Analysis of transport behavior in polymer powders

Vapor sorption studies on powder samples of glassy polymers have provided data which supplement results obtained on conventional film specimens and aid in the elucidation of glassy-state transport mechanisms. For uniform spherical particles of sub-micron size, sorption kinetics at very low activities of organic vapors follow a simple Fickian diffusion model. The short diffusion path in such samples allows determination of the very low diffusivities characteristic of the glassy state in experiments of conveniently short duration. Deviations from the Fickian, uniform-sphere model are observed in several circumstances: Particle size non-uniformity retards the approach to diffusion equilibrium. Sorption data at substantial vapor activities show an apparently similar slow approach to equilibrium which can be related to the contribution of a relaxation-controlled mode of sorption. The effects of particle non-uniformity and of relaxation processes can be distinguished by appropriate experimental design, and models for both have been developed. Sorption rate data obtained under Fickian diffusion conditions can be used to characterize particle size distribution. Sorption kinetics on uniform-sphere powders, conversely, can be analyzed through a diffusion-plus-relaxation model to distinguish and quantify the roles of the two transport modes more clearly than is possible with polymer film specimens. Polymer powder vapor solubility isotherms show significant variations with sample history which can be interpreted in terms of free volume changes and glassy state relaxations. This discussion, based on a study of vapor sorption by poly(vinyl chloride) samples, indicates that powder sorption measurements are also likely to be of general value in the study of other glassy polymers.     

Dual-mode free volume model for diffusion of gas molecules in glassy polymers

The development of a new model for the diffusion of gas molecules in glassy polymers is presented which utilizes concepts from free volume theory and relies on a dual-mode interpretation of sorptive dilation in glassy polymers. Three assumptions are made in the development of the model. First, the free volume available for molecular transport processes is taken as constant below the glass transition temperature. Second, two populations of gas molecules are assumed to exist—one which contributes to the maintenance of an iso-free volume state upon sorptive dilation and one which does not contribute owing to sorption into regions of unrelaxed volume. Third, the former population is assumed to be mobile while the latter is not. The resulting model predicts, at constant temperature, a diffusion coefficient that is independent of solute volume fraction. This is in contrast to the widely used dual-mode sorption model with partial immobilization for gas transport in glassy polymers which leads to a diffusion coefficient that is dependent on solute mole fraction through the molar gas concentration. The new model is used to interpret gas transport data from permeation experiments for carbon dioxide, methane, and ethylene in three polycarbonates. © 1997 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 35: 1737–1746, 1997     

Effect of penetrant-induced isothermal glass transition on sorption,dilation, and diffusion behavior of polybenzylmethacrylate/CO2

The effect of a penetrant-induced isothermal glass transition on sorption, dilation, and diffusion behavior was studied in a single experimental run for CO₂ in cast polybenzylmethacrylate films. The dual-mode type sorption isotherms below the glass transition temperature of the polymer changed to linear ones above a certain concentration. Meanwhile, partial molar volume of CO₂ determined from the dilation of the films above the concentration gave a value very close to the one reported for rubbery polymers, and diffusion coefficients became less concentration-dependent. The results were conformable to the concept of unrelaxed volume in glassy polymers. © 1996 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.     

Sorption of methanol/MTBE and diffusion of methanol in 6FDA–ODA polyimide

This article discusses the diffusion and solubility behavior of methanol/methyl tert‐butyl ether (MTBE) in glassy 6FDA–ODA polyimide prepared from hexafluoroisopropylidene 2,2‐bis(phthalic anhydride) (6FDA) and oxydianiline (ODA). The diffusion coefficients and sorption isotherm of methanol vapor in 6FDA–ODA polyimide at various pressures and film thicknesses were obtained with a McBain‐type vapor sorption apparatus. Methanol/MTBE mixed‐liquid sorption isotherms were obtained by head‐space chromatography and compared with a pure methanol sorption isotherm obtained with a quartz spring balance. Methanol sorption isotherms obtained with the two methods were almost identical. Both methanol sorption isotherms obeyed the dual‐mode model at a lower activity, which is typical for glassy polymer behavior. The MTBE was readily sorbed into the polymer in the presence of methanol, but the MTBE sorption isotherm exhibited a highly nonideal behavior. The MTBE sorption levels were a strong function of the methanol sorption level. Methanol diffusion in the polymer was analyzed in terms of the partial immobilization model with model parameters obtained from average diffusion coefficients and the dual‐mode sorption parameters. Simple average diffusion coefficients were obtained from sorption kinetics experiments, whereas the dual‐mode sorption parameters were obtained from equilibrium methanol sorption experiments. An analysis of the mobility and solubility data for methanol indicated that methanol tends to form clusters at higher sorption levels. © 2000 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 38: 2254–2267, 2000     

Simulation of experimentally observed dilation phenomena during integral gas sorption in glassy polymers

A molecular modeling investigation of dilation effects induced by sorbed gas molecules in two glassy polymers is presented. As experimental reference, integral sorption of CO₂ and CH₄ was measured for polysulfone (PSU) and a polyimide (6FDA‐TrMPD, PI4) at 308 K and a pressure of 10 bar. Simultaneously, the gas induced swelling effect was measured with a dilatometer based on a capacitive distance sensor recorded. The experimental evidence of the (on the observed time scale and concentration levels) elastic nature of the gas induced dilation is supported by the dilation and contraction behavior observed in molecular dynamics (MD) simulations of respective detailed atomistic packing models. These models were constructed in accordance with gas concentration levels obtained from the experimental sorption results. Quantitative deviations between simulated and measured dilations are discussed as a consequence of an anelastic response of the polymer matrix which is too fast to be resolved in the experiments whose kinetics is dominated by diffusional processes. In the simulation, the initial insertion of penetrant molecules into equilibrated packing models “circumvents” the slow diffusional process of the experiment and allows a reasonable representation of the dilation process as well as a closer investigation. Our simulation approach reveals a different behavior for PSU and PI4 on the corresponding time scale. Most likely, the different chain mobility of the two polymers is responsible for the respective response to the inserted amount of gas molecules which is discussed in terms of the different chain mobilities of the two polymers. © 2007 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 46: 59–71, 2008     

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

A 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.     

A Self-Consistent Model for Sorption and Transport in Polyimide-Derived Carbon Molecular Sieve Gas Separation Membranes

Demand for energy-efficient gas separations exists across many industrial processes, and membranes can aid in meeting this demand. Carbon molecular sieve (CMS) membranes show exceptional separation performance and scalable processing attributes attractive for important, similar-sized gas pairs. Herein, we outline a mathematical and physical framework to understand these attributes. This framework shares features with dual-mode transport theory for glassy polymers; however, physical connections to CMS model parameters differ from glassy polymer cases. We present evidence in CMS membranes for a large volume fraction of microporous domains characterized by Langmuir sorption in local equilibrium with a minority continuous phase described by Henry's law sorption. Using this framework, expressions are provided to relate measurable parameters for sorption and transport in CMS materials. We also outline a mechanism for formation of these environments and suggest future model refinements.     

Pure hydrocarbon sorption properties of poly(1-trimethylsilyl-1-propyne) (PTMSP), poly(1-phenyl-1-propyne) (PPP), and PTMSP/PPP blends

Propane and n-butane sorption in blends of poly(1-trimethylsilyl-1-propyne) (PTMSP) and poly(1-phenyl-1-propyne) (PPP) have been determined. Solubilities of propane and n-butane increased as the PTMSP content in the blends increased. This result is consistent with the higher free volume of PTMSP-rich blends and the better thermodynamic compatibility between PTMSP and these hydrocarbons. Propane and n-butane sorption isotherms were well described by the dual-mode model for sorption in glassy polymers. PTMSP/PPP blends are strongly phase-separated, heterogeneous materials. A noninteracting domain model developed for sorption in phase-separated glassy polymer blends suggests that sorption in the Henry's law regions (i.e., the equilibrium, dense phase of the blends) is consistent with the model. However, Langmuir capacity parameters in the blends are lower than predicted from the domain model, suggesting that the amount of nonequilibrium excess free volume associated with the Langmuir sites depends on blend composition. © 1996 John Wiley & Sons, Inc.     

Gas sorption environments in poly(2,6‐dimethyl‐1,4‐phenylene oxide) by xenon‐129 nuclear magnetic resonance: Effects of processing

One‐ and two‐dimensional xenon‐129 nuclear magnetic resonance (¹²⁹Xe NMR) experiments were performed on a series of poly(2,6‐dimethyl‐1,4‐phenylene oxide) (PXE) samples to characterize the sorption environments and the relative mobility of xenon in the samples. Samples of PXE in sealed NMR tubes pressurized with xenon were studied as a function of temperature, pressure, and processing. In a dense cast film of PXE, the shift relative to the free gas resonance is smaller than that observed for typical glassy polymers, indicating a higher free volume environment. Solubility rises rapidly as temperature decreases. The lower shift and rapid increase in solubility with decreasing temperature are consistent with a relatively high free volume environment for gas sorption. If PXE is antiplasticized, the shift is slightly larger, the increase in signal intensity with decreasing temperature is smaller, and the line widths are greater. This sample is a better packed glass with less free volume and slower diffusion. Samples of PXE produced by rapid precipitation have broad lines and even lower shifts corresponding to a wide distribution of higher free volume environments. The appearance of two lines at low temperatures is consistent with the presence of a bimodal distribution of environments similar to what has been observed with positron annihilation lifetime spectroscopy. The resonance closest to the free gas resonance is associated with very large free volume elements relative to those of traditional glassy polymers. In two‐dimensional experiments, there is a rapid exchange of xenon by diffusion between the two environments, indicating the close spatial proximity of the environments. Two‐dimensional experiments and one‐dimensional progressive saturation experiments reflect a rapid exchange of xenon between the sorbed state and the free gas resonance for the precipitated samples. At low temperatures, the high field peak exchanges more rapidly with the free gas. © 2002 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 40: 1965–1974, 2002     

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.     

Relaxation phenomena in dense,glassy membranes prepared from a polymer solution,monitored by changes in dielectric properties

The drying of a solvent-cast polymer film is monitored in a non-invasive way, by measuring the changes in time of dielectric properties, using interdigitated or comb electrodes. Experimentally, the vitrification of the polymer solution is observed at a distinct time, followed by the slow evaporation of solvent from the glassy state. As the solvent diffusion coefficient is a strong function of the concentration in the polymer film, removal of residual solvent proceeds at a self-decreasing rate. With a simple model, solvent mass transfer coefficients in the glassy state are determined from the experiments. It is shown that volume relaxation may be slow compared to the drying process, when preparing thin solvent-cast membranes and using solvents that diffuse relatively fast through the glassy polymer film.     

The effect of film thickness and sample history on the parameters describing transport in glassy polymers

The effect of a characteristic dimension of a glassy polymeric specimen on the kinetic mechanism controlling unidirectional absorption of organic vapors and liquids in glassy polymers is demonstrated by comparing n-alkane absorption experiments in microspheres, spheres, films, and sheets of polystyrene. Absorption in submicron microspheres is controlled by Fickian diffusion whereas, under otherwise identical boundary conditions, films (ca. 75μm thick) and spheres (ca. 184 μm in diameter) sorb according to Case II absorption kinetics. Thinner films (35 μm thick) sorb by Super Case II kinetics and relatively thick sheets (2000 μm thick) sorb initially by Case II kinetics but, at long times, diffusion through the outer swollen region contributes significantly to the overall resistance to mass transfer and the rate of absorption decreases progressively with time. p]The rather short experimental times, afforded by the exceedingly small mean diameter of the narrowly distributed microsphere powder sample, permitted convenient characterization of the effects of preswelling, sorption-desorption cycling, and annealing on the kinetics and apparent equilibria of sorption. History effects were quite dramatic and were related to glassy state relaxations initiated by the various thermal and swelling histories imposed upon the glassy microspheres. p]Prediction of sorption and permeation behavior in membranes, from kinetic and equilibrium parameters determined experimentally on film and powder samples, requires explicit recognition of these dimensional and history effects. These effects do not appear to be related to any special properties of this polymer-penetrant system.     

Model for sorption of mixed gases in glassy polymers

A model is presented for analysis of the sorption of mixed gases in glassy polymers at concentrations below which significant plasticization occurs. The well-known dual-mode sorption model comprised of a Henry's law term and a Langmuir isotherm term, which has been used extensively for interpretation of single-component gas sorption data, forms the basis for the analysis of binary mixtures discussed here. Measurements using pure gases provide dual mode parameters which can then be used to predict the resultant sorption isotherms for binary mixtures of any of the pure gases. The proposed analysis is based upon recognition that the Langmuir component of the overall sorption concentration should be governed by competition between the two penetrants for the fixed unrelaxed volume in the polymer, which is believed to be the locus of the Langmuir capacity. This effect may result in a significant depression of the measured sorption of similar penetrants competing for the limited Langmuir capacity. A numerical example is considered which illustrates the range of behavior expected for CO₂ and CH₄ in polycarbonate. Deviations from the theoretical predictions of the simple dual-mode model for binary systems are discussed in terms of plasticizing effects on the Henry's law constant and the Langmuir affinity constant. The analyses proposed here are of direct and critical interest to the applied problems of migration of trace contaminants in glassy polymers and analysis of barrier packaging for foods since all of these applied problems involve mixed-penetrant sorption. Specifically, it is predicted that the presence of residual monomers or solvents in glassy polymers can produce both anomolously low Langmuir sorption affinity constants and sorption enthalpies compared with the residual-free case.     

Gas sorption in polymers based on bisphenol-A

Equilibrium gas sorption measurements for CO₂, CH₄, and N₂ were made with three polymers based on bisphenol-A, namely a polyhydroxyether, a polyetherimide, and a polyarylate. These data plus previous results for two other bisphenol-A polymers, polycarhonate and polysulfone, were analyzed using the dual-mode sorption model and the more recent gas-polymer-matrix model. The models were compared on the basis of physical interpretations of the resulting parameters. The Langmuir capacity from the dual-model model was related to the unrelaxed volume of the glassy polymer. The Henry's law sorption parameter from the dual-mode model was related to the internal pressure of the polymer and to its tensile stress at yield. The work suggests a means for estimation of gas sorption levels from thermal and mechanical properties of the polymer.     

Dual-mode sorption and transport of sulfur dioxide in kapton polyimide

The kinetics and equilibria of SO₂ sorption in Kapton polyimide film have been studied at temperatures from 25 to 55°C and equilibrium sorption pressures up to 0.76 atm. The data are described well by the dual-mode model of sorption and transport in glassy polymers. The assumption of “partial immobilization” is required to correlate the transport data: the mobility of the Langmuir component of the sorbed population relative to the Henry's-law component is close to zero at 25°C, and increases to roughly 5% of the Henry's-law component mobility at 55°C. The heat of sorption is anomalously low, suggesting the presence of residual solvent in the film, a suggestion confirmed by annealing studies. However, the study also shows that partial removal of the residual has a relatively minor effect on the measured SO₂ sorption level and transport rates at 55°C.     

Interval kinetic gravimetric sorption of acetone in random copolymers of poly(ethylene terephthalate) and poly(ethylene 2,6-naphthalate)

Interval sorption kinetics of acetone in solvent cast films of random poly(ethylene terephthalate)-co-(ethylene 2,6-naphthalate) (PET-co-PEN) are reported at 35°C and at acetone pressures ranging from 0 to 7.3 cm Hg. Polymer composition is varied systematically from 0% to 50% poly(ethylene 2,6-naphthalate). Equilibrium sorption is well described by the dual-mode sorption model. Interval sorption kinetics are described using a two-stage model that incorporates both Fickian diffusion and protracted polymer structural relaxation. The incorporation of low levels of PEN into PET significantly reduces the excess free volume associated with the glassy state and, for these interval acetone sorption experiments in ∼ 5 μm-thick films, decreases the fraction of acetone uptake controlled by penetrant-induced polymer structural relaxation. © 1999 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 37: 2973–2984, 1999     

Dual-mode analysis of subatomspheric-pressure CO2 sorption and transport in Kapton H polyimide film

Sorption kinetics and equilibria as well as permeabilities and diffusion time lags for CO₂ in Kapton polyimide film have been studied at temperatures from 35 to 55°C and pressures up to 0.78 atm. The sorption/desorption cycles indicate that the diffusivity of CO₂ increases with increasing local penetrant concentration in the polymer. Both the permeability and time lag decrease with increasing upstream CO₂ pressure. All of these results are described well by theoretical expression based on the dual-mode theory of sorption and transport in glassy polymers.