## Minimization of the Free Energy under a Given Pressure

## by Natural Iteration Method

### Naoya Kiyokane

1;*_{and Tetsuo Mohri}

2;3
1
Division of Materials Science and Engineering, Graduate School of Engineering, Hokkaido University, Sapporo 060-8628, Japan
2_{Division of Materials Science and Engineering, Faculty of Engineering, Hokkaido University, Sapporo 060-8628, Japan}

3_{Research Center for Integrative Mathematics, Hokkaido University, Sapporo 060-8628, Japan}

Natural iteration method within the Cluster Variation Method is coupled withzero potentialto minimize the free energy under the constraint of a constant pressure. The calculated results reproduce the ones obtained by the conventional method in which a gradient of the free energy curve is estimated at each atomic distance. It is noted that the computation time is much reduced in the present scheme and, therefore, it has a potential applicability in the future calculations with a free energy containing a larger number of variables.

[doi:10.2320/matertrans.MBW201011]

(Received October 29, 2010; Accepted November 26, 2010; Published February 25, 2011) Keywords: cluster variation method, natural iteration method, zero potential, simple cubic lattice

1. Introduction

Among various approaches to calculate conﬁgurational
entropy, Bragg-Williams approximation1) _{has been often}
employed to calculate phase equilibria and phase diagrams
for binary and multicomponent systems.2–5)The advantages
of Bragg-Williams approximation are summarized as its
physical transparency, mathematical simplicity, and
expan-sibility to multicomponent systems. However, the order of
phase transition has been often misled by Bragg-Williams
approximation, which is ascribed to the oversimpliﬁed free
energy expression. In order to improve this approximation, it
is necessary to consider the atomic correlation which plays an
essential role in the conﬁgurational entropy.

Cluster Variation Method (hereafter abbreviated as CVM),
which was devised by Kikuchi,6) _{includes wide range of}
atomic correlations and has been accepted as one of the most
reliable theoretical tools to approximate the conﬁgurational
entropy. A free energy of the CVM is expressed as a function
of a set of cluster probabilities, which is formally written as

F¼ fðfxig;fyijg;fzijkg; Þ ð1Þ

wherei,jandkspecify the atomic species,xi,yij, andzijkare

cluster probabilities of a point, pair and three-body clusters, respectively, for a conﬁguration speciﬁed by subscript(s). When a free energy of CVM is only described as a set of point cluster probability, fxig, which corresponds to the

Bragg-Williams approximation, the minimization of the free energy is a simple problem. However, a more general expression given by eq. (1) is cumbersome since it involves a number of cluster probabilities and they are not mutually independent in the thermodynamic conﬁguration space. As an eﬃcient method to minimize the free energy of CVM, Kikuchi developed Natural Iteration Method7)(hereafter abbreviated as NIM).

Newton-Raphson method has been widely applied to minimize a free energy. However, in order to minimize the

free energy of the CVM by using this method, it is necessary
to convert the cluster probabilities to correlation
func-tions8–10)_{which form a set of independent variables, followed}
by the calculations of the ﬁrst and second order derivatives of
the free energy with respect to correlation functions. Since
CVM free energy is a multivalued function of correlation
functions the second order derivative is represented by a
matrix, while matrix inversion required by the
Newton-Raphson method is not trivial. An alternative way, NIM, is a
special iteration procedure unique to the CVM. NIM requires
neither the correlation functions nor the inversion of the
derivative matrix. It is further noted that the free energy
always decreases as the iteration proceeds and the
conver-gence of the iteration is guaranteed.7)_{Hence, a search of free}
energy minimum is easier compared to Newton-Raphson
method and NIM has been widely used for the minimization
of free energy of the CVM.

In order to formulate the internal energy, phenomenolog-ical atomic potentials are often employed. For example, Lennard-Jones type pair potential in which two potential parameters are determined from experimental cohesive energy and lattice constant is applied to calculate phase boundaries and it is conﬁrmed that the calculated phase diagram well reproduces the experimental one.11)Hence, in the present study, we employed a Lennard-Jones type pair potential (hereafter abbreviated as L-J potential) of which details will be described shortly.

Unlike the conventional procedure, to minimize the free energy solely with respect to cluster probabilities, the employment of L-J potential demands additional minimiza-tion with respect to volume, since the internal energy is explicitly given as a function of volume (equivalently atomic distance). Hence, the constancy of the pressure is imposed as an additional constraint. The conventional minimization procedure for such a free energy proceeds with two steps; one is the minimization of the free energy for each atomic distance, and the second one is the calculation of a pressure from the gradient of minimized free energy at each atomic distance. The procedure is repeated until the imposed

*_{Graduate Student, Hokkaido University}

pressure is attained. Such two-folds minimization process demands heavy computations. In order to reduce the amount of calculations, we propose an alternative procedure of minimization of the free energy under the constraint of a constant pressure in the present report. The core of the procedure is to employ NIM coupled withzero potential.

The organization of the present report is as follows. In the next section, the conﬁgurational free energy within the pair approximation of the CVM is introduced for the sake of completeness. The present minimization procedure and the conventional procedure are also summarized in the next section. The results are demonstrated and discussed in the third section followed by the conclusions in the last section.

2. Theoretical Calculation

2.1 Conﬁgurational free energy within cluster variation method

[image:2.595.104.232.71.137.2]For the sake of simplicity, the ordered phase on a simple cubic lattice is considered in the present report. A and B atoms are arranged alternatively in the nearest neighbor lattice points and the stoichiometric composition of the present ordered phase is ﬁfty atomic percent. The entire lattice is divided into two sublattices and as shown in Fig. 1 where ðÞ-sublattice is deﬁned as the preferential lattice point for AðBÞ atom. Then, the long range order (hereafter abbreviated as LRO) parameter,, is deﬁned as

¼x_{A}x_{A}; ð2Þ

wherex_{A} is the cluster probability of ﬁndingAatom in the
sublattice designated by.

The accuracy of the conﬁgurational entropy is improved by introducing wider range of atomic correlations. However, the pair approximation is suﬃcient to develop a new minimization procedure under a given pressure, and the conﬁgurational entropy of pair approximation is given as

S¼kB 5

2 X

i

Lðx_{i}Þ þ5

2 X

i

Lðx_{i}Þ 3X
i;j

Lðy_{ij} Þ þ2

( )

ð3Þ

where kB is the Boltzmann constant, yij is pair cluster

probabilities of ﬁnding atomic arrangements on the sub-lattices speciﬁed by subscripts and superscripts, respectively, andLoperator is deﬁned as

LðxÞ xlnxx: ð4Þ

The derivation of eq. (3) has been amply demonstrated in the previous publications12)and is not repeated here. Note that the conﬁgurational entropy, the internal energy and the free energy are deﬁned per lattice point throughout the present publication.

The internal energy in the present study is assigned by the sum of nearest neighbor atomic pair interaction energies and is described as

EðrÞ ¼3X i;j

eijðrÞ y

ij ð5Þ

where 3 is one half of the coordination number for a simple cubic lattice,eijthe nearest neighbor atomic pair interaction

energy between speciesiandj, andris the nearest neighbor atomic distance. It is noted that the nearest neighbor atomic distance is equivalent to the lattice constant for the simple cubic lattice. Hence, together with the conﬁgurational entropy given by eq. (3), the Helmholtz energy is expressed as

F¼3X i;j

eijðrÞy

ij

kBT 5

2 X

i

Lðx_{i}Þ þ5

2 X

j

Lðx_{j}Þ 3X
i;j

Lðy_{ij} Þ þ2

( )

ð6Þ

whereT is the temperature. To preserve the symmetry, the subscript of the point cluster probability onsublattice was transformed fromiin eq. (3) to jin eq. (6).

For the nearest neighbor pair potential,eijðrÞ, the L-J type

potential is employed, which is described as

eijðrÞ ¼eij

r ij

r

m

m

n r

ij

r

n

ð7Þ

where e

ij and rij express the depth of the potential and

equilibrium atomic distance, respectively, and the exponents m and n are related with stiﬀness of the lattice against the compression and expansion, respectively. In the present study, m¼8 and n¼4 are assigned and the values employed for e

ij and rij are summarized in Table 1.

Throughout the present manuscript, the energy and the
atomic distance are normalized with respect toe_{AB}andr_{AB} ,
respectively. Because je_{AB}j is larger than je_{AA}j and je_{BB}j,
ordering reaction can be guaranteed.

2.2 Minimization procedure

Two minimization procedures under the constraint of a constant pressure are discussed in this section. One is the conventional procedure in which a gradient of the free energy curve is calculated at each lattice constant. The other is the present procedure which employszero potentialwithin NIM. All of numerical calculations in the present report are performed by a workstation, Intel Xeon 5160.

2.2.1 Conventional minimization procedure

In the conventional procedure, grand potential, , is formulated by the Legendre transformation on the Helmholtz energy, F, with respect to the number of particles, and is given as

α sublattice β sublattice

[image:2.595.306.549.85.137.2]Fig. 1 Ordered phase in the simple cubic structure.

Table 1 Lennard-Jones parameters employed in the present study.

ij e

ij r

ij

AA 0.85 0.95

AB 1.00 1.00

[image:2.595.308.549.357.396.2]ðT;V;figÞ ¼F X2

i

ixi ð8Þ

where i is the chemical potential of i species and V is

the volume which is equivalent to r3 _{for a simple cubic}

lattice. Based on the grand potential, the pressure, p, is derived as

p @

@V

T;fig

¼ dr

dV @

@r

T;fig

¼ 1

3r2

@

@r

T;fig

:

ð9Þ

The minimization procedure is as follows. Firstly, in
addition to the temperature and a set of chemical potential,
the pressure, pin and the atomic distance,rin, are speciﬁed.
Throughout the course of this study in (out) in the
super-scripts indicatesinput (output) value. Secondly, the nearest
neighbor atomic interaction energies between speciesiandj
are calculated by substituting rin _{into eq. (7). Thirdly, the}
grand potential is minimized by applying NIM, and the
iteration is repeated until the following criterion of
con-vergence,

c

X

i

jx_{i}inx_{i}outj þX

j

jx_{j}inx_{j}outj; ð10Þ

becomes less than 1012 _{for all the point clusters. When}
c does not satisfy the convergence criterion, the input
of atomic distance is slightly changed and the second
step and the third step are performed, using the output
cluster probabilities as the input cluster probabilities.
From these procedures, the gradient of the grand potential,

=rjr¼rin, is obtained for a given temperature and the

chemical potentials. By substituting the gradient of the
grand potential into eq. (9), the pressure at the speciﬁed
atomic distance, pout_{, is obtained. Then, the iteration}
proceeds until a convergence criterion for the pressure,

P jpinpoutj; ð11Þ

becomes less than107_{.}

2.2.2 Present minimization procedure

Zero potential13) _{is employed in this calculation. By}
Legendre transformation of the grand potential with respect
to the volume, zero potential is obtained as

ZðT;p;figÞ ¼þpV¼F X2

i

ixiþpV: ð12Þ

By substituting eq. (6) into eq. (12), zero potential of the present study is formulated. It is noted that the zero potential of the present calculation is also the function of the atomic distance since the internal energy is described as L-J type potentials.

The minimization procedure is as follows. Firstly, the
temperature, chemical potentials and pressure are speciﬁed.
Secondly, one gives initial guess for cluster probabilities, and
the atomic distance, rout_{, is calculated by substituting given}
cluster probabilities into ð@Z=@rÞT;p;fig¼0. Then, the

min-imization of the zero potential is performed in the same manner as the second and third steps of the conventional procedure described in the section 2.2.1, using rout as rin. These steps are iterated until the following criterion of the convergence is satisﬁed,

c

X

i

jx_{i}inx_{i}outj þX

j

jx_{j}inx_{j}outj

þX

i;j

jy_{ij} iny_{ij}outj<1012:

ð13Þ

The summary of these procedures is shown in Fig. 2.

3. Results and Discussion

Shown in Fig. 3 is the temperature dependences of the LRO parameter, in eq. (2), at a ﬁxed zero pressure and chemical potentials. The minimization of the free energy is performed by applying the present procedure described in section 2.2.2. Since LRO parameter behaves very sensitively to the change of the temperature around a transition temper-ature, we perform careful numerical calculations with high accuracy, and conﬁrm that the LRO parameter successively decreases as the temperature increases. This indicates that the order of the transition is of the second order, which agrees with the previous results.14,15)Hence, the transition temper-ature, kBTC¼0:449, is determined as the temperature at whichðTÞfalls zero, where the temperature is normalized by the L-J parameter,e

AB.

By repeating the same procedure for diﬀerent chemical potentials and pressure, one can derive a phase diagram. Shown in Figs. 4(a) and (b) are calculated phase boundaries at the normalized pressure p¼0:0 and p¼0:05, respec-tively. Squares in Fig. 4 are determined by the present procedure, while cross marks () are obtained by the conventional procedure described in section 2.2.1. One can conﬁrm that these two results coincide satisfactorily. Hence, the validity of the minimization procedure of the present method is conﬁrmed.

The computation time is much reduced in the present scheme. The present procedure requires about ﬁve hours to

0
,
,
=
∂
∂
*i*
*p*
*T*
*r*
*Z*
µ
*i*
*p*

*T*, , µ *x _{i}*α

*in*,

*x*β

_{j}*in*,

*y*αβ

_{ij}*in*

*out*

*r*
*out*
*ij* *r*
*e*
0
,
,
=
∂
∂
*i*
*p*
*T*
*ij*
*y*
*Z*
µ
αβ
*out*
*ij*
*out*
*j*
*out*

*i* *x* *y*

*x*α , β , αβ
if ∆*c*>ε

Fig. 2 Minimization procedure. In the present calculation, as the
con-vergence condition1012_{is assigned to}_{"}_{.}

0.00 0.20 0.40 0.60 0.80 1.00 0.00

Temperature, *kBT*

LR

O parameter

,

### η

0.20 0.40 0.60 0.80 1.00

[image:3.595.308.546.74.137.2] [image:3.595.341.514.193.314.2]determine one point in the phase boundary. While, the conventional procedure takes more than thirty times of the present one. This is because the conventional procedure requires minimization of the free energy with respect to cluster probabilities for each speciﬁed atomic distance. In contrast, the present scheme is very eﬀective for the case in which the atomic interaction energy depends on the atomic distance.

The transition temperature at p¼0:05 at the stoichio-metric composition is slightly higher than the one atp¼0:0. This is because the eﬀective nearest neighbor pair interaction energy, v2¼eAAþeBB2eAB, at p¼0:05 is higher than

the one atP¼0:0as shown in Fig. 5.

For the sake of simplicity, the conﬁgurational entropy in
the present report is given as the pair approximation within
the CVM. In order to improve the accuracy, it is necessary to
consider wider range of atomic correlations. However, the
number of variables in a free energy drastically increases and
the computational burden becomes heavier with the
intro-duction of a wider range of atomic correlations through larger
clusters. For instance, when the conﬁgurational entropy is
described as the cubic approximation for the simple cubic
lattice,14)_{the free energy is a function of 280 kinds of cluster}
probabilities. It is expected that the computation time is much
reduced by applying the present minimization procedure.

One of the deﬁciencies of the CVM is the fact that the local lattice distortion is not introduced. This is because the conﬁgurational entropy within the CVM is calculated on the assumptions that an atom always occupies a Bravias lattice point and that Bravias lattice uniformly expands or contracts despite the fact that an atom is displaced from a Bravias

lattice point16–18) _{depending upon the diﬀerence in atomic}
size of constituent elements. In order to introduce local
atomic displacement, Kikuchi and his coworkers devised
Continuous Displacement Cluster Variation Method19–21)
(hereafter abbreviated as CDCVM). In this method,
addi-tional points, which are termed quasi lattice points, are
introduced around a Bravias lattice point and an atom can be
displaced to those points. Displaced atoms in the CDCVM
are regarded as diﬀerent atomic species and the entropy
arising from the freedom of displacements is calculated by
mapping it onto conﬁgurational entropy of a multicomponent
system for which atoms are located only at Bravais lattice
point. Hence, the number of cluster probabilities far more
increases and the minimization of free energy becomes
furthermore cumbersome. Due to the computational burden,
CDCVM has not been well employed in the phase equilibria
calculations, however the present minimization procedure
may apply to the CDCVM and reduces the computation time.
Application of the present scheme to CDCVM is left for the
future investigation.

4. Conclusion

The minimization procedure of the free energy under the constraint of a constant pressure is investigated in the present work. This is performed by applying Natural Iteration Method coupled with zero potential. The calculated results reproduce the ones obtained by the conventional method in which a gradient of the free energy curve is estimated at each lattice constant, and the computation time is much reduced. The present scheme has a potential applicability for the minimization of a free energy with a larger amounts of cluster probabilities typically faced in Continuous Displacement Cluster Variation Method with less computational burden.

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0.350 0.400 0.450 0.500

0.30

Concentration,

*x*

_{A}T

emperature,

*T*

*kB*
(a)

0.350 0.400 0.450 0.500

0.30

Concentration,

*x*

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emperature,

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(b). Squares are obtained by the present procedure, while cross marks () are by the conventional procedure.

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