Orientational disordering in crystals—II

J Phys Chem Sohds Vol 41. pp. 693-707 Pergamon Press Ltd, 1980 Pnnted m Great Britam

ORIENTATIONAL

DISORDERING

IN CRYSTALS-II

A MODEL FOR THE EVALUATION OF THERMODYNAMIC PROPERTIES OF ORIENTATIONAL DISORDERING IN CRYSTALS CHODZIWADZIWA

Chemistry

C. MJOJOt

Department, Faculty of Science, Australian National Box 4, G.P.O., Canberra, ACT 2600, Australia

University,

(Received 23 January 1979; accepted in revised form 2 August 1979) Ahstrati-A statistical mechanical model for the evaluation of thermodynamic properties of orientational transitions in crystals developed by Aston is generalised by taking into account the existence of more than two allowed molecular orientations (n z 2) in the rotator crystal phase explicitly. An orientational transition work function is derived in order to generate full thermodynamic functions leading to a formal expression for

the configurational energy change in terms of transition enthalpy change and an entropy integral. The value of n for camphor predicted by this equation compares very well with the one computed from another independent method and that reported by previous workers. A general scheme, based on both spectroscopic and thermodynamic data, by which n and phase eigenstates may be computed including the case where configurational and thermal properties are strongly coupled is presented. An analytical expression is obtained for the relative potential energy barrier vclfin a single-lattice model which parallels that defined for a double sub-lattice model of positional disorder by Pople and Karasz and their followers. It is shown that the &ue qfr,,,_..for a given value of ?IIS not uniquely defined for all substances of other thermodynamic parameters. INTRODUCTION

which may not have common

values

greater than the rotation frequency. Thus in both the liquid and the plastic crystal phase the molecule acquiring rotational energy does not have the time to perform sufficient rotations for the energy of the new state to become definite (a consequence of the Uncertainty Principle). In view of this it is perhaps not surprising that where pronounced quantum effects apply, such as is the case for the low temperature orientational transition of methane reported by Yamamoto et al. [4], the model of orientational disorder proposed to fit experimental data is neither of the Pauling type nor the Frenkel type. However the treatment to be presented in this paper is consistent with the Frenkel model of orientational disorder. Pople and Karasz [5] have shown from a twosublattice model of positional disorder for melting transitions that whether or not an orientational transition which gives rise to the plastic crystal phase will occur depends on the relative energy barriers for the orientational and positional disordering of a molecule in the solid state. The present intention is to apply a similar approach to that of Pople and Karasz to a single-lattice model developed in paper I of this series and to generate thermodynamic functions of the orientational transition taking into account the frequency changes of the oscillating molecules at the transition as proposed by Aston [6] thereby introducing some quantum features to the problem of orientational disordering in crystals.

Solid-state molecular reorientation is generally described according to two models, the Frenkel model and the Pauling model. The former views each molecule as capable of taking only certain orientations, rather than being in a continuous state of rotation as suggested by the latter. In most cases the Frenkel model is currently considered to be the most realistic [1,2]. Each molecule is assumed to spend most of its time in one of the energetically distinguishable allowed orientations. The fundamental principle assumed is that the favourable orientations in the rotator phase are those for which the effective point symmetry of the molecule is the same as the symmetry of the lattice site [3] (the site group), in the space group of the rotator phase structure. The present paper proceeds to show that this requirement pertaining to molecular symmetry may be of secondary importance in determining the thermodynamic properties of the transition. Pauling’s model is quantum mechanical. The question therefore arises: why are quantised rotations not to be expected in the plastic crystal phase for the majority of compounds even when the orientational disorder may be large? In a vibrating solid the barriers to rotation must change with the phase of the intermolecular vibration. An analogous barrier to rotation exists even in the liquid. Except in the case of very small molecules like HCl in solution in non-polar solvents, we do not find evidence of quantised rotation because the collision frequency between molecules is

A MODEL FOR ORIENTATIONAL ORDER-DISORDER TRANSITIONS IN MOLECULAR CRYSTALS

The model for orientational order-disorder transitions associated with plastic crystals has already been

t Present address: Chemistry Department. Chancellor College, Umversity of Malawi, P.O. Box 280,Zomba, Malawi. 693

CHODZIWADZIWA C. MJOJO

694

alluded to in paper I. In particular, the assembly configurational partition function is obtained from eqn (15) of paper I by setting p = 1, and therefore N,,, = N, and is given by the following equation. N!

Q, =

iNQl x exp { -@@DIRT)

(1)

where, from eqn (18) of paper I (E(a))

= AE”a,(l

- a,)

and AE” = NE”. Also the relations between this model and the Pople-Karasz [5] and Amzel-Becka [7] models have already been discussed in paper I. The partition function for the assembly is again written in the form

Q = (QJN.Q, where Q, is the molecular partition function. The function of Q, will be to account for all thermal (as opposed to configurational) properties associated with the orientational transition. The effect of lattice vibrations on order-disorder transitions in plastic crystals received the attention of Aston[6]. He allowed for thermal expansion by assuming that the IR and Raman frequencies observed above and below the transition are ascribed to molecules undergoing torsional lattice vibrations independently. This hypothesis is modified by assuming that these frequencies could be attributed to both intra- and intermolecular vibrations where the latter contain both the translational and torsional lattice vibrations. This assumption can be justified by observing that the volume of expansion and entropy change at the orientational transition are of the same order of magnitude as those observed at the melting transitions of substances without plastic crystal phases. This means that the translational lattice vibrations may be significantly affected also. Further a molecule with a general shape may be distorted in the orientationally ordered phase in order to acquire the geometry best adapted to packing in the crystal. Thus thevolume expansion at the transition would allow the molecule to relax in the plastic crystal phase with consequent changes in the force constants for intramolecular vibrations. These changes in the force constants would in turn be manifested in the thermal transition energy. It is assumed that these intra- and intermolecular vibrations can be represented by the Einstein harmonic oscillators. Thus, on the basis of an Einstein model, the disordered crystal consists of a locally distorted lattice where the molecule in the reference orientation occupies a cell of the same size and is

oscillating with the same frequency in the ith mode as it did before the transition. At the same time the molecule in any of the other orientations occupies an expanded cell and consequently oscillates with a different frequency from the one before the transition for the same mode. The observed IR and Raman spectra in the disordered phase therefore consists of the resultant of the averaging of the effects due to the two types of molecules, the reference-type and the disordered-type of molecules. The effect of the disorder, in this model, is to shift the frequency of the ith mode for those molecules transforming from the reference orientation to any of the other molecular orientations available in the plastic crystal phase. Thus the assembly molecular partition function for n molecular orientations relative to its value in the ordered state is defined as follows.

Qm(~,)/Qm(l) = n(Qbi)N"l.(Qii)N~l gi/c~i)N.(3) L

QLi and Qzi are molecular partition functions for the ith mode of lattice vibrations corresponding to the ordered and disordered phases respectively. Equation (3) simplifies to

Qm(~lYQm(l) = ~{Q2QL}""-""

(4)

which reduces to the result obtained by Aston when n = 2. The assembly partition function is given by the following equation.

Qb,YQU) = Q,(ol).Q,(al)lQ,(l)QE(1) (5) THE TRANSITION

WORK FUNCTION

To obtain full thermodynamic functions for a phase transition it is necessary to express the work done by the thermodynamic system undergoing the transition in terms of configurational and molecular parameters. This will be done by examining the condition of equilibrium between two co-existing phases which is customarily written for one-component systems as follows [8] - S,(cc) dT + V,(a) dP = -&,(/I) dT + V,(p) dP (6) where the subscript m has been used to denote molar quantities. However, if the two phases co-exist over a temperature range such as is the case for nonisothermal phase transitions (variously called i,transitions, transitions of second order or continuous transitions) the condition of equilibrium becomes -

P

ri s r,

S&)dT

+

s P”

v,(a) dP P

7i

v,(fl)dP (7) s PO r, where T, and TAare the initial and final temperatures for the nonisothermal transition. If the simplifying =-s

%,,(fi)dT+

695

Orientational disordering in crystals-II assumption is made that molar quantities of the two co-existing phases do not vary much during the transition, eqn (7) takes the following form PAV,

= AS,AT,,

+ P”AVm;

Consequently, from eqn (8), the specific heat of thermal expansion during the transition is given by the following equation. aw

AT,, = TA - T,.

_

aT

03)

This equation gives the work done by the crystal in raising its potential energy from that of the low temperature phase to that of the upper temperature phase, where P”AVm is the work done against the atmosphere during the transition. So far the internal work done by the crystal has been inaccessible and the correction term to energy has generally been the P” AV, term only. A remarkable observation is that as AT,, + 0 (isothermal transitions, transitions of first order or discontinuous transitions), PAV-+ P" AVm. Thus in strictly isothermal transitions, the total work done by the crystal is relatively small and equals the work done by the crystal against the atmosphere. Since all transitions occur over a finite temperature range, where the transition temperature range becomes smaller in the more isothermal transitions, eqn (8) is general and may be used in conjunction with the Clausius-Clapeyron equation [8] in both isothermal and non-isothermal phase transitions. As an illustration, the orientational phase transition in D-camphor is considered using the following data: A.T,, N 23 K (estimated from reported C,-curves [9]), AH,,, = 2.7Kcalmol-’ [9] (1 cal = 4.1845), T, = 242K [lo] and AV, = 10.2cm3mol-’ [ll]. When the appropriate substitutions are made eqn (8) gives

= AS,,,.

Equation (8) is generalised for any two temperatures T, and T, where T, < Tp as follows AW= AS(a,)AT,B

(10)

where AS(a,) is the entropy change between the two temperatures. The transition work function relative to its value at absolute zero may be definqd as follows @(al) = TAS(a,)

(11)

where AS(o,) = S(o,) - S(1). De notation A@z,(a,) is used to denote a change in the work function at the orientational transition. PHASE EIGENSTATE The

Gibbs free energy function is obtained from eqn (1) after applying the Stirling approximation. G(a) - G(1) = RT [ui In (al) + (1 - o,)ln{(l

- al)/@ - l)}l + @(a,) (12)

+ AE”a,(l - ul) + (1 - a,)AG, where

PAV,

= 260 + 0.25calmol-’

where, obviously, the second term is negligible. This gives a total transitional internal crystalline pressure of 1.1 x 103atm (latm = 1.013 x 105Nm-‘) for solid camphor at the l-transition. The P” AV, term in (8) is dropped henceforth. From the fore-going discussion it is clear that the work done associated with the less energetic isothermal melting transitions observed in plastic crystals must be negligible. This is a justification for the neglect of work terms in paper I of this series. However, work terms may not be negligible in orientational transitions of the I-type. In the case of camphor, the internal work done by the crystal during the orientational transition accounts for 10% of the total transition energy. Further, since the transition entropy AS,,, is generally expressible in terms of configurational and molecular parameters, eqn (8) provides a route through which the transition work function may be defined. Note that the work done during the transition is by definition

AG, = RT f: In {Q;,/Q;,}.

(13)

The term (1 - a,) AG, constitutes a coupling between configurational and thermal properties of the assembly. Stationary conditions for eqn (12) with respect to u1 are defined in the following equation. (1 - 28,) = A&/BE”

(14)

where AE,, is the thermal energy of the transition. Clearly (1 - 28,) > 0 (since the energy terms are both positive) and therefore the stationary value til <) (always). This result predicts an inverted distribution of molecules over the number of distinguishable orientations in the plastic crystal phase for the case with n = 2. This implies a modification of the molecular potential surface during the orientational transition leading to a deeper minimum for the nonreference orientation for the case with n = 2. To avoid this complication, the alternative expression for defining stationary conditions obtained through ignoring the work function is adopted. RTln{(l

- sl)/(n - 1)8,} = {AE”(l - 28,) - AG,}/T

(15)

696

CHODZIWADZIWA C. MJOJO

The stationary condition is satisfied by 8, = 1 and T = 0 K where AG, = 0 by definition. If 6, = l/n, then it is necessary that T+ co. For a given finite temperature T > 0 K values of 8, are stationary and depend on the magnitudes of the orientational interaction constant and the molecular free energy. The parameter 8, < 1 is thus an explicit function of T and therefore characterises the equilibrium state of the thermodynamic system with respect to the orientational number of allowed mollcular orientations, disordering process. For this reason 8, is referred to as the phase eigenstate in the present treatment. Means of getting at these phase eigenstates for finite temperature T > 0 K are devised below. THERMODYNAMIC FUNCTIONS FOR ORIENTATIONAL DISORDERING IN MOLECULAR CRYSTALS

Following Aston the molecular free energy is constructed from the Einstein free energy function as follows.

and therefore C,(B,) < C,(Bi)/2 generally. Equation (23) is a consequence ofthe transition enthalpy function H(B1) = E(8,) + aq6,)

where the function is defined relative to its value at absolute zero. The configurational specific heat function may be obtained from eqn (20) as follows

{C”(~,)- C”(l )>c,o =

2RTx{Sii (16)

If it is possible to ascribe specific frequencies to lattice vibrations, eqn (16) may be expressed in terms of the Debye and Einstein functions for the lattice and intramolecular vibrations respectively (see [12]). Applying the appropriate thermodynamic relations to (12) and setting cri = 8, the following thermodynamic functions are obtained.

G(B,) - G(1) = RT [8,ln(8,)

TRZInZ {(l - B,)/(n - 1)&i) {RT/B,(l - d,) - 2AE”)

- Sm,jln{l - d,)/(n - l)d,j

{RT/B,(l - 8,) - 2AE”)

-

Consequently the specific heat associated with the orientation transition is not a simple sum of contributions from configurational and thermal changes. For convenience let I’ = R In ((1 - d,)/(n - 1)&r}and A = 1 {Sdi - SLi}.Three distinct types of orientational transitions emerge from eqns (20) and (21).

+ (1 - 8,)ln{(l

- B,)/(n - l)}] + (1 - 8,) [8, AE”

+ (Nh/2)C{ wzi - c&} + 1 {ALi - Aki}] + @(8,)

(17)

I

i

H(B,) - H(1) = (1 - 8,) [B,AE” + (Nh/2)1 {w$ - w;,} + c {ELi - E;,}] i I s(al)

- S(1) = -R[8,ln(6,)

+ (1 - d,)ln((l

- B,)/(n - l)}] + (1 - B,)C{S$

C,(B,) - C,(l) = T[Rln ((1 - &,)/(n - 1)8,) - 1 {SAi - S~,}]‘/{RT/&,(l + (1 - B,)c{c;i C,(B,) - C,,(l) = 2?[Rln

- CLi}

+ CD@,)

- S&}

’

C&B,) - C,(B,) = T[Rln ((1 -‘B,)/(n - l)B,)} - c{Sz, - Sbi}]2/{RT/d,(1 - Cmi} + A&)

where Cmi, Emi and & are the Einstein thermodynamic functions; AS(B,) = S(B,) - S(1) and is given by eqn (19). Note that C,(l) = C,(l) = 0 at absolute zero. Also from (20) and (21) CJB,) = 2C,(d,) + AS(B)

(23)

(19)

(20)

((1 - $,)/(n - l)d,} - c {SL, - S~i)]Z/{RT/d,(l

i

(18)

- c?i) - 2AE”j

- 8) - 2AE”)

+ 2(1 - B,)c {CAi - Cki} + AS&)

+ (1 - B,)~{C;,

(25)

where the subscript CFG denotes configurational. It is noteworthy that in defining the configurational specific heat function, the terms suppressed include the coupling term

AG, = (Nh/2)C {o:, - WLi} + 1 {Ai, - ALi).

(24)

(21) - 8,) - 2AE”) (22)

I <
691

Orientational disordering in crystals--II plastic crystal phase characterised by larger amplitude

torsional oscillations or a larger occupation fraction in the reference orientation (8, > l/n). Such orientational transitions may be regarded as restricted by a strong coupling between intra- and inter-molecular vibrations and fall within the regime of elementary excitations in molecular crystals. Because of the relatively low characteristic temperatures for intermolecular vibrations associated with molecular crystals, orientational transitions of this type in these solids would tend to occur at relatively lower temperatures. They may however occur at low and moderate temperatures in inorganic crystals. Due to the considerable influence of phonon modes, the resultant disordering process in this case may depart from the Frenkel model. I- - A: Orientational transitions in the semi-classical regime This is in the strongly coupled limit with respect to configurational and molecular parameters where the orientational disordering process manifests both orientational and vibrational features with comparable intensity. This type of orientational disordering process is expected to be common in molecular crystals transforming at low temperatures; and in inorganic crystals transforming at low and moderate temperatures. I- >>A: Orientational transitions in the classical regime This is a perfect Frenkel-type orientational disordering process where configurational features dominate the character of the transition. In this case the coupling between internal and external vibrational modes is weak and the quasiparticle aspect alluded to above is therefore less prominent. Such transitions are expected to occur within the classical limit with respect to the external modes whose contribution to terms related to A generally vanish. The high energy internal modes are least perturbed during the transition leading to an overall small contribution to A. In the limit A + 0 eqn (21) takes the following form.

S(l/n) - S(1) = Rln(n) + ((n - l)/n}C{szi 1

- Smi}. (29)

Equations (17)-(21) give the transition thermodynamic functions for the phase stable between the perfectly ordered solid (obtainable only at 0 K) and the partially disordered solid with the phase eigenstate 8, at the transition temperature. This part of the transition is henceforth referred to as the premonitory transition region. The thermodynamic functions for the final orientational disordering process are obtained by subtracting the appropriate thermodynamic functions for the premonitory transition region (eqns 17-19) from those of the total transition (eqns 27-29) as follows. G(l/n) - G(B,) = -RT (8, In (6,) + (1 - 8,)ln{(l

- B,)/(n - 1)

+ In(n)} + [n{l - n(1 - B,)d,} - l]AE”/n* + (nB, - l)AG,/n

+ A@,n(B,)

(30)

H(l/n) - H(8,) = [n{l - n(1 - 8,)8,} - l]AE”/n’ + {W, - 1)/n} LNh12)Cwli - wL~}

I

+ C{ELi- Eki}l + A@z,(8,) (31) I

S(l/n) - S(6,) = R [In(n) + d, In (8,) + (1 - 8,)ln{(l

- d,)/(n - l)}]

+ {(n0, - l )/n> C

Is2

-

Xni}

(32)

i

where AQZI(B1)= AS,(d,)AT,, and is the work done during the orientational transition. A&,(B) is the transition entropy and is given by eqn (32). The transition temperature T, is defined as the temperature at which the partially disordered solid in the phase eigenstate c?~ (hereafter designated as the transition phase eigenstate) is in equilibrium with the completely C,@d- C,(l) = 2TR21n2 disordered solid in the limiting phase eigenstate 8, = l/n. Both sides of eqn (30) must equal zero at the - R [8,ln8, + (1 - B,)ln transition temperature. Figures 1 and 2 summarise the (26) properties of this equation with the work terms i(l - W(n - 1111. neglected. It is found that the transition becomes more Because of contributions from work terms the experimental specific heat function is given by eqn (26) isothermal with increase in n or the closer 8, approaches its limiting value of l/n for a fixed value of instead of (25). This observation is called upon later. AC,. However the effect of the variation d on the free Thermodynamic functions for the total orientaenergy is more pronounced at large separation of 8, tional transition are as follows. values. Hence the effect is less obvious from Fig. 2 G(l/n) - G(1) = -RTln(n) where the 8, values are close to each other. Also the + {(n - 1)/n} {AE”/n + AG,} + @(l/n) (27) more isothermal transitions generally occur at lower temperatures with a common value of AG,,,. The H(l/n) - H(1) = {(n - 1)/n} [AE”/n observation that solid state transitions become more isothermal with large values of n may correspond to + (Nh/2) T Iw:i - 4d~ the observation made by Amzel and Becka that transitions with n > 2 are always first order. + 1 {Eli Eai}] + @(l/n) w3) iRT,t,(l

_

8,)

_

2

AE,,j

698

CHODZIWADZIWAC. MJOJO

1.25

ml

cal

x(

mole

T

)-1 K

02E

* I

o.oc -025

-0.50

-0.75

-1.00

300

500

400

600

700

000

T x(K)-’ Fig. 1. Properties

of the free energy AG(d) = G(l/n) - G(B) where 8, = 0.95, and n is varied. The transition occurs along the line AG = 0. Values of parameters used: AE” = 6.34 x 10’ cal mol- ‘, C,,,,,= 221 cn- ’ and rAi = 141 cm-‘. From eqns (17), (27) and (30) it is clear that only a fraction of AC, makes a contribution to the thermodynamic properties of the orientational disordering process. The remainder of A$, per se contributes to the background thermodynamic properties in the temperature region of the transition. The following background free energy function is obtained if it is assumed that the perturbed and unperturbed vibrational modes of the solid during the transition may be enumerated independently.

{G(d,) - G(l)),,

= 6, AG, +,gr

- c(1)}Bc = ;

DETERMINATION OF THE NUMBER OF MOLECULAR ORIENTATIONS AND THE TRANSITION PHASE EIGENSTATE The

(hwmj/2

+

&jI

(33)

where fl= NrB,, r is the number of atoms in the constituent molecule, the index j counts the unperturbed modes and BG denotes background. The background specific heat function is of particular interest. I’(dI)

where the subscript v or p to the specific heat is dropped since the result is independent of orientational work terms. Equation (34) provides a method for baseline extrapolation on the C, data to obtain the orientational specific heat C&B,). However the task of calculating the background specific heat function may be too demanding of computer time depending on the magnitude of N.

cmj

transition

enthalpy

AH(8,) = T,AS,(B,), and

therefore from eqn (32) AH@,) = T,[{(nB, - 1)/n} 1 {Szi - SLi} + R{ Inn 4 d, lnc?, + (1 - 6,)ln{(l

- B,)/(n - l)}]. (35)

Substituting for AE“/p from (15) into (30) and setting AG(B,) = 0, the following expression is obtained for AC,.

j=l

AC, = [n’(l - 28,){A@,,(8,) - RT,lnn}

- 8,X {A$ - Ski} + AG,,,[Rln{l

-RT,({n(l

- Br)/(n - 1)8,}

- c {Szi - S;,}]/{RT/8,(1

- 8,) - 2AE”)

- n8:) - 1) lnc?,

- {n’(l - 8,)’ - n + 1) (34)

x In ((1 - d,)/(n - l)))]/(&,

- l)*. (36)

Orientational disordering in crystals-II

699

2.

l-

o

-xAG(b) T

Cdl -1 (n%leK 1

-0.

-

1.

0.: 2

-0.:3.

-0. Lb.

- 0. 5

- 0, 6

- 017--

200

240

220

260

200

TX(K)-’

Fig. 2. Properties of the free energy AG(B) = G(l/n) - G(B) where n = 6 and 8, is varied. The transition occurs along the line AG = 0.

Equations (35) and (36) facilitate the determination of d, and n once AG,, AH(B,) and AcD,,(B,) are known

via an iterative process as follows. (i) Compute iteratively the initial value of n from (35) with 6, = 1. (ii) With the value of n compute iteratively the new value of 8, from (36). (iii) Compute iteratively the value of n from (35) using the new value of 8,. (iv) Return to (ii) until a satisfactory degree of selfconsistency is achieved between the two eqns (35) and (36). The final value of n is then converted into an integer either by dropping the fractional part or by rounding off provided the corresponding configurational energy does not exceed the one deduced from calorimetric measurements discussed in the next section. The value of n so found is hereafter denoted by A in correspondence with 8,. In order to get at the specific heat function during the transition it is necessary to allow for the increase in the relative abundance of the plastic crystal phase during the transition. To do this it is convenient to assume that the two coexisting phases (the partially disordered phase in the transition eigenstate in equilibrium with the plastic crystal phase) may be represented by a series of partially disordered phases with progressively higher degrees of orientational disorder (i.e. values of d, smaller than that of the transition eigenstate) until the plastic crystal phase is

reached. If the transition eigenstates for the intermediate phases are denoted by d;, then the specific heat functions for the transition are given by eqns (20) and (21) with 6, = 8;. Values of 8, are determined from eqn (36) with T, = 7’ where T, < T’ I 7” and n = ri (fixed). CONFIGURATIONAL

AND THERMAL PROPERTIES

The difficulties likely to be encountered when an attempt is made to talk in terms of configurational and thermal properties as separable entities have already been alluded to above. For simplicity, eqn (20) is regarded as being predominantly configurational; and the energy obtained through the integration of this equation with respect to temperature will be assumed to give the configurational energy change associated with the transition. Similarly eqn (22) gives the thermal energy change during the transition. From eqn (23), the configurational energy change is given as follows: AE,(d,) = + AH@,) - t/r’

AS(B,)dT

T,

(37)

whereAH(b,)is the transition enthalpychange and the entropy integral is evaluated numerically. Equation

(37) is useful for obtaining the configurational energy change from calorimetric data. There is need to clarify

700

CHODZIWADZIWA C. MJOJO

what the parameters T, and 7” stand for. Conventionally TL is understood to mean the temperature corresponding to the maximum on the C,-curve of a i,transition. However the C,-curve normally returns to the baseline extrapolation at temperatures higher than the temperature with the maximum value of C,. Since interest is in the total enthalpy change, the integration in (37) runs from the temperature the C,-curve departs (T,) to the temperature where it rejoins (q) the baseline extrapolation. Illustratively, the configurational energy change for the I-transition in D-camphor is computed using the data reported by SchHfer and Wagner [9]. From their C,-curve for D-camphor, AT,, z 23 K. Further, their Fig. 8 shows that the total entropy varies more or less linearly from about 49calK-‘mol-’ at the transition temperature to 60 cal K-i mol- ’ at the end of the transition. They also determined the Debye temperature to be 48 K and assumed a value for the characteristic temperature for intramolecular vibration of 120K. This gives 18calK-1mol-’ and 10calK-lmol-l for the entropies of lattice and intramolecular vibrations respectively in the temperature region of the transition. Thus the orientational entropy varies from about 21 calK-‘mol-’ to about 32calK-‘mol-’ during the transition. The integral is thus -,620calmol-‘. This gives AE,(B,) = lOOOcalmol-’ which is about 37% of the total enthalpy change. Using the relation A&(b) = RT, Inn and T, = 242 K the largest possible value of n = 8. It is interesting to note that Schafer and Frey [ 111 deduced the value of II = 6 for LX-camphor from their specific heat measurements at low temperatures. Note that the isotropy of the plastic crystal phase requires a common value of n for both D- and DLcamphor [13]. Also the value of n = 6 is deduced for D-camphor from another independent approach below. Further, it is noteworthy that the value of n estimated by the use of eqn (37) and frequency assignments for intramolecular vibrations made in the mid-fifties is good enough as a starting point for the application of symmetry considerations (D-camphor is believed to have a hexagonal lattice [lo]) leading to the reported value. The fact that the i-transition in Dcamphor is predominantly thermal in character confnms the observation made by Schafer and Frey [ 111. Another view of the nature of coupling between configurational and thermal properties associated with orientational disordering may be obtained by examining eqns (31) and (32). The configurational energy and entropy changes may be written as follows: AE,(B,)

= AH,@,)

- q AEfh(B1) - A@,,@,) (38)

AS,(B,) = AS&?,)

- Y]A&,,

(39)

where AH,(B,) = H(l/Tz) - H(B,), A&,,(&,) = S(l/ii) - S(ti,) and r~ = (ri8i - 1)/r?. The subscripts c, m and th are used here to denote configurational, molar and

thermal quantities. The parameter n measures the proportion of thermal properties associated with the orientational disordering. Clearly, separability of configurational and thermal properties of the transition occurs when q = 1. This condition is obtained if ti is large and 8, 1~ 1. If, however, d, < 1.0 the separability may not exist even if ti is large. When 9 < 1, then the remainder of the thermal entropy (1 - n)A&,(Bi) and energy (1 - ~)AE,,,(Bi) make contributions to the background thermodynamic properties. COMPARISON WITH OTHER METHODS The

treatment by Aston [6] is given in terms of the degree of long-range order s which takes the value of unity at OK in the ordered phase, and zero in the disordered phase. The present treatment uses the fractional long range order a2 (Sin most formulations; e.g. Refs. [5,7], but S has been reserved for entropy in the present treatment) which takes the value of unity at 0 K in the ordered phase, and l/n in the plastic crystal phase. It can be seen that g1 = {l + (n - l)s)/n such that 0, = s = 1 when T= 0 K; and u, = l/n, s = 0 when T > TA.Thus the equivalent in our model of the quantity (1 - sz)/4 appearing in the energy term for 2n orientations in the Aston paper is (2n -

1) (1 + s [(2n - 2) - (2n - l)s]}/(2n)2

which returns to (1 - s2)/4 when n = 1. Thus our model reduces to that of Aston for one of the special cases. A method that has become useful for the estimation of the number of molecular orientations in the plastic crystal phase is due toGuthrie and McCullough [3]. This method has already been alluded to above and rationalises the transition entropy from primarily a configurational point of view using symmetry arguments. Thus this method by itself is generally not acceptable. More satisfactory is a combination of the Guthrie-McCullough method and the present model where the former specialises on rationalising n^ or deducing a more acceptable value of n from configurational data generated by the latter. Another method is due to Reynolds [14] and uses empirical intermolecular potentials. This method is based on the assumption that the configurational and thermal entropies are corrected via volume changes; and a self consistent calculation is adequate. Using this method Reynolds has calculated the entropy change in the plastic crystal phase transition of diazabicyclo[2,2,2] octane (DABCO) to be 60% due to entropy of expansion and 40”;, due to configurational entropy change. This calculation depends on the separability of the configurational and thermal entropies and indicates that in DABCO, the observed transition entropy of 7.2calK-‘mol-’ may be

701

Orientational disordering in crystals-II

from so that 2.9 cal K- ’ mol- 1 is configurational disordering. The result gives a value of 4 for n in the plastic crystal phase and therefore n = $ This implies that A&,(&,) = 57cal K-’ mole1 (not 4.3 cal K- ’ mol- ‘, where the latter is obtained from subtracting the configurational entropy from the transition entropy). It is interesting to note the similarity between the proportions of configurational entropy predicted for camphor and DABCO, and from totally different approaches. divided

CONFIGURATIONAL ENERGY CHANGES DUE TO ORIENTATIONAL DISORDERING FROM EXPERIMENTAL TRANSITION ENERGIES

An alternative approximate method which may be used in the calculation of configurational thermodynamic properties from thermal data alone is now presented. In principle a C,-curve can be converted to a C.-curve by the fundamental thermodynamic identity C, = C, - a=VTjK,

(40)

and the area between the C,-curve maximum and the baseline extrapolation would give the transition energy at constant volume, A&(&,). In practice, however, compressibility data are very difficult to obtain and for systems for which this type of data is not available the following approximate relation due to Nernst and Lindemann is commonly used. C, = C, - B(T/T,)C,z

(41)

where p = O.O214Kcal-‘. Lord et al.[l5] have empirically demonstrated that this expression is applicable to molecular crystals. Although eqn (41) is simple and widely used in the literature concerned with the study of plastic crystals (based on a common reference to Lord’s work) there appears to be some conflict in its application apparently due to some indecision on what values of T, are to be used-the orientational melting or the positional melting temperature. In applying eqn (41) to the C, data obtained through an orientational disordering transition, it is assumed that eqn (40) applies during the transitions. The problem simply amounts to demonstrating that quantities of the kind (dV/t?T), may be evaluated at the same pressure’ for the coexisting phases during the transition. Note that, from eqn (8), A&AT,, > 0 (always) during a thermal cycle. This implies that the internal pressure P > 0 for the orientational transition and P < 0 when the transition is reversed. Further, the positive pressure during the transition is exerted by the rotator sub-region with a larger specific volume on the surrounding non-rotator sub-region leading to thermal expansion. The negative pressure on reversing the transition refers to the pressure experienced by the surrounding rotator sub-region exerted by the enclosed non-rotator sub-region leading to thermal contraction. An important review pertaining to the interaction between phase sub-regions is found in JPcsVol. 41, No. 7-B

Ref. [16]. It has been shown above that the internal pressure associated with orientational disordering in D-camphor is about lo3 atm. Thus once this pressure is developed through the spontaneous creation of a rotator domain within the solid during the transition, the crystal relaxes through thermal expansion to minimise the free energy -(&4/W), = P = 0. A general and useful discussion on thermal expansion is found in Ref. [17]. Therefore, for cooperative transitions conducted quasi-statically, the internal pressure is constant throughout the crystal so that (W/‘laT),, may be evaluated at the same pressure (-0) for both phases. Equation (41) is now expanded into its components in order to obtain the contribution to the specific heat at constant volume due to the orientation of the molecules during the transition, C,(a). Let C, = C,(vib) + C,(a)

(42)

C, = C,(vib) + C,(a)

(43)

and assume C,(vib) = C,(D) + C,(I) where C,(D) represents the contribution to the specific heat from translational vibrations; C,(I), that from internal degrees of freedom of the molecules; and C,(a), that from the orientational degrees of freedom during the transition. Then putting (42)-(&l) in (41) and solving for C,(a) after ignoring small terms some of which cancel one another the following equation is obtained. C,(a) = [Ci(vib) + (Tm/BT){C,(c) + C,(vib) + T,/4fiTJ Ill2 - C,(vib) - T,I%bT.

(45)

Equation (45) facilitates the conversion of a C,(ajcurve into a C,(a)-curve. Integration of the area under the C,(a)-curve should give the transition energy at constant volume. However, it is necessary to decide what T, stands for at this stage. It is assumed that T, is the temperature at which the “orientational melting” is complete which should coincide with the temperature at which C,(u) is a maximum for an ideal I-transition. This temperature has already been called the final transition temperature Tl. This conclusion is arrived at by noting that the dominant contribution to the specific heat in the transition temperature region will be due to the orientational part of the specific heat, C,(u). Also the quantity C,(vib) in eqn (45) simply arises from the coupling term (2/?T/T,)C,(u)C,(vib), since the vibrational contributions to the specific heat have been cancelled out in the derivation of the equation. It is interesting to note that the coupling term is analogous to the one alluded to elsewhere above.

CHODZIWADVWA C. MJOJO

IQ2

In order to evaluate C,(vib) around the transition temperature which, it will be recalled, contains translational and intramolecular components it is proper to assume the high temperature limit of the Debye heat capacity for the intermolecular vibrations, especially for molecular crystals where the characteristic temperature is relatively small. This however will not be true for the high energy intramolecular vibrations. In this case the contribution to the specific heat should be done by a frequency assignment for each of the internal degrees of freedom and their thermodynamic properties evaluated from tabulated Einstein thermodynamic functions. However rough estimates for C,(vib) may be achieved by the following relation which assumes a small difference between C, and C, and that torsional oscillations are not significantly excited at lower temperatures before the transition. C’,(vib) 1: C,{ 1 - PTCJT,)

(46)

where T, is the ordinary melting point (assuming that the melting point of the ordered solid can be approximated by the observed melting point occurring above the plastic crystal phase). In some circumstances the C,-curve may not be available; but the experimental transition energy and the temperatures T, and T, characterising it may be known. In this case the form of C,(a) has to be assumed in order to calculate the transition energy at constant volume. Suppose C, = a + bT and let C,(o) = b(T- TJ such that T, < TI T,, then eqn (45) takes the form C,(g) = [Cz(vib) + (TJPT) x {2AH(a)(T-

T,)/AT: + C,(vib)

+ T,/4pT}]“’

- C,(vib) - T,@/3T

(47)

and the area under C,(a) gives the transition energy at constant volume. In obtaining CJu) a standard empirical formula which expresses C, as a power series in T has been used, where the formula has been used up to the first term only. Camphor is again used as an example with experimental data reported by Schafer and Wagner [9]. From their Table 6 C,(vib) = 17.61calK-‘mol-i at lOOK. Figure 3 depicts a C,(a)-curve obtained through eqn (47) when AH(a) = 27OOcalmol-’ and AT,, = 16K” (corresponding to the base of a right triangle that approximates the shape of the C,-curve). AE,(a) was estimated using Simpson’s rule to be 8701~1 mol- ‘. Thus about 32% of the experimental transition entropy in D-camphor is configurational. This is in reasonable agreement with the results reported above for D-camphor and by Reynolds for DABCO. The configurational transition energy of 870 cal mol- ’ gives a value of 6 for n in the plastic crystal phase of D-camphor. Further the thermal contribution to the background energy is about 364 cal mol - i. Table 1 summarises the property of eqns (46) and (47).

RELATIVE

ENERGY

PARAMETERS

It is now possible to discuss both the orientational and positional melting transitions together using thermodynamic functions derived for both orientational and positional disordering in papers I and II of this series. The enthalpy equations for the two types of disorder are now used in conjunction with the experimental transition temperatures and the readily calculable entropy functions to evaluate relative energy parameters. The positional mean disordering energy AE:,, has already been computed in paper I. To compute AE” without further information the approximation d, = 1.Ois made and eqns (28) and (29) are used to obtain AE” = {rt’/(n - l)}{T,AS(a) - (n - l)AE,JnJ

(48)

where AS(a) = .S(l/n) = S(1) and work terms have been ignored. It is convenient to re-write eqn (48) as follows AE” = T,{n’/(n - 1)) {AS(a) - (n - l)RAE,,,/n}

(49)

where AE,, = 1 {E(xi,) - E(xki)} 1 and E(x,~) = x,,,~[) + {exp (x,J - l} -i

I; X mt =

hw,JkT,

The relative disordering energy veff = AE”jAE& is now introduced. This parameter corresponds to the v of Pople and Karasz. In the present model v,rr is a measure of the relative energy for the orientation of a molecule on the one hand and simultaneous creation of a fluidised vacant site and diffusion of the molecule to the vacant site on the other hand. The latter situation simply corresponds to the creation of a gaslike molecule according to the Significant Structure of Liquids theory. Using eqn (29) of Paper I and eqn (49) above I.‘,rr=‘Vn,p,e,x)(T,IT,)

(50)

where ‘I’ = {xn2/(n - 1)) (AS(a) - (n - l)R AE,,,/n}/{AS(p) - R[t - E(x)11 T, is the melting temperature and x is a vibrational parameter. Note that vcfl.is generally defined in terms of configurational energies as the thermal energies are subtracted out. It is mteresting to note the direct correspondence between v,,r and the ratio x/T,. This ratio between transition temperatures has been used extensively byAmze1 and Becka in place of v to predict the behaviour of plastic crystals within the double

Orientational

340

I-

3oc

l.

260

I.

disordering

703

in crystals--II

18C

100.

60. C,(O)with

0

4

8

T,=TA

s

12

AT x (K)-’ Fig. 3. Specific heat curves from which AE,(d) is calculated using numerical integration when C,(vib) = 17.61 cal K-’ mol-‘. BE,(B) obtained by the use of T, is greater than the one obtained using TL. The latter is assumed to apply. Table 1. Properties of eqns (46) and corresponds to CDpbye + C,. C:(vib) C:(vib) is reasonable and improves as value of n 1s obtained with

(47) using C, and C,(vib) data from Table 6 of Ref. [9] where C,(vib) is generated by eqn (46). Note that agreement between C,(vib) and T increases. AE,,(u) decreases with increase in C,(vib). The best integral C,,(vib) values obtained at lower temperatures and equals 6

T

CP

80

17.68 21.08 24.55 28.09

100 120 140

C,(vib)

14.68 17.61 20.28 22.90

lattice model. The present model has the advantage that velf is readily calculable without any parameterfitting since all the relevant parameters can be determined directly from both theory and experiment. The critical relative disordering energy is by definition

(v,rf)e= {AE”IAGJ or = T,

(51)

C:(vib)

16.50 18.98 21.13 22.86

A&(a)

890 870 840 820

n

6.4 6.1 5.7 5.5

and is given by Y in eqn (SO). This parameter corresponds to the case where both orientational and positional disordering processes occur at the same temperature; i.e. the ordinary melting transition for substances without plastic crystal phases. To examine the properties of eqns (50) and (51) the simplifying assumption A& _ 0 is made, i.e. a perfect

CHODZIWADZIWA C. MJOJO

704

Frenkel-type transition (50) and (5 1) become V eff =

xT,n’Rln

where AG, = 0. Then eqns

W/(n

-

l)T,dh

(v,~& = Xn2R ln (@‘(fi - 1)4(~, x)

4

(52) (53)

where &,x)

= AS(p) - XR(4 - E(x)}.

The properties of eqns (52) and (53) for fixed values of T,IT,, p and &(p, x) (where the data for methane have been used with respect to these parameters) and varying n are summarised in Fig. 4 and Table 2. Thus for every value of n there are unique values for v,rf and (~,rr)~for a given substance. The values of the relative parameters for a given value of n are not uniquely defined for all substances which may not have

common values of T,/T,, p and c#& x). From Table 2, for n = 2 the value of v,rr = 0.037; and for n = 10, the value of this parameter is 0.340. The critical relative parameters (~,rr)~ for n = 2, 10 are 0.164 and 1.510 respectively. The two-sublattice model gives the critical value of 0.325 for n = 2 (Pople and Karasz); and 0.50 for n = 60 (Amzel and Becka). It is noted that the relative parameters vary more rapidly in the singlelattice model. From eqn (52) the following result is obtained for the melting temperature in plastic crystals. T, = n’(1 - P)AE,(a)/(n

- 1) (54)

x vC.rA(P, x)

The quantity &,x) is of the order of R for plastic crystals since AS(a) (the entropy of fusion) is of the order of R, x 2 0.1 for substances such as methane and

350

300

250

V,,, x JO2 200

750

100

50

0

Fig. 4. Properties of eqns (52) and (53) for fixed values of TJT,, p and +(p,x) and

n

is varied.

Orientational

disordering

705

in crystals--II

Table 2. Relative energy parameters for a number of molecular orientations using positional disordering parameters of methane (v

n

veff

2

0.037

0.164

10

0.340

1.510

20

0.838

3.722

E(x) = 3 in the classical limit. Therefore the melting temperature is directly proportional to the orientational configurational energy and inversely proportional to v,rf for given values of n and p. Thus the melting temperature is higher if A&(a) is large and v,rr is small. However eqn (54) is simply an analytical expression standing for a well known principle-that phase transitions in the solid elevate the melting temperature roughly in proportion to the entropy changes associated with them. This then is established as the explanation for the high melting temperature generally observed in plastic crystals. TRANSITION TEMPERATURE OF A GROUP OF PLASTIC CRYSTALS

As a follow up to eqn (54) the variation of transition temperatures for a group of plastic crystals have been examined. It is possible to expect a general trend in the transition temperature of plastic crystals by making the following observations. The value of p will not vary much between substances with plastic crystal phases since the magnitudes ofentropiesof fusion are generally similar. Thus the most important parameters are n, vefr and A&(o) as these parameters would be expected to vary markedly, but systematically, between different substances. Figure 5 is a plot of transition temperature for a group of plastic crystals tabulated in Appendix 1. The reference to the numbering system is also given in this appendix. Substances with a lower entropy of orientational transition occupy the lower end of the plot (e.g. methane); and those with a higher entropy of transition, the upper end (e.g. camphoric anhydride). Also molecules in the camphor group (16, 38, 48, 50, 51) and the methane group (l-3) form their own patterns. The methane group is particularly interesting. In this case the melting temperature decreases with molecular mass, while the plastic crystal phase transition temperature increases. The pattern with respect to the plastic crystal phase transition has been predicted by Yamamoto et al. [4] who attribute it to unusual quantum effects at the transition. It is seen that there is some correlation between the melting point and the orientational transition temperature. The correlation is partly governed by the constraint T,/T, > 1. Thus there is a forbidden region on the right side of the cluster of points bounded by a curve (not drawn) which passes through points T, = TA. This observed correlation implies that the

eff

1

c

proportionality between T, and A&(o) does not vary much from compound to compound in this group. CONCLUDING REMARKS

A number of important observations emerge from the present model and are now summarised. (1) The characterisation of the dynamics of orientational disordering in terms of the Frenkel or Pauling models depends on the extent to which a re-orientating molecule may be regarded as an independent particle in a dynamic crystal lattice. Both the Frenkel and Pauling models view the re-orientating molecule as an independent particle in the crystal lattice and therefore bear a fundamental classical feature of the transition. Further, the absence of any quantisation in the re-orientation process associated with the Frenkel model makes the model strictly classical. On the other hand, the coupling between the internal and external phonon modes leading to re-orientationphonon coupling imparts a quasi-particle status to the re-orientating molecule in the dynamic crystal lattice. Under these circumstances phenomenological descriptions based on the Frenkel and Pauling models may not apply. A general two-phase model that incorporates the re-orientation-phonon coupling AS0

A45

A39

A49

A41

A”

A46

Al2

A”’

T,x(K)”

A44

A23

A=

A’6

300

A25 A27

A=

A42

A43 A39 A40

24

150-

A4

!&

# 100

Fig. 5. The variation crystals

(see Appendix

I

L

I

I

300

I

,

T, x(K)-’

of T,, with T, in a group of plastic 1 for a guide to numbering scheme).

706

CHODZIWADZIWA C. MJOJO

explicitly bears a fundamental quantum feature of orientational disordering. The absence of this type of coupling, for example, in the molecular field method due to Yamamoto et al. [4] provides a source of weakness for that scheme, particularly when dealing with low temperature transitions. A useful review of models of orientational disorder in crystals is presented in Ref. [18]. (2) One of the important results generated by this model is eqn (37). This equation is a consequence of the introduction of the orientation work function and appears to have a general form. The equation applies in both perfect Frenkel-type and phonon-dominated orientational transitions. For example eqn (23) still applies even when C, and C, are given by eqns (25) and (26) respectively. The reasonable agreement between the configurational energy computed from eqn (37) and that predicted independently from eqn (47) is very encouraging. Note that the use of the expression A&(a) = RT,ln (n) is an approximation since from eqn (32) the configurational entropy change may be less than R In (n) if 8, < 1.A detailed application of the present model is the subject of a subsequent paper. The orientational transition work function plays an important role in determining the thermodynamic properties of the transition to the extent that phase eigenstates determined with work terms included lead to the possibility of coupling between orientational disordering and a reconstructive transition (a modification of the molecular potential energy surface). This model is in fact more realistic as orientational transitions often are accompanied with reconstructive transitions [lo] and is investigated further in a subsequent paper. Configurational changes are generally of secondary importance as far as transition thermodynamic properties are concerned. However configurational properties alone are sufficient in defining the pattern of orientational disordering in two component systems [ 131. (3) One of the parameters involved in the study of orientational disordering is n. The limitations of the Guthrie-McCullough method [3] for the determination of n have already been alluded to elsewhere and were also noted by Amzel and Becka [7] who attempted to improve the method by using the ratio TJT, in conjunction with n values to predict transition entropies. According to the present method the modified Guthrie-McCullough method still overestimates n values. For example using Table 1 of Amzel and Becka [7], the experimental transition entropy of the hexafluoride W’F6 is 7.81 calK-’ mol-’ and This A&(a) < AH(a)/2 = T, = 264.7 K. gives and therefore n < 7. The value of n l.O3kcalmol-‘, predicted by Amzel and Becka is 14. This value was arrived at by assigning to the molecule the D,,, and C,

sets of orientations which give n = 6 and n = 8 respectively. The thermodynamic configurational analysis above would seem to exclude the set of orientations generated by the C, subgroup. A more realistic modification to the Guthiie-McCullough method is provided by Yamamoto et al. [4] where the molecular symmetry subgroups are classified in terms of their associated potential energies thereby providing a means of selecting the effective orientation symmetry. This, in effect, implies that the scheme for the selection of the effective orientation symmetry based on symmetry considerations alone is not yet fully developed.

Acknowledgements-Discussions with Prof. A. N. Hambly and I. G. Ross and Dr. P. A. Reynolds and remarks made by Prof. J. G. Aston and Dr. R. C. Brown proved useful. I thank Mr. L. P. Chirwa for carrying out some of the computations and Mrs. A. Mtenje for her patience and skillful typing. Paper I and some parts of Paper II were written whilst the author was supported by a fellowship under the Commonwealth Postgraduate and Fellowship Plan in Australia and a supplementary scholarship by the Australian National University. I thank the Commonwealth Government of Australia and the ANU for these awards.

1.

2. 3. 4. 5. 6. 7. 8. 9.

REFFRENCES Darmon I. and Brot C., J. Molec. Cryst. 2, 301 (1967). Fyfe C. A., Faraday Transactions II lo,1633 (1974); 10, 1642 (1974). Guthrie G. B. and McCullough J. P., J. Phys. Chem. Solids 18(l), 53 (1961). Yamamoto T., Kataoka Y. and Okada K., J. Chem. Phys. 66, 2701 (1977). Pople J. A. and Karasz F. E., J. Phys. Chem. Solids 18,28 (1961); Karasz F. E. and Pople J. A., J. Phys. Chem. Solids 20, 294 (1961). Aston J. G., J. Chem. Therm. 1, 241 (1969). Amzel L. M. and Becka L. N., J. Phys. Chem. Solids 30, 521 (1969); 30, 2495 (1969). Knox J. H., Molecular Thermodynamics, p. 43. Wiley (1978). Schlfer K. L. and Wagner U., Z. Electrochem. 62,328 (1958).

10. Mjojo C. C., J.C.S. Faraday Trans. II 75, 692 (1979). 11. Schlfer K. and Frev_ 0.. Z. Electrochem. 56(9), 882 (1952). 12. McClelland B. J., Statistical Thermodynamics, p. 115. Chapman and Hall and Science Paperbacks (1973). 13. Mjojo C. C., J.C.S. Faraday Trans. il75, 667 (1979). 14. Revnolds P. A.. Molec. Phvs. 28(3). 633 (1974). 15. Loid R. C., Ahiberg J. E. a&?An&ws DI H., J. Chem. Phys. 5, 649 (1937). 16. Ubbelohde A. R., British J. Appl. Phys. 7, 313 (1956). 17. Reissland J. A., The Physics of Phonons, p. 123. Wiley (1973). 18. HullerA. and Press W., The Plastically Crystalline State, (Editor: Sherwood J. N.I. Chap. 10. Wiley (1979). 19 Aston J. G., Physics and Chenustry of the Organic Solid State, Vol. 1. (Editors, Fox D., Labes M. M. and Weissberger A.). Interscience, New York (1963).

OrIentational

disordering

707

in crystals-II

APPENDIX Guide to the numbering No.

i: $3 4 $2 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

of molecules in Substance

methane monodeuteromethane deuteromethane cyclopentane hexafluoroethane decalluorocyclopentane 2,2-dichloropropane 2,2,3_trimethylbutane cyclopentane t-butylethylene 2,2_dimethylbutane norbonane cyclohexane tetramethylmethane cyclobutane camphene 2,2,3,3_tetramethylbutane 1,l -dimethylcyclohexane methanol 1,3-cyclohexadiene cyclohexylfluoroform 1-chlorocyclopentane dodecafluorohexane cis-1,2-dimethylcyclohexane tetranitromethane t-butyl chloride

Fig. 5 No.

32 33 34 35 36 +Z f39 40 41 42 43 $44: 46 41 48 49

Substance cyclohexane t-butyl iodide cyclopentanol t-butyl bromide 2-chloro-2-nitropropane I-chlorocyclohexane methyl chloroform cyclohexanone carbon tetrachloride succinotrile 2-bromo-2-nitropropane camphor cyclohexanol cyclohexyl cyanide 2,3-dichloro-2,3-dimethylbutane 2,2-dinitropropane pivahc acid tetrakis-methane hexachloroethane carbon tetrabromide 1,Ccyclohexanedione borne01 2,3-dibromo-2,3-dimethylbutane camphorimide camphoric anhydride

tFrom our own measurements. All data (except those labelled t) are from a table m Ref. [19]. TransItion temperatures given by defining a range have been modified to the average of the two extreme numbers. $Two transitions occur before the melting transition and the rotational transition has been assumed to be one occurring at the lowest temperaturefollowmg the pattern observed in camphor [l 11; except for the case of the methane group where the rotational transition has been explicity associated with the intermediate transition temperature [19].