in the calculations of the free energies of ordered and disordered binary alloys, the effects of vibrational entropy and local lattice distortion are not taken into account. Therefore, it is of great significance to apply the present CD scheme of the CVM to the order-disorder phase transition of the binary alloys and investigate the effects of continuous atomic displacements on the phase stabilities. For simplicity, we consider bee stoichiometric alloys with B2 structure. We take two kinds of sublattices named L and R, and use 1 and 2 for A and B atoms. The point probability functions are denoted by fiL (rl) and fir (r0, and the pair distribution functions by gij (rt,r2), in which the first argument rl is on L, and the second one r2 on R. respectively. The reduction relations among the point and pair probability functions are given as

fiL(rl)=~dr2Zgij~l,r2), fiR(r2)=fdrl~-~.gij(rt,r2)i,j=l.2. (57) J i

The probabilities of i-atom on L and j-atom on R are written as XiL and XjR, respectively. They are written in terms of the point distribution functions and further of the pair functions, as in the treatments of fcc binary alloys. The fraction of i-atom in the alloy is

_f,1 x Pi = 2 k iL + XiR)" (58)

We may define the long-range order variable~ as

~ = I(XlL --X~R). (59)

For the ordered phase with B2 structure, the compositions of two sublattices are given by

X1L-----Pl +~ X2L =P2--~ xtR = P, - { X2R = 92 +{ (60)

For B2 ordered alloys, the point distribution fir) obeys the 4-fold symmetry, on the x, y and z crystalline axes. In view of this, we write the constraint terms C~x and C~, and the Lagrange multipliers ct as

CaxiL =- I drl°txiL (rl){fiL ( r l ) - f iL (R4xrl )} = l an {c,x¢ (~ ) - ~x~L ( R4xr~ )}f ~t (n ) - I an,4x~L (ra )AL (n ) (61)

where AxiL(r,)-- axiL(r, )--0%L(R4xF,) .

Then, the entire constraint terms are Cax + Ca~ for which Cax =- ~ CaxiL + F. C~jR = J drlf dr2 ~, {AxiL (rl )+ dxjR(r2 )Jgij (rl,r2 ) (62)

j ij and

Caz =- I drl l dr2 Z {AziL (rl)+ AzjR (r2 )}g ij (rl ,r2 ). (63) ij

The energy of the alloy with N lattice points is given simply by

E = 4~drlfdr2~eij(rl,r2)go.(rl,r2), (64) N 0"

where the energy between i and j atoms is written as Eli(r1#2). The entropy expression is given for the pair approximation as

7 ( 2 2 ]/ 2 2 S _ S * ~2 Sdrli~=lL[fij(rl)l+~drlX[fjr(r2) _4Sdrl~dr2~XL[gij(rl,r2)] ' (65)

kN kN j=l J i=lj=l

where S* is an additive constant resulting from the conversion from sums to integrals. The chemical potential terms Cg ofbcc alloys are symmetrized as


C/2 1 - fdrlldr2•{itl+l.t2}gij(rl,r2). N 2 ij

Then the grand potential of the system is written as

f l g - - ~ f = fl N kN fl - 6{Cax +

Further adding the normalization condition, the explicit form of ~is

flg=4flJdrlJdr2~gij(rl,r2)gij(rl,r2) -2flfdrlJdr2~..{gi +12j}gij(rl,r2) - - ~J v

tb .(n)l+ [&(r2) +4Iarllar2Z Z L[go(n,r2)] i=lj=l

+ fl2tl-jdrlJdr2~..gij(rl,r2) t -4fdrl[dr2Z..{Axil(rl)+ Axjr(r2)lgij(rl,r2) t q j lj

-4I drll dr2 ~, lAzil(rl) + Azjr(r2)lgij(rl,r2) • iJ

When we minimize the grand potential~with respect to gij(rl,rE), we obtain

flO¢ =- 4fl~ij(rl,r2)_71n{fil(rl)f jr(r2)} + 41n{gij(rl,r2)}_ l Og/j (rl, r2 )

- 4{Axi ! (r 1 ) + Axj r (r 2)}- 4{Azi I (r 1) + Azj r (r 2)}- ,82 = 0,

o r

go(rl,r2)=expI fl~4 - fl6ij(ri,r2)+ ~ {ui +/aj }]{fil(rl)f jr(r2)}7/8

(66)

(67)

S • (68)

kN

(69)

xexp{Axil(rl)+Axjr(r2)+Azil(rl)+Azjr(r2)}, i,j=l,2 (70)

When these equations are satisfied and the free energy is a minimum, the Lagrange multiplier for normalization X in eq. (68) again leads to as in eq. (44)

Z g ~ - - = ~.. (71)

N The minor iteration for bcc alloy is similar as that for fcc alloy, but one must solve both A×iL(r), AziL(r)

and AxjR(r), AzjR(r), self-consistently. We introduce the H functions for each sublattice defined by

H i - , • ( f*iL (R4xrl)'] iLt'~4xrl)=ePln~ f---*i; (r,----) ~ ' (72)

where ~p is the damping factor. Using the H functions, one can solve

AAxiL (rl) = I ~HiL (R4xrl)+ 2HiL (R4x2 rl)+ HiL (R4x3 rl)} . (73)

The similar iterations can be done by changing x to z for AAz, and also for the R sublattice as those for L. As one of example calculations, we take the following two sets of energy parameters:

el 10 = e220 = 3.0, el20 = 5.0 fe 110 = e220 = 3.0, el20 = 5.0 (74)

rll 0 =r120 =r220 =1.0 ' ) ( r l l0 = 1.0, r120 =1.05, r220 =1.10

The interaction energies are chosen to be appropriate for the ordering systems, and the energy parameters are taken to be

c'-= {2e120 - (ell 0 + e220)}/4 = 1.0. (75) In Fig. 7(a), we present the calculated order-disorder transition temperatures of B2 alloys using

parameter values of eq. (74). The upper curve is calculated by conventional CVM, while the lower one by the pair approximation of the CD scheme of CVM. In Fig. 7(b), we present the transition temperature curves with


second choice of the size parameters: ril0=l.0, r 120=1.05 and r 220=1.1. It can be seen in Fig. 7(b) that the transition temperatures become asymmetric and show skewed structure. One can then come to the conclusion that the critical temperature of order-disorder phase transformation of bcc alloys is reduced drastically by amount o f - 40%, when the continuous atomic displacements arising from the thermal lattice vibration and the size-misfit effects of the constituent atoms are taken into account.

In the light of the present theoretical findings of the order-disorder phase transition of the B2 ordered alloys, we briefly comment on the previous theoretical calculations and comparison with the experimental results. It has been pointed out that when the interaction potentials calculated by the first principles electronic theory are used together with the conventional CVM in constructing the free energy, the order-disorder transition temperature is estimated much higher than experiments and the area of single phase region is narrower than experiments [95Hor,88Moh,89Ter]. It is interpreted as caused by neglecting the local distortion of the lattice due to the size difference of the constituent atoms in the alloys. So far, the first principles phase diagram calculation has been extensively performed for noble metal based alloys. For instance, the characteristic topological feature of Cu-Au system has been reproduced well [95Hor,88Moh,89Ter]. However, the order-disorder transition temperature is greatly overestimated. This tendency results from the over-stabilization of the ordered phases due to the neglect of lattice distortion and thermal vibration effects [92Kik,97Kik,94Fin]. The present calculations using the CD scheme of CVM, although for bcc alloys, clearly shows that the order-disorder transition temperature are reduced significantly by taking into account the continuous displacement of atoms around the reference lattice sites. In addition to the above mentioned noble metal based alloys, it has also recently been pointed out that large discrepancy arises in the congruent temperature of L10 FePd alloy, when neglecting the effects of the continuous atomic displacements [39].

3.4 Surface Segregation of Binary Alloys

In this subsection, the CD treatment of CVM is applied to surface segregation of the fcc binary alloys [80Bal,99Tre]. This requires extending the single-component treatments to a binary system in addition to treating the surface. As a typical example, we consider the (001) surface of the binary fcc alloys. To start off, we solve the bulk system for a fixed concentration xB and at fixed temperature T. We use an extended Einstein model, and we allow the continuous atomic displacement, specified by r, for the "center" atom. It will be assumed that an atom on a lattice point other than the center is a mixture of A and B with the ratio XA:XB. The chemical potential ~t(T) and the interatomic distance rij are then determined. Since the surface is in equilibrium with the bulk and is a small portion of the system, the bulk ~t(T) is taken to be the same throughout the surface. The objective is to determine the surface composition XAa(j2,Jz) as a function ofj~z/ah, j2=1 (for A) or 2 (for B atoms). The z axis is perpendicular to the surface for a semi-infinite medium z~_0. The z coordinate of each (001) plane is an integer multiple of the unit ah--ao/2 with a0 being the cube edge of the fee lattice; "j" is referred to as the center atom and "k" to the atom at the other end of the pair. Together with XAB(j2,Jz), rij also varies with jz.

The energy of a species on a plane at Z=jz*ah is a functional of {XAa(j2,jz)}. Hence, the grand potential D is a functional of {XAB(j2,jz)}. In what follows, Nz is the number of lattice points on a (001) plane and R(m) is the coordinate of a lattice point at m that goes over the entire system, while XAB(j2,m)----Xas02,jz) is a function of jz only. Then, after a bit of algebra, we obtain the free energy of the system as

fl~,{xA B 0.2, jz)} __ ~{X~B 02 , J'z )} N z k T

1 jz max = 2 fl ~, Y.Y:Y~drf(kz,k2,r~r(j2,k2,ir-R(m~)XAB(k2,m)

jz=O j2m k2 / <76, + jz~=O Idr[ f (Jz ' j2 ' r ) ln f ( j z ' j2 ' r ) - f ( l z '12 ' r ) ]+l

j z n ~ x . , [


When we minimize [3t~ {XAB(j 2j1)} with respect to the point probability variables f0z,j2,r), the equilibrium state with the minimum value of the grand potential ff~ can be found as

flrgt({XXB(J2, Jz )}) ___ 1 fly, E 6(J2, k2,lr - R(m~)XAB(J2,m)+ In f(Jz, j2,r) - fl2(Jz)= 0. (77) .c91e(J'z,j2,r ) z m j2

This leads to the point probability function in the following form:

f ( j z , j2 , r )=exp[ f l2 ( j z ) - l fl~m~j2~(j2,k2,,r-R(m~cAB(J2,rn)]. (78)

When eq. (77) is satisfied, eq. (76) can be simplified as jzmax . .

~{xAB(J2,Jz)} = Z 2(Jz). (79) jz=O

From f(jz, j2, r) in eq. (78), the surface composition XAB(k2, jz) Can be determined as XAB (k2, Jz ) = I dr f(Jz, k2, r). (80)

Substituting eq. (80) into (79), it can be solved numerically using the normalization condition. We now apply the CD treatment outlined above to the surface segregation of CuAu(001) alloy. Discrete

sums over the mesh points are used for the integration calculations [97Kik]. Convergence of the equilibrium lattice constants with respect to the number of the mesh points is tested by calculating the bulk-lattice constants of CuAu alloys. Use are made of the many-body potential parameters in Table 3 with nrad=2-5. The bulk composition is chosen to be CA=CB=0.1-0.9. The calculated lattice constants for T=600K, 1000K and 1400K are presented in Table 4 with nrad=4 and nrad=2. The values of the chemical potential are also listed. As ~t increases, Au composition increases. The results of Table 4 show that the thermal expansion is negative for nrad=2. It is positive for nradD4. The size of the integration mesh of nrad=2 seems to be too large and not appropriate for the CD treatment.

The many-body potentials [93Cle] are equivalent to the energetics based on the second moment approximation of the TB electronic theory. The energy is given by

Ecoh = ~xi(EAi(r i )+ ERi(ri)), (81) i

2 • . . ri j EAi =- Z xj¢o(i,J) e x p - 2 . 0 qo(t,J ~ - 1 , (82)

L J¢i

ERi= ~.xjAo(i,j)exp -POO,J ~ - 1 , (83) j;~t

where the summations are taken over surrounding atoms j (¢ / ) . In these equations, A0, {0, P0, q0, and r0 are constants as given in Table 3. The minimum of interaction energies are at r0=2.5562/~ for Cu and r0=2.8843 .A, for Au, respectively.

The calculated surface compositions of the CuAu(001) alloys are presented in Fig. 8. The surface layer is numbered 0, and the number increases away from the surface. Fig. 8 shows the Au density XAu =0.5 as the difference from the bulk. The larger atoms are pushed away from the surface. We have checked that the plottings of log(x(Au)-0.5) lie nicely in lines expect for the topmost surface layer. It can thus be concluded that the surface composition varies exponentially with a wavy modification. Fig. 8 also shows that the Au enrichment occurs at the first surface layer. Au oscillatory behavior is obtained for the composition profile depending on the temperature. It is interesting to note that the segregated species has larger cohesive energy between the constituent atoms when it is in a pure metal. The composition modulations of surface layers of the bulk binary alloys can be used to predict those of the nano-scale materials.

46 R. KIKUCHI AND K, MASUDA-JINDO

3.5. Solid-Liquid and Liquid-Gas Phase Transformations

The statistical mechanical understanding of melting of solids are of great significance [00Kik,00Ros]. The liquid state to be outlined in this subsection is to be combined with the solid phase which may be treated with the CD formalism of the CVM. In the previous subsections 3.1-~3.3, we introduced a way to treat continuous motion of atoms away from its reference lattice point. There is another kind of continuous movement of atoms in space, that leads to the liquid phase. For treating the liquid phase of materials, the concept of continuous movement of atoms in space is also of great significance. In this subsection, we outline the procedure for treating the liquid phase on the basis of the CD treatment of the CVM.

The solid-liquid phase transitions may be treated by "lattice theory", a special scheme of the CD treatment of CVM. It has been formulated in conjunction with the premelting phenomena of the grain boundaries in metals using the lattice-gas model and nine-site cluster approximation of the CVM [80Kik]. At a temperature far below the melting temperature Tin, a gradual but clear transition has been found between the low and high-temperature structures of the boundary. Rigorously speaking, however, the melting transitions to the liquid phases must be treated in the limit of zero lattice constants. Such a treatment has been tried and successfully used for the liquid-gas phase transformation, as discussed below.

For simplicity, for treating the gas-liquid phase transformation we start with a simple cubic lattice of the lattice constant a and consider a model that atoms and vacancies are distributed over the cells. When we let a decrease to infinitesimal, keeping the density of atoms in space fixed, atoms can occupy any point in this continuous space, and the model represents the gas or the liquid state [00Kik]. When the lattice constant becomes small, the interatomic interaction becomes long-range. The common thought of treating a long-range interaction is to make the basic cluster three dimensional large.

The entropy expression for this case can be written, using the technique of the Correlation Correction Factor [94Kik], as

exp(S/k s) = W~,,,G2_ce,,G3_c,#G4_ce,t, (84) N!

Wcell =- {Cell} N (85)

,.-,r {Cell J}N {Cell k}N ] (86)

~ H [ N!{Pairjak}N{Pairkbm}N{Pairmej}N ], G3-cell [ {Cell J}N {Cell k}N {Cell m}N {3 - pts j akb me}N (87)

f {~m}{n}{Akmn}{Ajkm}{Ajmn}{Ajnk } G4-ce. -- n l N, {km}{kn}{kj'}{mn}{nj}{jm}{Ouartet jkmn} J " L

(88)

This expression is written for a small but finite size lattice constant, and the space is divided into N cells. Each cell is either occupied by an atom or vacancy. The first factor is for permutation of occupied and vacant cells among N cells. The product factor is the Correlation Correction Factor for the pairs, and goes over all j-k cell pairs; a is the vector connecting two cells j and k. It turns out that the pair approximation of [exp(S/kB)=WccuG2.ceu] does not lead to any phase changes. Also, the triplet approximation does not lead to a stable liquid phase.

Then, the quartet approximation using all shapes and all orientations of 4-ceil clusters is applied. When the grand potential is minimized, the 4-cell probability is derived as

g4(a,b,e)= g~(a'b)g3(a'b+e)g3(a+b'e)g~'(b'c) exp(-flz(lalblel)). (89) g(a)g(a + b)g(bXa + b + e)g(b + e)g(e)

When the 4-body potential z(a,b,c) vanishes, (89) reduces to the 4-body superposition relation derived by Fisher and Kopeliovich [60Fis]. The set of equations derived by minimizing the grand potential has the form


g(rt_, ) = exp{- fiB(r,2) - pl,2 (r,2)},

g~(a,b) : g(a)g(a + b)g(b)exp{- flg/(a,b)- Plut(a,b)},

(90)

f g~ (a,b + C) pl/n (a,b) ---/9 J dc[l - g(c) - g(b + c) - g(a + b + c) ÷

g(a) (91) + ga(a,b + c) + ga(b,c) _ ga(a,b,c) 1 .

g ( a+b) g(a+b) g3(a,b) " These are a part of the set of equations to be solved for the pair distribution function g(rl2). This set needs two tiers of interactive procedures and forms the bottleneck in the numerical computation and has not been successfully solved. For obtaining an approximate solution, we introduce an additional simplification on equations, and obtained a pressure versus density isotherm of the van der Waals type.

For the numerical calculation of the pressure P and chemical potential ~t of the system, we use the relations

t ip= lnu+ In ,o- /9 J'da[g(ayl)]+ l o 2 j'da j 'db[3- g(a) + g(b)g(c)

-(g(a)g(b)+ (b)g(c) + g(c)g(a~i+ 393 (a, b )], (92)

and

~P=p-P~ Sda[g(a)-l]+-~v. 3 j'da ~db[2-(g(a)+g(b)+g(e))+g3(a,b)] , (93)

for the triplet case without the 3-body interaction. The pressure P versus density (r0 3) values calculated by quartet treatment of CD-CVM formalism are

shown in Table 5. The critical temperature in the quartet approximation is found to be about kT/e=l.7. This value can be favorably compared with the value of 1.3 by Hansen and McDonald [76Han] calculated for Ar. The liquid phase is also confirmed by the radial distribution function g(r). Thus, it may be concluded that the quartet formulation leads to the stable liquid phase.

4. Conclusions

The continuous displacement treatments of the CVM have been applied to the calculations of phase diagrams of binary alloy systems. We have worked out with the binary alloy systems with phase separating imcractions in the fcc and bcc structures. The phase-separating diagrams of a number of model systems are calculated using both pair and quasi-chemical tetrahedron cluster approximations. The CD treatments of the CVM have also been applied to investigate the order-disorder phase transition of bcc alloys using the pair approximation. In general, it has been shown that the phase-transition temperatures are reduced drastically when the continuous atomic displacements are taken into account. Therefore, the large discrepancy between the experimental and theoretical phase diagrams [95Hor,88Moh,89Ter] calculated so far by the conventional CVM and ab initio electronic theory can be removed by applying the CD scheme of the CVM.

In addition, the CD treatment of CVM has also been applied to study the surface segregation phenomena of fcc binary alloys as well as the gas-liquid phase transition of a mono-atomic molecular fluid. In both calculations, the CD treatment of CVM is successful and provides us the simple physical insights into the problems.

References

35Bet 35Kir 41Tak

H.A. Bcthe, Proc. Roy. Soc., AI50, 552 (1935). J.G. Kirkwood, J. Chem. Phys., 3,300 (1935). Y. Takagi, Proc. Phys.-Math. Soc. Jap. 23, 44 (1941).


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73Lar 74Kik 76Han 80Bai 80Kik 82Gra 84Bur 84Kan 88Kik 88Moh

89Ter 90Fin 90Hol 9 l ind

92Bic 92Kik 93Ast 93Cle 94Fin 94Fon 94Kik 94Ono 95Hor 95Slu 96Mas

97Kik 97Vin 98Mas 98Vin 99Kik 99Tre

00Kik 00Moh 00Ros 00Shi 01Moh

Y.-Y. Li, J. Chem. Phys. 17, 447 (1949). R. Kikuchi, Phys. Rev., 81,988 (1951). J. A. Barker, Proc. Roy. Soc. A216, 45 (1953). J. Hijmans and J. de Boer, Physica 21,471 (1955). J. Hijmans and J. de Boer, Physica 22, 408, 429 (1956). I. Z. Fisher and B. L. Kopeliovich, Soy. Phys., Dokl. 5, 761 (1960). R. Kikuchi, Prog. Theor. Phys., 35, 1 (1966). "Thermal Expansion: Metallic Elements and Alloys" (Termophysical properties of matter Vol. 12), edited by Y. S. Touloukian, R. K. Kirby, R. E. Taylor and R D. Desai (IFI/PLENUM, New York-Washington) 1972. E Larch and J.W. Calm, Acta Met., 21, 1051 (1973). R. Kikuchi, J. Chem. Phys., 60, 1071 (1974). J. E Hansen and I. R. McDonald, Theory of Simple Liquids (New York: Academic), (1976) C. A. Balseiro and J. L. Mor~n-L6pez, Phys. Rev. B21,349 (1980). R. Kikuchi and John W. Cahn, Phys. Rev. B21, No. 5 1893 0980). D. Gratias, J. M. Snchez, and D. de Fontaine, Physica A 113, 315 (1982). B. Burton and R. Kikuchi, Phys. Chem. Minerals, 11,125 0984) J. Kanamori, J. Phys. Soc. Jpn. 53, 250 (1984). R. Kikuchi and B. R Burton, Physica B150, 132 (1988). T. Mohri, K. Terakura, T. Oguchi and K. Watanabe, "Phase Transformation '87" (Ed.G.W. Lorimer, The Institute of Metals, 1988) 433. K. Terakura, T. Mohri and T. Oguchi, Mat. Sci. Forum., 37, 39 (1989). A. Finel, V. Mazauric and E Ducastclle, Phys. Rev. Lett., 65, 1016 (1990). J. M. Holender, Phys. Rev. B41, 8054 (1990). G Inden and W. Pitsch, "Atomic Ordering," in Phase Transformations in Materials (R Hassen, ed.), pp.497-552, VCH press, New York (1991). A. Finel, "Statistics and Dynamics of Alloy Phase Transformations" (RE.A. Turchi and A. Gonis, eds.), NATO ASI Series, Plenum Press, New York. C. Bichara, S. Crusius, and G. Inden, Physica B179, 221 (1992); ibid B182, 42 (1992). R. Kikuchi and A. Beldjenna, Physica A182, 617 (1992). M. Asta, D. de Fontaine, and M. van Schilfgaarde, J. Mater. Res. 8, 2554 (1993). E Clerli and V. Rosato, Phys. Rev. B48, 22 (1993). A. Finel, Prog. Theor. Phys. Suppl., 115, 59 (1994). D. de Fontaine, Solid State Physics, 34, 73 (1979); ibid, 47, 33 (1994). R. Kikuchi, Prog. Theor. Phys. Suppl., 115 1 (1994). H. Onodera, T. Abe, and T. Yokokawa, Acta Metall. Mater. 42, 887 (1994). T. Horiuchi, S. Takizawa, T. Suzuki, T. Mohri, Metall. Mater. Trans. 26A, 11 (1995). M. H. E Sluiter, K. Esfarjani and Y. Kawazoe, Phys. Rev. Lett., 75, 3142 (1995). K. Masuda-Jindo, R. Kikuchi and R. Thomson, "Theory and Applications of the Cluster Variation and Path Probability Methods", edited by J.L. Moran-Lopez and J. M. Sanchez, Plenum Press, New York, 299 (1996). R. Kikuchi and K. Masuda-Jindo, Comp. Mater. Sci., 8, I (1997). V. L. Vinograd, S. K. Saxena and A. Putnis, Phys. Rev. B56, 11493 (1997). K. Masuda-Jindo and R. Kikuchi, Surf. Sci., 399, 160 0998). V. L. Vinograd and A. Putnis, Phys. Chem. Minel. 26, 135 (1998). R. Kikuchi and K. Masuda-Jindo, Comp. Mat. Sci., 14, 295 (1999). G Treglia, B. Legrand, F. Ducastelle, A. Saul, C. Gallis, I, Meunier. C. Mottet and A.Senhaji, Comp. Mat. Sci., 15, 196 (1999). R. Kikuchi and C.M. van Baal, Modell. Simul. Mater. Sci. Eng., 8, 251 (2000). T. Mohri, Modell. Simul. Mater. Sci. Eng., 8, 239 (2000). Y Rosenfeld, Phys. Rev. Lett., 84, No. 19, 4272 (2000). M. Shimono and H. Onodera, Phys. Rev. B61, 14271 (2000). T. Mohri, Y. Chen and T. Atago, to be published (2001).


Table 1. Number of small mesh points inside an nb sphere and the mesh size b/a when the sphere redius is 0.4a

n 2 3 4 5 Number of points 33 123 257 51 b/a=O.4/n 0.2 0.133 0.1 0.08

Table 2. Parameters for the phase diagram of FCC binary alloys.

Clio e12o e22o rHo r~2o 1"22o a 3.00 1.00 3.00 1.00 1.00 1.00 b 3.00 1.00 3.00 1.00 1.005 1.01 c 3.00 1.00 3.00 1.00 1.01 1.02 d 3.00 1.00 2.90 1.00 1.01 1.02 e 3.00 1.00 2.90 1.00 1.00 1.00 f 3.00 1.00 2.739 1.00 1.03 1.06

Table 3. Parameter values for the many body potentials, Cu-Cu row is for the alternative choice. Cu-Au values are for the L12 structure.

Ao(eV) ~0(eV) po q0 ~(O) Cu-Cu 0.0905 1.243 10.68 2.32 2.5562 Cu-Au 0.1539 1.5605 11.05 3.0475 2.641 Au-Au 0.2061 1.790 10.229 4.036 2.8843 Cu-Cu* 0.0855 1.224 10.960 2.278 2.5562

Table 4. Interatomic distance r0(13) in the bulk CuAu alloy with Xcu=0.5

T(K) nrad=4 1400 2.79011657 1000 2.78744563 600 2.78108819

-0.055010 -0.054485

nrad=2 i ~ 2.79703355 -0.058681 2.83741632 0.078752

-0.053285 2.85369436 -0.086063

Table 5. The pressure p as a function of density (r0 "3) calculated by quartet treatment of CD-CVM using the Lermard-Jones potential (kT/c=l.5)

r0 "3 0.05 0.1 0.15 0.25 0.3 0.35 0.4 0.45 0.48 0.5 0.56 0.58 0.6 Pr03/e 0.07 0.115 0.14 0.17 0.16 0.14 0.13 0.121 0.12 0.121 0.135 0.155 0.165


a)

b )

• • • • t • • • j • •

" : ". ". i , .~i ." ." : : ' . : : : : : : :..: :.."

t • • , • .

• ! • • •

• • • •

• . ~ : . . .

• • • • * ° • * * • •

• • •

• t •

Q

• • • • • • • •

o o • , t , • * • . •

• . . . . . . . . . ' . . ' . . : : j . ~ . . . . . . e . - , . . • •

• • • •

• • • • • •

a ~-

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ooooo ooooo

0 0 0 0 0 0 - 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Fig. 1 The concentric circles of radius nb, b being an appropriate length: 8n points on the n-th circle.


a) Z

Y

b)

Fig. 2 Sketch of the cubic mesh points within the sphere used for 3D discrete CVM (a). Relative atomic displacements in the CD-CVM scheme in bcc lattice (b).

52 R. KIKUCHI AND K. MASUDA-J INDO

a)

b)

R2z

R2x Fig. 3 Twofold rotation symmetries R2×, R2y and R2z of the fee lattice (a), to be treated as Lagrange multiplers in the CD treatment of CVM. (b) shows the tetrahedron symmetry in the quasi-chemical approximation.

t g

~ 1.06

1.05

0 1.04 ",~

¢~ 1.03 X

1.02

~ 1.01

[-" 1.0

a)

12-6 L.-J

0.05 0.1 0.15 kT/~ --*

1.025

1.020

1.015

1.010

1.005

1.000

0.995

1 b) o pair /

D qch tetrahedron I - - - MD O,~

exp "" / / B ]

0."

/ z l ~ ' l

i ~ s J ~

Cu I I I I

200 400 600 800 Temperature (K)

1.025

1 .020

1.015

1.010-

1.005 -

1 . 0 0 0

0.995

c) o/I

Au I I I I

200 400 600 800 Temperature (K)

Fig. 4 Thermal lattice expansions calculated by CD scheme of CVM for the crystals with 12-6 type of Lennard-Jones potential (a), Cu (b) and Au (c). For Cu (b) and Au (c) metals, the electronic many body potentials are used.


[ . - -

.

5'4 ............. *'"

.

,

8-4

O ~ 0 0.2 0.4 0.6 0.8 1.0

X 8

Fig. 5 Phase diagrams of the four cases of Table 1 with an exception b/a=0.075(0.08) for n=5, 2, 4 and 3. The upper two curves are calculated by 8-4 and 10-5 types of Lennard-Jones potentials.

5.5

5.4

5.2

5.1

5 0.1

a)

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

XB---~

5.5

t ~o

5.4

5.3

5.2

5.1

5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

X B --~

Fig. 6 Phase diagrams of size-different binary parameters presented in Table 2. The curves in 6(a) from bottom to top are (a), (b) and (c) in Table 2, while the curves in 6(b) from bottom to top are (e), (d), (a) and (f) in Table 2, respectively.

0.9

a)

R. KIKUCHI AND K. MASUDA-JINDO 54

7

6

5

--~4

3

2

1

0 0,0

b) 7 .

6-

5

?-.4

3

2

I

0 0.0 0'.2 ' 0'.4 ' 0'.6 ' 0'.8 ' 1.0 012 014 016 018 1.0

XB_..~ XB---~

Fig. 7 The order-disorder phase diagrams of B2 alloys. The upper and lower curves are calculated by conventional CVM and CD scheme of the CVM, respectively.

0.25 . . . . , . . . . , . . . . , . . . . , . . . . , . . . . , . . . .

d:Z

<

0.2

0.15

0.1

0.05

-0.05

-0,1

O 600K --fi3-- 1000K

, - -×- - 1400K

. . . . I . . . . I . . . . I . . . . I . . . . I . . . . I . . . .

0 1 2 3 4 5

Surface Layer

Fig. 8 Calculated surface compositions of CuAu alloy calculated by CD scheme of CVM..

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Cluster Variation Method in the Computational Materials Science