ISMAT ABU-ISAAND MALCOLMDOLE
2668
Specific Heat of Synthetic High Polymers, XII,
Atactic and
Isotactic Polystyrene'
by Ismat Abu-Isa and Malcolm Dole Departmsnt of Chemistry and Materkls Research Center, Northwestern University, Evanston, Illinois (Received March 4, 1966)
The specific heats of atactic, amorphous isotactic, and annealed semicrystalline isotactic polystyrene have been measured over the temperature range -50 to 280'. The heat of fusion of 100% crystalline isotactic polystyrene was estimated to be 23.0 2.0 cal. g.-'. The data are interpreted in terms of the encraty function which is c D / Tor dS/dT. The increment in encraty at the glass transition temperature is a function only of the fraction of amorphous content and is independent of the tacticity of the sample. In the liquid range above the melting point there is similarly no difference between the atactic and isotactic samples to be seen within the rather large experimental fluctuations of this work. Equations for the encraty and its temperature variation in terms of the Eyring hole theory of the liquid state are developed. The hole contribution to the encraty, theoretically, would pass through a maximum at the low temperature of 175°K. if equilibrium and the liquid state could persist to that temperature. The Ehrenfest relation seems to be valid for the increments in encraty, compressibility, and thermal volume coefficient of expansion.
*
I. Introduction After making a specific heat study of atactic and isotactic polypropylene,lI2 we became interested in a similar investigation of atactic and isotactic polystyrene. Accordingly, this research was initiated, but on completion of the measurements we learned of a similar investigation by Karasz, Bair, and O'Reilly3 and by mutual agreement decided on simultaneous publication. 11. Experimental A . Materials. The polystyrene samples were kindly supplied without antioxidants or other additives by Drs. R. F. Boyer and F. L. Saunders of The Dow Chemical Co. The annealed isotactic polystyrene was prepared in this laboratory by heating the amorphous (quenched) isotactic sample to 212" and allowing it to cool slowly to room temperature in a stepwise fashion over a period of 72 hr. The properties of the samples are collected in Table I. B. Calorimetric Measurements. In the early stages of this research our calorimetric equipment had to be moved from its original location to a laboratory in the The Journal of P h y s k d Chemistry
new wing of the Technological Institute. We used this opportunity to rewire much of the equipment and Table I: Properties of Polystyrene Samples EP 134-
r
Polymer type
Isotactic Quenched Crystalline
[SI,
toluene at 25O, dl. g. -1 Density, 25O Gradient tube Flotation Relative crystallinity, % X-Ray Density Max. melting point, "C. Molecular weight [VI = 1.10 X 10-4M0.7" Impurities in isotactic sample, p.p.m.
Annealed
2.3 1.056
8.6
Ti 31
0.88
1.072 1.074 40 35.6
9.0
x
Styron 690 Atactic
1.051 1.077 42 236
2 . 4 X 106
106
Fe < 5
0
C1 (100
(1) The previous publication of this series: I. Abu-Isa, V. A. Crawford, A. R. Haly, and M. Dole, J. Polymer Sci., C6, 149 (1964). (2) R. W. Wilkinson and M. Dole, ibid., 58, 1089 (1962). (3) F. E. Karasz, H. E. Bair, and J. M. O'Reilly, J. Phys. Chem., 69, 2657 (1965).
SPECIFIC HEATOF SYNTHETIC HIGHPOLYMERS
2669
to install a new higher capacity metal mercury diffusion pump. The watt-hour meter was recalibrated as before4 using ice in the calorimeter. In Aug. 1963 the calibration yielded an average value of 12.680 cal./rev. as compared to 12.697 obtained in Nov. 1952. Thus, there is no evidence of any drift with time in the readings of the watt-hour meter. The measurement of the electrical energy input is probably by far the most accurate and reliable part of the experimental procedure. As before, the heat capacity of the empty calorimetric system was determined by measuring the total heat capacity of the calorimeter filled with synthetic sapphire, A1203.6 A minor modification was introduced of using just enough aluminum oxide so that its total heat capacity was approximately equal to the total heat capacity of the polymer samples to be studied.
'I8 0.30
I
-40
0
80
40
3
~
120 160 Temp., OC.
200
240
I
280
I
32(
Figure 1. Specific heat of atactic polystyrene: circlw, data of this paper; squares, data tabulated by Warfield and Petree.'
0.65
III. Results A. Speci$c Heats and the Encraty. In Table I1 constants are tabulated of the linear equation C, =
A
+ Bt
(1)
where c, is the specific heat in cal. g.-l deg.-l, t is the temperature in "C.,and A and B are empirical constants. Also listed in the table are the temperature ranges over which the linear equation represents the data within the experimental uncertainty of about ~ 0 . 5 % . Figures 1-3 illustrate the specific heat as a function of temperature of the atactic, amorphous isotactic, and annealed isotactic polystyrene. In Figure 4 the specific heat of a semicrystalline sample of the isotactic polystyrene as received is plotted along with data of Dainton, Evans, Hoare, and Melia.6 The agreement between the values obtained in the two different laboratories is excellent ; however, our values
-20
20
60
Temp.,
100
140
180
220
O C .
Figure 2. Specific heat of amorphous isotactic polystyrene. 1.60
1
1.40
f' 1.20 ti
4
7 1.00 ti 0.80
8 0.60 0.40
Table Il : Constants of the Linear Specific Heat Equation for Polystyrene: cp = A Bt
+
A , cal. deg.-1 B, cal. deg.-2 Polymer
g. -1
g.-1
X 10'
Below the glass transition 1.024 Atactic 0.2595 Amorphous isotactic 0.2599 1.022 Semicrystalline isotactic 0.2654 1.045 Annealed isotactic 0.2570 1.082
Temp. range, "C.
-50t065 -50 to 60 -4Oto70 25 to 75
Above the glass transition and below the melting point Atactic 0.3705 0.607 105 to 275 Annealed isotactic 0,2771 1.160 120 to 175 Annealed isotactic
Above the melting point 0.3506 0.651
245 to 280
60
Figure 3.
100
140 180 220 Temp., OC.
260
300
Specific heat of annealed isotactic polystyrene.
of the specific heat of annealed isotactic polystyrene are somewhat lower than those of Dainton, et al. Another way of expressing the results is in terms of the encraty, L. The encraty was defined by Dole, (4) A. E. Worthington, P. C. Marx, and M. Dole, Rev. Sci. Instr., 26, 698 (1955). (6) D. C. Ginnings and G. T.Furukawa, J . Am. Chem. Sac., 75, 622 (1963). (6) F. S. Dainton, D. M. Evans, F. E. Hoare, and T. P. Melia, P o l y mer, 3, 286 (1902).
Volume 69, Number 8 August 1966
ISMAT ABU-ISAAND MALCOLM DOLE
26 70
et aZ.,1 as the first derivative of the entropy, S, with respect to temperature or as the negative of the second derivative of the free energy, G, with temperature or simply as c,/T; thus (per gram of polymer) L
=
cP/T = (dS/dT), = -(d2G/dT2)
(2)
For comparison purposes, we have plotted in Figure 5 the encraty of the atactic and carefully annealed isotactic polystyrene. I n Figure 6 the encraty of the quenched amorphous isotacric polystyrene is plotted along with that of the atactic. B. The Heat of Fusion. If the total heat required to melt the semicrystalline polymer at the maximum melting temperature T,, U t * , is known, then the true heat of fusion of 100% crystalline polymer at T,, AHfm,can be calculated from the equation AHf" = A H f * / X
r!
t
0*42
I -30
-10
10
30 50 Temp., O C .
70
90
110
Figure 4. Specific heat of semicrystalline isotactic polystyrene: circles, data of this paper; triangles, data of Dainton, et aZ.6 L
(3)
I
where z is the fraction of crystallinity. If we let H , equal the enthalpy per gram of polymer at the maximum melting temperature, T,, then the true heat of fusion at the temperature T,, 242", can be calculated from the equation AHrm = ( H m
- HT i-
S , ~d,TLT
/-T
where HT is the enthalpy per gram at a selected temperature, XT is the weight fraction of crystallinity at that temperature, and C,,L and C,,C are the estimated specific heats of the liquid and crystalline phases of the semicrystalline polymer, respectively. In this work, all the enthalpies of the isotactic polystyrene were calculated with reference to the enthalpy at the absolute zero of temperature using the data given by Dainton, et aL18up to the lowest temperature at which our work began. Thus, in the case of the annealed isotactic sample, we took the enthalpy at 294.11"K. to be 43.28 cal. g.-I and added our measured increments in H to this value to obtain enthalpies at higher temperatures. For the heat of fusion calculations only differences of enthalpies appear in the equation; hence, the reference value is immaterial in this case. For the atactic reference enthalpy we adopted the value 24.59 cal. g.-l a t 219.24"K. taken from the tabulation of Warfield and Petree.' I n order to calculate AHfmfrom eq. 4, it is necessary to know C ~ , Land cP,c in addition to the measured enthalpy difference H m - HT and the degree of crystallinity zT. This treatment also assumes the validity of the two-phase model of semicrystalline polymers. The Journal of Physical Chemistry
9.0 I
1
0
1
100 Temp., O C .
I
1
200
300
Figure 5. Encraty of atactic (solid circles) and annealed isotactic (open circles) polystyrene.
For C ~ , Lwe used the equation for the atactic noncrystalline sample valid above Tg given in Table I1 and for cP,c the equation for the carefully annealed isotactic sample valid below T,. From Table I1 it can be seen that below T, all samples have specific heat equations which extrapolate to values a t 200" that are within 2% of each other. At 0" the maximum spread is 3%. Thus, we feel that the equation adopted for cP,c is reliable. With respect to the equation for c p , ~the specific heat values above the melting point for the atactic and isotactic samples are the same within the rather wide experimental fluctuations (7) R. W. Warfield and M. E. Petree, J. Polymer Sci., 55, 497 (1961).
SPECIFIC HEATOF SYNTHETIC HIGHPOLYMERS
12.0
2X
2671
/ I 1
ll.ol
I
3
5
I
0 a c
8 10.0-
22 .
J
.j
21.
9.0
1
80 I
I
- 40
0
I
40 Temp., OC.
80
120
Figure 7.
100
120
140 160 Temp., "C
180
200
220
AH? calculated from eq. 4.
Figure 6. Encraty of atactic (solid circles) and quenched isotactic polystyrene (open circles).
of about 5%. In fact, none of the data reported in this paper demonstrate any difference between the atactic and isotactic samples below T, or above T,. Hence, for this reason we believe that our use of the equation for the specific heat of atactic polystyrene above T,to represent the specific heat of the amorphous component of the semicrystalline solid above T, is justified. The validity of eq. 4 can be tested by calculating Mimfrom different values of H,. This has been done over the temperature range 100 to 210" and the results plotted in Figure 7. A constant value of AHfmshould be obtained, and this can be seen to be the case from about 115 to 160". Below 115" complications due to the glass transition and above 160" to melting are probably reducing the validity of the equation; certainly the rapid drop from 180" to higher temperatures must be the result of melting. In eq. 4 we assumed xT to be constant at 0.42 (see Table I) which is obviously incorrect in the temperature range of melting. As a result of this treatment we adopt 23.0 cal. g.-l as the heat of fusion of 100% crystalline isotactic polystyrene at the maximum melting point. This value is somewhat higher than that obtained by Dedeurwaerder and Oth,8 namely 19.20 cal. g.-l, obtained by the melting point d e pression by diluents method. The heat of fusion calculations depend on the value chosen for x, in this work 0.42 for the carefully annealed polystyrene. The latter was calculated using 0.9515, the specific volume of our atactic sample, for the specific volume of the completely amorphous isotactic polymer and 0.8968 cm.s g.-I for the specific volume of the completely crystalline polystyrene.8 If we had used
0.9493 for the amorphous component as estimated from the data of Fox and F l ~ r y the , ~ crystallinity of the highly annealed isotactic sample would have been 0.396 and the estimated heat of fusion 24.4. The Dow Chemical Co. sample of crystalline isotactic polystyrene had a crystallinity of 0.40 on the basis of their X-ray estimates while our density data resulted in the value 0.356. If we correct the heat of fusion by the ratio of 0.356/0.40 or 0.89, the value 20.5 cal. g.-l results. These different values of AHf illustrate its dependence on the estimated fraction of crystallinity. However, although there is excellent agreement in the case of polyethylene between the calorimetrically determined AHrand that found by the diluent method, there are several examples of discrepancies between these two methods in the case of other polymers, discrepancies which are much higher than that reported here.l0 Knowing AHfm, the fraction of crystallinity, x, can be calculated at various temperatures above T, by rearrangement of eq. 4. Of course, this calculation will yield 0.42 over the range of temperatures for which A H r m was calculated to be 23.0 cal. g.-l, but it is interesting to see how x changes at other temperatures. The results are plotted in Figure 8 where the variation of x with temperature has the expected shape. We are uncertain as to whether the slight decrease in z as the temperature rises from 170 to 200" is real or not. The apparent decrease in x may be due to slightly incorrect coefficients in the empirical equations used (8) R. Dedeurwaerder and J. F. M. Oth, Bull. SOC. chim. Belges, 7 0 , 37 (1961); J . chim. phys., 56, 940 (1959). (9) T. G Fox and P. J. Flory, J. Polymer Sci., 14, 315 (1954). (10) See the review by M. Dole, FoTtschT. Hochpolymer. Forsch., 2. 221 (1960).
Volume 65, Number 8 August 1566
ISMAT ABU-ISAAND MALCOLM DOLE
2672
for the specific heats of the crystalline and amorphous phases. I n concluding this section on the results it should be pointed out that at high temperatures, 200", our data are about 2% below those of Karasz, et al., with the discrepancy increasing with temperature. This disagreement may result from inaccurate calibration of the empty calorimeter at the high temperature; however, over the range 100 to 200" there is an approximate constant difference of about 2%. Theoretically, there should be no discrepancies because, in the work of Karasz, et al., and this work, synthetic sapphire was the standard substance used for calibration purposes (Karasz, et al., directly measured the heat capacity of the empty calorimeter but checked their calibration with measurements on A403). If time permits in the future, we may repeat a few of our measurements using the same N.B.S. standard sample of atactic polystyrene as that used by Karasz, et al. Except for the differences in the absolute magnitude of the specific heats, our results agree in general very well with those of Karasz, et al.
IV. Discussion A . The Temperature Range below T,. By examination of Figure 5 it can be seen that, within the limits of the fluctuations in the data exhibited there, no significant difference can be seen between the specific heats of the atactic or isotactic polystyrene below the glass transition temperature range. This conclusion is different from that of Wunderlich, Jaffe, and Stahl,ll who compared the specific heats of the 43% isotactic polystyrene obtained by Dainton and co-worker@ with values for atactic polystyrene tabulated by Warfield and Petree' and found a significant difference in the specific heats beginning at about 220°K. and increasing up to the glass transition temperature. However, these differences are probably due to experimental error inasmuch as the Wariield and Petree' tabulated values for atactic polystyrene do not agree with ours below T,; see Figure 1. With rising temperature, specific heat measurements gradually become more difficult to carry out. From a number of lines of evidence polystyrene undergoes some sort of an intramolecular transition in the neighborhood of 50". Wunderlich and Bodily12 observed two small peaks at 53 and 26" in their differential thermal analysis study of atactic polystyrene. A heating rate faster than the cooling rate was required for the observation of these peaks; their heating rates were 3" min.71 to 9" minew1,much greater than the heating rate of the experiments of this research. Kilian and Boueke13 calculated from an XThe Journal of Physical Chemistry
0 80
100
120
140 160 180 Temp., "C.
200
220
240
Figure 8. Crystallinity of annealed isotactic polystyrene.
ray analysis of atactic polystyrene that the Bragg intramolecular interference distance, presumably the distance between the phenyl groups, went through a maximum at about 50". Moraglio, Danusso, Bianchi, Rossi, Liquori, and Quadrifoglio14 studied the 50" transition in both bulk and molecularly dispersed phases from the standpoint of dilatometric, internal pressure, and optical density measurements. They concluded that the transition was the result of intramolecular motions; hence, the evidence is strong that such an effect exists. We were unable to observe anything abnormal in the specific heat measurements in the 50" temperature range, except perhaps a slight change in the slope of the encraty-temperature plot at 50" for annealed isotactic polystyrene. However, the effect is hardly greater than the experimental error. Probably the transition occurs so gradually over a range of temperature that it cannot be detected in specific heat measurements of the usual type. B. The Glass Transition Temperature Range. The encraty-temperature plots illustrate rather well the changes in the entropy-temperature coefficient, (dS/ dT),, as the polymer rises slowly in temperature through the glass transition (Figures 5 and 6). We define the glass transition temperature, T,, as the temperature midway in the rise of the encraty, i.e., the temperature at which AL in the glass transition range is equal to one-half of its total magnitude. Somewhat to our surprise we found that Tg for the annealed isotactic polystyrene was identical with that for the highly amorphous atactic sample. Also to our surprise the quenched, amorphous, isotactic polystyrene exhibited a T , about 9" lower in temperature (11) B. Wunderlich, M. Jaffe, and M. L. Stahl, KoZloid-Z., 185, 65 (1962). (12) B. Wunderlich and D. M. Bodily, J . A p p l . Phgs., 35, 103 (1964). (13) H.G.Kilian and K, Boueke, J . Polymer Sci., 58, 311 (1962). (14) G.Moraglio, F. Danusso, U. Bianchi, C. Rossi, A. M. Liquori, and F. Quadrifoglio,Polyner, 4, 445 (1963).
SPECIFIC HEATOF SYNTHETIC HIGHPOLYMERS
than the atactic sample. The glass transitions and the increments in L at T , are collected in Table 111. Also in Table I11 are values of z and AL/(1 - z) in order to see if the increment in L is directly proportional to the fraction of amorphous polystyrene. For Table III: Glass Transition Temperatures of Polystyrene and Encraty Increments at T,
Polymer
Atactic Quenched isotactic Annealed isotactic
r,,
AL, oal. dag. -2 g. -1 oc.
93 84 92
x
104
1.77 1.69 1.04
2
0 0.086 0.420
AL/(l - 21, cal. deg. -2 g.-' x 10'
1.77 1.85 1.79
this table we have computed x in terms of the equation
and as before taken the measured v of the atactic sample, at 25", 0.9515, for v, instead of the value 0.9493 cm.3 g.-1 quoted by Dedeurwaerder and Oth* from the work of Fox and F10ry.~ Considering the uncertainty in estimating AL from the data, the constancy of the ratio AL/(1 - x) is rather remarkable. At any rate this study demonstrates that there is no essential difference in this respect between the atactic and amorphous isotactic polystyrenes and also shows that the properties of the amorphous component with respect to AL are not affected by the presence of a crystalline phase, at least up to about 40% crystallinity. This conclusion agrees with that recently announced by O'ReiUy and Karasz.aJ5 However, T, is different for the amorphous isotactic and atactic samples. T , for the isotactic is 9" lower than the atactic T,; this result is in the same direction as the difference in T , between the isotactic and atactic polymethacrylates tabulated by Boyer16 from data of Shetter. l7 As the amorphous isotactic polystyrene becomes partially crystalline, T , rises so that at 40% crystallinity the T , values for the atactic and isotactic samples coincide. The increment in hL at T , from Table 111 can be taken to be 1.8 x 10-4 tal. deg.-2 g.-l at 920. ~c~ is Of converted to a "p Per equal to 3.4 cal. deg.-' mole-'. Some years ago, Wunderlich'8tabu'ated many s i d r increments for a variety of glass-forming substances and found that ACD values averaged to 2.7 f 0.5 cal. deg.-l mole-l. Our value of 3.4 is somewhat higher than this average. Prigogine and Defaylghave derived the equation
2673
A@ - = - AaV Aa AL
(6)
where A a , Ap, and AL are called the configurational thermal expansion, the configurational compressibility, and (in our language) the configurational encraty. This relation may also be obtained by combining two equations of Ehrenfest and indeed is called the Ehrenfest equation for a second-order phase change.20 A recent thermodynamic discussion of eq. 6 is that of Goldstein.21 It is important to know whether eq. 6 is verified or not. In the case of atactic polystyrene we now have accurate data for AL, A a , and V ; the chief uncertainty is in the value of A& Hellwege, Knappe, and Lehrnannz2have published many curves showing the variation of the volume of atactic polystyrene with temperature and pressure. They quote Ap as being equal to 8.0 X cm.2/ktorr (or 2.51 X 10-4 ~m.~/cal.)at T,. However, the compressibility begins to rise rather rapidly as T , is approached, and if this'rise in p is added to the discontinuous rise in p at T, as Goldstein has done, Ap becomes equal to 6.2 X ~m.~/cal.Calculated values for the leftand right-hand side of eq. 6 are given in Table IV. Because of the considerable uncertainty in the value of &, it is impossible to come to an exact conclusion regarding eq. 6, but it does seem to be nearly verified, Table IV : Test of Eq. 6 for Atactic Polystyrene" Author of Apvalua
AB, om.a/cal.
deg. cm.a/cal.
VAa/AL
Hellwege, et al. Goldstein
2.51 X 6.20 x 10-4
0.81 1.99
1.67 1.67
a
1.8
A a = 3.11 X
x
AB/Aa,
deg.-l; lo-* cal. deg.-z g.-1.
V
=
0.966 cm.a g.-l;
AL =
C . The Temperature Range between T , and T,. We now turn to a consideration of the specific heats and encraties between T , and the melting range. In the case of the atactic sample, L decreases slowly (16) J. M.O'ReUy and F. E. Karasa, 148th National Meeting of the American Chemical Society, Chicago, Ill., Sept. 1964. (16) R. F.Boyer, Rubber Reu., 36, 1303 (1963). (17) J. A, Shebkr, J . polyner &i., B1, 209 (1963). (18) B. Wunderlich, J . Phys. Chem., 64, 1052 (1960). (19) I. Prigogine and R. Defay, "Chemical Thermodynamics," translated by D. H. Everett, Longmans, Green and CO., London, 1954, 297* (20) See ref. 19,p. 306. (21) M. Goldstein, J. chenz. Phys., 39, 3369 (1963). (22) K. H. Hellwege, W. Knappe, and P. Lehmann, Kolloid-Z., 183, 110 (1962).
Volume 69,Number 8 August 1966
ISMAT ABU-ISAAND MALCOLM DOLE
2674
and nearly linearly from a peak at about 100" up to the highest temperatures studied, but the fluctuations in the data become greater, the higher the temperature. The other polystyrene samples also exhibit a peak in L at about 100" except for the amorphous isotactic sample in which L has a maximum about 10" lower in temperature. The isotactic samples do not exhibit as pronounced a peak as does the atactic, as is to be expected. In terms of the Einstein specific heat function, the encraty for a single vibrational mode per molecule (at constant volume) is equal to2a k-2
z3ez hv (eZ - 1)2
(7)
where k and h are the Boltzmann and Planck constants, v is the vibrational frequency, and z is equal to hv/kT. The variation of L with temperature at constant volume is
which goes through a maximum when 3 z - ze' -e"-l-2+2
(9)
For the maximum, z is equal to 2.576. Knowing the latter and the temperature of the peak (loo"), we can calculate the frequency that, according to the above equation, would give such a maximum if it were the only frequency operative. The result turns out to be 668 cm.-1. This means that the dominant f r e quencies determining the specific heat up to this temperature have wave numbers equal to or less than 668. The G H bending and the C-C stretching frequencies equal to lo00 cm.-1 give L values that peak about 285". Thus, the dominant frequencies in the glass transition range are the acoustical frequencies and the chain twisting and bending frequencies. The reason that L decreases as T rises above the peak temperature is that the cumulative L for the lower frequencies decreases faster than the L for the high frequencies increases. The peak in L is not to be construed as the peak due to the Einstein function. For the low frequency chain twisting and bending motions the peaks come at a much lower temperature than at loo", in fact, a t 40°K. as demonstrated by the data of Dainton, Evans, Hoare, and MeIh6 According to the Eyring hole model of the liquid state as developed by Hirai and E y r i ~ ~for g ~the ~ glass transition, the rise in cp at Tgresults from the requirement of supplying energy to form the greater number The Jour& of Physical Chemistry
of holes at the higher temperatures. Hirai and Eyring's equation for the hole contribution to the heat capacity per mole of repeating units as expressed in the form given by Wunderlichl8is ACp = E=--Eh exp(- @T) RT2
where E , is the internal latent heat of vaporization at T,, approximately 7320 cal. mole-1 of repeating units as deduced by Wunderlich from molar cohesion energy data, and €h is the "molar excess energy over the no hole" configuration. If eq. 10 is divided by T to obtain the hole encraty, A h , and if the latter is then differentiated with respect to T to obtain its temperature coefficient, we find
which demonstrates that the hole contribution to the encraty will go through a maximum at eh/RT equal to 3. I n this work a maximum in the encraty appeared at about 102"; see Figures 5 and 6. This is not the maximum predicted by eq. 11 because eq. 11 is based on the assumption of equilibrium which does not exist in the glassy state. However, it is possible to calculate the temperature where the maximum in L due to the Eyring hole mechanism would occur provided €h is known. From the jump in the specific heat at T,, €h can be calculated from eq. 10. We found 1044 cal. mole-' which is slightly less than that estimated by Wunderlich from specific heat data on low molecular weight polystyrenes. Inserting this value for €h into eq. 11, the value of T necessary to make eh/RT equal to 3 iS only 175"K., far below T,. This calculation assumes that €h remains constant with temperature, but a lower value of €h would make T, still lower. This value of 175°K. is approximately the temperature where the rise in entropy with temperature due to the Eyring hole mechanism would have its greatest value if equilibrium could be attained at all temperatures and if the material were in the liquid state at that temperature. One could conceive of this temperature as being close to that temperature below which the configurational entropy of the liquid due to holes approaches zero. Our value of Eh is much less than values estimated by Hirai and Eyring who deduced, by assuming a universal value for the fraction of free volume at T , that Eh,/RTg is approximately equal to 4. In our case this ratio is more nearly equal to 1.5. Wunderlich has B. Wunderlich, ikfakromoi. Chem., 34, 29 (1959). (24) N.Hirai and H. Eyring, J . Polyner SCi., 37, 51 (1959).
(23) M. Dole and
SPECIFIC HEATOF SYNTHETIC HIGHPOLYMERS
tabulated many values of this ratio ranging all the way from 1.5 for polyvinyl chloride to 4.6 for ethyl alcohol. Most polymers have values around 2.0. D. The Gibbs-DiMarzio Theory. It is interesting to calculate, as Karasz, et u Z . , ~ have done, the temperature, T2,to which the glass transition would have to be lowered so that the entropy per gram of the liquid phase becomes equal to that of the crystal. This is the temperature at which the liquid should have a zero configurational entropy according to the theory of Gibbs and DiMarai0.2~ Recently, Adam and GibbsZ6have shown how the Gibbs-DiMarzio theory can be adapted to give a theoretical explanation of the Williams, Landel, and Ferry empirical equation by adapting a value of T , - T2equal to about 50". As stated above, T z is to be considered the temperature at which the entropy of the supercooled (but not glassy) liquid becomes equal to that of the 100% crystalline phase. This temperature is readily calculated from the equation
where ASrm is the entropy of fusion a t T, of the 100% crystalline phase in units of cal. g.-l deg.-l. The other symbols are the same as those of eq. 4. Using the same values of c p , and ~ cP,c as used in the calculation of the heat of fusion, we find TZto be 261°K. which is 105" below T,. This value confinas the conclusion of Karasz, et aL, that T g - TZas estimated from calorimetric data is considerably greater than the 50" average found in the treatment of Adam and Gibbs. The ratio T$T2 is 1.40, which is close to several values
2675
of this ratio2' for other substances tabulated by Adam and Gibbs, hevea rubber, glycerol, and propylene glycol, for example. Bestul and Changz7have pointed out that if the jump in specific heat per mole of chain beads a t T, is constant1* and if ASo, the excess entropy at T,, is also a universal constant as demonstrated by Bestul and Chang (within a factor of about 2), then T J T z must be a universal constant (assuming Acp to be independent of temperature between T, and Tz). Thus, our ratio of 1.40is in line with the observations of Bestul and Chang and of Adam and Gibbs, but the difference, T , - Tz, is too large for agreement with the Adam and Gibbs average value by about a factor of 2. Obviously, if T J T Z is a universal constant, T , - TZcannot be because T, varies widely from substance to substance. Acknowledgments. We are indebted to Drs. R. A. Boyer and F. L. Saunders of the Dow Chemical Co. for supplying us with the samples of polystyrene and for X-ray crystallinity and other data for the samples. We are grateful to Dr. F. E. Karasz and co-workers of the General Electric Co. for an opportunity of seeing their data in advance of publication and for fruitful discussions with Dr. D. M. Bodily and Dr. F. E. Karasz. This research was supported by the Advanced Research Projects Agency of the Department of Defense through the Northwestern University Materials Research Center. (26) J. H.Gibbs and E. A. DiMawio, J. Chem. Phys., 28, 373,807 (1968). (26) G. Adam and J. H. Gibbs, ibid., 43, 139 (1965). We are indebted to Profeasor Gibbs for the opportunity of seeing this paper in advance of publication. (27) A. B. Bestul and 9. S. Chang, ibid., 40, 3731 (1964).
Voluma 69,Number 8 Augwt 1966
