Viscoelastic diffusion

A constitutive equation is proposed for the viscoelastic diffusion flux, and this equation is used to calculate sorption curves for differential step‐change sorption experiments. The predictions of the theory are shown to be consistent with eight experimental observations that report a wide range of diffusional behaviors for polymer–solvent systems. © 2001 John Wiley & Sons, Inc. J Polym Sci Part B: Polym Phys 39: 1529–1547, 2001     

Transport of toluene/methyl ethyl ketone mixtures in polyisobutylene

In this study, the diffusion behavior of methyl ethyl ketone (MEK)/toluene mixtures, with MEK molar fractions (x₁) in the liquid source lower than 0.2, in polyisobutylene (PIB) was investigated with vapor‐sorption Fourier transform infrared attenuated total reflectance (FTIR‐ATR) spectroscopy. FTIR‐ATR Fickian diffusion models for both binary and ternary systems were used to determine diffusion coefficients. Obtained diffusion coefficients for MEK from the binary diffusion model did not agree with those determined previously for the diffusion of MEK/toluene mixtures, with x₁ > 0.2, in PIB. When the ternary diffusion model was used, the main‐term and cross‐term diffusion coefficients of MEK were comparable. © 2001 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 39: 908–914, 2001     

On the role of solute solvation and excluded-volume interactions in coupled diffusion

Coupled diffusion is observed in multicomponent liquid mixtures in which strong thermodynamic interactions occur. This phenomenon is described by cross terms in the matrix of multicomponent diffusion coefficients. This paper reports a theoretical analysis on the relative role of thermodynamic factors and Onsager cross-coefficients on cross-diffusion coefficients relevant to ternary mixtures containing macromolecules or colloidal particles in the presence of salting-out conditions. A new model based on frictional coefficients between solvated solutes is reported. This model predicts that the Onsager cross-coefficient is negative and contributes significantly to cross-diffusion coefficients even at infinite dilution for solutes with a large difference in size. These predictions are consistent with recent experimental results. The role of preferential solvation and excluded-volume interactions on the thermodynamic factors are also examined. Excluded-volume interactions are introduced through the use of the McMillan-Mayer thermodynamic framework after emphasizing some important aspects of diffusion reference frames and thermodynamic driving forces. Finally, new expressions for cross-diffusion coefficients are proposed.     

Measurement of diffusion coefficients in multicomponent gas mixtures by the reversed-flow technique

Summary The reversed-flow method for measurement of gas diffusion coefficients in binary mixtures is now extended to simultaneous determination of effective diffusion coefficients for each substance in a multicomponent gas mixture. The method is applied to six ternary mixtures, each consisting of two gaseous hydrocarbons and H₂, He or N₂. The results are in agreement with a limiting case of the Stefan-Maxwell equations.     

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

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

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     

Homopolymer and small‐molecule tracer diffusion in a gyroid matrix

Forced Rayleigh scattering was used to measure the tracer diffusion coefficients of the photochromic dye tetrathioindigo (TTI) and a 1,4‐polyisoprene (PI) homopolymer (8000 g/mol) in a poly(styrene‐b‐isoprene) (SI) diblock copolymer matrix that formed a bicontinuous gyroid microstructure. The diblock copolymer contained 63% polystyrene (PS) by volume and had a total molecular weight of 21,300 g/mol. Rheology and small‐angle X‐ray scattering confirmed that the diblock copolymer microphase‐separated into the bicontinuous gyroid over the temperature range 60–230 °C, where the sample disordered. For both the TTI and PI tracers, two distinct modes of transport were observed. The faster mode displayed a temperature dependence consistent with diffusion within a PI matrix, whereas the slower mode had a temperature dependence more similar to diffusion within PS. The fast diffusivities were both over an order of magnitude lower than in a corresponding PI homopolymer matrix. For TTI, this was attributed to the preferential selectivity of the dye for PS and, therefore, an averaging of the mobility between the PS and PI domains. The slow mode was consistent with a small fraction of the TTI dye molecules becoming trapped within the much slower PS domains. For the PI tracer, the reduction in the diffusion coefficient for the fast mode was attributed to a combination of the tortuosity of the struts, the suppression of constraint release within the diblock matrix, and additional friction due to the presence of some styrene segments within the PI domains. The inevitable presence of grain boundaries or defects within the matrix interrupted the percolation of the PI struts, thereby forcing some of the PI tracers to diffuse through PS. Consequently, the slow mode was attributed to the diffusion through these defects, where the PI diffusion was retarded by both the increased segmental friction and the thermodynamic barrier to entering the PS domains. © 2001 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 39: 843–859, 2001     

Limiting law for distinct diffusion coefficients in ionic solutions

The limiting laws for the distinct diffusion coefficient and the corresponding Onsager phenomenological coefficient are obtained from the cluster theory of electrolyte conductance. Results are given for single electrolytes and common-ion mixtures with unrestricted charges and friction coefficients in both cases.     

Numerical calculation of the Onsager transport coefficients for isothermal binary electrolytes

The Ebeling-Falkenhagen diffusion equations are applied to calculate the Onsager transport coefficients as well as the electrical conductances, transference numbers, and mutual diffusion coefficients for isothermal binary electrolytes. For this purpose the hierarchy of diffusion equations is closed on the level of the binary distribution functions by the superposition approximation. The resulting system of common differential equations is solved by numerical methods. Hydrodynamic interactions are taken into account up to first order. Some results are given for symmetrically charged binary electrolytes with hard-core ions (restricted primitive interaction model). The model parameters (Bjerrum parameter and Debye screening length) are chosen to represent strong electrolytes up to the molar region.     

Hydrogen uptake mechanism of a silicone‐rubber DEB getter mixture

The kinetics of the hydrogen getter 1,4‐bis(phenylethynyl)benzene (DEB) blended with carbon‐supported Pd (DEB‐Pd/C) dispersed uniformly in silicone [DEB‐Pd/C‐poly(dimethyl siloxane)] were studied with a thermogravimetric method as a function of the hydrogen pressure and temperature. A diffusion‐controlled reaction model was developed to explain the experimental results. The diffusion coefficient, solubility coefficient, and permeability of hydrogen through silicone rubber were determined. © 2001 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 39: 425–431, 2001     

Analysis of solvent diffusion in high‐density polyethylene using NMR imaging techniques

Two different samples of high‐density polyethylene (HDPE) were studied. One (isotropic) was extracted from the material core, whereas the other (anisotropic) involved two sides that were in contact with the injection mold. Using radiofrequency field gradients, it was observed by NMR microscopy that these two sides favor toluene penetration into the material. Solvent diffusion in both samples could be successfully modeled, as demonstrated by the comparison between experimental NMR images and simulated images. Weight measurements appear to be consistent with the quantitative conclusions derived from NMR microscopy data. © 2001 John Wiley & Sons, Inc. J Polym Sci Part B: Polym Phys 39: 2781–2792, 2001     

Water sorption and permeation in chitosan films: Relation between gas permeability and relative humidity

The water sorption of chitosan has been studied at 20 °C. Water transport is governed by a Fickian process for relative humidities lower than 0.4, and in that range of partial pressures, the diffusion coefficient is concentration‐dependent. At a higher activity, anomalous diffusion is observed. The sorption isotherm is well described by the Guggenheim‐Anderson‐de Boer (GAB) model, and the clustering phenomenon observed at high relative pressures can be studied with the parameters of this model. The water permeability coefficient greatly increases with the relative pressure, and the water plasticization effect leads to a loss of the gas barrier properties under wet conditions. © 2001 John Wiley & Sons, Inc. J Polym Sci Part B: Polym Phys 39: 3114–3127, 2001     

Gas sorption,diffusion, and permeation in poly(dimethylsiloxane)

The permeability of poly(dimethylsiloxane) [PDMS] to H₂, O₂, N₂, CO₂, CH₄, C₂H₆, C₃H₈, CF₄, C₂F₆, and C₃F₈, and solubility of these penetrants were determined as a function of pressure at 35 °C. Permeability coefficients of perfluorinated penetrants (CF₄, C₂F₆, and C₃F₈) are approximately an order of magnitude lower than those of their hydrocarbon analogs (CH₄, C₂H₆, and C₃H₈), and the perfluorocarbon permeabilities are significantly lower than even permanent gas permeability coefficients. This result is ascribed to very low perfluorocarbon solubilities in hydrocarbon‐based PDMS coupled with low diffusion coefficients relative to those of their hydrocarbon analogs. The perfluorocarbons are sparingly soluble in PDMS and exhibit linear sorption isotherms. The Flory–Huggins interaction parameters for perfluorocarbon penetrants are substantially greater than those of their hydrocarbon analogs, indicating less favorable energetics of mixing perfluorocarbons with PDMS. Based on the sorption results and conventional lattice solution theory with a coordination number of 10, the formation of a single C₃F₈/PDMS segment pair requires 460 J/mol more energy than the formation of a C₃H₈/PDMS pair. A breakdown in the geometric mean approximation of the interaction energy between fluorocarbons and hydrocarbons was observed. These results are consistent with the solubility behavior of hydrocarbon–fluorocarbon liquid mixtures and hydrocarbon and fluorocarbon gas solubility in hydrocarbon liquids. From the permeability and sorption data, diffusion coefficients were determined as a function of penetrant concentration. Perfluorocarbon diffusion coefficients are lower than those of their hydrocarbon analogs, consistent with the larger size of the fluorocarbons. © 2000 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 38: 415–434, 2000     

Diffusion behavior in polymer–clay nanocomposites

This paper reviews our previous studies on the diffusion behavior in polymers clay nanocomposites. A geometric model for predicting the effective diffusivity through this type of systems as a function of clay sheets orientation, volume fraction, polymer clay interaction, and aspect ratio is proposed. Model predictions are compared to the effective diffusivity generated using random walk simulations as well as with predictions obtained from already existing theoretical models. Fair agreement is found between the model prediction and the results of numerical simulations. With respect to the already existing theoretical models, the present mathematical derivation seems more adequate to describe diffusion behavior in conventional nanocomposites systems (i.e. when fillers present very low values of volume to surface ratio). Experimental diffusion tests are discussed and interpreted with the aid of the proposed model. In addition to the aspect ratio and clay concentration, the polymer clay interactions as well as the sheets orientation are the factors controlling the barrier properties of polymer‐layered silicate nanocomposites. Good agreement was found in the case of samples containing exfoliated clay, whereas the model fails in the case of micro‐composites, in which the inorganic lamellae are agglomerated in clusters. © 2005 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 44: 265–274, 2006     

Dynamic evolution of gases in the γ‐ and helium‐ion radiolyses of solid polymers

The dynamic evolution of gaseous hydrogen, methane, and carbon dioxide in the γ‐ and ⁴He‐ion radiolyses of solid polymers was investigated. The polymers used include low‐density and high‐density polyethylene, polypropylene, polystyrene, poly(methyl methacrylate), Nylon 11, Nylon 6, and poly(dimer acid‐co‐alkyl polyamine). An inline quadrupole mass spectrometer was utilized to monitor the dynamic profiles of the gases produced in the radiolysis. One‐ and two‐dimensional numerical diffusion models were developed to simulate and extract optimum diffusion coefficients and gas yields from the experimental dynamic gas profiles. It was found that the dynamic evolution of molecular hydrogen from the bulk polymer is controlled by its diffusion in most cases, such as CO₂ in poly(methyl methacrylate). In the γ radiolysis of some polymers such as low‐density polyethylene and polypropylene, the dynamic evolution of methane is only partially controlled by the diffusion process, and some other postirradiation process is a factor. It is concluded that the simulation method developed in this article is helpful in understanding and predicting the mechanisms of gas evolution in the radiolysis of solid polymers. © 2001 John Wiley & Sons, Inc. J Polym Sci B Part B: Polym Phys 39: 1449–1459, 2001     

Styrene and isoprene friction factors in styrene–isoprene matrices

Monomeric friction factors, Ξ, for polystyrene (PS), polyisoprene (PI), and a polystyrene–polyisoprene (SI) diblock copolymer have been determined as a function of temperature in four poly(styrene-b-isoprene-b-styrene-b-isoprene) tetrablock copolymer matrices. The Rouse model has been used to calculate the friction factors from tracer diffusion coefficients measured by forced Rayleigh scattering. Within the experimental temperature range the tetrablock copolymers are disordered, allowing for measurement of the diffusion coefficient in matrices with average compositions determined by the tetrablock copolymers (23, 42, 60, and 80% styrene by volume). Remarkably, for a given matrix composition the styrene and isoprene friction factors are essentially equivalent. Furthermore, at a constant interval from the system glass transition temperature, T_g, all of the friction factors (obtained from homopolymer, diblock copolymer, and tetrablock copolymer dynamics) agree to within an order of magnitude. This is in marked contrast to results for miscible polymer blends, where the individual components generally have distinct composition dependences and magnitudes at constant T − T_g. The homopolymer friction factors in the tetrablock matrices were systematically slightly higher than those of the diblock, which in turn were slightly higher than those of the homopolymers in their respective melts, when all compared at constant T − T_g. This is attributed to the local spatial distribution of styrene and isoprene segments in the tetrablocks, which presents a nonuniform free energy surface to the tracer molecules. © 1998 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 36: 3079–3086, 1998     

Viscosity effects in cobaloxime‐mediated catalytic chain‐transfer polymerization of methacrylates

The effect of bulk viscosity on the cobaloxime‐mediated catalytic chain‐transfer polymerization of methacrylates at 60 °C was investigated by both the addition of high molecular weight poly(methyl methacrylate) to methyl methacrylate polymerization and the dilution of benzyl methacrylate polymerization by toluene. The results indicate that the bulk viscosity is not directly linked to the chain‐transfer activity. The previously measured relationship between chain‐transfer‐rate coefficient and monomer viscosity therefore probably reflects changes at the molecular level. However, the results in this article do not necessarily disprove a diffusion‐controlled reaction rate because cobaloxime diffusion is expected to scale with the monomer friction coefficient rather than bulk viscosity. Considering the published data, to date we are not able to distinguish between a diffusion‐controlled reaction rate or a mechanism directly affected by the methacrylate substituent. © 2002 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 40: 782–792, 2002; DOI 10.1002/pola.10152     

Numerical simulation of coupled‐stress case II diffusion in one dimension

This article is concerned with the robust and efficient numerical simulation of case II diffusion, which constitutes an important regime of solvent diffusion into glassy polymers. Even in the one‐dimensional case considered here, the numerical simulation of case II diffusion is made difficult by the extreme nonlinearities and coupling in the governing model equations due to a nonlinear flux law needed to produce sharp solvent fronts, a concentration‐dependent relaxation time of the polymer used to model the glass‐rubber transition, and coupling between the diffusion and deformation phenomena. Having an efficient and accurate solution to such equations is central to advancing a clear understanding of the meaning of such models. The difficulties due to coupling and nonlinearities are highlighted by the consideration of a specific, normalized, one‐dimensional case II diffusion model laid out in a general framework of balance laws. Issues such as the stiffness of the spatially discrete differential algebraic equations obtained from the finite element discretization of the governing equations and their bearing on the choice of time‐stepping schemes are discussed. The key requirements of numerical schemes, namely, robustness and efficiency, are addressed by the use of an implicit, adaptive, second‐order backward differentiation formula with error control for time discretization. Error control is used to maximize the step size to satisfy a target error and the radius of convergence requirements while nonlinear algebraic equations are solved at each time step. An example of an initial boundary value problem is solved numerically to show that the chosen model reproduces case II behavior and to validate that the stated objectives for the numerical simulation are met. Finally, the features and numerical implementation of this model are compared with those of a closely related case II diffusion model due to Wu and Peppas. © 2003 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 41: 2091–2108, 2003     

Calculation and experimental verification of concentration profiles for selected species at the interphase generated by diffusion in polymer pairs

A method for calculating diffusion rates for individual species in concentrated regime is outlined. The effects of monomeric friction coefficient, Flory–Huggins thermodynamic interaction parameter, individual species molecular weights, local molecular weights distribution, and local T_g are precisely calculated. The method is used to calculate individual concentration profiles generated by diffusion of multicomponent polymer blends, and experimentally tested. Polystyrene with a bimodal molecular weight distribution is allowed to diffuse in a blend of polyphenylene oxide and polystyrene. Local physical properties change markedly along the interdiffusion path and, therefore, this is a demanding test for the proposed calculation method. The simulated concentration profiles are compared with results obtained by using two independent experimental techniques: Raman spectroscopy and dynamic mechanical analyzer (DMA). The total polystyrene (PS) concentration profiles, calculated using the proposed method, agree well with Raman spectroscopy results. Simulated DMA results—which are sensitive to the PS species molecular weight distribution—obtained using the concentration profiles, calculated for each PS molecular weight species agree well with the experimental DMA results. Calculations based on average molecular weights give incorrect results. © 1999 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 37: 3097–3107, 1999     

Study of the self‐diffusion of poly(ethylene glycol)s in poly(vinyl alcohol) aqueous systems

We have measured the self‐diffusion coefficients of a series of oligo‐ and poly(ethylene glycol)s with molecular weights ranging from 150 to 10,000, in aqueous solutions and gels of poly(vinyl alcohol) (PVA), using the pulsed‐gradient spin‐echo NMR techniques. The PVA concentrations varied from 0 to 0.38 g/mL which ranged from dilute solutions to polymer gels. Effects of the diffusant size and polymer concentration on the self‐diffusion coefficients have been investigated. The temperature dependence of the self‐diffusion coefficients has also been studied for poly(ethylene glycol)s with molecular weights of 600 and 2,000. Several theoretical models based on different physical concepts are used to fit the experimental data. The suitability of these models in the interpretation of the self‐diffusion data is discussed. © 1999 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 37: 2396–2403, 1999