Diffusion In Glassy Polymers Research Articles

Correlations with and prediction of activation energies of gas permeation and diffusion in glassy polymers

Three types of novel correlations for activation energies of gas permeation E P and diffusion E D in amorphous glassy polymers are considered and their application for prediction of the E P and E D values for different gases are examined. The first one is based on application of the group contribution method. Combined consideration of the equation of free volume and Arrhenius equation results in the correlation of E P and E D with free volume V f and fractional free volume (FFV). At last, the correlations between E P and the permeability coefficient at a certain reference temperature P( T ref), as well as E D versus D( T ref), are based on the fulfilment of the so-called compensation effect between activation energies and preexponential factors in activated processes. Examples of applicability of the correlations considered and recommendations for their use in prediction of the E P and E D values are given for transport of various gases in glassy polymers and separately in amorphous glassy polyimides.

Diffusion in glassy polymers — relaxation and antiplasticization

The applications of free-volume concepts to describe diffusion in polymer-solvent systems is reviewed. It is shown how the Vrentas-Duda free-volume model for diffusion in polymer solutions above the glass transition temperature can be modified to describe diffusion of low molecular weight species in glassy polymers. The free-volume model for the diffusion of a trace amount of solvent in a glassy polymer is evaluated utilizing data for toluene-polystyrene. The inverse gas chromatography technique using a capillary column coated with a polymer film was employed to obtain diffusivity data for the toluene-polystyrene system over a temperature range traversing the glass transition temperature. Finally, it is discussed how free-volume concepts can be extended to describe antiplasticization behavior which occurs when molecular motions in glassy polymers are retarded by the addition of certain low molecular weight species.

Modeling of penetrant diffusion in glassy polymers with an integral sorption deborah number

AbstractA mathematical model was developed to explain the anomalous penetrant diffusion behavior in glassy polymers. The model equations were derived by using the linear irreversible thermodynamics theory and the kinematic relations in continuum mechanics, showing the coupling between the polymer mechanical behavior and penetrant transport. The Maxwell model was used as the stress–strain constitutive equation, from which the polymer relaxation time was defined. An integral sorption Deborah number was proposed as the ratio of the characteristic relaxation time in the glassy region to the characteristic diffusion time in the swollen region. With this definition, an integral sorption process was characterized by a single Deborah number and the controlling mechanism was identified in terms of the value of the Deborah number. The model equations were two coupled nonlinear differential equations. A finite difference method was developed for solving the model equations. Numerical simulation of integral sorption of penetrants in glassy polymers was performed. The simulation results show that (1) the present model can predict Case II transport behavior as well as the transition from Case II to Fickian diffusion and (2) the integral sorption Deborah number is a major parameter affecting the transition. © 1993 John Wiley & Sons, Inc.

Numerical simulation of anomalous penetrant diffusion in polymers

AbstractThis work introduces a new numerical algorithm that can be used to analyze complex problems of penetrant transport. Penetrant transport in polymers often deviates from the predictions of Fick's law because of the coupling between penetrant diffusion and the polymer mechanical behavior. This phenomenon is particularly important in glassy polymers. This leads to a model consisting of two coupled differential equations for penetrant diffusion and polymer stress relaxation, respectively. If the polymer relaxation is the rate‐limiting step, both the concentration and stress profiles are very steep. A new algorithm based on a finite difference method is proposed to solve the model equations. It features the development of a tridiagonal iterative method to solve the nonlinear finite difference equations obtained from the finite difference approximation of the differential equations. This method was found to be efficient and accurate. Numerical simulation of penetrant diffusion in glassy polymers was performed, showing that the integral sorption Deborah number is a major parameter affecting the transition from Fickian to anomalous diffusion behavior. © 1993 John Wiley & Sons, Inc.

A hole‐filling theory of anomalous diffusion in glassy polymers. Effects of microvoids

AbstractNon‐Fickian diffusion is often observed at low concentrations and without any obvious irreversible changes. This contradicts some of the predictions of the existing theories, which in general work quite well. For these cases, it is suggested that the filling of microvoids give rise to such effects. The effective diffusivity and a memory‐dependent source/sink term are obtained for describing the overall transport in heterogeneous media under local volume averaging. Some comparisons with experiments are shown. This mechanism is confined to low concentrations and supplements the existing theories which are effective at higher concentrations. © John Wiley & Sons, Inc.

Diffusion in glassy polymers: A model using a homogenization method and the effective medium theory

Glassy polymers are considered as inhomogeneous with regions in which the gas sorption follows Henry's law and others where it follows Langmuir's law. It is assumed that the linear dimensions of these regions are small compared with the macroscopic length of interest but large compared with the mean free path of the penetrant gas molecules. Applying an homogenization method it is shown that the average flux is directly proportional to the concentration gradient in the polymer. This relationship can be expressed in terms of an effective diffusion coefficient Deff, which depends on the details of the microstructure. Deff is evaluated in the framework of the effective medium theory and compared with experimental data for diffusion of five vapors in ethylcellulose.

Free Boundary Problems in Controlled Release Pharmaceuticals. I: Diffusion in Glassy Polymers

This paper formulates and studies two different problems occurring in the formation and use of pharmaceuticals via controlled release methods. These problems involve a glassy polymer and a penetrant, and the central problem is to predict and control the diffusive behavior of the penetrant through the polymer. The mathematical theory yields free boundary problems which are studied in various asymptotic regimes.

The imbibition of carbon disulphide by poly(methyl methacrylate) latex particles

A light scattering study of the imbibition of carbon disulphide by poly(methyl methacrylate) latex particles is presented. It is shown that, by monitoring the time dependence of the scattered light intensity at a fixed angle, it is possible to determine penetrant diffusion coefficients and polymer segmental relaxation constants using a two stage model. It is found that the penetrant diffusion coefficients are typical of those for diffusion in glassy polymers and the segmental relaxation constants are independent of colloid particle size.

An application of an homogenization method to a model of diffusion in glassy polymers

AbstractThe sorption of gases in polymers below their glass‐transition temperature Tg is known in many cases to be described by the “dual sorption” theory, according to which the gas is held in accordance with both the Langmuir and Henry's laws. Based on this theory, expressions for the “effective diffusion coefficient” in the glassy polymers have been obtained by investigators in the past, notably by Paul and Koros.2 The present analysis regards the glassy polymers as inhomogeneous with regions on which the gas sorption follows the Langmuir law. Assuming that the linear dimensions of these regions, which are often referred to as “microvoids” (although they are not space filled by vacuum), are small compared to the macroscopic length of interest but large compared to the mean free path of the penetrant gas molecules, we derive a rigorous relation between the average flux and the concentration gradient in the polymer and show that this relation can be expressed in terms of an “effective diffusion coefficient” Deff which depends on the details of the microstructure, i.e., the size, shape and spatial distribution of the “microvoids.” This expression for Deff is shown to reduce to that of Paul and Koros2 in two situations: (1) when the “voids” consist of slabs running parallel to the concentration gradient, and (2) when the “voids” are spherical and the temperature of the polymer is not too different from Tg. The results of the present study lead to an alternative procedure for interpreting the experimental data on sorption and permeation which may have some advantages over the procedure currently employed. Finally, the analysis presented here is also applicable to polymers containing adsorptive fillers.

Diffusion in polymer‐solvent systems

AbstractRecent advances in the area of molecular diffusion in polymer‐solvent systems will be reviewed. Alfrey's classification scheme for the diffusion of solvent molecules in polymers will be used as a starting point. It will be shown that the various regions on Alfrey's diagram are distinguished by the ratio of two characteristic times, a characteristic relaxation time for the polymer‐solvent system and a characteristic diffusion time. Fickian diffusion is realized when this characteristic ratio is a small number, and anomalous behavior occurs when these two characteristic times are the same order of magnitude. Alfrey's “null” region corresponds to those conditions where this dimensionless ratio is much greater than one. The different manifestations of anomalous diffusion with polymer‐solvent systems will be discussed, including case II transport as defined by Alfrey, Gurnee, and Lloyd. Finally, diffusion in polymer‐solvent systems above the glass transition temperature where classical diffusion theory is applicable will be considered. Emphasis will be placed on the demonstration of the ability of theoretical methods based on free volume concepts to predict the concentration and temperature dependencies of diffusion coefficients. The extension of this free volume theory to describe diffusion in glassy polymers will also be demonstrated. Experimental results for several polymer‐solvent systems which exhibit different types of transport behavior will be used to illustrate the correlative and predictive capabilities of these recently developed theories.

Theoretical models for diffusion in glassy polymers. II

AbstractThe author refines and generalizes a model for diffusion in glassy polymers which he previously introduced. The model unifies many diverse observations by explicity formulating the common property of a glassy polymer in all its various modes, namely the finite relaxation time due to its slow response to changing conditions. An integral approximation method is used to study the motion of the penetrant front and the glass‐gel interface and a useful polynomial approximation method is introduced for use in special simple situations.

Theoretical models for diffusion in glassy polymers

AbstractA model is derived which incorporates and unifies many of the diverse observations occurring in diffusion in glassy polymers. This unification is made possible by explicitly formulating the common property of a glassy polymer in all its various modes, namely the finite relaxation time due to its slow response to changing conditions. The application and use of the model in various situations is discussed.

Gaseous Diffusion in Glassy Polymers

A model for gaseous diffusion in glassy polymers is developed with a view to accounting for the observations made in dual sorption and certain other phenomena in polymers below their glass transition temperature. In this paper a preliminary study of the effects of both the immobilizing mechanism and the generalized diffusion mechanism on travelling waves and the diffusive wavefronts is made.

A deformation model for Case II diffusion

Case II diffusion in glassy polymers is considered in terms of a deformation model. The process is controlled by the mechanical response of the glass just ahead of the sharp front to an osmostic swelling stress. The important factor governing the establishment of the sharp solvent front is taken to be the concentration dependence of the creep response time. These ideas are set in the context of existing models for diffusion in glassy polymers, and experimental evidence is presented to support the deformation mechanism.

A model incorporating reversible immobilization for sorption and diffusion in glassy polymers

AbstractA model incorporating reversible, bimolecular immobilization for diffusion and sorption in glassy polymers is developed. Sorption is considered to occur by two distinct mechanisms: ordinary diffusion‐controlled sorption and sorption resulting from the immobilization of diffusing gas molecules by prexisting sites in the polymer. Expressions are obtained for equilibrium sorption, transient sorption, and time lag. The effects of kinetic parameters of the model are illustrated and discussed.

Gas transport in glassy polymers

AbstractSorption kinetics and equilibria for CO2 in polycarbonate were studied over a range of temperatures from 35° to 65°C and pressures from 3 to 22 atm. The dual mode sorption model of Vieth et al. was used to test the data, and a comparison was made with previous work on glassy polystyrene. Further evidence of the validity of generality of the model in providing a consistent picture of small‐molecule diffusion in glassy polymers was thereby provided. The technical feasibility of membrane separation of H2 from a H2/CO2 mixture by selective permeation through polycarbonate was examined at 50°C.

Quantitative analysis of gaseous diffusion in glassy polymers

AbstractThe current theoretical treatment of diffusion of gases in glassy polymers is based on the “dual sorption” model, but also includes certain other important assumptions, at least some of which cannot be fully justified a priori. They require experimental validation, which, however, is not possible by the procedures used or proposed so far. Methods suitable for this purpose are discussed here.

Effect of immobilizing adsorption on the diffusion time lag

AbstractA “dual sorption” model has been proposed by Michaels, Vieth, et al. to explain extensive equilibrium sorption data for several gases in some glassy polymers. To explain data on sorption kinetics, it was further postulated that one of the sorption modes immobilizes the gas molecules. Stated mathematically, this model leads to a modified form of Fick's second law. Both normal and desorption time lags for diffusion have been computed here for this model of diffusion in glassy polymers. The computed time lags are shown to be dependent on the boundary concentrations used in permeation. Experimental measurement of these time lags would be a sensitive and critical test to ascertain the validity of this theory.

Diffusion in glassy polymers. I

AbstractSynopsisThe diffusion of six solvents in three crosslinked, glassy epoxy polymers is studied. Case II swelling and Fickian sorption are observed as two simple limiting cases. The mechanism of diffusion changes from one limit to another as the nature of the solvent or the crosslink density of the polymer is altered. With mixed solvents, properly chosen, a superposition of Fickian diffusion and case II swelling is observed.

Diffusion in Glassy Polymers. I. Effects of Initial Concentration upon the Sorption of Organic Vapors in Polymers

Diffusion in Glassy Polymers. I. Effects of Initial Concentration upon the Sorption of Organic Vapors in Polymers