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Approximate solutions to the cluster variation free energies by the variable basis cluster expansion
Computational Materials Science 122 (2016) 301–306
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Approximate solutions to the cluster variation free energies by the variable basis cluster expansion J.M. Sanchez a,⇑, T. Mohri b a b
Institute for Computational Engineering and Sciences and Texas Materials Institute, The University of Texas at Austin, Austin, TX 78712, USA Institute for Materials Research, Tohuko University, Sendai, Japan
a r t i c l e
i n f o
Article history: Received 22 April 2016 Received in revised form 25 May 2016 Accepted 26 May 2016
Keywords: Cluster variation method Cluster expansion
a b s t r a c t In this paper we obtain approximate solutions to the Cluster Variation free energy by carrying out a cluster expansion of the probabilities appearing in the free energy functional in terms of concentration-dependent basis functions, and by truncating the expansion at different cluster levels prior to minimization. We show that a significant improvement over the Bragg–Williams approximation can be achieved by truncating the expansion of the cluster probabilities at relatively small clusters, thus dramatically reducing the number of equations that need to be solved in order to minimize the free energy. Furthermore, the free energy functional in the Cluster Variation Method offers a wellcontrolled case study to infer the effects of truncating the expansion of the energy of alloy formation in the commonly used Cluster Expansion method, versus the effects of truncating the expansion when using a concentration-dependent basis. Examples of the approach are given for simple Ising models for fcc- and bcc-based prototype alloy systems. Ó 2016 Elsevier B.V. All rights reserved.
1. Introduction Since its inception by Kikuchi in 1951, the Cluster Variation Method (CVM) has been widely used to accurately account for the effect of short-range order in the free energy of binary and multicomponent alloys [1]. While the accuracy of the CVM generally increases with the size of the largest cluster included in the approximation, so does the computational effort involved in the minimization of the free energy functional. With increasing cluster sizes, the minimization needs to be carried out with respect to a relatively large number of configurational variables, a number that grows exponentially with cluster size. Within mean-field theories, at the other extreme of computational convenience is the commonly used Bragg–Williams approximation. While a significant body of research and applications have shown that the shortcomings of the Bragg–Williams approximation can be compensated with other terms in the configurational free energy, it is apparent that the use of increasingly accurate ab initio methods to calculate energies of formations calls for a similar increase in the accuracy of the configurational entropy. A central task in the configurational theory of alloys is the description of functions of configurations in terms of a complete ⇑ Corresponding author. E-mail address: [email protected] (J.M. Sanchez). http://dx.doi.org/10.1016/j.commatsci.2016.05.035 0927-0256/Ó 2016 Elsevier B.V. All rights reserved.
set of configurational variables. A widely used approach to describe functions of configuration, formalized by Sanchez et al. [2], is the so-called Cluster Expansion (CE) method in which the functions are expanded in terms of pair and multisite correlation functions. These correlation functions form a complete and orthonormal basis in configurational space [2]. Subsequently, it was shown that the complete and orthonormal basis set introduced in Ref. [2], which will be referred here as the SDG basis, is a particular case of an infinite number of closely related complete and orthonormal basis sets in configurational space [3,4]. Presently, the CE in the SDG basis is the method of choice to obtain effective cluster interactions from the energies of a set of ordered compounds, which are typically calculated by means of some implementation of Kohn–Sham equations in Density Functional Theory. The approach is particularly appealing since, for the case of the energy of alloy formation, and after truncating the expansion at some maximum cluster size, the method casts the energy in the form of an Ising-like model with constant expansion coefficients. Such expansion coefficients are commonly referred to as Effective Cluster Interactions (ECIs). However, and despite the unquestionable success of the CE method in the SDG basis for the parametrization of the energy of alloy formation, the validity of truncating the expansion remains, at least at a fundamental level, an unresolved issue. In particular, if the energy has a nonlinear dependence in the concentration of the alloy, like it is the
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case for a regular solution model or for Fridel’s square band model for transition metal alloys, it is straightforward to show that the CE in the SDG basis does not converge [4]. The existence of an infinite number of basis sets provides a straightforward path to address non-linear concentration dependences in the energy of the system [3,4]. The method consists in matching the basis sets to the overall concentration of the alloy under consideration, and is akin to doing the expansion in a canonical ensemble as first proposed by Asta et al. [5]. This particular implementation of the cluster expansion will be referred as the Variable Basis Cluster Expansion (VBCE) [4]. Unlike the expansion in the SDG basis, the ECIs in the VBCE are concentration dependent. Although most current applications of the CE method are aimed at characterizing the energy of alloy formation, the method was initially motivated by the need to develop efficient algorithms to minimize the CVM free energy functional [7]. Such early studies were aimed at the calculation of prototype alloy phase diagrams based on Ising-like models for the description of the energy of formation, and on the CVM for the treatment of the configurational entropy [7–10]. The approach adopted in these early studies was to describe the probabilities appearing in the CVM configurational entropy in terms of a cluster expansion using the SDG basis functions, the expectation values of which are the so-called correlation functions. Since for a given cluster approximation the correlation functions form a complete and orthogonal basis set, they naturally constitute a full set of independent variables for the free energy minimization. In this paper we formulate the minimization of the CVM free energy in terms of a VBCE of the cluster probabilities, and show the effects of truncating the expansion at different cluster levels. In particular, the CVM free energy functional offers a well-controlled case study to investigate the truncation of the CE in the SDG basis versus the truncation of the CE in the Variable Basis. Our main result, to be presented in the following sections, is that truncating the CE of the probabilities in the SDG basis leads to large errors in the CVM free energy as we move away from the 50/50 concentration and, eventually, leads to negative values of the probability distributions. On the other hand, truncation of the CE in the Variable Basis gives a sequence of free energies that converge uniformly towards the full CVM free energy. The organization of the paper is as follows: we begin with a review of the CVM in Section 2 and place the emphasis on the expansion of the cluster probabilities in the different basis sets. An explicit example of how to connect the CE in the SDG basis with the expansion in the Variable Basis is presented in Section 3 for the tetrahedron approximation in the fcc lattice. We also present examples of approximate solutions to the standard Ising model for several CVM approximation in the fcc and bcc lattices. We conclude in Section 4 with a summary and observations on the proposed set of approximate solutions to the CVM free energy. 2. The cluster variation method The CVM, based on the variational principle of classical statistical mechanics, provides approximations to the exact configurational free energy functional in terms of a cumulant expansion of cluster entropies [1,11–13]. The general form of the CVM free energy functional is:
FðfX g gÞ ¼ hEi kB T
X X ag X g ð~ rg Þ log X g ð~ rg Þ g
ð1Þ
~ rg
where hEi is the expectation value of the configurational energy, X g ð~ rg Þ is the probability of observing a finite cluster of type g in rg , and the ag are geometric coefficients. the configuration ~ As mentioned in the Introduction, the energy of the alloy can always be expanded in any one of an infinite number of complete
and orthonormal basis sets. The basis functions in question are given by [3,4]:
Uax ð~ rÞ ¼
Y ðrp xÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 p2a
ð2Þ
where a ¼ fg; p; mg stands for a cluster of type g, with its center of mass located at point p in the lattice, and where m labels all distinct orientations of the cluster obtained by applying the symmetry operations of the point group of the lattice. For a given value of x in Eq. (2), the scalar product of two functions of configuration is defined as:
hf ; gix ¼
X ~ r
elhri 2 coshðlÞ
N
f ð~ rÞgð~ rÞ:
ð3Þ
with l such that tanhðlÞ ¼ x and with hri the point correlation given by:
hri ¼
N 1X rp N p¼1
ð4Þ
It follows from the completeness of the basis functions fUax g, that the expectation value of the energy can be written as:
hEi ¼
X g
xg Jg ðxÞzgx
ð5Þ
where xg is the number of clusters of type g per lattice point, zgx is the expectation value of the basis function Ugx , and J g ðxÞ is the ECI for cluster g obtained by projecting the energy E onto the basis function Ugx [3,4]. Fom Eqs. (2) and (3) we see that a choice of a fixed basis with a value of x ¼ 0 corresponds to the commonly used SDG basis. On the other hand, in the VBCE, we use, for each configuration ~ r, a basis such that x ¼ hri. Several methods [1,11–13] have been developed to calculate the coefficients ag , with the approach introduced by Barker [11] being perhaps the simplest. In Barkers’s approach, one starts by selecting a maximum cluster, or clusters, to be used in the approximation, say g0 . The coefficient for the maximum cluster ag0 equals the number of clusters of type g0 per lattice point, i.e. ag0 ¼ xg0 . The remaining coefficients for clusters g g0 are given by the following recursive formula:
ag ¼ xg
X
M gg ag0 0
ð6Þ
g 0 g
where M gg stands for the number of cluster of type g contained in a cluster of type g0 . The main computational task in implementations of the CVM is the minimization of the free energy functional, Eq. (1), with respect to the cluster probabilities X g ð~ rg Þ. In order to facilitate the minimization step, it is convenient to introduce a set of independent variables to describe all the cluster probabilities involved in a given CVM approximation. Such full set of independent variables can be identified quite straightforwardly by cluster expanding the probability distributions X g ð~ rg Þ. As mentioned in the Introduction, the CE method was initially developed to expand the cluster probabilities in terms of the expectation values of the correlation functions in the SDG basis [7]. In such a basis, the expansion is: 0
X g ð~ rg Þ ¼
X ag0 # ag
hX g ; /0ag0 i0 /0ag0 ð~ rg0 Þ
ð7Þ
with the coefficients hX g ; /0ag0 i0 taking a particularly simple form [7]:
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hX g ; /0ag0 i0 ¼
1 2
jgj
z0g0 ;
ð8Þ
and where jgj stands for the number of points in cluster g, and z0g is
and
The elements of the matrix VðgÞ are given by: ðgÞ
Vi;j ¼
1 X 2jgj aj # ag
/0aj ð~ rgi Þ
ð10Þ
The completeness of the set fUgx g implies that the basis functions for different values of x are related by a linear transformation of the form [3,4]: 0
0
x x Bx;x g;g0 ¼ hUg ; Ug0 ix0
0 0 ~ Z x ¼ Bx;x ! Z x
r
~ r
where J2 and are, respectively, the effective interaction and correlation function for nearest-neighbor pairs. Consistent with the Ising model, J2 is constant and the nearest-neighbor pair correlation, z02 , is given in the SDG basis. Since the minimization of the free energy will be carried at a fixed value of x, the term involving X 1 ðrÞ is constant. For the sake of completeness, the matrices VðgÞ for g ¼ 2 (nearest-neighbor pair) and g ¼ 4 (regular tetrahedron), which are needed for the representation of the probability distributions in the SDG basis (see Eq. (9)), are given in Table 2 for all disr. tinguishable configurations ~ Calling ~ Z 0 ¼ f1; x; z0 ; z0 ; z0 g the vector of correlation functions in 2
ð12Þ
where B ¼ fBg;g0 g and ~ Z x ¼ f1; z1x ; z2x ; . . . ; zjx ; . . .g. Combining Eq. (9) with Eq. (12), we arrive at an expansion of the cluster probabilities which, in the variable basis, is given by:
3
4
the SDG basis, and ~ Z x ¼ f1; 0; z2x ; z3x ; z4x g that in the Variable Basis, it follows from Eq. (12) and Table 1 that they are related as follows:
(
ð11Þ
Thus, it follows that the expectation values of the basis function Ugx , namely zgx , for different values of x are also related and, in matrix form, given by:
x;x0
~ r
z02
ð9Þ
~ with X g ¼ fX g ð~ rg1 Þ; X g ð~ rg2 Þ; . . . ; X g ð~ rgi Þ; . . .g ~ Z 0g ¼ f1; z01 ; z02 ; . . . ; z0j ; . . .g.
! X X X 2 X 4 ð~ rÞlogðX 4 ð~ rÞÞ 6 X 2 ð~ rÞlogðX 2 ð~ rÞÞ þ 5 X 1 ðrÞlogðX 1 ðrÞÞ ð14Þ
the expectation value of the basis function U0g . For ease of notation we rewrite Eq. (7) as:
~ X g ¼ VðgÞ~ Z 0g
F ¼ 6J 2 z02 kB T
Zx ¼
1; 0;
z02 x2 z03 3xz02 þ 2x3 z04 4xz03 þ 6x2 z02 3x4 ; ; 3 2 1 x2 ð1 x2 Þ ð1 x2 Þ2
n 3 Z 0 ¼ 1; x; z2x ð1 x2 Þ þ x2 ; z3x ð1 x2 Þ2 þ 3z2x xð1 x2 Þ þ x3 ; o 3 2 z4x ð1 x2 Þ þ 4z3x xð1 x2 Þ2 þ 6z2x x2 ð1 x2 Þ þ x4
ð15Þ
x;x0
x ~ X g ¼ VðgÞ B0;x ~ Zg
ð13Þ
with ~ Z gx ¼ f1; 0; z2x ; . . . ; zjx ; . . .g. Note that the second component of x ~ Z g , which corresponds to the point cluster, vanishes since in the
variable basis cluster expansion the quantity x ¼ tanhðlÞ is always matched to the expectation value of the point correlation hrp i ¼ z01 . The representations of the X g ðrg Þ given by Eqs. (9) in the SDG basis or by Eq. (13) in the variable basis, provide the functional dependence of the cluster probabilities on the independent variables z0g or zgx , needed to carry out the minimization of the free energy functional FðfX g gÞ. Both representations of the probabilities, or for that matter of any other function of configuration, are equally valid unless the expansion is truncated. As we shall see with a simple example in the next section, truncation of the cluster expansion at a given cluster size in the SDG basis leads to significantly different results than the truncation of the cluster expansion in the variable basis at the same cluster size. 3. Truncated cluster expansions and the CVM We begin this section with a simple example of the procedure used to truncate the CE of the probability distributions in the CVM free energy functional. For that purpose, we consider the tetrahedron approximation for the Ising model with nearest neighbor interactions in an fcc lattice. The complete set of clusters in this approximation are the empty cluster (g ¼ 0), the point (g ¼ 1), the nearest-neighbor pairs (g ¼ 2), the regular triangle (g ¼ 3), and the tetrahedron (g ¼ 4). Table 1 lists all the geometric parameters needed to set up the CVM free energy functional, namely, xg ; M gg0 , and ag , together with 0
the elements Bx;x g;g0 of the transformation matrix relating the basis functions for two different concentrations x and x0 . From Table 1 it follows that the CVM free energy functional is:
)
Thus, the minimization of the free energy at a constant value of @F x involves solving three coupled equations, namely, @z 0 ¼ 0 in the g
SDG basis or, equivalently,
@F @zgx
¼ 0 in the variable basis, for g equal
2, 3 and 4. Of interest here is the truncation of the cluster expansion of the probabilities X g ð~ rÞ in the concentration-dependent basis, which is accomplished by setting the correlations zgx for selected cluster equal to zero. This approximation to the CVM free energy functional effectively lowers the number of equations that need to be solved in the minimization step. In the tetrahedron approximations there are only two levels of truncation or cutoff to the expansion of X 4 ð~ rÞ to be considered: (1) setting both z3x ¼ 0 and z4x ¼ 0, thus reducing the minimization step to solving only one equation and (2) setting only z4x ¼ 0, which requires solving two equations. The free energy of formation for the Ising model with J 2 < 0 at 50/50 concentration (x ¼ 0) and at a reduced temperature kB T=jJ 2 j ¼ 12:5 is shown in Fig. 1 as a function of the number of correlations included in the CE of the probability X 4 ð~ rÞ. In Fig. 1, the number of correlations equal to zero corresponds to the Bragg– Williams approximation, while the two possible levels of truncation and the full CVM solution in the tetrahedron approximation correspond, respectively, to a number of correlations equal to one, two and three. Recall that the temperature-composition phase diagram for this system consists of a miscibility gap with a critical point in the Bragg–Williams approximation at x ¼ 0 and kB T c =jJ 2 j ¼ 12. We see from Fig. 1 that most of the improvement of the CVM over the Bragg–Williams approximation is already achieved by the lowest level of truncation (i.e. by setting z3x ¼ z4x ¼ 0 and considering z2x as the only variable for minimization). As a curiosity, we note that for x ¼ 0 and with the cutoff z3x ¼ z4x ¼ 0 it is possible to find an analytical expression for z2x as a function of temperature that minimizes the free energy. The fact that a basis for any value of x is related to the SDG basis by the linear transformation Bx;0 trivially implies that the cluster expansions in all basis sets should be strictly equivalent. Based on this equivalence, it has been noted that, in carrying out cluster
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J.M. Sanchez, T. Mohri / Computational Materials Science 122 (2016) 301–306
Table 1 Geometric parameters characterizing the tetrahedron approximation of the CVM in the fcc lattice. g
g
g
g
g
xg
M1
M2
M3
M4
ag
Bx;x g;0
Bx;x g;1
0 1
– 1
– 1
– –
– –
– –
–
1
0
2
6
2
1
–
–
6
3
8
3
3
1
–
0
0
0
ðx0 xÞ
5
1x02 1x2
1 ð1x2 Þ2
3
2
4
6
4
1
2
12
pffiffiffiffiffiffiffiffiffi 2ðx0 xÞ 1x02 1x2 pffiffiffiffiffiffiffiffiffi 3ðx0 xÞ2 1x02
ðx0 xÞ2 1x2 ðx0 xÞ3 ðx0 xÞ4 2 ð1x2 Þ
0
0
Bx;x g;3
Bx;x g;4
0 0
0 0
0 0
1x02 1x2
0
3ðx0 xÞð1x02 Þ 3
3
ð1x2 Þ2 pffiffiffiffiffiffiffiffiffi 4ðx0 xÞ3 1x02 2 ð1x2 Þ
ð1x2 Þ2
4
0
Bx;x g;2
ð1x2 Þ2 6ðx0 xÞ2 ð1x02 Þ 2 ð1x2 Þ
0 3 02 2
0
1x 1x2
3
4ðx0 xÞð1x02 Þ2 2 ð1x2 Þ
1x02 1x2
2
Table 2 Matrices VðgÞ relating the cluster probabilities to the correlation functions z0g in the SDG basis for the tetrahedron approximation of the fcc lattice.
g
i
~ ri
di
2jgj V i;0
2jgj V i;1
2jgj V i;2
2jgj V i;3
2jgj V i;4
2 2 2 4 4 4 4 4
1 2 3 1 2 3 4 5
f1; 1g f1; 1g f1; 1g f1; 1; 1; 1g f1; 1; 1; 1g f1; 1; 1; 1g f1; 1; 1; 1g f1; 1; 1; 1g
1 2 1 1 4 6 4 1
1 1 1 1 1 1 1 1
2 0 2 4 2 0 2 4
1 1 1 6 0 2 0 6
– – – 4 2 0 2 4
– – – 1 1 1 1 1
ðgÞ
ðgÞ
ðgÞ
ðgÞ
0.16
ðgÞ
zx3 zx4 0
z20
z4x 0 Full Tetrahedron
0.14 0.12 0.1 12
13
14
15
16
kB T J2
expansions, it is generally more convenient to use the SDG basis introduced in Ref. [2] than the concentration-dependent basis [6]. However, while the equivalence argument is valid for the full cluster expansion, for both finite clusters and infinite lattices, it breaks down if the expansion is truncated at any level. For the case of the tetrahedron approximation, we see from Eq. (15) that, for example, setting z3x ¼ z4x ¼ 0, implies that the correlations in the SDG basis are given by z03 ¼ 3z2x xð1 x2 Þ þ x3 and 0 x 2 2 4 z4 ¼ 6z2 x ð1 x Þ þ x . In fact, a non-zero value of z2x in the concentration-dependent basis gives rise to non-zero values of the correlation functions z0g in the SDG basis for an infinite set of clusters g that contain a nearest-neighbor pair. Figs. 2–4 show the correlation functions for pair, triangle and tetrahedron clusters in the SDG basis as a function of the reduced temperature kB T=jJ 2 j at x ¼ 0:2 for the fcc Ising model (J 2 < 0) in the tetrahedron approximation. Shown in the figures are the results for the two levels of cutoff of the expansion in the concentration-dependent basis, as well as the full CVM solution. From Fig. 2 we see that the CVM solution obtained by truncating the CE of the probability X 4 ð~ rÞ at the lowest levels already gives values of the pair correlation function z02 in good agreement with the correct CVM value. The non-zero values of the triangle and
Fig. 2. The nearest-neighbor pair correlation, z02 , as a function of the reduced temperature kB T=jJ 2 j for the Ising model with J 2 < 0 at x ¼ cA cB ¼ 0:2, in the full tetrahedron approximation of the CVM (red), and for cases where the VBCE of the probabilities are truncated at the triangle (green) and the tetrahedron (blue) clusters. In the Bragg–Williams approximation, not shown, the pair correlation equals 0.04.
0.08 z3x z4x 0 zx4 0 Full Tetrahedron
z30
Fig. 1. Free energy of formation in units of jJ 2 j for the Ising model with J 2 < 0 at x ¼ cA cB ¼ 0 as a function of the number of correlations included in the VBCE of the probabilities appearing in the tetrahedron approximation of the CVM.
0.06
0.04 12
13
14
15
16
kB T J2 Fig. 3. The correlation function for the nearest-neighbor regular triangle, z03 , as a function of the reduced temperature kB T=jJ2 j for the Ising model with J2 < 0 at x ¼ cA cB ¼ 0:2, in the full tetrahedron approximation of the CVM (red), and for cases where the VBCE of the probabilities are truncated at the triangle (green) and the tetrahedron (blue) clusters. In the Bragg–Williams approximation, not shown, the correlation for the regular triangle equals 0.008.
J.M. Sanchez, T. Mohri / Computational Materials Science 122 (2016) 301–306
305
0.08 zx3 zx4 0 z4x 0
z40
Full Tetrahedron
0.04
0. 12
13
14
15
16
kB T J2 Fig. 4. The correlation function for the regular tetrahedron, z04 , as a function of the reduced temperature kB T=jJ 2 j for the Ising model with J 2 < 0 at x ¼ cA cB ¼ 0:2, in the full tetrahedron approximation of the CVM (red), and for cases where the VBCE of the probabilities are truncated at the triangle (green) and the tetrahedron (blue) clusters. In the Bragg–Williams approximation, not shown, the correlation the tetrahedron equals 0.0016.
Fig. 5. Free energy of formation for the Ising model (J 2 < 0), on the bcc lattice in the Octahedron approximation of the CVM, as a function of the number of correlations included in the SDG and in the VB expansions of the cluster probabilities. Also shown are the free energies in the Bragg–Williams and in the full Octahedron-CVM approximations.
Fig. 6. Free Energy of formation for the fcc Ising model (J2 < 0) as a function of number of clusters kept in the expansion of the 13–14 points cluster probabilities of the CVM. Also shown are the free energies in the Bragg–Williams and in the full 13– 14 point cluster CVM approximations.
tetrahedron correlations in the SDG basis shown in Figs. 3 and 4 illustrate the in-equivalence of the truncated cluster expansion in different basis sets. The inappropriateness of truncating the cluster expansion in the SDG basis for x – 0 is further illustrated in Fig. 5 for the bcc Ising model (J 2 < 0) in the Octahedron approximation of the CVM [7].
Fig. 7. Free energy of formation for the fcc Ising model (J 2 < 0) as a function of concentration at kB T=jJ 2 j for the 13–14 points cluster approximation of the CVM with the cluster probabilities truncated at the nearest-neighbor pairs, compared to the Bragg–Williams approximation and the full 13–14 points cluster CVM approximation.
Fig. 5 shows the energy of formation obtained by truncating the expansions of the cluster probabilities in both the SDG and the concentration-dependent basis, together with the free energy of formation obtained in the Bragg–Williams and in the full CVMOctahedron approximations, for x ¼ 0:2 and at a reduced temperature of kB T=jJ 2 j ¼ 8:5. We see from the figure that truncating the expansion of the probabilities in the SDG basis at the pair level results in a free energy of formation that is in fact less accurate than that in the Bragg–Williams approximation, while truncating the expansion in the concentration-dependent basis gives a sequence of reasonably accurate values that converge uniformly to the full CVM solution. More importantly, it can be shown that truncating the expansion in the SDG basis will invariably lead to negative values of the probabilities X g ð~ rÞ for values of x that depart sufficiently from the 50/50 composition (x ¼ 0). As a final example, we consider the fcc Ising model (J 2 < 0) in the 13–14 point cluster approximation of the CVM. The approximation combines a 13 point cluster formed by a lattice point surrounded by its 12 nearest neighbors and the 14 point fcc unit cell [14]. The minimization of full CVM approximation requires solving 741 equations, which correspond to the total number of cluster correlations. Fig. 6 shows the free energy of formation as a function of the number of clusters kept in the cluster expansion, for x ¼ 0 and kB T=jJ 2 j ¼ 12:5. At the lowest level of truncation, i.e. keeping only nearest-neighbor pairs in the expansion of the cluster probabilities, the error relative to the full CVM approximation is 3.5%. Thus, the truncation of the probabilities in the concentration-dependent basis is particularly attractive for such large cluster since it results in relatively accurate configurational free energies with a significant reduction in the computational effort. As mentioned, the truncation of the cluster probabilities in the concentration-dependent basis, unlike that in the SDG basis, is valid for the entire concentration range. The free energy in the 13–14 points approximation, at a reduced temperature kB T=jJ 2 j is depicted in Fig. 7 for the case in which only nearest-neighbors correlations are included in the expansions of the probabilities. Also shown are the free energies in the Bragg–Williams approximation and in the full 13–14 points cluster CVM approximation. 4. Conclusions By means of several examples we have shown that the computational task of minimizing the CVM free energy functional can be significantly reduced by expanding the cluster probabilities in a
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concentration-dependent basis, and truncating such an expansion at different cluster levels. Significantly, most of the improvement afforded by the CVM over the Bragg–Williams approximation is achieved by truncating the expansion at the lowest levels, which typically keeps only pair correlations as the independent variables for the minimization step. Furthermore, we have used the expansion of the cluster probabilities in conjunction with the CVM to establish important differences between truncated cluster expansions in the SDG and in the concentration-dependent basis sets. In particular, the analysis shows that, despite the fact that all basis sets are related by a linear transformation, their equivalence does not hold when the expansions are truncated at any level.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
R. Kikuchi, Phys. Rev. 81 (1951) 988. J.M. Sanchez, F. Ducastelle, D. Gratias, Physica 128A (1984) 334. J.M. Sanchez, Phys. Rev. B 48 (1993) 14013. J.M. Sanchez, Phys. Rev. B 81 (2010) 224202. M. Asta, C. Wolverton, D. de Fontaine, H. Dreyssé, Phys. Rev. B 44 (1991) 4907. T. Mueller, G. Ceder, Phys. Rev. B 82 (2010) 184107. J.M. Sanchez, D. de Fontaine, Phys. Rev. B 17 (1978) 2926. J.M. Sanchez, D. de Fontaine, Phys. Rev. B 21 (1980) 216. J.M. Sanchez, D. de Fontaine, Phys. Rev. B 25 (1982) 1759. T. Mohri, J.M. Sanchez, D. de Fontaine, Acta Metall. Mater. 33 (1985) 1171. J.A. Barker, Proc. R. Soc. A 216 (1953) 45. T. Morita, J. Phys. Soc. Jpn. 12 (1957) 753. T. Morita, J. Math. Phys. 13 (1972) 115. A. Finel, Ph.D. Dissertation, Paris IV, 1987.
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Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Approximate solutions to the cluster variation free energies by the variable basis cluster expansion J.M. Sanchez a,⇑, T. Mohri b a b
Institute for Computational Engineering and Sciences and Texas Materials Institute, The University of Texas at Austin, Austin, TX 78712, USA Institute for Materials Research, Tohuko University, Sendai, Japan
a r t i c l e
i n f o
Article history: Received 22 April 2016 Received in revised form 25 May 2016 Accepted 26 May 2016
Keywords: Cluster variation method Cluster expansion
a b s t r a c t In this paper we obtain approximate solutions to the Cluster Variation free energy by carrying out a cluster expansion of the probabilities appearing in the free energy functional in terms of concentration-dependent basis functions, and by truncating the expansion at different cluster levels prior to minimization. We show that a significant improvement over the Bragg–Williams approximation can be achieved by truncating the expansion of the cluster probabilities at relatively small clusters, thus dramatically reducing the number of equations that need to be solved in order to minimize the free energy. Furthermore, the free energy functional in the Cluster Variation Method offers a wellcontrolled case study to infer the effects of truncating the expansion of the energy of alloy formation in the commonly used Cluster Expansion method, versus the effects of truncating the expansion when using a concentration-dependent basis. Examples of the approach are given for simple Ising models for fcc- and bcc-based prototype alloy systems. Ó 2016 Elsevier B.V. All rights reserved.
1. Introduction Since its inception by Kikuchi in 1951, the Cluster Variation Method (CVM) has been widely used to accurately account for the effect of short-range order in the free energy of binary and multicomponent alloys [1]. While the accuracy of the CVM generally increases with the size of the largest cluster included in the approximation, so does the computational effort involved in the minimization of the free energy functional. With increasing cluster sizes, the minimization needs to be carried out with respect to a relatively large number of configurational variables, a number that grows exponentially with cluster size. Within mean-field theories, at the other extreme of computational convenience is the commonly used Bragg–Williams approximation. While a significant body of research and applications have shown that the shortcomings of the Bragg–Williams approximation can be compensated with other terms in the configurational free energy, it is apparent that the use of increasingly accurate ab initio methods to calculate energies of formations calls for a similar increase in the accuracy of the configurational entropy. A central task in the configurational theory of alloys is the description of functions of configurations in terms of a complete ⇑ Corresponding author. E-mail address: [email protected] (J.M. Sanchez). http://dx.doi.org/10.1016/j.commatsci.2016.05.035 0927-0256/Ó 2016 Elsevier B.V. All rights reserved.
set of configurational variables. A widely used approach to describe functions of configuration, formalized by Sanchez et al. [2], is the so-called Cluster Expansion (CE) method in which the functions are expanded in terms of pair and multisite correlation functions. These correlation functions form a complete and orthonormal basis in configurational space [2]. Subsequently, it was shown that the complete and orthonormal basis set introduced in Ref. [2], which will be referred here as the SDG basis, is a particular case of an infinite number of closely related complete and orthonormal basis sets in configurational space [3,4]. Presently, the CE in the SDG basis is the method of choice to obtain effective cluster interactions from the energies of a set of ordered compounds, which are typically calculated by means of some implementation of Kohn–Sham equations in Density Functional Theory. The approach is particularly appealing since, for the case of the energy of alloy formation, and after truncating the expansion at some maximum cluster size, the method casts the energy in the form of an Ising-like model with constant expansion coefficients. Such expansion coefficients are commonly referred to as Effective Cluster Interactions (ECIs). However, and despite the unquestionable success of the CE method in the SDG basis for the parametrization of the energy of alloy formation, the validity of truncating the expansion remains, at least at a fundamental level, an unresolved issue. In particular, if the energy has a nonlinear dependence in the concentration of the alloy, like it is the
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case for a regular solution model or for Fridel’s square band model for transition metal alloys, it is straightforward to show that the CE in the SDG basis does not converge [4]. The existence of an infinite number of basis sets provides a straightforward path to address non-linear concentration dependences in the energy of the system [3,4]. The method consists in matching the basis sets to the overall concentration of the alloy under consideration, and is akin to doing the expansion in a canonical ensemble as first proposed by Asta et al. [5]. This particular implementation of the cluster expansion will be referred as the Variable Basis Cluster Expansion (VBCE) [4]. Unlike the expansion in the SDG basis, the ECIs in the VBCE are concentration dependent. Although most current applications of the CE method are aimed at characterizing the energy of alloy formation, the method was initially motivated by the need to develop efficient algorithms to minimize the CVM free energy functional [7]. Such early studies were aimed at the calculation of prototype alloy phase diagrams based on Ising-like models for the description of the energy of formation, and on the CVM for the treatment of the configurational entropy [7–10]. The approach adopted in these early studies was to describe the probabilities appearing in the CVM configurational entropy in terms of a cluster expansion using the SDG basis functions, the expectation values of which are the so-called correlation functions. Since for a given cluster approximation the correlation functions form a complete and orthogonal basis set, they naturally constitute a full set of independent variables for the free energy minimization. In this paper we formulate the minimization of the CVM free energy in terms of a VBCE of the cluster probabilities, and show the effects of truncating the expansion at different cluster levels. In particular, the CVM free energy functional offers a well-controlled case study to investigate the truncation of the CE in the SDG basis versus the truncation of the CE in the Variable Basis. Our main result, to be presented in the following sections, is that truncating the CE of the probabilities in the SDG basis leads to large errors in the CVM free energy as we move away from the 50/50 concentration and, eventually, leads to negative values of the probability distributions. On the other hand, truncation of the CE in the Variable Basis gives a sequence of free energies that converge uniformly towards the full CVM free energy. The organization of the paper is as follows: we begin with a review of the CVM in Section 2 and place the emphasis on the expansion of the cluster probabilities in the different basis sets. An explicit example of how to connect the CE in the SDG basis with the expansion in the Variable Basis is presented in Section 3 for the tetrahedron approximation in the fcc lattice. We also present examples of approximate solutions to the standard Ising model for several CVM approximation in the fcc and bcc lattices. We conclude in Section 4 with a summary and observations on the proposed set of approximate solutions to the CVM free energy. 2. The cluster variation method The CVM, based on the variational principle of classical statistical mechanics, provides approximations to the exact configurational free energy functional in terms of a cumulant expansion of cluster entropies [1,11–13]. The general form of the CVM free energy functional is:
FðfX g gÞ ¼ hEi kB T
X X ag X g ð~ rg Þ log X g ð~ rg Þ g
ð1Þ
~ rg
where hEi is the expectation value of the configurational energy, X g ð~ rg Þ is the probability of observing a finite cluster of type g in rg , and the ag are geometric coefficients. the configuration ~ As mentioned in the Introduction, the energy of the alloy can always be expanded in any one of an infinite number of complete
and orthonormal basis sets. The basis functions in question are given by [3,4]:
Uax ð~ rÞ ¼
Y ðrp xÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 p2a
ð2Þ
where a ¼ fg; p; mg stands for a cluster of type g, with its center of mass located at point p in the lattice, and where m labels all distinct orientations of the cluster obtained by applying the symmetry operations of the point group of the lattice. For a given value of x in Eq. (2), the scalar product of two functions of configuration is defined as:
hf ; gix ¼
X ~ r
elhri 2 coshðlÞ
N
f ð~ rÞgð~ rÞ:
ð3Þ
with l such that tanhðlÞ ¼ x and with hri the point correlation given by:
hri ¼
N 1X rp N p¼1
ð4Þ
It follows from the completeness of the basis functions fUax g, that the expectation value of the energy can be written as:
hEi ¼
X g
xg Jg ðxÞzgx
ð5Þ
where xg is the number of clusters of type g per lattice point, zgx is the expectation value of the basis function Ugx , and J g ðxÞ is the ECI for cluster g obtained by projecting the energy E onto the basis function Ugx [3,4]. Fom Eqs. (2) and (3) we see that a choice of a fixed basis with a value of x ¼ 0 corresponds to the commonly used SDG basis. On the other hand, in the VBCE, we use, for each configuration ~ r, a basis such that x ¼ hri. Several methods [1,11–13] have been developed to calculate the coefficients ag , with the approach introduced by Barker [11] being perhaps the simplest. In Barkers’s approach, one starts by selecting a maximum cluster, or clusters, to be used in the approximation, say g0 . The coefficient for the maximum cluster ag0 equals the number of clusters of type g0 per lattice point, i.e. ag0 ¼ xg0 . The remaining coefficients for clusters g g0 are given by the following recursive formula:
ag ¼ xg
X
M gg ag0 0
ð6Þ
g 0 g
where M gg stands for the number of cluster of type g contained in a cluster of type g0 . The main computational task in implementations of the CVM is the minimization of the free energy functional, Eq. (1), with respect to the cluster probabilities X g ð~ rg Þ. In order to facilitate the minimization step, it is convenient to introduce a set of independent variables to describe all the cluster probabilities involved in a given CVM approximation. Such full set of independent variables can be identified quite straightforwardly by cluster expanding the probability distributions X g ð~ rg Þ. As mentioned in the Introduction, the CE method was initially developed to expand the cluster probabilities in terms of the expectation values of the correlation functions in the SDG basis [7]. In such a basis, the expansion is: 0
X g ð~ rg Þ ¼
X ag0 # ag
hX g ; /0ag0 i0 /0ag0 ð~ rg0 Þ
ð7Þ
with the coefficients hX g ; /0ag0 i0 taking a particularly simple form [7]:
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hX g ; /0ag0 i0 ¼
1 2
jgj
z0g0 ;
ð8Þ
and where jgj stands for the number of points in cluster g, and z0g is
and
The elements of the matrix VðgÞ are given by: ðgÞ
Vi;j ¼
1 X 2jgj aj # ag
/0aj ð~ rgi Þ
ð10Þ
The completeness of the set fUgx g implies that the basis functions for different values of x are related by a linear transformation of the form [3,4]: 0
0
x x Bx;x g;g0 ¼ hUg ; Ug0 ix0
0 0 ~ Z x ¼ Bx;x ! Z x
r
~ r
where J2 and are, respectively, the effective interaction and correlation function for nearest-neighbor pairs. Consistent with the Ising model, J2 is constant and the nearest-neighbor pair correlation, z02 , is given in the SDG basis. Since the minimization of the free energy will be carried at a fixed value of x, the term involving X 1 ðrÞ is constant. For the sake of completeness, the matrices VðgÞ for g ¼ 2 (nearest-neighbor pair) and g ¼ 4 (regular tetrahedron), which are needed for the representation of the probability distributions in the SDG basis (see Eq. (9)), are given in Table 2 for all disr. tinguishable configurations ~ Calling ~ Z 0 ¼ f1; x; z0 ; z0 ; z0 g the vector of correlation functions in 2
ð12Þ
where B ¼ fBg;g0 g and ~ Z x ¼ f1; z1x ; z2x ; . . . ; zjx ; . . .g. Combining Eq. (9) with Eq. (12), we arrive at an expansion of the cluster probabilities which, in the variable basis, is given by:
3
4
the SDG basis, and ~ Z x ¼ f1; 0; z2x ; z3x ; z4x g that in the Variable Basis, it follows from Eq. (12) and Table 1 that they are related as follows:
(
ð11Þ
Thus, it follows that the expectation values of the basis function Ugx , namely zgx , for different values of x are also related and, in matrix form, given by:
x;x0
~ r
z02
ð9Þ
~ with X g ¼ fX g ð~ rg1 Þ; X g ð~ rg2 Þ; . . . ; X g ð~ rgi Þ; . . .g ~ Z 0g ¼ f1; z01 ; z02 ; . . . ; z0j ; . . .g.
! X X X 2 X 4 ð~ rÞlogðX 4 ð~ rÞÞ 6 X 2 ð~ rÞlogðX 2 ð~ rÞÞ þ 5 X 1 ðrÞlogðX 1 ðrÞÞ ð14Þ
the expectation value of the basis function U0g . For ease of notation we rewrite Eq. (7) as:
~ X g ¼ VðgÞ~ Z 0g
F ¼ 6J 2 z02 kB T
Zx ¼
1; 0;
z02 x2 z03 3xz02 þ 2x3 z04 4xz03 þ 6x2 z02 3x4 ; ; 3 2 1 x2 ð1 x2 Þ ð1 x2 Þ2
n 3 Z 0 ¼ 1; x; z2x ð1 x2 Þ þ x2 ; z3x ð1 x2 Þ2 þ 3z2x xð1 x2 Þ þ x3 ; o 3 2 z4x ð1 x2 Þ þ 4z3x xð1 x2 Þ2 þ 6z2x x2 ð1 x2 Þ þ x4
ð15Þ
x;x0
x ~ X g ¼ VðgÞ B0;x ~ Zg
ð13Þ
with ~ Z gx ¼ f1; 0; z2x ; . . . ; zjx ; . . .g. Note that the second component of x ~ Z g , which corresponds to the point cluster, vanishes since in the
variable basis cluster expansion the quantity x ¼ tanhðlÞ is always matched to the expectation value of the point correlation hrp i ¼ z01 . The representations of the X g ðrg Þ given by Eqs. (9) in the SDG basis or by Eq. (13) in the variable basis, provide the functional dependence of the cluster probabilities on the independent variables z0g or zgx , needed to carry out the minimization of the free energy functional FðfX g gÞ. Both representations of the probabilities, or for that matter of any other function of configuration, are equally valid unless the expansion is truncated. As we shall see with a simple example in the next section, truncation of the cluster expansion at a given cluster size in the SDG basis leads to significantly different results than the truncation of the cluster expansion in the variable basis at the same cluster size. 3. Truncated cluster expansions and the CVM We begin this section with a simple example of the procedure used to truncate the CE of the probability distributions in the CVM free energy functional. For that purpose, we consider the tetrahedron approximation for the Ising model with nearest neighbor interactions in an fcc lattice. The complete set of clusters in this approximation are the empty cluster (g ¼ 0), the point (g ¼ 1), the nearest-neighbor pairs (g ¼ 2), the regular triangle (g ¼ 3), and the tetrahedron (g ¼ 4). Table 1 lists all the geometric parameters needed to set up the CVM free energy functional, namely, xg ; M gg0 , and ag , together with 0
the elements Bx;x g;g0 of the transformation matrix relating the basis functions for two different concentrations x and x0 . From Table 1 it follows that the CVM free energy functional is:
)
Thus, the minimization of the free energy at a constant value of @F x involves solving three coupled equations, namely, @z 0 ¼ 0 in the g
SDG basis or, equivalently,
@F @zgx
¼ 0 in the variable basis, for g equal
2, 3 and 4. Of interest here is the truncation of the cluster expansion of the probabilities X g ð~ rÞ in the concentration-dependent basis, which is accomplished by setting the correlations zgx for selected cluster equal to zero. This approximation to the CVM free energy functional effectively lowers the number of equations that need to be solved in the minimization step. In the tetrahedron approximations there are only two levels of truncation or cutoff to the expansion of X 4 ð~ rÞ to be considered: (1) setting both z3x ¼ 0 and z4x ¼ 0, thus reducing the minimization step to solving only one equation and (2) setting only z4x ¼ 0, which requires solving two equations. The free energy of formation for the Ising model with J 2 < 0 at 50/50 concentration (x ¼ 0) and at a reduced temperature kB T=jJ 2 j ¼ 12:5 is shown in Fig. 1 as a function of the number of correlations included in the CE of the probability X 4 ð~ rÞ. In Fig. 1, the number of correlations equal to zero corresponds to the Bragg– Williams approximation, while the two possible levels of truncation and the full CVM solution in the tetrahedron approximation correspond, respectively, to a number of correlations equal to one, two and three. Recall that the temperature-composition phase diagram for this system consists of a miscibility gap with a critical point in the Bragg–Williams approximation at x ¼ 0 and kB T c =jJ 2 j ¼ 12. We see from Fig. 1 that most of the improvement of the CVM over the Bragg–Williams approximation is already achieved by the lowest level of truncation (i.e. by setting z3x ¼ z4x ¼ 0 and considering z2x as the only variable for minimization). As a curiosity, we note that for x ¼ 0 and with the cutoff z3x ¼ z4x ¼ 0 it is possible to find an analytical expression for z2x as a function of temperature that minimizes the free energy. The fact that a basis for any value of x is related to the SDG basis by the linear transformation Bx;0 trivially implies that the cluster expansions in all basis sets should be strictly equivalent. Based on this equivalence, it has been noted that, in carrying out cluster
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Table 1 Geometric parameters characterizing the tetrahedron approximation of the CVM in the fcc lattice. g
g
g
g
g
xg
M1
M2
M3
M4
ag
Bx;x g;0
Bx;x g;1
0 1
– 1
– 1
– –
– –
– –
–
1
0
2
6
2
1
–
–
6
3
8
3
3
1
–
0
0
0
ðx0 xÞ
5
1x02 1x2
1 ð1x2 Þ2
3
2
4
6
4
1
2
12
pffiffiffiffiffiffiffiffiffi 2ðx0 xÞ 1x02 1x2 pffiffiffiffiffiffiffiffiffi 3ðx0 xÞ2 1x02
ðx0 xÞ2 1x2 ðx0 xÞ3 ðx0 xÞ4 2 ð1x2 Þ
0
0
Bx;x g;3
Bx;x g;4
0 0
0 0
0 0
1x02 1x2
0
3ðx0 xÞð1x02 Þ 3
3
ð1x2 Þ2 pffiffiffiffiffiffiffiffiffi 4ðx0 xÞ3 1x02 2 ð1x2 Þ
ð1x2 Þ2
4
0
Bx;x g;2
ð1x2 Þ2 6ðx0 xÞ2 ð1x02 Þ 2 ð1x2 Þ
0 3 02 2
0
1x 1x2
3
4ðx0 xÞð1x02 Þ2 2 ð1x2 Þ
1x02 1x2
2
Table 2 Matrices VðgÞ relating the cluster probabilities to the correlation functions z0g in the SDG basis for the tetrahedron approximation of the fcc lattice.
g
i
~ ri
di
2jgj V i;0
2jgj V i;1
2jgj V i;2
2jgj V i;3
2jgj V i;4
2 2 2 4 4 4 4 4
1 2 3 1 2 3 4 5
f1; 1g f1; 1g f1; 1g f1; 1; 1; 1g f1; 1; 1; 1g f1; 1; 1; 1g f1; 1; 1; 1g f1; 1; 1; 1g
1 2 1 1 4 6 4 1
1 1 1 1 1 1 1 1
2 0 2 4 2 0 2 4
1 1 1 6 0 2 0 6
– – – 4 2 0 2 4
– – – 1 1 1 1 1
ðgÞ
ðgÞ
ðgÞ
ðgÞ
0.16
ðgÞ
zx3 zx4 0
z20
z4x 0 Full Tetrahedron
0.14 0.12 0.1 12
13
14
15
16
kB T J2
expansions, it is generally more convenient to use the SDG basis introduced in Ref. [2] than the concentration-dependent basis [6]. However, while the equivalence argument is valid for the full cluster expansion, for both finite clusters and infinite lattices, it breaks down if the expansion is truncated at any level. For the case of the tetrahedron approximation, we see from Eq. (15) that, for example, setting z3x ¼ z4x ¼ 0, implies that the correlations in the SDG basis are given by z03 ¼ 3z2x xð1 x2 Þ þ x3 and 0 x 2 2 4 z4 ¼ 6z2 x ð1 x Þ þ x . In fact, a non-zero value of z2x in the concentration-dependent basis gives rise to non-zero values of the correlation functions z0g in the SDG basis for an infinite set of clusters g that contain a nearest-neighbor pair. Figs. 2–4 show the correlation functions for pair, triangle and tetrahedron clusters in the SDG basis as a function of the reduced temperature kB T=jJ 2 j at x ¼ 0:2 for the fcc Ising model (J 2 < 0) in the tetrahedron approximation. Shown in the figures are the results for the two levels of cutoff of the expansion in the concentration-dependent basis, as well as the full CVM solution. From Fig. 2 we see that the CVM solution obtained by truncating the CE of the probability X 4 ð~ rÞ at the lowest levels already gives values of the pair correlation function z02 in good agreement with the correct CVM value. The non-zero values of the triangle and
Fig. 2. The nearest-neighbor pair correlation, z02 , as a function of the reduced temperature kB T=jJ 2 j for the Ising model with J 2 < 0 at x ¼ cA cB ¼ 0:2, in the full tetrahedron approximation of the CVM (red), and for cases where the VBCE of the probabilities are truncated at the triangle (green) and the tetrahedron (blue) clusters. In the Bragg–Williams approximation, not shown, the pair correlation equals 0.04.
0.08 z3x z4x 0 zx4 0 Full Tetrahedron
z30
Fig. 1. Free energy of formation in units of jJ 2 j for the Ising model with J 2 < 0 at x ¼ cA cB ¼ 0 as a function of the number of correlations included in the VBCE of the probabilities appearing in the tetrahedron approximation of the CVM.
0.06
0.04 12
13
14
15
16
kB T J2 Fig. 3. The correlation function for the nearest-neighbor regular triangle, z03 , as a function of the reduced temperature kB T=jJ2 j for the Ising model with J2 < 0 at x ¼ cA cB ¼ 0:2, in the full tetrahedron approximation of the CVM (red), and for cases where the VBCE of the probabilities are truncated at the triangle (green) and the tetrahedron (blue) clusters. In the Bragg–Williams approximation, not shown, the correlation for the regular triangle equals 0.008.
J.M. Sanchez, T. Mohri / Computational Materials Science 122 (2016) 301–306
305
0.08 zx3 zx4 0 z4x 0
z40
Full Tetrahedron
0.04
0. 12
13
14
15
16
kB T J2 Fig. 4. The correlation function for the regular tetrahedron, z04 , as a function of the reduced temperature kB T=jJ 2 j for the Ising model with J 2 < 0 at x ¼ cA cB ¼ 0:2, in the full tetrahedron approximation of the CVM (red), and for cases where the VBCE of the probabilities are truncated at the triangle (green) and the tetrahedron (blue) clusters. In the Bragg–Williams approximation, not shown, the correlation the tetrahedron equals 0.0016.
Fig. 5. Free energy of formation for the Ising model (J 2 < 0), on the bcc lattice in the Octahedron approximation of the CVM, as a function of the number of correlations included in the SDG and in the VB expansions of the cluster probabilities. Also shown are the free energies in the Bragg–Williams and in the full Octahedron-CVM approximations.
Fig. 6. Free Energy of formation for the fcc Ising model (J2 < 0) as a function of number of clusters kept in the expansion of the 13–14 points cluster probabilities of the CVM. Also shown are the free energies in the Bragg–Williams and in the full 13– 14 point cluster CVM approximations.
tetrahedron correlations in the SDG basis shown in Figs. 3 and 4 illustrate the in-equivalence of the truncated cluster expansion in different basis sets. The inappropriateness of truncating the cluster expansion in the SDG basis for x – 0 is further illustrated in Fig. 5 for the bcc Ising model (J 2 < 0) in the Octahedron approximation of the CVM [7].
Fig. 7. Free energy of formation for the fcc Ising model (J 2 < 0) as a function of concentration at kB T=jJ 2 j for the 13–14 points cluster approximation of the CVM with the cluster probabilities truncated at the nearest-neighbor pairs, compared to the Bragg–Williams approximation and the full 13–14 points cluster CVM approximation.
Fig. 5 shows the energy of formation obtained by truncating the expansions of the cluster probabilities in both the SDG and the concentration-dependent basis, together with the free energy of formation obtained in the Bragg–Williams and in the full CVMOctahedron approximations, for x ¼ 0:2 and at a reduced temperature of kB T=jJ 2 j ¼ 8:5. We see from the figure that truncating the expansion of the probabilities in the SDG basis at the pair level results in a free energy of formation that is in fact less accurate than that in the Bragg–Williams approximation, while truncating the expansion in the concentration-dependent basis gives a sequence of reasonably accurate values that converge uniformly to the full CVM solution. More importantly, it can be shown that truncating the expansion in the SDG basis will invariably lead to negative values of the probabilities X g ð~ rÞ for values of x that depart sufficiently from the 50/50 composition (x ¼ 0). As a final example, we consider the fcc Ising model (J 2 < 0) in the 13–14 point cluster approximation of the CVM. The approximation combines a 13 point cluster formed by a lattice point surrounded by its 12 nearest neighbors and the 14 point fcc unit cell [14]. The minimization of full CVM approximation requires solving 741 equations, which correspond to the total number of cluster correlations. Fig. 6 shows the free energy of formation as a function of the number of clusters kept in the cluster expansion, for x ¼ 0 and kB T=jJ 2 j ¼ 12:5. At the lowest level of truncation, i.e. keeping only nearest-neighbor pairs in the expansion of the cluster probabilities, the error relative to the full CVM approximation is 3.5%. Thus, the truncation of the probabilities in the concentration-dependent basis is particularly attractive for such large cluster since it results in relatively accurate configurational free energies with a significant reduction in the computational effort. As mentioned, the truncation of the cluster probabilities in the concentration-dependent basis, unlike that in the SDG basis, is valid for the entire concentration range. The free energy in the 13–14 points approximation, at a reduced temperature kB T=jJ 2 j is depicted in Fig. 7 for the case in which only nearest-neighbors correlations are included in the expansions of the probabilities. Also shown are the free energies in the Bragg–Williams approximation and in the full 13–14 points cluster CVM approximation. 4. Conclusions By means of several examples we have shown that the computational task of minimizing the CVM free energy functional can be significantly reduced by expanding the cluster probabilities in a
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concentration-dependent basis, and truncating such an expansion at different cluster levels. Significantly, most of the improvement afforded by the CVM over the Bragg–Williams approximation is achieved by truncating the expansion at the lowest levels, which typically keeps only pair correlations as the independent variables for the minimization step. Furthermore, we have used the expansion of the cluster probabilities in conjunction with the CVM to establish important differences between truncated cluster expansions in the SDG and in the concentration-dependent basis sets. In particular, the analysis shows that, despite the fact that all basis sets are related by a linear transformation, their equivalence does not hold when the expansions are truncated at any level.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
R. Kikuchi, Phys. Rev. 81 (1951) 988. J.M. Sanchez, F. Ducastelle, D. Gratias, Physica 128A (1984) 334. J.M. Sanchez, Phys. Rev. B 48 (1993) 14013. J.M. Sanchez, Phys. Rev. B 81 (2010) 224202. M. Asta, C. Wolverton, D. de Fontaine, H. Dreyssé, Phys. Rev. B 44 (1991) 4907. T. Mueller, G. Ceder, Phys. Rev. B 82 (2010) 184107. J.M. Sanchez, D. de Fontaine, Phys. Rev. B 17 (1978) 2926. J.M. Sanchez, D. de Fontaine, Phys. Rev. B 21 (1980) 216. J.M. Sanchez, D. de Fontaine, Phys. Rev. B 25 (1982) 1759. T. Mohri, J.M. Sanchez, D. de Fontaine, Acta Metall. Mater. 33 (1985) 1171. J.A. Barker, Proc. R. Soc. A 216 (1953) 45. T. Morita, J. Phys. Soc. Jpn. 12 (1957) 753. T. Morita, J. Math. Phys. 13 (1972) 115. A. Finel, Ph.D. Dissertation, Paris IV, 1987.