J. Phys. Chem. 1083, 87, 5041-5045
5041
Pressure- and Temperature-Dependent Homodyne Photon Correlation Studies of Liquid o-Terphenyl in the Supercooled State 0. Fytas, Th. Dorfmuller, University of BielefeM, facuity of Chemistry, BlelefeM, West Germany
and C. H. Wang* Department of Chemistry, University of Utah, &It Lake City, Utah 84 112 (Received: April 1 1, 1983; In f i n d Form: June 6, 1983)
The homodyne photon correlation functions of the light scattered from o-terphenyl in the supercooled state are studied as a function of pressure and temperature. A t all temperatures and pressures studied, the shape of the time correlation functions obtained in both VV and VH configurations are nonexponential. The mean relaxation times obtained from VV and VH correlation functions are identical. From the temperature and pressure dependence of the mean relaxation time, it is found that, with increasing pressure, the height of the activation barrier increases and the free volume decreases. Both effects have been incorporated into the Vogel-WLF equation, and an extended version of this equation has been proposed.
Introduction “decoupling” of the orientational and the center of mass motion may take place at different states in the presIn a previous paper, we investigated the orientational sure-temperature (P,T) plane can be answered by using dynamics of o-terphenyl above Tg(=243 K) at 1 atm by the pressure as one additional parameter. To obtain the monitoring the anisotropic component of the scattered information about the size of the moving unit involved in light in the time range of 10°-lO+ s.l Three important relaxation, we choose to study the photon correlation features which emerged from this study are the following: spectra of the supercooled o-terphenyl liquid as a function (1) the measured photon correlation functions of the poof temperature and pressure. The o-terphenyl (1,2-dilarized (VV) and depolarized (VH) components of the phenylbenzene) liquid is very convenient for the light scattered light cannot be represented by a single exposcattering study because of the high glass temperature nential nor by a bimodal distribution, but they can be range and its high stable scattered intensity. Furthermore, described by the Williams-Watts distribution expression liquid o-terphenyl does not have the complication of the exp[-(t/~~)B]; (2) the mean relaxation times derived from connected chain segments associated with polymeric liboth the isotropic and the anisotropic components of the quids; the experimental results should be of great theoscattered light are nearly equal; (3) no angular dependence retical interest, since a great deal of information about the for the mean relaxation times is observed either for the dynamics of the molecular motion in a supercooled liquid VH or for the W component. Similar experimental resulta is expected to be related to the molecular shape and the supporting this physical picture are also found in a-pheability of the molecules to undergo configurational changes nyl-o-cresol.2 These results are attributed to the coupling in the liquid state. of the rotational and the localized translational motion of molecules in the viscoelastic state of the supercooled liquid. Experimental Section The nonexponential correlation function is believed to be Photon correlation functions at different temperatures due to a complex cooperative process which takes place between -18 and 35 O C and presures in the range of 1-1250 in the supercooled state rather than in distribution of bar were taken at a scattering angle of 90° by using the environments which results in a distribution of relaxation apparatus described el~ewhere.~ The light source was an times. argon ion laser operating at X,= 514.5 nm and at a power The hydrostatic pressure is expected to have important up to 400 mW. Both the incident and the scattered light effects on the dynamics of molecular motion for liquids were polarized perpendicular (V) to the scattering plane. in the supercooled state. Recent investigations of photon This geometry gives the VV spectral component of the correlation spectra of bulk polymers above Tgas a function of temperature have given rise to very interesting r e ~ u l t s . ~ , ~ scattered light. The single clipped photocount autocorrelation functions were measured with a 96-channel These measurements yield the activation volume and the Malvern correlator. At 15.9 “C and at a pressure lo00 bar, activation energy; these are very useful parameters to the anisotropic (VH) component was taken to compare characterize the relaxation process. The experiments show with the VV scattered component of the scattered light. that the mean relaxation times exhibit an exponential The W and VH correlation functions were found to have dependence with pressure. The activation volume is found similar shapes, similar to that found in the experiment at to be temperature dependent, apparently due to the en1 atm. The o-terphenyl sample was the same as the one hanced cooperativity of the motion at lower temperature^.^ used in previous light scattering study.’p5 The questions such as how large is the “moving” unit involved in the relaxation process and whether or not any Data Analysis a n d Results (1)G. Fytas, C. H. Wang, H. Lilge, and Th. Dorfmuller, J. Chem. The measured single-clipped autocorrelation function Phys., 75, 4247 (1981). Gk@)(t)= N(nk(0)n ( t ) )in a homodyne light scattering (2) C. H. Wang, R. J. Ma, G . Fytas, and Th. Dorfmuller, J. Chern. experiment has the form Phys., 78, 5863 (1983). (3)G. Fytas, A. Patkowski, G. Meier, and Th. Dorfmuller, Macromolecules, -15, 870 (1982). (4) G. D. Patterson, J. R. Stevens. and P. J. Carroll, J. Chern. Phys., 77,622 (1982). 0022-3654/83/2087-5041$01.50/0
( 5 ) Y. Higashigaki and C. H. Wang, J. Chem. Phys., 74,3175 (1981).
(6)G.Fytas and Th. Dorfmuller, Mol. Phys., 47, 741 (1982).
@ 1983 American Chemical Society
5042
The Journal of Physical Chemistty, Vol. 87,No. 24, 1983
Gk(2)(t)= A[1
+ bk"'(t)12]
(70/~)r(~1)
I
,
\\
\
'
1250 bar
\
500
750
05-
1000
'.
3'
, \>,
'\
'.\
\
.
.\.
,\-
(2)
where 0 (0 < /3 I 1) can be considered either as a measure of the width of the distribution of the relaxation times or as a deviation from the exponential shape. Recent theoretical efforts based upon the idea of strong dynamic cooperativity are being directed to justify the nonexponential shape.8 The mean relaxation time T is the area under g(')(t) and is given by t=
. .
(1)
where g(')(t)is the desired normalized correlation function of the scattered electric field, k is the clipping level, ( n ) k and ( n )are respectively the mean clipped and unclipped counts per sample time A7, N is the total number of accumulation, and t (=IA7) is the delay time, I being the channel number. The amplitude b is a parameter which depends mainly on the clipping level, the coherence area, the sample time, and the relaxation strength (i.e., the fraction of the VV scattered intensity associated with the slow density fluctuations), etc., and can be obtained by the data fitting process. The base line A ( = N (nk)( n ) )can be either computed or measured a t large delay times. The fitting of the time correlation functions is carried out in terms of the resulting unnormalized intensity correlation (Gk(2)(t) - A)/A using a fixed base line A, essential in order to reduce the number of unknown parameters. Fitting the time correlation function with a floating base line, especially for broad distributions, can seriously affect the true shape of the g(')(t),thus making the results unreliable.'q2 As pointed out above, the time correlation functions for glass-forming systems do not exhibit a single exponential shape but rather they can be well represented by the empirical Williams-Watts function given by7 g(')(t) = exp[-(t/~~]fi]
Fytas et al.
TABLE I: Mean Relaxation 7 Obtained from the Homodyne Photon Correlation Function of o-Terphenyl Plbar 1
(3)
where r(pl)is the gamma function. In general, the nonexponential time correlation function g")(t)covers several decades of delay time. If g(')(t)is measured with a linearly spaced delay time correlator with a limited number of delay channels, the desired composite correlation functions can the be formed by splicing together sections of g ( l ) ( t ) obtained with different vlaues of A7. In the case of oterphenyl, we started the matching procedure from the section with the largest delay time A7 using the computed base line A and adjusting only the amplitude b in eq 1. Assuming the applicability of eq 2 with fi N 0.55, we have found that the measured base line a t about 250 delay channels away from the last section taken with A7 = rO/1O is, within the experimental error (about 0.1%), equal to the calculated base line given by N ( nk)(n). We use either the computed or the measured value of A in eq 1 and then fit the measured correlation functions in the form
250
500
750
1000
[ ( G k W )- A)/A]'/' = b1/2 e x p [ - ( t / ~ ~ ) ~ l (4) Figure 1 shows three normalized correlation functions (Gk(')(t)- A)/(Ab) a t different temperatures (11.1, 19.9, and 29.7 " C ) a t 1250 bar and three correlation functions a t -12 "C but a t different pressures (500,750, and 1000 bar) obtained by splicing two correlation functions with an increment delay time A7 differing by a factor of 5-8. The solid lines represent the fits of the correlation functions to eq 4. A fit to eq 4 for the section of correlation function obtained with r7,, lying in the range 2.5-3.5 (7= 92Ar and r being the average line width computed (7) G . Williams and D. C. Watts, T r a m . Faraday Soc., 66, 80 (1980). (8) S. Bozdemir, Phys. Status Solidi B , 103,459; 104, 37 (1981).
1250
temp/"C -17.7 -13.8 -8.8 -5.2 -1.4 -13.0 -9.0 -5.3 - 1.4 2.3 5.9 -5.5 -1.5 2.3 6.1 10.1 12.9 0.4 4.5 8.2 12.1 15.9 12.5 7.4 12.1 15.9 19.7 23.9 28.0 11.1 15.9 19.9 24.7 29.7 34.6
7I S
P
0.12 7.8 X 8.0 X 1.3 x 5.3 x 10-5 0.40 4.7 x lo-* 3.5 x 10-3 9.1 x 10-4 2 . 3 x 10-4 3.5 x 10-5 0.19 2.0 x 2.6 x 10-3 5.5 x 10-4 1.7 X l o + 5.5 x 10-5 0.17 2.4 X lo-* 4 . 9 x 10-3 7.8 x 10-4 1.7 X 3.2 x 10-5 8.7 X lo-* 1.8 x l o - * 2.0 x 10-3 7.1 x 10-4 8.5 x 10-5 2.3 x 10-5 0.41 2 . 6 X lo-* 6 . 9 x 10-3 1.05 x 10-3 9.1 x 10-5 2.0 x 10-5
0.53 0.57 0.57 0.57 0.54 0.52 0.53 0.55 0.53 0.54 0.57 0.53 0.56 0.59 0.59 0.54 0.55 0.59 0.58 0.57 0.56 0.58 0.60 0.59 0.54 0.58 0.57 0.54 0.59 0.56 0.58 0.56 0.55 0.61 0.62
by the comulant method) leads to the same value of the relaxation time P and distribution parameter /3. In the present case, for /3 2 0.5, the fit of the time correlation function to eq 4 containing those parameters (b, 0,and 7 0 ) using either one section or the composite correlation function gives the same value of /3 and 7 . Nevertheless,
The Journal of Physical Chemistry, Vol. 87, No. 24, 1983 5043
Liquid o-Terphenyl in the Supercooled State
,
1
- 10
lo
TlOC
30
Flgure 2. Mean relaxation time measured for o-terphenyl plotted as a function of temperature for different pressures. The pressure value indicated by the numbers. The solid lines represent fRs of eq 5 assuming T o = 219 K, b = dTldP = 0.02 K/bar.
in order to ensure a unified data analysis, we have always measured two sections of the correlation function, as outlined above. The values of t and /3 of o-terphenyl at various temperatures and hydrostatic pressures are listed in Table I. The mean relaxation time i V H derived from the depolarized component of the scattered light at 15.9 "C and at the pressure of 1000 bar is almost the same as t obtained from the polarized component under the same conditions, a result already observed in a-phenyl-o-cresol2 and o-terphenyl' a t 1 bar. For this reason, in the next section we shall make no distinction between r Vand ~ 7 and shall discuss the result using the relaxation time t,as obtained from the polarized component of the scattered light.
Discussion In Figure 2, the mean relaxation time t is plotted vs. the temperature a t different pressures. The Arrhenius fit to the t values in this temperature range at 1 bar gives an apparent activation energy of 66 kcal/mol, which is very close to the activation energy previously reported for the orientational relaxation time of o-terphenyl' and is consistent with the high activation energy value found in other glass-forming molecular liquids above but near Tg.1-3 However, the Arrhenius fit gives a preexponential factor amounting to 5.4 x s. Clearly the parameters obtained by using the simple arrhenius equation do not adequately describe the physics of the motion of the molecules in the supercooled liquid. Alternatively, it is wellknown that the temperature dependence of the viscosity of a glass-forming liquid is better represented by the empirical Vogel or Fulcher equation: i = t oexp(B/(T - To)] (5) The Vogel-Fulcher equation can be derived from the free volume model of Turnbull and Coheng by assuming that the free volume varies linearly with temperature. If one assumes that the nonexponential relaxation is associated with the defect diffusion, according to Bordewijk,l0 the parameter 7o involved in the Williams-Watt equation is inversely proportional to the defect diffusion coefficient, which in turn is inversely proportional to the shear viscosity. Thus, one expects eq 5 to be applicable for the light scattering relaxation time. The applicability of the Vogel-Fulcher equation to correlate the t data has been experimentally verified in several supercooled liquids of polymeric1' and nonpolymeric1,2molecules. In this work, (9)0.Turnbull and M. H Cohen, J. Chem. Phys., 34,120 (1961);52, 3038 (1970). (10)P.Bordewijk, Chem. Phys. Lett., 32,592 (1975).
LOO
800
1200 P I bar
Flgure 3. Mean relaxation time measured for o-terphenyl plotted as a function of pressure for different temperatures. The temperature values are indicated by the numbers.
we have found that eq 5 can also be used to represent the relaxation times given in Table I, provided that the parameters B and Toare allowed to be pressure dependent. However, due to a relatively narrow temperature range which is covered in the photon correlation measurements, the value of the paramter To given in eq 5 cannot be otained precisely. To overcome this difficulty, we have fitted the data to eq 5 by also including the interferometric depolarized Rayleigh times5 (70-150 "C) to the relaxation times tVH obtained from the correlation functions of the depolarized scattered light (-16 to 9 "C).' The value of To= 219 K obtained is then used fist as a fixed parameter in the fitting of eq 5 to the relaxation times t(T). For the experimental relaxation times at 1 bar, we obtained i = (1.2 f 1) X s and B = 919 f 35 K. The values of the parameters Toand B can be discussed in terms of the hole energy and the free volume expansion coefficient in the free volume model.12 The pressure dependence of t at 10 different temperatures is shown in Figure 3. We observe that (within experimental errors), log t increases linearly with increasing pressure in the range of 1-1250 bar. If the relaxation process is considered as an activated process, the activation volume AV* which reflects the volume requirements for the relaxation is obtained as AV* = 2.303RT(a log 7/dP)T (6) Using eq 6 and the experimental slopes given in Figure 3, we find that the activation volume AV* is a function of temperature; it decreases from 375 cm3/mol at -15 "C to 230 cm3/mol at 30 "C. The decrease of activation volume with increasing temperature is in qualitative agreement with the light scattering results found in amorphous polymer^.^^* Comparison of AV* with the value of 219 cm3/mol reported for the molar volume of melt o-terphenylI3 suggests that roughly one molecular unit is involved in the observed structural relaxation process in the supercooled liquid state. This is consistent with the previous assignment of the relaxation process associated with the photon correlation spectrum to the localized center of mass motion of one single molecule which is strongly coupled to the molecular ~rientation.',~!~ However, as shown in Figure 4,the decrease of AV* with increasing temperature is not linear. This has been attributed to the decrease of the cooperativity which is grossly represented (11)C. H.Wang, G. Fytas, D. Lilge, and Th. Dorfmuller, Macromolecules, 14, 1363 (1981). (12)I. Sanchez, J. Appl. Phys., 45,4209 (1974). (13)T.N.Andrews and A. R. Ubhelade, Proc. R . SOC.London, Ser. A, 228,435 (1955).
5044
The Journal of phvsical Chemisfty, Vol. 87, No. 24, 1983
LO 0
Fytas et al. r -
I
'
I
I
2001
, -10
, 0
j
,
,
,
10
20
30
TI
k
-201
0
O C
Flgure 4. Activation volume AV' calculated according to eq 6 for o-terphenyl plotted vs. temperature. The solid line represents the fit of A V * = a R T / (T - T o )to the experimental data.
by T/(T - TO).l4J5Similar behavior was found for the a relaxation in poly(ethy1 a ~ r y l a t e ) polystyrene,16 ,~ and poly(methy1 acrylate)." To incorporate the temperature and pressure dependence of the relaxation time, we now modify eq 5 by assuming that B is linearly dependent on
P t = t oexp
(",'
;:}
-
(7)
Equation 7 shows that AV* is dependent on temperature (Figure 4) but is independent of pressure (Figure 3). Fitting the T(TQ)data of Table I to eq 7 using To= 219 K yields to= (1.1f 0.1) X s, B = 1131 f 35 K, and a = 0.71 K/bar. Equation 7 assumes t oand To to be independent of pressure. The effect of allowing To to depend on the pressure will be considered later. In eq 7 the parameter B is related to the free volume model according to the Cohen-Turnbull model by B = 9.07 (yV*/Vm)To;the parameter To is related to the energy required to form a "mole of holes" by E h = 2RT0 and the parameter a to the average hole volume by v h = 0.22aR/(yV/V,).3J2 The factor y(V/Vm) (where V, is the volume of the mobile unit, V is the minimum volume of the hole necessary for mobility, and y is a numerical factor having a value between l/z and 1) is a measure of the ability of the molecule to use the available free volume. Using the foregoing relations we obtain y( V /V,) = 0.57 E h = 0.86 kcal/mol, and v h = 23 cm3/mol. By taking V v h and y = 1, we obtain the volume of the mobile molecule V, to be equal to 39 cm3/mol. One notes that this volume is nearly equal to the hydrodynamic volume (=37 cm3/mol) for the orientation of the o-terphenyl molecule previously obtained from the viscosity dependence of the orientational time under stick boundary condition^.^ Although this agreement may be fortuitous, it shows that the two results are self-consistent and meaningful. Consider next the pressure dependence of T, It is worth mentioning that, if eq 7 were applied to each set of isobar data (Table I), the result would yield a tovalue which shows a slight decrease as the pressure is increased. This is due to the neglect of the pressure dependence for To. Nevertheless, eq 7 provides only a crude representation of the t(T,P) data as the fit is not sensitive to the value of To,due to the relative narrow temperature and pressure ranges covered by the photon correlation measurements.
-
~
(14)G. Adam and J. H. Gibbs, J. Chem. Phys., 43, 139 (1965). (15)A. A. Miller, Macromolecules, 11, 859 (1978). (16)G.D.Patterson, P. J. Carroll, and J. R. Stevens, J.Polym. Sci., Polym. Phys. Ed., in press. (17)G. Fytas, A. Patkowski, G. Meier, and Th. Dorfmuller, J. Chem. Phys., in press.
LOO
800
1200 P I bar
Flgure 5. Temperature vs. pressure plot for a given value of the relaxation time for o-terphenyl for various r values (the isokinetic curves).
For this reason, the effect of the pressure dependence of To is smeared out and does not appear to significantly affect the values of T ~ B, , and a. We could relax the pressure dependence of the ro parameter and also consider fitting the data by allowing To to be pressure dependent. Shown in Figure 5 is the locus of equal values of t in the pressure-temperature (P,T) plane for various values of t. The slope of these cures (dT/dP) is found equal to 0.02 f 0.002 K/bar for t = 10-1 s and appears to increase slightly for shorter t values. On the basis of the free volume model, the condition for constant relaxation time is dT/@ = bf/af (bf and af denote the free volume compressibility and the expansion coefficient, re~pectively).'~ This result is equivalent to the condition describing a second-order phase transition.18 Assuming dT/@ = 0.02 K/bar, we obtain To(P)= To + b P with To = 219 K and b = 0.02 K/bar. If this result is now incorporated into eq 5 and then carries out the fit of the relaxation times t at different pressures according to the equation, we obtain that, within the experimental uncertainty, t odoes not depend on pressure and the parameter B increases with increasing pressure; the pressure dependence obtained for B is, however, less than that obtained by keeping To pressure independent. This result suggests that allowing the pressure dependence of B alone is not sufficient to adequately describe the pressure and temperature dependence of t(T,P).I4 It is necessary also to allow the parameter To to be pressure dependent. In short, instead of eq 7, a better equation to describe the pressure and temperature dependence of i should be t = t oexp
[
T
