464 BROCKMEIER, McCoy, MEYER
Macromolecules
Gas Chromatographic Determination of Thermodynamic Properties of Polymer Solutions. I. Amorphous Polymer Systems N. F. Brockmeier,*l" R. W. McCoy,lb and J. A. Meyerlb Research and Decelopment Department, Amoco Clzemiculs Corporation, and Research and Decelopment Department, Standard Oil Cornpan),, (Indiana), Standard Oil Research Center, Naperaille, Illinois 60540. Receiced April 12, I972
ABSTRACT: A gas chromatographic (gc) method has been developed to measure solubility isotherms of a low molecular weight solvent in a high polymer. This method depends on the elution of the injected solvent on a concentration plateau of the same solvent in nitrogen carrier gas. The measured solubility is used to calculate the vapor-solid equilibrium ratio (Q = a ~ / w ~along ) , with the solvent-polymer interaction parameter X, as a function of solution composition. Based on the FloryHuggins and Maron theories, a mathematical correlation permits computation of Q values, usually with less than 5 deviation from those measured experimentally. The following polymer-solvent systems have been studied: polystyrene with benzene and ethylbenzene at 120" and with ethylbenzene at 185", polyethylene with n-decane at 185", and atactic polypropylene with hexane at 80". The computed i2 values range from about 4 to 5 .
T
he urgent need for reliable process design data has led us t o develop a new gas chromatographic (gc) method for measuring the thermodynamic properties of polymer solutions and a mathematical model for correlating the results. Several investigators*-; have reported their gc measurements of activity coefficients and vapor-polymer equilibrium ratios at infinite dilution. By contrast, our method yields equilibrium values using finite concentrations of solvent vapor in the carrier gas.6 We have determined distribution isotherms, activity coefficients rationalized by weight fraction (Q value), and polymer-solvent interaction parameters for polystyrene, amorphous polyethylene, and atactic polypropylene with selected hydrocarbon vapors constituting u p to 60 mol of the carrier gas. I n the limit of infinite dilution, our D values and interaction parameters essentially agree with those reported in the literature. 8 , 5 Although the gc method has been used for some time in measuring the activity coefficient of a vapor over a low molecular weight stationary phase,7j8 it has only recently been applied to a polymeric stationary phase. I n 1969, the activity coefficients of acetic acid, butyl alcohol, and hexadecane were determined a t infinite dilution in a polyamide of 50,100 molecular weight coated on Chromosorb packing. 2a The equations for calculating a more useful form of the activity coefficient have been derived in another gc study. Patterson, et u I . , ~ divide the activity coefficient by weight fraction of solvent to yield a logarithm that does not approach minus infinity for a polymer. They also report values for the interaction parameter of n-dodecane in branched polyethylene and n-decane in linear polyethylene at high temperatures (110-190'). In another investigation, gc was used to measure the activity coefficients a t infinite dilution for polyisobutylene and polystyrene with 13 different hydro(a) Amoco Chemicals Corp.; (b) Standard Oil Go. (a) 0. Smidsrpd and J. E. Guillet, Macromolecules, 2, 272 (1969); E. Guillet and A. N. Stein, ibid., 3, 102 (1970). D . Patterson, Y . B. Tewari, H. P. Schreiber, and J. E. Guillet, ibid., 4, 356 (1971). (4) W. E. Hammers and C. L. DeLigny, Recl. Trati. Chim. PayS-BaS, 90,9 12 (1971). (5) R. D. Newman and J. M. Prausnitz, J . Phys. Chem., 76, 1492 11972). -,(6) N. F. Brockmeier, R. W. McCoy, and J. A. Meyer, Macromolecules, 5, 130(1972). (7) F. I. Stalkup and H . A. Deans, AIChE J., 9, 106 (1963). (8) K. T. Koonce, H. A. Deans, and R. Kobayashi, ibid., 11, 259 ( 1965). \ -
carbon solvents.: Both polar and nonpolar solvents were included at temperatures of 25,150,175, and 200". Theory Two factors have been important in facilitating the application of gc to polymer thermodynamics in solutions of finite concentrations: appropriate theoretical equations and polymer coating procedure. The first of these is discussed here, while the second is covered in the Experimentdl Section. The theoretical relations for computing the desired isotherms from gc data have been developed in a general form in a series of articles by Conder and P ~ r n e l l . g - ~They ~ demonstrated the utility of their equations by accurately measuring the activity coefficients of n-hexane in squalane and n-heptane in di-nnonyl phthalate at carrier gas concentrations ranging from zero to 70 mol %. l 2 Their technique of elution on a plateau of finite concentration was the one chosen for all of our work. Their key equation yields the solubility isotherm: q(P) moles of hydrocarbon solvent per unit mass of polymer in the stationary phase at mean pressure P. To calculate a n accurate value for q(P), corrections must be made to the experimental values for pressure, flow rate, and feed gas composition. These corrections will be discussed only briefly here because they habe been derived in detail in the literature." The true value of the mole fraction of solvent vapor above the stationary phase is determined from
+
=
a h
(1)
where y o is the mole fraction measured at the detector outlet, j is a compressibility correction to compensate for the column pressure gradient, and a is a correction for gas nonideality. To correct for the pressure drop required to force the carrier gas through the column, a mean pressure for the isotherm is defined by
P
=
poJa4
(2)
The form of the J function is defined as (3) (9) J. R. Conder and J. H . Purnell, Trans. Faradan?' Soc., 64, 1505 (1968). (10) J. R. Conder and J. H. Purnell, ibid., 64,3100 (1968). (1 1) J. R. Conder and J. H. Purnell, ibid., 65,824 (1969). (12) J. R . Conder and J. H. Purnell, ibid., 6 5 , 839 (1969).
AMORPHOUS POLYMER SYSTEMS465
Vol. 5, NO. 4 , Jdy-August 1972 The arithmetic mean pressure in a column isp0/J2l. Accurate measurements of inlet and outlet column pressures are required for eq 2 and 3. With the inclusion of a term to compensate for gas nonideal P-V behavior, the full compressibility correction takes the form (4) where 8 2 2 is the second virial coefficient of the solvent vapor at the column operating temperature, T. Values of B2? are either estimated from a correlation in the literatureI3 or obtained from tabulated results.14 Sorption into the solid phase can have a significant effect, causing the gas velocity in the column to differ at each end. A common measure of sorption is the solvent distribution coefficient, given approximately by
somewhat for the special case of a nearly straight distribution isotherm. With a straight isotherm, the contribution to retention time a t the inlet of the gc column (high c ) is exactly offset by the loss a t the outlet (low c ) . The pressure gradient must be nearly linear ( p t / p o< 1.7). We were able to tolerate about 5 error caused by changing concentration, so the limit set by eq 11 was extended t o 0.08. Beyond this limit, the curvature of the distribution isotherms contributed more than 5 error. Equation 10 provides the key information from which equilibrium values are calculated. One equilibrium ratio commonly used in work with polymers is
The weight fraction of solute in the polymer phase is (13)
where tR and tar are, respectively, the retention times of the solvent and of an air peak. One of the relationships needed to correct the retention volume for sorption is bnm = 1
+ k(1
-
Jnmyo)
(6)
The composition correction factor for gas nonideality is defined as
Calculations demonstrate that b11/b3*is within 1 % of unity if the pressure ratio over the column is less than 1.1, a fact that simplifies some of the computations. The solvent concentration in the gas above the polymer is calculated from a virial equation of state, dropping all terms after the second virial coefficient. This approximation introduces less than 1 2 error at our conditions. Solving this equation for concentration gives
The effect of sorption causes a difference between the retention times of air (nonsorbed) and of hydrocarbon. This is expressed classically as a difference in retention volumes (9) The temperature ratio, T/TF,is included to correct the flow rate, F ( y ) , measured a t the flowmeter temperature t o that of the column. It has been shown" that the distribution isotherm a t pressure P is given by the integral
For our method, elution on a plateau of constant solvent concentration in the carrier, the above integral is evaluated by finding the area under the curve generated by plotting (VR V~r)/(l - $) us. concentration. The criterion for constant concentration as given by Conder and Purnell isll where 0.01 is the experimental uncertainty in VR. If reliable data are to be obtained a t high solvent mole fraction, the column pressure drop must be held to a very low value. However, this restriction o n pressure drop can be relaxed (13) I
