VI. - THEORETICAL MODELS FOR ORDERING AND KINETICSTHE CLUSTER VARIATION METHOD

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VI. - THEORETICAL MODELS FOR ORDERING

AND KINETICSTHE CLUSTER VARIATION

METHOD

Ryoichi Kikuchi

To cite this version:

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THE CLUSTER VARIATION METHOD

Ryoichi KIKUCHI

H u g h e s liesearch Litboratories, Malibu. C'al~Cc)rn~a 90265. L.S.A.

RCsum6. - La mtthode Cluster Variation (CV) constitue une hierarchie d'approximations ana- lytiques utilisee pour I'analyse des phenomtnes cooptratifs. On Ccrit I'tnergie libre en fonction d'un ensemble de variables (qui specifient l'ittat du systtme), et I'on minimise I'tnergie libre en vue de dkterminer I'etat le plus probable, c'est-a-dire, I'etat d'tquilibre d u syst6me. La methode de Bragg et Williams et la mCthode quasi chimique peuvent &tre considerees comme cas particuliers de cette hierarchie d'approximations. A titre d'exemple, on discute le systkme Cu-Au dans le detail, et I'on compare les diagrammes d'equilibre theoriques obtenus par diffkrentes methodes. On presente Cgalement la mtthode d'ItCration Naturelle, algorithme indispensable a la r6solution des tquations non lineaires simultanees qui resultent de la minimisation de I'energie libre par la mtthode CV. Enfin, o n decrit la methode Puth Probability qui constitue une extension dans le temps de la

mithode CV.

Abstract. - The Cluster Variation method is a hierarchy of closed form approximations of cooperative phenomena. It writes the free energy in terms of an appropriately chosen set of variables (which specify the state of the system) and minimizes the free energy to find the most probable. i.e., the stable, state of the system. The Bragg-Williams approximation and the quasi-chemical method use the same conceptual approach. The Cu-Au alloy is discussed as an example in detail, and theoretical phase diagrams of the system by different methods are compared.

An explanation is given for the Natural Iteration method, which is an indispensable accompanying mathematical tool in solving the equations which result from minimizing the free energy in the CV method. The Path Probability method, which is a time-dependent version of the CV method, is discussed.

1. Introduction. - The cluster variation (CV) method is a hierarchy of closed form approximations of cooperative phenomena. It includes the Bragg- Williams approximation and the Bethe approximation. Historically, the concept of the general CV approach was conceived as a generalization of Takagi's formulation of the Bethe approximation.

The quasi-chemical method of Guggenheim is also regarded as a generalization of Bethe's approximation. If we interpret the CV method in a narrow sense, it is superior to, i.e., more efficient than, the corresponding quasi-chemical method. In a wider interpretation of the CV method, however, it can include the quasi- chemical method as a special case.

The methods mentioned so far are for equilibrium states. The general CV consideration is of such a character that it can be generalized to time-dependent cases. In order to distinguish the latter extension of the CV thinking, we will call the time-dependent version the path probability method, which we discuss in Section 7.

2. The Bragg-Williams approximation. - In order to present the basic concept of the CV method without unnecessary complication, in this and the following three sections, we will use the Ising model terminology.

The Ising model is equivalent to the 50-50 composition binary alloy ordering system of two sublattices. The advantage of the Ising model is that it is not necessary to use sublattices.

In a system of N spins, the number of

+

spins and that of

-

spins are written as x, N and x, N , respec- tively. The interaction potential is such that it is

- E for a ( + +) pair or a (- -) pair, and

+

E for a

(

+

-

) pair.

The Bragg-Williams (B-W) approximation is sum- marized in the following expression of the free energy F of the system :

where E(xl, x,) is the energy of the system characte- rized by the parameters (x,, x,). The Boltzmann factor exp[- E(xl, x,)/kT] is proportional to the probabi- lity that one of the states specified by (x,, x,) appears in the equilibrium state. T o multiply the Boltzmann factor by the combinatorial factor (which is the number of ways of randomly distributing x, N

+

spins and x, N - spins over N lattice points) makes the right-hand side proportional to the probability that

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C7- 308 R. KIKUCHI

any one of the states specified by ( x , , x,) appears in the equilibrium state.

The approximate nature of the B-W method lies in writing E(x,, x 2 ) as

E ( x , , x , ) = a,N

2

cij xi x j ( 2 . 2 ) i = 1 j = 1

where

c l l = s 2 , = - s and E , , = s , , =

+

E ( 2 . 3 )

and 2 o is the coordination number of the lattice. The energy expression ( 2 . 2 ) is approximate, because the number of nearest-neighboring (i. j ) pairs is in general not equal to xi xi. However, it may be noted that the combinatorial factor in ( 2 . 1 ) is exact.

3. The pair approximation. - The approximation in the B-W treatment originated in the energy expression. There is, however, a way to write the energy expression exactly. For that purpose we introduce variables y i j which indicates the probability of finding an (i, j) pair on any nearest-neighboring bond. Then the energy expression corresponding to ( 2 . 2 ) is

and now this energy expression is exact.

The free energy F corresponding to ( 2 . 1 ) is then written in the form

e-F1kT = Max SZ (

vij

) exp[- E { y i j ) / k T ] ( 3 . 2 )

( ~ 8 , )

where SZ { yij ) is the number of ways of distributing { y i j ) pairs over the o N nearest-neighboring bonds. It is convenient to introduce S { y i j ) and write the weight factor SZ ( yij j as

52 { Yij

1

= exp[S { Yij Ilk]. ( 3 . 3 )

We then call S { y,, } the entropy of the system specified by the variable set { yij }. Using ( 3 . 3 ) in ( 3 . 2 ) , we can rewrite

This expression is exact for a linear system (o = I),

but is approximate for w

>

1. The formulation based on the SZ f y j j ) expression of ( 3 . 6 ) is called the pair method. The form of SZ ( y i j ) in ( 3 . 6 ) was first derived by Takagi [I]. It was derived later by several different authors [2-51. For the reader of this article, references

[2] and [3] are particularly recommended for additional helpful information.

The free energy F { y i j ) based on the set of variables ( y i j ) is written using E { yij ) of ( 3 . 1 ) and

derived from ( 3 . 6 ) . The values { y$) ) for the equilibrium of the system are calculated as those which minimize F { y i j ). We will explain the mathematical steps in Section 5.

The equilibrium state thus calculated using l2 { yij ) in ( 3 . 6 ) agrees, except for special cases, with that which Bethe [6] has derived using a different approach before Takagi [ I ] , and thus the word pair approximation is often used interchangeably with the Bethe approximation.

4. Tetrahedron approximation for f.c.c. - The scheme of finding the equilibrium state as the most probable state is a variational problem. In the B-W

method of Section 2, which we may call the point approximation, the free energy is written as F ( xi ) in terms of the two point variables ( x , and x,), and in Section 3 F { pij } uses four pair variables, y , , ,

y l z, y z , .and y2,. We can expect F ( y i j ) to give a better approximation than F f xi } because the former is formulated with a larger number of variables.

e-F1kT = exp[- Min F { yij ] / k T ] (3.4)

where

F { y i j } = E ( ' y i j j - T S { y i j ) ( 3 . 5 )

FIG. 1. - The basic tetrahedron i-j-k-/in the f.c.c. structure.

can be called the free energy of the system specified by the variable set { y i j j .

When we formulate the free energy using the pair

variables { y,, ), we write the weight factor SL ( yij ) as For an f.c.c. structure, a still better approximation is obtained when we use a tetrahedron made of four

[II

( x i N ) !] nearest-neighboring points, as shown by i-j-k-1 in

i figure 1. The probability variables for the tetrahedron

Q

t

vij

1

=

n n

( y i j N ) !

"

N ! O w ' '

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in (3.1). The entropy is derived by the following weight factor :

The derivation of this expression is found in references [2] and [3].

At this point we show the comparison of the three approximations we have discussed so far. The problem is the Ising model in the f.c.c. lattice (o = 6), and figure 2 shows the specific heat c per spin nor- malized to the Boltzmann constant k. It is empirically accepted (there is no rigorous proof) that as the approximation improves, the Curie temperature Tc decreases and also the discontinuity in cat Tc increases. Based on these rules, we can see how the approximation improves as we go from the point (B-W) to the pair and then to the tetrahedron approximation. The bestxstimate of Tc based on the high and low temperature series expansions

[ A

is also indicated in figure 2. At the exact Tc, it is expected that the specific heat c goes to infinity.

The Curie poinl kTC/c calculated by drfferent methods Method

-

Bragg Williams Pair Approximation Double-Tetrahedron [9] Tetrahedron Approximation [2]

Octahedron plus Tetrahedron [8]

Octahedron plus Double-Tetrahedron [9] (( Exact )) [7]

The approximations using clusters larger than the tetrahedron, as listed in table I, are not merely of interest in getting more accurate treatments, but are also of great significance because they are now open- ing up a way of taking into account interactions of longer range than those of the nearest neighbors. Work in this; area is now in progress.

It is worth commenting on the quasi-chemical method. A tetrahedron approximation was proposed by several authors [lo-121 as a generalization of the quasi-chemical method interpretation of the pair a ~ ~ r o x i m a t i o n .

.

A Their treatment is based on the same Results of a closed form approximation better than _wij,,_of_{this section, but starts with the}_Q_{factor of} the tetrahedron have recently been reported [8]. This _{the form}

approximation is based on a combination of a tetrahedron and an octahedron. A still better approxima-

tion has also been calculated by Sanchez at UCLA [9]

[I!

(xi N ) !]

using a combination of an octahedron and a double- ₅₂_{_{~ i j k l . )}₌

[

n

_{( w i j k i}_{N ) !:} _N_!'

I

( 4 . 2 )

tetrahedron. The numbers are listed in table I. i.j,k,l

0

0 2 4 6 8 10 12 14

r = k TIC

-

THE 6-W APPROXIMATION

---

THE PAIR APPROXIMATION

THE TETRAHEDRON APPROXIMATION

---

THE WSlTlON OF THE BEST ESTIMATE OF THE CRITICAL POINT

FIG. 2. - The specific heat c per lattice point for the f.c.c. cubic lattice (Ising model) against 7 = kT/c.

As will be shown in Section 6,a in ( 4 . I) gives a better approximation than ( 4 . 2 ) . From the point of view presented at the very beginning of the Introduction, we can interpret that ( 4 . 2 ) is simply one of the possible forms of 52 in the CV hierarchy.

5. Natural iteration method. - In the example of the tetrahedron approximation in Section 4 , the number of variables wijk, we work with is 24 = 16.

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C7-310 R. KIKUCHI

In the tetrahedron approximation, when the free for the ordered state, whatever the temperature may energy is minimized with respect to the probability be. Compared with such an easy choice of the initial variables wiik1, we obtain the following set of equa- values for the NI method, the Newton-Raphson (NR)

tions [I31 : iteration method requires a more accurate guess of

the starting values. For example, fbr very high tempe-

( ~ i j Yik Yil Yjk Yjl ykr)'" ratures, (5.5~2) gives good starting values for the NR

xi x , xk x,)'/? method; temperature may be lowered step by step

in the NR procedure; however, right below the

for i7 k 9

'

= and 2 . (5 '

transition temperature, the xi and yii change so rapidly

In this equation and in the rest of the paper, we use for a decrement the it is

usually difficult to find the starting values of xi and

/?

=

(kT)-'

( 5 . 2 ) yij for the N R method.

The variable I is a Lagrange multiplier for the normalization :

and when the free energy is a minimum,

1

is equal to the free energy per lattice point. The .parameter eijfl is the energy per tetrahedron, and xi and y i j are those defined in Sections 2 and 3, respectively. The relations (5.1) are accompanied by the following reduction relations :

The NI method works as follows. We fix a temperature, and then give a set of guess values f xi ) and ( y i j ). Using these values on the right-hand side of (5. l), we calculate wij, e-BY2. Then

gAi2

is evaluated from the normalization (5.3), and thus the output set f wij,, ) is obtained. This output set leads to { xi ) and { j l i j } from (5.4), which are used as the second

input set of the iteration.

The NI method has the following noteworthy features :

(1) The iteration always keeps xi, y i j and wijkl positive, however small some of them may become.

(2) The iteration always converges whatever initial (naturally positive) values of x i and yij it starts with.

(3) At each step of the iteration, the free energy always decreases. This can be proved analytically. (4) As the iteration progresses, xi and y i j approach their respective converged values exponentially. Using this property, it is possible to stop the iteration at a certain stage and extrapolate the xi and y i j to the converged values with great accuracy.

For the Ising model calculation, for example, we can always start with the initial values

6. Phase diagram calculation.

-

The CV method, together with the NI technique, is particularly useful in computing phase diagrams.

The conventional method of calculating phase equilibrium of two coexisting phases is to calculate the free energy as a function of the composition, and then determine a common tangent BF, as shown in figure *3. Instead of following this procedure, the CV-NI method finds the equilibrium value of the grand potential G defined as

whei.e pi is the chemical .potential of the i-th species. In the CV-NI method, is minimized with respect to the state variables keeping pi (and also temperature) fixed rather than keeping the composition fixed. Then the diagram corresponding to the conventional h-c diagram of figure 3 is the

G-p

diagram illustrated in figure 4 (for the binary case). Points of the same name correspond in the two figures. It is noteworthy that the two points B and F at which the common tangent touches the curve become one point written as B(F) in figure 4. Since it is much easier to compute

a point at which two curves cross than to draw a common tangent, the G-,u construction in figure 4 is

superior (with less labor and leading to more accuracy) to the common tangent method of figure 3. Figures 3 and 4 are illustrations for the case of separation of two phases. When an ordered phase is

I J

COMPOSITION, c

for the disordered state, and _FIG._{3. -The} _F_VS._c_{curve for the two-phase separation case.}

The two states B and F at the common tangenf coexist. Points C

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CHEMICAL POTENTIAL. 9

FIG. 4. -The G vs. p curve corresponding to figure 3. Here p is actually p = p,

-

p2 of (6.1). The letters correspond to those in figure 5. Note that B and F become the same point. Also note

that the spinodals C and G arc the tips of thorns.

involved, the shape of the curves is somewhat different; however, it remains the same that the two coexisting phases are determined either from the common tangent construction like the one in figure 3

or as the point where two curves cross like the B(F) in figure 4.

We can calculate a phase diagram of the Cu-Au system using the tetrahedron approximation of (4.1) and can compare the diagram with those calculated by other methods. Figures 5 through 8 are these results. Historically, the first attempt to calculate the Cu-Au diagram was done using the B-W approxima- tion by Shockley [14]. His curves are reproduced in figure 5, in which the Cu3Au and the CuAu phases overlap and share only one peak. It has been known that the pair approximation does not lead to any

-

B - W PHASE DIAGRAM

. . .

Tc FOR A 3 B

----

Ti FOR A 3 B . AB

stable ordered phase. Figure 6 shows the result [lo] due to the SL { wij,, } expression of (4.2), which was derived by extending the quasi-chemical thinking. Figure 6 can separate the Cu3Au and CuAu phases, but they are distinctly separated down to T = 0.

Golosov et al. [15], in Russia and van Baal [la inde- pendently used the SZ { wij,, } of (4.1) and derived the

FIG. 6. -Phase diagram of Cu-Au type system calculated by the tetrahedron approximation using Eq. (4.2), Ref. (101.

0 0 0 25 0 5 0 ? 5 1 0

A _A36 _A6 _A83 B

CONCENTRATION B

FIG. 7. - Phase

the tetrahedron

diagram of Cu-Au type system calculated by approximation using Eq. (4. I), Refs. [13], [IS]

and 1161.

0

0 20 00 60 80 100

ATOMIC PER CENT Au

FIG. 8. - Phase diagram of Cu-Au system calculated by the

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C7-312 R. KIKUCHI

diagram in figure 7 The Cu3Au and CuAu phases touch, as experiments show. The anisotropic four-

bod! interactions suggested by van Baal 1161 can make the Cu3Au side and CuAu, side asymmetric, and the result recently completed 1 1 1 is shown in figure 8. An experimental diagram taken from Hansen [18]

is drawn in figure 9 for comparison. The agreement between the curves in figures 8 and 9 is good. Compa- rison of the five figures 5 through 9 is a clear demons- tration how each approximation behaves.

FIG. 9. - Experimental Cu-Au phase diagram, Ref. 1181.

The basic reason why the tetrahedron approximation is needed for the f.c.c. ordered phases is the geometry of the lattice [19]. In this lattice, two nearest neighbors of a lattice point can be nearest neighbors of each other. This situation makes the pair approximation very poor.

7. The path probability method.

-

As we mentioned in Section 2, exp(- FIkT) is the relative probability that the system is found in any one state specified by the assigned variables ({ yij ) or { wijkl }, for example). The equilibrium state is the most probable state, and the mathematical device of statistical mechanics finds the state which maximizes the probability of occurrence, exp(- FIkT).

In kinetic problems, we can use a similar proba- bilistic reasoning to derive how the system changes [20]. When the state of a system at time t is specified by, for example, a set of the pair variables { yij(t) ), we formulate the probabiIity of a path going from the set { yi,(t)

j

to another set ( ykl(t

+

At) ) in a short time interval At. In writing the path probability P,

we introduce a set of new variables which we call the path variables { YijVk,(t, t

+

At) ). We define

oNYij,,,(t, t

+

At)

to be the number of pairswhich was i - j at t and has changed to k - I at t

+

At ( o N being the total number of pairs in the system). When At is infinitesimally small, it is safe to assume that at most only one of the second subscripts k and I is different from the first subscripts i and j. In YijVil, for example, j changes into 1, while i remains unchanged; further, where atoms migrate via the vacancy mechanism, one of the changing subscripts ( j or 1 in this example) represents a vacancy.

The path probability P is made of two factors. The first factor

P,

corresponds to S1 ( yij ) in (3.2), and has the following form in the pair approximation :

The variable Xi,,(t, t

+

At) is a generalization of x ,

in (3.6) and is the probability that the configuration of a lattice point changes from i to k in At (i = k is also allowed). As in (3.6), the coordination number is written as 2 a, and N is the number of lattice points in a system. The second factor

P,

in the path probability is the product of jumping probabilities and factors containing activation energies of atomic jumps; it can be regarded as the a priori probability factor in the same sense that the Boltzmann factor in (3.2) is the apriori probability for a state of energy E. When the initial state { yiJ(t) ) is given, the state

{ ykl(t

+

At) ) into which the system changes can be found by maximizing the path probability

P { Yij,,(t, t

+

At) ) = PI

P,

with respect to Yij,, under the constraints :

Applications of the path probability method (PPM) are reported in this conference in two papers by Sato and Kikuchi, one on diffusion and ionic conductivity in solid electrolytes and the other on relaxation of order in an ordered binary alloy.

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References [ I ] TAKAGI, Y., Proc. Phys. Math. Soc. Japan 23 (1941) 44.

[2] KIKUCHI, R., Phys. Rev. 81 (1951) 988. 131 BARKER, J. A,, Proc. R . SOC. A 216 (1953) 45.

[4] HIJMANS, J., DE BOER, J., Physicu 21 (1955) 471, 485 and 499.

[5] GUWENHEIM, E. A,, Mixtures (Oxford University Press) 1952. 161 BETHE, H. A,, Proc. R. SOC. A 150 (1935) 552.

[7] Phase Transition and Critical Phenomena, Vol. 111, Ed. C. Domb and M. S. Green (Academic Press, New York) 1974, p. 425.

[8] AGGARWAL, S. K. and TAMAKA, T., Bull. Am. Phys. Soc. 22

(1976) 376.

191 SANCHEZ, J. M., private communication. [lo] YANG, C. N., J. Chem. Phys. 13 (1945) 66 ;

LI, Y. Y., J. Chem. Phys. 17 (1949) 447. [Ill HILL, T. L., J. Chem. Phys. 18 (1950) 988.

[12] GUGGENHEIM, E. A. and MCGLASHAN, M. L., Proc. R. SOC.

A 206 (1951) 335.

[I31 KIKUCHI, R., J. Chem. Phys. 60 (1974) 1071. [I41 SHOCKLEY, W., J. Chem. Phys. 6 (1938) 130.

[I51 G o ~ o s o v , N. S., P o ~ o v , L. E. and PUDAN, L. Ya., J. Phys. & Chem. solid^ 34 (1973) 1149 and 1157.

[I61 VAN BAAL, C. M., Physica 64 (1973) 571.

1171 DE FONTAINE, D. and KIKUCHI, R., Fundamentol Calculations

of Coherent Phase Diagrams, National Bureau of Standards Conference on Phase Diagrams, January 10-12, 1977, Gaithersburg, Maryland.

1181 HANSEN, M., Constitution of Binary Alloys (McGraw-Hill, New York) 1958.