Order Parameter Functional Method in the Theory of Atomic Ordering

A. I. GUSEV and A. A. REMPEL: Order Parameter Functional Method 43

phys. stat. sol. (b) 131, 43 (1985)

Subject classification: 1.2; 22

Institute of Chemistry, Academy of Sciences of the USSR, Ural Scientific Centre, Sverdlovskl)

Order Parameter Functional Method in the Theory of Atomic Ordering BY A. I. GUSEV and A. A. REMPEL

An order parameter functional method is proposed which permits the description of structural order-disorder-type phase transitions and combines features of the cluster variation method and static concentration waves method. The method proposed ia employed to describe the atomic ordering in a nonstoichiometric niobium carbide.

CTpywypHbIe (Pa30~b1e nepexomI Tnna I I O ~ R ~ O H - ~ ~ C I I O ~ R ~ O H EZ coqeTarouuii3 oco6eH- npen.TIO)€CeH MeTOH @yHKqHOHaJIa IIapaMeTpOB IIOpHnHa, II03BOJIfUOIIIkffi OnllCbIBaTb

HOCTU MeTOHa BapllaukiH HJIaCTepOB kf MeTOAa CTaTkfUeCKkfX KOHUeHTpaUUOHHbIX BOJIH. PaCCMOTpeHO IlpkfMeHeHEle IIpe&TlOHceHHOI'O MeTOAa RJIR OIlllCaHllR aTOMHOrO yIlOpHAO- 'XeHHR B HeCTeXkfOMeTpLl~eCKOM ~ap6nge H E I O ~ H R .

1. Introduction One of the fundamental problems of the statistical theory of atomic ordering is to construct the thermodynamic potential of a system of interacting particles. Since the exact partition function of a three-dimensional Ising lattice is difficult t o calculate, recourse is normally made to approximate methods. The most common technique of describing structural phase transitions in the various systems is the mean-field approximation, in which the interaction of every single particle with the other particles is described with the aid of the mean field determinable from the self-consistence condition. It has turned out that an efficient tool to describe structural order-disorder phase transitions in substitutional and interstitial solid solutions is the static concentration wave method [l] which is one of the mean-field approximation versions. The gist of this method lies in developing the occupation probability for each of the nonequivalent sublattices, n( r ) = ( c ( r ) ) , as a Fourier series in superstructural vectors k, (r is a vector that determines the position of an Ising lattice site).

Different approximate methods are used to take into account the correlations in structural order-disorder phase transitions. The techniques involved differ from one another in the way the thermodynamic characteristics of a system of N interacting particles are represented in terms of those of the finite groups of these particles, which form some geometrical figures, i.e., clusters. One of the most promising and frequently used cluster approximations is the cluster variation method devised by Kikuchi [2, 31 and refined by Hijmans and de Boer [4]. Owing to sequential allowance for clusters of different size and for their interrelation, and also for the many-particle correlations in the atomic arrangement on the sites of the figures, the Kikuchi approximation is capable of yielding tjhe most exact, according to [5, 61, results for transition temperatures. According to [7], the cluster variation method is a generaliza- tion of the entire quasi-chemical trend in the investigation of atomic ordering phenom- ena.

l ) Pervomaiskaya 91, 620219 Sverdlovsk, USSR.

44 d. I. GUSEV and A. A. REMPEL

The present paper proposes an order parameter functional method of describing atomic ordering, which allows structural order-disorder phase transitions to be described not only qualitatively but also quantitatively. The method proposed combines features of the cluster variation method, where the interaction of particles within a cluster and their many-particle correlations are taken into account exactly, and those of the static concentration waves method, which permits a detailed account of the lattice symmetry and, accordingly, the long-range interaction in the crystal.

2. Basic Equations of the Mcthod

In the cluster variation method [2 to 41 the crystal is described as a set of s-type figures with i-configurations and the sequence of essential figures {s } required to describe the crystal involves a basis cluster and overlap figures [4] (what is implied by figures is different combinations of crystal lathice sites). Each i-configuration of figure of s-type is specified by a probability P!?) equal to the ratio of the number of figures of s-type with i-configuration to the total number of figures of s-type in the crystal. Because of symmetry, some configurations in the lattice may be equivalent ; we denote the number of equivalent i-configurations of an s-type figure (or the multiplicity of the i-configuration) by A?). The definition of Py) and Ap) entails the usual probability normalization condition C Ap)Pp' = 1. According to [4], the free

energy of a crystal composed of N atoms in the cluster variation method is represented as

z

n

where y@) is a coefficient that takes into account the reevaluation of the number of microstates as a result of the overlap of figures, and EJS) is the energy of s-type figures with i-configurations. In the cluster variation method the energies &(iS) are treated as model parameters and are assumed to be known. Therefore, the free energy of the crystal depends only on the probabilities Pi'"), which are variable.

*4n essential point in the cluster variation method is the choice of a basal cluster figure. The latter should correspond to the geometry of the crystal lattice, allow for the mode of the interparticle interactions in the crystal, and be sufficiently large in size to allow for the most important correlations in the lattice. For the ordering to be correctly described with the help of this method, the minimum size of the basis cluster in the ideal case is limited by the unit cell. For complicated lattices whose unit cells contain several dozens of atoms, il description of the ordering by such a cluster meets with insurmountable computational difficulties. The latter may be avoided if the atomic ordering is described using B cluster smaller in size and with allowance for long-range interactions in the crystal. To this end, the cluster probabilities, which are independent variables, should be related to the long-range order parameters. This can be done with the help of the atomic distribution function n(r) of the crystal lattice.

In the static concentration wave method [l] the atomic distribution in substitutional or interstitial solutions is described by means of the distribution function n(r) which represents the probability of an atom of a given species being located at an Ising lattice site r . In a disordered crystal the probabilities are equal for all sites r and coincide with the relative fraction of sites occupied by atoms of a given species, i.e., n(r ) = c. The spatially periodic modulation of the atomic distribution in crystal lattice sites, i.e., the deviation of the probability n ( r ) from its value in the case of a statistical distribution, is described by a standing static plane wave with a wave

Order Parameter Functional Method in the Theory of Atomic Ordering 45

vector k,. Therefore, the distribution function may be represented in terms of the relative fraction of occupied sites and the superFosition of static plane waves

n( r ) = c + C q[y (ks ) exp (ik,r) + y * ( k s ) exp (-iksr)l , (2) where exp ( ik,r) is a static plane wave of amplitude qy(ks), q is a long-range order parameter, y(ks) a coefficient which is chosen so that the long-range order parameter in a completely ordered state is equal to unity, and the summation is taken over all nonequivalent superstructural vectors involved in the first Brillouin zone of the disordered crystal. As is clear from (2), the number of long-range order parameters is determined by the number of superstructural wave-vector stars. In turn, the distribution function n(r ) on the set of Ising lattice sites takes on a number of different values which exceeds the total number of long-range order parameters by unity.

Let us see in which fashion the cluster probabilities may be expressed in terms of a distribution function. If x ( ~ ) is a number which indicates by which factor the number of s-type figures exceeds the number of lattice sites, then the total number of s-type figures in a crystal with N sites is equal to N d S ) . Therefore, the probability of an s-type figure of i-configuration may be represented as the mean probability for the entire lattice,

where Pi;) is the probability that an s-type figure of i-configuration will form at the place of the j-th s-type figure. The summation in (3) is carried out over all s-type figures present in the crystal. If the ordering of the atoms of a given species in the crystal is described by a distribution function taking on t values, which signifies that the Ising lattice is broken up into t sublattices, then the situation for specified values of long-range order parameters ql, qz, ... , qt-l will be as follows. To any Ising lattice site Y there will correspond one of the t values of the function n( r ) , depending on to which of the t sublattices the site of interest belongs. Since the distribution function describes the probability that some lattice sites will be occupied by atoms of a given species irrespective of other lattice sites, the probability of a configuration in the general case will be a sum of the products of the occupation probability of the sites involved in the figure,

Here 1 = 01, @, y , ... , r is the species of the atoms, Us), y") stands for the notation

and number of s-type figure sites occupied by atoms of species 1 (c Lis) = R(s) is I=a

the total number of lattice sites involved in an s-type figure), and n2(rjkl(#)) denotes the values of the distribution function of species 1 atoms on s-type figure sites occupied by species 1 atoms and corresponding to the positions of any one of the t sublattices. The summation over k in (4) arises from the fact that there exist several different ways of carrying the i-configuration of an s-type figure into the j-th s-type figure in the crystal; the number of modes of superposition coincides with the multiplicity of the configuration, A?). It can be readily verified that in the absence of long-range order, when q = 0 and n(r ) = c , (3) and (4) describe the probabilities Pp) by a

46 A. I. GUSEV and A. A. REMPEL

binomial dist,ribution. Since the distribution function n( r ) is directly related to the order parameters q, the probability Pis), which according to (3) and (4) is expressed in terms of the distribution function, is an order parameter functional. Thus, the pobabilities of all figures of a sequence {s} may be represented as functionals determinable by summation over an infinite number of crystal lattice sites. It is evident that the order of such a functional is equal to the number of sites involved in an s-type figure.

Representing the probabilities of figures with the help of (3) and (4) allows the free energy of a crystal whose lsing lattice during ordering breaks down into t sublattices to be written in the general form in terms of the distribution function. In order to take the long-range correlations into account correctly, the entropy term in (1) should be written in the form

t

S = - N k , c C nz(r ) In nl(r ) , r l = u

(5)

where the summation is performed over all Ising lattice sites r . With allowance for (1) and (3) t o (5) , the free energy of the crystal may be represented in the general form in terms of the distribution function

n y(s) N,Jo &) T

F = c - C s!) C C fl % t ( r j k z ( s ) ) + N ~ B T C C ni(r) ln ni(r) (6)

Equation (6) thus obtained for the free energy of a crystal with any degree of ordering is a mathematical formulation of the method proposed to describe atomic ordering. A major feature of the method proposed, which has been developed on the basis of two methods, viz., the static concentration wave method and the cluster variation method, is that the probabilities of different figures (clusters) are represented in terms of the distribution function, thus allowing the long-range correlations in the crystal to be taken into account implicitly.

For the most common ordering of atoms of two species (or atoms and vacancies) in two sublattices, when n,(r) + ns(r ) = 1 according to the normalization condition for any arbitrarily chosen lsing lattice site r , (6) may be represented as

s=a d " ) i G S j-1 k = l z=a p)=o r Z=a

+ NkBT c {n,(r) In na(r) + r1 - n&)l In [1 - nu(r)l) (7) r

By explicitly taking into account in (6) and (7) the values of the distribution function expressed in terms of long-range order parameters and by subsequently mini- mizing the free energy F of the crystal with respect to these parameters, it is possible to obtain all the principal characteristics of an order-disorder phase transition, viz., the transition temperature as well as the behaviour of the free energy, entropy, and long-range order parameters during the transition and thus the kind of the phase transition.

3. Structural Order-Disorder Phase Transition in a Nonstoichiometric Niobium Carbide

Phase t,ransitions such as ordering in nonstoichiometric compounds result from the redistribution of interstitial atoms and structural vacancies in nonmetal sublattice sites. The existence of ordered nonstoichiometric compounds has been established for many transition-metal carbides and nitrides, for example V,C, [8], V,C, [9, lo],


Nb4N3 [ll]. I n [12 to 161 it has been found experimentally that a structural order- disorder phase transition occurs in a nonstoichiometric niobium carbide NbC, in the composition range NbCo,a-NbC&,aa over the temperature interval between 1258 and 1355K. The ordered Nb,C, phase that arises has monoclinic symmetry and belongs to the space group C2/m. A calculation performed in [15] has shown that for an arbitrary composition c (for NbC, c = y) and arbitrary long-range order parameters q,, q2, and q3 the distribution function of carbon atoms, n( r ) , written in the rectangular coordinate system of the initial f.c.c. crystal structure and describing the ordering in the niobium carbide, has the form

1 1 4n

where q, yI, zI are coordinates that locate the Ising lattice sites r (in the present case the nonmetallic f.c.c. sublattice is an Ising lattice). Equation (8) manifests that in the general case the ordering in the niobium carbide is described by three long- range order parameters. For an ideal ordered crystal, the composition of the ordered niobium carbide phase corresponds, according to calculation, to Nb,C,. In the case when the long-range order parameters are equal to one another, q, = q2 = q3 = q, the nonmetallic sublattice of the disordered niobium carbide during ordering breaks down into two sublattices, that of carbon atoms and that of vacancies, and, accordingly, the distribution function takes on two values: n, = y + 116 7 (probability of finding a carbon atom in a carbon sublattice site) and n2 = y - 51617 (probability of finding a carbon atom in a vacancy sublattice site). It should be noted that, in accordance with the distribution functions n ( r ) found when considering any six nonmetal sublattice sites generating an octahedron, five of them will be the sites of the carbon sublattice, and one of them the site of the vacancy sublattice.

To describe a structural order-disorder phase transition in NbC, with the help of the order parameter functional method proposed, it is necessary to choose a basis cluster. As demonstrated in [12, 141, for f.c.c. nonstoichiometric compounds with NaCl structure i t is physically justifiable to choose the basis figure to be an octahedron formed by six nonmetal sublattice sites; a metal atom is located at the centre of the octahedron (Big. 1). Choosing such a basis cluster allows all crystal lattice sites to

P5 p6 p7 p8 p9 Fig. 1. Nonequivalent configurations of an octahedral basis cluster. o metal, 0 C, ~3 vacancy


be taken into account. The sequence of special figures {s}, required for a correct description of the crystal and found according to the procedure suggested in [a], comprises a nonmetal sublattice site (s = a), a pair of adjacent nonmetal sublattice sites (bond) separated by a distance equal to the octahedral edge (s = b), and an octahedron made up of six nonmetal sublattice sites (s = c). The relationship between the special figures of the sequence {s} is established with the aid of a matrix I U ( ~ ) ( ~ ) [ whose elements show how many m-type figures an s-type figure contains. To take into account the revaluation of the number of microstates as a result of the overlap of the figures, coefficients y(s) are introduced which may be found by solving a linear system composed of equations such as

Here x ( ~ ) is a number which shows how many times the number of m-type figures is larger than that of Ising lattice sites, N (x@) is equal to the ratio of the number of m-type figures which simultaneously contain any arbitrarily chosen lattice site to the number of sites in an m-type figure). The matrix elements ( U @ ) ( ~ ) I and the values of dm) and y(s) for a NaC1-type lattice are presented in Table 1.

T a b l e 1

Matrix lu(m)(a)l for the lattice of NaC1-type

figure figure of s-type d m ) of m-type

a b e

a 1 2 6 1 b 0 1 12 6 C 0 0 1 1

The sites of the same s-type figure may either be occupied by atoms or be vacant. Therefore, to each figure there may correspond several versions of the relative positions of occupied and vacant sites, i.e., several i-configurations with probabilities P',"). Configurations that coincide with each other after symmetry operations have been applied to them are equivalent ; the multiplicities of equivalent configurations, A?), are given in Table 2. Bor a figure a (site) two configurations are possible: an atom- occupied site (configuration probability Pg)) and a vacant site (configuration prob-

T a b l e 2 Multiplicities A?) of i-configurations of s-type figure

S i

0 1 2 3 4 5 6 7 8 9

a 1 1 b 1 2 1 c 1 6 3 1 2 8 1 2 1 2 3 6 1


ability Pp)). The probabilities Pib), Pib), and Pp) correspond to a figure b with the following configurations: C-C, C - 0 , and 0-0 (C is a nonmetal sublattice site occupied by an atom; 0 is a vacancy). For a basis cluster as an octahedron ten nonequivalent configurations are possible : a complete cluster (all the six sites are occupied by atoms), a cluster with one vacancy, etc., up to a fully defective cluster (with all the six sites being vacant), to which there correspond the probabilities from Pp) to

v

P t ) (Fig. 1). Taking into account the sequence of figure { s } chosen and the calculated values of " -

the distribution function n ( r ) of carbon atoms, let us see how the method proposed may be used to describe a structural order-disorder phase transition in a nonstoichiometric niobium carbide. For this purpose we use (7). In the case under consideration the ordering of carbon atoms and vacancies occurs in two sublattices and therefore in (7 ) we have /3 = 0 and n,(r) = n( r ) , where n(r) is the distribution function of carbon atoms. As has already been stated, the energies s(j) of s-type figures with different i-configurations are parameters of the theory and do not depend on the probabilities of these configurations, Pis), i.e., on the composition and degree of order of the compound concerned, and therefore the energy of the crystal changes only when the probabilities Pp) vary. It follows from the representation of the free energy that the first summand in (7) has the meaning of an enthalpy. This enables us to find the energies of the clusters if for some state of the crystal we know its enthalpy of formation and the probabilities Pi'"). It is most convenient to take this state to be a disordered state in which the probabilities Pis) are defined by the binomial distribution,

( 6 )

Cg:) (1 - y w - g i ), (10) 1

p;>)bin = _.___ 0 c 1:;)

j=1

where qi(6) is the number of vacancies in the i-configuration of an s-type figure, and o) is the number of nonequivalent configurations of an s-type figure that have an equal number of vacancies, qis).

The results of the numerous experimental and theoretical papers [17, 181 indicate that the C-C, Go, and 0-0 interaction energies are small in carbides. For this reason, these bonding energies as well as the energies of the C (occupied by a carbon atom) and - (vacant) sites were assumed in the present paper to be negligibly small compared with cluster energies. The functional form of the dependence of the enthalpy of formation AH&98 on the composition of the disordered carbide NbC, [19], and the functional dependence of the heat capacity C, on the temperature and composition of the disordered niobium carbide [16, 201 are known. In accordance with the approximation adopted, these data allow the energies of the clusters, E?), to be found from the equation

It was also assumed in the calculation that the energies of clusters that have an equal number of vacancies but different arrangement are equal to each other. By solving a system of linear equations such as (ll), written for different compositions of NbC,, it has been possible to find the temperature dependences of the energy of clusters with different number of vacancies.

If the calculated values n, and n2 of the distribution function n(r) and the approximations made are taken into account explicitly, then (7) for the free energy of the 4 physica (b) l 3 l j l


crystal of a non-stoichiometric niobium carbide with any degree of ordering becomes

where is the number of carbon atoms occupying the positions of the carbon sublattice in a c-type figure (octahedron), and yp) is the number of vacancies occupying the positions of the vacancy sublattice in a c-type figure.

4. Computation Results

It follows from (12) that the free energy of the crystal in the ordering model under discussion is determined by its composition y, temperature T , and long-range order parameter q. For a given composition and temperature, when the values of the cluster energies &(iC)(T) are known, minimization of the free energy F with respect to the long-range order parameter permits the determination of all the necessary characteristics of the crystal in the equilibrium state, viz., the long-range order parameter q, free energy F , and entropy 8. All numerical calculations associated with the minimization of the free energy and with the determination of the characteristics of the equilibrium state of a nonstoichionietric niobium carbide were carried out according to specially developed programs using BESM-6 and ES-1060 computers.

According to computation results, the long-range order parameter decreases slowly with increasing temperature and goes abruptly to zero when the order-disorder transition temperature is reached. At the same transition temperature Ttrans the temperature dependence of the free energy F ( T ) exhibits a kink, and the temperature dependence of configurational entropy X( T) displays a pronounced jump typical of a first -order phase transition. This is entirely consistent with experimental evidence [12, 15, 161, according to which the ordering in NbC, is really a phase transition of the first kind. Fig. 2 and 3 present the temperature dependence of q, P, and S calculated for the niobium carbide NbC0.83; similar 1,emperature curves of q, P, and S were obtained for other compositions of the niobium carbide NbC, undergoing order-

300 600 900 7200 1500 1800 2700 T(KI - TIK)+

Fig. 2 Pig. 3 Fig. 2. Jump of the long-range order parameter rl during order-disorder phase transition in niobium carbide NbC0.83

Fig. 3. Variation of the free energy P and entropy S of niobium carbide NbC0.83 during ordering


ing. The results of the computation also indicate that the region of homogeneity of the ordered phase is somewhat wider than follows from neutron diffraction data [12, 151.

The calculated transition temperature values give a fairly good fit to the Ttrans values determined experimentally on an HT-1500 CETARAM calorimeter using the DTA method : For the carbides NbC0.81 and NbC0,83 the estimated values of Ttrans are equal to 1300 and 1304K, respectively, whereas the experimental values of this transition temperature are 1258 and 1304 K, respectively. For the niobium carbide NbCo,s3 the estimated values of the heat of the order-disorder phase transition Nb,C,- NbC0.83 is AHtrans = 3.3 kJ/mol. Experimental dat on the heat of the order-disorder transition have been reported only for one of the nonstoichiometric transition-metal compounds, viz., vanadium carbide ; the experimental value for the V,C,-VCO.S~ transition is equal to AHtrans = (1.7 f 0.7) kJ/mol [21], which coincides in order of magnitude with the heat of the order-disorder transition for the niobium carbide.

On the whole, the calculation of the fundamentally true values of the principal characteristics of the order-disorder phase transition in a nonstoichiometric niobium carbide corroborates the correctness of the major premises of the order parameter functional method proposed and its applicability for describing the ordering in interstitial and substitutional solid solutions. It is worth noting that when using the proposed method to describe atomic ordering in metallic alloys it is abolutely necessary that the energy of pairwise interactions is taken into account.

References [l] A. G. KHACHATIJRIAN, Theory of Structural Transformations in Solids, John Wiley & Sons,

[2] R. KIKUCHI, Phys. Rev. 81, 988 (1951). [3] M. KURATA, R. KIKUCHI, and T. WATARI, 5. chem. Phys. 21, 434 (1953). [4] J. HIJMANS and 5. DE BOER, Physica (Utrecht) 21, 471 (1955). [5] V. G. VAKS and V. E. SCHNEIDER, phys. stat. sol. (a) 35, 61 (1976). [6] V. G. VAKS and V. I. ZINENKO, Solid State Commun. 39, 643 (1981). [7] N. S. GOLOSOV, Izv. vuzov, Ser. fiz. No. 8, 64 (1976). [S] C. H. DE NOVION, N. LORENZELLI, and P. COSTA, C. R. Acad. Sci. (France) B %63,775 (1966). [9] 5. D. VENABLES, D. KAHN, and R. G. LYE, Phil. Mag. 18, 177 (1968).

New York 1983.

[lo] 5. BILLINGRAM, P. S. BELL, and M. H. LEWIS, Phil. Mag. 26, 661 (1972). [ll] N. TERAO, Japan 5. appl. Phys. 4, 353 (1965). [12] A. A. REMPEL and A. I. GUSEV, Ordering in Nonstoichiometric Niobium Monocarbide, In-

[13] A. A. REMPEL, A. I. GUSEV, V. G. ZUBKOV, and G. P. SHVEIKIN, Dokl. Akad. Nauk SSSR

[14] A. I. GUSEV and A. A. REMPEL, phys. stat. sol. (a) 84, 527 (1984). [15] A. I. GUSEV and A. A. REMPEL, Fiz. tverd. Tela 26, 3622 (1984). [ l6] A. A. REMPEL, S. Z. NAZAROVA, and A. I. GUSEV, phys. stat. sol. (a) 86, K11 (1984). [17] H. 5. GOLDSCHMIDT, Interstitial Alloys, Butterworths, London/Washington 1967. [18] L. E. TOTH, Transition Metal Carbides and Nitrides, Academic Press, Inc., New York/London

[19] E. J. HUBER, E. L. HEAD, C. E. HOLLEY, E. K. STORMS, and N. H. KRIKORIAN, J. phys.

[20] P. V. GELD and F. G. KUSENKO, Izv. Akad. Nauk SSSR, Ser. Metallurgia i toplivo No. 2,

[21] W. S. WILLIAMS, High Temp.-High Press. 4, 627 (1972).

stitute of Chemistry, Sverdlovsk 1983 (in Russian).

275, 883 (1984).

1971.

Chem. 65, 1846 (1961).

79 (1960).

(Received March 15, 1985)

4'

Documents

Order Parameter Functional Method in the Theory of Atomic Ordering