New methods for the determination of the parameters of a concentration‐dependent diffusion law for molecular penetrants from transient permeation or sorption data

New methods were developed for the determination of the plasticization parameters and the limit diffusion coefficient of a penetrant whose diffusivity varies exponentially with its concentration. No specific computer softwares is required for their use and they are user friendly. The method using differential permeation data is based on the correlations between the slope of the reduced permeation flux versus time plot at the inflexion point, and the two key parameters of the concentration‐dependent diffusivity laws. For the transient sorption method, the slope of the penetrant mass uptake versus square‐root of time curve leads to the same parameters of the diffusion law via similar correlations.     

Pressure and temperature dependence of the gas‐transport properties of dense poly[2,6‐toluene‐2,2‐bis(3,4‐dicarboxylphenyl)hexafluoropropane diimide] membranes

The gas‐transport properties of poly[2,6‐toluene‐2,2‐bis(3,4‐dicarboxylphenyl)hexafluoropropane diimide] (6FDA‐2,6‐DAT) have been investigated. The sorption behavior of dense 6FDA‐2,6‐DAT membranes is well described by the dual‐mode sorption model and has certain relationships with the critical temperatures of the penetrants. The solubility coefficient decreases with an increase in either the pressure or temperature. The temperature dependence of the diffusivity coefficient increases with an increase in the penetrant size, as the order of the activation energy for the diffusion jump is CH₄ > N₂ > O₂ > CO₂. Also, the average diffusion coefficient increases with increasing pressure for all the gases tested. As a combined contribution from sorption and diffusion, permeability decreases with increases in the pressure and the kinetic diameter of the penetrant molecules. Even up to 32.7 atm, no plasticization phenomenon can be observed on flat dense 6FDA‐2,6‐DAT membranes from their permeability–pressure curves. However, just as for other gases, the absolute value of the heat of sorption of CO₂ decreases with increasing pressure at a low‐pressure range, but the trend changes when the feed pressure is greater than 10 atm. This implies that CO₂‐induced plasticization may occur and reduce the positive enthalpy required to create a site into which a penetrant can be sorbed. Therefore, a better diagnosis of the inherent threshold pressure for the plasticization of a glassy polymer membrane may involve examining the absolute value of the heat of sorption as a function of pressure and identifying the turning point at which the gradient of the absolute value of the heat of sorption against pressure turns from a negative value to a positive one. © 2003 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 42: 354–364, 2004     

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

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

The 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.     

Sorption and transport of linear alkane hydrocarbons in biaxially oriented polyethylene terephthalate

Equilibrium sorption and uptake kinetics of n‐butane and n‐pentane in uniform, biaxially oriented, semicrystalline polyethylene terephthalate films were examined at 35 °C and for pressures ranging from 0 to approximately 76 cmHg. Sorption isotherms were well described by the dual‐mode sorption model. Sorption kinetics were described either by Fickian diffusion or a two‐stage model incorporating Fickian diffusion at short times and protracted polymer structural relaxation at long times. Diffusion coefficients increased with increasing penetrant concentration. n‐Butane solubility was lower than that of n‐pentane, consistent with the more condensable nature of n‐pentane. However, n‐butane diffusion coefficients were higher than those of n‐pentane. Infinite‐dilution, estimated amorphous phase diffusion and solubility coefficients were well correlated with penetrant critical volume and critical temperature, respectively. © 2001 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 39: 1160–1172, 2001     

Quantitative approach for the prediction of preferential sorption in the case of pervaporation of a physico-chemically similar binary mixture

Selective transport by pervaporation of physico-chemically similar and polar components through a polar membrane is complicated due to competitive sorption and diffusion phenomena. In the present study the mechanism of preferential sorption has been predicted for such a system, methanol-ethylene glycol-cellophane. The mechanism has been explained in terms of intercrystalline swelling of the polymer matrix in presence of increasing methanol concentration in the feed. The reduction in preferential sorption at increasing wt% of methanol in the feed may be due to the increasing accessibility of the membrane towards ethylene glycol. This phenomenon has been quantitatively explained by considering a non-linear dependence on concentration of the binary liquid-polymer interaction parameter. Theoretical sorption data have been derived from the Flory-Huggins thermodynamics by using the swelling equilibrium condition. The coupling and plasticization phenomena in sorption are explained in terms of liquid-polymer interaction parameters. The theoretical results show good agreement with previously published experimental data.     

The concentration dependence of the diffusion and permeability in a homogeneous membrane

For the sorption and diffusion coefficient dependence on the concentration of the penetrant the transport properties of a homogeneous medium are calculated. The diffusion current is assumed to be proportional to the negative gradient of the chemical potential. This is in contrast with the first Fick's law that assumes this current to be proportional to the negative gradient of the concentration of the penetrant. The difference between the two cases depends on the concentration dependence of the sorption coefficient. In a homogeneous membrane the chemical potential formulation leads to an equation which is very similar to the Fickian expression. The apparent diffusion coefficient, however, depends not onlly on the transport resistance but also on the deviation of the sorption coefficient from constancy.     

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.     

Gas transport properties of poly(2,2,4,4-tetramethyl cyclobutane carbonate)

Gas transport properties of semicrystalline films of poly(2,2,4,4-tetramethyl cyclobutane carbonate) (TMCBPC) were studied. Permeability coefficients for He, O₂, N₂, CH₄, and CO₂ at 35°C for pressures between 1 and 20 atm are reported as well as sorption isotherms for N₂, CH₄, and CO₂ at the same conditions. The permeability coefficients for TMCBPC are larger than corresponding values for the aromatic bisphenol-A polycarbonate (PC) and tetramethyl bisphenol-A polycarbonate (TMPC), even though the TMCBPC films are semicrystalline. These results are explained on the basis of the larger free volume available for permeation in this polymer. Significant TMCBPC plasticization by CO₂ was also observed and this causes typical time-dependent behavior. The plasticization process starts at very low pressures compared with the behavior of aromatic polycarbonates PC and TMPC. This early onset of plasticization seems to be related also to the larger free volume in the amorphous phase of TMCBPC which favors high gas sorption. The diffusion coefficients for TMCBPC are also larger than those reported for the aromatic polycarbonates PC and TMPC. Ideal gas separation factors were found to follow the usual trend; that is, as permeability increases, the ideal separation factor decreases. © 1993 John Wiley & Sons, Inc.     

Permeation of water through polar and nonpolar polymers and copolymers: Determination of the concentration‐dependent diffusion coefficient

The diffusion and permeation properties of liquid water through different polar and nonpolar polymers and copolymers were studied with a highly sensitive permeameter. The transient permeation fluxes through the polar polymer films could be fitted well only with an exponential equation for the diffusivity concentration dependence; this empirical exponential equation represented the diffusion plasticization effect of water on the materials. For the hydrophobic polyolefins, this exponential equation was no longer valid, and another form of the equation was empirically found to account for the reduction of the water diffusivity with the extent of the permeation. Such a negative plasticization effect might be attributed to the formation of water clusters in the polyolefins. The values of the diffusion coefficient of water in the dry polar polymers were smaller than those in dry polyolefins, but the opposite behavior was found for the permeability because it was much more favorable for water sorption in the polar polymers than in the hydrophobic polyolefins. For the ethylene–vinylacetate copolymers, the plasticization effect of water on its own diffusion was negative for the sample with a low vinyl acetate (VA) content; it became nil at 19 wt % VA and positive at higher VA contents. This increase in the extent of the water sorption with the increase in the VA content led to a steady increase in the water permeability in the poly(ethylene‐co‐vinylacetate) copolymers. © 2000 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 38: 1998–2008, 2000     

Diffusion of organic vapors at low concentrations in glassy PVC, polystyrene, and PMMA

Diffusion coefficients of various C₁ to C₆ organic vapors, at concentrations 0.5 wt. percent, have been determined by gravimetric sorption rate measurements on emulsion and suspension-polymerized powder samples of PVC, polystyrene, and PMMA. Fickian diffusion kinetics were observed at the lowest concentrations, with a second-stage, relaxation-controlled sorption appearing at higher concentrations. In conjunction with published data for diffusivities of fixed gases in these polymers, the results indicate that diffusivity decreases exponentially, and that diffusion activation energy (E_D) increases linearly, with increasing diameter of “spherical” penetrant molecules (e.g., the noble gases, CH₄, SF₆, CCl₄, and neopentane). Much of the observed scatter in these correlations is attributable to uncertainty in the molecular diameters. For C₄ and larger n-alkanes and other elongated or flattened molecules, diffusivities are higher, and E_D lower, than for spherical molecules of similar molar volume. This finding suggests that anisometric molecules are oriented and move along their long axes during diffusion through the glassy polymer matrix. Correlations of diffusivities with molecular dimensions suggests that transport of anisometric molecules is governed by a diameter smaller than the mean (equivalent sphere) diameter but larger than the minimum dimension of their extended-chain conformation. Among the three polymers studied, diffusivity of each penetrant, at a given temperature, decreases in the order polystyrene> PVC ≥ PMMA.     

Relation between amorphous structure of polymers and penetrant diffusion: A molecular dynamics simulation

Molecular dynamics simulation has been performed for studying the relation between amorphous structure of polymers and penetrant diffusion. The self-diffusion coefficients of O₂ and He in various polymer models, which differ from each other in view of the amorphous structure, were calculated above their glass transition temperatures. The amorphous structure was characterized by considering the percolation of the unoccupied volume. A good correlation was found between the self-diffusion coefficients and the number of clusters in the unoccupied volume at the critical point of the percolation. Based on the simulated cluster size distribution at the critical point, we defined a parameter into which effects of both the amorphous structure and the penetrant size are well incorporated. It was confirmed that the penetrant diffusion is intimately associated with the amorphous structure of polymers.     

Solid-state 13C NMR evidence for gas–polymer interactions in the carbon dioxide–poly(vinyl chloride) system

The presence of small amounts of CO₂ in poly(vinyl chloride) (PVC) results in increased main-chain molecular motions of the polymer as measured by the carbon rotating-frame relaxation rate. This effect increases with increasing gas concentration. Since molecular motions of the polymer and the diffusion coefficient of the gas are related, the latter must be concentration dependent. Main-chain motions of PVC also increase upon exposure to CO₂ followed by degassing. This result is interpreted in terms of the effect of the penetrant gas on the interchain packing in amorphous PVC. These results cannot be reconciled with the dual-sorption-mobility model, which claims that gas molecules preferentially occupy preexisting sorption sites in a conditioned polymer with no perturbation of the polymer matrix.     

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.     

Sorption and diffusion of toluene in highly oriented polypropylene

The sorption and diffusion of toluene vapor at 30°C in polypropylene with draw ratios from 1 to 18 have been studied. Drawing leads to the transformation of the initially spherulitic material into the fibrous structure, with many taut tie molecules lying mainly on the outer boundary of the microfibrils. The free volume and hence the sorption sites are thereby reduced, and the microfibrils become less and permeable as the draw ratio increases. As a result, the equilibrium concentration and the zero-concentration diffusion coefficient drop by factors of 4 and 30, respectively. The diffusion coefficient increases exponentially with toluene concentration but the concentration dependence becomes weaker with increasing draw ratio, indicating that the severely constrained chain segments in the drawn samples have much less freedom to mix with penetrant molecules. Annealing relaxes the tie molecules and thus restores the sorption and diffusion properties to values corresponding to completely relaxed amorphous component, i.e., to values even higher than those of the undrawn but quenched material.     

Plasticization mechanism in polybenzimidazole membranes for organic solvent nanofiltration: Molecular insights from in situ FTIR spectroscopy

Despite substantial research efforts being devoted to design polymer membranes for organic solvent nanofiltration (OSN) exhibiting enhanced plasticization resistance, detailed studies of membrane structural stability in chemically challenging environments are rare. This study sets forth a multiscale method, which combines in situ FTIR measurements in the transmission mode with sorption and transport measurements, to investigate the molecular mechanism of polymer plasticization upon exposure to organic species. We recently reported that polybenzimidazole (PBI), a polymer that has been considered for OSN application, experiences severe plasticization upon exposure to methanol. FTIR measurements suggest that the mechanism of PBI plasticization relies on competitive hydrogen bonding. According to this mechanism, methanol molecules disrupt the hydrogen bonded network characteristic of dry PBI, by forming mutual polymer/penetrant hydrogen bonds, which enhances polymer chain mobility and favors polymer plasticization. The analysis of the isosteric heat of sorption supports the picture sketched above. The method developed in this study can be readily extended to many other systems of practical interest in membrane science, to shed fundamental light on the phenomenon of plasticization, whose molecular aspects are still largely unknown. Equally important, this study provides some useful guidelines to design OSN membrane materials.     

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

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

Mass and momentum transfer in polymers. III. Effects of liquid penetrants on tensile stress relaxation of amorphous polymers

Sorption, diffusion, swelling, and tensile stress relaxation measurements were made at room temperature (23°C) for the systems poly(n-butyl methacrylate) (PBMA) with liquid methanol and ethanol, and poly(methyl acrylate) (PMA) with liquid water. Stress relaxation curves for the fully swollen polymers could be superimposed approximately with those for the dry polymers by appropriate shifting along the long axes. For PMA–water the measured curve for stress relaxation with concurrent sorption could be predicted accurately by using a moving boundary theory with data measurements of stress relaxation of the unswollen and swollen polymer combined with sorption data. The modified moving boundary theory is generalized to include the effects of dimension changes through swelling and the larger effects of plasticization associated with sorption of liquids. This improved theory accurately predicts measured curves of stress relaxation with concurrent sorption for the PBMA–alcohol systems from individual stress relaxation, sorption, diffusion and swelling data. The general approach should be applicable to other amorphous polymer–liquid swelling agent systems. The anisotropic nature of swelling of polymer films and its effect on calculated diffusion coefficients are discussed briefly.     

Diffusion through rubbery and glassy polymer membranes

Mass transport of a number of organic vapors through polydimethylsiloxane films (PDMS) and carbon dioxide through a variety of polyimides based on a hexafluorotetracarboxylic acid unit (6FDA) were investigated. Vapor diffusion through PDMS films strongly depends on the concentration of the penetrant molecules in the network. For chloroform, increasing diffusivity at lower upstream activities occurs due to network plasticization, while a decreasing diffusion coefficient at larger concentration is supposed to stem from penetrant molecule clustering. The diffusion of carbon dioxide in 6FDA-based polyimides was modelled on a molecular basis. An exponential relation was found between Δc_p and the diffusion coefficient and the permeability, respectively. This relation holds also for on-chain modifications.