842
I n d . Eng. C h e m . Res. 1991,30, 842-851
Molecular Thermodynamic Model for Sorption and Swelling in Glassy Polymer-C02 Systems at Elevated Pressures Raymond G. Wissinger and Michael E. Paulaitis* Department of Chemical Engineering, University of Delaware, Newark, Delaware 19716
A molecular thermodynamic model is presented that can quantitatively predict both gas sorption and polymer swelling in glassy polymer-compressed gas systems. The model is based on a lattice theory of polymer solutions and the concept of order parameters to describe the glass state. Model parameters that characterize molecular interactions between gas molecules and polymer segments are determined by fitting data for gas sorption in the polymer melt, and the order parameters that characterize the glassy polymer are determined from measured gas solubilities at the glaas transition. These parameters are then used to predict gas sorption in the polymer glass and swelling behavior for both the liquid and glassy polymer. The predictions are compared to experimental data for poly(methy1 methacrylate) and polystyrene in the presence of COPa t temperatures from 33 to 65 “ C and pressures up to 175 atm. The model is also used to predict the depression in glass transition temperature for poly(methy1 methacrylate) as a function of the amount of C 0 2 sorbed.
Introduction The thermodynamic behavior of glassy polymer-compressed gas systems at elevated pressure has received considerable attention in recent years as an important factor in several new process applications such as the use of supercritical fluids to extract low-molecular-weight compounds from polymers (Dhalewadikar et al., 1987; Krishnamurthy and Chen, 1989) or to impregnate polymers with chemical additives (Sand, 1986; Berens et al., 1988),in the production of polymer foams (Spalding, 1988), and in the separation of gas mixtures using polymer membranes (Koros, 1985; Stookey et al., 1986). Many of these applications involve gas penetrants, such as COz, which can have high solubilities in glassy polymers, even at moderate pressures. These high solubilities can produce substantial polymer swelling (Fleming and Koros, 1986; Sefcik, 1986; Wissinger and Paulaitis, 1987; Kamiya et al., 1989) and large depressions in glass transition temperatures (Wang et al., 1982; Chiou et al., 1985; Sanders, 1988). Quantitative descriptions of sorption, swelling, and the glass transition are therefore essential for the design and operation of these processes. Experimental data for glassy polymer-compressed gas systems at elevated pressure are, however, often scarce or limited to a narrow range of operating conditions. Thermodynamic models that can predict both gas sorption and polymer swelling at elevated pressures are also needed. From a practical standpoint, these models should be applicable to both the polymer glass and the melt, be computationally simple, be reliable over wide ranges of temperature, pressure, and composition, and have few adjustable parameters. A variety of thermodynamic models have been applied to describing gas sorption in rubbery and liquid polymers. Fleming and Koros (1986) used the Flory-Huggins equation to correlate experimental data for COz sorption in silicone rubber at 35 “C and pressures up to 60 atm. Since COz is above its critical temperature at 35 “C, a hypothetical vapor pressure was required to calculate the activity of COz in the polymer. This hypothetical vapor pressure was estimated by extrapolating the actual vapor pressure curve to supercritical temperatures. While this approach may be reliable for temperatures just above the critical temperature of COz, it can lead to significant errors in gas solubilities at higher temperatures. Indeed, the problem of defining hypothetical states can be avoided altogether by using an equation of state to calculate ac0888-5885/91/2630-0842$02.50/0
tivities in both the gas phase and the polymer solution. Several equations of state that are derived from statistical thermodynamics of polymer solutions are available for such calculations; for example, models based on perturbed hard chain theory (Liu and Prausnitz, 1979; Ohzono et al., 19841, corresponding states (Cheng and Bonner, 1978), or lattice theories (Panayiotou, 1986; Kiszka et al., 1988; Beckman et al., 1990)have been applied to correlating gas solubilities in liquid polymers over wide ranges of temperature and pressure. A phenomenological model that has also been used extensively to describe gas sorption in glassy polymers is the dual-mode sorption model (Barrer et al., 1958; Michaels et al., 1963; Vieth et al., 1964). Dual-mode sorption is based upon the assumption of two distinct sorption mechanisms: gas dissolution in the bulk polymer, characterized by Henry’s law, and sorption in molecular-scale “microvoids”which are assumed to exist due to the nonequilibrium nature of the glassy polymer (Koros et al., 1981). This second sorption mode is described by an expression analogous to a Langmuir adsorption isotherm. The dual-mode sorption equation is
C=k$+-
CkbP 1 + bP
where C is the total concentration of sorbed gas, kD is related to a Henry’s constant, C g represents the microvoid sorption capacity, b describes the affinity of gas molecules for microvoids, and P is the gas pressure. The parameters hD, Cl,,and b are usually determined by fitting (1) to measured sorption isotherms. Correlations for estimating these parameters have also been developed on the basis of the physical properties of the gas and polymer. Although the dual-model sorption model accurately represents experimental data for a variety of polymer-gas systems (Sada et al., 1987; Koros et al., 1981, 1976; Vieth et al., 1966; Koros and Paul, 1978), there are notable disadvantages: the model has limited predictive capability, three adjustable parameters are required at each temperature, and, in its original form, the model neglects polymer swelling and the plasticization of the polymer glass by dissolved gases. Fleming and Koros (1986) and Kamiya et al. (1988a,b, 1989) have proposed ad hoc extensions of the dual-mode sorption model to account for swelling and plasticization; however, these extensions require additional, independent information on thermodynamic properties of 0 1991 American Chemical Society
Ind. Eng. Chem. Res., Vol. 30, No. 5, 1991 843 the polymer-gas system. For example, to describe polymer swelling, the partial specific volume of the gas dissolved in the polymer is required. An alternative model for gas sorption in glassy polymers is the gas-polymer-matrix (GPM) model (Raucher and Sefcik, 1983) which, unlike dual-mode sorption, is based upon the assumption that only one population of sorbed gas molecules exists. The sorbed gas molecules can, however, interact with the polymer chains to alter the molecular structure and properties of the glassy polymer. The concentration of gas sorbed in the polymer is given by
c=-
U P
1 + aC
where uo is the solubility coefficient in the limit of zero penetrant concentration and (Y is a parameter that describes the magnitude of gas-polymer interactions. The GPM model with two adjustable parameters provides an equally satisfactory representation of gas sorption data compared to the dual-mode sorption model, but also lacks predictive capability and does not describe polymer swelling. Vrentas and Vrentas (1989) recently developed a semiempirical model for calculating volumetric properties of glassy polymer-penetrant systems that is based on the GPM model and have shown that model predictions for the swelling of polycarbonate by compressed COz were in good agreement with experimental results. Several lattice theories of polymer solutions have also been developed to describe the thermodynamic behavior of glassy polymers. Gibbs and DiMarzio (1953) proposed a lattice model to describe the glass transition which assumes that the experimentally observed glass transition temperature can be related to a hypothetical transition temperature at which the polymer has zero configurational entropy. The model has been applied to predicting the effects of molecular weight (Gibbs and DiMarzio, 1958), diluent concentration (DiMarzio and Gibbs, 1963), and pressure (DiMarzio et al., 1976; Panayiotou and Vera, 1984) on the glass transition temperature by making the additional assumption that these variables affect the glass transition temperature in the same quantitative manner that they affect the hypothetical transition temperature. Nose (1971) and Simha et al. (Quach and Simha, 1972; McKinney and Simha, 1974; Simha, 19761 developed equations of state based on lattice theories to predict PVT and other thermodynamic properties of pure polymers in both the liquid and glass states. These models characterize the glass by “freezing in” the fraction of vacant sites in the lattice a t temperatures below the glass transition temperature of the polymer. As discussed below, this assumption arises naturally from the concept of order parameters applied to lattice theories for polymer solutions. The theoretical approach taken in this work has been to describe sorption and swelling in glassy polymer-compressed gas systems by incorporating order parameters into a lattice theory of polymer solutions. Model parameters that characterize molecular interactions between the gas molecules and polymer segments are determined by fitting data for gas sorption in the polymer melt. Order parameters that characterize the glassy polymer are determined from the measured gas solubility a t the glass transition. These parameters are then used to predict gas sorption in the polymer glass as well as the swelling behavior for both the liquid and glassy polymer. Our approach has several advantages over existing models for gas sorption and polymer swelling: (1) only two mixture parameters are required for a polymer-gas system and these parameters can be applied over a wide range of temperatures, pressures, and compositions; (2) sorption and swelling are
described in a consistent manner using one thermodynamic model; and (3) both the liquid and the glass are described with this one thermodynamic model.
Order Parameters The concept of order parameters provides a useful framework for describing systems that are not at equilibrium in the context of a thermodynamic model. The concept was first proposed by de Donder (1936) and later used by Prigogine and Defay (1954) to describe the departure of chemical reactions from equilibrium. Davies and Jones (1953) were the first to apply order parameters to describe the glass transition, and several other analyses of the glass transition based on order parameters have since followed (Staverman, 1966; Rehage, 1980; Kovac, 1981; Goldstein, 1973; DiMarzio, 1974; Gupta and Moynihan, 1976; Astarita et al., 1989). Order parameters are derived from the assumption that, in addition to the usual state variables (i.e., temperature and pressure), there exists a set of N order parameters GI, Zz, ..., 2 ), that describe the thermodynamic state of a system. The Zi cannot be adjusted experimentally and are therefore internal state variables of the system. Examples of order parameters or internal state variables include the extent of a chemical reaction in a reacting system or the degree of crystallinity in a semicrystalline polymer. In a glass, the order parameters presumably describe the morphology of the material. The change in Gibbs free energy of the system is formally be expressed in terms of the N order parameters as N
dG = -S d T + V dP + CAidZi i=l
(3)
The affinity, Ai, corresponding to order parameter Zi is defined by (4)
At equilibrium, the free energy is minimized with respect to each Zi and hence Ai = 0 for all i. Furthermore, the Zi will adjust to minimize the free energy at any equilibrium temperature and pressure; thus the free energy for equilibrium states of the system will be a function only of temperature and pressure. In the glass state, the system does not reach equilibrium and the free energy is not necessarily a t a minimum. The value of each Zi thus depends on the history of glass formation and characterizes the departure of the glass from an equilibrium state. One common assumption that is made to avoid consideration of relaxation kinetics in the glass is to treat the order parameters as constants that are “frozen in” at their equilibrium values at the glass transition (Astarita et al., 1989). This assumption allows a purely thermodynamic, rather than kinetic, treatment of the glass state. Classical thermodynamic treatments of the glass state do not provide molecular significance to the order parameters, although various intuitive arguments have associated these parameters for glassy polymers with such macroscopic quantities as free volume or excess entropy (Staverman, 1966). Statistical thermodynamics can, however, provide insights into the molecular nature of order parameters for polymer glasses. Simha (1976), for example, used the fraction of vacant sites in the lattice theory of Somcynsky and Simha as an order parameter to characterize the effect of pressure on the glass transition temperature for pure polymers. Roe (1977) used two order parameters (the fraction of vacant lattice sites and the number of polymer chains) in a one-dimensional lattice
844 Ind. Eng. Chem. Res., Vol. 30, No. 5, 1991 model to obtain an expression for the Gibbs free energy of a pure glass-forming material. We have extended this approach to more realistic lattice models with the goal of describing gas sorption and polymer swelling in glassy polymer-compressed gas mixtures. Two lattice models were initially considered: the SL model of Sanchez and Lacombe (1978) and the PV model of Panayiotou and Vera (1982a). The SL model is based on a "mean field" approximation and contains one order parameter: the fraction of vacant sites or holes in the lattice. The PV model is based on the "quasi-chemical" approximation (Guggenheim, 1952) which accounts for local ordering or nonrandom mixing and contains two order parameters: the fraction of holes in the lattice and the number of nearest-neighbor contacts between polymer segments on the individual lattice sites. Since the PV lattice model has the additional flexibility of a second order parameter to describe the glass state, this model was chosen for study in our work.
The PV Lattice Model for Polymer Solutions For the pure polymer, N polymer molecules each consisting of r segments are arranged with Novacant sites or holes on a lattice of coordination number z. Each segment is assumed to occupy one lattice site, and the connectivity of the segments is accounted for by defining an effective chain length, q zq = ( z - 2)r 2 (5) The total number of lattice sites is N, = ( N o+ rN) (6) and the lattice enery is given by E = Nllt (7) where N l l is the number of nearest-neighbor segment contacts and t is the interaction energy of this contact. Equation 7 assumes that the segment-hole and the holehole interactions have zero energy. The segment-segment interaction energy is also assumed to be temperature dependent; i.e.
+
= eo + Td (8) where to and tl are constants independent of temperature, pressure, and composition. Since the interaction energies for the nearest-neighbor contacts are not all equal, a nonrandom distribution of segments and holes is expected. From the "quasi-chemical" approximation (Guggenheim, 1952), the number of nearest-neighbor segment contacts per molecule is
where 0 is the fraction of total external contacts in the system defined by
and rlois a factor for nonrandom segment-hole contacts given by 2 (11) r10 = 1 + [l - 40(1 - O)(l with X = exp(t/kV. The standard derivation of the most probable values for Nll and No or equivalently the fraction of holes NO f o = No + rN
formally involves minimizing the free energy of the system with respect to these two parameters (McQuarrie, 1976). Thus, based on the definition of order parameters given in the previous section, these two quantities arise naturally as order parameters in the PV lattice model. Details of the derivation for both the pure-component and the mixture equations of state are given elsewhere (Panayiotou and Vera, 1982a; Wissinger, 1988). For mixtures, closed-form expressions for the equation of state and the chemical potentials can be obtained if the holes are assumed to be uniformly distributed throughout the lattice, while the segments of the different molecules have nonrandom distributions (Panayiotou and Vera, 1982a). The same order parameters are obtained for a binary mixture: the hole fraction and the number of nearest-neighbor segment contacts per molecule of component 1. The mixture equation of state is
where r, q, and 0 are determined by the following mixing rules:
r = &ri
(14)
q = Exigi
(15)
i i
and xi = Ni/N is the pole fraction of component i. The Leduced temperature, T = T / P ,and the reduced pressure, P = P / P , are obtained from the characteristic temperature, P,and pressure, P*: zt*/2 = P*vo = kT* (17) where uo is the lattice site volume. A constant value of z = 10 is assumed for the lattice coordination number, and uo is set equal to 9.75 cm3/mol as suggested by Panayiotou and Vera (1982b). The characteristic energy, E*,for a binary mixture is given by e* = tlitii + 712622 - tl1tl2A12(€11+ €22 - 2 4 (18) where tij is the interaction energy between nearest-neighbor segment contacts of species i and j , vi = qixi/q, and A12 is a factor for nonrandom contacts between segments of components 1 and 2 2 (19) A12 = 1 + [1 - 4tlltl2(1 - X i ~ ) ] l / ~ with A,, = exp[O(tll + t22- 2e12)/kT]. In the PV lattice model, a pure component is characterized by three parameters: the specific volume of a ~ , the two energy parameters to and 2. A segment, u * ~and binary mixture is characterized by two additional parameters since the interaction energy t12 is assumed to have the same temperature dependence as the pure component interaction energies; i.e. where the two mixtures parameters, and ti2, are assumed to be independent of temperature, pressure, and composition.
Sorption and Swelling Calculations The amount of C 0 2 (component 2) sorbed in the polymer (component 1) at a specified temperature and pressure
Ind. Eng. Chem. Res., Vol. 30, No. 5, 1991 845 is determined by equating COz fugacities in the pure gas phase and the polymer-rich phase. These calculations are based on the assumption that the solubility of the polymer in the compressed gas is negligible. The following algorithm was applied to calculate COz solubilities in liquid polymers (Wissinger, 1988): 1. Set T and P. 2. Solve the equation of state for the reduced volume of pure C02 in the gas phase. 3. Calculate the fugacity of pure C 0 2 in the gas phase. 4. Guess an initial value of x2. 5. Solve the equation of state for the reduced volume of the mixture. 6. Calculate the fugacity of COz in the polymer-rich phase. 7. Compare COz fugacities; if not equal, adjust x2 and return to step 5. After a solution for xz is obtained, the specific volume of the mixture, uBP,can be calculated and the percent volume change of the polymer determined from
Table I. Pure Component Equation of State Parameters comDonent u*.", cm3/g eo, cal/mol d, &/(mol K) r 0.871 159.90 0.019 3.89 co2 PMMA 0.783 221.46 0.131 7154O polystyrene 0.870 167.54 0.222 22081* "Based on a molecular weight of 90000. *Based on a molecular weight of 250000. 0.90
, Expt'l. ( 1
bar)
Calc'd. ( 1 bar) A
I ,
Expt'l. ( 1 00 bar)
,' , 0,
Calc'd. (100 bar) t
Expt'l. (200 bar) Calc'd. (200 bar)
,
,'* , '0
';ir 0.87 -
>
8
