3158
V. S. NANDAAND ROBERT SIMHA
Equation of State and Related Properties of Polymer and Oligomer Liquids1
by V. S. Nanda2 and Robert Simha Department of Chemistry, University of Southern California, Los Angeles, California 90007 (Receiaed M a y 4? 1964)
The cell theory has been applied earlier to chain liquids by using a modified harmonic approximation to the cell potential. We discuss here the equation of state and related thermodynamic expressions derived by means of the square well approximation which forms the basis of a fairly successful description of the liquid and glassy states in terms of the free volume concept. The theoretical equation of state is in satisfactory agreement with the experimental volume-temperature data for reduced volumes up to 1.2, as conipared with 1.1, obtained earlier. In terms of temperature, this extends, for polystyrene as an example, the range of the theory by about 150’. The variation of the cohesive energy density with temperature and molecular weight is also satisfactorily described. This permits a prediction of the cohesive energy density for a high polymer from the enthalpy of vaporization of a single oligomer without having recourse to swelling data. With respect to the internal pressure, however, no agreement between theory and experiment is found. Nevertheless, it is possible to establish the validity of the principle of corresponding states at low pressures with respect to such properties as the internal pressure and the compressibility.
I. Introduction Recently the principle of corresponding states has been shown to be valida for several series of oligomer and high polymer liquids a t atmospheric pressure. A comparison of the experimental reduced volumereduced temperature curves with the theoretical equation of state for chain liquids, obtained by Prigogine and c o - w o r k e r ~ as ~ , ~a generalization of the LennardJones-Devonshire (LJD) cell theory for monomer liquids, showed good agreement with experiment for 1 I P 5 1.1 for the cases considered. Actually, Prigogine, et ~ l . replace , ~ the LJD cell potential by a “modified” harmonic one as an approximation. In the study of monomer liquids, however, it is convenient to adopt the square well potential as an adequate approximation to the cell potential. This approximation, moreover, provides a basis for the so-called free volume theories, which have been quite successful in the description of the liquid and glassy states. In a later publication Prigogine, et al.,s in the deduction of the law of corresponding states for polymer liquids, also consider the square well potential as one of the simple alternatives to the actual cell potential. It appeared to us worthwhile to make a comparison of T h e J o u r n a l of Physical Chemistry
the .experimental results with the equation of state using this approximation in order to see whether a theoretical representation over a wider range of temperatures is possible. Finally, we have determined the theoretical values of the internal pressure (pi) and the cohesive energy density (c.e.d.) and compared these values with the experimental results reported by Gee and co-workers*J and some additional data available in the literature. While the theory seems adequate for the description of the variation of c.e.d. with molecular weight for a polymer series, and even for explaining variations of c.e.d. with temperature for a given member, no accurate representation of pi (1) This work was supported by Research Grant NsG-343 of the National Aeronautics and Space Administration t o the University of Southern California. (2) On leave of absence from the Physics Department, University of Delhi, Delhi 6, India. (3) (4) day (5)
R. Simha and A. J . Havlik, J . Am. Chem. Soc., 86, 197 (1964). I. Prigogine, N. Trappeniers, and V. Mathot, Discussions FaraSoc., 15, 93 (1953).
I. Prigogine, A. Bellemans, and C. Naar-Colin, J . Chem. P h y s . ,
26, 75 (1957).
(6) G. Allen, G. Gee, and G. J. Wilson, Polymer, 1 , 456 (1960). (7) G. Allen, G. Gee, D. Mmgaraj, D. Sims, and G. J. Wilson, ibid., 1 , 467 (1960).
3159
EQUATIOX OF STATE OF POLYMER AND OLIGOMER LIQUIDS
is obtained. ?\‘evertheless our investigation validates the principle of corresponding states with respect to internal pressure and compressibility.
Prigogine expre~sion.~From eq. 5 and 6 we obtain the ratio
11. Theoretical Expressions
It may be noted from eq. 5, 6, and 7 that p , , c.e.d., and
The relevant theoretical expressions on the basis of cell theory, using the square well potential, can be obtained from the coiresponding partition function5
y depend explicitly on
z
=
ArNg(N)(a-
0)3cN
exp{ [ - ~ q x e * /
(2kT)l[ A(u*Iv)4 - B(u*/u)2]
(1)
where A , is a geometrical constant, g ( N )a coinbinatory factor, 3c the tota,l number of volunie-dependent degrees of freedom per chain, and q x the corresponding number of intermolecular nearest neighbors on the quasi-lattice. If the coordination number z is set equal to 12, then A = 1,011 and B = 2.409. The other symbols have their usual meaning as in the cell theory of monomer liquids. However, the quanti ties e*, v*, and v refer here to the chain segment. The Helmholtz free energy F is given by
F
=
-kT In Z
=
-NkT[ln A , $- 1,” In g ( N ) ] ,3cNkT In (a -
cr)
+ N p e * / 2 [ A ( ~ * / v )-~ (2)
Making use of the thermodynamic relation p == - (bF/bV),, we get, after some simplification, the reduced equation of state in the form $P/T = (1 - 2-1/8g-1/3)-1 + (2V-2/F)(AP-2
- B/2)
(3)
where (as in ref. 3) 6 = pV*/(qxe*), P = V / V * , !F = Tck/(qxe*) while V and V* represent the volume and the characteristic volume parameter per chain, respectively. The internal energy U is derived from eq. 2 as
=
-P[b(F/T)/bT]v (Npe*/2)(A P4- BP-2)
(4)
whence p , = T(bp/dT)v
- ;P
=
APP2) ( 7 )
only.
HI. Equation of State I n Fig. 1 log P is plotted against log F with the aid of the equation of state (3) as curve 1. Here, because of the extremely small magnitude of 6,the left-hand side has been taken to be effectively zero. Since there is no independent knowledge of the parameters in eq. 3, the experimental reduced volunie-reduced temperature representation for n-paraffins, already reported in ref. 3, was shifted parallel to the two axes to give the best coincidence with the theoretical curve. This procedure naturally defines a new reduced coordinate system for the experimental data. The connection between the old and the new reduced variables is represented by the relations log
[P,,,/Vold]
=
0.0072; log
[!Fncw/!FOld]
=
0.1045 (8)
B(u*/v) 1
u=
y = p,/c.e.d. = 2 ( B - 2 A P - ’ 9 / ( B -
(bU/bV), =
[ N ~ ~ * / ( v * P ) ] ( B-P~-A~ P - 4 )
(5)
Xoting that for a “prisoner,” motion is free inside the well, we have
This result should be compared with the corresponding
I n order to show graphically the more extensive agreement with experiment obtained with the square well potential, the previous theoretical curve was shifted to the same extent along the log P and log !F axes as the experimental data, according to eq. 8. It may be noted that we have extended here the experimental region on the low temperature side by including a point for supercooled propane.8 This appeared of interest especially because the two superposed theoretical curves begin to show deviations just below the experimental range covered in ref. 3. The present theoretical equation provides a satisfactory representation of the low temperature region as well. The obP served range of validity is now approximately 1 < 1.20,compared with the previous range 1 5 P < 1.10. The experimental voluine-temperature results for other polymer series, viz., polystyrene, polydimethylsiloxane, and perfluorinated polyniethylenes, are exhibited as inserts in Fig. 1, along with the relevant region of the theoretical curve, labeled, respectively, as 2, 3, 4. The location of the experimental points has been determined in all cases by using the shift factors defined by eq. 8 for the n-parafin series. This procedure is in accord with the law of corresponding states. It should also be pointed out here that the shifting of the experimental data to the new coordi-
Vumber 2 1
.Vouember, 1964
