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Orientational disordering in crystals—I
.I. Phys Chem. Solids Vol 41, pp 685-691 Pergamon Press Ltd. 1980 Pnnted I,, Great Br,ta,n
ORIENTATIONAL
DISORDERING
IN CRYSTALS-I
A MODEL FOR THE EVALUATION OF THERMODYNAMIC PROPERTIES OF MELTING TRANSITIONS IN ATOMIC AND PLASTIC CRYSTALS CHODZIWADZIWA C. MJOJO~ Chemistry Department, Faculty of Science, Australian National University, Box 4, G.P.O., Canberra, ACT 2600, Australia (Received
23 January
1979; accepted in revised form 2 August 1979)
generalisation of two-sublattice models ofmeltmg transitions in atomic and molecular crystals (Lennard-Jones and Devonshire, Pople and Karasz; Amzel and Becka) has been attempted. The intention was to develop a scheme by which parameters, such as the number of sublattices and distinguishable molecular orientations can be selected. It is found that the choice of these parameters made by LennardJones and Devonshire, and by Pople and Karasz, protects the theory to some extent from some of the more diilicult physical issues. A single-lattice model also emerges as another possibility in the series of models that may be called upon. Thus a single-lattice model, based on concepts from Eyring’s Significant Structure Theory of Liquids for the melt, and standard thermodynamic functions for the solid, has been developed to evaluate thermodynamic properties of melting transitions in atomic and molecular crystals. Despite the remarkable simplicity of the model predicted values of entropy of fusion for both argon and methane are in good agreement with experimental data. Mean positional disordering energies for the same substances have also been computed. Abstraet-A
INTRODUCTION
There exists a functional relationship between the entropy of fusion and the fractional volume change at the melting point for simple substances (e.g. atomic crystals and plastic crystals) whose melting transitions involve positional disorder only [ 11. This observation has been the basis for treating the melting transition as a positional order-disorder phenomenon in a number of theoretical treatments. The two-sublattice model used by Lennard-Jones and Devonshire [2] is one of the few more successful treatments of the melting process of inert gases. This model has been extended by Pople and Karasz [3] to cover the case of molecular crystals with orientational disorder by allowing the molecules to take two distinguishable orientations in the plastic crystal phase. Various improvements have been suggested [4,5] to the Pople and Karasz model; but the most notable one is that due to Amzel and Becka [6]. In the treatment due toAmze1 and Becka an extension of the Pople and Karasz model to include the case ‘of the general number of distinguishable molecular orientations on a lattice site was attempted. Both the Pople-Karasz and Amze-Becka theories of the melting process contribute some insight into the problem of orientational disorder in crystals. This paper attempts to develop a more realistic model of melting transitions involving atomic and plastic crystals.
Despite the success of the two-sublattice models in affording both quantitative and qualitative descriptions of real systems in some accord with experimental data, a number of the assumptions made have been shown to be contrary to experimental observations [7]. Also these models assume that melting in both atomic and molecular crystals occurs predominantly via the migrations of constituent units from lattice sites to interstitial sites. Chadwick and Sherwood [8] observe, both from theoretical and experimental evidence, that vacancy migration is fundamentally responsible for self-diffusion phenomena in molecular crystals. Thus the application of a two-sublattice model to the melting phenomena in molecular crystals as proposed by Pople and Karasz, and Amzel and Becka may not be realistic. Helpful reviews on the theories ofmelting are presented in Refs. [9-121. Reference [12] provides the necessary background for the standard thermodynamic functions introduced in this paper. Would the theory be improved by postulating more than two sublattices? The generalisation of the theory to an arbitrary number of sublattices is examined first. The solid is then assumed to be represented by the configuration that maximises the partition function of the general model with respect to the number of sublattices. THE MULTISUBLATTICE
MODEL OF A LATTICE GAS
Suppose we have m sublattices of N lattice sites each tPresent address: Chemistry Department, Chancellor and N molecules with n allowed orientations. The College, University of Malawi, P.O. Box 280, Zomba, Malawi. ordered solid is defined as consisting of all the N 685 IPCS Vol 41, No. 1-A
686
CHODZIWADZIWA
molecules in one of the m sublattices and one of the n distinguishable orientations. Then the liquid and gaseous states will correspond to the situation when the N molecules are distributed equally over a given number of sublattices and orientations, the situation corresponding more and more to the gaseous state as m is increased. The plastic crystal phase then will be defined as that state of aggregation of matter in which all the molecules are predominantly in one of the m sublattices and are equally distributed over the n distinguishable orientations. Let p, stand for the fraction of molecules in the ith sublattice and uj for the fraction of molecules in the jth orientation. The crucial assumptions are those which relate the energies of molecules displaced from a site on the initial sublattice to a site on another sublattice, or else (on the same site) from a reference orientation to another orientation. Definitions are simple if the lattice is initially perfect, i.e. if all the molecules are on the same sublattice, in the same orientation. Let us now define E’ as the energy required to shift one molecule to a site on any other sublattice (such “other’ sublattices being thus degenerate); and E”, as the energy required to move one molecule from the reference orientation to any other orientation (these “other” orientations being thus degenerate). For higher degrees of disorder, decisions have to be taken about interactions between displaced molecules. The choice Pople and Karasz, and following them Amzel and Becka, made is to say that E’ and E” apply to all neighbouring pairs of molecules that are not in the same sublattice or orientation. However, there is no orientational disorder energy (E”) between molecules in different orientations but on different sublattices. This last provision means that orientational disorder carries no energy penalty if molecules are positionally disordered, and thus the desired element of cooperation is brought into the model. The average configurational energy per molecule then takes the form
C.
MJOJO
If this distinction is drawn, then a limiting case is E; = Es = 0. In either situation, expressed through (1) and (2), there should be equal populations on all sites or orientations other than the initial ones. Then since X:p, = 1, cu, = 1: (if
P, = (1 - p,)l(m - 1) g, = (1 - a,)/(n - 1) PROPERTIES
1) (3)
ci# 1).
OF THE MULTI-SUBLATTICE
MODEL
assembly partition function, customarily written on the basis of an Einstein model, has the form The
Q = (QAN.Q, where Q, is the molecular partition function and Q, =
1 exp{ - E(m,n, P. o)lk Ti
(4)
PJ
is the assembly configurational part; Q, = 1 for a perfectly ordered state. Assuming that Q, is independent ofdisorder (an assumption improved in Paper II of this series) equilibrium conditions associated with disordering depend solely on Q,. To evaluate Q, the Bragg-Williams approximation [12,13] is employed which simply requires replacing the argument of the exponential in eqn (4) by its average, and identifying the total number of terms in the sum as g(m, n, p, a), the number of ways of arranging particles for given values of m, n, p, CJ.Thus eqn (4) is replaced by
Q, = g(m, n, P, e)exp { - (E(m, n, P. a))lkTJ
(5)
where
&,n,p,a)
=
(N!}”
n (J”PiI!
fi
m
,vl iNPi} ! (N(l -
P,)) ! i=l,Fl {NPP,)!
(6)
An alternative view is to regard the initial positions and orientations as being so much deeper on the molecular potential energy surface that E’ and E” pertain only to displacements from the original lattice (occupation fractions pi and a,); for interactions between disordered molecules different quantities, say E; and E; must be used. Then the mean configurational energy per molecule has the form
= ? PS& +
ipf
i
n
+ f i>l
k>l
i
( l>l n
c P,P&, k>i
a,a,E” \
(2)
and (E(m, n, p, a)) is identified either with the expression (I) or with (2). Using eqn (3) to eliminate pi, CT,for i,j # 1 the more general case (eqn (2)) becomes
=
E’Np,(l x (mpf +
-2p,
p1) + E”No,U + 1)/b
-
aI)
1)
)E;N(~ - pl)‘(m - 2)/(m - 1)
+ *EE;N(I - G1)‘(mp: - 2p, + 1) x (n - 2)/(n - l)(m - 1).
(7)
Equations (6) and (7) reduce to the results of Amzel and Becka if m = 2, Ei = E’ and Ei = E”. Further it will be observed that the values of E& and E; are immaterial if m = 2 and n = 2 respectively. Also if m = n = 2, then the issue concerning possible improvements to the existing theory which initiated the
687
Orientational disordering in crystals-I definition of the elementary energy quantities is of no consequence. This case is of course the Pople-Karasz model. Thus it would seem that the fortuitous choice of these parameters which LJ-D and Pople and Karasz made protects the theory, to some extent, from one of the more difficult physical issues. These parameters were chosen arbitrarily and the choice of m = 2 was mainly encouraged by developments in the crystal statistics of alloys. Using eqn (3) again, and applying the Stirling approximation in eqn (6), the free energy is obtained as follows: A, = RT[p,ln(p,)
+ (1 - pr)ln’,(l
+ (m - 2 + Pi)ln{(m +a,ln(a,)+
- &)‘/(m - 1))
- 2 + ~,)/(m - l)}
(1 -a,)ln((l
-c,)/(n-
l)}]
+
(8)
Interest is in the stationary values of A,, and so eqn (8) may be differentiated with respect to m, n, p, and o. Illustratively, dA,/am = 0 if -(mx{E”a,
l)21n{(m-2+p,)/(m-
1)}
= (1 - PI) MA1 - PI) + (1 - 01) + )E;(l - o,)(n - 2)/(n - l)}]/kT
(9)
This condition is always satisfied for all values of m if p1 = 1;ifo i = l/n the plastic crystal phase is obtained. This result and the observations made by Chadwick et al. alluded to above offer an encouragement to proceed, in the next section, and employ the single lattice model (m = 1) in which there may be vacancies. However, the model that is developed shortly (m = 1) and the Pople-Karasz and Amzel-Becka (m = 2) models are merely the first two of a series of models which might be called upon.
fraction of neighbouring positions to a vacancy which are filled by molecules is VdK there are
moles of gas-like molecules. The remaining yV/Vmoles comprise solid-like molecules. It follows therefore that the molecular partition function is constructed from the Einstein partition function for solids to account for the solid-like molecules; and from the translational partition function for an ideal gas, for the gas-like molecules. The locked lattice vacancies present in solids at finite temperatures will be assumed to make a negligible contribution to the thermodynamic properties of the system, since according to this model such vacancies will be present in both the solid and liquid and only a very small fraction of them will become fluidised. Chen [15] has summarised the evidence for the structure of liquids based on the similarities between the thermodynamic properties of liquids and solids such as densities, specific heats and compressibilities. Further it is now well established from X-ray diffraction studies that nearest neighbours in the melt are at essentially the same distance as the solid. The significant structure model, on which the present theory is based, observes the similarity between liquids and solids, and views the melt as a distorted fluctuating lattice caused by the addition of holes to the solid during the melting transition. For the present purpose the assembly molecular partition function is defined as follows, Q,(P)/Q,(~)
= {Q,,(p).Q,,(p))IQ,,(l)
(10)
where Q,,(P)
= (QmiY”“”
Qnts(1) = (QmiF A SINGLE-LATTICE MODEL FOR POSITIONAL DISORDERING IN ATOMIC AND PLASTIC CRYSTALS
Having in mind the discussion above an attempt is made to discuss melting in a way consistent with a single-lattice model of positional order-disorder transitions in atomic and plastic crystals. The theory will be modelled around argon and methane; substances which have received extensive studies and have the relevant experimental data. From the Significant Structure Theory of Liquids [14] the concept is adopted that melting is to be seen as a consequence of the creation of fluidised vacancies leading to a volume expansion characteristic to the melting process. According to this model the fluidised vacancies are, on the average, of molecular size so that the number of moles of such vacancies per mole of occupied sites is (V - V,)/V, where I/ and V, are the molar volumes of liquid and solid respectively. It is argued that the energy for the creation of a lattice vacancy is equal to the energy of sublimation. Since the
and Q,(;(P) = {(277mkT)3’2(V {NV-
- t$)/h3}N(“-V~)‘V WV!
where the assembly molecular partition function has been defined relative to that of the positionally ordered solid at absolute zero including the zero point energy. Qmican be either the Einstein partition function or the Debye partition function, the subscript i denoting the ith mode in the case of the former or the mode corresponding to the Debye cut-off frequency in the latter case. It will be found later that both the Einstein and Debye thermodynamic functions lead to good agreement of the predicted properties of real systems with experimental data, though the Debye function appears to be a little better. Let x = 1 - V’K the number of moles of the gas-like molecules per mole of
688
CHODVWADZIWA C. MJOJO
substance. Then the relative free energy is given by the following expression.
This expression may be written in the form
!G,(P) - G,(l )}IRT= X{In (Q,,) + In (NA) - In [(27~nkT)~‘~~~/h~] - lj. (11) The problem of constructing the appropriate assembly configurational partition function is considered next. For one mole of substance the total number of lattice sites in the melt is given by N,,, = n, + N
T2
T,
(12)
where n, = N(I/- V,)/V,, the number of fluidised vacancies and T, is the melting point. Therefore p is defined as follows P = NIN,,, = YIV
(13)
and (1 - p) = nJN,,
= 1 - V’V
(14)
where p, a long range positional order parameter, is the fraction of lattice sites occupied by molecules on average. Note that L, the fraction of gas-like molecules, and (1 - p), the fraction of vacant sites, are numerically identical. Both quantities are retained, however, and will be used according to what aspect of the problem requires emphasis, i.e. the population of gas-like molecules or vacant sites. Therefore the assembly configurational partition function in the Bragg-Williams approximation takes the form
Qc ={pN,,,j
’
(No,)!
N,,,! ! { 1 - p)N,,,) !
($~)!ln-’
where E, = Eb” + E1.2’+ Ea’ + . . .
Thus, for a completely vacant lattice (p = 0),
(E(l,a))
GYP,1)) = N,,,{U - PE” + (1 -
+ (1 -
P)~E~’ +
= N,,,(l - p){(E;” - ,#f
+ 2Ei3’ +
’ -) + . . .).
-a,)
(18)
(E(P, a)) = (E(P, 1)) + (E(l,a)). free energy, after in eqn (15) is
G,(P, 0) - G,(L 1) = ET(l& + (1 - p)ln(l + (1 - a,)ln{(l
(19) applying
ln P
-p)> -or)/@
+ arlna, - 1))) + (E(p,c)) (20)
where & = 1 + nJN, the total number of moles of lattice sites. From (11) and (20), the Gibbs free energy of the assembly relative to the ordered solid (ignoring work terms for the present) is given by the following expression G(P, CJ)- G(f, 1) = RT((ln (Qmi) - In [(2mnkT)3%,/h3] - 1)~
P)‘J%*’
. .) + El” + Ea’ +
= NE”a,(l
where it has been assumed that E; (the interaction energy between molecules in the disordered orientations) is negligible. This model for orientational disordering has been used in the study of binary phase diagrams of plastic crystals comprising optically active quasi-globular molecules [ 161. Finally
Stirling’s approximation where ur = 1 for the orientationally ordered solid, and ur = l/n for the plastic crystal phase. The energy term (E(p, u)) has a component attributable to the existence of vacancies per se, plus the usual interaction term for the reorientational disordering process (if present). In respect of the vacancy term, let El” be the energy needed to create an isolated vacancy. Vacancy pairs, triplets, etc. introduce new energy terms. The number density of n- fold vacancy clusters is proportional to (1 - p )” Therefore the vacancy term which is designated (E(p, 1)) may be expanded as a power series in (1 - p):
(17)
E, may be identified as the sublimation energy per molecule. The next term in the series has an interesting form, as it seems to depend on the number of holemolecule pairs or gas-like and solid-like molecule pairs (N,,,(l - p)p). Thus this term may be assumed to include any interaction between vacancies and molecules, or gas-like and solid-like molecules in the Bragg-Williams approximation. Further, the term involving N,,,(l - p) may be associated with the assembly molecular partition function for gas-like molecules (eqn (10)) within the framework of the Significant Structure Theory of Liquids. The orientational disordering term assumes a contribution from a disordered molecule next to an ordered molecule (but not next to a vacancy), and so is given by
(15) The configurational
x exp ( -
(16)
. .)
+ Ati(~) + Ati( + Ml - P) x (AE, + pAE’,) + ur(1 - u,)AE”
(21)
Orientational disordering in crystals--l THERMODYNAMIC FUNCTIONS FOR THE MELTING TRANSITION IN ATOMIC AND PLASTIC CRYSTALS
where Ati(~) = LiPlnp + (1 - p)ln(l -P)> A$(cr) = u, ln(a,) + (1 - o,)ln x{(l - o,)/(n - 1)) u,,, = V/N, the molecular volume AE, = NE,,
sublimation
energy per mole
AE: = NE: AE”= NE”
and the truncated positional configurational energy defined in (16) has been used. Conditions for stationary states in (21) with respect to p are not of particular interest here since this equation is at best expected to apply for a narrow range of values of p - 1 and are therefore omitted in the present discussion. However those with respect to a, are given in the following equation. ln{(l - al)/(n - l)o,} = (1 - 2a,)AE”/R7:
(22)
In eqn (22), if n = 2, the condition of minimum free energy is satisfied by crl = f (complete orientational disorder) at all temperatures 7’( >O). This corresponds to the Pople-Karasz model. If n > 2 and c1 = l/n (complete orientational disorder), then it is necessary that T+ co. This is a consequence of the fact that E&’<
689
Although all thermodynamic functions are generated, no immediate use is made of the specific heat function. Also, the enthalpy change at the melting transition is not calculated, since such a calculation requires knowledge of the interaction constants or mean disordering energies. Instead, the enthalpy function is regressed on the calculated entropies to evaluate these disordering energies. This task will be undertaken in the next section. In atomic crystals, of course, only positional disorder is possible. Thus, provided the melting transition in the plastic crystal phase does not involve additional orientational disordering (i.e. a change in n), then the melting process in both atomic and plastic crystals can be described according to the following Gibbs free energy function. G(p) - G(1) = RT [{In (Q,,) - In [(2nmkT)3’2u,/h3] - 1;~ +
AG)l+
id1- P)(& + PAE:;.
The remaining thermodynamic (23);
(23)
functions follow from
u(,) - U(l) = &(l - p){AE, + PAE:; + RTx[3/2
- E(x)]
(24)
where E(x) =
3{x/2 + x/[exp(x) - 11); (;)x + 3D(x);
S(p) - S(1) = R
(2nm)3’2(ekT)5’2 Ph3
-S(x)+l)z-Ati(p)]
+O.
x = &/T x = OJT
1
(25)
where
S(x) =
-3ln -31n
[l - exp(-x)] [1 - exp(-x)]
C,(P) - C,(l)
where =
3Rx’exp(x)/[exp(x) - l12; 3R(4D (x)- 3x/ [exp (x) - 1 I);
x = 0JT x = eJT
= [AE, + p2AE; + p2RT
x (3 - E(x)}
“(‘)
+ 3x/[exp(x) - 11; + 4D(x);
x = eE/T x = 6JT
LYis the thermal expansivity defined by the relation c( = (l/V)W/aT QE and B. are Einstein and Debye
characteristic temperatures respectively; and D(x) is the Debye function. The Sacker-Tetrode equation has been used for the translational entropy of the gas-like
KP + ,#R/2 - C,(x))
(26)
molecules in eqn (25). Both the Debye function and the Sackur-Tetrode equation are defined in standard textbooks of statistical thermodynamics such as [12]. The application of either the Debye or Einstein functions in the equations above is simply a matter of choice. Note that work terms have been ignored in the present treatment. Justification for neglecting these terms for melting transitions of plastic crystals is provided in paper II of this series.
690
CHODZIWADZIWA C. MJOJO MEAN POSITIONAL DISORDERING ENERGIES IN ATOMIC AND PLASTIC CRYSTALS
Since two energy terms, namely AE, and AEZ, are involved in the positional disordering process at the melting point it is reasonable to introduce a new term, the effective positional disordering energy AE:,, defined as follows. AE:,, = AEJp + AE;.
(27)
Note that AE:,, is explicitly a function of volume, and therefore of temperature, from its dependence on p. With this modification the configurational energy becomes (28)
where the identity [, = l/p has been used. BE:,, at the melting point is given by the following equation. AE:,, = K/X)
[AS(P) - RX{+ - E(x))
where AS(p) is the entropy
1
(29)
change at the melting point
T,. From (29) it is observed that AE,,, + co when x = 0. This corresponds to the case of p = 1, since x = 1 - p, and is a consequence of cooperative phenomena. It should require a great deal of energy to move a molecule from a lattice site to another when all other molecules are occupying their sites without initially
requiring the creation of a vacant site (in which case p < 1) except at the surface. However surface effects are ignored in the pesent model. PREDICTED VALUES OF ENTROPY OF FUSION AND MEAN POSITIONAL DISORDERING ENERGIES
Values of entropy of fusion for argon and methane have been computed from eqn (25). Table 1 summarises the numerical results. Three calculations for each substance were done assuming the classical limit of theories of lattice vibrations, the Einstein and Debye functions. Considering that the model does not involve any parameter fitting the agreement between predicted and experimental values of argon and methane is reasonable. Further the quality of its predictions depends on the accuracy of the experimental data used. i.e. AE,K V,, 1)n, 11~and T,. Table 2 compares the mean positional disordering energies obtained for argon and methane. Although the sublimation energies AE, are very different (that of argon being much higher), the values of AE:,, are very similar and reflect the similarities in the values of p and T, observed for the two systems. AEI is negative in the case of argon; but it is positive in the methane case. In both cases A& < A&. Thus the interaction between holes and molecules, or gas-like and solid-like molecules, appears to be attractive in argon and repulsive in methane. Table 3 shows the data used in the computations. Acknowkdgements--I thank Prof. I. G. Ross for very useful discussions and for bringing certain aspects of the problem to my attention. I thank Dr. E. R. Smith for his critical remarks.
Table 1. Predicted entropies of fusion for argon and methane Reduced entropy of fusion AS(P Theory Present theory/classical Present theory/Einstein Present theory/Debye LJ-D[ll] HSE[ll] LHW [ll] Observed
Argon
Methane
1.52 1.73 1.74 1.7 1.0 1.64 1.71t
1.00 1.17 1.24
1.24$
tRef. [17]. IFrom heat of fusion data in Ref. [18]. Key. U-D = Lennard-Jones S. E. and Devonshire A. F. HSE = Hirschfelder J. O., Stevenson D. P. and Eyring H. LHW = Longuet-Higgins H. C. and Widom B. All these theories are reviewed, in Ref. [ 111.
Table 2. Mean positional disordering energy for argon and methane
UK) Argon Methane
83.96 90.65
P
AE,(kJ mol - ‘)
AE:(kJ mol- i)
0.89 0.92
79.019 9.209
- 76.42 2.80
AE~,,(kJ mol-‘) 12.25 12.77
Orientational
Table 3. A summary
disordering
in crystals-I
of values for parameters used in the calculation of entropies Ref. [ 141 except where otherwise noted).
T,(K) V, (cc/mole) v(cc/mole) AE, (kJ/mole) 0, (K) o,(K)
691 of fusion (all data from
Argon
Methane
83.96 24.98 28.03 79.019 60.0 t85
90.65 31.06 33.63 9.209 71.34 $137
tFrom Ref. [19]. 1 Calculated from spectroscopic data in Ref. [11] (which cross-reference to [20]; Y” was taken to be 95 cm- I). Yamamoto et ul. [21] have recently used the same value of OD for methane which they attrlbute to Bezuglyi et al [72]. REFERENCES 1. 2. 3. 4. 5. 6. I. 8. 9. 10. 11. 12. 13.
Kohler F., The Liquid State, Section 6.4, Verlag Chemie (1972). Lennard-Jones J. E. and Devonshlre A. F., Proc. Roy. Sot. A169, 317 (1939); A170, 464 (1939). Pople J.A. and Karasz F. E., J. Phys. Chem. Solids l&28 (1961); Karasz F. E. and Pople J.A.lbldZO, 294 (1961). Chandrasekhar S., Shashidar R. and Tara N., J. Molec. Cryst. and Liquid Cryst. 10, 337 (1970); 12, 245 (1971). Webster D. S. and Hoch M. J. R., J. Phys. Chem. Solids 32, 2663 (1971). Amzel L. M. and Becka L. N., J. Phys. Chem. Solids 30, 521 (1969); 30, 2495 (1969). Rowlinson J. S., J. Phys. Chem. Solids 18, 28 (1961). Chadwick A. V. and Sherwood J. N., Di@iisionProcesses, Vol. 2, Chap. 6, Gordon and Breach (1971). Ubbelohde A. R., The Molten State of Matter. Wiley (1978). Smith G. W., Advances tn Liquid Crystals, 1, 189 (1975). Bondi A., Physical Properties of Molecular Crystals, Llqurds and Glasses. Wiley (1969). Miinster A., Statistical Thermodynamics Academic Press Vol. 1. (1969) and Vol. 2 (1974). Fowler R. and Guggenheim E. A., Statistical Thermodynamics. Cambridge University Press, London (1949).
14.
15.
16. 17. 18. 19. ‘0.
21. 22.
Jhon M. S. and Eyring H., Physical Chemistry-An Advanced Treatise, Vol. VIIL4, Chap. 5. (Edited by H. Eyring, D. Henderson and W. Jost). Academic Press, New York (1971). Chen Sow-Hein, Physical Chemistry-An Advanced Treatise, Vol. VIIIA, Chap. 2. (Edited by H. Eyring, D. Henderson and W. Jost). Academic Press, (1971). MjoJo C. C., J.C.S. Faraday Transactions II, 75, 667 (1979). Flubacher P., Leadbetter A. J. and Morrlson J. A., Proc. Phys. Sot. (London) 78, 144Y (1961). CRC Handbook of Chemistry and Physics (CRC Press, 1973), 54th Edn. Hnschfelder J. O., Stevenson D. P. and Eyrmg H., J. Chem. Phys. 5, 896 (1937). Stller H. and Haulecler S., Proc AEA Con\. Inelustic Scattering of Neutrons tn Solids and Liquids, p. 281 (1962). Yamamoto T., Kataoka Y and Okada K., J. Chem. Phys. 66, 2701 (1977). Bezuglyi P. A., Burma N. G. and Minyafaev R. Kh.. Phys. Solid State 8, 596 (1966).
ORIENTATIONAL
DISORDERING
IN CRYSTALS-I
A MODEL FOR THE EVALUATION OF THERMODYNAMIC PROPERTIES OF MELTING TRANSITIONS IN ATOMIC AND PLASTIC CRYSTALS CHODZIWADZIWA C. MJOJO~ Chemistry Department, Faculty of Science, Australian National University, Box 4, G.P.O., Canberra, ACT 2600, Australia (Received
23 January
1979; accepted in revised form 2 August 1979)
generalisation of two-sublattice models ofmeltmg transitions in atomic and molecular crystals (Lennard-Jones and Devonshire, Pople and Karasz; Amzel and Becka) has been attempted. The intention was to develop a scheme by which parameters, such as the number of sublattices and distinguishable molecular orientations can be selected. It is found that the choice of these parameters made by LennardJones and Devonshire, and by Pople and Karasz, protects the theory to some extent from some of the more diilicult physical issues. A single-lattice model also emerges as another possibility in the series of models that may be called upon. Thus a single-lattice model, based on concepts from Eyring’s Significant Structure Theory of Liquids for the melt, and standard thermodynamic functions for the solid, has been developed to evaluate thermodynamic properties of melting transitions in atomic and molecular crystals. Despite the remarkable simplicity of the model predicted values of entropy of fusion for both argon and methane are in good agreement with experimental data. Mean positional disordering energies for the same substances have also been computed. Abstraet-A
INTRODUCTION
There exists a functional relationship between the entropy of fusion and the fractional volume change at the melting point for simple substances (e.g. atomic crystals and plastic crystals) whose melting transitions involve positional disorder only [ 11. This observation has been the basis for treating the melting transition as a positional order-disorder phenomenon in a number of theoretical treatments. The two-sublattice model used by Lennard-Jones and Devonshire [2] is one of the few more successful treatments of the melting process of inert gases. This model has been extended by Pople and Karasz [3] to cover the case of molecular crystals with orientational disorder by allowing the molecules to take two distinguishable orientations in the plastic crystal phase. Various improvements have been suggested [4,5] to the Pople and Karasz model; but the most notable one is that due to Amzel and Becka [6]. In the treatment due toAmze1 and Becka an extension of the Pople and Karasz model to include the case ‘of the general number of distinguishable molecular orientations on a lattice site was attempted. Both the Pople-Karasz and Amze-Becka theories of the melting process contribute some insight into the problem of orientational disorder in crystals. This paper attempts to develop a more realistic model of melting transitions involving atomic and plastic crystals.
Despite the success of the two-sublattice models in affording both quantitative and qualitative descriptions of real systems in some accord with experimental data, a number of the assumptions made have been shown to be contrary to experimental observations [7]. Also these models assume that melting in both atomic and molecular crystals occurs predominantly via the migrations of constituent units from lattice sites to interstitial sites. Chadwick and Sherwood [8] observe, both from theoretical and experimental evidence, that vacancy migration is fundamentally responsible for self-diffusion phenomena in molecular crystals. Thus the application of a two-sublattice model to the melting phenomena in molecular crystals as proposed by Pople and Karasz, and Amzel and Becka may not be realistic. Helpful reviews on the theories ofmelting are presented in Refs. [9-121. Reference [12] provides the necessary background for the standard thermodynamic functions introduced in this paper. Would the theory be improved by postulating more than two sublattices? The generalisation of the theory to an arbitrary number of sublattices is examined first. The solid is then assumed to be represented by the configuration that maximises the partition function of the general model with respect to the number of sublattices. THE MULTISUBLATTICE
MODEL OF A LATTICE GAS
Suppose we have m sublattices of N lattice sites each tPresent address: Chemistry Department, Chancellor and N molecules with n allowed orientations. The College, University of Malawi, P.O. Box 280, Zomba, Malawi. ordered solid is defined as consisting of all the N 685 IPCS Vol 41, No. 1-A
686
CHODZIWADZIWA
molecules in one of the m sublattices and one of the n distinguishable orientations. Then the liquid and gaseous states will correspond to the situation when the N molecules are distributed equally over a given number of sublattices and orientations, the situation corresponding more and more to the gaseous state as m is increased. The plastic crystal phase then will be defined as that state of aggregation of matter in which all the molecules are predominantly in one of the m sublattices and are equally distributed over the n distinguishable orientations. Let p, stand for the fraction of molecules in the ith sublattice and uj for the fraction of molecules in the jth orientation. The crucial assumptions are those which relate the energies of molecules displaced from a site on the initial sublattice to a site on another sublattice, or else (on the same site) from a reference orientation to another orientation. Definitions are simple if the lattice is initially perfect, i.e. if all the molecules are on the same sublattice, in the same orientation. Let us now define E’ as the energy required to shift one molecule to a site on any other sublattice (such “other’ sublattices being thus degenerate); and E”, as the energy required to move one molecule from the reference orientation to any other orientation (these “other” orientations being thus degenerate). For higher degrees of disorder, decisions have to be taken about interactions between displaced molecules. The choice Pople and Karasz, and following them Amzel and Becka, made is to say that E’ and E” apply to all neighbouring pairs of molecules that are not in the same sublattice or orientation. However, there is no orientational disorder energy (E”) between molecules in different orientations but on different sublattices. This last provision means that orientational disorder carries no energy penalty if molecules are positionally disordered, and thus the desired element of cooperation is brought into the model. The average configurational energy per molecule then takes the form
C.
MJOJO
If this distinction is drawn, then a limiting case is E; = Es = 0. In either situation, expressed through (1) and (2), there should be equal populations on all sites or orientations other than the initial ones. Then since X:p, = 1, cu, = 1: (if
P, = (1 - p,)l(m - 1) g, = (1 - a,)/(n - 1) PROPERTIES
1) (3)
ci# 1).
OF THE MULTI-SUBLATTICE
MODEL
assembly partition function, customarily written on the basis of an Einstein model, has the form The
Q = (QAN.Q, where Q, is the molecular partition function and Q, =
1 exp{ - E(m,n, P. o)lk Ti
(4)
PJ
is the assembly configurational part; Q, = 1 for a perfectly ordered state. Assuming that Q, is independent ofdisorder (an assumption improved in Paper II of this series) equilibrium conditions associated with disordering depend solely on Q,. To evaluate Q, the Bragg-Williams approximation [12,13] is employed which simply requires replacing the argument of the exponential in eqn (4) by its average, and identifying the total number of terms in the sum as g(m, n, p, a), the number of ways of arranging particles for given values of m, n, p, CJ.Thus eqn (4) is replaced by
Q, = g(m, n, P, e)exp { - (E(m, n, P. a))lkTJ
(5)
where
&,n,p,a)
=
(N!}”
n (J”PiI!
fi
m
,vl iNPi} ! (N(l -
P,)) ! i=l,Fl {NPP,)!
(6)
An alternative view is to regard the initial positions and orientations as being so much deeper on the molecular potential energy surface that E’ and E” pertain only to displacements from the original lattice (occupation fractions pi and a,); for interactions between disordered molecules different quantities, say E; and E; must be used. Then the mean configurational energy per molecule has the form
= ? PS& +
ipf
i
n
+ f i>l
k>l
i
( l>l n
c P,P&, k>i
a,a,E” \
(2)
and (E(m, n, p, a)) is identified either with the expression (I) or with (2). Using eqn (3) to eliminate pi, CT,for i,j # 1 the more general case (eqn (2)) becomes
=
E’Np,(l x (mpf +
-2p,
p1) + E”No,U + 1)/b
-
aI)
1)
)E;N(~ - pl)‘(m - 2)/(m - 1)
+ *EE;N(I - G1)‘(mp: - 2p, + 1) x (n - 2)/(n - l)(m - 1).
(7)
Equations (6) and (7) reduce to the results of Amzel and Becka if m = 2, Ei = E’ and Ei = E”. Further it will be observed that the values of E& and E; are immaterial if m = 2 and n = 2 respectively. Also if m = n = 2, then the issue concerning possible improvements to the existing theory which initiated the
687
Orientational disordering in crystals-I definition of the elementary energy quantities is of no consequence. This case is of course the Pople-Karasz model. Thus it would seem that the fortuitous choice of these parameters which LJ-D and Pople and Karasz made protects the theory, to some extent, from one of the more difficult physical issues. These parameters were chosen arbitrarily and the choice of m = 2 was mainly encouraged by developments in the crystal statistics of alloys. Using eqn (3) again, and applying the Stirling approximation in eqn (6), the free energy is obtained as follows: A, = RT[p,ln(p,)
+ (1 - pr)ln’,(l
+ (m - 2 + Pi)ln{(m +a,ln(a,)+
- &)‘/(m - 1))
- 2 + ~,)/(m - l)}
(1 -a,)ln((l
-c,)/(n-
l)}]
+
(8)
Interest is in the stationary values of A,, and so eqn (8) may be differentiated with respect to m, n, p, and o. Illustratively, dA,/am = 0 if -(mx{E”a,
l)21n{(m-2+p,)/(m-
1)}
= (1 - PI) MA1 - PI) + (1 - 01) + )E;(l - o,)(n - 2)/(n - l)}]/kT
(9)
This condition is always satisfied for all values of m if p1 = 1;ifo i = l/n the plastic crystal phase is obtained. This result and the observations made by Chadwick et al. alluded to above offer an encouragement to proceed, in the next section, and employ the single lattice model (m = 1) in which there may be vacancies. However, the model that is developed shortly (m = 1) and the Pople-Karasz and Amzel-Becka (m = 2) models are merely the first two of a series of models which might be called upon.
fraction of neighbouring positions to a vacancy which are filled by molecules is VdK there are
moles of gas-like molecules. The remaining yV/Vmoles comprise solid-like molecules. It follows therefore that the molecular partition function is constructed from the Einstein partition function for solids to account for the solid-like molecules; and from the translational partition function for an ideal gas, for the gas-like molecules. The locked lattice vacancies present in solids at finite temperatures will be assumed to make a negligible contribution to the thermodynamic properties of the system, since according to this model such vacancies will be present in both the solid and liquid and only a very small fraction of them will become fluidised. Chen [15] has summarised the evidence for the structure of liquids based on the similarities between the thermodynamic properties of liquids and solids such as densities, specific heats and compressibilities. Further it is now well established from X-ray diffraction studies that nearest neighbours in the melt are at essentially the same distance as the solid. The significant structure model, on which the present theory is based, observes the similarity between liquids and solids, and views the melt as a distorted fluctuating lattice caused by the addition of holes to the solid during the melting transition. For the present purpose the assembly molecular partition function is defined as follows, Q,(P)/Q,(~)
= {Q,,(p).Q,,(p))IQ,,(l)
(10)
where Q,,(P)
= (QmiY”“”
Qnts(1) = (QmiF A SINGLE-LATTICE MODEL FOR POSITIONAL DISORDERING IN ATOMIC AND PLASTIC CRYSTALS
Having in mind the discussion above an attempt is made to discuss melting in a way consistent with a single-lattice model of positional order-disorder transitions in atomic and plastic crystals. The theory will be modelled around argon and methane; substances which have received extensive studies and have the relevant experimental data. From the Significant Structure Theory of Liquids [14] the concept is adopted that melting is to be seen as a consequence of the creation of fluidised vacancies leading to a volume expansion characteristic to the melting process. According to this model the fluidised vacancies are, on the average, of molecular size so that the number of moles of such vacancies per mole of occupied sites is (V - V,)/V, where I/ and V, are the molar volumes of liquid and solid respectively. It is argued that the energy for the creation of a lattice vacancy is equal to the energy of sublimation. Since the
and Q,(;(P) = {(277mkT)3’2(V {NV-
- t$)/h3}N(“-V~)‘V WV!
where the assembly molecular partition function has been defined relative to that of the positionally ordered solid at absolute zero including the zero point energy. Qmican be either the Einstein partition function or the Debye partition function, the subscript i denoting the ith mode in the case of the former or the mode corresponding to the Debye cut-off frequency in the latter case. It will be found later that both the Einstein and Debye thermodynamic functions lead to good agreement of the predicted properties of real systems with experimental data, though the Debye function appears to be a little better. Let x = 1 - V’K the number of moles of the gas-like molecules per mole of
688
CHODVWADZIWA C. MJOJO
substance. Then the relative free energy is given by the following expression.
This expression may be written in the form
!G,(P) - G,(l )}IRT= X{In (Q,,) + In (NA) - In [(27~nkT)~‘~~~/h~] - lj. (11) The problem of constructing the appropriate assembly configurational partition function is considered next. For one mole of substance the total number of lattice sites in the melt is given by N,,, = n, + N
T2
T,
(12)
where n, = N(I/- V,)/V,, the number of fluidised vacancies and T, is the melting point. Therefore p is defined as follows P = NIN,,, = YIV
(13)
and (1 - p) = nJN,,
= 1 - V’V
(14)
where p, a long range positional order parameter, is the fraction of lattice sites occupied by molecules on average. Note that L, the fraction of gas-like molecules, and (1 - p), the fraction of vacant sites, are numerically identical. Both quantities are retained, however, and will be used according to what aspect of the problem requires emphasis, i.e. the population of gas-like molecules or vacant sites. Therefore the assembly configurational partition function in the Bragg-Williams approximation takes the form
Qc ={pN,,,j
’
(No,)!
N,,,! ! { 1 - p)N,,,) !
($~)!ln-’
where E, = Eb” + E1.2’+ Ea’ + . . .
Thus, for a completely vacant lattice (p = 0),
(E(l,a))
GYP,1)) = N,,,{U - PE” + (1 -
+ (1 -
P)~E~’ +
= N,,,(l - p){(E;” - ,#f
+ 2Ei3’ +
’ -) + . . .).
-a,)
(18)
(E(P, a)) = (E(P, 1)) + (E(l,a)). free energy, after in eqn (15) is
G,(P, 0) - G,(L 1) = ET(l& + (1 - p)ln(l + (1 - a,)ln{(l
(19) applying
ln P
-p)> -or)/@
+ arlna, - 1))) + (E(p,c)) (20)
where & = 1 + nJN, the total number of moles of lattice sites. From (11) and (20), the Gibbs free energy of the assembly relative to the ordered solid (ignoring work terms for the present) is given by the following expression G(P, CJ)- G(f, 1) = RT((ln (Qmi) - In [(2mnkT)3%,/h3] - 1)~
P)‘J%*’
. .) + El” + Ea’ +
= NE”a,(l
where it has been assumed that E; (the interaction energy between molecules in the disordered orientations) is negligible. This model for orientational disordering has been used in the study of binary phase diagrams of plastic crystals comprising optically active quasi-globular molecules [ 161. Finally
Stirling’s approximation where ur = 1 for the orientationally ordered solid, and ur = l/n for the plastic crystal phase. The energy term (E(p, u)) has a component attributable to the existence of vacancies per se, plus the usual interaction term for the reorientational disordering process (if present). In respect of the vacancy term, let El” be the energy needed to create an isolated vacancy. Vacancy pairs, triplets, etc. introduce new energy terms. The number density of n- fold vacancy clusters is proportional to (1 - p )” Therefore the vacancy term which is designated (E(p, 1)) may be expanded as a power series in (1 - p):
(17)
E, may be identified as the sublimation energy per molecule. The next term in the series has an interesting form, as it seems to depend on the number of holemolecule pairs or gas-like and solid-like molecule pairs (N,,,(l - p)p). Thus this term may be assumed to include any interaction between vacancies and molecules, or gas-like and solid-like molecules in the Bragg-Williams approximation. Further, the term involving N,,,(l - p) may be associated with the assembly molecular partition function for gas-like molecules (eqn (10)) within the framework of the Significant Structure Theory of Liquids. The orientational disordering term assumes a contribution from a disordered molecule next to an ordered molecule (but not next to a vacancy), and so is given by
(15) The configurational
x exp ( -
(16)
. .)
+ Ati(~) + Ati( + Ml - P) x (AE, + pAE’,) + ur(1 - u,)AE”
(21)
Orientational disordering in crystals--l THERMODYNAMIC FUNCTIONS FOR THE MELTING TRANSITION IN ATOMIC AND PLASTIC CRYSTALS
where Ati(~) = LiPlnp + (1 - p)ln(l -P)> A$(cr) = u, ln(a,) + (1 - o,)ln x{(l - o,)/(n - 1)) u,,, = V/N, the molecular volume AE, = NE,,
sublimation
energy per mole
AE: = NE: AE”= NE”
and the truncated positional configurational energy defined in (16) has been used. Conditions for stationary states in (21) with respect to p are not of particular interest here since this equation is at best expected to apply for a narrow range of values of p - 1 and are therefore omitted in the present discussion. However those with respect to a, are given in the following equation. ln{(l - al)/(n - l)o,} = (1 - 2a,)AE”/R7:
(22)
In eqn (22), if n = 2, the condition of minimum free energy is satisfied by crl = f (complete orientational disorder) at all temperatures 7’( >O). This corresponds to the Pople-Karasz model. If n > 2 and c1 = l/n (complete orientational disorder), then it is necessary that T+ co. This is a consequence of the fact that E&’<
689
Although all thermodynamic functions are generated, no immediate use is made of the specific heat function. Also, the enthalpy change at the melting transition is not calculated, since such a calculation requires knowledge of the interaction constants or mean disordering energies. Instead, the enthalpy function is regressed on the calculated entropies to evaluate these disordering energies. This task will be undertaken in the next section. In atomic crystals, of course, only positional disorder is possible. Thus, provided the melting transition in the plastic crystal phase does not involve additional orientational disordering (i.e. a change in n), then the melting process in both atomic and plastic crystals can be described according to the following Gibbs free energy function. G(p) - G(1) = RT [{In (Q,,) - In [(2nmkT)3’2u,/h3] - 1;~ +
AG)l+
id1- P)(& + PAE:;.
The remaining thermodynamic (23);
(23)
functions follow from
u(,) - U(l) = &(l - p){AE, + PAE:; + RTx[3/2
- E(x)]
(24)
where E(x) =
3{x/2 + x/[exp(x) - 11); (;)x + 3D(x);
S(p) - S(1) = R
(2nm)3’2(ekT)5’2 Ph3
-S(x)+l)z-Ati(p)]
+O.
x = &/T x = OJT
1
(25)
where
S(x) =
-3ln -31n
[l - exp(-x)] [1 - exp(-x)]
C,(P) - C,(l)
where =
3Rx’exp(x)/[exp(x) - l12; 3R(4D (x)- 3x/ [exp (x) - 1 I);
x = 0JT x = eJT
= [AE, + p2AE; + p2RT
x (3 - E(x)}
“(‘)
+ 3x/[exp(x) - 11; + 4D(x);
x = eE/T x = 6JT
LYis the thermal expansivity defined by the relation c( = (l/V)W/aT QE and B. are Einstein and Debye
characteristic temperatures respectively; and D(x) is the Debye function. The Sacker-Tetrode equation has been used for the translational entropy of the gas-like
KP + ,#R/2 - C,(x))
(26)
molecules in eqn (25). Both the Debye function and the Sackur-Tetrode equation are defined in standard textbooks of statistical thermodynamics such as [12]. The application of either the Debye or Einstein functions in the equations above is simply a matter of choice. Note that work terms have been ignored in the present treatment. Justification for neglecting these terms for melting transitions of plastic crystals is provided in paper II of this series.
690
CHODZIWADZIWA C. MJOJO MEAN POSITIONAL DISORDERING ENERGIES IN ATOMIC AND PLASTIC CRYSTALS
Since two energy terms, namely AE, and AEZ, are involved in the positional disordering process at the melting point it is reasonable to introduce a new term, the effective positional disordering energy AE:,, defined as follows. AE:,, = AEJp + AE;.
(27)
Note that AE:,, is explicitly a function of volume, and therefore of temperature, from its dependence on p. With this modification the configurational energy becomes (28)
where the identity [, = l/p has been used. BE:,, at the melting point is given by the following equation. AE:,, = K/X)
[AS(P) - RX{+ - E(x))
where AS(p) is the entropy
1
(29)
change at the melting point
T,. From (29) it is observed that AE,,, + co when x = 0. This corresponds to the case of p = 1, since x = 1 - p, and is a consequence of cooperative phenomena. It should require a great deal of energy to move a molecule from a lattice site to another when all other molecules are occupying their sites without initially
requiring the creation of a vacant site (in which case p < 1) except at the surface. However surface effects are ignored in the pesent model. PREDICTED VALUES OF ENTROPY OF FUSION AND MEAN POSITIONAL DISORDERING ENERGIES
Values of entropy of fusion for argon and methane have been computed from eqn (25). Table 1 summarises the numerical results. Three calculations for each substance were done assuming the classical limit of theories of lattice vibrations, the Einstein and Debye functions. Considering that the model does not involve any parameter fitting the agreement between predicted and experimental values of argon and methane is reasonable. Further the quality of its predictions depends on the accuracy of the experimental data used. i.e. AE,K V,, 1)n, 11~and T,. Table 2 compares the mean positional disordering energies obtained for argon and methane. Although the sublimation energies AE, are very different (that of argon being much higher), the values of AE:,, are very similar and reflect the similarities in the values of p and T, observed for the two systems. AEI is negative in the case of argon; but it is positive in the methane case. In both cases A& < A&. Thus the interaction between holes and molecules, or gas-like and solid-like molecules, appears to be attractive in argon and repulsive in methane. Table 3 shows the data used in the computations. Acknowkdgements--I thank Prof. I. G. Ross for very useful discussions and for bringing certain aspects of the problem to my attention. I thank Dr. E. R. Smith for his critical remarks.
Table 1. Predicted entropies of fusion for argon and methane Reduced entropy of fusion AS(P Theory Present theory/classical Present theory/Einstein Present theory/Debye LJ-D[ll] HSE[ll] LHW [ll] Observed
Argon
Methane
1.52 1.73 1.74 1.7 1.0 1.64 1.71t
1.00 1.17 1.24
1.24$
tRef. [17]. IFrom heat of fusion data in Ref. [18]. Key. U-D = Lennard-Jones S. E. and Devonshire A. F. HSE = Hirschfelder J. O., Stevenson D. P. and Eyring H. LHW = Longuet-Higgins H. C. and Widom B. All these theories are reviewed, in Ref. [ 111.
Table 2. Mean positional disordering energy for argon and methane
UK) Argon Methane
83.96 90.65
P
AE,(kJ mol - ‘)
AE:(kJ mol- i)
0.89 0.92
79.019 9.209
- 76.42 2.80
AE~,,(kJ mol-‘) 12.25 12.77
Orientational
Table 3. A summary
disordering
in crystals-I
of values for parameters used in the calculation of entropies Ref. [ 141 except where otherwise noted).
T,(K) V, (cc/mole) v(cc/mole) AE, (kJ/mole) 0, (K) o,(K)
691 of fusion (all data from
Argon
Methane
83.96 24.98 28.03 79.019 60.0 t85
90.65 31.06 33.63 9.209 71.34 $137
tFrom Ref. [19]. 1 Calculated from spectroscopic data in Ref. [11] (which cross-reference to [20]; Y” was taken to be 95 cm- I). Yamamoto et ul. [21] have recently used the same value of OD for methane which they attrlbute to Bezuglyi et al [72]. REFERENCES 1. 2. 3. 4. 5. 6. I. 8. 9. 10. 11. 12. 13.
Kohler F., The Liquid State, Section 6.4, Verlag Chemie (1972). Lennard-Jones J. E. and Devonshlre A. F., Proc. Roy. Sot. A169, 317 (1939); A170, 464 (1939). Pople J.A. and Karasz F. E., J. Phys. Chem. Solids l&28 (1961); Karasz F. E. and Pople J.A.lbldZO, 294 (1961). Chandrasekhar S., Shashidar R. and Tara N., J. Molec. Cryst. and Liquid Cryst. 10, 337 (1970); 12, 245 (1971). Webster D. S. and Hoch M. J. R., J. Phys. Chem. Solids 32, 2663 (1971). Amzel L. M. and Becka L. N., J. Phys. Chem. Solids 30, 521 (1969); 30, 2495 (1969). Rowlinson J. S., J. Phys. Chem. Solids 18, 28 (1961). Chadwick A. V. and Sherwood J. N., Di@iisionProcesses, Vol. 2, Chap. 6, Gordon and Breach (1971). Ubbelohde A. R., The Molten State of Matter. Wiley (1978). Smith G. W., Advances tn Liquid Crystals, 1, 189 (1975). Bondi A., Physical Properties of Molecular Crystals, Llqurds and Glasses. Wiley (1969). Miinster A., Statistical Thermodynamics Academic Press Vol. 1. (1969) and Vol. 2 (1974). Fowler R. and Guggenheim E. A., Statistical Thermodynamics. Cambridge University Press, London (1949).
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Jhon M. S. and Eyring H., Physical Chemistry-An Advanced Treatise, Vol. VIIL4, Chap. 5. (Edited by H. Eyring, D. Henderson and W. Jost). Academic Press, New York (1971). Chen Sow-Hein, Physical Chemistry-An Advanced Treatise, Vol. VIIIA, Chap. 2. (Edited by H. Eyring, D. Henderson and W. Jost). Academic Press, (1971). MjoJo C. C., J.C.S. Faraday Transactions II, 75, 667 (1979). Flubacher P., Leadbetter A. J. and Morrlson J. A., Proc. Phys. Sot. (London) 78, 144Y (1961). CRC Handbook of Chemistry and Physics (CRC Press, 1973), 54th Edn. Hnschfelder J. O., Stevenson D. P. and Eyrmg H., J. Chem. Phys. 5, 896 (1937). Stller H. and Haulecler S., Proc AEA Con\. Inelustic Scattering of Neutrons tn Solids and Liquids, p. 281 (1962). Yamamoto T., Kataoka Y and Okada K., J. Chem. Phys. 66, 2701 (1977). Bezuglyi P. A., Burma N. G. and Minyafaev R. Kh.. Phys. Solid State 8, 596 (1966).