J. Phys. Chem. 1082, 86,3045-3052
Conclusion Studies of the perchlorate anion Raman profiles have proved to be useful in the study of ion+lvent interactions. The use of the perchlorate anion in a ternary system of water and a second electrolyte has often been used to promote ion pairing of the second e l e ~ t r o l y t e . ~How,~~ ever, the assumption that the perchlorate anion is not also involved in ion pairing is not strictly true. The perchlorate anion has also been used as a "noninteracting" anion in (33)T.G. Chang and D. E. Irish, J. Solution Chem., 3, 161 (1974). (34)B.Balaeubrabmanyan and G. J. Janz, J. Am. Chem. SOC.,92,4189 (1970).
3845
the study of ligand-metal e q ~ i l i b r i a . It ~ ~is assumed that the perchlorate anion does not form metal-anion association. Such interactions cannot be neglected, and the perchlorate anion may play a greater role in the study of metal-ligand equilibria than has been previously considered. Certainly our component analysis indicates that association of the perchlorate anion to the group 1metals is small but significant. Acknowledgment. The ARGC are thanked for grants enabling the purchase and maintenance of the Raman spectrometer and for a fellowship for R.A. (35)L. Johansson, Coord. Chem. Rev., 12,241 (1974).
Test of the Entropy Basis of the Vogel-Tammann-Fulcher Equation. Dielectric Relaxation of Polyalcohols Near T, C. A. Angell" and D. L. Smlth Department of Chemisby, furdue University, West Lafeyette, Indiana 47907 (Received: October 13, 1981; I n Flnal Fwm: May 18, 1982)
We describe a family of hydrogen-bonded molecular liquids in which transport processes all appear to be strongly coupled and which exhibit relaxation times conforming to the Vogel-Tammann-Fulcher (VTF) equation 7 = T~ exp(B/ [ T - To]) over as much as 12 orders of magnitude in 7. For such systems a stringent test of the identity of the VTF equation Toand the Kauzmann isoentropy temperature TK can be made. New data on dielectric relaxation times extending into the glass transformation range are presented for three polyalcohols, and heat capacity data for crystallineand liquid ethylene glycol are also reported. Estimates of the Kauwnann temperature, TK,at which the internally equilibrated supercooled liquid would have the same entropy as the crystal, are made. Despite a threefold variation in TK,these are found to be close to the best-fit VTF equation Toparameter and even closer to the Tovalue obtained by adding the constraint B = 12.7T0to the analysis.
Introduction The temperature dependence of liquid transport properties,' particularly in the case of viscous liquids, is complex and at this time poorly understood. An empirical equation which has been given much attention is that first proposed for viscosity by Voge12and later applied in simplified form to various liquids by Tammann and Heme3 and, independently to molten oxides, by F ~ l c h e r . ~The VogelTammann-Fulcher (VTF) equation has the simple form 7 = A exp[B/(T - To)] log 7 = A ' + B ' / ( T - To)= A ' + ( B / 2 . 3 0 3 ) / ( T - To) (1) and is transformable into the Williams-Landel-Ferry (WLF) equation, well-known for its ability to describe It has also been successfully applied polymer fl~idities.~ to molecular liquids! ionic salts and solutions,' and metallic glasses.8 Davidson and Colegpointed out that Toin (1) J. R. Partington, 'An Advanced Treatise on Physical Chemistry", Longmans, Green and Co., London, 1951. (2)H.Vogel, Phys. Z., 22,645 (1921). (3)G.Tammann and G. Hesse, Z. Amrg. A&. Chem., 166,245(1926). (4)G. S.Fulcher, J. Am. Chem. SOC.,8,339 (1925). (5)M. L. Williams, R. F. Landel, and J. D. Ferry, J. Am. Chem. SOC., 77,3701 (1955). (6) . . (a) . . R. C. Makhiia and R. A. Stairs. Can. J. Chem.. 48.1214(1970): (b) M. P. Carpenter, D.B. Davies, and A. J. Matheson,'J. &em. Phys.; 46,2451 (1967). (7)(a) C. A. Angell, J.Phys. Chem., 68,1917(1964);70,2793(1966); (b) C. T.Moynihan, ibid., 70,3399 (1966).
alcohols and glycerol had the same value for dielectric relaxation and viscosity and that this value was close to the value of the Debye temperature OD determined from velocity of sound measurements. A more challenging correlation was emphasized by Adam and GibbslOJ' and, indirectly through the WLF equation, by Bestul and Chang.12 These authors noted that in several cases the temperature Toalmost coincided with a temperature (which we denote here TK)at which the liquid entropy curve extrapolated below the glass transition temperature would intersect the crystal entropy curve. That this alarming intersection would occur quite generally in supercooling liquids were it not for the timely intervention of the (nonequilibrium) glass transition (at which (dS/BT), for the liquid changes to much lower values) was first pointed out by Kauzmann13-hence, our choice of subscript for the extrapolated isoentropy temperature, TW Since Gibbs and Dimarzio14had earlier formulated an equilibrium theory for polymers (recently challenged by Gujrati and Gold~tein'~) which predicted an equilibrium (8) H.S. Chen and D. Turnbull, J. Chem. Phys., 48, 2560 (1968). (9)D. W.Davidson and R. H.Cole, J. Chem. Phys., 19,1484 (1951). (10)G. Adam and J. H.Gibbs, J. Chem. Phys., 43,139 (1965). (11)References 10 and 12 actually compare T,/To with T /TK.Direct comparisons of Toand TK are given in C. A. Angell and J. Rao, J. Chem. Phys., 57,470 (1972). (12)A. B. Bestul and S. S. Chang, J. Chem. Phys., 40,3731 (1964). (13) (a) W.Kauzmann, Chem. Reu., 43,219 (1948);(b) C. A. Angell, J. Chem. Educ., 47,583 (1970). (14)J. H.Gibbs and E. A. Dimarzio, J. Chem. Phys., 28,373 (1958).
0022-3654l82l2086-3845$0 1.2510 0 1982 American Chemical Society
k.
3848
-
phenomenon in which (dS/dT) changes value in associa, 0, it seemed tion with the condition Sex(=&, -J,S natural to interpret the glass transition as the kinetically determined reflection of the ideal (Sex= 0) glass transition phenomenon. Adam and Gibbs'O then formalized this notion with a theory for relaxation processes that yielded, for the temperature dependence of transition frequency, the expression
w(T) = A exp[-C/TS,]
(2)
where C is a constant, and S, is the configurational part of the total entropy, which we identify with the quantity Sex.By writing (3) it is found16that eq 1 is obtained as an exact relation if we use the description
ACp = D / T
(4)
and make the identification
To E TK
(5)
An examination of the data for a wide range of substances16J7shows that, in fact, the hyperbolic form of eq 4 gives the best simple description of the experimental quantity Cp(liquid) - C,(crystal), which is usually taken as a measure of ACT Several experimental studies supporting the identity, eq 4, were quickly reported,lg20 some of them involving concordant Tov ~ l u e sextracted from data on quite different transport properties. The Adam-Gibbs equation has the advantage of correctly predicting Arrhenius temperature dependences for transport in (a) the unusual case of liquids which show very small changes in heat capacity at the glass transition (e.g., the network liquids Ge02,21probably BeF2, Si02)and (b) liquids which have their structures (hence S,) frozen by paasage through the glass transition but still slowly respond to shear stresses or exhibit electrical conductance.= Such observations go far toward establishing TK as a thermodynamic limit for the liquid state%% in contrast with Tgwhich sets an experimental limit on the observation of liquidlike properties on the normal observational time scale. This harmonious picture was, unfortunately, shaken by the that in numerous cases the parameter (15)P. D. Gujrati and M. Goldstein, J. Chem. Phys.,74,2596(1981). (16)(a) C. A. Angell and R. D. Breesel, J.Phys.Chem., 76,3244(1972); (b) C. A. Angell and W. Sichina, Ann. N.Y. Acad. Sci., 279,53 (1976). (17)(a) V. P. Privalko, J. Phys. Chem., 84, 3307 (1980); (b) C. A. Angell, unpublished work. (18)A. A. Miller, J. Polym. Sci., Part A-2, 6,249 (1968). (19)C. A. Angell and C. T. Moynihan, Ed. G. Mamantov, Marcel Dekker, New York, June 1969. (20)C. A. Angell and J. C. Tucker, J. Phys. Chem., 78,278 (1974). (21)J. C. Tucker, Ph.D. Thesis, Purdue University, Lafayette, IN, 1974, data quoted in ref 16b. (22)0. V. Mazurin and L. N. Poteelueva, Fiz. Khim. Stekla, 4,570 (197RL ~ - -,. -.
(23)F.S.Howell, R. A. Bose, P. B. Macedo, and C. T. Moynihan, J. Phys.Chem., 78,639 (1974). (24)J. H. Gibbs in "Modern Aspects of the Vitreous State", J. D. McKenzie, Ed.,Butterworth, London, 1960,Chapter 7,p 152. (26)We should note, however, that Cohen and Greatmhave proposed, on the basis of a free volume cell percolation theory, that a first-order collapse to a low-entropy phase would intercede before TK is reached. This would be heralded by anomalous behavior in the fluctuation-sensitive derivative properties C, and KT in the liquid, of which there is no clear indication to date (even though, in some cases, TB- TKis only 20
K).
Angell and Smith
The Journal of Physical Chemistry, Vol. 86, No. 79, 1982
TABLE I:
ComDosition and
ethylene glycol propylene glycol sorbitol 15% PG/EG
T, 0.16 0.06 0.61 0.16
153 (extrap) 172 266 156
To,which had agreed with TK when obtained from measurements made in convenient ranges of viscosity and electrical resistance, became temperature dependent and fell increasingly below TKwhen measurements were extended to higher viscosity and resistance ranges at temperatures approaching TB' Different transport properties then yield differing Tovalues, and in many cases Arrhenius behavior was again found in the liquid near, but still above, T g(although at first sight it would be expected that the Adam-Gibbs equation would be most reliable near T J . By contrast, TK, which is obtained from heat capacity extrapolations, shows very little dependence on the range of data used in the analysis. If any trend exists, it is for TK to move to higher values the more low-temperature C, data are included in the analysis. Despite the existence of theoretical models for isotropic liquids"BB and polymersmwhich show how Toand TK can arise artifactually from extrapolation of functions which are actually continuous to 0 K, or terminate in a first-order transition, it is our prejudice that there is physical truth in the eq 5 identity. We suspect that the failures of eq 1 and 2, for which certain theoretical models have been advanced,11*26.29~30 are basically due to some form of decoupling of the different transport mechanisms at lower thermal excitation levels. To reestablish some confidence in eq 2, it is necessary to confirm it for liquids in which decoupling does not occur, so that eq 1yields the same To for different properties over the whole liquid range and, in particular, fita data obtained near Tgover many orders of magnitude in the property being measured. The latter condition would seem most likely to prevail in liquids in which all molecular segments are cross-connected to reduce or eliminate clustering and microheterogeneity. It seems significant that the two molecular liquids which a literature search identifies as conforming to eq 1 over at least a major part of the whole liquid range should be the multilaterally hydrogen-bonded polyhydric alcohols glycerole and 1,3-b~tanediol.~lThe polyhydric alcohols also have smaller compressibilities than other molecular liquids, implying relatively small mean square density fluctuations. On the basis of this observation, we have proceeded to measure relaxation times near Tgon three additional polyhyric alcohols, sorbitol (C6H8(OH)6, propylene glycol (1,2-propanediol, C3H6(OH),), and ethylene glycol (C2H,(OH),), and obtain thermodynamic data where needed in order to provide a family of compounds for which the equivalence of TK and To may be tested. The range of Tgcovered by these liquids is quite large, varying by more than 100 "C from 156 to 265 K. Experimental Section Materials. Sorbitol (C,H,(OH),), the largest polyhydric alcohol of this study, and one which has been the subject (26)W. T.Laughlin and D. R. Uhlmann, J. Phys.Chem., 76,2317
(19721. ~ -.-,. -
(27)J. H.Ambrus, C. T. Moynihan, and P. B. Macedo, J . Electrochem. SOC.,119,192 (1972). (28)H.Tweer, J. H. Simmons, and P. B. Macedo, J. Chem. Phys.,54, 1952 (1971). (29)M. H. Cohen and G. S. Grest, Phys.Reu. B,20, 1077 (1979). (30)A. A. Miller, Macromolecules, 11,859(1978). (31)P.W. Drake, J. F. Dill, C. J. Montrose, and R. Meister, J. Chem. Phys.,67,1969 (1977).
Test of the Entropy Basls of the VTF Equation
of previous studies in this laboratory,92fiwas obtained from Matheson Coleman and Bell. The sorbitol was used without further purification. When approximately 10 g of the crystals was melted in a small beaker under vacuum (100 OC, 1 mmHg), an opaque liquid was first obtained. After about 2 h in the vacuum oven, the liquid cleared; a similar observation on melting sorbitol is reported in the l i t e r a t ~ r e .A ~ ~Fischer titration to determine the water content of the sample was performed, and the result is reported in Table I along with the glass transition temperature as measured by DTA (heating rate of approximately 10 "C min-'). The retained water content is such that about 1 of every 50 hydroxyl groups in the liquid belongs to HzO. Propylene glycol (PG), the next largest molecule of this study, is easily supercooled. Unfortunately, the liquid cannot be crystallized mainly because, as normally obtained, it is a racemic mixture. The difficulty of one stereoisomer fitting into the other stereoisomer lattice precludes crystallization to a congruently melting crystal (indeed, no crystallization is observed at all) and therefore frustrates the determination of T2 Nonetheless, transport studies were carried out to check the general applicability of the VTF equation to this polyhydric alcohol, and to provide a basis for the extrapolation referred to next. The PG used was J. T. Baker USP grade. The glycol was first dried for 2 days over 4A Linde molecular sieve and then decanted to a distilling flask, and a vacuum distillation was performed, the middle boiling fraction being retained for use. The result of a Fischer titration and the glass transition as determined by DTA are given in Table I. During storage the glycols were kept in tightly stoppered bottles, the air space above the liquid being flushed with dry nitrogen. Ethylene glycol (EG) crystallizes easily and cannot be studied near Tg.However, admixture of as little as 15 mol % PG suppresses crystallization very effectively and allows the characteristics of EG near its Tgto be estimated by short extrapolations. The EG was Fisher-certified grade. As in the case of PG, the EG was first dried over molecular sieve and then vacuum distilled. The constanbboiling middle fraction was retained for use. The water content as determined by Fischer titration is given in Table I. Tgfor EG is estimated, using values for PG and for a 15 mol 9% solution of PG in EG, to be 154 K (see Table I). Dielectric Measurements. In principle, any zero-freas quency transport coefficient could serve to define To, long as the temperature dependence of the transport coefficient corresponds to the temperature dependence of the average structural relaxation time. In particular, it seems that, for associated liquids such as those of this study, the different relaxation times determinable for shear, volume, and polarization all have the same temperature d e p e n d e n ~ e . ~In~ this work the dielectric relaxation time has been used to follow the structural relaxation. The dielectric relaxation time can be measured a region in which, as with reasonable accuracy near Tg, noted above, the VTF equation often fails. The dielectric cell used for the measurements has been described before.% The Berberian-Cole and WayneKerr transformer ratio bridges used in the dielectric measurementa covered the frequency range of from 0.1 Hz to 1 (32)C. A. Angell, R. C. Stell, and W. Sichina, J . Phys. Chem., 86,1540 (1982);C. A. Angell and W. J. Sichina, to be submitted for publication. (33)A. Barkatt and C. A. Angell, J. Chem. Phys., 70, 901 (1979). (34)H.H.Strain, J. Am. Chem. SOC.,56, 1756 (1934). (36)G.E.McDuffie and T. A. LitoVitz, J. Chem. Phys., 39,729(1963). (36)I. M.Hodge and C. A. Angell, J . Chem. Phys., 67, 1647 (1977).
The Journal of Physical Chemistty, Vol. 86, No. 19, 1982 3847
30.00
W 20.00 D m
10.0c
.00 ,000
1 ,000
2.000
3.000
LOG FI HZ I
Y.000
6.b00
6.000
Flgure 1. Dielectric loss spectra for a solution of 85 mol % ethylene 15 mol % propylene glycol, at various temperatures. glycol
+
MHz and have also been described in the l i t e r a t ~ r e . ~ ~ When a residual conductivity greater than 10-l2Q-l cm-l could be detected, its contribution to the loss was subtracted out in the usual fashion. This correction was negligible except at the lowest frequencies but here was significant because of the importance attached to data very close to Tgin our analysis. The frequency of the dielectric loss maximum at constant temperature corresponds to a most probable rather than the average relaxation time for the liquid at that temperature. Provided the dispersion in d does not change with temperature, the most probable and the average relaxation times will have the same temperature dependence. Since it is the temperature dependence of the structural relaxation time that is of interest, a series of isothermal measurements of the loss were made and the resulting frequency of maximum loss and temperature recorded. An alternative way of determining the dielectric relaxation times is to make isofrequency measurements of the loss as the temperature is scanned. The advantage of the temperature scan measurement is that the lower-temperature relaxation times can be more easily studied than in the case of the isothermal loss measurements. Both types of measurements were made and are reported in the Results section. When the relaxation time from isofrequency measurements falls in a range covered by the isothermally determined relaxation times, no systematic difference in the values is observed. Calorimetric Data. Measurements on pure EG and EG + 15% PG solution were made by using a Perkin-Elmer DSC-2 differential scanning calorimeter equipped with a Perkin-Elmer scanning auto zero. The procedure for measuring the heat capacity has been adequately described in the literature.%
Results Dielectric Loss Measurements. The loss spectra were determined at several temperatures above T for each liquid studied. As an example, the isothermaf measurements for the 15 mol % PG in EG solution are shown in Figure 1. The shapes of the spectra appear to be nearly independent of temperature for the substances of this study. However, their half-widths vary markedly from one substance to another. We characterize them by the exponent p of the fractional exponential time-domain decay functi~n~~~~~ d t ) = exp[-(t/d@I
(6)
~~
(37)J. G.Berberian and R. H. Cole, Reu. Sci. Instrum., 40,811(1969). (38)M.Oguni and C. A. Angell, J. Chem. Phys., 73, 1948 (1980).
3848
Angell and Smith
The Journal of Physical Chemistry, Vol. 86, No. 19, 1982
40.00
1655K
i
I
1617
--r
I
30.00 4
.
Fl@w2. Didaric kss as a finctlon of temperatwe for different k e d scanning frequencles for 85 mol % ethylene glycol 15 mol % propylene glycol solution.
+
The relation of /3 to the half-width of the loss spectrum has been tabulated by Moynihan et aL4I For the spectra of Figure 1, /3 = 0.77 f 0.03 (cf. /3 = 1.00 for a single relaxation time). The broadest spectra were found for sorbitol. In this case the spectrum is severely distorted on the high-frequency side by the presence of a secondary relaxation (not discussed in this paper). The value of /3 for the sorbitol spectra is 0.50. This value may be compared with the values 0.6 and 0.55 obtained from analysis of the relaxation of the enthalpy, and of the hydrogen-bond equilibrium, respectively, during steady heating.33 For propylene glycol, the spectrum is narrower, with /3 = 0.75, as in Figure 1. It may be noted that values of 0.5-0.6are common for molecular liquids, and single relaxation times are rarely found, though they seem to be characteristic of monohydric a l ~ o h o l s .For ~ glycerol, the spectral width has been found to be weakly temperature d e ~ e n d e n t ~and " ~ best described by a Davidson-Cole distribution function with width parameter y = 0.56 near T increasing to 0.71 at high temperatures after a wide pfateau a t 0.61, in the range -5 to 0 "C. The equivalent values of /3 (eq 6) are 0.78 rising to 0.82 on the plateau. The maximum loss in all curves was used to mark the frequency f, characteristic of the structural relaxation for the liquid. The procedure for locating the maximum was as follows. One isothermal loss curve was chosen as a reference and the maximum of the loss marked on the curve. Others were then matched to the reference curve by shifting along the log frequency axis, and the maximum of the reference curve was used to locate the corresponding maximum on the new curve. The matching procedure was greatly facilitated by the nearly constant shape of the loss curves of this study. A comparable matching procedure cannot be used for the isofrequency temperature scans, shown in Figure 2, because of the non-Arrhenius temperature dependence of the relaxation time, which causes the low-frequency plots to be narrower in 1/T. A combination of the isofrequency and isothermally determined TD values from the data of Figures 2 and 3 is shown on an Arrhenius plot in Figure 3. Included in Figure 3 are corresponding data sets for the other liquids of this (39) G. Williams and D. C. Watts, Ram. Faraday Soc., 66,80 (1970). (40)C.A. Angell and J. Wong, 'Glass: Structure by Spectroscopy", Marcel Dekker, New York, 1976, Chapter 11. (41) C. T. Moynihan, L. P. Boesch, and N. L. Laberge, Phys. Chem. Glasses, 14, 122 (1973). (42) G. E. McDuffie and T. A. Litovitz, J. Chem. Phys., 37, 1699 (1972).
^
r? -
L.""
b l
IC
00 3
'zsc
3 k00
3 750
,ooc
4
Y
253
lCOO/T
,
I
4 550
4 75c
5 00:
5 25i
Flgure 3. (a) Dependence of the log (frequency at maximum dielectric loss) as a funcblon of reciprocal temperatwe for the liquids and solutions of this study and for glycerol (data of Howell, ref 23). Filled or boxed symbols dlstinguish constant-frequencyscans. The dashed lines are plots of eq 10 wlth parameters ghren in Table IV. (b) Arrhenius plot of part (a) data for glycerol converted to dielectric relaxation time (most probable value) combined wlth shear relaxation times calculated from viscosity data using G , values according to ref 47.
3 300
2 OCO 0751
\ 5
1.000
L
1.500
1.750
2.000
2.2%
2.530
2.750
2 . m
l O 0 / ~ ~ - T Io
Figure 4. Tests of the VTF equation on data for the solution 85 mol % EG 15 mol % PG. Insert: Plot of the resMuals as a function of frequency to test for systematlc departures from the VTF form of temperature dependence.
+
study and also data for glycerol taken from the recent study by which extends the frequency range of earlier s t u d i e ~ ~tov0.01 ~ ~ Hz. In judging the applicability of the VTF equation to the data, it is common to examine the log of the maximum loss frequency plotted against 1/(T - To). Scatter of the data points about the best-fit straight line then gives a rough (43) F.S.Howell, Ph.D. Thesis, The Catholic University of America, Washington, D.C., 1972.
The Journal of Physlcal Chemism, Vol. 86, No. 19, 1982 3849
Test of the Entropy Bask of the VTF Equation
T/K 100 110 120
3 h
300
200
150
0 .-4 c
1
I
o pure ethylene glycol
EG
9
+ EG + 15% PG (scaled to EGI
e
v
k-M5
(D
m
N
h
I
I
I
Log (T/K) Flgure 5. Heat capacity data for liquid and crystalline ethylene glycol, and for liquid and vitreous 85 mol % EG 15 mol % Po solution, as a functkn of kg TIK to perml assessment of vanishing excess entropy temperature T,. The entropy of fuslon for EO is shown as an area in the top left of the diagram.
visual indication of the goodness of fit. A plot for the 15 mol 7% solution of propylene glycol in the ethylene glycol is given in Figure 4. The best-fit VTF parameters are collected in Table 11. Systematic departures from the VTF equation, such as are often encountered near T , can be detected by plotting the residuals of the best-fit Jata points against the fitted log The plot for the EG + 15% PG solution is shown as the inset to Figure 4. No trend is apparent though the small number of points available decreases the ability to reveal any systematic departures should they exist. The limited number of relaxation times available from the dielectric measurements reflects the laborious procedure involved in obtaining each f m datum (each isothermal loss curve is made up of 25 points). For the purpose of obtaining more points for the VTF plot, the dc conductivity is attractive as a probe of the structural relaxation. The introduction of ions to make the liquid sufficiently conducting to perform measurements near Tg must not, of course, influence the temperature dependence of the structural relaxation time. Data are available for the case of alkali halides in glycerol46and are referred to in the Discussion section. Calorimetric Measurements. Data for EG (0) and its 15% PG solution (+) are presented in Figure 5 in the form C vs. T (log scale) suitable for graphical estimate of the d u z m a n n t e m p e r a t ~ r e . 'The ~ ~ molar C, of the 15% PG solution has been scaled onto the EG data by use of the factor 62/64 = (molecular weight of EG)/(mean molecular weight of the 15% solution). The heat of fusion was determined to be 42.1 cal g-l, in fair agreement with the literature value of 43.26 cal g-1.30 The latter, after conversion to the molar quantity and division by TF, is shown as an area, ASF in Figure 5. Matching this area to that between the liquid (extrapolated and crystal curves identifies TK ( A S F = STKTFACp below Tg) d log T). As seen in Figure 5, TK is found to be 115 f 5 K. The uncertainty estimate is based on estimates of the reliability of the value of ASP and of the extrapolation of the liquid heat capacity curve of the solution down to the temperature range of TK.
h
h
+
h
,
L
0
ri
X
1 40
fmS4
(44)P.B.Macedo and G. Napolitano, J . Chem. Phys., 49,1887(1968). (45)F. J. Bartoli, J. N. Bierch, N g u y e n - H u u - T o a n , and G. E. McDuffiie, J. Chem. Phys., 49,1916 (1968).
D.
4
W
0 rl
rlrl 0
5
P
0 rl
0
0 0
rl
rlrl
9
(D
m N mt-
m m
I
4
2
0
I
I
0 0
/$ I
co
N
rlrl
rl
m
o
rl
~
3850
Angel1 and Smith
The Journal of Physlcal Chemistry, Vol. 86, No. 19, 1982
Discussion The dielectric relaxation data extend to within a few degress of the glass transition temperature (Figure 3). With only a few 10’s of degrees of temperature increase above T , the dielectric relaxation time of the liquid has exceeded the 8 orders of magnitude frequency range of our combination of bridges. Although this is a larger range of transport parameters than usually used in tests of eq 1, for our purposes even wider ranges of both time and temperature are desirable, so we make recourse to measurements other than those of the dielectric relaxation. It has been found for the case of glycerol that the average dielectric and shear relaxation times are very similar, ~ ~ = 2.36 / Since 7 ~ the shear relaxation time T , is related to the shear viscosity q b p T, = v / G , (7)
in which the limiting high-frequency shear modulus G, has a weak linear temperature dependence:’ the VTF parameters for T,, hence to good approximation for TD, can be obtained by analysis of the shear viscosity. Fortunately, viscosity data are available for all the liquids of this study. Combination of dielectric and viscosity data extends the range of 7 over which constancy of VTF parameters can be tested to 12 orders of magnitude. In the case of glycerol for which the ultrasonic data have yielded an equation for G, vs. T14’ the shear relaxation time itself can be obtained over a wide range. Using viscosity data quoted in the CRC “Handbook of Physics and Chemistry”Fowe find T~ values which overlap TD, and extend the total range of 7 to 12 orders of magnitude. These are displayed in part b of Figure 2. The dielectric and shear time data have been fitted separately to the VTF equation. They yield, by linear regression analysis, To= 132 and 134 K, respectively. The viscosity data, before conversion to T ~yielded , To = 126. On the other hand, precise rD data extending to temperatures below Tgreported by Kaddour and Champane3P8 since submission of this article yield with excellent precision To = 119 f 1.3 K compared with a value of 124 f 2.4 K from analysis of shear viscosity values critically chosena from a number of literature sources. Again, the range of data fitted with this precision was very broad, in excess of 10 orders of magnitude for the dielectric data. The difference between these parameters and ours may be due to our choice of the most probable relaxation time (at e’’ max) rather than the parameter T~ of the ColeDavidson distribution fitted by Kaddour and Champaney. Before passing on to the results for the other liquids of this study, we note (Table 11) that the preexponent A, (which is very sensitive to the values of the other parameters) has a value which is not unacceptable for a parameter which represents the time between attempts at molecular rearrangement, modified by some unknown activation entropy term. The value for glycerol, 2.5 X s, which is actually the smallest among the liquids of this study, corresponds to an attempt frequency f = 1/(2.xA,), 2100 cm-l, some 4 times higher than the frequency of the broad intermolecular librational modes IR band at 500-700 cm-l. This implies a transition state somewhat “looser” than the normal structure, as seems reasonable. The (46)K. F. Herzfeld and T. A. Litovitz, ‘Absorption and Dispersion of Ultrasonic Waves”, Academic P r w , New York, 1959. (47) R. Meister, C. J. Marhoeffer, R. Scbiamanda, L. Cotter, and T. A. Litovitz, J. Appl. Phyr., 31, 854 (1960). (48) F. 0. Kaddour and D. C. Champaney, to be submitted for publication (reported by D. C. Champaney, University of E. Anglia, at Molten Salt Discussion Group Spring Meeting, Aberdeen, Scotland, March 29-30, 1982).
TABLE 111: Excess Heat Capacity and Entropy at T, AS(T )/
ACp(Tg)/
(cap deg-I (mol of beads)-’)
(cal deg-’ (mol of beads)-’)
T,/K 106
T,/K 63
152
125
0.84 1.06
2.92 3.42
5
172
122
0.89
3.42
6 12
194 282
135 236
0.96 1.10
3.25 4.12
liquid
beads
methanol ethylene glycol propylene glycol glycerol mannitol dulcitol
2 4
0.8
BestulChang av Wunderlich av
2.7
corresponding frequency from the analysis of Kaddour and Champaney is 2.87 X lo5 cm-l. For the remaining liquids, 7, cannot be assessed, but the viscosity data have been analyzed and the best-fit results for T,,, together with the T and T ranges fitted, are collected for comparison with the dielectric and other data in Table 111. For sorbitol, viscosity data covering 3 orders of magn i t ~ d yield e ~ ~a Toof 212 K, significantly lower than that from the dielectric data, 236 K (see Table 111),perhaps for the reason referred to above (see also discussion of the B parameter below). Combination of viscosity with enthalpy relaxation data at lower temperatures leads to To = 215 K.32 The same pattern is repeated for PG, whereas, for EG (represented at low temperatures by the 15% PG solution), dielectric and viscosity data yield the same To, though the viscosity fit is very uncertain because of the small range of data available (see Table 11). Note that, when differences in best-fit To between high-temperature (7)and low-temperature (TD) parameters exist, they are the reverse of the usual order. Thus, there is no tendency to return to Arrhenius behavior near Tgon the part of the liquids of this study. In fact, a weak trend in the opposite direction seems to exist. With the latter point established, we turn to comparisons with the TK obtained from calorimetric data. Again glycerol is the least ambiguous case because of the precision of the classical C, data49on which the analysis is based. T K obtained previously1’ by the graphical method illustrated in Figure 4 for the case of EG is 135 K, within 1 K of the VTF analysis value from the most probable relaxation time data. For sorbitol itself, the T K cannot be determined directly because the crystal form is partly disordered due to the existence of a sluggish A-type anomaly.32 The disorder generated in this transition cannot be fully annealed at low temperatures with the result that the sum of solid-state transition entropy entropy of fusion does not correctly yield the configurational entropy of the liquid. However, there are various isomers of sorbitol which do not have this problem. For instance, for the isomers mannitol and rhannitol, TKis found to be 236 f 10 K, which is the same as the best-fit Toof the VTF equation for T~ in sorbitol (Table 11). Another isomer of sorbitol, dulcitol, gives a lower value, 217 f 10 K, which may be preferable in view of the observation that T for sorbitol is about 20 K lower than for these isomers. d e value of Towhich best fits the
+
(49) G. S.Parks and H. M. H u f f ” , J. Phys. Chem., 31,1842(1927).
Test of the Entropy Basis of the
VTF
The Journal of Physical Chemistry, Vol. 86, No. 19, 1982 3851
Equation
data subject to the constraint B(sorbito1) = B(glycero1) is 224 K (see below). The thermodynamic properties of the sorbitol isomers are discussed in more detail elsewhere. The TK for ethylene glycol is found from Figure 5 to be 114 K f 5. The TK is 11 K below the best-fit value of To for the 15 mol % mixture of 126:; and the viscosity-based If we apply the constraint value for EG of To = 125:;;. B(PG) = B(glycerol), then the best-fit value of Tois 109 K, but the standard deviation is large. The B parameter is discussed separately below. The important finding on which we wish to focus attention is the consistent proximity of best-fit and constrained best-fit Toparameters from the dielectric relaxation data near Tgand the Kauzmann temperature TK, which is summarized in Table 11, last two columns. We believe that Table I1 provides the strongest evidence yet presented for the physical significance of the VTF equation and for the overall credibility of the thermodynamic limit concept for the liquid state based on entropy vanishing. This becomes even more convincing following the discussion of the B parameter given below. It may be noted further, for the case of glycerol, that precise conductivity measurements of dilute salt solutions have been made in the conductivity relaxation time range 104-104 s by Bartoli et al.45 The VTF analysis yields To = 121 K. The range of temperature over which To and T Kmay be correlated would be considerably extended if data for the last member of the (CHz),(OH), series, methanol (CH,OH), could be included. Methanol, like EG, is a readily crystallizing liquid and cannot easily be studied near T .50 Solution studies using 20% ethanol, analogous to our I$G + 15% PG measurements, could be performed, but they have not been included in this study. However, some high-frequency dielectric measurements for the range T = 104-10-' s have been performed by Denny and Cole5' (who also studied methanol + propanol solutions), and good calorimetric data are also available from Sugisaki et al.52 Hence, we include the results in Table 11,where they are seen, within their wide uncertainty limits, to substantiate the findings that we have presented. Fitting methanol-propanol mixture viscosity data to the VTF equation yields rather high values of To (85 K) coupled with very low B values that change little with composition. Methanol-ethanol mixtures, which would be more appropriate for an extrapolation to pure methanol, have yet to be studied. Thermodynamic State at T While the above correlations go far toward establiskng the plausibility of a configurational ground state for liquids, nothing has yet been stated about the thermodynamic condition of the liquid at the experimental glass transition temperature, where the relaxation time has some fixed value (-200 s as defined by DSC measurements at 10 "C min-'). The frequent observation that the Ehrenfest-type relation
at T was approximately constant at 0.8 cal deg-' (mol of beacfs)-'. It is of interest to compare the findings for the present compounds with this value. The "bead" concept55represents an attempt to define a unit rearrangeable element in a relaxing liquid and is unfortunately subject to much ambiguity56 (particularly when dealiig with tightly bound atomic groups, e.g., CO,2-, -S03H, whose constituents are nevertheless massive enough that their ground-state vibrational degrees of freedom are significantly excited at Tg). In our case, however, it is fairly clear that each carbon center and each OH group should be counted. On this basis we compare, in Table 111, the excess entropies at T gfor the liquids of this study and find a value indeed close to (a few percent above) the Bestul-Chang average. It should be noted that the Bestul-Chang value of S J T ) is obtained notwithstanding the finding (column 6, Table 111) that ACp at Tgfor our polyalcohols is considerably above the average value55of 2.7 cal deg-l (mol of beads)-'. This result is consistent with the common experience that hydrogen-bonded substances give large and easily detected thermal manifestations of T It is implied that Tg/Tois, in general, not a constant as Las often been suggested. Indeed, when inorganic network liquids, such as ZnClz and GeOz,which have very small AC, per bead values,6' are included in the survey, it is clear that T / T o can vary enormously (for GeOz it becomes impossibfe to obtain a good estimate of Tobecause of the long extrapolation below Tgwhich is req~ired).~'Surprisingly, Sexat Tg still proves to be quite close to the average value.58 B Parameter. It is clear from eq 1that in any analysis of the temperature dependence of viscosity or other transport properties the possible values of B and Togiven by the analysis will be correlated. Differentiation of eq 1 shows that, near Tg,Tois the dominant parameter, the slope of the Arrhenius plot being given by
dTg/dP = VgTgAa/AC,
(56) For example, we cannot in any way accept the recent assignment by V. P. Privalko, J . Phys. Chem., 84,3307 (1980), of 1 (rather than 3) bead(s) for the substance ZnClz;yet this assignment helped lead Privalko to the generalization that A C p ( T g )is particularly large for inorganic glasses. If the natural choice of 3 is made for the number of beads in ZnClz,then AC, per bead is only 1.9 cal deg-' (mol of beads)-', well below the normal value found for organic liquids, and also below the value of -2.4 cal (mol of beads)-' found for chalcogenide glasses. This finding is consistent with ZnClpbeing a mild case of the small AC tetrahedral network liquid-of which GeOz is the best example for wkch data are available. For GeOz AC per mole of beads (3 beads mol-') is only 0.42 cal deg-' (mol of beads)-9although Sexat TBis still close to (though now a little below) the 'universal value". (57) C. A. Angell and J. C. Tucker, to be submitted for publication. (58) It is ironical that, before the exceptionally small values of AC, for inorganic network glasses had been found, one of us had used the observed large Tg/Tovalues for these substances to argue that their residual entropies should, in general, by unusually large (C. A. Angell, J. Am. Ceram. SOC., 51, 117 (1968).
(8)
is obeyed i m p l i e ~ ~that, ~ ~ "for individual substances, Sex is a constant at Tgwhen the glass is formed at different preasures. Bestul and Chang12made a broader observation: they compared different substances on a "per bead" basis, following Wunderlich,&and found that the excess entropy (50) M.Sugisaki, H.Suga, and S. Seki, Bull. Chem. SOC. Jpn., 41,2591 (1968). (51) D.J. Denney and R. H. Cole, J. Chem. Phys., 23, 1768 (1955). (52) M.Sugisaki,H.Suga, and S. Seki, Bull. Chem. SOC.Jpn., 41,2586, 2591 (1968). (53) M.Goldstein, J . Chem. Phys., 39, 3369 (1963). (54) M.Goldstein, J . Phys. Chem., 77, 667 (1973). (55) B. Wunderlich, J . Phys. Chem., 64, 1052 (1960).
E, = B [ T / ( T- To)]*
(9) The point to be noted though is that noise in the data which results in Tobeing falsely high at best fit will generate a concomitantly low#valueof the B parameter. In particular, small errors in Towill cause larger errors in B. In a family of compounds such as the present one, it would be reasonable to expect a regular trend in the B parameter, and erratic variations will point to probable inaccuracies in the corresponding Tovalues. In view of the extent of the data for glycerol and its internal consistency, it seems probable that the B value is reliable. Suspicion is therefore drawn to the smaller values for sorbitol and EG in Table 11. Printouts from the data analysis allow one to observe the variations of B associated with different choices of To: we find that, when B for sorbitol has the glycerol value,
J. Phys. Chem. 1982, 86, 3852-3855
3852
TABLE IV: Comparison of TK from Entropy Data with ToValues for VTF Equation Best-Fitting f, Data When Subject to Constraint BIT, = 12.7 T,/K sorbitol'"
268
glycerol ethylene glycol E G + 15%PG
153 (extr) 154
methanol 103 methanol + 20% propanol5' methanol + 1 2 % propanol" ethanol 97 propanolb 100
logA
B
TJK
16.9
2625
208
13.8
1741
137
TK/K -217 135 115
113
11.9
1427
10.8
921
72.5
921
72.5
64 11.3
73.5 70
'" These values for sorbitol in this table should be compared with those derived from a combination of viscosity and enthalpy relaxation data, t h e latter measured a t T, itself (see concurrent paper), These results suggest that the constraint BIT, = 1 2 . 7 m a y be past its limit for sorbitol and that T ocannot be less t h a n 2 1 5 K.
1740 K, To is some 12 K lower in value, 224 K, which is intermediate between the two isomer TK values and closer to the viscosity Tovalue. The dielectric loss curves were unusually broad for sorbitol, making the choice of peak frequency subject to some uncertainty, suggesting that the force-fitted Tovalue, 225 K, may be a more appropriate one. Likewise, the value of B for EG + 15% PG seems anomalous by comparison with that for PG. In this case, however, the data seemed particularly reliable, and it is possible that some solution property is being manifested. It is relevant that, in fitting Denny and Cole's TD data for methanol + propanol solutions to eq 1, we consistently obtained combinations of large To (-86 K, cf. TI( = 64 (methanol) and 70 (propanol))with small B. In an early study of electrical conductance and viscosity of aqueous solutions, in a range of transport parameters which yielded good agreement between Toand TK, it was noted that the parameter B was proportional to which is expected from eq 2 if C (eq 2) and D (eq 4) are constants. If we impose this condition on the present series using the glycerol values To = 137 K, B = 1740 to fix the
slope (at 12.7), we find a rather impressive result, summarized in Table IV. The agreement of the constrained-fit Tois now within the extrapolation errors of the TKvalue in all cases except methanol where a solid-solid phase transition complicates the TK assessment. The latter observation means that the relaxation time vs. temperature relations for this family of substances and their mixtures can be described, with some sacrifice in precision, by a simplified VTF-type equation with two free parameters TD = A, exp(12.7/[T/To - 11) (10) in which one of the free parameters is predictable within a very few degrees from thermodynamic data (in the case of pure compounds in the family). The sacrifice in precision is indicated for the worst cases by the dashed lines added to Figure 3, which are plots of eq 10 in the equivalent f, form with best-fitting To and preexponential parameters, as listed in Table IV.
Concluding Remark For simple liquid models it can be argued that excess entropy and free volume modes are equivalent,26even though it is still not clear what relation the limiting amorphous packing volume for an atomic system at TK has to the crystal volume at that tem~erature.6~ So far as can be told from computer experiments5g*@ (since such systems cannot be vitrified in the laboratory), a single-order parameter suffices to describe the glass transition, since the Prigogine-Defay ratio is unity within experimental error. In laboratory glasses, however, this is clearly not the therefore, that entropy should ~ a s e . It~ is~ convenient, ,~ turn out to be the important thermodynamic variable controlling relaxation times near Tgsince the zero point, or projected phase transition temperature, can be estimated from simple measurements (Figure 5), without solving the random close-packing problem. Acknowledgment. This work was supported by the National Science Foundation under solid state chemistry Grant Nos. DMR 77-04318A1 and DMR 8007053. (59)C. A. Angell, J. H. R. Clarke, and L. V. Woodcock, Adu. Chem. Phys., 48,397 (1981). (60) J. H. R. Clarke, J. Chem. SOC., Faraday Trans. 2,75,1371(1979).
Proton Medium Effects in 5-40 wt % Acetone-Water Mlxtures. An Investigation Using the Ferrocene Assumption James R. Waggoner and L. M. UukherJee"+ Chemlsby Department, Unlversliy of FlorMs, Gshesvllle.Fbrkk 3261 7 (Received: December 15, 1987; I n Final Form: April 5, 7982)
Estimates of the standard potentials of the ferrocene electrode in 5-40 wt% acetone-water mixtures, at 25 O C , were obtained from cyclic voltammetric studies. Proton medium effects (log evaluated from these potentials, referred to the normal hydrogen electrode (NHE) in the respective media, have been examined in an attempt to resolve earlier controversies on the proton affinities of such solvent mixtures. Introduotion Earlier studies1.2 concerning the solvation properties of the acetone-water mixtures and their proton affinities reveal striking differences of opinion. In 1969, Fong and Grunwaldl concluded from their work on 2,4,6,2',4',6'On leave from Banaras Hindu University, Varanasi, India.
hexanitrodiphenylamine (dipicrylamine) that wateracetone mixtures containing up to 24 Wt % of the latter favored solvation of the dipicrylamine anion by dispersion (1) D. Fong and E. Grunwald, J.Phys. Chem., 73, 3909 (1969). (2)D.Bax,C. L. de Ligny, and A. G. Remijnse, R e d . Trau. Chim. PQYS-BQS, 91,1225 (1972).
0022-3654/82/20853852$01.25/00 1982 American
Chemical Society
