Penetrant molecule models

In glassy polymers tire interactions of tire penetrant molecules witli tire polymer matrix differ from one sorjDtion site to anotlier. A limiting description of tire interaction distribution is known under tire name of tire dual-soriDtion model [, 60]. In tliis model, tire concentration of tire penetrant molecules consists of two parts. One obeys Henry s law and tire otlier a Langmuir isotlienn ... 

The above results indicate that the general characteristics of gaseous diffusion in glassy polymer membranes can be represented reasonably well in terms of the dualmode concept. The basic reason for the observed increasing D(C) function is seen to be the concurrent increasing proportion of less strongly sorbed (and hence more easily activated) penetrant molecules. The model is, no doubt, highly idealized, but is nevertheless shown to be physically reasonable and consistent with the correspond-... 

Rashid, M.H. 2005. Electrically assisted enhancement of human skin penetration by model molecules. PhD. thesis, University of Bradford, U.K. 

Generally, polymers that crystallize are not considered good candidates for membrane materials however, there are some exceptions [26, 27]. The presence of crystallinity reduces permeability [28,29] and good membranes should be capable of high fluxes. The usual physical picture is to think of a semicrystalline polymer in terms of a simple two-phase model one phase being amorphous and the other being crystalline. In the typical case, the crystals do not sorb or transmit penetrant molecules the following relationship has been proposed [28, 29] to describe the extent to which crystallinity reduces permeability from that if the polymer were amorphous... 

Many theoretical models have been proposed in the last six decades to describe the diffusion of small penetrant molecules in polymer matrices. According to the distance-time scale at wich the basic physico-chemical processes involved in these models have been set one could classify them in two main categories, namely microscopic and atomistic diffusion models. 

Since about 15 years, with the advent of more and more powerfull computers and appropriate softwares, it is possible to develop also atomistic models for the diffusion of small penetrants in polymeric matrices. In principle the development of this computational approach starts from very elementary physico-chemical data - called also first-principles - on the penetrant polymer system. The dimensions of the atoms, the interatomic distances and molecular chain angles, the potential fields acting on the atoms and molecules and other local parameters are used to generate a polymer structure, to insert the penetrant molecules in its free-volumes and then to simulate the motion of these penetrant molecules in the polymer matrix. Determining the size and rate of these motions makes it possible to calculate the diffusion coefficient and characterize the diffusional mechanism. 

To construct the model, it has been assumed that the amorphous polymer regions posses an approximately paracrystalline order with chain bundles locally parallel. A penetrant molecule may diffuse through the matrix of the polymer by two modes of motion, Fig. 5-2. 

A detailed discussion on how the equations of this model have been used to analyse the diffusion of simple gaseous penetrants in a series of polymers was given (46,47). The results reported show that the model allows a reasonable correlation of the experimental Ed with the diameters of the diffusant species. The main advantage of this classical molecular model is that the Ed can be estimated without using any adjustable parameters derived from correlation with experimetal diffusion data. However to use the equation proposed in the framework of this model to evaluate an Ed the knowledge of a series material, structural and thermodynamic parameters of the host polymer as well as the dimension and shape of the penetrant molecules must be known (45,46). The determination of these data, as far as they have not already been tabulated in some publication, depends on the availability of certain experimental data and is not always a simple task. 

Among the popular methods for interpreting the diffusion of small penetrants in polymers are the so called free-volume models (6,11,13,51-54). The basic assumption of these models is that the mobility of both polymer segments and penetrant molecules is primarly determined by the available free-volume in the penetrant polymer system. The free-volume of the polymer is regarded as an empty volume between the chains of the polymer. Similarly the free-volume of the penetrant can be regarded as the volume not occupied between the molecules of the penetrant. 

In order to develop a consistent free-volume diffusion model, there are some issues which must be addressed, namely i) how the currently available free-volume for the diffusion process is defined, ii) how this free-volume is distributed among the polymer segments and the penetrant molecules and iii) how much energy is required for the redistribution of the free-volume. Any valid free-volume diffusion model addresses these issues both from the phenomenologic and quantitative points of view such that the diffusion process is described adequately down to the microscopic level. Vrentas and Duda stated that their free-volume model addresses these three issues in a more detailed form than previous diffusion models of the same type. Moreover, it was stated that the model allows the calculation of the absolute value of the diffusion coefficient and the activation energy of diffusion mainly from parameters which have physical significance, i.e. so-called first principles . In the framework of this model the derivation of the relation for the calculation of the self-diffusion coefficient of the sol-... 

Somewhat closer to the designation of a microscopic model are those diffusion theories which model the transport processes by stochastic rate equations. In the most simple of these models an unique transition rate of penetrant molecules between smaller cells of the same energy is determined as function of gross thermodynamic properties and molecular structure characteristics of the penetrant polymer system. Unfortunately, until now the diffusion models developed on this basis also require a number of adjustable parameters without precise physical meaning. Moreover, the problem of these later models is that in order to predict the absolute value of the diffusion coefficient at least a most probable average length of the elementary diffusion jump must be known. But in the framework of this type of microscopic model, it is not possible to determine this parameter from first principles . 

Traditional approaches for the calculation of the phase equilibria and sorption of penetrant molecules in polymers are based on equation-of-state models [27,28,29], which take into account the PVT properties of both gas and polymer, and the activity coefficient models [30], which take into account the specific interactions between... 

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 ... 

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]. 

Brandt (13) used a simple molecular model that takes into account polymer structure in order to estimate E. The activated state in diffusion involves two polymer chains which bend symmetrically in order to create a passageway for the penetrant molecule (see Figure la). Cooperation of neighboring polymer... 

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... 

Pace and Datyner ( 5, 5) have also proposed a model for the absorption (solution) of small molecules in polymers applicable at temperatures above and below Tg, which incorporates the dual-mode sorption model for the glassy region. The presence of microvoids is assumed for rubbery polymers as well as for polymer glasses. "Hole filling" is suggested as an important sorption mode above as well as below Tg, with one crucial difference between the sorption mechanism in the rubbery and glassy regions hole saturation does not occur in the rubbery state because new microvoids are formed to replace those filled with penetrant molecules. 

The free-volume model proposed by Vrentas and Duda (67-69) is based on the models of Cohen and Turnbull and of Fujita, while utilizing Bearman s (7j0) relation between the mutual diffusion coefficient and the friction coefficient as well as the entanglement theory of Bueche (71) and Flory s (72) thermodynamic theory. The formulation of Vrentas and Duda relaxes the assumptions deemed responsible for the deficiencies of Fujita s model. Among the latter is the assumption that the molecular weight of that part of the polymer chain involved in a unit "jump" of a penetrant molecule is equal to the... 

A substantial number and variety of models of gas transport in polymers have been proposed during the last 20-30 years, in view of the great practical and scientific importance of this process. Molecular-type models are potentially most useful, since they relate diffusion coefficients to fundamental physicochemical properties of the polymers and penetrant molecules, in conjunction with the pertinent molecular interactions. However, the molecular models proposed up to now are overly simplified and contain one or more adjustable parameters. Phenomenological models, such as the dual-mode sorption model and some free-volume models, are very useful for the correlation and comparison of experimental data. 

Figure 1. Schematic illustration of how the results of three different types of calculations, each one providing a perspective at a different scale, can be combined synergistically, to construct a unified physical model for the transport of penetrant molecules in plastics.

The drastic differences of Dqj and from the values observed in PS, and the observation that the density was the only parameter entering the model of P D with a very different value for PVDC than for any of the non-barrier polymers studied, [13,14] indicate that the packing efficiency is the most important descriptive physical parameter both at the "global" and at the "intermediate" scales. The investigation of (i) the local distribution of unoccupied volume, and (ii) the MD trajectory of the penetrant molecule in the polymer matrix, will therefore be very useful. 

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