Solute and penetrant diffusion in swellable polymers. I. Mathematical modeling

A mathematical model was developed to describe diffusion of a penetrant and a solute in a swellable polymer slab. The model was applied to the case of a hydrophilic polymer loaded with a soluble bioactive agent, in which the penetrant (water) is sorbed and solute is desorbed. The model allows the incorporation of any appropriate form of the diffusion coefficients. A Fujita-type exponential dependence on penetrant concentration was chosen and shown to be adequate for prediction of a range of transport behavior. Dimensional changes in the sample were predicted by allowing each spatial increment to expand according to the amount of penetrant sorbed. During the initial period of release, the swelling was restricted to one dimension by the glassy core of the sample. At a later point in the process, the center of the sample had sorbed enough penetrant to plasticize it, and the sample relaxed to an isotropically swollen state; thereafter swelling was three-dimensional.     

Diffusion in glassy polymers. II

The Fickian diffusion coefficient of methylene chloride in a glassy epoxy polymer is calculated with the use of Crank's model of discontinuous change of D with concentration C. The diffusion constant is obtained as 1.93 × 10^?6 cm²/sec. The swollen layer behind the advancing solvent front is essentially in the rubbery state of the same polymer. The case II swelling by benzene is discussed in terms of a convective transport arising from the partial stress (internal) tensor of the penetrant. The superposition of Fickian and case II diffusion found with mixtures of methylene chloride and benzene is also discussed briefly.     

Effects of adsorbed polyelectrolytes on convective flow and diffusion in porous membranes

Poly(acrylic acid) and poly(styrene sulfonate) were adsorbed from aqueous solutions to track-etched mica membranes with pores of radius 290 to 1400 Å. Effects of the adsorbed polymers on momentum and mass transport within the pores were studied by measuring the decrease in both the hydraulic and diffusional permeabilities caused by the presence of the polymers. The diffusional permeability was determined by measuring the flux of a small, uncharged solute molecule (thiourea). Polymer-free, aqueous solutions of potassium chloride (KCl) were used in all transport experiments. The reduction in hydraulic permeability increased with a decrease in KCl concentration in the range 10^-2 to 10^-1 molarity (M), but was independent of electrolyte concentration below 10^-2 M, presumably because the small pores constrained expansion of the adsorbed polymer chains. Shear thickening effects, that is, a decrease in hydraulic permeability with increasing solvent velocity through the pores, was observed with both polymers at 10^-1 M KCl in pores of 600 Å or smaller. Effects of the polymer on diffusional permeability of thiourea, on the other hand, were relatively insensitive to electrolyte concentration. Perhaps the most significant result is that the reduction in diffusional permeability was substantially less than the reduction in hydraulic permeability at each pore size and electrolyte concentration, indicating that the blockage of momentum transport by these adsorbed polymers is greater than the blockage of diffusion of a small solute. Measurements of thiourea transport by simultaneous diffusion and convection over a range in Peclet numbers from -2 to +2 showed that thiourea was filtered by the polymer. This filtration was probably not due to steric limitations on the thiourea, rather it is likely that thiourea was excluded from the water which solvated densely packed regions of the polymer chains.     

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

A 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.     

Anomalous transport of penetrants in glassy polymers

Solutions are presented for the Fickian and non-Fickian equations describing the case of penetrant transport in a glassy polymer. Due to associated macromolecular relaxation, a sharp penetrant front is observed which separates the glassy core from the rubbery (gel-like) layer at the surface. Concentration profiles are compared and general comments about Fickian versus non-Fickian transport in polymers are made.     

The effect of polymer swelling and resistance to flow on solvent diffusion and permeability

The main purpose of this paper is to test the model of molecular sorption [Vesely D. Polymer 2001;42:4417-22] for Case II type diffusion by measuring the effect of sorption/swelling and resistance to flow through the swollen region on the mass transport of solvents in glassy amorphous polymer. The system of methanol and polymethylmethacrylate (PMMA) has been selected for easy comparison with the existing literature data.The weight loss of penetrant permeating through the polymer has been monitored using a permeability cell placed on a balance (gravimetry). The rate of diffusion and swelling has been measured using light microscopy on samples cut after different elapsed time exposure to the solvent.The contribution of polymer swelling and resistance to flow has been evaluated by comparing the mass transport during diffusion and permeation processes. It is shown that for thin films the thickness independent component of the mass transport process (swelling) makes a significant contribution to the diffusion rate. For thicker samples the thickness dependent component (the resistance to flow through the swollen polymer) dominates both, diffusion and permeation.     

Diffusion-controlled penetration of polymethyl methacrylate sheets by monohydric normal alcohols

The kinetics of ethanol, n-propanol, and n-butanol penetration, sorption, and dimensional swelling in 2 mm poly(methyl methacrylate) sheets were determined over the temperature range 50–95°C. At 50°C, Case II relaxation-controlled transport dominated the observed sorption and penetration kinetics for all three alcohols. At higher temperatures, diffusion of swelling penetrant to the relaxing boundary contributes increasingly to the observed sorption kinetics. In addition, as the temperature is raised, the completion of sorption lags significantly behind the penetration of the relaxing boundary to the sheet midplane. p]The activation energy describing low temperature penetration is significantly higher than the activation energy describing the temperature dependence of high temperature penetration. A distinct transition in the penetration kinetics is apparent for all three alcohols at approximatively 65°C. Independent Clash—Berg determinations of the T_g of the alcohol-swollen sheets indicate that the transition in behaviour is not related to a thermal transition in the polymer, but rather to the generation of diffusional resistance in the high temperature penetration experiments which is comparable to the otherwise rate-determining Case II relaxations dominant in low temperature penetration. At high temperatures, the overall activation energy reflects the combination of diffusional absorption and the more highly activated relaxation-controlled transport. At low temperatures, diffusion of penetrant to the relaxing boundary is rapid compared with the slow, rate determining relaxations and, therefore, the concentration of penetrant is everywhere uniform within the already swollen shell. The extra-ordinarily high apparent activation energy describing the temperature dependence of the initial sorption rate at low temperature reflects the endothermic enthalpy of sorption of alcohols in PMMA as well as the strong coupling between relaxation rate and the penetrant concentration driving the rate determining relaxations. p]Clash—Berg measurements of the temperature dependence of the ten second shear moduli in partially swollen sheets, completely swollen sheets, and unswollen sheets suggest a T_g of approximatively 40°C in the alcohol-swollen PMMA. Moreover, an analysis of the Clash—Berg measurements suggests that the properties of the swollen regions of partially penetrated sheets are identical to the properties of the completely swollen sheets.     

Partial molar volume of small molecules in glassy polymers

The sorption of atoms or molecules in glassy polymers is assumed to occur within a variety of sites belonging to the intermolecular volume and providing different space for the dissolved molecules. If the size of the small molecule is larger than the size of the site, the glassy polymer is elastically distorted during sorption of the solute molecules. Minimizing the total free energy yields the result that large sites are occupied first, giving rise to small volume changes only. By increasing the solute concentration, smaller sites have to be occupied as well and the corresponding volume changes are larger. Thus the molecules can be considered to act as probes for the intermolecular space. A quantitative analysis and comparison with experimental results provides information on the intermolecular space in a glassy polymer. Compared to the dual-sorption model, the model of this study is able to explain the nonlinear relationship between volume change of the polymer and the partial pressure of the solute. At large solute concentrations, swelling of the glassy polymer or its transformation into the rubbery state occurs, which gives rise to structural changes after desorption. © 1993 John Wiley & Sons, Inc.     

Effect of the morphology of hydrophilic polymeric matrices on the diffusion and release of water soluble drugs

Drug release mechanisms from, and diffusion processes in, hydrophilic crosslinked polymeric systems were investigated in two macromolecular states: in the glassy and rubbery states during the early part of countercurrent water diffusion, and in the rubbery state after thermodynamic equilibrium between the network and the surrounding dissolution medium (water) was attained. Dilute, aqueous poly(vinyl alcohol) (PVA) solutions containing theophylline were crosslinked with glutaraldehyde. The crosslinking ratio, X, varied between 0.01 and 0.20 moles glutaraldehyde per mole of PVA repeating unit. Theophylline release from these rubbery matrices was followed as a function of time. It was determined that, within the range of crosslinking ratios studied, the crosslinked macromolecular structure affected the solute diffusion process. Theophylline release from crosslinked PVA slabs, which were originally dehydrated at 30°C, was also measured. The drug release process was significantly impeded in these systems, especially for samples with crosslinking ratio X ≥ 0.10. This behavior was explained in terms of relaxation of the macromolecular chains and possible existence of ordered chain structures. Glass-to-rubber transitions, a result of the countercurrent diffusion of water into the originally dried (glassy) polymer, shifted the fractional release of theophylline from a f(t^1/2) to a f(tⁿ) time dependence, with n taking values between 0.50 and 0.76. This type of release behavior indicates anomalous diffusion mechanisms. These results may be helpful in the development of swelling controlled drug delivery systems.     

Mathematical model for the prediction of the overall profile of in vitro solute release from polymer networks

The loading of solutes onto and their release from hydrogel-based devices can be better understood when they are treated as a partition phenomenon. Partition activity (alpha) is a parameter that determines the existence of partition phenomena. It expresses the physical chemical affinities of the solute between the solvent and hydrogel phases. When alpha=0, there is no release of the solute from the hydrogel; however, if alpha>0, there is partitioning of the solute between the solvent and the hydrogel phases, and release of the solute from the hydrogel can be observed. The mathematic model proposed here predicts the overall release profile of vitamin B(12), methylene blue (MB), and acid orange 7 (AO) from semi-interpenetrating network (semi-IPN) hydrogels composed of PNIPAAm and PAAm. Experimental release tests demonstrated that alterations on variables of the system change both the released fraction and the release rate of such solutes, confirmed by the changes on values of alpha (an equilibrium parameter) and k(R) (an kinetic parameter). The modeling of solute release describes the alpha effects on release of the solute from polymer networks. The solute release mechanism is viewed here as a diffusional transport process and as a partition phenomenon. The partitioning of the solutes occurs between the solvent phase and the hydrogel phase, and the possible physical chemical affinities of the solute between hydrogel and solvent are considered.     

Induction and measurement of glassy-state relaxations by vapor sorption techniques

Recent gravimetric studies of the sorption of organic vapors by poly(vinyl chloride) and polystyrene powders have demonstrated several features which promise to be generally useful in studying the structure and properties of the glassy state. The uptake of vapor can be significantly altered by prior thermal or vapor treatment of the polymer, apparently reflecting changes in the microvoid content or free volume of the polymer. Fickian sorption in sufficiently fine powders proceeds to equilibrium in a few minutes. Upon exposure of a polymer powder to an appreciable pressure of vapor, both a rapid Fickian sorption and a slower, relaxation-controlled sorption are observed. Superposition of these processes leads to widely varied sorption kinetics; a model comprising Fickian diffusion and first-order relaxation terms accurately describes the data and allows estimation of equilibrium and rate constants for both processes. After prolonged exposure, removal of a swelling vapor induces a slow reconsolidation of the polymer structure; this deswelling relaxation can be monitored by the decreasing amounts of vapor sorbed in repeated brief exposures to low vapor pressures, and can also be described by a first-order relaxation model. In this regard, the penetrant vapor serves as a molecular probe, monitoring glassy-state relaxation occurring in the absence of penetrant. The same, presumably true equilibrium is ultimately reached both by swelling from a low free-volume state and by consolidation from a preswollen state of high free volume. The rates of both swelling and consolidation relaxations appear to be retarded by the presence of low concentrations of vapor in the polymer, suggesting that vapor molecules may preempt some of the free volume required for relaxation.     

Modelling of solute transport across a temperature-sensitive polymer membrane

An important target of many controlled release systems is to properly modify the drug release behaviour in response to some external stimuli like temperature or pH changes. The copolymer gels made up by poly(N,N-dimethylaminoethyl methacrylate (DMAEMA)-co-acrylamide (AAm)), being thermosensitive, are suitable systems for the thermocontrol of the solute release. Modifying the ratio DMAEMA/AAm, one may select the copolymer systems showing desired swelling properties. In order to better design such systems and in order to understand better the solute transport through the copolymer membrane, a mathematical model has been developed. The model employs suitable flux equations for the water uptake or release and for the solute (hydrocortisone) diffusion. With the condition of knowing the copolymer swelling kinetics, the model is able to describe in a reasonably good manner the solute transport across a DMAEMA-AAm membrane undergoing step temperature changes.     

The effect of plasticization on the transport of gases in and through glassy polymers

The effects of plasticization on the transport of gases and vapors in and through glassy polymers are examined from the viewpoint of the “dual-mode” sorption model with partial immobilization. The analysis assumes the existence of two penetrant populations with different mobilities in the Henry's law and Langmuir domains of the glassy polymers. These mobilities are characterized by their mutual diffusion coefficients D_D and D_H. The plasticization of the polymer by penetrant gases is reflected in the concentration dependence of D_D and D_H. Expressions for the effective (apparent) diffusion and permeability coefficients are derived assuming that D_D and D_H are exponential functions of the penetrant concentration in the polymers. The results of this study are compared with a similar analysis which assumed the existence of a single mobile penetrant population. The present analysis provides information on the effects of plasticization on the penetrant transport in the Henry's law and Langmuir domains separately. The effects of antiplasticization or clustering of penetrant molecules on the effective diffusion and permeability coefficients are also examined.     

The solution and transport of gases in poly(N-butyl methacrylate) in the glass transition region. I. Absorption-desorption measurements

Solubility coefficients, S, and diffusion coefficients, D, have been determined for ethane and n-butane in poly(n-butyl methacrylate) (PnBMA) by the microbalance technique in the temperature range from ?14 to 50°C, which encompasses the glass transition of the polymer (22–35°C). S and D for ethane were found to be independent of penetrant pressure and concentration at all temperatures studied No transition to “dual-mode” sorption behavior, as reported for a number of penetrants in glassy polymers, was observed with ethane, even at the lowest experimental temperature. Plots of log S and log D versus 1-T, the reciprocal absolute temperature, were linear for the ethane-PnBMA system and did not exhibit discontinuities in the glass transition region. The above results suggest that the same mechanism of solution and transport of ethane in PnBMA is operative both above and below the glass transition of the polymer under the experimental conditions. This behavior is attributed to the low “excess” free volume of glassy PnBMA, as indicated by the small difference between the coefficients of thermal expansion of this polymer in its rubbery and glassy states. Possible conditions for the appearance of dual-mode gas sorption are discussed. A similar study with the n-butane-PnBMA system showed that the polymer was plasticized by the penetrant below 20°C, due to the higher solubility of n-butane compared with that of ethane in PnBMA.     

An asymptotic analysis of polymer desorption and skinning

During desorption of penetrant‐saturated polymers, a glassy skin can form at the exposed surface. The associated dynamics are not purely Fickian due to viscoelastic relaxation effects in the polymer. A model is presented which captures these nonlocal effects. The motion of the glass‐rubber interface and the accumulated desorbed flux are calculated. The model also describes trapping skinning, where an increase in the driving force reduces the amount of penetrant released.     

Phenomenology and mechanisms of unidimensional stress-dependent micromolecular transport in a stiff-chain polymer. III. Effect of modification of the polymer

A detailed study of the kinetics and mechanism of micromolecular transport in cellulose acetate films containing 2.0 acetate groups per glucose unit (CA-2.0) is reported. The polymer was prepared by controlled hydrolysis of CA-2.45 films studied in preceding articles. The same series of simple liquid penetrants varying from weak swelling agent to good solvent of the polymer was used. As before, measurement of rates of penetration along the polymer film confined between glass plates was supplemented with information on penetrant distribution profiles in the polymer film and on the corresponding deformation and structural relaxation of the swelling polymer, deduced from refractive index and birefringence profiles, respectively. Transport was studied in (a) unoriented CA-2.0 films and (b) uniaxially oriented films with penetration normal and parallel to the orientation axis. This was equivalent to varying the viscoelastic polymer properties affecting transport, under otherwise identical experimental conditions. The results complemented and extended those previously obtained with CA-2.45 in interesting ways and were successfully interpreted on the basis of a previously developed theoretical model designed to represent the influence of (a) the stress generated by the constraints imposed on the swelling polymer, and (b) the viscoelastic response of the latter thereto, on the transport mechanism. It was shown that the observed differences in transport mechanism in CA-2.45 and CA-2.0 are primarily related to the corresponding changes in the sorptive capacity of the polymer for the relevant penetrant rather than the chemical constitution of the latter. The most striking result in this respect was that the remarkable kinetic pattern (which involved a drastic change from Case I kinetics for penetration across, to Case II kinetics for penetration along, the axis of orientation) exhibited by oriented CA-2.45 film penetrated by the strong swelling agent of the series of penetrants used, namely methylene chloride, was reproduced here for the penetration of acetone, which occupies the slot of strong swelling agent in the case of CA-2.0. © 1997 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 35 : 2593–2607, 1997     

Theoretical models for diffusion in glassy polymers. II

The 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.     

Sharp fronts due to diffusion and stress at the glass transition in polymers

We derive and analyze a model for sharp fronts in glassy polymers. We take the major effect of a diffusing penetrant on the polymer entanglement network to be the inducement of a differential viscoelastic stress. This couples diffusive and mechanical processes through a viscoelastic response where the strain depends upon the amount of penetrant present. Analytically, the major effect is to produce explicit delay terms, via a relaxation parameter, to account for the fundamental difference between a polymer in its rubbery state and the polymer in its glassy state, namely the finite relaxation time in the glassy state owing to slow response to changing conditions. We produce concentration profiles in good agreement with observations on sharp front formation. In addition the model can account for the phenomenon of sorption overshoot.     

Sorption of a finite amount of swelling solvent in a glassy polymer

We study the time history of a diffusing front when a polymer is exposed to a finite amount of penetrant which becomes used up. A class of polymers is considered for which slow molecular relaxation occurs only at or near the glass-gel interface with instantaneous relaxation both ahead of and behind the progressing front. We show that the position of the penetrant front versus time undergoes a long smooth transition from standard Fickian t^1/2 behavior to exponential time decay onto a final equilibrium position attained when all the penetrant is used up.