Solvent diffusion in amorphous glassy polymers

In this article, a mathematical model is proposed for predicting solvent self‐diffusion coefficients in amorphous glassy polymers based on free volume theory. The basis of this new model involves consideration of the plasticization effects induced by small molecular solvents to correctly estimate the hole‐free volume variation above and below the glass‐transition temperature. Solvent mutual‐diffusion coefficients are calculated using free volume parameters determined as in the original theory. Only one parameter, which can be predicted by thermodynamic theory, is introduced to express the plasticization effect. Thus, this model permits the prediction of diffusion coefficients without adjustable parameters. Comparison of the values calculated by this new model with the present experimental data, including benzene, toluene, ethyl benzene, methyl acetate, and methyl ethyl ketone (MEK) in polystyrene (PS) and poly(methyl methacrylate) (PMMA), has been performed, and the results show good agreement between the predicted and measured values. © 2000 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 38: 846–856, 2000     

Fickian diffusion in glassy polymer-solvent systems

A free-volume theory is developed for the analysis of Fickian diffusion processes in glassy polymers. Equations are presented for the prediction of mutual diffusion coefficients in concentrated glassy polymer-penetrant systems. The concentration dependence of the mutual diffusion coefficient is dependent on how much free volume the solvent contributes to the system and on how the addition of solvent affects the densification of the polymer matrix. © 1992 John Wiley & Sons, Inc.     

Self‐diffusion of small molecules into rubbery polymers: A lattice free‐volume theory

In the framework of the free‐volume (FV) theory, a new equation was derived for the evaluation of self‐diffusion coefficients of small molecules in polymers above the mixture glass transition temperature. The derivation of the equation turned out to be straightforward once the equivalence between the free volume and the unoccupied volume given by thermodynamic lattice theories is assumed. A parameter evaluation scheme is proposed, which is substantially simpler compared with the conventional Vrentas–Duda approach, even without losing generality. The key assumption is discussed, and its consistency is verified from a numerical viewpoint. A comparison with experimental solvent self‐diffusion coefficients for several solvent/polymer binary systems confirmed that the proposed theory presents good correlative ability over wide temperature and composition ranges. Moreover, the introduced thermodynamic foundation allows one to easily include the pressure effect too. In the frame of the proposed lattice free volume theory, the sizes of the polymer jumping units decrease with temperature and increase with pressure. Such behavior converges with theoretical expectations and opens the way for a predictive FV theory. © 2010 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 48: 529–540, 2010     

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     

Understanding the effect of skin formation on the removal of solvents from semicrystalline polymers

The effect of glassy skin formation on the drying of semicrystalline polymers was investigated with a comprehensive mathematical model developed for multicomponent systems. Polymers with high glass‐transition temperatures can become rubbery at room temperature under the influence of solvents. As the solvents are removed from the polymer, a glassy skin can form and continue to develop. The model takes into account the effects of diffusion‐induced polymer crystallization as well as glassy–rubbery transitions on the overall solvent content and polymer crystallinity. A Vrentas–Duda free‐volume‐based diffusion scheme and crystallization kinetics were used in our model. The polymer–solvent system chosen was a poly(vinyl alcohol) (PVA)–water–methanol system. The drying kinetics of PVA films were obtained by gravimetric methods with swollen films with known water/methanol concentrations. The overall drying behaviors of the polymer system determined by our model and experimental methods were compared and found to match well. © 2005 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 43: 3191–3204, 2005     

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.     

Velocity‐correlation analysis in polymer–solvent mixtures

The subject of this article is the combined interpretation of intradiffusion and mutual‐diffusion data for polymer–solvent mixtures in terms of integrals over velocity self‐correlation functions and velocity cross‐correlation functions. The combination of mutual‐diffusion, intradiffusion, and activity data allows the evaluation of velocity‐correlation coefficients (VCCs) and distinct‐diffusion coefficients in systems containing one monodisperse solute. This study is the first attempt to extend these approaches to polymers that are polydisperse solutes. Because of the polydispersity, this correlation analysis may become critical for polymers. Its application to polydisperse samples requires the reduction of intradiffusion and mutual‐diffusion coefficients to the same average. After such a reduction, the VCCs and distinct‐diffusion coefficients are evaluated for a homologous series of poly(ethylene glycol)s (PEGs). Attractive PEG–PEG interactions depend on the chain length and concentration of PEG. In this analysis, network formation in PEG–water systems appears to be a smooth process. © 2001 John Wiley & Sons, Inc. J Polym Sci Part B: Polym Phys 40: 43–51, 2002     

Prediction and estimation of solvent diffusivities in polyacrylate and polymethacrylates

The relationship between polymer side‐chain length and the hole free volume that is effective for solvent diffusion was investigated for polyacrylates and polymethacrylates on the basis of free‐volume theory. Measurements of a polymer's viscoelasticity and solvent diffusivity provided experimental evidence for polymer segment mobility, and the results indicated that hole free volume in a linear polymer increases with hydrocarbon side‐chain length. Because the molecular mechanisms of polymer viscoelasticity and diffusivity are identical, the free‐volume parameters obtained for polyacrylates and polymethacrylates by measuring the polymer viscoelastic‐temperature dependence can reliably be used in predicting the solvent diffusion coefficient. © 2003 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 41: 1393–1400, 2003     

Predicting the proton conductivity of perfluorosulfonic acid membrane via combining statistical thermodynamics and molecular dynamics simulation

The electrochemical properties of a perfluorosulfonic acid (PFSA) membrane are estimated using a combination of molecular dynamics simulation and statistical thermodynamic model. We obtain all parameters in an ionic conductivity model from an atomistic simulation and remove all adjusted model parameters. From a microscopic point of view, the hydrated PFSA membrane shows micro‐phase segregation which separated into hydrophilic and hydrophobic phases. Our present work originates with this phenomenon and we treat this phase segregation as if it is a continuous phase for each of which the proton (H⁺) is transported inside the PFSA membrane/solvent (water and alcohols) mixture. The chemical potential for a given system is estimated using a molecular simulation technique to predict the van der Waals interaction energy between the polymer and solvent. In addition, the self diffusion coefficients are calculated from the molecular dynamics simulation. We study various polymer/solvent compositions to understand the concentration dependence of self diffusion coefficient. Our self diffusion coefficients and also the predicted final ionic conductivity agree well with previously reported experimental data. © 2011 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 49: 1455–1463, 2011     

Prediction of solvent diffusion coefficient in amorphous polymers based on an equation‐of‐state

This study develops a modified free‐volume model to predict solvent diffusion coefficients in amorphous polymers by combining the Vrentas–Duda model with the Simha–Somcynsky (S‐S) equation‐of‐state (EOS), and all the original parameters can be used in the modified model. The free volume of the polymer is estimated from the S‐S EOS together with the Williams‐Landel‐Ferry fractional free volume, and the complex process of determining polymer free‐volume parameters in the Vrentas–Duda model and measuring polymer viscoelasticity can be avoided. Moreover, the modified model includes the influence of not only temperature but also pressure on solvent diffusivity. Three common polymers and four solvents are employed to demonstrate the predictions of the modified model. The calculation results are generally consistent with the experimental values. It is reasonable to expect that the modified free‐volume model will become a useful tool in polymer process development. © 2006 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 44: 1000–1009, 2006     

Solubility and diffusivity of methylmethacrylate and butylacrylate monomers in a MMA–BA copolymer

Mutual diffusion coefficients and sorption isotherms of methyl methacrylate (MMA) and butyl acrylate (BA) monomers in methyl methacrylate‐butyl acrylate copolymer (MMA‐BA) have been measured by gravimetric sorption. MMA is found to have higher solubility and diffusion rates in the copolymer than BA. Sorption data for MMA were interpreted using classical Flory‐Huggins thermodynamic theory with a constant interaction parameter (χ). A modified version of this theory has been applied to correlate the sorption data of BA, which exhibit a temperature and concentration‐dependent χ parameter. For MMA, the isotherm data reveal enhanced polymer‐solvent interactions with increasing temperature, while for BA the data indicate a drive toward phase separation with increasing temperature. Despite the difference in thermodynamic behavior, both monomers are found to exhibit Fickian diffusion and the diffusivity data are correlated reasonably well with the Vrentas‐Duda free volume theory. Some deviation between the free‐volume correlation and the experimental data is observed at the lowest temperature and BA concentration examined. © 2007 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 45: 1996–2006, 2007     

Atomistic packing models for experimentally investigated swelling states induced by CO2 in glassy polysulfone and poly(ether sulfone)

Atomistic packing models have been created, which help to better understand the experimentally observed swelling behavior of glassy polysulfone and poly (ether sulfone), under CO₂ gas pressures up to 50 bar at 308 K. The experimental characterization includes the measurement of the time‐dependent volume dilation of the polymer samples after a pressure step and the determination of the corresponding gas concentrations by gravimetric gas‐sorption measurements. The models obtained by force‐field‐based molecular mechanics and molecular dynamics methods allow a detailed atomistic analysis of representative swelling states of polymer/gas systems, with respect to the dilation of the matrix. Also, changes of free volume distribution and backbone mobility are accessible. The behavior of gas molecules in unswollen and swollen polymer matrices is characterized in terms of sorption, diffusion, and plasticization. © 2006 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 44: 1874–1897, 2006     

Self‐diffusion of styrene, 2,2′‐azobisisobutyronitrile,and polystyrene in bulk polymerization by pulsed‐gradient spin‐echo nuclear magnetic resonance spectroscopy

The self‐diffusion of styrene, polystyrene, and 2,2′‐azobisisobutyronitrile has been determined in the bulk polymerization of styrene with pulsed‐gradient spin‐echo nuclear magnetic resonance at 25 °C. Data on small molecules are discussed with respect to recent diffusion models. They can fit self‐diffusion coefficient data of small molecules in dilute or semidilute polymer solutions; in concentrated solutions, however, there is a breakdown. A semiempirical model based on scaling laws is used to describe the self‐diffusion of styrene and 2,2′‐azobisisobutyronitrile over the whole range of concentrations studied. The dependence of the polystyrene self‐diffusion coefficient on the polymer concentration is described with a stretched exponential function, D = D₀ exp(?αc^ν), where α depends on the molecular weight of the polymer and ν depends on the kind of solvent. © 2003 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 41: 1605–1614, 2003     

Finite concentration inverse gas chromatography: Diffusion and partition measurements

Inverse gas chromatography (IGC) is a very fast, accurate, and reliable technique to measure diffusion coefficients. This technique however, has been limited to measurements in the infinite dilution region, i.e., in the region of negligible amount of solvent in the polymer. We have extended the scope of inverse gas chromatography to measure diffusion coefficients at finite concentrations of the solvent. This involves doping the carrier gas with a solvent of interest to achieve finite concentrations of solvent in the carrier gas and hence in the polymer. The carrier gas is passed through a saturator maintained at constant temperature to achieve this purpose. Diffusion coefficients for polyvinyl acetate–toluene, and polystyrene–toluene systems were determined at finite concentrations. The results were compared with the traditional gravimetric sorption and piezoelectric sorption measurements reported in the literature. The data are in excellent agreement with the values reported, correlate well with the Vrentas–Duda free volume theory, and can also be predicted from infinitely dilute data using the free volume theory. © 1997 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 35 : 1279–1290, 1997     

用渗流理论计算高分子溶液中溶剂的自扩散系数

基于微观渗流理论建立了溶剂小分子在高分子溶液中的自扩散模型,并据此模型对不同温度和浓度下的聚苯乙烯(PS)-苯、PS-甲苯、PS-乙苯和PS-四氢呋喃4个体系中小分子的自扩散系数进行了关联,计算出在不同温度下溶剂分子扩散所需的临界浓度。结果表明,在PS玻璃化温度以下,本模型对于温度和浓度具有很好的适用性和关联精度。     

Impact of average free‐volume element size on transport in stereoisomers of polynorbornene. I. Properties at 35 °C

This article demonstrates that transport of gases through glassy polymers is significantly influenced not only by the absolute amount but also by the distribution of free volume. Two stereoisomers of polynorbornene with nearly equivalent total free volume, but markedly different average free‐volume sizes, were evaluated. The free‐volume element size was probed with positron annihilation lifetime spectroscopy, wide‐angle X‐ray scattering, gas sorption, and molecular modeling. The permeation, sorption, and diffusion of light gases were measured in each stereoisomer at 35 °C. All analytical techniques indicated that one isomer (labeled as Architecture II) had a larger average free‐volume element size but fewer elements. This isomer also had a very slightly higher bulk density (1.000 vs 0.992 g/cm³ for the other stereoisomer). Architecture II also had gas sorption and diffusion coefficients that were two to three times those of the less dense counterpart. These differences have been attributed to differences in the free‐volume element size available within the polymer matrix. © 2003 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 41: 2185–2199, 2003     

Diffusion in polymer—solvent systems. I. Reexamination of the free-volume theory

The free-volume theory describing diffusion in polymer–solvent systems is reexamined. Calculation of the specific free volume for such systems is discussed, and equations are presented for the determination of the self-diffusion coefficients of the polymer and the solvent. Conditions under which the mutual diffusion coefficient can be deduced solely from free-volume considerations are clarified, and a more general version of the free-volume diffusion theory proposed by Fujita is presented. The further restrictions needed for the theory of Fujita are discussed, and it is concluded that these additional restrictions are responsible for failures of the Fujita theory.     

Effect of blend composition on diffusivity and solubility of small molecules in polystyrene/poly(vinyl methyl ether) blends

The capillary column inverse gas chromatography technique was used to determine diffusivity and solubility data for several solvents in polymer blends composed of polystyrene and poly(vinyl methyl ether) (PVME). Diffusivity behaved as expected, increasing as the concentration of PVME increased in the blend. Knowing only the free‐volume parameters for the pure polymers, the free‐volume theory was successfully applied to predict the dependence of the diffusion coefficients on the blend composition. Transport in blends above the glass transition temperature is controlled by free volume, and the effect of concentration fluctuations is minimal at the temperatures studied. Experimental data show an increase in the partition coefficient of some solvents in the blends with respect to the pure polymers. © 2007 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 45: 2071–2082, 2007     

Diffusion in gas separation membrane materials: A comparison and analysis of experimental characterization techniques

Typically, materials with high-performance transport properties such as zeolites, carbon molecular sieves, or hyper rigid polymers are inherently difficult or impossible to characterize by steady-state membrane permeation experiments used for conventional polymers. Diffusion coefficients determined by transient sorption, a measurement easily performed on brittle media, are analyzed here and compared to those determined by steady-state permeation/sorption and transient permeation for a glassy polymer and a carbon molecular sieve. Average and local diffusion coefficients are extrapolated to zero upstream partial pressure to eliminate effects caused by concentration dependence. Good agreement between the techniques was observed for the glassy polymer. On the other hand, carbon molecular sieves, possessing a more complex morphology, exhibit a greater difference in diffusion coefficients determined by the various techniques. Nevertheless, comparison of the analysis techniques is shown to provide potentially valuable insights into the morphological features of such carbon molecular sieves. © 1998 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 36: 1747–1755, 1998     

Dual-mode free volume model for diffusion of gas molecules in glassy polymers

The development of a new model for the diffusion of gas molecules in glassy polymers is presented which utilizes concepts from free volume theory and relies on a dual-mode interpretation of sorptive dilation in glassy polymers. Three assumptions are made in the development of the model. First, the free volume available for molecular transport processes is taken as constant below the glass transition temperature. Second, two populations of gas molecules are assumed to exist—one which contributes to the maintenance of an iso-free volume state upon sorptive dilation and one which does not contribute owing to sorption into regions of unrelaxed volume. Third, the former population is assumed to be mobile while the latter is not. The resulting model predicts, at constant temperature, a diffusion coefficient that is independent of solute volume fraction. This is in contrast to the widely used dual-mode sorption model with partial immobilization for gas transport in glassy polymers which leads to a diffusion coefficient that is dependent on solute mole fraction through the molar gas concentration. The new model is used to interpret gas transport data from permeation experiments for carbon dioxide, methane, and ethylene in three polycarbonates. © 1997 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 35: 1737–1746, 1997