Concentration dependence model polymer system

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 ... 

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... 

The Rouse and Zimm models provide little direct help in dealing with t](y) since each predicts a viscosity which is independent of shear rate. The principal interest here is in concentrated systems where entanglement effects are prominent. Nevertheless, shear rate can influence the viscosity of polymer systems at all levels of concentration, including infinite dilution (307) and melts with M < Mc (308, 315). It is therefore essential to identify the causes of shear rate dependence in systems of isolated or weakly interactions molecules in order to separate intramole-... 

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. 

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,... 

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). 

Despite the large number of analytical solutions available for the diffusion equation, their usefulness is restricted to simple geometries and constant diffusion coefficients. The boundary conditions, which can be analytically handled, are equally simple. However, there are many cases of practical interest where the simplifying assumptions introduced when deriving analytical solutions are unacceptable. For example, the diffusion process in polymer systems is sometimes characterized by markedly concentration-dependent diffusion coefficients, which make any analytical result inapplicable. Moreover, the analytical solutions being generally expressed in the form of infinite series, their numerical evaluation is no trivial task. That is, the simplicity of the adopted models is not necessarily reflected by an equivalent simplicity of evaluation. 

Several interesting theoretical papers have appeared dealing with molecular dynamics and excimer formation in polymer systems. Frank and coworkers have developed a model to describe the transport of electronic excitation energy in polymer chains. The theory applies to an isolated chain with a small concentration of randomly placed chromophores, and a three-dimensional transport model was used to solve the problem which is based on a diagrammatic expansion of the transport Green function. (The Green function is related to time-dependent and photostationary depolarization and to transient and steady-state trap fluorescence.) The analysis is shown to be... 

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]. 

Fujita s model is valid for penetrant/polymer systems with diffusion coefficients that exhibit a strong concentration dependence, such as organic vapors in amorphous polymers, (20,22,24-27), but fails to describe the difference between water in poly(vinyl acetate) and in poly(methyl acrylate) (28). This may be due to the hydrogen-bonding nature of water rather than to a failure of the model. Fujita viewed his theory as inappropriate for small penetrant molecules, whose diffusion is largely independent of concentration, because the critical hole size for such penetrants is... 

Vrentas and Duda s theory formulates a method of predicting the mutual diffusion coefficient D of a penetrant/polymer system. The revised version ( 8) of this theory describes the temperature and concentration dependence of D but requires values for a number of parameters for a binary system. The data needed for evaluation of these parameters include the Tg of both the polymer and the penetrant, the density and viscosity as a function of temperature for the pure polymer and penetrant, at least three values of the diffusivity for the penetrant/polymer system at two or more temperatures, and the solubility of the penetrant in the polymer or other thermodynamic data from which the Flory interaction parameter % (assumed to be independent of concentration and temperature) can be determined. An extension of this model has been made to describe the effect of the glass transition on the free volume and on the diffusion process (23.) ... 

Caroline and co-workers have recently reported measurements of translational diffusion coefficients in solutions of PS in two mixed-solvent systems at or near theta conditions. In the solvent CCb-methanol (85), they observed the diffusion theta state, defined when the coefficient y of Equation 41 equals 0.5, to occur at 25°C and a volume fraction of CCI4, (fyCCU = 0.8025. In this system there is strong preferential adsorption of the polymer for CCI4, and it is not possible to define a true theta state such that y = a = V2 and A2 = 0 simultaneously. Under diffusion theta conditions, the concentration dependence of Dt apparently is closely described by the Pyun-Fixman hard-sphere model. In the mixed solvent benzene—2 propanol, polystyrene exhibits a true theta condition at T = 25.5°C and (benzene) = 0.04. Frost and Caroline confirmed that y = 0.5 within experimental error in this system (86) and report that values of the parameter fcf are scattered between the extreme values corresponding to the predictions of Yamakawa (and Imai) and the soft-sphere model of Pyun-Fixman (or the Freed theory). 

The constancy of the effective diffusion coefficient In the substrate transport with reaction model. Equation 26 versus Equation 27, depends on two factors. First, if the volume fraction of polymer is not constant with time, radial position in the gel, or extent of reaction, then D is influenced by the relation given in Equation 40. Dooley ct al. (11) present an example of this in their study. Second, if the substrate s diffusion coefficient in the solvent alone is dependent on substrate concentration at the range of concentrations in the reaction system, then D is Influenced similarly according to Equation 39. The assumption of a constant diffusion coefficient in the substrate transport with reaction model must always be justified. 

This model has been tested primarily on polymer systems (a more detailed treatment will be presented in a publication in preparation) but appears to explain to a great extent the observed features of FE, in particular, the time dependence of EE and PIE. In terms of applications involving adhesive failure, such a model would be useful to relate the measured FE characteristics to fracture phenomena of interest i.e., the fragment species, density of trapped electrons, initial concentrations (before decay) and their rate of production all should be closely related to processes occurring at the crack tip. Although the FE model and characteristics outlined above have been observed in a limited number of situations, there appears to be evidence for considerable generality. 

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