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.     

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.     

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.     

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.     

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.     

The evolution of steep fronts in non-fickian polymer-penetrant systems

We adapt a recently proposed model for non-Fickian diffusion of penetrants into polymers and use it to study a drug-delivery problem. The model modified Fick's diffusion equation by the addition of stress-induced flux. A stress evolution equation incorporating aspects of the Maxwell and Kelvin-Voigt viscoelastic stress models completes the model. The relaxation time in the polymer is taken as a function of the penetrant concentration. The system is studied under the assumption that the diffusivity is large. Singular perturbation techniques are used to show that the concentration and stress evolve diffusively for small time, but exhibit steep fronts in a narrow region within the domain for larger time. These predictions are verified numerically for specified parameter values. Finally, the equations are studied in the steady state and are found to predict the evolution of shocks.     

Strain and temperature accelerated relaxation in polycarbonate

The nonlinear viscoelastic behavior of glassy polymers and its relationship to ductile yielding is studied by single- and double-step stress relaxation experiments. In the latter case a small stress relaxation step is superimposed on a specimen at an elevated state of temperature or strain. The results show that the changes in the relaxation behaviors in the two cases closely parallel each other. The relaxation behavior at strains near yield closely approximates that at low strain but near Tg. The small strain relaxation response can be described well by a Kohlrausch-Williams-Watts (KWW) type function. The interpretation of these data in terms of a coupling model which includes the KWW form is discussed.     

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.     

Polymer structure and the compensation effect of the diffusion pre-exponential factor and activation energy of a permeating solute

In the present work, the relation between the pre-exponential factor and the apparent activation energy of diffusion, ln D(0) = alpha + betaE(D), so-called compensation effect, is re-examined and critically discussed for diffusion of gases in rubbery and glassy polymers. In principle, the above equation could be derived from the enthalpy-entropy compensation in the framework of the transition state theory. However, one should consider the influence of the jump length term contained in the pre-exponential factor, which may be affected by permeating species and polymer properties. We found that parameter alpha depends on penetrant size and polymer properties, such as local chain mobility and free volume. This can be interpreted by the fact that the jump length is affected by both penetrant and polymer properties. Finally, methods for estimating the jump length are discussed.     

Viscoelastic characterization of polymers using instrumented indentation. I. Quasi‐static testing*

The use of instrumented indentation to characterize the mechanical response of polymeric materials was studied. A model based on contact between a rigid probe and a linear viscoelastic material was used to calculate values for the creep compliance and stress relaxation modulus for two glassy polymeric materials, epoxy and poly(methyl methacrylate), and two poly(dimethyl siloxane) (PDMS) elastomers. Results from bulk rheometry studies were used for comparison with the indentation stress relaxation results. For the two glassy polymers, the use of sharp pyramidal tips produced responses that were considerably more compliant (less stiff) than the rheometry values. Additional study of the deformation remaining in epoxy after indentation creep testing as a function of the creep hold time revealed that a large portion of the creep displacement measured was due to postyield flow. Indentation creep measurements of the epoxy with a rounded conical tip also produced nonlinear responses, but the creep compliance values appeared to approach linear viscoelastic values with decreasing creep force. Responses measured for the unfilled PDMS were mainly linear elastic, with the filled PDMS exhibiting some time‐dependent and slight nonlinear responses in both rheometry and indentation measurements. © 2005 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 43: 1794–1811, 2005     

Sorption studies that reflect sequential physical changes in polymer-liquid systems from saturation to virtual dryness in a glassy state

The changes in macromolecular architecture that occur sequentially when a polymer is allowed to swell to saturation in a test-liquid and then evaporated from its gel-saturated state through its rubbery transition and finally down to a glassy state at virtual dryness are described in detail. The influence of molecular structure of the sorbed liquid and temperature on the kinetics of evaporation during each part of the above cycle is discussed. The relevance of these results with respect to the models proposed by polymer physicists to describe thermoreversible gelation and polymer relaxation in the glassy state is also discussed.     

Theoretical models for diffusion in glassy polymers

A 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.     

Synthesis and properties of polyisocyanurate networks based on 2,4-toluene diisocyanate and poly(oxytetramethylene) glycol

With the use of the method of bulk polymerization, network polyisocyanurate polymer materials based on poly(oxytetramethylene) glycol and 2,4-toluene diisocyanate have been synthesized for the first time. To analyze their performance and relaxation mechanism, the mechanical characteristics of polyisocy-anurates are studied. The above polymers show a quasielastic mechanical behavior, even though their elastic modulus is characteristic of the temperature interval where all traditional polymers experience the transition from the glassy state to the rubbery state, and demonstrate a well-pronounced viscoelastic behavior. The proposed procedure for the construction of master curves shows that, in a certain temperature interval, shift factor remains invariable; in other words, it does not depend on temperature.     

Local Transient Jamming in Stress Relaxation of Bulk Amorphous Polymers

Bulk amorphous polymers become stretched and parallel-aligned under loading stress, and their intermolecular cooperation slows down the subsequent stress relaxation process. By means of dynamic Monte Carlo simulations, we employed the linear viscoelastic Maxwell model for stress relaxation of single polymers and investigated their intermolecular cooperation in the stress relaxation process of stretched and parallel-aligned bulk amorphous polymers. We carried out thermal fluctuation analysis on t...     

Constrained Rouse model of rubber viscoelasticity

In this work we use a new approach to investigate the equilibrium and linear dynamic-mechanical response of a polymer network. The classical Rouse model is extended to incorporate quenched constraints on its end-boundary conditions; a microscopic stress tensor for the network system is then derived in the affine deformation limit. To test the model we calculate the macroscopic stress in equilibrium, corresponding to the long-time limit of relaxation. Particular attention is paid to the treatment of compressibility and hydrostatic pressure in a sample with open boundaries. Although quite different in general, for small strains the model compares well with the classic equilibrium rubber-elasticity models. The dynamic shear modulus is obtained for a network relaxing after an instantaneous step strain by keeping track of relaxation of consecutive Rouse modes of constrained network strands. The results naturally cover the whole time range--from the dynamic glassy state down to the equilibrium incompressible rubber plateau.     

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     

Shear stress relaxation and physical aging study on simple glass-forming materials

Relaxation and aging behaviors in three supercooled liquids: m-toluidine, glycerol, and sucrose benzoate have been studied by shear stress relaxation experiments in the time domain above and below their nominal glass transition temperatures. For the equilibrium state, the current study provides new data on the behavior of organic complex fluids. The shape of the relaxation function as characterized by the stretching exponent beta is discussed considering that a time-temperature master curve can be constructed even though the beta's for the individual response curves at each temperature vary systematically. In the nonequilibrium state, isothermal physical aging experiments at different glassy structures reveal that the effect of the aging process on the mechanical shear relaxation in these simple glass formers is similar to that observed in polymeric and other systems. Departure from the Vogel-Fulcher-Tamman behavior after the samples have aged back to equilibrium in the glassy state is observed for m-toluidine and, less strongly, for glycerol but not for sucrose benzoate. An inherent structure-based energy landscape concept is briefly discussed to account for the slow dynamics during the physical aging process.     

A quantitative model for the specific volume of polymer–diluent mixtures in the glassy state

A mathematical model to describe the specific volume of glassy mixtures of a polymer and a low molecular weight diluent or additive is presented. The model is based on understandable physical assumptions and relies on parameters that can be determined experimentally or estimated from methods available in the literature. The predictions of the model show good agreement with the experimental data for mixtures of four polymers with diluents that in the pure state are liquid, glassy, or crystalline. The observed negative departure from volume additivity, as defined by simple additivity of the specific volume of the pure glassy polymer and the pure amorphous diluent, is the result of the relaxation of the excess volume of the glassy mixture relative to the equilibrium state caused by mixing two components with different glass transition temperatures. © 1998 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 36: 1037–1050, 1998     

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.     

PHASE INVERSION AND ITS MECHANICAL MODEL STUDIES FOR THE TWO-PHASE POLYMER BLEND SYSTEMS (Ⅱ)——DYNAMIC VISCOELASTIC RESPONSE

The dynamic viscoelastic response of the two-phase polymer blend systems shows the characteristics of the thermorheologically complex materials. In this paper theoretical equations for describing the dynamic viscoelastic response of such polymer blend systems have been established by means of the mechanical modeling technique. The dynamic viscoelastic response of the blend systems at any blend composition can be predicted theoretically by using the equations established, provided that the dynamic viscoelastic response of the two pure components and the mechanical model parameters are known in advance. Thus, we provide an effective method for studying the dynamic mechanical properties and the molecular relaxation characteristics of the two-phase polymer blend systems.