The swelling interface number as a criterion for prediction of diffusional solute release mechanisms in swellable polymers

When a glassy polymer containing a uniformly dispersed solute is brought in contact with a penetrant, solute diffusion will be associated with the transport mechanism and penetration velocity of the penetrant in the polymer. Analysis and prediction of mechanisms of diffusional solute release may be obtained through a new dimensionless number, the swelling interface number, Sw, which compares the relative mobilities of the penetrant and the solute in the presence of macromolecular relaxations in the polymer. It is shown that a sufficient and necessary criterion for time-independent diffusional solute release rates from these swellable systems is that the Sw be smaller than 10^?2. The swelling interface number Sw may be related to easily determined structural and thermodynamic parameters of the solute/polymer/penetrant system. Preliminary experimental results of dynamic water swelling of poly(2-hydroxyethyl methacrylate-co-methyl methacrylate) and diffusional release of theophylline from initially glassy copolymers show that decreasing values of Sw are related to increased pseudo-case-II transport kinetics of the solute.     

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.     

A generalization of dual mode transport theory for glassy polymers

The polymer matrix, divided in a number of cells in which the penetrant molecules can be sorbed and migrate, is considered. Each cell has been assigned an effective energy value that obeys a particular distribution. The effective diffusion coefficient and its concentration and temperature dependence are determined. The origin of sorbed penetrant mobility is studied. Using a delta-Dirac distribution for the site's energetic values, the model is reduced in the appropriate limit (low pressure) to other formulations of the dual transport model. More general results, allowing the site's energetic values to be drawn from a Gaussian distribution, are also given. © 1994 John Wiley & Sons, Inc.     

An Experimental and Theoretical Investigation into the Diffusion of Olefins in Semi‐Crystalline Polymers: The Influence of Swelling in Polymer‐Penetrant Systems

A comprehensive dynamic diffusion model is developed to calculate the diffusion coefficients of low molecular weight penetrants (i.e., α‐olefins) in semi‐crystalline polyolefins from dynamic sorption measurements. The model also takes into account the extent of polymer swelling on the penetrant diffusion flux, resulting in a moving boundary value problem. The free volume theory is employed to calculate the dependence of the diffusion coefficient on the penetrant concentration. The solubilities and diffusivities of ethylene and propylene in semi‐crystalline high density polyethylene films were measured at different temperatures and pressures, using a Rubotherm® magnetic suspension microbalance operated in series with an optical view cell for the measurement of the degree of polymer swelling. It is shown that model predictions are in excellent agreement with the experimental dynamic measurements on the mass uptake of the sorbed species. Moreover, it is shown that the proposed model can predict correctly the diffusion coefficient of α‐olefins in semi‐crystalline polyolefins.

     

The influence of transverse differential swelling stresses on the kinetics of sorption of penetrants by polymer membranes

The influence of transverse differential swelling stresses on the kinetics of sorption of a penetrant in a polymer membrane exhibiting linear viscoelasticity is described by a model developed from the much simpler one of Crank. Sorption and transverse swelling kinetic curves are computed numerically. The character of absorption and desorption curves is examined systematically mainly as a function of (i) the magnitude of the stresses set up and of the stress-dependence of the diffusion coefficient, (ii) the relative rates of stress relaxation and of diffusion, and (iii) the degree of plasticization or “softening” of the polymer by the penetrant. It is shown that important general features of experimental sorption kinetic curves can be reproduced satisfactorily under well defined conditions. Attention is also given to transverse swelling kinetic curves. Their correlation with the corresponding sorption curves is examined briefly but systematically and discussed with reference to experimental data.     

Mass and momentum transfer in polymers. I. Effects of equilibrium vapor sorption on tensile creep of an amorphous polymer

Diffusion and tensile creep measurements were made for systems of poly(-n-butyl methacrylate) (PBMA) and sorbed ethanol, MEK, or benzene at 23°C. Rates of penetration of an inert spherical indenter into PBMA also were investigated and compared with the tensile creep behavior of the polymer. Creep measurements for various volume fractions of penetrant sorbed at equilibrium revealed that master curves, resulting from a time-concentration superposition procedure, could be constructed for each penetrant. At long times, these master curves, particularly that for ethanol, show deviations from the corresponding time-temperature superposition master curve. These deviations are interpreted in terms of probable long-range entanglement coupling governed in part by the partially specific nature of polymer-penetrant interactions. Parameters calculated by a free-volume theory, describing both diffusion and tensile-creep data, indicate that MEK is a more efficient plasticizing agent than the other penetrants and requires less local free volume for diffusion. Analysis in terms of the free volume concept was not attempted for the case of ethanol, where specific polymer-penetrant interactions are more important.     

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 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.     

Determination of penetrant solubility from transient permeation measurements

The permeability coefficient for the transport of a gas, vapor, or liquid through a polymer film is the product of the penetrant solubility and a diffusion coefficient. A transient permeation experiment known as the time-lag technique can be used to separate this product, provided the diffusion coefficient is independent of penetrant concentration. In this well-known experiment the polymer is initially free of penetrant. A new transient permeation experiment where the polymer is initially saturated with penetrant is suggested here. A general mathematical proof is given to show that by using the results form these two transient experiments which have different initial conditions one can determine the penetrant solubility no matter how the diffusion coefficient depends on penetrant concentration. Also one can determine two different concentration averaged diffusion coefficients from the results.     

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.     

Statistical mechanical model of diffusion of complex penetrants in polymers. I. Theory

A previously developed model of simple penetrant diffusion is extended to encompass complex penetrants of idealized molecular shape, characterized by dimensions of length, width, and thickness. Expressions are obtained for D(0,T), the diffusion coefficient at zero penetrant concentration (c), and the fractional increase in D(0,T) as a function of c and temperature (T). The model predicts that D(0,T) will exhibit Arrhenius behavior at temperatures well above T_g and gives the limiting activation energy as a function of penetrant thickness and the polymer energy/distance constants used previously. For T_g < T ? T_g + 150 K the model requires two new disposable parameters, in addition to the jump-length parameter of the simple penetrant theory. These parameters, however, have precise physical meanings (all are lengths) and together with the penetrant dimensions and polymer constants determine the absolute magnitude of the diffusion coefficient as well as its relative dependence on c and T. For T ? T_g + 40 the relative concentration dependence may be calculated in terms of the penetrant dimensions and polymer constants only.     

Disentanglement and reptation during dissolution of rubbery polymers

The dissolution mechanism of rubbery polymers was analyzed by dividing the penetrant concentration field into three regimes that delineate three distinctly different transport processes. The solvent penetration into the rubbery polymer was assumed to be Fickian. The mode of mobility of the polymer chains was shown to undergo a change at a critical penetrant concentration expressed as a change in the diffusion coefficient of the polymer. It was assumed that beyond the critical penetrant concentration, reptation was the dominant mode of diffusion. Molecular arguments were invoked to derive expressions for the radius of gyration, the plateau modulus, and the reptation time, thus leading to an expression for the reptation diffusivity. The disentanglement rate was defined as the ratio between the radius of gyration of the polymer and the reptation time. Transport in the second penetrant concentration regime was modeled to occur in a diffusion boundary layer adjacent to the polymer-solvent interface, where a Smoluchowski type diffusion equation was obtained. The model equations were numerically solved using a fully implicit finite difference technique. The results of the simulation were analyzed to ascertain the effect of the polymer molecular weight and its diffusivity on the dissolution process. The results show that the dissolution can be either disentanglement or diffusion controlled depending on the polymer molecular weight and the thickness of the diffusion boundary layer. © 1996 John Wiley & Sons, Inc.     

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     

Water transport in a polyketone terpolymer

Water transport in a polyketone terpolymer was analyzed performing both sorption and permeation experiments. Water vapor sorption tests were conducted at four temperatures (35, 45, 55, and 65°C) and at several activities. The analysis of sorption isotherms revealed the occurrence of water clustering. A reduction of the endothermicity of mixing as the amount of sorbed water increased was observed which is consistent with significant association of penetrant molecules in the polymer. Permeation experiments performed at 35°C at upstream pressures ranging from 4 to 25 Torr showed evidence of a reduction in water diffusivity as function of sorbed water concentration which is a typical indication of penetrant aggregation. © 1995 John Wiley & Sons, Inc.     

Size effect on competition of two diffusion mechanisms for drug molecules in amorphous polymers

Diffusion of drug molecules in polymer materials is of great importance in controlled drug release, and the investigation of the mechanism of drug release from the polymer matrix would help us to understand the release behavior of the controlled release system. In this work, molecular dynamics simulations were employed to investigate the diffusion mechanisms of penetrant molecules with different sizes in poly(lactic acid-co-ethylene glycol) (PLA-PEG). The size effect on the diffusion mechanism of penetrant molecules in polymer matrixes was discussed in detail. A competition mechanism in a two-step diffusion process-(1) motion within the cavities (free volumes), and (2) jumps between cavities or movement of the cavity itself originated from the wriggling of the polymer chains-was observed, and the contributions of these two factors to the diffusion coefficient were successfully separated. With the medium volume of penetrant molecules (e.g., benzene), a competition between these two steps was observed. Step (2) controlled the diffusion when penetrant molecules became bigger.     

Solute diffusion in swollen membranes VII. Diffusion in semicrystalline networks

A model is developed to express the solute diffusion coefficient through semicrystalline polymeric networks. The crystallites create impermeable diffusional barriers around the amorphous regions. Solute diffusion is determined by applying a transport model to the amorphous phase and incorporating the crosslinked polymer structure characteristics. This model is tested with theophylline and vitamin B₁₂ permeation experiments through semicrystalline poly(vinyl alcohol) membranes prepared by annealing of amorphous PVA membranes. The degree of crystallinity varies between 23.1 % and 40.5 % on a dry basis. The solute diffusion coefficients correlate well with various parameters of the model.     

渗透物在致密聚合物膜中扩散的分形介质模型

利用自由体积理论讨论了渗透物分子在致密聚合物膜内的扩散机理, 提出了“扩散通道”的概念, 建立了渗透物在致密聚合物膜中扩散的分形介质模型, 考虑了自由体积分布对扩散过程的影响. 根据建立的模型, 渗透物在膜内的扩散是由在“扩散通道”上的一系列跳跃构成的. 根据致密膜内扩散通道的关联长度ξ(p)与膜厚L的关系, 可以把扩散分为正常扩散、 过渡扩散和分形扩散三部分, 给出了扩散相图, 提出并解释了分形渡越现象.     

Phenomenology and mechanisms of unidimensional stress-dependent micromolecular transport in a stiff-chain polymer. I. Unoriented cellulose acetate

The longitudinal penetration of micromolecular liquid solvents (acetone, dioxane) or swelling agents (methylene chloride, methanol) into unoriented (unstretched) cellulose acetate film has been studied in detail by a variety of techniques, including observation of visible penetrant fronts, birefringence profiles, colored tracer microdensitometry, and microinterferometry. The results, in conjunction with those of Part II, provide a fuller picture of the relevant phenomenology than was previously available, leading to further insight into the mechanism of micromolecular transport in stiff-chain polymers and its dependence on the nature of the penetrant and the structural changes of the swelling polymer. © 1992 John Wiley & Sons, Inc.