Some time ago, Nanda and Simha (1) dis- cussed equation of state data on n-paraffins. (2) and on several amorphous and crystalliz- able polymers (3) from two ...
Kolloid-Z. u. Z. Polymere 2 5 1 , 4 0 2 - 4 0 8
(1973)
.From the Division o/Macromolecular Science, Case Western Reserve University, Cleveland, Ohio 49106 (U.S.A.)
Pressure-volume-temperature properties of amorphous polymers: empirical and theoretical predictions By Robert Simha,
P. S. W i l s o n ,
and Olagolce O l a b i s i
With 6 figures and 5 tables
1. Introduction Some time ago, N a n d a and S i m h a (1) discussed equation of state data on n-paraffins (2) and on several amorphous and crystallizable polymers (3) from two points of view. One, in order to explore the validity of a principle of corresponding states and to obtain a reduced p - V - T surface. The other was to examine the position of the cell theory (4) at elevated pressures, examined previously in detail at atmospheric pressure (4, 5). Since that time, a hole theory has been formulated (6) which describes successfully the reduced isobar for zero (i. e., atmospheric) pressure. Furthermore, accurate isotherms for two polymers, polystyrene and poly(omethyl styrene) have been obtained recently and found to be in good accord with theoreticyl predictions (7). The purposes of this paper are three-fold. In the analysis of experimental data in refs. 2 and 3, and other data as well, the usefulness of the essentially empirical Tait equation with the single temperature dependent parameter B has been extensively demonstrated (1, 2, 7, 8). However, _Nanda and S i m h a (1) committed an oversight and misinterpreted the definition of the tabulated relative volume changes in ref. 3. Hence the parameter B must be recomputed at all temperatures and for all polymers studied by Hellwege et al. (3). The revised results will be expressed in terms of simple exponential relationships between B and T. Secondly, we compare experimental p - V - T data in the liquid state with the theoretical isobars and isotherms and evaluate the eharacteris: tic pressure (p*), volume (V*) and temperature (T*) parameters. Finally, we present in graphical form a series of reduced isochores and a simple interpolation formula for the
(ReceivedAugust 28, 1972)
atmospheric pressure isobar, in order to facilitate future applications of the theory. The equation of state and related configurational thermodynamic functions in the hole theory do not appear as respective single equations, as for example, in the cell theory. Instead, the two equations involving the variables of state and the hole fraction must be considered simultaneously because of the extremum condition for the latter. Atthough the requisite results can be readily obtained by numerical methods, the diagrams to be presented will make comparisons between experiment and theory and theoreticalpredictions considerably more convenient.
2. The Tait relation The expression to be considered is (1, 2, 7, 8): 1 - Y(~, T)/V(O, T) = C • In [1 + p/B(T)] [1]
with C = 0.0894. Although some theoretical rationalization of the behavior of C and B can be given (9, 10), eq. [1] must be considered as essentially empirical. We shall recompure B ( T ) from Hellwege et al.'s (3) data on polystyrene, poly(methyl methacrylate) and poly (vinyl chloride), but postpone a reconsideration of the results for a high pressure and a low pressure polyethylene, also investigated by the above authors. The results in ref. 1 must be revised, because the tabulated entries (3) of [V (200, T) - - V (]9, T ) ] / V (0,293) were misread as the volume changes relative to V (0, T), which enter into eq. [1]. In this manner, the original table 1 (1) is now replaced by table 1 below. Again, B is satisfactorily independent of p (although ~ trend is noticeable in the last column) and an average value can be defined.
Simha et al., Pressure-volume-temperature properties o/amorphous polymers Table 1. Variation of Tait parameter B (kbar) with pressure for high polymer liquids p/0.9806 (Bar)
Polystyrene t = 162.3 ~
200 400 600 800 1000 1200 1400 1600 1800 2000
1.22 1.21 1.21 1.23 1.24 1.24 1.25 1.25 1.25 1.24
Poly(methyl methaPoly(vinyl crylate) chloride) t = 129.7~ t=97~ 1.62 1.62 1.62 1.61 -------
1.95 1.98 2.00 2.03 2.05 2.05 2.07 2.08 ---
403
T h e differences b e t w e e n t h e results o f t h e t w o e q u a t i o n s for p o l y s t y r e n e do n o t e x c e e d 4 . 5 % (at 195.6 ~ A l t h o u g h t h e discussion in s u b s e q u e n t sections will be c o n c e r n e d w i t h t h e liquid s t a t e solely, we i n c l u d e for t h e sake o f c o m pleteness t a b l e s 3 a n d 4 p e r t a i n i n g t o t h e g l a s s y state. T h e f o r m e r illustrates a r e a s o n able c o n s t a n c y o f B. T h e l a t t e r s u m m a r i z e s t h e r e v i s e d a v e r a g e d B's. T h e following r e l a t i o n s m a y be given, a l t h o u g h t h e r e are only three temperatures available at best: Polystyrene: B = 3337 exp (--3.30 • 10-~t)
Table 2 exhibits the averaged values of B in t h e l i q u i d t e m p e r a t u r e r a n g e o f t h e t h r e e p o l y m e r s . T h e l o w e r r a n g e o f t e m p e r a t u r e is l i m i t e d b y t h e o n s e t o f t h e glass t r a n s i t i o n process a t e l e v a t e d pressure, w h i c h affects
[3a]
Poly(methyl methacrylate) : B = 3717 exp (--3.96 • 10-~t)
[3b]
Poly(vinyl chloride) : B = 3751 exp (--2.4I X 10-~t)
[3el
Table 2. Averaged experimental and computed Tait parameters B (bar) in the liquid range Polystyrene, t ~ Experimental B Eq. [2a]
248.9 851.3 868
229.0 959.3 943
202.8 1067 1051
Poly(methyl methaerylate) t ~ Experimental/~ Eq. [2b]
139.3 129.7 119.8 1506"(5) 1618"(4) 1717"(2) 1510 1610 1721
Poly(vinyl chloride, t ~ Experimental B Eq. [2cI
97.0 90.6 82.0 2025*(8) 2128"(5) 2207*(2) 2035 2110 2215
1 7 8 . 7 162.3 1161 1232 1161 1243
145.2 1325 1334
136.7 126.0 115.4 1385"(7) 1438"(5) 1519"(3) 1382 1445 1509
* Parentheses indicate number of data points available for averaging. All data were used in other cases. the behavior of the B-parameter over a range of several hundred bars at a given temperat u r e (7). Also s h o w n are v a l u e s c o m p u t e d b y m e a n s o f t h e following e q u a t i o n s : Polystyrene: B = 2435 exp (--4.144 X 10-~t)
[2a]
Poly(methyl methacrylate) : B = 3850 exp (--6.720 • 10-3t)
[2b]
Poly(vinyl chloride) : B ~ 3522 exp (--5.654 X 10-~t)
[2 c]
where t is the temperature in degrees Celsius. A v e r y s a t i s f a c t o r y r e p r e s e n t a t i o n results f r o m eq. (2) w i t h m a x i m u m d e v i a t i o n s o f 1 . 9 6 % , 0 . 0 0 5 % , a n d 0 . 8 % , r e s p e c t i v e l y , for the three systems, and the largest deviation a t 248.9 ~ F o r a p o l y s t y r e n e s a m p l e w i t h a s l i g h t l y h i g h e r glass t e m p e r a t u r e , Quach a n d S i m h a (7) o b t a i n e d t h e r e l a t i o n B = 2169 exp (--3.31, • 10-~t)
[2']
W e n o t e t h a t for p o l y s t y r e n e a t 21 ~ B - - - - 3 2 5 0 b a r w a s o b t a i n e d earlier (7), c o m p a r e d w i t h 3124 b a r a t 20 ~ in t a b l e 4. Similarly, B = 2888 b a r a t 64.1 ~ (7), as c o m p a r e d w i t h 2733 b a r at 60.5 ~ here. W e c o n c h l d e , a t least for n o t t o o e l e v a t e d t e m p e r a t u r e s , t h a t t h e Tait r e l a t i o n w i t h a u n i v e r s a l v a l u e o f C, is q u i t e s a t i s f a c t o r y . Table 3. Variation of Tait parameter B (khar) with pressure for high polymer glasses at 20 ~ p/0.9806 (Bar)
Polystyrene
200 400 600 800 1000 1200 1400 1600 1800 2000
3.15 3.15 3.13 3.11 3.13 3.13 3.13 3.12 3.11 3.08
Poly(methyl Poly(vinyl methachloride) crylate) 3.48 3.46 3.44 3.45 3.44 3.43 3.42 3.42 3.41 3.40
3.56 3.58 3.61 3.61 3.59 3.58 3.58 3.57 3.55 3.53 if6*
Kolloid-Zeitschri/t und Zeitschri/t /'~r Polymere, Band 251. Heft 6
404
Table 4. A v e r a g e d e x p e r i m e n t a l values of Tait param e t e r B (bar) in t h e glassy range Temp. (~
20.0
51.0
60.0
60.5
100.0
Polystyrene Poly(methyl methyerylate) Poly(vinylchloride)
3124
--
--
2733
--
3435 3575
-3318
2931 --
---
2503 --
More recently its applicability to argon and methane has been demonstrated in detail (11).
log~ 0.05
3. Theoretical equation of state
The hole theory yields the following equations in terms of reduced variables of state =
p/p*, ~
=
257/~' = [1 - - 2-~l~
v/v*, T
=
~/T*:
-1
+ ((2y/T) (yV) -2 [1.011
(y?)-2 - 1.2045].
[3]
The equation to be satisfied b y the hole fraction (1 -- y) for the infinite polymer is (6): (s/3c) [1 + y-~ln(1 - - y)] = ( y / 6 T ) ( y ? ) - ~ • [2.409 -- 3.033(yV) -2] + [ 2 - 1 / 6 y ( y V ) - l / 8 - 1/3] X [1 -- 2-~16y(yV)-l/'] -1
[4]
where s/3c is t h e n u m b e r of external degrees of freedom per chain segment.
The results for different values of this ratio are superimposable (6) and we adopt s/3 c = 1. This has to be kept in mind when the characteristic parameters for different polymers are compared with each other. Equation (4) represents the minimization condition for the Helmholtz free energy. Eq. [3] and [4] will now be applied to the experimental data of Hellwege et al. (3). The procedure to be followed is to obtain the volume and temperature parameters V* (cc/g) and T* (~ b y a comparison of the experimental isobar at atmospheric pressure with eq. [r for p ---- 0, where y is obtained b y numerical solution of eq. [4]. The results are shown in fig. 1, where the solid line represents the theory on which experimental points are superimposed. To extend the range of coordinates, we have included experimental data for an isotactic poly(methyl methaerylate) (12) and an oligomer, n-docosane (12, 13). Thus V* and T* are now known. These results confirm once more the very good agreement observed earlier (6, 7). Next, appropriate isotherms are computed by the solution of eq. [3] and [4] for a series of reduced temperatures. The experimental
0"00
rl
-i.5
i
[
-1,# lo~ "T -'I,3
i
-~,2
Fig. i. R e d u c e d volume as a function of reduced temperature. Line, eqs. [3] a n d [4] for 25 --> 0 a n d (s/3c) ~ 1. Points, e x p e r i m e n t a l a t a t m o s p h e r i c pressure. Polymers: crosses, styrene (3); open triangles, atactic met h y l m e t h a c r y l a t e (3); open squares, vinyl chloride (3); solid triangles, isotactic m e t h y l m e t h a e r y l a t e (12). Oligomer: solid a n d open circles, n-docosane (12, 13)
compressibility factors can be cast at this stage in the semi-reduced form p [/T. The ratio between these and the theoretical reduced compressibility factor, eq. [3], yields p*. Depending on the experimental accuracy and the adequacy of the theory, these ratios Should be independent of V at any T. In practice we m a y take a single or several isotherms over a range of reduced densities and compute an average p* (7). The final results of these computations are shown in figs. 2 and 3. The most extensive experimental information is available for polystyrene and it will be seen that theory and experiment are in good accord, except for the highest temperature isotherm, (A). The agreement for isotherms B - H is comparable with that obtained for polystyrene and poly(orthomethyl) styrene b y Quach and Simha (7), whose measurements extended only to 195.6 ~ The discrepancy for isotherm A (248.9 ~ is much larger than in any material previously studied and may be due to either experimental reasons or inadequancy of the theory at elevated temperatures and pressures. The results for poly(methyl methacrylate) are exhibited in fig. 3. The agreement between theory and experiment is quite saris1) W e t a k e this o p p o r t u n i t y to correct ~ mislabelling of t h e abscissa in fig. 13 of ref. 7. T h e n u m b e r s should r e a d from 0.93 to 1.04 r a t h e r t h a n t h e original 0.92 to 1.03.
Nimha et al., Pressure-volume-temperature properties o] amorphous polymers
.92
.94
.96
.98
1,00
than to minimize the deviations b y empirical means. The Success of the theoretical equation of state in its present formulation and without recourse to additional empirical parameters or adjustments, is gratifying. We have previously commented on its deficiencies in respect to temperature derivatives, the internal pressure in particular (7). The thermal expansivity is considered in the next section. In table 5 are summarized the characteristic parameters derived from the measurements in ref. 3, to which are added those for polymers of styrene, o-methylstyrene (7), and isotactic methyl methaerylate (12). I t should be noted that the differences in the p* for the two polystyrene samples are now reduced, as was to be expected from the corrected results in Section 2; these corrections do not affect V* and T*. However, we have slightly modified the values given originally [see table 7, ref. (7)]. According to the theoretical definitions of the reduction parameters (4) and considering that V* is expressed in table 5 as a specific volume, the characteristic compressibility factor for infinite chains should obey the relation
1.02
V~
Fig. 2. Reduced compressibility factor as a function of reduced density. Lines, eqs. [3] and [4]; points, styrene (3), between 115.4 and 248.9 ~
q.O,
PMMA
7=.o3~6g~_ Z ~03389
~--.0.~27~ //
PVC
2.0 r 0.0995
.96
.97
.98
.99
"I.00
~/~
.97
".90
.9g
"1.00
405
]lO?
Fig. 3. Reduced compressibility factor as a function of reduced density. Lines, eqs. [3] and [4] ; methyl methacrylate, between i09.3 and 139.3 ~ (3); vinyl 6hloride, between 82.0 and 97.0 ~ (3)
factory, with a maximum deviation in the reduced compressibility factor of about 5%. The theory is also in good agreement with the limited data for poly(vinyl chloride). I t is clear from these results that improved fits could be obtained b y means of adjustments in the characteristic pressure io* with temperature. However, our aim here is to explore the position of the theory rather
p* V*/T* = l?(c/s)/Mo
[5]
where M o is the molecular weight of the segmental unit. I t is characteristic of the hole theory, in contrast to the cell theory, that the "flexibility" ratio c/8 appears explicitly in the equation of state, eqs. [3] and [4], even for the infinite polymer. No allowance for possible differences between different polymers has been made, and c/s = 1/3 has been assumed throughout. As is indicated in table 5 of ref. 6, the choice of c/s affects primarily T* through an inverse relationship. tIence we have obtained apparent values of the characteristic parameters and differences in T* have been made to reflect variations in characteristic segmental energies solely. In the units chosen, eq. [5] then reduces to (I~*V*/T*) M0 = 27.7
[5']
It is seen from table 5, that this relation is approximately obeyed for poly(methyl methacrylate) and the styrene derivatives, if the average molecular weight per backbone carbon is substituted for M 0. For polyvinyl chloride, on the other hand, a molecular weight of about 4t would have to be used. Although one can not make too much of this observation, it is qualitatively in the right direction, because of the comparatively small size of the vinyl chloride unit. We have
406
Kolloid.Zeitschri]t und ZeitschriJt ]i~r Polymere, Band 251 9HeJt 6
Table 5. Characteristic parameters for high polymers Polymer Poly(o-methyl styrene) (7) Polystyrene (3) Polystyrene (7) Poly(methyl methacrylate) (3) Poly(vinyl chloride) (3) Isotactic poly(methyl methaerylate) (12)
p* (bar)
V* (ce/gm)
T* (~
Tg (~
7,458 7,638 7,453 9,303 10,345
.9762 .9625 .9598 .8350 .7105
12,740 12,700 12,680 11,890 11,320
404 362 365 376* (386)** 349
33.8 30.1 29.4 32.7 20.3
10,088
.8160
11,170
320
36.9
(p*V*/T*) M o
* From interseetion of V -- T lines at atmospheric pressure. ** From extrapolation of Tg -- p line to atmospheric pressure, included in table 5 t h e glass transition t e m peratures, Tg, and a parallel t r e n d with T* will be discerned. Actually, t h e ratios T g / T * v a r y f r o m 2.8 to 3.2 • 10 -2. Strict c o n s t a n c y would i m p l y Tg to be a corresponding t e m perature, and as a consequence a c o n s t a n c y of the p r o d u c t ~ T at T = Tg (14).
4. N u m e r i c a l R e s u l t s
The theoretical isobar for /5 ~ 0 in fig. 1 can be represented b y a simple i n t e r p o l a t i o n equation. An a p p r o p r i a t e least square analysis leads to the e q u a t i o n In V = -0.10335 + 23.8345 • ~m or In V = --0.10335 + (2/3) a • T;
1.65 < r~ X 10e < 7.03
[6]
with a s t a n d a r d deviation of 0.16 ~ where represents the t h e r m a l e x p a n s i v i t y (I/V) (OV/OT)~o. A similar equation, based on an empirical m a s t e r curve o b t a i n e d (5) prior to the f o r m u l a t i o n of t h e t h e o r y , was given earlier (15). T a k i n g into consideration the t r a n s f o r m a t i o n s between the different coordinate systems e m p l o y e d (5, 6), this equation would r e a d In V = -0.1069 + 24.757 • ~3/~
theory, in spite of the latter's b e t t e r prediction of numerical values. This indicates, as has already been p o i n t e d out in connection w i t h the internal pressure (7), t h a t an improv e m e n t is to be sought p r i m a r i l y in the t e m p e r a t u r e derivatives of the configurational p a r t i t i o n function. Otherwise, and as an empirical device, t h e formal a g r e e m e n t in c~ can be m a d e perfect b y slight a d j u s t m e n t s in V* a n d T*. This was the early p r o c e d u r e in applying the t h e o r y of the Brussels school (4) to liquid h y d r o c a r b o n s (17). I t will be recognized, of course, t h a t the prediction of second derivatives of extensive t h e r m o d y n a mic quantities places quite a strain on an analytical t h e o r y of the liquid state. Eq. (6) represents in a n y ease a useful i n t e r p o l a t i o n formula, which permits a r a p i d estimation of the v o l u m e and t e m p e r a t u r e parameters, or at least of their averages, b y means of a plot of In V vs. Ts/~. A b o v e t h e u p p e r limit indicated, the e x a c t In ~ increases more r a p i d l y t h a n according to eq. [6]. H o w e v e r , t h e l a t t e r should be satisfactory in a range of a c t u a l t e m p e r a t u r e s from a b o u t 273 to 573 ~ for a value of T * - 1.2 x 104. Fig's. 4 and 5 exhibit a series of r e d u c e d isoehores in density intervals of 0.01. T h e t e m p e r a t u r e range is selected so as to en7
6 J I j J r j / showing the close a g r e e m e n t with eq. [6]. 5 t f f Eq. [6] predicts an increase of ~ with temJJ1 J j jt p e r a t u r e , as has been actually observed b y several authors for b o t h high a n d low moleFj f.-1 . j . / ~J cular weight liquids. I n connection with a T 3 ~. careful c o m p a r a t i v e analysis of the e q u a t i o n 2 /f ~ j of state of n-alkanes, Patterson and B a r d i n (16) h a v e considered the relationship bet w e e n the two dimensionless quantities 3.0 3.1 3.2 3.3 3.~ 3.5 3.5 3.7 3.8 3.9 /4.0 14.1 - - d ( ~ - i ) / d T and ~ T. T h e cell t h e o r y yields a positive slope in such a plot which is in qualitative accord with e x p e r i m e n t and Fig. 4. Reduced theoretical isochores, low to moderate opposite to t h a t p r e d i c t e d b y the hole densities with increments of 0.01
407
Simha et al., Pressure-volume-temperature properties o/ amorphous polymers
0"30I
0.2@ • eoa
101
0
3,1
3.2 3.3 3..9 3.5 3.6 3,7
yxlO2
3.8 .3.9 4.0 &l
Fig. 5. Reduced theoretical isochores, moderate to high densities with increments of 0.0t
compass the liquid range of polymers with glass transitions between about 50 and 150 ~ at atmospheric pressure, whereas t h e densities correspond to pressures up to about 2 kbar. These curves were constructed as cross plots of isotherms computed by means of eqs. [3] and [4]. A combination of densitytemperature measurements at atmospheric pressure and a single datum at elevated pressure provide an estimate of the eharacteristic p* with the aid of these graphs, and thus at least an approximate equation of state. A rougher estimate, requiring only atmospheric pressure data, results from an interpolation by means of the last column in table 5. I t would be possible, of course, to derive simple polynominal relationships for the isochores, but this would not appear to be too fruitful. Finally, we revert to the T a i t relation eq. (1), which m a y alternatively be written in reduced coordinates. Theoretical expressions for the two parameters C and B, based on the cell model with a square well cell potential, have been previously derived by evaluating the expressions (9) P-oC = =
_
(o#IOP)T/(o2~/o?~)~ ; -(Or
• ?oC - f~.
W]
I t could be shown t h a t C is indeed a but slowly varying function of T. I t is possible to carry out the analogous computation on the basis of the hole theory, eqs. [3] and [4]. In view of the satisfactory agreement shown in figs. 2 and 3 and previously (7), we feel justified in omitting this effort. Instead we calculate an apparent /?, b y assuming the constant yalue of C ----- 0.0894 and employing the theoretical equation of state. The result, fig. 6, shows t h a t B is exponentially dependent on T, in accord with the form of eq. [2].
9
30
I 32
9
3#
I , I 36 38 7 x 10 3
/40
I 0-2
t 4~z/-
:Fig. 6. Reduced Tait parameter /~ as a function of reduced temperature T. Solid line computed from eqs. [3] and [4] with C = 0.0894. Solid circles, polystyrene (3); crosses, poly(vinyl chloride) (3); open triangles, poly(methyl methacrylate) (3); open circles, polystyrene (7); solid triangles, poly(o-methyl styrene) (7)
Also indicated are the experimental values obtained in Section 2, which have been reduced by the pressure and temperature parameters obtained from the theory and listed in table 5. Considering the problems of experimental and numerical accuracy in the evaluation of empirical and theoretical expressions, the scatter around the solid line is all one m a y expect as an expression of a principle of corresponding states. Acknowledgements
Acknowledgement is made to the National Science :Foundation under Grant GK-20653 and the donors of the Petroleum Research Fund, administered by the American Chemical Society, for support of this research. We also wish to express ore' appreciation to )~Ir. D. Nutter for his assistance as a Marbon Fellow in the Division's summer honor undergraduate research program. Notes added in proo/
1. The characteristic parameters for several polymers and oligomers, and a correlation between the characteristic compressibility factor p * V * / T * and T* over a wide range have been established1). 2. A detailed analysis of experimental and theoreticM thermal expansivities at atmospheric pressure for a variety of polymers and oligomers has been carried
out2).
Summary
Equation of state data on several amorphous polymers, viz., polystyrene,poly(methylmethacrylate) and poly(vinyl chloride) are considered from two points of view. :Firs~ in terms of the empirical Tait equation applied to the liquid as well as the glassy state with a single disposable and temperature dependent parameter. These results supersede and extend an earlier 1) O. Olabisi and R. Simha, in preparation. 2) p. S. Wilson and R. Simha, in preparation.
408
Kolloid.Zeitschri/t und Zeitschrifl liar Polymere, Band 251 9Heft 6
analysis. Good agreement between independent investigators is obtained for polystyrene. Secondly, the hole theory of Simha-Somcynslcy is applied to the liquid state of these polymers. The isobar at atmospheric pressure and the compressibility factor p V / T up to pressures of about 2 kbar are analyzed in detail and the numerical values of the characteristic volume (V*), temperature (T*), and pressure (p*) parameters are established. The evaluation of the theoreticM expressions requires the numerical solution of a transcendental equation for the hole fraction as a function of the reduced volume and temperature, which arises from the extremum condition on the partition function. At atmospheric pressure, this computation is avoided and the evaluation facilitated by means of a simple and accurate interpolation formula for the theoretical volume-temperature relation. I n this manner, V* and T* are readily obtained. A series of theoretical reduced isochores are presented in graphical form, which encompass a change in density of 15% and a temperature range of at least 200 degrees. A single pressure datum then provides an estimate of io* and thus of the complete equation o~state. Finally we note that the reduced Tait parameter B, as computed from the theory, exhibits the exponential variation with temperature, observed experimentally.
Zusammen]assung Die Zustandsgleiehung einer ~eihe yon amorphen Polymeren, n~m]ieh Polystyrol, Polymethylmethaerylat und Polyvinylchlorid wird yon zwei versehiedehen Standpunkten untersucht. Zun~ehst mittels einer empirischen Gleiehung yon Tait mit einem einzigen willkiir]iehen und temperaturabh~ngigen Parameter. Diese wird auf den flfissigen sowie den glasigen Zustand angewendet. Die Ergebnisse ersetzen und erweitern die gesultate einer friiheren Untersuchung. Eine befriedigende Ubereinstimmung zwischen den experimentellen Werten verschiedener Authoren fiir Polystyrol wird festgestellt. Sodann wird die Fehlstellentheorie yon Simha-Somcynsky auf den fiiissigen Zustand dieser Polymeren angewendet. Die Isobars bei atmospherischem Druek und der Kompressibilit~tsfaktor p V / T bis zu einem Maximaldruck yon ca. 2 kbar werden ausfiihrlich untersucht und die eharakteristischen Zustandsparameter V*, T* und p* bestimmt. Die Auswertung der Theorie erfordert die numerische L6sung einer transzendentMen Gleiehung fiir die L6oherzahl als Funktion yon reduziertem Volumen und Temperatur, we]che aus der Extremalbedingung fiir das Zustandsintegral folgt. Eine einfache Interpolationsformel stellt die theoretisehe Isobars fiir :p = 1 bar mit groBer Genauigkeit dar. Auf diese Weise wird die obige ~echnung vermieden und die praktisehe Auswertung der Theorie und somit yon V* und T* erleichtert. I n graphischer Form wird ein Netz
yon reduzierten theoretischen Isochoren angegeben, das einer Dichte~nderung von 15% und einem TemperaturintervM1 yon 200 ~ entspricht. Eine einzelne Druckangabe geniigt sodann fiir eine AbschS~tzung yon p* und demgem~g der vollst~ndigen Zustandsgleiehung. Schlieglich wird gezeigt, dab der reduzierte Tait Parameter/~, mittels der Theorie bestimmt, eine exponentielle Temperaturabh~ngigkeit aufweist, wie aueh experimentell gefunden wird.
Ite]~rencss 1) Nanda, V. S. and It. Simha, J. Chem. Phys., 41, 3870 (1964). 2) Cutler, W. G., It. H. McMickle, W. Webb, and R. W. Schiessler, J. Chem. Phys., 29, 727 (1958). 3) Hellwege, K. H., W. Knappe, and P. Lehmann, Kolloid-Z. u. Z. Polymere 183, 110 (1962). 4) Prigogine, I., N. Trappeniers end V. Mathot, Discussions Faraday Soc. 15, 93 (1953); 1. Prigogine, The Molecular Theory of Solutions. (Amsterdam 1957). 5) Simha, R. and A. J. Havl#:, J. Amer. Chem. Soc. 86, 197 (1964); Nanda, V. S. and R. Simha, J. Phys. Chem. 68, 3158 (1964). 6) Simha, It. and T. Somcynsky, Macromols 2, 342 (1969). 7) Qua&, A. and R. Simha, J. App]. Phys. 42, 4592
(1971).
8) Wood, L. A. and G. M. Martin, J. I~es. Natl. Bur. Stand. (U.S.) 68A, 259 (1964). 9) Nanda, V. S. and It. Simha, J. Chem. Phys. 41, 1884 (1964). 10) Curro, J. G., J. Chem. Phys. 56, 5739 (1972). 11) Jain, S. C. and V. S. Nanda, J. Phys. C: Solid State Phys. 4, 3045 (1972). 12) Quach, A., P. S. Wilson, and It. Simha, unpublished. 13) Orwoll, 1?. A. and P. J. Flory, J. Amer. Chem. Soc. 89, 6814 (1967). 14) Simha, R., and It. F. Boyer, J. Chem. Phys. 37, 1003 (1962). 15) Simha, R. and C. E. Weil, J. Macromo]. SciPhys. B 4 (1), 215 (1970). 16) Patterson, D. and J. M. Bardin, Trans. Farad Soc. 66, 321 (1970). 17) Simha, It. and S. T. Hadden, J. Chem. Phys. 25, 702 (1956).
Authors' addresses: Professor Dr. R. Simha, P. S. Wilson, O. Olabisi Division of Maeromoleeular Science Case Western Reserve University Cleveland, Ohio 44106, U.S.A.
