416
A. QUACHAND ROBERT SIMHA
Statistical Thermodynamics of the Glass Transition and the Glassy State of Polymers by A. Quach and Robert Simha” Division of Macromolecular Science, Case Western Reserve University, Cleveland, Ohio 44106 (Received September $4, 1071) Publication costs asaiated by the Petroleum Research Fund and the National Science Foundation
The hole theory of Simha and Somcynsky is applied to an analysis of the liquid-glass boundary and to the equation of state in the region between the glass transition and the 0-relaxation. Two systems already studied experimentally are considered, namely, polystyrene and poly (o-methylstyrene). The liquid-glass boundary relations are investigated under two sets of conditions corresponding to a low- (LPG) and a high-pressure glass (HPG). The former is formed by cooling the liquid a t atmospheric pressure, whereas the latter is obtained by pressurizing the liquid isothermally. The equation of state is analyzed for LPG only. The link between the conventional thermodynamic relations, experiment, and the statistical theory is formed by identifying the vacancy fraction 1 - y appearing in the latter with the ordering parameter 2 introduced in t h e thermodynamic theory. For LPG, ye, the value of y along the boundary, is indeed found to be constant for both polymers. For HPG, 1 - y, is a decreasing function of pressure, as should be expected. The equation dT,/dP = (bT,/W)z (bT,/bZ)p X dZ/dP is tested by evaluating the product on the right-hand side by a combination of the statistical theory with experiment. An equation of state for LPG is first computed entirely from theory by assuming that a single constant parameter, y = yg, characterizes not only the liquid-glass boundary line, but the glassy region as well. This results in too low a thermal expansivity, as had been noted earlier by Somcynsky and Simha for several other polymers a t atmospheric pressure. Hence, within the frame of the hole theory, y cannot remain constant in the glass but is a function of T and P. It differs, of course, from the function derived by maximization of the configurational partition function of the liquid and is obtained here from experiment. Thus, additional constants enter into the equation of state of the glass, which cannot be obtained solely from the properties of the liquid and the liquid-glass boundary line. On approaching this line, however, the above function reduces to a single constant, viz., yg.
+
I, Introduction The concept of internal ordering parameters (&, ZZ, . . .), which assume fixed values at the glass transition temperature, to describe the glass transition process as well as the glassy state, has been proposed by several authors. By applying the formalism of de Donder,l Prigogine and Defayz obtained for a system characterized by a single ordering parameter, 2, the following identity connecting the changes (A) in the thermodynamic derivatives at the transition AKA& = VT(ACY)~
Here V and T are the volume and temperature a,t the transition point and K, CY, and CP are, respectively, the compressibility, thermal expansivity, and heat capacity. Meixners and Davies and Jones4 showed that the effect of pressure on the glass transition temperature (T,) can be described by the relations (hTg/bP)z = TVAa/ACp (hTg/hP)z = AK/ALX
(2)
(3)
Equation 3 was also obtained by Gee5 using a slightly different approach. Moreover, if entropy S instead of volume V is chosen as the dependent variable, his procedure readily leads to eq 2. The Jouvnal of Physvsical Chemistry, Vol. 713, No. 8,1079
By equating the right-hand side of relations 2 and 3, eq 1 is recovered, which provides the basis of a rigorous test for a one-parameter theory. When two or more parameters are involved, we have4$6
(4) but nothing is asserted about dTg/dP. Goldstein? introduced the concepts of excess volume, entropy, and enthalpy of the liquid over the glass and showed that if either of the latter two is the determining factor, eq 2 holds, while eq 3 is obeyed if the excess volume is the pertinent quantity. The current experimental evidence is contradictory. Analyses of some authors6,7J have shown that the inequality (4)rather than the equality (1) holds, implying the existence of more than one parameter, while Breuer (1) Th. de Donder and P. Rysselberghe, “Affinity,” Stanford University Press, Menlo Park, Calif., 1936. (2) I. Prigogine and R. Defay, “Chemical Thermodynamics,” Longmans, Green and Co., London, 1954. (3) J. Meixner, C . R . Acnd. Sci., 432 (1952). (4) R . 0. Davies and G . 0. Jones, Advan. Phys., 2 , 370 (1953). (6) G . Gee, Polymer, 7, 177 (1966). (6) A. J. Staverman, Rheol. Acta, 5, 283 (1966). (7) M. Goldstein, J. Chem. Phys., 39, 3369 (1963). (8) J. M. O’Reilly, J. Polym. Sci., 57, 429 (1962).
GLASSTRANSITION AND THE GLASSY STATE
417
eq 3 holds for LPG but not for HPG." If indeed a single ordering parameter characterizes the transition, then T, = T,(P, Z), andl0 dTg/dP = (bTg/bP)z
I" 1
Figure 1. Diagram of cycles to measure T, as a function of pressure P.
and RehagesTobservedthat for polystyrene eq 2 and 3 are not obeyed but the equality (1) holds. They concluded that a single parameter is adequate to describe the liquid-glass transition behavior. The conflicting conclusions probably arise in part from the fact that in tests of eq 2 and 3 the requirement of maintaining Z constant has been disregarded. Very few literature data concerning the change of Tg as a function of P satisfy this condition. Bianchi, et al.,'" have carried out experiments in such a way that they expected Z to remain as close to constant as possible. Figure 1 recapitulates the cycle proposed by these authors. The polymer melt at point C is cooled at a fixed rate and atmospheric pressure to point A, where a pressure P is applied to compress the glass to point G. The pressure is then kept constant while the glass is heated isobarically to point D, with a transition point F. By repeating the experiments with different pressures, the locus defined by the dashed line BF is obtained. We shall refer to the glass formed by cooling the melt a t atmospheric pressure as low pressure glass (LPG). A different way to form a glass is defined by the path CDEH. In this case the transition point is E and the liquid-glass boundary is marked by the dashed line BE which departs significantly from line BF. A third kind of experiment commonly adopted is to pressurize the melt at point I isothermally. The corresponding liquid-glass transition boundary is not shown in the graph but is expected to differ from both lines BE and BF. The glass formed by the last two types of experiments will be referred to as high pressure glass (HPG). It is not implied, of course, that the dashed portions in Figure 1 are necessarily straight lines. Obviously, only the low pressure glass can possibly satisfy the condition of constant 2, We have recently reported P V T results for polystyrene (PS) and poly (o-methylstyrene) (PoMS) derived from experiments of the third and first type.ll I n this work line GF was obtained by cross-plotting and extrapolating isothermal data rather t'han directly from isobaric measurements. This is acceptable as long as the transition region itself is avoided. It was shown that
+ (bTg/bZ)p X dZ/dP
(5)
where (bT,/dP)z = A x / A a , as is seen from eq 3. Since (dT,/bZ)p should be positive, if Z is a measure of disorder, the relation dT,/dP = hrc/ha! only holds if dZ/dP 0. A purely thermodynamic theory need and does not specify the physical significance of the ordering parameter or parameters. These parameters have been variously associated with such quantities as a free volume, and a single or a spectrum of relaxation time constants. We shall bring molecular theory to bear on first, the properties of the liquid-glass transition line, and second, the equation of state of the low pressure glass. The basis for this is the hole theory of Simha-Somcynsliy,12 which has been successfully used to discuss the equation of state of amorphous polymer systems above the glass temperature.11i12 I n addition to the characteristic volume (V*), temperature (T*),and pressure (P*)quantities, this theory contains an additional parameter pertinent for the present discussion, namely y, the fraction of occupied sites in the quasilattice. It is suggestive then to identify the vacancy fraction 1 - y with the thermodynamic parameter 2. At equilibrium, y is determined as a function of V and T by the maximization of the partition function and this fixes then by assumption the value of y = ye at the glass point. On this basis, Somcynsky and Simha13have compared theoretical predictions with experimental thermal expansivities a, at the glass temperature and atmospheric pressure for a series of polymers. We shall extend this line of investigation by a detailed examination of the two polymer systems mentioned above. To begin with, however, we analyze the liquid-glass transition lines. According to the concepts reviewed above and the postulated relationship between Z and y, the magnitude of yg at the observed T, and Pgis determined entirely by equilibrium relations. These are briefly recapitulated in section 11,and applied to the transition line in section 111. In section IV, an equation of state for the glassy state is developed.
11. Theory The equation of state expressed in reduced variables (P = P/P*, 7 = V/V*, and = T/T*) isI2 (9) H.Breuer and G. Rehage, Kolloid-2. 2. Polym., 216-217, 159 (1967). (10) U. Bianchi, A . Turturro, and G. Basile, J. Phys. Chem., 71, 3555 (1967). (11) A. Quach and R . Simha, J . A p p l . Phys., 42,4592 (1971); see also A. Quach, Ph.D. Dissertation, Case Western Reserve University, Cleveland, Ohio, 1971. (12) R.Simha and T. Somcynaky, Macromolecules, 2, 342 (1969). (13) T.Somcynsky and R. Simha, J. A p p l . Phys., 42,4545 (1971). The Journal of Physical Chemistry, Vol. '76,No. 23, 1978
418
A. QUACHAND ROBERT SIMHA
pF/p
=
[1-2-'/6y(y7)-1/"--' (2y/
+
F ) (y P) --2 [1.011(g P) --2
.08
IIO
- 1.2045]
(6) For the equilibrium liquid y is obtained as a function of 7 and p from the minimization of the Helmholtz free energy by the solution of the following equation for the infinite chain
+ In(1 - y)/y] = [2-'/6g(y17)-1/8- '/a] [ l - 2-"6y(yP)-"3]-1 + (y/61)(y7)-2 X
(s/3c)[l
E2.409 - 3.033(yP)+]
X
.Ob
.04
-
-
HI0
-.06
HI0
PJ
.oz 0
(7)
where s and 3c are, respectively, the number of segments and external degrees of freedom per chain. As previously, we adopt the value of unity for the ratio s/3c. This particular choice affects the numerical values of the reducing parameters, but not the generality of the At a prescribed temperature, the relationship between y and P is most easily computed from eq 7 by Newton's iteration method. However, at constant pressure, p must be eliminated from eq 6 and 7 before the same method can be used.
111. Analysis of the Liquid-Glass Boundary Line Our previous experimental results for the effect of pressure on T, and the three characteristic parameters for the two polymers are summarized in Table I. We emphasize that the numerical values of the reducing parameters P*, V*, and T* to be used are those derived from the equation of state of the liquid. By means of these and of eq 6 and 7, the vacancy fraction 1 - g, a measure of an unoccupied volume, is computed as a function of the transition pressure P,, corresponding to a transition temperature T,. The results for both polymers and the two types of glasses appear in Figure 2. For LPG, 2 = 1 - y is practically independent of P , i.e., dZ/dP =0, although in the case of polystyrene there is a slight trend of increasing 2 with pressure. This may be attributed to the difficulty of extrapolating isobars in the glassy state to locate the transition temperatures of the low pressure glass. There is considerable curvature in the lines for the low pressure region (see ref 11, Figure 5). Therefore, the two temperatures closest to T , were disregarded, resulting in a somewhat lower value of T, (see ref 11, Figure 8). Since 2 = 1 - y increases linearIy with T as will be shown below, a lower T , would result in a lower value of 2 at low pressure and this is actually observed in Figure 2. Another fact to support the above argument is that if 2 is to change at all, it should be decreasing with increasing pressure. In any case, the change is small enough to be disregarded for our purposes and a horizontal line was drawn, representing the average value. The fact that dZ/dP = 0 for the low pressure glass is in accord with the concept that 2 is frozen in the glassy state and does not vary with T and P. On the other hand, it is possible that y remains constant only along the boundary line (see section The Journal of Physical Chemistry, Vol. 76,No. 8,197.9
-
1200
600 Pp
I800
, EAR
Figure 2. Ordering parameter 2 identified with vacancy fraction (I - y) of the Simha-Somcynsky theory as a function of transition pressure Pg for low and high pressure glasses.
IV). The identification of 1 - y with Z thus provides a physical interpretation as well as a numerical value for the latter. Moreover it attests once more to the validity of the theoretical equation of state, eq 6 and 7. Finally, it permits a prediction of T, as a function of P, once the glass temperature a t atmospheric pressure and hence yB are known. For the high pressure glasses, on the other hand, Z decreases with increasing pressure and dZ/dP is negative. Physically, this is reasonable because the number of vacancy sites at the glass transition is expected to be reduced a t eIevated pressure. Now that we have shown that with 2 = 1 y, dZ/dP C 0 for a high pressure glass, we proceed to compute the derivative (bT,/bZ)p from the molecular theory and examine the validity of eq 5. Figure 3 represents the reduced temperature as a function of 2 = 1 - y for a given reduced pressure f3 in the region of the glass transition, as calculated from the equation of state, eq 6 and 7. The relation is linear with a slope which increases with f3. This suggests that T, should be more sensitive to experimental conditions such as the cooling rate at elevated pressures. With the aid of the results in Figures 2 and 3, the correction term for variable 2 on the right-hand side of eq 5 can now be computed. For computational convenience, dZ/dP = -dy/dP was obtained by fitting a least-squares parabola through the points in Figure 2 rather than by analytical evaluation of the derivative from eq 7. The results are shown and compared with the experimental values of dT,/dP in Figure 4. Except for polystyrene at low pressures, we feel that the extent of agreement observed is quite good, considering the uncertainties involved in computing a, and K~ of the HPG by means of an asymptotic value of the Tait parameter B.11 The residual deviations observed in Figure 4, however, may well be real, and clearly similar analyses of additional polymer systems are highly desirable. These should show whether not only similar magnitudes but also the identical sign of the departures prevail, when the thermodynamic eq 5 is
-
419
GLASSTRANSITION AND THE GLASSY STATE Table I : Glass Transition Temperatures, Average Pressure Coefficients, and Characteristic Parameters for Polystyrene and Poly(o-methylstyrene)ll (Tg)P-o,
dTg/dP, deg/kbar (HPG)
"K (HPG)
374 404
PS PoMS
( Tg)P-O , OK
(LPG)
74.2 73.0
365 404
31.6 34.2
4 .O
*0
IbIDI
3.6
>
3.2-
I+
2.8
2.4
.os
.03
.07
dTg/dP, deg/kbar (LPG)
.09
2.1-y
Figure 3. Reduced temperature P as a function of ordering parameter 2 = 1 - y a t a series of pressures in the region of the glass transition, eq 6 and 7.
P* x 10-8,
v* x
10,
T* X 10-4,
bars
cma/g
OK
7.453 7.458
9.598 9.762
1.268 1.274
where (yo, is the thermal expansivity of the glass a t atmospheric pressure. From Shishkin's densification data,l6 df/dT was estimated to be about 3.3 X deg-l for poly~tyrene.~ Equation 8 appears to underestimate significantly dT,/dP. Equation 9 is quite satisfactory, at least in the low pressure range, with deviations now in the opposite direction from those arising in the molecular theory. On basic grounds and because of the nature of the information required in the application of eq 9, we believe our approach to the thermodynamics of the glass transition t o be preferable in the cases of both LPG and HPG. A final decision regarding the validity of eq 1 and thus of a one-parameter description of the transition boundary requires a knowledge of ACp on the identical sample with identical thermal history. Lacking this information and as a first approximation, we use the value of 0.0525 cal/g-deg for PS given by Wunderlich and Jones,16and obtain at P -+ 0 AKAC~/[TV(ACY)'] cz 1.5
(10)
Although the ratio does not turn out to be unity, this cannot yet be considered as an argument against the adequacy of the one-parameter concept. Final judgment should be based on appropriate heat capacity measurements a t both atmospheric and elevated pressures. 600
1200 P,
~
IV. Equation of State (LPG)
1800
0AR
Figure 4. Comparison of various expressions (see text) for the pressure coefficient of T,of high pressure glasses.
combined with the statistical theory, eq 6 and 7. Such results, of course, have a bearing also on the question of one us. two (or more) ordering parameters. I n Figure 4 there are also plotted the results of two other proposed modifications of eq 3. One, due to Bianchi, l4 introduces the temperature dependence of the volume V, a t T,, wiz. dT,/dP
=
AK/(AcY - d In V,/dT)
(8)
On the other hand, taking the point of view that in thc isothermal experiments a degree of compression f occurs prior to glass formation, Gee6 proposed the expression dT,/dP = AK/(w
- a,,,+ df/dT)
(9)
I n Figure 5 are shown the isobars a t atmospheric pressure for PS and PoMS in the region between the @-relaxationand the glass transition temperature, that is, between T, (see Table I) and about 0.7 X T,." Also indicated are the theoretical results from eq 6 with the assumption of constant y = y, and P = 0. For PS and PoMS the numerical values from Figure 2 are 0.940 and 0.925, respectively. The predicted expansion coefficients are smaller than those observed. This is in accord with previous findings and possible reasons for the discrepancy have been suggested. l 3 Within the frame of the hole theory, a single frozen parameter is not sufficient to describe the PVT properties of the glass. Here we wish to explore quantita(14) U.Bianchi, J . Phys. Chem., 69, 1497 (1965). (15) N. I. Shishkin, So*.Phys.-Solid State, 2 , 322 (1960). (16) B. Wunderlich nnd L. D. Jones, J . Macromol. Sci., Phys., 3 , 67 (1969).
The Journal of Phyaicat Chemistry, Vol. 76, N o . 3,1978
420
A. QUACHAND ROBERT SIMHA
.02-
250
Figure 5. Reduced volume-temperature curve at atmospheric pressure. ( A ) polystyrene, (0)poly(o-methylstyrene). Other lines, theoretical, eq 6 with constant y = yg.
tively one particular suggestion, namely, a nonvanishing temperature and pressure dependence of the vacancy fraction (1 - y) in the glassy state. The functional relationship will be derived from experiment, ie., from a series of isobars. This is accomplished by intersecting at a given pressure the experimental V-T curve with a net of theoretical lines computed from eq 6 with appropiately chosen values of y. The result for PS at atmospheric pressure is exhibited in Figure 6 and compared with the horizontal line, y = yB, and the curve extrapolated from the liquid region by means of eq 6 and 7. As was to be expected, the actual result is intermediate between these two extremes, I n Figure 7, the values of 1 - y for the two glasses are plotted as a function of the temperature difference T,(P) - T for three pressures. We note that the relationship is linear in T at a given P and may be expressed in the form Y
- Ye
=
[ T m
- TIA(P)
(104
The numerical values of the coefficient A ( P ) = - (by/ dT)p are shown in Table I1 for P = 0, 400, and 800 bars. They can be represented by the equations
PS
A(P) = 1.36 x 10-4
P ~ M S A(P) = 1.82
x
- 1.52 x
+
10-7~ 7.2 X 10-l1P2
+
10-4 - 1.44 x 1 0 - 7 ~ 5.9 X 10-l1P2 (lob)
Equation 10 represents a variation of the vacancy fraction 1 - y which is intermediate between the equilibrium function of the supercooled liquid and the value 1 - yB “frozen” at T,. I n combination with the hole theory, eq 6, and the pressure dependence of T,, which may be derived either directly from experiment or from the theory by means of low pressure data, The Journal of Physical Chemistry, Vol. 76,No. 3,197.9
/
/
/
//
300
350 T, O K
400
450
Figure 6. Variation of vacancy fraction 1 - y with temperature a t atmospheric pressure for polystyrene glass, Dashed line, supercooled liquid, eq 6 and 7. Solid line a, awumption of constant y = yg, b, from eq 6 and experimentJal volume-temperature data.
s >I
0
40
120
80 1s-T,
160
‘K
Figure 7. Difference in vacancy fraction y - yg = Z, - Z as a function of temperature and pressure, derived from ey 6, pressure dependence of glass transition temperature and value of y a t Tg.
it provides an equatjion of state for LPG above the @relaxation temperature of the tm7o polymers investigated. Besides the parameters P*, V*, T*, yg> and Tg(P) which are characteristic of the liquid and liquidglass boundary line, there appears, in the pressure range explored, also one of the derivatives (by/bT)p or ( b ~ / d P at ) ~T = T,(P). Appropriate experimental information for our two polymers required to make similar analyses on HPG is not available.
421
GLASSTRANSITION AND THE GLASSYSTATE
Table 11: Numerical Values for A ( P ) = P , bars
0 400 800
- (ay/i)T)p,
eq 10a
r - A ( P ) X 104, deg-s-PS PoMS
1.36 0.869 0.609
1.82 1.34 1.05
V. Conclusions The statistical hole theory of Simha and Somcynsky can be applied to the liquid-glass transition boundary with the vacancy fraction playing the role of a frozen parameter in a limited sense, Le., remaining constant along the boundary in the case of LPG. For HPG, this fraction is shown to decrease with increasing pressure, since the glassy region is reached from the liquid under different conditions of pressure and temperature. The computed decrease leaves a residual discrepancy between the predictions of the thermodynamic one-parameter theory and experiment. I n order to interpret the equation of state of the LPG within the frame of the hole theory, the vacancy fraction must be permitted to vary with temperature and pressure, converging, however, to a constant value
at the liquid-glass boundary. On a different basis involving kinetic arguments, an analogous conclusion regarding the existence of several internal parameters and the reduction to a single one has been reached by Breuer and RehageaS The usual concept of ordering parameters presumes the existence of quantities which are frozen throughout the glassy region. I n this sense the quantity y, which is constant only along the boundary, cannot be considered as such a parameter. Possibly an appropriate two-parameter theory of the equilibrium liquidla could be formulated to be in accord with the above concept. On the other hand, one might speculate that this concept requires modification. Our approach has been applied to polymer glasses. Since not only the thermodynamic but the statistical relations as well are not specific to high-molecularweight systems, it mould be important to examine lowmolecular-weight glasses along the same lines. Acknowledgment. Acknowledgment is made to the donors of the Petroleum Research Fund, administered by the American Chemical Society, and to the National Science Foundation under Grant GK-20653 for support of this research.
The J O U Tof ~Physical ~ Chemistry, Vol. 76,No. 3, 19W
