Polymer concentration dependent sorption

Pressure-dependent sorption and transport properties in polymers can be attributed to the presence of the penetrant in the polymer. Crank (32) suggested in 1953 that the "non-ideal" behavior of penetrant-polymer systems could arise from structural and dynamic changes of the polymer in response to the penetrant. As the properties of the polymer are dependent on the nature and concentration of the penetrant, the solubility and diffusion coefficient are also concentration-dependent. The concentration-dependent sorption and transport model suggests that "non-ideal" penetrant-polymer systems still obey Henry s and Fick s laws, and differ from the "ideal" systems only by the fact that a and D are concentration dependent,... 

C. Concentration-Dependent Sorption and Transport in Glassy Polymers... 

The basic difference between Mconcentration-dependentM and dual-mode, models is in their assumption about penetrant-polymer interactions. Concentration-dependent sorption and transport models are based on the assumption that the concentration-dependence of the solubility and diffusion coefficients arises... 

For a classical diffusion process, Fickian is often the term used to describe the kinetics of transport. In polymer-penetrant systems where the diffusion is concentration-dependent, the term Fickian warrants clarification. The result of a sorption experiment is usually presented on a normalized time scale, i.e., by plotting M,/M versus tll2/L. This is called the reduced sorption curve. The features of the Fickian sorption process, based on Crank s extensive mathematical analysis of Eq. (3) with various functional dependencies of D(c0, are discussed in detail by Crank [5], The major characteristics are... 

The sorption of a weak electrolyte by a charged polymer membrane is another case where Nernst + Langmuir-like dual mode sorption, involving the undissociated and dissociated species respectively, may be expected. The concentration of each species in solution follows, of course, from the dissociation constant of the electrolyte. The sorption isotherms of acetic acid and its fluoroderivatives have been analysed in this manner, and the concentration dependence of the diffusion coefficient of acetic acid interpreted resonably successfully, using Nylon 6 as the polymer substrate 87). In this case the major contribution to the overall diffusion coefficient is that of the Nernst species consequently DT2 could not be determined with any precision. By contrast, in the case of HC1, which was also investigated 87 no Nernst sorption or diffusion component could be discerned down to pH = 2 and the overall diffusion coefficient obeyed the relation D = DT2/( 1 — >1D), which is the limiting form of Eq. (25) when p — 00. 

The discussion directly following Eq (6) provides a simple, physically reasonable explanation for the preceding observations of marked concentration dependence of Deff(C) at relatively low concentrations. Clearly, at some point, the assumption of concentration independence of Dp and in Eq (6) will fail however, for our work with "conditioned" polymers at CO2 pressures below 300 psi, such effects appear to be negligible. Due to the concave shape of the sorption isotherm, even at a CO2 pressure of 10 atm, there will still be less than one CO2 molecule per twenty PET repeat units at 35°C. Stern (26) has described a generalized form of the dual mode transport model that permits handling situations in which non-constancy of Dp and Dh manifest themselves. It is reasonable to assume that the next generation of gas separation membrane polymers will be even more resistant to plasticization than polysulfone, and cellulose acetate, so the assumption of constancy of these transport parameters will be even more firmly justified. 

Section IIA summarizes the physical assumptions and the resulting mathematical descriptions of the "concentration-dependent (5) and "dual-mode" ( 13) sorption and transport models which describe the behavior of "non-ideal" penetrant-polymer systems, systems which exhibit nonlinear, pressure-dependent sorption and transport. In Section IIB we elucidate the mechanism of the "non-ideal" diffusion in glassy polymers by correlating the phenomenological diffusion coefficient of CO2 in PVC with the cooperative main-chain motions of the polymer in the presence of the penetrant. We report carbon-13 relaxation measurements which demonstrate that CO2 alters the cooperative main-chain motions of PVC. These changes correlate with changes in the diffusion coefficient of CO2 in the polymer, thus providing experimental evidence that the diffusion coefficient is concentration dependent. 

A number of attempts have been made to explain the nonlinear, pressure-dependent sorption and transport in polymers. These explanations may be classified as "concentration-dependent (5) and "dual-mode (13) sorption and transport models. These models differ in their physical assumptions and in their mathematical descriptions of the sorption and transport in penetrant-polymer systems. 

Experimental results presented in this work and in the literature are inconsistent with the assumptions and the physical interpretations implicit in the dual-mode sorption and transport model, and strongly suggest that the sorption and transport in gas-glassy polymer systems should be presented by a concentration-dependent model ... 

Nonlinear, pressure-dependent sorption and transport of gases and vapors in glassy polymers have been observed frequently. The effect of pressure on the observable variables, solubility coefficient, permeability coefficient and diffusion timelag, is well documented (1, 2). Previous attempts to explain the pressure-dependent sorption and transport properties in glassy polymers can be classified as concentration-dependent and "dual-mode models. While the former deal mainly with vapor-polymer systems (1) the latter are unique for gas-glassy polymer systems (2). 

The concentration-dependent models attribute the observed pressure dependence of the solubility and diffusion coefficients to the fact that the presence of sorbed gas in a polymer affects the structural and dynamic properties of the polymer, thus affecting the sorption and transport characteristics of the system (3). On the other hand, in the dual-mode model, the pressure-dependent sorption and transport properties arise from a... 

Through the uptake of a substance in a polymer matrix, time dependent changes in the polymer matrix can take place, particularly at high concentrations. As a consequence, the diffusion coefficient can be time- as well as concentration-dependent. One observes such behavior for example by the sorption of substances that lead to swelling at temperatures below the Tg. After a relatively rapid approach to an apparent state of equilibrium one observes a slow change towards the actual equilibrium (Fig. 9-2e). These two-step processes are caused by a gradual loosening of the cohesive forces between the macromolecules. 

Recently Koros and Hopfenberg have considered explicitely the effect of dual mode sorption on the local effective concentration dependent diffusion coefficient for low activity penetrant migration in glassy polymers. They showed that... 

The models most frequently used to describe the concentration dependence of diffusion and permeability coefficients of gases and vapors, including hydrocarbons, are transport model of dual-mode sorption (which is usually used to describe diffusion and permeation in polymer glasses) as well as its various modifications molecular models analyzing the relation of diffusion coefficients to the movement of penetrant molecules and the effect of intermolecular forces on these processes and free volume models describing the relation of diffusion coefficients and fractional free volume of the system. Molecular models and free volume models are commonly used to describe diffusion in rubbery polymers. However, some versions of these models that fall into both classification groups have been used for both mbbery and glassy polymers. These are the models by Pace-Datyner and Duda-Vrentas [7,29,30]. 

Chapter 4(71) focuses on the characterization of sorption kinetics in several glassy polymers for a broad spectrum of penetrants ranging from the fixed gases to organic vapors. The sorption kinetics and equilibria of these diverse penetrants are rationalized in terms of the polymer-penetrant interaction parameter and the effective glass transition of the polymer relative to the temperature of measurement. The kinetic response is shown to transition systematically from concentration independent diffusion, to concentration dependent diffusion, and finally to complex nonFickian responses. The nonFickian behavior involves so-called "Case II" and other anomalous situations in which a coupling exists between the diffusion process and mechanical property relaxations in the polymer that are induced by the invasion of the penetrant (72-78). ... 

SOLVENT MOBILITIES. One check on the physical significance and the reliability of the data representing the concentration dependence of the diffusion coefficient is to convert these results to solvent mobilities. The values should increase rapidly with increasing concentration and extrapolate to the self-diffusion coefficient for toluene. The procedure for carrying out the calculations was outlined in previous publication (11) and is repeated here in a brief form for convenient reference. The diffusion coefficient obtained directly in the vapor sorption experiment is a polymer, mass-fixed, mean diffusion coefficient, D, in the sorption interval. Duda et. al. (12) have shown that, if the concentration interval is small, the true diffusion coefficient, D, is simply related to the mean diffusion coefficient at a prescribed intermediate concentration in the interval ... 

The fit of these expressions to experimental results is very good. At low pressure regimes, the fit was shown to be even better than that of dual sorption expressions. Except for these regimes, the two models seem to do equally well in describing sorption and permeability data. Concentration dependent diffusivity and permeability have been considered before mainly for vapors. The new aspect of the matrix model is that it broadens these effects to fixed gases. The important difference between the matrix and dual sorption models is in the physical picture they convey of gas transport and interaction with the polymer. Additional experimental evidence will be needed to determine the preference of these different physical representations. 

Time Dependence. Time dependence may be observed in the diffusion of organic vapors in polymers below their glass transition temperature (Tg) (5). At these temperatures, the rate of diffusion is comparable with the rate of motion of the polymer segments. As a result, the value of the diffusion coefficient attained at a given concentration in an element of the polymer will depend on the time for which this concentration has existed at the element. D has more time in which to approach its equilibrium value in thicker films. Therefore, sorption proceeds more rapidly the thicker the film, and the reduced sorption curves do not coincide as required by Equation 1 describing Fickian diffusion. 

For systems based on the above two epoxy polymers, the concentration dependence of the parameter logVq represents a nonmonotoni-cally increasing deviation from the additive value. Hence, for small additions of one polymer to another a decrease of Vq is observed in terms of the absorption process this can be explained by the decrease of the system sorption activity, probably due to closer packing of the polymer mixture. 

The following two subsections provide physical imeipretations of the forms of sorption isotherms and concentration-dependent diffusion coefficients observ for rubbery and glassy polymers, respectively. These sections are not required for simulation of module operations if a compirae set of empirical pressure, temperature, and composition-dependent permeation data are available. A sinqde polynomial 6t of permeability data as a function of all operating variables would suffice for design simulation. 

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