Transient permeation of organic vapors through elastomeric membranes

The permeation of benzene and acetone vapors through sulfur-cured natural rubber was studied by the time-lag method. The experimental results were analyzed by a method suggested by Meares. The zero concentration diffusion coefficient D₀ was obtained by the early-time method. The Frisch time-lag equation was utilized to estimate both the solubility coefficient s and the additional parameter b required to define the concentration dependence of the diffusion coefficient: D(c) = D₀ exp {bc}. This form of concentration dependence was manifested by the corresponding permeability coefficient values. At low entering penetrant pressure, where the transport coefficients are constant, indirect evidence was obtained that D₀ is the mechanistically correct diffusion coefficient. The solubility coefficient values calculated for benzene vapor in natural rubber are in reasonable agreement with published equilibrium sorption data for a similar rubber compound. At higher entering penetrant pressures, average diffusion coefficients obtained at steady state tended to be larger than the corresponding average diffusion coefficients derived from the time lags. This same effect has been detected by other experimental approaches. Permeation experiments designed for this rapid method of analysis appear capable of yielding information consistent with that obtained by more time-consuming traditional methods.     

Determination of penetrant solubility from transient permeation measurements

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

Kinetic analysis of transient permeation curves

The kinetic analysis of transient permeation curves on a systematic basis is suggested as a method for the study of non-ideal membrane—penetrant diffusion systems, analogous and complementary to the method of time-lag analysis which we have previously developed. Experimental data on N₂ or propane permeating through porous graphite pellets of varying non-homogeneity are reported and used for a preliminary application of these ideas. Previous work on early-time permeation kinetics is considerably extended here and late-time permeation kinetics is considered for the first time. The results obtained show that the proposed method promises to play an important role in the study of non-ideal diffusion systems, when an adequate theoretical background has been built up.     

Effect of pressure on gas permeability coefficients. A new application of “free volume” theory

The present work is a continuation of a general study of the effect of pressure on gas and vapor permeation through nonporous polymeric membranes. Permeability coefficients have been measured for 1,1-difluoroethylene (C₂H₂F₂) and fluoroform (CHF₃) in polyethylene at penetrant pressures up to 35 atm and at temperatures between -18 and 70°C. The permeability coefficient P? for the 1,1-difluoroethylene—polyethylene system was found to increase with increasing pressure differential Δp across the membrane. Isothermal plots of log ΔP versus Δp are generally linear and can be represented by empirical relations of the form ΔP = P(0)exp{m Δp}, where P(0) and m are constants. The slope m of these isotherms decreases with increasing temperature. Plots of log P? versus Δp for the fluoroform—polyethylene system are also linear, but exhibit negative slopes, i.e., P? decreases with increasing Δp. An extension of Fujita's “free volume” theory of diffusion in polymers shows that the dependence of P? on pressure reflects how the free volume of the polymer is affected by this pressure. An increase in the penetrant pressure may result in two opposing effects: (a) the concentration of the penetrant dissolved in the membrane is increased, thereby increasing the free volume, and (b) the hydrostatic pressure on the membrane is also increased, which causes a decrease in the free volume. If the overall effect is an increase in the free volume of the polymer, then P? will also increase, and vice versa.     

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     

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

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

Ethylbenzene solubility,diffusivity, and permeability in poly(dimethylsiloxane)

The pure‐gas sorption, diffusion, and permeation properties of ethylbenzene in poly(dimethylsiloxane) (PDMS) are reported at 35, 45, and 55 °C and at pressures ranging from 0 to 4.4 cmHg. Additionally, mixed‐gas ethylbenzene/N₂ permeability properties at 35 °C, a total feed pressure of 10 atm, and a permeate pressure of 1 atm are reported. Ethylbenzene solubility increases with increasing penetrant relative pressure and can be described by the Flory–Rehner model with an interaction parameter of 0.24 ± 0.02. At a fixed relative pressure, ethylbenzene solubility decreases with increasing temperature, and the enthalpy of sorption is −41.4 ± 0.3 kJ/mol, which is independent of ethylbenzene concentration and essentially equal to the enthalpy of condensation of pure ethylbenzene. Ethylbenzene diffusion coefficients decrease with increasing concentration at 35 °C. The activation energy of ethylbenzene diffusion in PDMS at infinite dilution is 49 ± 6 kJ/mol. The ethylbenzene activation energies of permeation decrease from near 0 to −34 ± 7 kJ/mol as concentration increases, whereas the activation energy of permeation for pure N₂ is 8 ± 2 kJ/mol. At 35 °C, ethylbenzene and N₂ permeability coefficients determined from pure‐gas permeation experiments are similar to those obtained from mixed‐gas permeation experiments, and ethylbenzene/N₂ selectivity values as high as 800 were observed. © 2000 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 38: 1461–1473, 2000     

Estimation of gas transport coefficients from differential permeation, integral permeation, and sorption rate data

Experimental methods for studying the transport of gases in polymers may be divided into three categories: integral permeation rate measurement, in which the cumulative amount of a penetrant that has passed through a membrane is determined; differential permeation rate measurement, in which the rate of penetration through a membrane is measured directly; and sorption rate measurement, or determination of the cumulative amount of a penetrant absorbed in a polymer sample. This paper reviews commonly used techniques for estimating diffusion coefficients from transport data of all three types. Several new estimation formulas are presented, and the relative merits of different measurement and estimation methods are discussed. A general relationship between the traditional time lag method for integral rate data analysis and a recently developed moment method for differential rate data analysis is established, extending the applicability of the moment approach to the analysis of non-ideal transport in membranes of arbitrary geometry and composition.     

Computer‐Aided Estimation of Diffusion Coefficients in Non‐Solvent/Polymer Systems

A novel method for estimating the mutual and self‐diffusion coefficients of a non‐solvent/polymer system is proposed in this work. The idea is to study the evaporation process from non‐solvent/solvent/polymer systems as a one‐dimensional numerical experiment and to use polymer solution weight versus time data to fit the unknown parameters of the diffusion‐coefficient correlations based on free‐volume theory. For this purpose, the evaporation process is modeled as a coupled heat‐ and mass‐transfer problem with a moving boundary, and the Galerkin finite‐element method is used to solve simultaneously the non‐linear governing equations. This method is successfully applied to the estimation of water–cellulose acetate diffusion coefficients and is valid over the whole range of temperatures and concentrations for practical applications in membrane technology. Additionally, there is a detailed discussion on if water affects the morphology of the final cellulosic membrane by studying the concentration profiles of the constituents of the casting solution.     

Diffusion in inhomogeneous films and membranes

In a number of different contexts one is interested in the diffusion of a penetrant into an inhomogeneous film or membrane. We review exact results in the theory of permeation through membranes containing fixed inhomogeneities. We also present the available exact results on permeation, time lag, and sorption for a sufficiently dilute penetrant in an inhomogeneous slab or film. We discuss and compare the kind of information provided by permeation and sorption studies using a variety of examples of simple types of inhomogeneities.     

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.     

Diffusion of Methanol in Polydimethylsiloxane

The permeation and sorption of methanol in polydimethylsiloxane at 10 and 30°C. has been measured and the results analyzed to determine the concentration dependence of the steady-state diffusion coefficient D which is found to decrease as the total concentration C is increased. An analysis of the isotherms indicates that clustering of the methanol occurs in the polymer, becoming more predominant as the concentration is increased. A polymerization model used to describe the shape of the D versus C curve for water in polydimethylsiloxane has been modified and applied with some success to describe the shape of the isotherm and the D versus C curve for methanol. The linearity of the permeation rate with relative pressure in this and a number of water—polymer systems is briefly commented on.     

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

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

     

Anomalous penetrant diffusion as a probe of the local structure in a blend of poly(ethylene oxide) and poly(methyl methacrylate)

Pulse field gradient (PFG) NMR measurements have been made to study the diffusion of diethyl ether in blends of poly(ethylene oxide) (PEO) and poly(methyl methacrylate) (PMMA). The blends have 10–30 wt % PEO, a composition range within which these materials are amorphous glasses. The diffusion of diethyl ether through the blends is quite rapid, with diffusion constants in the range of 10^?7 to 10^?8 cm²/s. In PFG NMR experiments, the apparent diffusion constant depends on the timescale over which diffusion is observed. The values decrease to a plateau as the time increases, this being the signature of tortuous diffusion. Tortuous diffusion is usually observed in heterogeneous systems in which there are regions that support fast diffusion and regions that support slow diffusion or act as barriers. In these blends, PEO is known to undergo rapid segmental motion typical of a rubbery state well below the glass transition, whereas the segmental motion of PMMA is slower by many orders of magnitude. Mobile PEO provides a pathway for the diffusion of structurally similar diethyl ether, whereas solid‐like PMMA acts as a barrier. The size of the domains can be estimated either from a lattice model or from equations for tortuous diffusion. Micrometer sizes are indicated that are unexpectedly large, given the size of the polymer chains and the size of the concentration fluctuations, both of which are thought to be in the tens of nanometers. The lattice model and the equations for tortuous diffusion assume a random dispersion of impenetrable or less penetrable objects. This may not be the appropriate morphology for the diffusion pathway. Recently, large sizes have been indicated by PFG NMR experiments, in which a penetrant is thought to diffuse in a curvilinear fashion. In these blends, the pathway for diethyl ether is along the PEO backbone. A plot of the logarithm of the mean‐square displacement versus the logarithm of time has a slope of about 0.6, close to the value of 0.5 for pure curvilinear diffusion. Exponents with values in this range can also be associated with diffusion in a fractal space, which, in this situation, still consists of mobile PEO. © 2004 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 42: 1053–1067, 2004     

Diffusion in glassy polymers. III

The diffusion studies of several solvents in epoxy polymer reported by Kewi and Zupko in Part I of this series are explained with the solution obtained from the generalized diffusion equation which includes the internal stress contribution. The rate of permeation of a penetrant through a polymer film and the time lag needed to reach steady state are also given for the generalized diffusion equation.     

Disentanglement and reptation during dissolution of rubbery polymers

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

Solubility,diffusivity, and mobility of n-pentane and ethanol in poly(1-trimethylsilyl-1-propyne)

The diffusion coefficient of ethanol and of n-pentane in PTMSP, at 27°C, was measured as a function of concentration up to a penetrant content of about 12% by weight, for polymer samples obtained through different processes; differential sorptions and desorptions with vapor phases were considered. In the case of ethanol a nonmonotonous behavior was observed for the diffusivity, while in the case of n-pentane the same property was found to monotonously decrease with increasing the penetrant content. The sorption isotherms were also reported, indicating that n-pentane exhibits a typical dual mode behavior, while ethanol follows an unusual s-shape curve. The chemical potential of the dissolved penetrants, calculated directly from the isotherms, shows the very different importance of the energetic interactions of the two penetrants with the polymer units. In spite of the remarkably different concentration dependencies observed for both solubility and diffusivity of the two penetrants, the mobility factors are in both cases monotonously decreasing with the penetrant concentration, and follow very similar trends. The significant differences observed for the concentration dependence of the diffusion coefficients are, thus, associated to the thermodynamic contributions, which are very different for n-pentane and ethanol. Different polymeric films, obtained through different solvent evaporation processes, show quite different solubility, diffusivity and mobility for both ethanol and n-pentane. On the other hand, the ratio between the mobility of the two penetrants as well as the slope of mobility as function of the concentration remains the same for all the different samples inspected. © 1997 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 35 : 2245–2258, 1997     

Correlation of Activation Energies of Gas Diffusivity and Local Matrix Mobility in Polycarbonate/POSS Nanocomposites

According to basic phenomenological models describing the solution‐diffusion based mechanism of penetrant diffusion in dense polymers, a connection between the diffusive transport of gas molecules in a polymeric matrix and the molecular mobility of that matrix on a certain length scale is, in principle, established for a long time. However, experimental data directly showing this correlation are rare. The investigation of a series of nanocomposites based on a polyhedral oligomeric silsesquioxane (POSS) and a polycarbonate matrix allows a systematic change of the molecular mobility on a local length scale (β‐relaxation) and of the corresponding activation energy E_A, both determined by broadband dielectric spectroscopy. Independently, activation energies of penetrant diffusion (E_D) of these nanocomposites were determined for N₂, O₂, CO₂, and CH₄ and a clear linear correlation between the two activation energies was established for the first time. © 2013 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2013 , 51, 1593–1597     

Modelling of the Potentiometric Response of ENFETs Based on Enzymatic Multilayer Membranes

Based on the modified site‐binding model, the response of urea‐sensitive enzymatic field effect transistors (ENFETs) is fitted. The effect of two types of surface sites (silanol and amine sites) and of additional polymeric permselective membrane on the sensitivity of the biosensor is discussed. It is found that the dependence of the slope of the sensor response versus pUrea, on the buffer concentration and on the diffusion of the urea in the enzymatic membrane can be translated by a parameter α that depends on the membrane composition. Moreover, the enhancement of the sensor sensitivity by deposition of additional permselective membranes can be illustrated by a parameter δ that depends on the association of additional permselective membranes and a buffer; the higher values of parameter δ are obtained with a Nafion membrane associated with a phosphate buffer.     

Sorption and diffusion of methyl substituted benzenes through cross-linked nitrile rubber/poly(ethylene co-vinyl acetate) blend membranes

The diffusion and sorption of methyl substituted benzenes through cross-linked nitrile rubber/poly(ethylene co-vinyl acetate) (NBR/EVA) blend membranes has been studied. The influence of blend composition, cross-linking systems, temperature and size of penetrants on the transport behaviour has been analysed. It was observed that as the EVA content increases in the blends, the solvent uptake decreases. An increase in the penetrant size also decreases the solvent uptake. The diffusion experiments were carried out in the temperature range 23–75 °C. As temperature increases the equilibrium uptake also increases. The transport coefficients namely diffusion coefficient, sorption coefficient and permeation coefficient have been calculated. The sorption data has been used to estimate the activation energies for permeation and diffusion. The van’t Hoff relationship was used to determine the thermodynamic parameters. The affine and phantom models for chemical cross-links were used to predict the nature of cross-links. Models for permeability were used and the theoretical values compared with the experimental results.