Mathematical Modelling of the Permeation of Gases in Polymers (original) (raw)
Related papers
“Transition-Site” Model for the Permeation of Gases and Vapors through Compact Films of Polymers
The theory of the permeation of effectively spherical molecules of gases and vapors (permeants) through films of compact amorphous solids such as polymers is developed using the activated-jump model (AJM). The conventional view is that permeation has to be analyzed as the resultant of sorption (solution) and diffusion effects. By contrast, in the present article, it is proposed that permeation may be viewed as a simple or fundamental process. This is suggested by a number of experimental observations: (a) the permeation correlations of Szwarc (1955-1956); (b) the "Permachor" concept of Salame (1961)(1962)(1963)(1964)(1965)(1966)(1967)(1968)(1969)(1970)(1971)(1972)(1973); (c) the "ideal" permeation behavior of water vapor through moderately polar polymers; (d) the absence of any effect of oxidation on the water-vapor permeability of polyethylene (PE); and (e) the "isokinetic" correlations between the Arrhenius parameters for permeation. The conventional AJM for diffusion is analyzed using the principle of microscopic reversibility, which shows that the average jump is characterized by a "transition site" L at its midpoint, analogous to the transition state in chemical reactions. For amorphous solids, these transition sites would be structural features, distributed at random and with their axes pointing at random. This leads to the present transition-site model (TSM) of permeation, where, at the steady state, a certain fraction of these sites will be transiently occupied by molecules of the permeant in equilibrium with the free molecules at that level. The concentration of these free molecules corresponds to the thermodynamic activity at that point, that is, for gases and vapors, the partial pressure. The ratedetermining step of the permeation process is then taken to be the release of the permeant molecule from the transition site according to classical transition-state theory. Using an idealized cubic-lattice model for the distribution of the transition sites, this is shown to lead to the observed proportionalities of the permeation rate to the area of the film, the pressure difference across it, and the reciprocal of the film thickness. It also accords with the observed Arrhenius-type dependence of the permeability coefficient on temperature, where the Arrhenius parameters relate to the thermodynamic parameters for the transfer of the permeant molecule from the gas phase and its insertion in the transition site. The Arrhenius parameters from the literature (Polymer Handbook) for 16 homopolymers-NR, PA 11, PC, PDMB, PDMS, three PEs (HDPE, LDPE, and hydrogenated polybutadiene), PETFE, PEMA, PET, PP, PTFE, PVAC, PVBZ, and PVC-with 16 "simple" permeants-H 2 , He, CH 4 , Ne, N 2 , CO, O 2 , HCl, Ar, CO 2 , SO 2 , Cl 2 , Kr, SiF 4 , Xe, and SF 6 as well as H 2 O vapor-are used as the dataset. These Arrhenius parameters are first discussed in relation to isokinetic behavior. They are then correlated according to the TSM theory with the van der Waals molecular diameter of the permeant G , and its absolute entropy S 0 . With certain exceptions, linear correlations are obtained with the 10 smaller-molecule permeants (He to CH 4 ) 981 that show that they use the same set of transition sites, below and above the glass transition temperature, with each polymer; the permeant molecules evidently behave here as "hard spheres," regardless of their other chemical characteristics. This enables estimates to be made of the four characteristic parameters for the polymer: the intersite spacing (equivalent to the lattice parameter of the idealized model and to the jump length of the AJM); the size of the transition-site aperture, L ; the force constant associated with expansion of the aperture by the permeant molecule; and the entropy increment also associated with this expansion. For most of the systems, the sitespacing is of the order of 10 nm, and the aperture L is about 200 pm. The theory provides a molecular basis for the interpretation and design of the permeation characteristics of polymers.
Rate type equations for the diffusion in polymers: Thermodynamic constraints
AIChE Journal, 1993
ABSTRACT Conditions imposed by the second law of thermodynamics on viscoelastic rate type constitutive equations for the diffusive mass flux are considered. The analysis of three different rate type models proposed in the literature points out that presently physically unrealistic predictions are possible in desorption processes. The thermodynamic analysis of such models, based on the entropy inequality and on the stability requirement of the equilibrium states, leads to precise relationships among relaxation times, diffusion coefficients, and entropy equations of state. In particular, the analysis shows that relaxation times and diffusion coefficients cannot be simply constant numbers. When the thermodynamic constraints imposed on the constitutive equations are introduced, the models do not show physically unrealistic behaviors any more; Fickian diffusion close to the pure penetrant or pure polymer regions is also recovered. Finally, it is shown that the stability requirement for the equilibrium states may introduce very rigid requirements for the model feasibility, well beyond what appears explicitly from the kinetic equations alone.
Computer simulation study of the permeability of driven polymers through porous media
Physical Review E, 1995
A computer simulation model is used to study the permeability of polymer chains driven by a biased How field through a porous medium in two dimensions. The chains are modeled by constrained selfavoiding walks, which reptate through the heterogeneous medium with a biased probability imposed by the driven field. A linear response description is used to evaluate an effective permeability. The permeability o. shows an unusual decay behavior on reducing the porosity p,. We find that the permeability decreases on increasing the bias above a characteristic value 8,. This characteristic bias shows a logarithmic decay on reducing the porosity, 8,-y ln(1p,), with y =0.35. The permeability decays with the length (L,) of the chains; at low polymer concentration it shows a power-law decay, o.-I.. . the exponent o. ' is nonuniversal and depends on both the porosity as well as the biased field (cr =1.64-3.73). We find that the biased field 8 and porosity p, affect the conformation of the chains. The radius of gyration R~o f the chains increases with increasing biased field in high porosity, while it decreases on decreasing the porosity at high field bias. In high porosity and low polymer concentrations, the radius of gyration shows a power-law dependence on the chain length, R~-L, , with v depending on the biased field {v=0.84-0.94). In order to explain the deviations from the Darcy Law for the polymer Aow, a plausible nonlinear response theory via a power-law response formula is suggested; we point out the associated complexities involved in addressing the How problems in driven polymers.
Chemical Engineering Science, 2005
Transport properties are important information not only for industrial equipment design but also for many research areas. While there is a well-developed theory for gases at low densities, there is no established theory to calculate diffusion coefficients for dense fluids, especially for polymeric solutions. Recently, a database of 96 self-diffusion coefficient data points were obtained from molecular dynamics (MD) simulations for freely jointed Lennard-Jones chains (LJC) with lengths of 2, 4, 8 and 16 at reduced densities ranging from 0.1 to 0.9 and in the reduced temperature interval of 1.5 to 4. These data were used to develop an equation that correlates MD self-diffusion coefficient points with an overall absolute average deviation of 15.3%. The aim of this work is to show that this equation can be used to calculate diffusivities of pure liquids and liquid mixtures, including polymeric solutions. The proposed equation is used for correlating self-diffusion coefficients for 22 pure real substances and then for predicting mutual diffusion coefficients for 12 binary liquid mixtures. The proposed equation is also used to calculate mutual diffusion coefficients for polymeric systems as: polystyrene-toluene at 110 • C, poly(vinyl acetate)-toluene at 35 • C, and poly(vinyl acetate)-chloroform at 35 and 45 • C. Results show that the model developed here seems to be a promising approach for correlating mutual diffusion coefficients not only for small-molecule systems but also for polymer-solvent systems. One advantage of the equation proposed here is that the parameters have physical meaning and most of them can be estimated without any information on binary diffusion data.
Diffusion and permeation are discussed within the context of irreversible thermodynamics. A new expression for the generalized Stokes-Einstein equation is obtained which links the permeability to the diffusivity of a two-component solution and contains the poroelastic Biot-Willis coefficient. The theory is illustrated by predicting the concentration and pressure profiles during the filtration of a protein solution. At low concentrations the proteins diffuse independently while at higher concentrations they form a nearly rigid porous glass through which the fluid permeates. The theoretically determined pressure drop is nonlinear in the diffusion regime and linear in the permeation regime, in quantitative agreement with experimental measurements.
Unified altered free volume state model for transport phenomena in polymeric media
Pure and Applied Chemistry, 1983
An Altered Free Volume State (AFVS) model has been proposed for analysing and correlating a variety of transport phenomena in poliiieric media. The key concept is the calculation of the alteration of the free volume state of the parent medium with respect to a carefully defined reference state. This approach enables the prediction of the influence of alteration of many variables including the changes in the physico.-chemical structural attributes of the polymeric systems on a surprisingly large variety of transport phenomena. The successful application of this unified model has been demonstrated by analysing exhaustive experimental data. INTRODUCT ION A considerable effort has been spent in analysing and modelling transport processes involving viscous flow, diffusional transport, theniial conduction, electrical conduction etc. in poliieric media. Apart from a significant number of experimental investigations which exists, a large number of predictive or correlative equations, which are based on either molecular or phenomenological considerations have been proposed. In most such
A mathematical model for a dissolving polymer
AIChE Journal, 1995
In certain polymer-penetrant ,systems, nonlinear viscoelastic effects dominate those of Fickian diffusion. This behavior is often embodied in a memory integral incorporating nonlocal time effects into the dynamics; this integral can be derived from an augmented chemical potential. The mathematical framework presented is a moving boundary-value problem. The boundary separates the polymer into two distinct states: glassy and rubbery, where different physical processes dominate. The moving boundary condition that results is not solvable by similarity solutions, but can be solved by perturbation and integral equation techniques. Asymptotic solutions are obtained where sharp fronts move with constant speed. The resultant profiles are quite similar to experimental results in a dissolving polymer. It is then demonstrated that such a model has a limit on the allowable front speed and a self-regulating mass uptake.
Macromolecular Theory and Simulations, 1995
The measured transient permeation kinetics of acetic acid from a water-acetic acid mixture through poly(viny1 alcohol) films could not be accounted for by the Fick law with a constant diffusion coefficient. A new calculation procedure was developed on the basis of simulation results of the Fickian kinetics in which the diffusion coefficient was assumed to increase exponentially with the local permeant concentration. A fast and reliable fitting procedure, which was set up on a Personal Computer, involves an iterative numerical adjustment of (a) the value of the limiting diffusion coefficient D* (i. e., difusion of the permeant in dry polymer) using the onset part of the permeation rate followed by (b) the value at the upstream face of the plasticization term (argument of the exponential function) using the shape of the experimental curve. The values obtained from fitting the model to the transient kinetics showed that the limiting diffusion coefficient increases drastically, but the plasticization term changes little, with increasing temperature. 0 1995, Huthig &
Gas permeation through a polymer network
Journal of Physics-condensed Matter, 2005
We study the diffusion of gas molecules through a two-dimensional network of polymers with the help of Monte Carlo simulations. The polymers are modeled as non-interacting random walks on the bonds of a two-dimensional square lattice, while the gas particles occupy the lattice cells. When a particle attempts to jump to a nearest-neighbor empty cell, it has to overcome an energy barrier which is determined by the number of polymer segments on the bond separating the two cells. We investigate the gas current JJJ as a function of the mean segment density rho\rhorho, the polymer length ell\ellell and the probability qmq^{m}qm for hopping across mmm segments. Whereas JJJ decreases monotonically with rho\rhorho for fixed ell\ellell, its behavior for fixed rho\rhorho and increasing ell\ellell depends strongly on qqq. For small, non-zero qqq, JJJ appears to increase slowly with ell\ellell. In contrast, for q=0q=0q=0, it is dominated by the underlying percolation problem and can be non-monotonic. We provide heuristic arguments to put these interesting phenomena into context.