J . Phys. Chem. 1990, 94, 3 152-3 156
3152
TABLE II: LLE Parameters4 of the Present Activity Coefficient Model for Different Binary Mixtures system @,I" a,,, x 103
s
methanol + heptane -1070.9 methanol + cyclohexane -980.29 formic acid + benzene -748.15 phenol + octane -1077.0 methanol + carbon disulfide -797.20 nitrobenzene + hexane -975.97 I ,3-dihydroxybenzene + benzene -937.02
1.0243 1.0191
0.97983 1.0031 1.0186
1.0027 1.0032
0.12 0.12 1.7
7.3 0.45 2.0 5.5
LLE, because it always satisfies inequality.29 The present activity coefficient theory is tested versus experimental LLE data.I2 For the variation of the isothermal compressibilities with temperature, which is needed in this calculation, a three-parameter equation'j is used. The temperature and pure component volume dependence of parameters and a 1 2for a number of organic binary systems ( 12) Liquid-Liquid Equilibrium Data Collecrion; DECHEMA: Frankfurt, 1979. (13) Brostow. W.: Maynadier, P. High Temp. Sci. 1979, / I , 7. (14) McQuarrie, A. Statistical Mechanics, Harper and Row: New York, 1975. (15) Haile, J. M., Mansoori, G. A,, Eds. Molecular-Based Study of Fluids; American Chemical Society: Washington, DC, 1983. (16) Mazo, R. M. J . Chem. Phys. 1958, 29, 1122. (17) Hamad, E. Z.; Mansoori, G. A.; Ely, J. F. J . Chem. Phys. 1987,86,
1478.
(18) Wilson, G. M. J . Am. Chem. Soc. 1964, 86, 127.
exhibiting LLE are represented by the following expressions. a21 = ( U * / U I ) exp(a,,,/kT) (31) = 0'20(4/~2) (32) In which a210 and a I Z O are constants independent of temperature and composition. The experimental LLE data is fitted to the activity coefficient equation with eqs 31 and 32 for a2,and c y l 2 by minimizing the difference between the activities in the two liquid phases a12
where superscripts (1) and (2) are for phases 1 and 2, respectively. and aI2, and optimum Table I1 shows values of parameters a z I O S for a number of binary systems. The variation of the compositions of two LLE systems (methanol heptane and formic acid benzene) with temperature is shown in Figures 4 and 5. According to these figures the agreement with the data is satisfactory. The largest deviation of theory from experimental data occurs near the upper critical solution temperature. Overall the present technique is capable of formulating analytic expressions for activity coefficients in mixtures. For this purpose it is necessary to define closure expressions for the cross direct correlation function integrals. Application of the activity coefficient expression resulting from this technique for vapor-liquid and liquid-liquid equilibria calculations has been as successful as the other activity coefficient expressions available.
+
+
Acknowledgment. This research is supported by the Gas Research Institute Contract No. 5086-260-1244.
Relationship between Microscopic and Macroscopic Orientational Relaxation Times in Polar Liquidst Amalendu Chandra and Biman Bagchi* Solid State and Structural Chemistry Unit, Indian Institute of Science, Bangalore 560 012, India (Received: May I , 1989; In Final Form: October 12, 1989)
Microscopic relations between single-particle orientational relaxation time ( T , ) , dielectric relaxation time ( T ~ ) and , many-body orientational relaxation time ( T ~ of) a dipolar liquid are derived. We show that both T~ and T~ are influenced significantly by many-body effects. In the present theory, these many-body effects enter through the anisotropic part of the two-particle direct correlation function of the polar liquid. We use mean-spherical approximation (MSA) for dipolar hard spheres for explicit numerical evaluation of the relaxation times. We find that, although the dipolar correlation function is biexponential, the frequency-dependent dielectric constant is of simple Debye form, with T~ equal to the transverse polarization relaxation time. The microscopic T~ falls in between Debye and Onsager-Glarum expressions at large values of the static dielectric constant.
Introduction
The relationship between single-particle orientational relaxation time, T,, and many-body orientational relaxation time, T ~ of, dipolar molecules in a dense liquid has been a subject of much discussion in the recent past.'-4 Another related question is the relationship between T , and dielectric relaxation time T~ of a pure dipolar liquid.5-" It is clear that the dielectric relaxation time contains some amount of many-body effects, although the precise , T~ have never been made clear. relations between T,, T ~ and Following Kivelson and Madden,Iv2 we shall call a relation between a microscopic relaxation time, such as T " , and a macroscopic relaxation time, such as T M ,a macro-micro relation. The first such relation was proposed by D e b ~ e ' who , ~ used his continuum 'Contribution No. 567 from Solid State and Structural Chemistry Unit. 0022-3654/90/2094-3 1 52$02.50/0
theory of dielectric relaxation to propose the following relation n2 + 2 7, = (1) co 2TD
+
where co is the static dielectric constant and n is the refractive ( 1 ) Kivelson, D.; Madden, P. Mol. Phys. 1975, 30, 1749. (2) Madden, P.: Kivelson, D. Adu. Chem. Phys. 1984, 56, 467. (3) Evans, M . ; Evans, G. J.; Coffey, W. T.; Grigolini, P. Molecular Dynamics and Theory of Broad Band Spectroscopy; Wiley: New York, 1984; Chapter 3. (4) Berne, B. J . Chem. Phys. 1975, 62, 1154. Berne, B.; Pecora, R. Dynamic Light Scattering; Wiley: New York, 1976. (5) Hubbard, J. B.; Wolynes, P. G. J . Chem. Phys. 1978, 69, 998. (6) Hill, N. E. In Dielectric Properties and Molecular Eehauiour; Hill, N. E., Vaughan, W. E., Prince, A. H., Davies, M., Eds.; van Nostrand: London, 1989.
0 1990 American Chemical Society
The Journal of Physical Chemistry, Vol. 94, No. 7, 1990 3153
Relaxation Times in Polar Liquids index. The Debye form is inadequate for strongly polar sol~ e n t s . ~ Glarum’O .~J~ modified eq 1 for strongly polar liquids and his expression is given by 2eo t, TU = 7D 3€0 where e, is the infinite-frequency dielectric constant. Equation 2 is based on Onsager’s model of static dielectric constant and we shall refer to eq 2 as the Onsager-Glarum expression. Equation 2 predicts that, for large to. 7,, = 0 . 6 7 ~ In ~ . fact, the OnsagerGlarum expression is a limiting form of a more general relation derived by Powlesll several years ago. This form is given by
+
7,
-( + ):
= 2to
t,
3t0
where g is the well-known Kirkwood’s g factorI2which is a measure of short-range correlations in the dense dipolar liquid; these correlations are neglected in a Onsager-type theory. Thus, Powles’ modification takes into account the effects of the short-range correlations. Equation 2a reduces to eq 2 when g = I . An analytic expression for g can be given in terms of an integration over the anisotropic part of the radial distribution function of the dipolar liquid.2 Madden and Kivelson’-2have shown that 78 and 7u are related by the equation
where p = (kB7“)-l, k B is Boltzmann constant, and T i s the temperature. p is the magnitude of the dipole moment of liquid molecules and po is the equilibrium density of the liquid. g’ is the dynamic coupling parametere2 I n this work we present, for the first time, an analytic expression for the dynamic coupling parameter, g’, in terms of the two-particle direct correlation function by comparing eq 2b with our microscopic expression for ‘D.
It is clear that, in a dense, strongly polar liquid, intermolecular interactions play an important role in determining the rate of orientational relaxation. It is equally clear that the microscopic structure of the dense liquid is important. In this paper, we present, for the first time, relations between T ” , 7 M , and 7 D which are derived from a microscopic theory that pays proper attention to the intermolecular interactions and also to the microscopic structure of the polar liquid. Especially, we show that neither eq 1 nor eq 2 provides correct description at large e@ We also find that, although the dipolar correlation function is biexponential, the frequency-dependent dielectric function has the simple Debye form with 7 D equal to the transverse polarization relaxation time of the dipolar liquid. The question of the correct relationship between a macroscopic correlation function, containing many-body effects, and a microscopic correlation function dominated by single-particle motion has also been raised.’-3 For a dipolar liquid, we define the macroscopic correlation function by the following equation N
Theory The generalized Smoluchowski equation for the time dependence of the position- and orientation-dependent density fluctuation, Gp(r,w,t),is given by
a
-Gp(r,w,t) = DRVu2bp(r,w,t)+ DTV26p(r,w,t)at
DR( $ ) V > l d r ‘ I
dw’ c(r,w,r’,o’) 6p(r’,w’,t) -
\
N
cM(t) = ( E c i ( o ) C G j ( t ) ) I=‘
Kivelson and Madden’,2 have shown that, if C,(t) is single exponential, then CM(t) is biexponential. Moreover, these authors concluded that the longitudinal and transverse components of cM(t) separately must have the same form as the single-particle correlation function; the only difference may be that the parameters are different. The calculations presented in this paper agree with this conclusion. The new aspect of this work is that we present explicit analytic forms of the relaxation parameters in terms of molecular quantities and that we consistently include the effects of the translational modes in the relaxation of the wave-vectordependent correlation functions. The frequency-dependent dielectric constant e(w) may be calculated by using either the Fatuzzo-Mason relation” or by using more general linear response relations recently given by Loring and M ~ k a m e 1 . I ~Thus, it is straightforward to relate the dielectric relaxation time to the macroscopic relaxation time if the correlation function c M ( t ) is known completely. Our calculation is based on a generalized Smoluchowski e q ~ a t i o n ’ ~for - ’ ~the time dependence of the position- and orientation-dependent density field. This Smoluchowski equation or its different forms have been used previously in the study of and solvation polarization relaxation in dense dipolar also in the study of the dynamics of the liquidsolid transition^.'^,^^ The Smoluchowski equation contains a mean-field force term because of intermolecular interactions between dipolar molecules and contains contribution from both rotational and translational motions. In the limit of long wavelength, the translational contribution to the orientational correlation functions disappears but it is significant in intermediate-wavelength processes and for that reason we have retained this term. Our calculations show that, even for a system of dipolar hard spheres, cM(t) is biexponential with one component which relaxes with a transverse polarization time constant which is of the same order of magnitude as the orientational relaxation time 7,. The second component, however, relaxes much faster with a time constant equal to the longitudinal relaxation time. We find numerically that the second, faster component makes a relatively small contribution to CM(t). The frequency-dependent dielectric constant, however, is found to have a simple Debye form with a relaxation time equal to the transverse polarization relaxation time. Thus, at least in this model, we can define a unique Debye relaxation time which can be related to other relaxation times of the polar liquid.
j=l
(3)
where Ci is a unit vector with the orientation of the dipole i. The microscopic correlation function is defined by Cu(t) = ( c i ( O ) * c i ( t ) ) (4) (7) Farber, H.; Petrucci, S. In The Chemical Physics of Soluarion, Parr B; Degonadze, R. R., Kalman, E., Kornyshev, A. A,, Ulstrup, J., Eds.; El-
sevier: Amsterdam, 1986; pp 433-436. (8) Scaife, B. K. P. In Molecular Relaxation Processes; Chemical Society Special Publications No. 20; Academic Press: New York, 1966. (9) Bottcher, C. J . F.; Bordewijk, P. Theory of Electric Polarization, 2nd ed.; Elsevier: Amsterdam, 1978; Vol. 2. (IO) Glarum, S. H. In Dielectric Properties and Molecular Behaviour; Hill, N. E., Vaughan. W. E., Prince, A. H., Davies, M., Eds.; van Nostrand: London, 1969. ( 1 1) Powles, J. G.J . Chem. Phys. 1953,Zl.633. Deutch, J. Faraday Sym. Chem. SOC.1977,1 1 , 26. See also ref 2, pp 492 and 522. (12) Kirkwood, J . G. J . Chem. Phys. 1939,7, 911.
where DR and DT are the rotational and translational diffusion coefficients of the liquid molecules, respectively, and 0 and V u are the usual spatial and angular gradient operators. c(r,w,r’,w’) is the two-particle direct correlation function’* that depends both on position ( r ) and orientation ( w ) of the two molecules. The validity of this Smoluchowski equation has been discussed elsewhere16 where we have also presented a simple derivation of the GSE from time-dependent density functional theory. As we (13) Fatuzzo, E.; Mason, P. Proc. Phys. Soc. London 1967,90, 74. (14) Loring, R. F.; Mukamel, S. J . Chem. Phys. 1987,87, 1271. (15) Calef, D.; Wolynes, P. J . Chem. Phys. 1983,78, 4145. (16) Chandra, A,; Bagchi, B. Chem. Phys. Lett. 1988,151, 47. (17) Bagchi, B.; Chandra, A. Proc. tndian Acad. Sei. (Chem. Sei.) 1988, 100, 353. (18) Bagchi. B. J . Chem. Phys. 1985,82, 5677. (19) Bagchi, B. Phys. Lett. 1987, 121, 29.
3154 The Journal of Physical Chemistry, Vol. 94, No, 7, 1990 discuss below, eq 3 has all the correct limiting behavior except that it neglects the inertial effects. As mentioned earlier, translational contribution is absent for c(w). Next, we use eq 5 to derive expressions for C,(t), CM(t),and 4w).
A. Correlation Functions. Let us define the following correlation functions, C/,(k,t), by
C/m(k,t) = ( Y/m*(k) Y/m(k,t))
(6)
Chandra and Bagchi integrations over the pair correlation function, h(k,w,w’). In the last step, the pair correlation function h(k,o,w’) is expanded in spherical harmonics (see ref 20 for such expansions). These steps are straightforward to carry out and we obtain the following expression of the orientational correlation function J.(k,t) J.(k,t) = Ale-‘/’l(k) + AZe-f/V(k) (14) where
where the spherical harmonics Y,,(k,t) are defined by4 N
Y/m(kJ) = Cexp(ik*r,(f)) Y/m(wa(t)) UE
I
(7)
.
c
and (...) denotes average over an equilibrium ensemble of homogeneous states of the liquid. For convenience we work in the Fourier space and a Fourier transform of a functionfir) is defined as f(k) = l d r exp(ik.r) f(r) Next we define a reduced correlation function #(k,t) by
- -
The correlation functions cM(t)and CJt) are now obtained by taking the limits k 0 and k 03 (with & = 0), respectively, of the function J.(k,t). In the next step, we expand the fluctuation in the density field in the spherical harmonics Gp(k,w,t) = Ca/,(kJ) Y/m(w)
As mentioned earlier, eq 15 and 16 have been derived by using the spherical harmonic expansion of the two-particle pair correlation function, h(k,w,,w’), of dipolar hard spheres.z0 The collective orientational correlation function cM(t)is given by cM(f)= Iim $(k,t) = Al(k=O)e-‘/‘MI Az(k=O)e^’/’M2 (19)
I,m
+
k-0
(9) where
It is straightforward to show that the correlation function J.(k,t) is given by
The direct correlation function c(k,o,w’) can also be expanded in terms of the spherical harmoniesZo C(k,w,w’) =
C c(lII2m;k)Y/m(w) Y I , ~ ( ~ ’ )
(1 1)
1lh
where k is taken parallel to the intermolecular axis. Substituting eqs 9 and 1 1 in eq 5, we obtain the following equation for the time derivative of alm(k,t)
a
Za/,(k,t) = - [ D ~ / ( l + l )+ D ~ k ’ ] a / ~ ( k , + t)
The equation of motion for the correlation function relevant for dielectric relaxation is obtained by setting 1 = 1 in eq 12. A considerable simplification of eq 12 results if we use MSA for c(k,w,w’).20MSA predicts that the only nonvanishing c(ll,lz,m;k)’s are c(000;k), c( 110,k), and c( 1 1 1,k). Equation 12 then reduces to a simpler equation of the following form
Thus we see that cM(t)is biexponential with two time constants given by (20) and (21). We show below that T M I is the transverse polarization relaxation time constant and T M Z is the longitudinal relaxation time. Equations 14, 20, and 21 constitute an important result of this paper. The new features about these expressions are that the wave-vector-dependent relaxation times are expressed in closed form in terms of the components of the two-particle direct correlation function. The complete form of J.(k,t) is also presented (eqs 14-18). B. Dielectric Constant t(w). In this subsection, we first derive microscopicexpressions for frequency- and wave-vector-dependent dielectric constant c(k,w). We then take the proper limit ( k 0) to obtain t(w). Formal expressions for the longitudinal dielectric constant cL(k,w)and transverse dielectric constant eT(k,w) were recently presented by Loring and MukamelI4 in terms of the polarizabilities aLand aT. They are given by
-
tL(k,w) = [l tT(k.W) = [ 1
- 4saL(k,w)]-’ 4Ta~(k,U)]
(22) (23)
whereI4
P
al(k,w) = ; [cML(k,o) - iWcML(k,W)] Equation 13 is a linear equation which can be solved easily to obtain a,(k,t). In order to obtain the desired correlation function $(k,t), we need to evaluate the same time correlation function (alm(k,t)alm(-k,t)).These correlation functions can be obtained by using the expansion of 6p(k) as given in eq 9. The calculation of the same time autocorrelation of alm(k)then involves simple (20) Gray, C. G.; Gubbins, K.E.Theory ofhfolecular Fluids;Clarendon: Oxford, U.K.,1984; Vol. I. ( 2 1 ) Zwanzig, R. J . Chem. Phys. 1963, 38, 2766.
P
ar(k,w) = ;[CMT(k,O) - i d ~ ~ ( k , O ) ]
(24) (25)
with cML(k,W) = k M ( k , w ) * i
(26)
(27) cMT(k.W) = fi*CM(k,U)*k where k is a unit vector parallel to k and ti is a unit vector perpendicular to k. CM(k,t)is a tensor correlation function defined as
Relaxation Times in Polar Liquids
The Journal of Physical Chemistry, Vol. 94, No. 7, 1990 3155
CM(k,t) = pz(SP(k) GP(-k,t))
(28)
and cM(k,w)is the Laplace transform of the function, CM(k,t), p is the magnitude of the dipole moment of the liquid molecules. The polarization fluctuation, SP(k,t), is defined as GP(k,t) = Jdw $ ( w ) bp(k,w,t)
PURE DIPOLAR LIQUID
p0 -’=0.8
/
(29)
Now, the time evolutions of the longitudinal and transverse polarization correlation functions are given byi6J7 (PL(k) PL(-k,t)) = (PL(k) PL(-k))e-fl’L(k)
(30)
(PT(k) PT(-k,t)) = (PT(k) PT(-k))e-‘/‘T(k)
(31) The longitudinal and the transverse relaxation times areI6J7
___ _______-_
-__
- - - -----
0.2 00
1.0
-
ONSAGER -GLARUM
LO
8.0
120
160
200 240
2130 320
36.0 4 0 0
Eo-
(33) where p ’ = oT/(20Ruz) and u is the diameter of the solvent molecules. The dimensionless parameter p’is a measure of the relative importance of the translational modes. cA(k)and cD(k) are the anisotropy components of the direct correlation function, c(k,u,u’). In terms of c(ll12m;k),T~ and T~ can be written as
We next combine eqs 24-27 and 28 with 32 and 33 to obtain 1 iuTL(k) tL.(k,w) = 1 - a(k) iwrL(k)
+
+
(37)
Figure 1. Dependence of dielectric relaxation time, T D (given by eq 42) on the static dielectric constant eo of the dipolar liquid. The solid line gives the result from the present theory. Predictions of Debye (eq 1 of the text) and Onsager-Glarum (eq 2) are also plotted for comparison.
time constant. It contains many-body effects through its dependence of the c( l l l ;k=O) component of the two-particle direct correlation function (see eq 42 and 35). This result is to be contrasted with the lattice model calculations of Zwanzig2’ and of Loring and MukamelI4 where no such Debye form could be recovered. C. Relations between Different Relaxation Times. From eqs 14 and 19, we see that Cu(t) is a single exponential with a time constant T~ and cM(t) is biexponential with time constants 7 ~ 1 and Equation 41 shows dielectric constant e ( w ) is single . 20, 21, and 42 exponential with a time constant T ~ Equations give the desired relations between 7” (=1/(2&)), 7 ~ 1 T, ~ and~ , TD’
An interesting consequence of the preceding analysis is that we can combine eq 2b with eqs 20 and 42 to obtain an analytic expression for the dynamic coupling parameter, g’. This is given by g’ =
3
-e0(to
PP2Po
where
( + ;;
- 1) 1
-c(
1 1 1;O)
1
(43)
We can further simplify this expression by relating c( 111;O) to the static dielectric constant by using eq 36 and the OrnsteinZernike relation.
-
In the limit k 0, cL(w)= tT(u) = t ( ~ ) .In this limit, a relation between a(k) and b(k), defined by eqs 38 and 39, was earlier obtained by Fulton,z2 given by b = a/(l - a )
-
(40)
However, note that eq 40 is valid only in the limit k 0. For the nonzero finite value of wave vector k , the longitudinal and and eq 40 does not hold. transverse parts of e(k) are not In the limit k 0, we obtain the following expression for the frequency-dependent dielectric constant
-
t(W)
= 1
to- 1 +1 + iwTD
(41)
where = TT(k-0) = eOTL(k=O)
(42) Thus, the present theory predicts a single Debye form with T~ given by eq 42. Note that T~ is not the single-particle rotational TD
(22) Fulton, R. L. Mol. Phys. 1975.29.405; J . Chem. Phys. 1975,63,77. ( 2 3 ) Chandra, A.: Bagchi, B. J . Chem. Phys. 1989, 90, 1832.
Numerical Results Since we now have microscopic expressions of the orientational relaxation times, it is instructive to check the reliability of the continuum theoretic expressions of Debye (eq 1) and of Glarum (eq 2). In order to carry out numerical calculations, we need values of the anisotropy components, c( 110; k=O) and c( 1 11; k=O) for a dipolar liquid. We have used the MSA solution for dipolar hard spheresz0 with the Percus-Yevick equation for a hard-sphere reference system. The MSA is known to provide semiquantitatively accurate results for not too strongly polar liquids. Moreover, the results obtained here do not depend critically on the use of MSA. In Figure 1, we have plotted the dependence of Debye relaxation time (which is equal to T~~ in the present theory) against the static dielectric constant of the dipolar liquid for all the three expressions. At low values of the static dielectric constant, all three expressions agree with each other. As the value of to is increased, the three predictions deviate from each other. The Debye equation is known to be inaccurate’at large eo because it neglects the reaction field effects. The continuum model prediction (eq 2, referred to as Onsager-Glarum expression) correctly predicts a saturation in the dependence of T~ on eo, but the saturation is predicted much too early. In fact, the microscopic theory presented here predicts that the real behavior falls in between the two continuum pre-
J . Phys. Chem. 1990, 94, 3 156-3 160
3156
dictions. In fact, the functional dependence of T~ on the properties of the dipolar liquid agrees with that given by Powles" with the Kirkwood g factor is given by eq 43.
equation. We are presently carrying out a detailed analysis of the full nonlinear Smoluchowski equation. We also presented the first microscopic calculation of the frequency-dependent dielectric constant, t(u). Surprisingly, t(u) is of simple Debye form, even when the relevant dipolar correlation function is bie~ponential.~~ The transverse and longitudinal time constants are expressed in terms of the anisotropic part of the two particle direct correlation function. Important future extensions of this work would be to consider nonspherical dipolar molecules and also the use of a more accurate representation of thc direct correlation function than MSA. We are currently working on these problcms.
Conclusion In this paper, we presented the first microscopic relations be, T~ of a dipolar liquid. tween the relaxation times T", T ~ and Unlike previous continuum model approximations, these relations are valid over a large polarity range. Interestingly, our microscopic relation between T~ and T, interpolates between the expression from Debye theory, valid at low dielectric constant, and the Onsager-Glarum expression. We also find that the assertion of Kivelson and Madden's2on the similarity between C,(t) and CM(t) holds exactly in the prsent case as CM(t)is biexponential with both the longitudinal and transverse components decaying single exponentially with different time constants. However, this may be because of the Markovian nature of the linearized Smoluchowski
~~
_____~
(24) Thi:. simple Ilcbyc form arises principally from the neglect of nonMiirkoviaii cllcct:. in cq 5. We have shown elsewhere that a non-Markovian gcticr:ili/:ilioii 01' Ihc lhcory presented here can lead to non-Debye behavior o l ( ( ~ 3 ) . (Ihgchi. H.:Chandra, A. Phys. Reo. Left., in press.)
Investigation of the Phase Transition Behavior in Solid Cycloheptane Julian Haines and D. F. R. Gilson* Department of Chemistry, McGill University, 801 Sherbrooke St. W., Montreal, Quebec H3A 2K6, Canada (Received: September 18, 1989)
The phase transition behavior in solid cycloheptane has been investigated by differential scanning calorimetry and vibrational spectroscopy (infrared and Raman). The spectra of four stable phases and of two glassy crystalline phases have been obtained. The spectrum of the stable, low-temperaturephase, phase IV, is very different from the spectra of the other phases, indicating that this is the only ordered phase. The observed low-frequency Raman spectra are consistent with isotropic reorientation and anisotropic reorientation, related to a molecular pseudorotation process, occurring in phases I and I1 and phase 111, respectively. The spectra of the glassy crystalline phases indicate that these phases possess static disorder, which is related to that of the corresponding high-temperature phases.
Introduction The existence of order-disorder phase transitions in cyclic hydrocarbons has been well d0cumented.I The members of the homologous series CnHzn,with the exception of cyclopropane, C3H6,which has no out-of-plane vibration, exhibit one or more solid-state transitions. Cycloheptane, with four out-of-plane vibrations, has four solid phases2 The transitions occur at 134.8, 198.2, and 212.4 K with entropies of transition of 36.81, 1.46, and 2.12 J K-' mol-', respectively. Cycloheptane melts at 265.1 K with an entropy of melting of 7.09 J K-I mol-I. These results indicate that phases I, 11, and 111 (numbered starting from the high-temperature phase as phase I) are disordered. Considerable supercooling of the disordered phases has been o b ~ e r v e d . ~ In - ~ particular, upon slow cooling, the supercooling 111 and 111 IV transitions of phase I 1 through both the I1 was observed. Differential thermal analysis (DTA) r e s ~ l t s ~ . ~ indicated that a "glassy crystal" was formed, which above its Tg at 100 K, transformed irreversibly to phase IV, the stable, lowtemperature phase. Glassy crystals corresponding to phases I and 111 can also be formed and exhibit glass transitions at 100 and 96 K, respectively. Molecular motion in solid cycloheptane has been studied by proton spin-lattice relaxation and second-moment meas~rements.~
-
-
( I ) Parsonage, N. G.; Staveley, L. A. K . Disorder in Crystals; Clarendon: Oxford, 1978. (2) Finke, H . L.; Scott, D. W.; Gross, M . E.; Messerly, J. F.; Waddington, G. J . A m . Chem. SOC.1956, 78, 5469. (3) Adachi. K.;Suga, H.; Seki, S. Bull. Chem. SOC.Jpn. 1970.43, 1916. (4) Suga, H.; Seki, S . J . Non-Cryst. Solids 1974, 16, 195. (5) Brookeman, J . R.; Rushworth, F. A. J . Phys. C 1976, 9. 1043.
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A reduction in the second moment from the rigid lattice value was observed between 120 and 134.8 K, at which point a marked decrease occurred. This corresponds to the IV 111 transition and was interpreted in terms of the onset of a molecular pseudorotation process, for which a barrier of 7.1 kJ mo1-I was obtained. A further reduction in second moment was observed at I1 transition and was associated with the onset of the I l l isotropic rotation and lattice diffusion. Above the I1 I transition, an extremely narrow line was observed, indicating a significant increase in the rate of lattice diffusion. A study of the 13C spectrum of cycloheptane did not indicate the freezing out of motion down to 88 K.6 The vibrational spectrum of gaseous and liquid cycloheptane has been studied6$' and has been interpreted in terms of a twist-chair structure with C, molecular symmetry. The same investigation presented a detailed study of the mechanism of the pseudorotation process. No studies appear to have been reported on the vibrational spectra of the solid phases of cycloheptane, and such a study could provide significant information on both the stable and glassy crystalline phases, as well as the transitions between them.
-
-
-
Experimental Section Cycloheptane (Wiley Organics, purity 99.8%)was used without further purification. Cycloheptane (Aldrich Chemical Co., purity 98%), purified by fractional distillation, was used for some of the Raman experiments. (6) Bocian, D. F.; Strauss, H. L. J . Am. Chem. SOC.1977, 99, 2876. ( 7 ) Bocian, D. F.; Strauss, H . L. J . Am. Chem. SOC.1977, 99, 2866.
0 I990 American Chemical Society
