INFLUENCE OF RHEOLOGICAL PROPERTIES IN MASS TRANSFER PHENOMENA: SUPER CASE II SORPTION IN GLASSY POLYMERS

The sorption rate of swelling penetrants in glassy polymers has been considered as controlled both by the swelling kinetics and the penetrant diffusion through the swollen layer. The stress exerted on the glassy core at the moving boundary is the driving force for the swelling, and results from an osmotic stress and a differential swelling stress contribution. During the sorption process, the osmotic stress at the moving boundary decreases, due to the increasing diffusion resistance; the differential swelling stress, on the contrary, increases giving rise to an acceleration of the front velocity (Super Case II).

The particular case of negligible diffusion resistance in the swollen region is here considered in more detail. It is shown that the rheological properties of both swollen and glassy phases crucially enter the mass transport problem; thestress relaxation in the swollen region must be taken into account in order to obtain a thickness dependent Super Case 11 effect     

The effect of penetrant activity and temperature on the anomalous diffusion of hydrocarbons and solvent crazing in polystyrene part I: Biaxially oriented polystyrene

The effects of temperature and penetrant activity on the sorption kinetics and equilibria of a series of alkanes in glassy, biaxially oriented polystyrene were studied. Normal isomers of pentane, hexane, and heptane cause crazing of polystyrene film samples at high penetrant activities (> 0.85). Crazing kinetics are identical to the kinetics of Case II transport. Transport of these normal hydrocarbons in glassy polystyrene in the temperature range 25 to 50°C is markedly non-Fickian; limiting Case II transport is observed at activities in exces of 0.6. Sorption appears to be controlled by highly activated relaxation processes including primary bond breakage at these high penetrant activities. Fickian diffusion behavior is approached, however, as penetrant activity is reduced. Sorption of the branched isomers of these compounds does not result in polymer microfailure. The sorption kinetics of the branched isomers, although time dependent, appear to be controlled primarily by thermally activated diffusion rather than large scale polymer relaxations which control Case II transport.     

A theory of case II diffusion

A theory is proposed to explain the transport behaviour of organic penetrants in glassy polymers in terms of two basic parameters: the diffusivity of the penetrant, D, and the viscous flow rate of the glassy polymer, $\text{1}\text{η}{}_0$. The rate controlling process for transport in these systems is considered to be diffusion of solvent down an activity gradient coupled with time-dependent mechanical deformation of the polymer glass in response to the swelling stress. The theory combines these two factors and is able to predict a wide range of observed transport phenomena from Fickian diffusion kinetics at one extreme to so-called Case II and Super-Case II behaviour at the other. The existence of a sharp front separating swollen and unpenetrated polymer is shown to result from the concentration dependence of the viscous flow rate.     

Solute and penetrant diffusion in swellable polymers. VII. A free volume-based model with mechanical relaxation

Penetrant transport through and solute release from continuously swelling polymers is viewed as a process associated with major structural changes in the polymer morphology. Changes in the diffusivities of penetrant and solute reflect a free volume mechanism for transport. The polymer is initially glassy with a uniform dispersion of solute. After the system is placed in contact with a thermodynamically good penetrant, a glassy/rubbery phase transition occurs at a well defined swelling interface. The Fickian equations with concentration-dependent diffusivities and moving boundaries are solved simultaneously in polymer-fixed coordinates. A constitutive relation is used to describe the effect of macromolecular relaxations on the rate of volume expansion as the polymer swells. The penetrant fractional uptake, solute fractional release, sample dimensions, swelling front position, and instantaneous swelling interface number are determined and related to the nature of the swelling process.     

A Model for the Diffusion of Moisture in Adhesive Joints. Part I: Equations Governing Diffusion

A methodology is proposed to relate the diffusion coefficient of small penetrant molecules in polymers to temperature, strain, and penetrant concentration. The approach used is based on well-known free volume theories. It is assumed that the transport kinetics is governed by the constant redistribution of the free volume, caused by the segmental motions of the polymeric chains. An expression for the diffusion coefficient is inferred from the temperature, strain, and penetrant concentration dependence of the free volume. The stress dependence of solubility is predicted from the Hildebrand theory. It is shown that the resulting constitutive equations exhibit many features desirable for joint durability studies. Finally, a non-Fickian driving force arising from differential swelling is included in the governing equations.     

Experimental and constitutive analyses of the stress relaxation behavior of glassy polymers

The physical mechanism underlying the mechanical behavior of glassy polymers has been studied over decades but remains a long-standing issue. A consensus view achieved is that the yield, flow, and stress relaxation behaviors are due to structural relaxation in the polymer mainly caused by chain conformation transitions. This is the key physical idea behind the many existing elastic–plastic constitutive models for glassy polymers. In this paper, such a constitutive model was employed for predicting and analyzing the stress relaxation of a glassy polymer. It is found that the model works well in predicting the pre-yield stress relaxation but significantly underestimates the post-yield stress relaxation. As considering the chain conformation transition alone leads to a dilemma for the model to concurrently represent the yield/flow and stress relaxation behaviors, the model was extended to incorporate an additional structural relaxation mechanism assumed to originate from the dissociation of weak linkages in the chain network. The extended model succeeds in concurrently representing the yield/flow, and stress relaxation behaviors in the whole deformation region, of which the reasons were analyzed. The knowledge revealed in this paper is instructive and may shed new light on understanding the structural relaxation and mechanical behavior of glassy polymers.     

A novel transport model for sorption and desorption of penetrants in dense polymeric membranes

The transport of penetrants in polymeric membranes is assumed to be a process of mixing of penetrant and polymer molecules, in which a creep strain is induced concurrently. Based on the assumption and combined with the mass conservation equation and phenomenological diffusive flux expression, a new transport model of penetrant in polymeric membrane is established, in which the total change of excess Gibbs free energy is considered as a sum of three parts calculated by the modified Scatchard-Hildebrand model, Flory-Huggins theory and linear viscoelastic theory, respectively. The partial difference equations in the model are solved by implicit finite difference method of Crank-Nicolson form, and the model parameters are optimized by simplex algorithm. The model is used to calculate various diffusions (Fickian diffusion, and non-Fickian diffusion) of ethanol in polyphthalazine ether sulfone (PPES), and polyphthalazine ether sulfone ketone (PPESK) membranes at 293.15, 298.15, and 303.15 K. The calculated results are in agreement with the experimental data and the maximal relative deviation is no more than 10.54% and the average relative deviation is 4.35%.     

Sorption and diffusion of alcohols in amorphous polymers

The effects of polymer composition and penetrant molecular size on the solubility and diffusivity of alcohol vapors in a series of well characterized isoprene-methyl methacrylate copolymers and their corresponding homopolymers has been investigated at room temperature. The rate of sorption behavior changes progressively from Fickian to non-Fickian, to Case II to “Super Case II” transport with increasing methyl methacrylate (MMA) content in the polymers. The equilibrium solubility of the alcohols increases linearly with increasing penetrant molecular size for polymers which are above their glass transition temperature and decreases for polymers which are below their T_g. The solubility also initially increases as an approximately linear function of MMA content in the copolymers. At about 55 mole percent MMA, the sorbed concentration either levels off or passes through a maximum depending on the size of the penetrant. The apparent “diffusion coefficients” (D) decrease with increasing molecular volume of the penetrants. An exponential dependence was found between these two variables for PMMA. These “diffusion coefficients” also decrease exponentially with increasing MMA content in these polymers. However, at 55 mole percent MMA the copolymer undergoes a rubber to glass transition at the temperature of the experiments. On this basis, it is suggested that the hindered chain segmental motion contributes to the sorption process in addition to strictly thermodynamic considerations. Free volume theory can be used to explain the mechanism of diffusion through the rubbery polymers while the “hole” theory can be applied to explain the transport of the penetrants through the glassy polymers.     

Strain effects in the mass flux of methanol in poly(methyl methacrylate)

Diffusion of organic solvents into glassy polymers often results in a phase transformation of the hard, solid polymer into a swollen, rubbery material. During the sorption, internal stresses exist in the swollen and glassy parts of the polymer and are thought to contribute significantly to the “anomalous” diffusion observed in many penetrant–polymer systems. In this investigation, isothermal sorption data for the methanol–poly(methyl methacrylate) system have been obtained on plates ranging in thickness from 1/32 to ¼ in. The results show features characteristic of both a strain-dependent diffusion coefficient and of a stress gradient contribution to the mass flux. An attempt to reproduce these results by combining a strain-dependent diffusion coefficient model with a stress-induced contribution to the flux is presented.     

Sorption and transport of benzene in poly(ethylene terephthalate)

The kinetics and equilibria of benzene sorption in poly(ethylene terephthalate) were measured at 40°C, 50°C, and 60°C, with benzene activities ranging from 0.02 to 0.3. At most experimental conditions, diffusion was found to be Fickian; however, evidence of non-Fickian transport was found at the highest activity levels. Values of the diffusion coefficient of benzene range from 10^-14 cm²/s at 40°C to 10^?12 cm²/s at 60°C in the limit of low concentrations. Nonlinear isotherms observed for benzene sorption were successfully interpreted in terms of the dual mode model for sorption in glassy polymers, whereby the sorbed penetrant exists as two populations: one sorbed according to Henry's law and the other following a Langmuir isotherm. Non-Fickian transport data were correlated with a model that superimposes diffusion of both the Henry's law and Langmuir populations (the “partial immobilization” model) upon first-order relaxation of the polymer matrix.     

Stress‐induced densification of glassy polymers in the subyield region

Constitutive equations are derived for the viscoelastic response of amorphous glassy polymers in the region of subyield deformations. The model treats an amorphous polymer as a composite material consisting of an ensemble of flow units, immobile holes, and clusters of interstitial free volume moving through a network of long chains to and from voids. Changes in macropressure lead to an increase in the equilibrium concentration of interstitial free volume that, in turn, induces diffusion of free‐volume elements from holes. The mass flow results in dissolution of voids that is observed as time‐dependent densification of a glassy polymer. It is demonstrated that the model correctly predicts stress relaxation and a decrease in the specific volume observed in uniaxial tensile and compressive tests on polycarbonate at room temperature. © 1999 John Wiley & Sons, Inc. J Appl Polym Sci 74: 1705–1718, 1999     

Influence of copolymer composition on non-fickian water transport through glassy copolymers

The dynamic swelling behavior of poly(2-hydroxyethyl methacrylate-co-methyl methacrylate) and poly(2-hydroxyethyl methacrylate-co-N-vinyl-2-pyrrolidone) in water was followed at 37°C. The results were analyzed in terms of a simple non-Fickian transport equation, which expresses the fractional penetrant uptake as an exponential function of the diffusion time. The exponent n, which indicates Fickian or non-Fickian transport mechanism, was correlated to the content of the more hydrophilic component of the copolymer. Photomicrographs obtained with polarized light offer new information about the position and movement of the penetrant front in glassy, hydrophilic polymers.     

Effective medium approximations for penetrant sorption in glassy polymers accounting for excess free volume

An accurate determination of a penetrant volume fraction in a swollen polymer is of crucial importance in a range of different technologies. Using optical methods, such as in-situ spectroscopic ellipsometry, it is possible to extract the thickness and refractive index of dry and swollen polymer films. The volume fraction of the penetrant can then be calculated from the change in thickness, or from the refractive index using effective medium approximations. For thermodynamically equilibrated and ideal swollen rubbery polymers, these calculations yield accurate results. However, for glassy polymers the influence of the excess free volume trapped within the polymer network during vitrification is rarely taken into account. In this work we investigate the effect of excess free volume in the calculations of penetrant volume fraction in a model glassy polymer – penetrant system. The influence of the excess free volume is included by extrapolating the properties of an equilibrium polymer matrix from above its glass transition temperature. The error between the approaches that do, and do not, take account for the non-equilibrium of the glassy polymer is quantified and the implications for other systems are discussed. The errors are shown to be very significant, especially when the dry polymer has a large excess free volume. Such materials are particularly relevant in membrane applications.     

溶剂在高分子膜中传递现象研究的进展

溶剂在高分子膜中吸收和解吸的传递机理非常复杂,对其传递过程的研究有助于膜分离过程的开发、膜材料的选择。本文对溶剂在高分子膜中传递现象研究的进展作简要介绍。     

Evaluation of a sorption equation for polymer–solvent systems

The predictive capabilities of a proposed sorption equation for describing the sorption behavior of glassy polymer–penetrant systems is evaluated. Factors determining the shapes of isotherms for glassy polymer–penetrant systems are considered. Data–theory comparisons are presented for three glassy polymer–penetrant systems. © 1994 John Wiley & Sons, Inc.     

Anomalous penetrant transport in glassy polymers: 4. Stresses in partially swollen polymers

The equilibrium swelling process and stress evolution of glassy polymer systems undergoing Case-II penetrant transport were examined using a rubber elasticity theory. Consideration of coupling between stresses and penetrant concentration permitted calculation of equilibrium stress and concentration profiles for a variety of molecular and geometric factors.     

Solvent osmotic stresses and the prediction of Case II transport kinetics

The rate controlling step in Case II transport kinetics is the swelling which occurs at the internal moving boundary. A physical model describing the swelling kinetics of glassy polymers in liquids is presented here. Using a thermodynamic argument, the stress induced by the penetrant on the glassy matrix is evaluated in terms of the penetrant concentration. The velocity of the swelling front is expressed in terms of the solvent stress, using the same functional relationship which gives the mechanical craze propagation rate, in terms of the mechanical stress. The resulting model permits the prediction of the kinetics of the swelling front from an independent characterization of mechanical properties.     

Bending-beam study of water sorption by thin poly(methyl methacrylate) films

A bending-beam technique has been employed to study the effects of film thickness (7–55 μm) and rate of cooling during film preparation (～ 6°C/h to a dry ice quench) on sorption characteristics of water by poly(methyl methacrylate) films coated on thin fused quartz beams (? 84 μm thick). In each experiment, the curvature of a polymer coated beam exposed to liquid water was monitored as a function of time by a low power laser pointer. With the use of a transport model which considers the sorption process as the linear superposition of contributions from Fickian diffusion and a first-order polymer molecular relaxation process, the beam curvature data were analyzed to determine the governing transport kinetics and associated transport parameters such as water diffusion coefficient and relaxation rate constant. From curvature analysis for thin films (7–13 μm in thickness), it was found that water diffusion proceeds at early times in a Fickian-like manner with a diffusion constant of 2–4 × 10^?9 cm²/s. At later times, significant relaxation contributions lead to non-Fickian diffusion behavior, an effect that is more pronounced as the film thickness or sample cooling rate decreases. In addition, sorption of water was found to reduce the film stress (initially tensile at ? 10⁸ dyn/cm²) at a rate that increases with sample cooling rate. The high initial film stress not present in free-standing films may account for the relatively higher diffusion coefficient (～ 2 × 10^?8 cm²/s) found here for very thick (55 μm) PMMA coatings. Because the bending-beam technique uses coated samples, it is especially well suited for studying penetrant transport into polymer coatings.     

Energetics of gas sorption in glassy polymers

The molar enthalpy for gas sorption in glassy polymers at a fixed concentration, often called the isosteric enthalpy of sorption, exhibits a clearly discernable minimum when plotted as a function of penetrant concentration. This unexpected behaviour has been observed in several glassy polymer systems including poly(ethylene terephthalate), polyacrylonitrile and polycarbonate. The behaviour can be modelled by analysing the temperature dependence of the various equilibrium parameters comprising the so-called dual mode sorption model for gas sorption in glassy polymers. The fundamental significance of the various enthalpies describing the temperature dependence of the Henry's law solubility constant, the Langmuir affinity constant and the Langmuir capacity constant are included in the discussion. Provision is made for non-ideal vapours and gases by introduction of the compressibility factor in the expression for the isosteric enthalpy. Application of relationships for calculating both the isosteric and the isothermal enthalpies of sorption is made to the case of CO₂ in poly(ethylene terephthalate) in the temperature range 35° to 115°C. These results and analyses complement the wealth of equilibrium and transport data which are consistent with the dual mode sorption model for penetrant sorption in glassy polymers.     

THE EFFECT OF TEMPERATURE TREATMENT ON PENETRANT TRANSPORT IN COAL

The macromolecular structure of coals thermally treated at 35^°C, 100^°C and 150^°C was investigated by dynamic penetrant transport in thin coal sections. The effects of temperature, carbon content in coal, and penetrant type on the transport mechanism were investigated. Dynamic swelling studies showed that penetrant transport into coal is a function of the average molecular weight between crosslinks, M_c. The penetrant transport mechanism at low activity is Fickian diffusion. The transport mechanism deviates from Fickian diffusion to anomalous transport, when the carbon content in coal and penetrant activity increase. Variations of the diffusion coefficients and relaxation constants were determined using a diffusion/relaxation coupled model.