On the diffusion of gases in partially crystalline polymers

The diffusion of gases through partially crystalline polymers is studied. The effective diffusion coefficient D^eff is obtained as the result of the averaged superposition of two fundamental mechanisms, namely, diffusion through the crystallites is considered to be zero, and diffusion through the rubbery fraction of the polymer obeys a Fujita-like free-volume theory. The predicted D^eff is compared with experimental data of Kreituss and Frisch. The behavior of the diffusion coefficient in terms of concentration and crystalline fraction is satisfactorily explained through the model.     

Diffusion in glassy polymers: A model using a homogenization method and the effective medium theory

Glassy polymers are considered as inhomogeneous with regions in which the gas sorption follows Henry's law and others where it follows Langmuir's law. It is assumed that the linear dimensions of these regions are small compared with the macroscopic length of interest but large compared with the mean free path of the penetrant gas molecules. Applying an homogenization method it is shown that the average flux is directly proportional to the concentration gradient in the polymer. This relationship can be expressed in terms of an effective diffusion coefficient D_eff, which depends on the details of the microstructure. D_eff is evaluated in the framework of the effective medium theory and compared with experimental data for diffusion of five vapors in ethylcellulose.     

Effect of microheterogeneities on the diffusion of gases through glassy polymers

The diffusion of gases through glassy polymers is studied and the effective diffusion coefficient D^eff is represented as the result of the superposition of two fundamental mechanisms, namely slipping and hopping. D^eff is calculated by a two-point correlation method. Comparisons are made with experimental data of Meares for diffusion coefficients of Kr, O₂, He, and A in poly(vinyl acetate) in the glassy state. Good fits are obtained and yield significant parameters.     

An application of an homogenization method to a model of diffusion in glassy polymers

The sorption of gases in polymers below their glass-transition temperature T_g is known in many cases to be described by the “dual sorption” theory, according to which the gas is held in accordance with both the Langmuir and Henry's laws. Based on this theory, expressions for the “effective diffusion coefficient” in the glassy polymers have been obtained by investigators in the past, notably by Paul and Koros.² The present analysis regards the glassy polymers as inhomogeneous with regions on which the gas sorption follows the Langmuir law. Assuming that the linear dimensions of these regions, which are often referred to as “microvoids” (although they are not space filled by vacuum), are small compared to the macroscopic length of interest but large compared to the mean free path of the penetrant gas molecules, we derive a rigorous relation between the average flux and the concentration gradient in the polymer and show that this relation can be expressed in terms of an “effective diffusion coefficient” D_eff which depends on the details of the microstructure, i.e., the size, shape and spatial distribution of the “microvoids.” This expression for D_eff is shown to reduce to that of Paul and Koros² in two situations: (1) when the “voids” consist of slabs running parallel to the concentration gradient, and (2) when the “voids” are spherical and the temperature of the polymer is not too different from T_g. The results of the present study lead to an alternative procedure for interpreting the experimental data on sorption and permeation which may have some advantages over the procedure currently employed. Finally, the analysis presented here is also applicable to polymers containing adsorptive fillers.     

Computer simulation of three‐arm star polymers

We show that Shaffer's version of the bond fluctuation model can be used to simulate three‐arm star polymers. We report a simulation study of both single stars and melts of star polymers with arm lengths up to 90 monomer units (approximately twice the entanglement crossover length for linear chains). Center‐of‐mass self‐diffusion of single stars is Rouse‐like (D ˜ N^–1). Due to a limited range of molecular weights we cannot distinguish between a power‐law and an exponential dependence of the star‐melt self‐diffusion coefficient on arm length.     

Diffusion of radioactively tagged 1,1-diphenylethane and n-hexadecane through rubbery polymers: Dependence on temperature and dilution with solvent

The diffusion of 1,1-diphenylethane in trace amounts through eight rubbery polymers has been studied by radioactive tagging of this penetrant with ¹⁴C. For several polymers, the dependence on temperature and on dilution (swelling) by untagged diphenylethane was investigated. In the diluted systems, tagged n-hexadecane was also used as a trace penetrant. The temperature and concentration dependences were interpreted rather successfully in terms of the free volume. In comparing different polymers, with a 4000-fold range of diffusion coefficients, the translatory friction coefficient of 1,1-diphenylethane was found to be proportional to that of n-hexadecane to the power 1.06. This is interpreted qualitatively by the free volume concept to indicate a slightly less efficient mobility mechanism for the diphenylethane.     

Water-absorption properties of an acetal copolymer

The kinetics of sorption from the liquid phase to equilibrium and desorption were studied over the temperature range 0–80°C. Equilibrium uptake was found to increase linearly with concentration in this range. Sorption-desorption kinetics showed the diffusion coefficients to decrease with increasing concentration, although the extent of this dependence did not appear in itself to be temperature-dependent. The apparent diffusion coefficient obeyed the law D = D₀ exp {? E/RT} over the temperature range studied, giving E = 9.9 kcal./mole and D₀ = 0.45 cm.² sec.^?1. These values are compared with corresponding values for other polymers.     

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 camphorquinone in uniaxially drawn polycarbonate films

Employing the laser-induced holographic grating relaxation technique, we have measured tracer diffusion coefficients of a phtochromous dye, camphorquinone, in uniaxially drawn polycarbonate films as a function of stretch ratio. Anisotropy in the tracer diffusion coefficient has been observed with D_∥ greater than D_⟂ by at least a factor of 4 for the film stretched to the stretch ratio δ = 2.3. The diffusion coefficient along the direction of stretch D_∥ increases significantly with increasing δ, whereas D_⟂ decreases slightly with increasing δ. The stretch ratio dependence of D_∥ and D_⟂ is interpreted according to a modified free volume theory. The strain rate and stretch temperature dependence of the anisotropic tracer diffusion coefficient has also been investigated. © 1992 John Wiley & Sons, Inc.     

Effect of crosslink density on the properties of monolithic polymer matrices: Reversed-phase gas-chromatography study

Monolithic polymer matrices of different natures and different degrees of crosslinking have been synthesized in capillaries with an inner diameter of 100 μm. The properties of the monolithic matrices are characterized by reversed-phase gas chromatography. Solubility coefficient S, Flory-Huggins parameter gC₁₂^∞, and reduced Flory-Huggins parameter gC₁₂^∞ are evaluated. For all tested sorbates, the values of S depend on the degree of crosslinking of the polymer, which is characterized by parameter gC₁₂^∞. In the case of all monolithic polymer matrices under study, the logarithm of the solubility coefficient plotted as a function of the squared critical temperature of sorbate is described by a straight line, a circumstance that is likewise typical of linear polymers. Parameter D/d _f ², which characterizes the rate of diffusion of low-molecular-mass compounds in the monolithic matrix, is calculated. For both polar and nonpolar polymers, the dependence of D/d _f ² on the degree of crosslinking follows an extremum pattern.     

Mutual diffusion and thermodynamics in the blends of polystyrene and tetramethylbisphenol-A polycarbonate

A study was made of miscible polymer blends of deuterated polystyrene (d-PS) and tetramethylbisphenol-A polycarbonate (TMPC). The Flory interaction parameter χ was obtained from the relation between mutual and tracer diffusion coefficients, D? and D^*, which were measured by forward recoil spectrometry. The temperature dependence of diffusion at PS weight fractions ω of 0.25 and 0.5, and the composition dependence at temperatures 45°C above the glass transition temperature, T_g, were investigated. A stronger dependence of χ on both temperature (at ω = 0.5) and composition was observed in comparison with other miscible binary polymer blends involving PS. Analysis using the generalized lattice-fluid model of Sanchez and Balazs¹ showed that the incorporation of a significant specific interaction is needed to explain the temperature dependence of χ. The diffusion coefficients obtained in the one-phase region were extrapolated to the two-phase region, and these were compared with the effective diffusion coefficient extracted from phase separation dynamics measured by light scattering.² A significant discrepancy between the extrapolated and effective diffusion coefficients was observed. © 1995 John Wiley & Sons, Inc.     

Diffusion in concentrated dispersions: a study with fiber-optic quasi-elastic light scattering (FOQELS)

A fiber-optic, quasi-elastic light-scattering instrument is described using single-mode fiber optical components, including a novel slanted exit face optode. The setup operates with homodyne signal detection. It enables the characterization of diffusion processes in concentrated dispersions up to volume concentrations of 50%.The performance of the instrument is exemplified with results obtained from latex spheres with diameters of 226 nm and 404 nm at volume fractions from =0.01 to =0.5. The correlation functions are analyzed according to the second order cumulants method and the Contin-procedure yielding an average and a distribution function of the short-time self-diffusion coefficient,D _eff ^s , respectively.At high ionic strength the concentration dependence ofD _eff ^s /D ₀ is found to be in close agreement with theoretical predictions based on a multi-body interaction model of hard spheres up to =0.45. With decreasing ionic strength the negative slope of the virial expansion tends to increase, presumably due to enhanced repulsive electrostatic interactions.The described technology offers new experimental means for on-line remote control sensing of particle size in concentrated disperse systems.Presented at the 34. Hauptversammlung der Kolloidgesellschaft e.v., Bochum, Oct. 1–4, 1989     

NMR self-diffusion study of polyethylene and paraffin melts

The self-diffusion coefficient D of paraffin and polyethylene melts—covering the range between N = 19 and 10³ where N is the number of monomeric units—was measured by the pulsed-magnetic-field-gradient NMR method for diffusion times between 3 ms and 1 s. For the paraffins, D is proportional to N^?2 though the molecular weights are smaller than the critical molecular weight for entanglement. In polyethylene, melts a strong dependence of the diffusion coefficient on the diffusion time is observed, whereas no such dependence is found in paraffin melts. A mathematical formalism for describing spin-echo attenuation in terms of a velocity autocorrelation function is shown to yield qualitative agreement with the experimental results.     

A model for predicting solvent self-diffusion coefficients in nonglassy polymer/solvent solutions

Cohen-Turnbull diffusion theory is used to develop a model for predicting solvent self-diffusion coefficients D₁ in nonglassy polymer/solvent solutions. Polymer molecules are envisioned as hindering solvent mobility by reducing the average free volume per unit mass in the system and through the lower mobility of polymer segments relative to solvent molecules. The concentration dependence of D₁ predicted by the model is in reasonable agreement with data for the solvents heptane, hexadecane, benzene, cyclohexane, and decalin in polyisobutylene (PIB), and for toluene in polystyrene, poly(methyl mothacrylate), and PIB. Although none of the data is for high concentrations of polymer (volume fractions ?≥0.9) it is anticipated the model will be less representative in this regime where the assumptions in its development are unsure. The model also demonstrates the correct temperature and concentration dependence of the apparent activation energy for diffusion. The only experimental data needed to use the model are the viscosity and critical volume of the pure solvent, and the specific volume of both the solvent and mixture. No binary transport data are required.     

Effect of the solvent upon the diffusion coefficient of polydisperse polystyrene and styrene-acrylonitrile copolymer in dilute solution

A laser homodyne spectrometer was used to obtain translational diffusion coefficients for dilute polystyrene and styrene-acrylonitrile copolymer solutions at room temperature. Data were obtained in the concentration range from 0.01 to 2.0 g polymer per 100 cm³ solution for polystyrene in benzene and in decalin; and for copolymer in dimethyl formamide, in methyl ethyl ketone, and in benzene. The samples were polydisperse polystyrenes of weight average molecular weights between 80,000 and 350,000 and polydisperse copolymers of weight average molecular weights between 200,000 and 800,000. The SAN copolymers were random copolymer samples containing 24% by weight acrylonitrile. For each of the systems investigated the concentration dependence of the diffusion coefficient was linear over the concentration range studied, and was expressed as D(c) = D₀(1+k_Dc). Values of D₀ could be explained with a modified Kirkwood-Riseman expression. Values of the parameter k_D obtained from the slopes could be interpreted using the two-parameter theory approach as suggested by Vrentas and Duda. The value of k_D is positive for high-molecular-weight polymers and negative for low-molecular-weight polymers. For a particular polymer, the molecular weight at which k_D changes sign is greater for poor solvents than for good solvents. Observed values of D₀ were 1 × 10^?7 to 7 × 10^?7 cm²/sec.     

Diffusion of water in solutions of alkali metal bromides,tetraalkylammonium bromides and ammonium bromide

The diffusion coefficient of water D _w in aqueous solutions of the alkali metal bromides, tetraalkylammonium bromides (methyl, ethyl, n-propyl, and n-butyl) and ammonium bromide at 25°C is reported for concentrations up to 2 mol-dm^–3. In addition, values for D _w in 2 mol-dm^–3 solutions of CsBr, KBr, NaBr, LiBr, and fully deuterated methanol, acetonitrile, and acetone have been measured for temperatures in the range 5 to 50°C. The concentration dependence of the relative water diffusion coefficient D _w /D _o , where D _o is the self-diffusion coefficient of water, has been analyzed in terms of an equation analogous to the Jones-Dole equation for relative viscosity. The B-coefficient for diffusion is well correlated with the viscosity B-coefficient. For the structure-breaking electrolytes CsBr and KBr, D _w /D _o decreases rapidly with increasing temperature, whereas for the structure-makers NaBr and LiBr, the temperature dependence of D _w /D _o has the same sign but is much smaller in magnitude. For the nonelectrolyte solutions, the structure-making effect decreases with increasing temperature and the temperature coefficient of D _w /D _o is positive. It is apparent that, when diffusion of the solvent is being considered, the temperature must be taken into account in the classification of an electrolyte as a structure-breaker or structure-maker.     

Influence of the structure and morphology of ultrathin poly(3-hydroxybutyrate) fibers on the diffusion kinetics and transport of drugs

Ultrathin fibers of a biodegradable polymer poly(3-hydroxybutyrate) with an encapsulated drug (dipyridamole, 0–5% of poly(3-hydroxybutyrate) mass) are obtained by electrospinning. Introduction of the drug substantially affects the geometric shape and crystallinity of individual filaments as well as the total porosity of the fibrillar film on their basis. As follows from the SEM data, in the absence of the drug or at its low concentration (<3%), poly(3-hydroxybutyrate) fibers appear as ellipse-like fragments alternating with cylindrical ones. At a higher content of the drug (3–5%), the abnormal ellipse-like structures are practically absent and the fiber acquires the cylindrical shape. A set of morphological and crystallinity characteristics of the fibers determines the absorption of water and the rate of the diffusion transport of the drug as well as the corresponding profiles of its controlled release. A simplified model of drug desorption from the fibrillar film is advanced which considers two sequential stages of the process: (i) diffusion of the drug in the polymer fiber with coefficient D _f ~ 10^–12 cm²/s and dimeter φ_f ~ 2–4 μm and (ii) transport of the drug in the interfibrillar porous space filled by solvent with diffusion coefficient D _w = 5.5 × 10^–6 cm²/s. Using the characteristics of porosity, crystallinity, and geometry of the fibers and diffusion effective coefficients D _eff calculated from the profile of drug release, it is shown that the limiting stage of the transport of the drug is its diffusion in the volume of the cylindrical fiber. The model makes it possible to turn from the experimental values of D _eff to partial diffusion coefficients D _f and to calculate the kinetic profile of drug release with allowance made for the above-listed factors.     

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.     

Molecular modeling studies of polymeric gas separation and barrier materials: structure and transport mechanisms

Molecular modeling studies have been completed on cis-PTBA(poly(tert-butylacetylene)) and Sixef44 polyimide, two glassy polymers that can be used to form gas separation membranes. The modeling studies show that polymer backbone bond rotations in PTBA are not thermally allowed. This leads to a helical structure for the cis-PTBA chains which pack as if the helices were rigid rods. Here, polymer free volume is formed by the interstitial space between adjacent helices, and gas transport occurs via continuous diffusion through the resulting channel-like free volume. On the other hand, Sixef44 exhibits a flexible polymer backbone, which leads to the formation of irregular voids. In this case, gas molecules are free to move within the voids, but transport occurs only by hopping to an adjacent void, or by void diffusion. In either case, gas transport is closely coupled to polymer backbone motion. Thus, these studies suggest two different types of free volume and gas transport mechanisms. The diffusion mechanism in glassy polymer membranes will depend on the nature of the free volume (e.g. the type of chain packing), and the polymer backbone chain flexibility.