Gas/polymer matrix model

The gas-polymer-matrix model for sorption and transport of gases in polymers is consistent with the physical evidence that 1) there is only one population of sorbed gas molecules in polymers at any pressure, 2) the physical properties of polymers are perturbed by the presence of sorbed gas, and 3) the perturbation of the polymer matrix arises from gas-polymer interactions. Rather than treating the gas and polymer separately, as in previous theories, the present model treats sorption and transport as occurring through a gas-polymer matrix whose properties change with composition. Simple expressions for sorption, diffusion, permeation and time lag are developed and used to analyze carbon dioxide sorption and transport in polycarbonate. 

In Section I we introduce the gas-polymer-matrix model for gas sorption and transport in polymers (10, LI), which is based on the experimental evidence that even permanent gases interact with the polymeric chains, resulting in changes in the solubility and diffusion coefficients. Just as the dynamic properties of the matrix depend on gas-polymer-matrix composition, the matrix model predicts that the solubility and diffusion coefficients depend on gas concentration in the polymer. We present a mathematical description of the sorption and transport of gases in polymers (10, 11) that is based on the thermodynamic analysis of solubility (12), on the statistical mechanical model of diffusion (13), and on the theory of corresponding states (14). In Section II we use the matrix model to analyze the sorption, permeability and time-lag data for carbon dioxide in polycarbonate, and compare this analysis with the dual-mode model analysis (15). In Section III we comment on the physical implication of the gas-polymer-matrix model. 

Another model for the sorption and transport of gases in glassy polymers at super atmospheric pressures is the gas-polymer-matrix model, proposed by Raucher and Sefcik (1983). The premise of this model is that the penetrant molecules exist in the glassy polymer as a single population and that the observed pressure dependence of the mobility is completely due to gas-polymer interactions. In the mathematical representation of this model the following expression for sorption and transport is used ... 

Gas-polymer-matrix model, 687 Gaussian or random-flight statistics, 246 Gel layer, 697... 

An alternative model known as the gas polymer matrix theory has been proposed, which assumes that there is a continuous variation in the state of the dissolved gas molecules rather than the two distinct states as proposed in the dual-mode theory. 

Fourth, the model of a rigid cage for a bimolecular reaction in the polymer matrix helps to explain another specific feature. This model explains the simultaneous increase in activation energy and preexponential factor on transferring the reaction from the liquid (Eh At) to solid polymer matrix (Es, As). In the nonpolar liquid phase / obs = E = gas but in the polymer matrix [3,21] it is... 

The sorption and diffusion behaviour of gas mixtures is of particular interest from the point of view of membrane gas separation, which is steadily gaining in importance by virtue of its low energy requirements. On the basis of the dual mode sorption model, one may reasonably expect that sorption of a binary gas mixture A, B in the polymer matrix will exhibit little gas-gas interaction and hence will tend to occur essentially additively. In the Langmuir-like mode of sorption, on the other hand, there will be competition between A and B for the limited number of available sites. These considerations led 67) to the following reformulation of Eqs. (8) and (9)... 

Sefcik M. D., Raucher D. The Matrix Model of Gas Sorption and Diffusion in Glassy Polymers, to be published... 

The susceptibility tensors measure the macroscopic compliance of the electrons. Since the second order polarization is a second rank tensor, SHG is zero in a centrosymmetric or randomly oriented system. To make the material capable of SHG, the NLO dopants must be oriented noncentrosymmetricaly in the polymer matrix (2-3). When modeling the poled, doped films using a free gas approximation, the poled second order susceptibilities are given by (2.19)... 

In the following chapter we present the matrix model of gas sorption and diffusion in glassy polymers which is based on the observation that gas molecules interact with the polymer, thereby altering the solubility and diffusion coefficients of the polymer matrix. 

Both the matrix-model and the dual-model represent the experimental data satisfactory (Fig. 1). After modeling sorption measurements in several gas-polymer systems we have observed no systematic differences between the mathematical descriptions of the two models. 

The first attempts in the direction of simulating theoretically at an atomistic level the diffusion of simple gas molecules in a polymer matrix were made more than two decades ago (100). But, the systematic development of ab initio computer simulations of penetrant diffusion in polymeric systems dates only from the late 80 s (101-104). At the beginning of the 90 s it was achieved to simulate some qualitative aspects such as the diffusion mechanism, temperature, and pressure dependence of diffusion coefficients (105-109). The polymers chosen for investigation mainly fell into two categories either they were easily described (model elastomers or polyethylene) or they were known to have, for simple permanent gases like H2, 02, N2, H20 or CH4,... 

Barbari, Koros and Paul (1988) compared the two models ("dual-mode" and "matrix") on the basis of their experimental data. They state that both models give a good description of the experiments yet the dual mode model has their preference, since it has simple physical interpretations of the parameters and can be related rather well to gas and polymer characteristics. The parameters of the matrix model do, however, not follow any consistent trend. 

The mechanism of transport by pervaporation can be described in the light of the sample diffusion model [159], which comprises the following steps (a) evaporation of the analyte into the air gap (b) sorption into the membrane on the sample side (c) diffusion of the sorbed component through the polymer matrix and (d) desorption into a liquid or gas phase on the acceptor side. The last three steps are also included in industrial pervaporation processes. 

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. 

When two gas species are present, competition can restrict both the solubility within the polymer matrix and the amount adsorbed in the Langmuir free volume. Competition in the former case is best modelled by adjustments to based on Equations (11.20) or (11.22). To accoimt for changes to the occupancy of the Langmuir sites for a binary mixture of gases A and B, the mobile concentration of gas A becomes [22] ... 

It has been shown in a previous section that, in most cases of practical interest, the rate of gas permeation through nonporous polymer membranes is cOTitrolled by the diffusion of the penetrant gas in the polymer matrix. Many theoretical models have been proposed in the literature to describe the mechanisms of gas diffusion in polymers on a molecular level. Such models provide expressions for gas diffusion coefficients, and sometimes also for permeability coefficients, derived from free volume, statistical-mechanical, energetic, structural, or other considerations. The formulation of these coefficients is complicated by the fact that gas transport occurs by markedly different mechanisms in rubbery and glassy polymers. 

The Cussler model focuses on the diffusion of small gas molecules through a polymer matrix which is partly filled of impermeable flakes which are oriented perpendicular to the direction of diffusion. Therefore, the diffusion process is mainly related to three factors the tortuous wiggles to get around the flake, the tight slits between the flakes, and the resistance of going from the wiggle to the slit. This model proposes that the diffusion depends on the volume fraction of the impermeable filler and the aspect ratio. Then, a permeability model of Cussler can be obtained by multiplying the diffusion by the appropriate solubility as following equation ... 

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