J. Phys. Chem. 1980, 84, 859-863
micelles, decreasing [17] a t 6 = 0 in eq 1 means decreasing Y, Le., an approach toward a spherical shape. Such a calculation also leads to unusual results and change of shape cannoi, also account for the low [VI. Na,SO,, on the other hand, may have imposed further hydration to the head group and has increased [TI. A t a higher salt concentration a salting-out type phenomenon may have dehydrated the micelles and reduced [VI. The latter is in line with the observed clouding of T X 100 at lower temperaNa2S04. t ~ r with e ~ increasing ~ The partial molal volume above the cmc at a particular temperature has been observed to be almost constant. The volume contribution at micellar point and above is therefore the same. Since the excess volume P is negative, some kind of organization of the water molecules by the poly(oxyethy1eneoxide) chains around the micelles has been revealed. Hydrogen bonding of water through such chains can be a very common occurrence. Each oxygen center has roughly been observed to fix, on the average, 2-3 water molecules. Entrapping, over and above hydrogen bonding, is therefore possible in the network of the many flexible head groups on the outer mantle of the micelles. With increasing temperature, although the partial molail volume has increased, VE has decreased. Dehydration of the water embedded chains might have brought them closer (since the hydration barrier is reduced) and the increment due to the temperature rise has compensated for this effect with a net decrease of VE. Above 45 “C this is not orderly. It points to the complex nature above 35 “C (cf. Table I). The activation parameters for solution fluidity are in favor of more ordering of the e n v i r ~ n m e n tboth ; ~ ~ A € P and
a59
AS* have higheir magnitudes in the presence of T X 100 than in its absence. Acknowledgment. Thanks are due to the authorities of the Jadavpur University for the use of laboratory facilities by S. Ray and A. M. Biswas.
References and Notes (1) A. Helenius and K. Slmons, Biochim. Biphys. Acta, 415, 29 (1975). (2) S. Razin, Biochim. Biophys. Acta, 265, 241 (1972). (3) R. A. Deems, B. R. Eaton, and E. A. Dennis, J . Bioi. Chem , 250, 9013 (1975). (4) T. G. Warner and E. A. Dennis, J . Bo/. Chem., 250, 8003 (1975). (5) E. A. Dennis, ,Arch. Biochem. Biophys., 158, 485 (1973). (6) R. J. Robson and E. A. Dennis, Biochim. Biophys. Acta, 5016, 513 (1978). (7) E. H.Crook, D. 6.Fordyce, and G. F. Trebbi, J. Phys. Chem., 67, 1987 (1963). (8) A. Ray and G. Nemethy, J. Am. Chem. Soc., 03, 6787 (1971). (9) R. J. Robson end E. A. Dennis, J . Phys. Chem., 81, 1075 (1977). (10) C. Tanford and J. A. Reynolds, Biochim. Siophys. Acta, 45’7, 133 (1976). (11) C. Tanford, Y. Nozaki, and M. F. Rohde, J . Phys. Chem., 81, 1555 (1977). (12) L. M. Kushner and W. D. Hubbard, J. Phys. Chem., 58, 1163 (1954). (13) A. A. Ribeiro and E. A. Dennis, Biochemistry, 14, 3746 (1975). (14) G. 6.Jeffery, Proc. R. Soc. London, Ser. A , 102, 161 (1923). (15) R. Simha, J . Phys. Chem., 44, 25 (1940). (16) F. Podo, A. Ray, and G. Nemethy, J. Am. Chem. SOC.,05, 6164 (1973). (17) S.P. Moulik, lbctrochim. Acta, 16, 981 (1973). 118) S. P. Moulik and D. P. Khan. Carbohvd. Res.. 36. 147 11974). i19i V. Vand, J. Rbys. Colloid Chem., 52,’277 (1948). (20) (a) H. Fricke, Phys. Rev., 24, 575 (1924); (b) ibid., 26, 682 (1925). 121) H. C. Thomas and A. Cremers. J . Phvs. Chem., 74, 1072 (1970). (22) F. Perrin, J . Phys. Radium, 7, 1 (1936). (23) K. Shinoda, T. Nakagawa, 6.Tamamushi, and T. Isemura, “Colloidal Surfactants: Some PhysicochemicalProperties”, Academic Press, New York, 1963. (24) S. P. Moulik and D. P. Khan, Indian J . Chem., 16A, 16 (1978). >
,
Viscous Liquids and the Glass Transition. 9. Nonconfigurational Contributions to the Excess Entropy of Disordered Phases P. D. Gujrati’ and Martin Goldstein” Division of Natural Science and Mathematics, Yeshiva University, New York, New York 10033 (Received August 27, 1979)
An analysis of calorimetric data on crystal, liquid, and glassy phases of a single substance permits an estim.ation of the relative contributions of configurational and nonconfigurational (e.g., vibrational) factors to the excess entropy of the supercooled liquid at Tg. Calorimetric data on three substances were previously analyzed; we report here an analysis of data on eight additional substances. We find in the 11 substances that there is an average decrease of the entropy difference between glass and crystal on cooling from Tgto the vicinity of 0 K of 5.8 J/(K mol), representing a fraction, on the average, of 0.30 of the excess entropy at Tg. A comparable analysis on orientationally disordered crystals below the “glass” transitions shows that these substances show larger absolute and relative losses: 7.9 J/(K mol) and 0.58, respectively. It is concluded that communal entropy in the liquid state does not play a major role in the glass transition.
Introduction In an earlier paper from this laboratory1 we considered two questions: (1)How much of the excess entropy A S of the amorphous phase at Tgmay arise from nonconfigurationa] degrees of freedom (e.g., vibrational differences between amorphous and crystalline phases)? + Physics Department, Carnegie -Mellon University, Pittsburgh, P A 15213.
0022-3654/80/2084-0859$0 1.OO/O
(2) How much of the “configurational” specific heat AC, a t T gis truly configurational? We showed that the first question could be answered if specific heat daki on crystal and glass phases were available from Tgto the vicinity of 0 K, and the second if specific heat data on glasses of different fictive or structural temperatures were available over the same range. Data of the latter sort had been obtained for six substances by Chang and associates a t the National Bureau of Standards.2-6 Three of these substances had also been 0 1980 American
Chemical Society
860
The Journal of Physical Chemistry, Vol. 84, No. 8, 1980
TABLE I: Substances Studied substance 2-butanethiol butene-1 ethanol glycerol isopentane isopropylbenzene 3-methyl-1-butanethiol 2-methylpentage 3-me th y lpe nt ane 2-methyl-1-pro panethi n-propyl alcohol sulfuric acid trihydrate
ref 11 12 13 14 15 16 17 18
18,19 20 21 22
studied in the crystalline state. The conclusions were (1) in the three substances studied the range of the nonconfigurational contribution to the excess entropy A S was from 10 to 4070,and ( 2 ) that the nonconfigurational contribution to AC, averaged about 45%, with a range, in the six substances studied, of from 20 to 85%. It was pointed out that the Gibbs-DiMarzio theory of the transition is of dubious applicability to any substances for which the nonconfigurational contribution to AC, is significant. In a second paper it was pointed out how the driving force for a second-order transition of the type postulated by Gibbs and DiMarzio to circumvent the Kauzmann paradox can arise from these nonconfigurational contributiom8 While the analysis of the importance of nonconfigurational contributions to AC, appears more fundamental to understanding the nature of the glass transition, obtaining calorimetric data of the type needed requires unusual accuracy and care. As Chang and associates are no longer engaged in this type of research, data on additional glass-forming substances may be long in coming. The analysis of the contributions of such factors to the excess entropy A S of the amorphous phase has some bearing on the glass transition, and some also on the thermodynamics of the crystal-liquid transition. As our original study reported such an analysis on only three substances, and as data on additional substances appeared to be available in the literature, we decided to extend our analysis of A S to as many substances as possible. We were able to find data on 8 additional glass forming molecules, so that we have been able to perform the analysis on a total of 11 substances, respresenting a much more reliable sample of glass-forming liquids. Independently, Johari has been making a similar analysis of a class of substances known as “glassy” plastic c r y s t a l ~ . ~Comparison J~ of Johari’s results and ours sheds new light on the role of communal entropy in the glass transition, a point to be discussed later in this paper.
Substances Studied To perform the analysis, calorimetric data on C, of crystal and supercooled liquid states were needed from the vicinity of 0 K to the melting point for the crystal and from Tgto the melting point for the liquid. C, of the glass was needed from the vicinity of 0 K to T,. Further, the enthalpy of fusion at the melting point was required. In Table I we give a list of substances for which we found data. We choose not to extrapolate specific heats below the lowest temperature at which direct measurements have been obtained in this analysis although some workers in this field have extrapolated specific heats to T = 0 K from their measured values. The differences between glassy and crystal specific heafs a t low temperatures is of great significance from a fundamental point of view,23but they
Gujrati and Goldstein
make only negligible contributions to the excess entropy. Errors in the absolute entropy of the glass phase due to irreversible processes that take place in the glass transition range have been considered in our earlier paper, and were shown to be neg1igible.l
Data Analysis We will describe in this section the method of curve fitting we used to analyze the various sets of data on the specific heat. This method differs from the procedures of other workers in this kind of analysis who have used polynomial approximations, with the constants chosen by least-squares m e t h o d ~ . ~ ~ These J ~ J ~ procedures J~ required different polynomials to be used over different temperature ranges, with some subjective judgment used to fit the two polynomials in the temperature region of overlap. We preferred a method we found was both more objective and faster to apply. The method chosen was a “piecewise exact” curve-fitting procedure described below. Let us assume that the function f(x) takes the values f(xo),f(xl), ...,f(x,) at almost equally spaced points xo,xl, ..., x,. Let us also assume that f(x) is not a singular function and that it does not oscillate widely between any two consecutive points or that it does not have a strong variation between any two consecutive points. Thus, in a sense, f(x) is a well-behaved function over the range R = (xo,xn), Let A, be a representative value for the distance between any two consecutive points xo,xl,...,I,. Our aim is to interpolate f(x) over the range R and extrapolate on either side of R over a distance not exceeding A,. As will be apparent below, our method will not be suitable for extrapolation beyond a distance A,. Let us divide the range R into sets of overlapping subranges R1, R2, RB,... defined by Ro:(xOX1,*..,Xm) R1:(~1,~2,-~,x,+i) R~:(x~,x~,*..J~+z)
Rn-m+l:(xn-m,x,-,+l,...,Xn)
(1)
for any integer m > 1. Now, we fit a polynomial Pl(,) (x) of order m over the subrange R,. Since each range, R, has exactly (m 1)points at which the values of the function f(x) is known, the above polynomial P,(x) (in what follows, for simplicity) we will suppress the superscript (m)on P,(x) takes the same value as the function f(x) at these points. Thus, if we approximate f(x) by P,(x)in that range R,, it is an “exact” fit since the error is identically zero:
+
P,(x,+,) = f(x,+J)
j = 0,1, ..., m
(2)
We note that for any pair of consecutive points ( x , , ~ , + ~ ) , we will obtain, in general, more than one polynomial. We
may, therefore, take an average of Q,(x) of these polynomials which will be assumed to approximate f(x) between x,and x , + ~ .It should be noted that Q l ( x ) defined in the domain (x,,x,+J is such that Q,(x,) = f W (3) QC(xC+i) = f(x,+i)
(4)
Q,hJ= Q,-ibJ
(5)
Also Thus, Q,(x) and Q,-l(x) are continuous approximations of
The Journal of Physical Chemistry, Vol. 84, No. 8, 1980 861
Viscous Liquid!; and the Glass Transition
TABLE 11: Substances with Data for Full Temperature Range‘
--
AS,/
substance
TE
TL
Tm
AS,
AS,
butene ethanol glycerol isopenlane isopropylben zene 3-methyl-1-butanethiol 2-methylpentane sulfuric acid {xihydrate ( H2S0,.3H,O)
60 97 185 65 126 100 78 157
13.70 13.73 10.2 15.60 14.53 11.66 11.93 16.0
87.83 159.0 291.8 113.4 177.1 139.6 119.55 236.72
19.80 16.58 26.92 17.84 14.87 21.51 25.02 34.81
12.29 11.64 21.30 13.76 8.47 13.93 17.78 25.48
a
T,/T,
AS,
43.81 31.00 63.95 45.40 41.36 53.05 52.39 102.22
0.68 0.61 0.64 0.58 0.71 0.72 0.65 0.66
AS,
AS,/ AS,
0.45 0.54 0.43 0.39 0.35 0.40 0.48 0.34
0.62 0.70 0.79 0.77 0.57 0.65 0.71 0.73
AS,-
AS,
7.51 4.94 B.62 4.08 6.40 ’7.58 ‘7.24 9.33
In 1,his a n d in the following tables temperatures are given in K, entropies in J/(K mol).
f(x) a t x,. However, their slopes at this point need not, in general, match: Ql’(xJ f - Ql-i’(xJ (6) Thus, we conclude that the set (Qo,QI,...) provides us with a continuous exact fit to the data over the range R. Before we proceed further, let us digress for a moment and describe what kind of data fitting we require. We are given the specific heat values at certain values of the temperature. From these, we want (i) to interpolate for the specific heat at intermediate temperatures, (ii) to extrapolate for the specific heat over a short range on either side of the given range of temperature, and (iii) to calculate the entropy over this extended range. Thus, our interest is eventually in the integration of the function f(x)/x (Le., of the specific heat divided by the temperature). This implies that the error due to any “discontinuity” in the slope (eq 6) would be reduced after integration. Thus, the error introduced in the evaluation of the entropy would be much less than the error in the evaluation of the specific heat. As a test for the applicability of our method, we first have applied it to various mathematical functions f(x). We chose n 1points xo, xl, ..., x , in the range R and evaluated f(x,) at all these points, i = 0, 1, ..., n. We chose various values of m .> 1 and computed the average polynomials Q,(x), i = 0, 1,...,n - 1. We used these average polynomials in their appropriate ranges of definition to estimate f(x) at any point in these ranges. We found that the difference
+
I&,(%)
- f(x)l
x E (x,,x,+i)
(7)
was hardly appreciable in all the cases we studied. Thus, the set (Q,(x))is indeed a very good approximation of f(x) in the range R : ( x o , x , ..., , x,). We have also noted that for extrapolation IQo(x) - f(x)l
x E (x: - Ax,xo)
We now replaced f(x,) by “fx,) in the above test and calculated the average polynomials Q l ( x ) . We have again found that the differences x E
lQ,(x, - f(x,)l
@O(d
x
-
lQfl-i(x) - f(x)l
X:
(x,,x1+1)
E ( 5 0 - Ax,x) E ( x , , ~ , + A,)
were within our confidence limits. This provided further assurance about the validity of our method of analysis. As a final test we compared our results with some examples of specific heat data which had been fitted with smoothing polynomials over limited ranges of tempera~ u ~ ~ . ~ We - ~again J ~ found J ~ Jno~substantial differences between these results. From the point of view of the computational time on the machine and the accuracy of the results, it is obvious that we cannot, in general, choose a large value of m (the order of the polynomial) for data fitting. We have found that in our method m = 2, that is, parabolic fits over subranges defined by only three points gave very good results. Therefore, throughout our analysis, we have taken rn = 2. There were a few instances where two data points x , and x,+i were very close, i.e. Xl+l
- x,
