Temperature and concentration dependence of diffusion coefficients in ethylene–propylene-diene copolymers

Self-diffusion and partition coefficients were measured for two commercial ethylene–propylene-diene copolymers (EPDM) and five solvents at infinite dilution using inverse gas chromatography. Mutual diffusion coefficients for solvents in EPDM also were measured for finite concentration using gravimetric sorption for three of the solvents. From the inverse gas chromatography experimental values for self-diffusion coefficients were obtained. Free-volume parameters were obtained through regression of the self-diffusion coefficient as a function of temperature. Mutual diffusion coefficients as a function of concentration were predicted using free volume theory and compared with experimental data obtained using gravimetric sorption. © 1998 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 36: 1713–1719, 1998     

Atomistic packing models for experimentally investigated swelling states induced by CO2 in glassy polysulfone and poly(ether sulfone)

Atomistic packing models have been created, which help to better understand the experimentally observed swelling behavior of glassy polysulfone and poly (ether sulfone), under CO₂ gas pressures up to 50 bar at 308 K. The experimental characterization includes the measurement of the time‐dependent volume dilation of the polymer samples after a pressure step and the determination of the corresponding gas concentrations by gravimetric gas‐sorption measurements. The models obtained by force‐field‐based molecular mechanics and molecular dynamics methods allow a detailed atomistic analysis of representative swelling states of polymer/gas systems, with respect to the dilation of the matrix. Also, changes of free volume distribution and backbone mobility are accessible. The behavior of gas molecules in unswollen and swollen polymer matrices is characterized in terms of sorption, diffusion, and plasticization. © 2006 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 44: 1874–1897, 2006     

Impact of average free‐volume element size on transport in stereoisomers of polynorbornene. I. Properties at 35 °C

This article demonstrates that transport of gases through glassy polymers is significantly influenced not only by the absolute amount but also by the distribution of free volume. Two stereoisomers of polynorbornene with nearly equivalent total free volume, but markedly different average free‐volume sizes, were evaluated. The free‐volume element size was probed with positron annihilation lifetime spectroscopy, wide‐angle X‐ray scattering, gas sorption, and molecular modeling. The permeation, sorption, and diffusion of light gases were measured in each stereoisomer at 35 °C. All analytical techniques indicated that one isomer (labeled as Architecture II) had a larger average free‐volume element size but fewer elements. This isomer also had a very slightly higher bulk density (1.000 vs 0.992 g/cm³ for the other stereoisomer). Architecture II also had gas sorption and diffusion coefficients that were two to three times those of the less dense counterpart. These differences have been attributed to differences in the free‐volume element size available within the polymer matrix. © 2003 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 41: 2185–2199, 2003     

Solvent diffusion in amorphous polymers: Polystyrene–solvent systems

The inverse gas chromatography (IGC) technique was used to obtain the partition and diffusion coefficients of solvents in polystyrene over a wide range of temperatures. Infinite dilution experiments were performed with three solvents: toluene, benzene, and hexane. Finite concentration data were measured for the polystyrene–toluene system at various concentrations from 110 to 180 °C. For the finite concentration region, the modified capillary column model used by Tihminlioglu and Danner (J Chromatogr A 1999, 845, 93–101) was used to calculate diffusion and thermodynamic data. Finite concentration thermodynamic data were also calculated with the retention theory approach and compared with the capillary column model. The experimental IGC results are in good agreement with data from other experimental techniques. © 2000 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 38: 1965–1974, 2000     

Application of free‐volume theory to self diffusion of solvents in polymers below the glass transition temperature: A review

Theories based on free‐volume concepts have been developed to characterize the self and mutual‐diffusion coefficients of low molecular weight penetrants in rubbery and glassy polymer‐solvent systems. These theories are applicable over wide ranges of temperature and concentration. The capability of free‐volume theory to describe solvent diffusion in glassy polymers is reviewed in this article. Two alternative free‐volume based approaches used to evaluate solvent self‐diffusion coefficients in glassy polymer‐solvent systems are compared in terms of their differences and applicability. The models can correlate/predict temperature and concentration dependencies of the solvent diffusion coefficient. With the appropriate accompanying thermodynamic factors they can be used to model concentration profiles in mutual diffusion processes that are Fickian such as drying of coatings. The free‐volume methodology has been found to be consistent with molecular dynamics simulations. © 2011 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys, 2011     

Water sorption and transport in blends of poly(vinyl pyrrolidone) and polysulfone

Water sorption and transport properties for a series of miscible blends of hydrophobic bisphenol A polysulfone and hydrophilic poly(vinyl pyrrolidone) are reported. Study was restricted to blends that remained homogeneous after exposure to liquid water. The solubility of water in the blend films increased with increasing hydrophilic polymer content. Equilibrium sorption isotherms show dual-mode behavior at low activities and swelling behavior at high activities. The sorption kinetics are generally Fickian for blends containing 20% poly(vinyl pyrrolidone) or less, but exhibit two-stage behavior in blends containing 40% poly(vinyl pyrrolidone). Diffusion coefficients extrapolated to zero concentration decrease with increasing poly(vinyl pyrrolidone) content, owing to a decrease in the fractional free volume. However, the diffusion coefficient becomes a greater function of activity as the composition of hydrophilic polymer in the blend is increased, due to plasticization of the material by large levels of sorbed water. Permeability coefficients generally decrease with increasing poly(vinyl pyrrolidone) content for blends containing 20% poly(vinyl pyrrolidone) or less because the decrease in the diffusion coefficient is greater than the increase in the solubility coefficient. Blends containing 40% poly(vinyl pyrrolidone) have permeability coefficients greater than those of polysulfone due to high water solubility. The permeability coefficients depend on water concentration in approximately the same way for all blends. © 1997 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys, 35: 655–674, 1997     

Toluene diffusion in butyl rubber

The sorption isotherm and the polymer mass-fixed diffusion coefficients, D, for toluene in butyl rubber have been measured by the incremental sorption method to concentrations of 130%, corresponding to a solvent volume fraction of 0.578. The increase in D with concentration is strongly exponential to a concentration of 30% and then begins to level out. Since the nature of the dimensional change occurring in vapor sorption was not known, the values of D were converted to solvent self-diffusion coefficients, D₁, assuming both swelling in the thickness direction (1D) and isotropically (3D). The free volume (FV) theory of Fujita was fitted to the resulting D₁ with the zero concentration diffusion coefficient and the self-diffusion coefficient of toluene as limiting values leaving only a single arbitrary parameter. In this form the FV theory was able to describe the trend of the experimental D₁ for the 1D and 3D cases equally well. Values of D were back-calculated from the FV relations for the 1D and 3D cases for comparison with the experimental results and with the diffusion coefficient determined by immersion in toluene. These comparisons favor the assumption that swelling is isotropic. It appears that the simple free volume relation is capable of providing a satisfactory representation of the experimental data with only a single fitting parameter, although there are moderate quantitative discrepancies. © 1994 John Wiley & Sons, Inc.     

Solubility and diffusivity of methylmethacrylate and butylacrylate monomers in a MMA–BA copolymer

Mutual diffusion coefficients and sorption isotherms of methyl methacrylate (MMA) and butyl acrylate (BA) monomers in methyl methacrylate‐butyl acrylate copolymer (MMA‐BA) have been measured by gravimetric sorption. MMA is found to have higher solubility and diffusion rates in the copolymer than BA. Sorption data for MMA were interpreted using classical Flory‐Huggins thermodynamic theory with a constant interaction parameter (χ). A modified version of this theory has been applied to correlate the sorption data of BA, which exhibit a temperature and concentration‐dependent χ parameter. For MMA, the isotherm data reveal enhanced polymer‐solvent interactions with increasing temperature, while for BA the data indicate a drive toward phase separation with increasing temperature. Despite the difference in thermodynamic behavior, both monomers are found to exhibit Fickian diffusion and the diffusivity data are correlated reasonably well with the Vrentas‐Duda free volume theory. Some deviation between the free‐volume correlation and the experimental data is observed at the lowest temperature and BA concentration examined. © 2007 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 45: 1996–2006, 2007     

Swelling equilibrium and sorption kinetics of urethane-modified bismaleimides elastomer

The swelling equilibrium and diffusion kinetics in various solvents of the maleimide-terminated polyurethanes (UBMIs) and of the triol and tetraol-crosslinked polyurethanes (PU) were studied. The polymer volume fraction of the UBMIs at swelling equilibrium is much higher than that of the tetraol-crosslinked PU networks for the same type of polyol used in the PU. It was explained by the high functionality of the UBMIs produced in the network structure. Furthermore, the molecular weight between crosslinks (M_c) has been calculated from the swelling model and the results exhibit good agreement with the proposed network structure. The early time sorption kinetic data were obtained to investigate the diffusion mechanism of the solvent in the networks. The solubility, diffusion coefficients, and permeability of the solvent in UBMI networks were found to be lower than in the multiol-crosslinked PU networks. © 1997 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 35: 1747–1755, 1997     

Water sorption and transport in blends of polyethyloxazoline and polyethersulfone

Water sorption and transport properties for a series of homogeneous blends of hydrophobic polyethersulfone and hydrophilic polyethyloxazoline are reported. Only blends that remained homogeneous after exposure to liquid water were studied in detail. Equilibrium solubility of water in the blend films increases with increasing hydrophilic polymer content. For all materials, equilibrium sorption isotherms show dual-mode behavior at low water vapor activities and swelling behavior at high activities. The sorption/desorption kinetics for PES are generally Fickian, but two-stage behavior is evident in blends containing 10 and 20% polyethyloxazoline. Diffusion coefficients decrease with increasing polyethyloxazoline content, owing to a decrease in the fractional free volume. For all materials, the diffusion coefficient shows a positive dependence on water vapor activity or concentration due to plasticization of the material by high levels of sorbed water, but it becomes a greater function of activity as the composition of hydrophilic polymer in the blend is increased. Since the decrease in the diffusion coefficient is greater than the increase in the solubility coefficient, the permeability coefficient decreases with increasing hydrophilic polymer content. © 1997 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 35 : 993–1007, 1997     

Diffusion of methyl ethyl ketone in polyisobutylene: Comparison of spectroscopic and gravimetric techniques

In this study, vapor sorption FTIR–ATR (Fourier Transform Infrared Attenuated Total Reflectance) spectroscopy was combined with a conventional gravimetric sorption balance to examine diffusion in polymers. Mutual diffusion coefficients of methyl ethyl ketone in polyisobutylene were measured using both methods at various penetrant activities and temperatures in the range 40–60°C. Actual penetrant concentrations were determined from the sorption balance. The diffusion coefficients from the two techniques agree very well with each other. In addition, the diffusivity data from both techniques could be correlated successfully as a function of temperature and concentration with the Vrentas and Duda free-volume model. © 1997 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 35 : 1261–1267, 1997     

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     

Phase equilibrium and diffusion of solvents in polybutadiene: A capillary‐column inverse gas chromatography study

The capillary‐column inverse gas chromatography method was used to measure the diffusion and partition coefficients of ethylbenzene, styrene, and acrylonitrile in polybutadiene (PBD) at infinite dilution of the solvents. Experiments were performed over a temperature range of 50–125 °C. At temperatures well above the glass‐transition temperature of PBD, the diffusivities were correlated using an Arrhenius expression. The Arrhenius parameters in turn were intercorrelated and shown to be a function of the occupied volume, thus providing a method for predicting the diffusion of other solvents in the same polymer. Further, the activation energy was predicted using the Duda‐Vrentas free‐volume approach. The activation energy thus obtained was compared with the activation energy of the Arrhenius approach. The weight‐fraction activity coefficient data were compared to the predictions of the group contribution, lattice‐fluid equation‐of‐state, and the UNIquac Functional‐group Activity Coefficient (UNIFAC) free‐volume models. © 2002 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 40: 1046–1055, 2002     

Sorption and diffusion of oxygen in plasticized ethyl cellulose films of varying degrees of substitution

Plasticized films cast from ethyl cellulose were examined to evaluate the effect of the degree of substitution, DS, and the plasticizer content on the sorption and diffusion of oxygen. Sorption and permeation measurements were performed over a temperature range of 25–65°C on three different types of ethyl cellulose in the DS range 1.7–2.5 that had been plasticized with organic esters of comparatively low molecular weight. Sorption coefficients were determined by the pressure decay method, and permeability coefficients were measured independently according to ASTM D-1434-66. The diffusion coefficients were calculated assuming Fickian transport, and were compared to the values directly obtained from the evaluation of the sorption kinetics. The permeability coefficient indicates that there is a significant improvement of the barrier properties of the materials when the DS is reduced and when the plasticizer content is at the absolute minimum required. It was found that the variation in the magnitude of the permeability coefficient is related to the value of the diffusion coefficient, which is governed by the chemical composition of the mixtures. In contrast, the solubility of oxygen was determined by the physical state of the polymer matrix and increased rapidly at temperatures significantly below the glass transition temperature. Using an ergodic model, the diffusion coefficients obtained were related to the size distribution of microvoids in the materials and relative values for the diffusion coefficient were computed as a function of DS and temperature. The model calculates the concentration (number per volume) of voids that are large enough to be occupied by a penetrant molecule. It was assumed that the unoccupied volume fraction as a function of the cohesive energy density follows a Boltzmann distribution. The cohesive energy density and the unoccupied volume fraction of the polymer-plasticizer mixtures were calculated by fitting the Simha-Somcynsky equation of state to pressure-volume-temperature data. © 1997 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys, 35: 639–653, 1997     

Prediction of solvent diffusion coefficient in amorphous polymers based on an equation‐of‐state

This study develops a modified free‐volume model to predict solvent diffusion coefficients in amorphous polymers by combining the Vrentas–Duda model with the Simha–Somcynsky (S‐S) equation‐of‐state (EOS), and all the original parameters can be used in the modified model. The free volume of the polymer is estimated from the S‐S EOS together with the Williams‐Landel‐Ferry fractional free volume, and the complex process of determining polymer free‐volume parameters in the Vrentas–Duda model and measuring polymer viscoelasticity can be avoided. Moreover, the modified model includes the influence of not only temperature but also pressure on solvent diffusivity. Three common polymers and four solvents are employed to demonstrate the predictions of the modified model. The calculation results are generally consistent with the experimental values. It is reasonable to expect that the modified free‐volume model will become a useful tool in polymer process development. © 2006 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 44: 1000–1009, 2006     

Prediction and estimation of solvent diffusivities in polyacrylate and polymethacrylates

The relationship between polymer side‐chain length and the hole free volume that is effective for solvent diffusion was investigated for polyacrylates and polymethacrylates on the basis of free‐volume theory. Measurements of a polymer's viscoelasticity and solvent diffusivity provided experimental evidence for polymer segment mobility, and the results indicated that hole free volume in a linear polymer increases with hydrocarbon side‐chain length. Because the molecular mechanisms of polymer viscoelasticity and diffusivity are identical, the free‐volume parameters obtained for polyacrylates and polymethacrylates by measuring the polymer viscoelastic‐temperature dependence can reliably be used in predicting the solvent diffusion coefficient. © 2003 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 41: 1393–1400, 2003     

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     

Gas permeability and hydrocarbon solubility of poly[1‐phenyl‐2‐[p‐(triisopropylsilyl)phenyl]acetylene]

The effects of film thickness, physical aging, and methanol conditioning on the solubility and transport properties of glassy poly[1‐phenyl‐2‐[p‐(triisopropylsilyl) phenyl]acetylene] are reported at 35 °C. In general, the gas permeability coefficients are very high, and this polymer is more permeable to larger hydrocarbons (e.g., C₃H₈ and C₄H₁₀) than to light gases such as H₂. The gas permeability and solubility coefficients are higher in as‐cast, unaged films than in as‐cast films aged at ambient conditions and increase to a maximum in both unaged and aged as‐cast films after methanol conditioning. For example, the oxygen permeability of a 20‐μm‐thick as‐cast film is initially 100 barrer and decreases to 40 barrer after aging for 1 week at ambient conditions. After methanol treatment, the oxygen permeabilities of unaged and aged films increase to 430 and 460 barrer, respectively. Thicker as‐cast films have higher gas permeabilities than thinner as‐cast films. Propane and n‐butane sorption isotherms suggest significant changes in the nonequilibrium excess free volume in these glassy polymer films due to processing history. For example, the nonequilibrium excess free volume estimated from the sorption data is similar for as‐cast, unaged samples and methanol‐conditioned samples; it is 100% higher in methanol‐conditioned films than in aged, as‐cast films. The sensitivity of permeability to processing history may be due in large measure to the influence of processing history on nonequilibrium excess free volume and free volume distribution. The propane and n‐butane diffusion coefficients are also sensitive to film processing history, presumably because of the dependence of diffusivity on free volume and free volume distribution. © 2000 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 38: 1474–1484, 2000     

Self‐diffusion of small molecules into rubbery polymers: A lattice free‐volume theory

In the framework of the free‐volume (FV) theory, a new equation was derived for the evaluation of self‐diffusion coefficients of small molecules in polymers above the mixture glass transition temperature. The derivation of the equation turned out to be straightforward once the equivalence between the free volume and the unoccupied volume given by thermodynamic lattice theories is assumed. A parameter evaluation scheme is proposed, which is substantially simpler compared with the conventional Vrentas–Duda approach, even without losing generality. The key assumption is discussed, and its consistency is verified from a numerical viewpoint. A comparison with experimental solvent self‐diffusion coefficients for several solvent/polymer binary systems confirmed that the proposed theory presents good correlative ability over wide temperature and composition ranges. Moreover, the introduced thermodynamic foundation allows one to easily include the pressure effect too. In the frame of the proposed lattice free volume theory, the sizes of the polymer jumping units decrease with temperature and increase with pressure. Such behavior converges with theoretical expectations and opens the way for a predictive FV theory. © 2010 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 48: 529–540, 2010     

Effect of blend composition on diffusivity and solubility of small molecules in polystyrene/poly(vinyl methyl ether) blends

The capillary column inverse gas chromatography technique was used to determine diffusivity and solubility data for several solvents in polymer blends composed of polystyrene and poly(vinyl methyl ether) (PVME). Diffusivity behaved as expected, increasing as the concentration of PVME increased in the blend. Knowing only the free‐volume parameters for the pure polymers, the free‐volume theory was successfully applied to predict the dependence of the diffusion coefficients on the blend composition. Transport in blends above the glass transition temperature is controlled by free volume, and the effect of concentration fluctuations is minimal at the temperatures studied. Experimental data show an increase in the partition coefficient of some solvents in the blends with respect to the pure polymers. © 2007 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 45: 2071–2082, 2007