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.     

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.     

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.     

Size effect on competition of two diffusion mechanisms for drug molecules in amorphous polymers

Diffusion of drug molecules in polymer materials is of great importance in controlled drug release, and the investigation of the mechanism of drug release from the polymer matrix would help us to understand the release behavior of the controlled release system. In this work, molecular dynamics simulations were employed to investigate the diffusion mechanisms of penetrant molecules with different sizes in poly(lactic acid-co-ethylene glycol) (PLA-PEG). The size effect on the diffusion mechanism of penetrant molecules in polymer matrixes was discussed in detail. A competition mechanism in a two-step diffusion process-(1) motion within the cavities (free volumes), and (2) jumps between cavities or movement of the cavity itself originated from the wriggling of the polymer chains-was observed, and the contributions of these two factors to the diffusion coefficient were successfully separated. With the medium volume of penetrant molecules (e.g., benzene), a competition between these two steps was observed. Step (2) controlled the diffusion when penetrant molecules became bigger.     

Differential permeation — Part I: A method for the study of solvent diffusion through membranes

We describe the differential permeation method for the study of the diffusion of solvents from a liquid (or liquid mixture) through flat or tubular membranes. This method consists of measuring the transient permeation rates through the membrane when one of its faces is suddenly put into contact with the liquid medium. The change in the transient rate with time is analyzed by numerical best fitting methods to determine the Fickian diffusion coefficient. A simplified equation is proposed for the fitting of the response of a tubular membrane. Deviations from the Fickian transport mechanism with concentration-independent diffusion coefficient can be evidenced and eventually analyzed by using other mechanistic models.     

Simulation and modelling of transient permeation of organic solvents through polymer films in the case of a concentration dependent diffusion coefficient

The measured transient permeation kinetics of acetic acid from a water-acetic acid mixture through poly(vinyl alcohol) films could not be accounted for by the Fick law with a constant diffusion coefficient. A new calculation procedure was developed on the basis of simulation results of the Fickian kinetics in which the diffusion coefficient was assumed to increase exponentially with the local permeant concentration. A fast and reliable fitting procedure, which was set up on a Personal Computer, involves an iterative numerical adjustment of (a) the value of the limiting diffusion coefficient D^* (i.e., difusion of the permeant in dry polymer) using the onset part of the permeation rate followed by (b) the value at the upstream face of the plasticization term (argument of the exponential function) using the shape of the experimental curve. The values obtained from fitting the model to the transient kinetics showed that the limiting diffusion coefficient increases drastically, but the plasticization term changes little, with increasing temperature.     

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     

Transport of heavy organic vapors through silicone rubber

Permeability, solubility, and diffusion coefficients have been determined for halothane (CF₃CHClBr) and methoxyflurane (CHCl₂CF₂OCH₃) in silicone rubber at temperatures from 17 to 60°C and at relative pressures from 0.05 to 0.96. The solubility of both penetrants in silicone rubber is a strong function of penetrant concentration (or relative pressure), and can be represented satisfactorily by the Flory-Huggins relation with single values of the interaction parameter χ. The solubility coefficients decrease with increasing temperature at constant pressure. Mutual diffusion coefficients exhibit maxima when plotted against penetrant concentration; these maxima are attributed to the mass flow of polymer together with dissolved penetrant. Intrinsic diffusion coefficients increase linearly with increasing concentration. The energies of activation for diffusion are low, probably because of the ease of segmental motion about the Si? O linkage. The diffusivity data are examined in terms of Fujita's “free volume” model and of transition-state theory. Permeability coefficients for the two penetrants are large, of the order of 10^?6–10^?5 cm³(STP)-cm/(sec-cm²-cm Hg), and increase markedly with increasing concentration or decreasing temperature. This behavior is regarded as a consequence of the low energies of activation for diffusion.     

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.     

Transport of organic vapors and liquids in poly(vinyl chloride)

This paper reviews research since 1980 on the equilibria and kinetics of transport of small organic molecules in rigid and plasticized PVC. The forms of both the solubility isotherms and the sorption kinetics are shown to change as the PVC/penetrant system undergoes a glass-rubber transition with an increase of either temperature or penetrant concentration. The isotherms are of “dual-mode” form (concave to the activity axis) for the glassy state, and show an inflection to Flory-Huggins form when the penetrant concentration exceeds C_g, the transition composition at the experimental temperature. The solubility at a given penetrant activity is governed primarily by the PVC/penetrant interaction parameter, χ. Sorption kinetics are Fickian for conditions producing small changes of concentration in either the glassy or rubbery state. For sorption into initially unplasticized PVC, kinetics are anomalous if the final penetrant concentration is between about C_g/2 and C_g, and Case II if C_g is exceeded. The magnitude of the Fickian diffusion coefficients depends largely on the geometric factors of molecular size and shape of the penetrant; this dependence is much steeper in the glassy than in the rubbery state. Recent results show that carbon dioxide displays both high diffusivity and substantial solubility in PVC under high pressure; this combination makes compressed CO₂ uniquely useful in accelerating the absorption of low-molecular-weight additives into PVC.     

Measurement of sorption and diffusion in nonporous membranes by transient permeation experiments

The knowledge of sorption and diffusion of liquids or vapors and their mixtures in nonporous membranes facilitates systematic membrane selection and process design. A novel experimental technique for the measurement of sorption and diffusion in nonporous membranes is described. An experimental apparatus has been designed for carrying out the transient permeation experiments. A general time-lag analysis procedure has also been developed in order to obtain the sorption and diffusion parameters from the transient permeation data. The effects of concentration dependent diffusivities, polymer swelling and a thermodynamic correction factor have been included in the analysis. Transient permeation experiments have been carried out to determine the sorption and diffusion of acetone in polydimethylsiloxane (PDMS). The results obtained are in good agreement with data from the literature, indicating that the proposed technique is suitable for the measurement of sorption and diffusion in nonporous membranes.     

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.     

Free-volume estimates in heterogeneous polymer systems. I. Diffusion in crystalline ethylene–propylene copolymers

Apparent thermodynamic diffusion coefficients were obtained from carbon tetrachloride, benzene, and n-hexane sorption-desorption kinetics in crystalline and amorphous ethylene-propylene copolymers (with propylene content from 1 to 70 wt. %, and crystallinity from 0% to 77%), in high-density and low-density polyethylene, and in polypropylene. The dependence of the diffusion coefficient vs. concentration curves on crystallinity and propylene content in the copolymers is reported. The diffusion coefficient at zero penetrant concentration increases with decrease in crystallinity. The apparent diffusion activation energies in the temperature interval investigated (25 to 75°C) are independent of temperature and are constant for crystalline copolymers. A modified Fujita-like free-volume theory for diffusion in crystalline polymer systems is introduced, and the theoretical estimates of diffusion coefficients show satisfactory agreement with experiment.     

Characterization of water vapor transport in glassy polyacrylonitrile by combined permeation and sorption techniques

Diffusion time lags, steady state permeabilities and sorption/desorption kinetics are reported for water vapor in biaxially oriented, solvent cast polyacrylonitrile (PAN) films. A wide range of vapor activities was studied at 15°C, 30°C, and 45°C. The transient and steady state permeation behavior at low and intermediate upstream vapor activities suggests that Fickian transport occurs under most of the conditions studied. Specifically, time lags predicted by Fick's law using the concentration-dependent diffusion coefficient derived from steady state permeation measurements agree reasonably well with experimentally measured values in most cases. p]Integral sorption/desorption kinetics at low and intermediate vapor activities also appear to be Fickian with a concentration-dependent diffusion coefficient. The form of the concentration dependency, evaluated from the “long time” solution of the diffusion equation for sorption experiments, is consistent with the form established for the diffusion coefficient from the steady-state permeation data. The diffusion coefficient exhibits a maximum near the concentration at which clustering is initiated. Presumably, the effective diffusion coefficient of water increases initially due to plasticizing or dual mode sorption effects associated with gap filling in the glassy matrix. As clustering becomes significant, the effective mobility of water is substantially reduced; therefore, the diffusion coefficient decreases at higher activities as clustering becomes the dominant mode of sorption. p]A tendency of the “early time” sorption/desorption kinetic data to exhibit concavity to the square root time axis at high activities suggests that time-dependent reductions in the diffusion coefficient may be occurring. Such reductions could be related to the kinetics of cluster formation at the higher vapor activities during sorption and to slow polymer consolidation during desorption. Any such non-Fickian effects, related to chain segment relaxations occurring over time scales similar to those of a diffusional jump, appear to be of importance only at short times. The short time nature of any such processes is suggested by the fact that diffusion coefficients evaluated from the “long time” solution to the diffusion equation for sorption are consistent with coefficients evaluated from steady state permeation data, in which case all time-dependent relaxation phenomena should be absent.     

On the unusual solvent retention and the effect on the gas transport in perfluorinated Hyflon AD membranes

Dense membranes of Hyflon AD 60X were prepared by the solvent evaporation method and by melt pressing. The diffusion coefficient, solubility and permeability of the membranes were measured for six permanent gases using time lag and steady state permeation measurements. The thermal properties were determined by Differential Scanning Calorimetry (DSC) and the solvent content was measured gravimetrically and was estimated by the Fox equation. It was found that unusually strong solvent retention in the solution-cast membrane leads to considerable plasticization of the polymer, to possible foam formation upon drying and, most important, to significant changes in the permeation properties. The residual solvent increases the diffusion coefficient and permeability of the larger gas species up to almost one order of magnitude, and it reduces the permselectivity. For most gas species the solubility is about two times higher in the solvent-free melt-pressed film than in the solution-cast film. The relation between the residual solvent and the membrane properties is discussed.     

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.     

Simultaneous nonlinear diffusion of a solvent and organic penetrant in a polymer

We obtain the exact solution of a fairly large class of somewhat simplified problems involving the simultaneous nonlinear Fickian diffusion of a poor solvent and a dilute organic penetrant, such as a plasticizer, in a semi-infinite polymer slab immersed in the solvent bath. Such problems can also provide useful estimates during certain time regimes of release of drugs from implants, and pesticides and pheromones from certain slow or controlled-release polymeric devices, insofar as these are affected by an environmental solvent bath. We assess the effect of various simplifications made (e.g., neglect of the penetrant concentration dependence in the diffusion coefficient, cross diffusion of solvent and penetrant, and a variety of boundary conditions) by solving exactly some additional specialized cases.     

Comparison of permeation resistance of protective gloves to organic solvents with ISO, ASTM and EN standard methods

In this study, the permeation resistance of nitrile and neoprene gloves to benzene and 1,2-dichloroethane was investigated using two permeation cells according to the ISO 6529, ASTM F739, and EN 374-3 standard test methods. The permeability coefficients were found to significantly increase with the flow rate of the collection medium. The appropriate flow rate of nitrogen collection should be higher than 75 and 150 mL/min for the ISO 6529 and ASTM F739 cells, respectively. For an open-loop system, the permeability coefficient of the ISO 6529 cell was obviously greater than that of the ASTM F739 cell, and the difference between these two cells was statistically significant. On the other hand, the breakthrough time was about 20–30 min for either the ISO 6529 or ASTM F739 cell at different flow rates of nitrogen collection. Fick's diffusion coefficient and solubility of permeant in the polymer glove can be specified as alternative test results for standard methods.     

Diffusion and solubility of mineral oils through ethylene-vinyl acetate copolymer

This paper reports a study of mineral oil diffusion through a filled ethylene-vinyl acetate crosslinked polymer, together with some comparisons with aliphatic linear hydrocarbons. Permeation was monitored by classical gravimetric measurements leading to values of solubility and diffusion coefficient at several temperatures ranging from 20 to 120 °C. Diffusion coefficients display a change in activation energy at at ca. 70 °C for mineral oils but not for simple hydrocarbons. The values obtained were discussed regarding available structure-diffusivity relationships and diffusion models derived from free volume theory. A relationship between penetrant evaporation temperature and its diffusivity was observed and tentatively justified.