Diffusion of low molecular weight substances into a laminar film. I: Rigorous solution of the diffusion equation in a composite film of multiple layers

A rigorous solution of the diffusion of penetrant into a laminar film comprised of multilayers of m components is presented by an orthogonal-expansion method. As the simplest practical cases of m ? 2 and m ? 3, with stepwise distribution of both diffusion coefficients and solubility coefficients at the boundary between respective layers, the diffusion properties in the transient state are analyzed in detail. That is, changes in the penetrant concentration distribution within the laminar film and the total amount of penetrant sorbed within the film both with time after exposing the film to an atmosphere of fixed penetrant concentration are calculated for A–B as well as A–B–A type layer arrangements. The calculation is performed while keeping (L₂/L₁) at a constant value of 2.0 but varying the diffusion coefficient ratio of (D₁/D₂) from 10² to 10^?2, and/or varying the solubility coefficient ratio of (S₁/S₂) from 1 to 10, where L₁ and (L₂ ? L₁) are the thickness D₁ and D₂ are the diffusion coefficients of penetrant, and S₁ and S₂ are the solubility coefficients in the A-component and B-component, respectively. The sorption curves deviate considerably from those of Fickian curves of homogeneous film with (D₁/D₂) ? 1 in their respective ways. The results obtained here can be applied to the diffusion in a single component polymer film having a surface layer with different diffusion properties from that of the inner side of the film caused by differing distributions in molecular orientation or degrees of crystallinity.     

Molecular dynamics simulations to compute diffusion coefficients of gases into polydimethylsiloxane and poly{(1,5‐ naphthalene)‐co‐[1,4‐durene‐2,2′‐bis(3,4‐dicarboxyl phenyl)hexafluoropropane diimide]}

Diffusion coefficients of N₂, O₂, CO₂ and CH₄ at 298 K in polydimethylsiloxane (PDMS) and poly{[(1,5‐naphthalene)‐co‐[1,4‐durene‐2,2′‐bis(3,4‐dicarboxyl phenyl)hexafluoropropane diimide]} (6FDA‐1,5‐NDA) polymers have been estimated using molecular dynamics (MD) simulations. Estimated diffusion coefficients in PDMS decrease systematically with increasing size of the penetrant gas molecules following the experimental observations. For 6FDA‐1,5‐NDA polymer, diffusion coefficients decrease in the same order of magnitude, but differ in their sequential order, due to varying side group interactions of the polymer with the gaseous molecules. Cohesive energy density, solubility parameter and free volume of the polymers were determined using MD simulations. Reliability and accuracy of the simulations have been tested typically with the computed values of the diffusion coefficient of O₂ in PDMS polymer, which compare well with the literature data. X‐ray scattering profiles of 6FDA‐1,5‐NDA have been generated to understand the interrelationship between the morphology and diffusion coefficients. The radial distribution function was evaluated to find the contribution of atoms that are important in understanding the molecular interactions during gas diffusion in polymers. Copyright © 2007 Society of Chemical Industry     

Solubility and diffusion behavior of compressed CO2 in polyurethane oligomer

In CO₂-assisted polyurethane (PU) foaming, the solubility and diffusion coefficient of CO₂ is vitally important to the cell nucleation and growth. This work is aimed at the effect of molecular weights (M _n) and crosslinking densities (V _e) on the solubility and diffusion coefficient of CO₂ in PU oligomers. A series of PU oligomers with different M _n and V _e were synthesized. The solubility and diffusivity of CO₂ in PU oligomers were measured experimentally in the temperature from 80 to 140 °C and with pressures up to 15 MPa. It was shown that the solubility and diffusion coefficients of CO₂ was decreased 20.5 and 21.0%, respectively, with the M _n increasing from 5864 to 153,754 g mol⁻¹ at 80 °C, 15 MPa. The solubility and the diffusion coefficient also decreased 11.1 and 38.0% as the V _e was increased from 64 to 1493 mol m⁻³. Furthermore, the diffusion mechanism of CO₂ in PU oligomers was explored via molecular dynamics simulations. The results indicated that the calculated diffusivity of CO₂ showed the same changing trend as the experimental values, and the smaller M _n or crosslinking degree contributed to an increase in fractional free volume and stronger polymer–CO₂ interactions. © 2018 Wiley Periodicals, Inc. J. Appl. Polym. Sci. 2019 , 136, 47100.     

Gas Diffusion and Solubility in Poly(organophosphazenes): Results of Molecular Simulation Studies

A combination of quantum chemistry, molecular dynamics, and Monte Carlo methods have been used to investigate gas diffusion and solubility in three isomeric poly[di(butoxyphosphazenes)] and in amorphous and crystalline states of poly[bis(2,2,2-trifluoroethoxyphosphazene)] (PTFEP). In this review of recently published studies reported from our laboratory, conclusions are reached in regards to the relationship between polymer structure and gas diffusion and sorption in poly(organophosphazenes). These conclusions also serve to validate our current understanding of the nature of gas transport in other polymers. Specifically, gas diffusivity has been shown to increase with increasing side-chain and main-chain mobility as determined from vectorial autocorrelation function analysis; however, high diffusivity is accompanied by a loss in diffusive selectivity resulting in decreasing permselectivity with increasing permeability. Simulation of crystalline supercells of PTFEP indicate that gas diffusion is unrestricted in the crystalline state as has been reported only for a few other polymers, principally poly(4-methyl-1-pentene). Gas solubility in poly(organophosphazenes) correlates well with gas condensability as measured by the Lennard–Jones potential well depth parameter, ɛ/k. Exceptions are cases where specific interactions can occur between gas molecules and the polymer chain such as is the case of CO₂ and PTFEP. High-level ab initio calculations of the interaction of CO₂ with low-molecular-weight fluoroalkanes indicate the presence of a weak quadrupole–dipole interaction. Association of CO₂ with the trifluoromethyl groups of the trifluoroethoxy side chain of PTFEP has been confirmed by radial distribution function (RDF) analysis of MD trajectories. Comparison between solubility coefficients obtained from Grand Canonical Monte Carlo (GCMC) simulations of amorphous cells with experimental values of microcrystalline PTFEP indicates that gas solubility in polyphosphazenes such as PTFEP that exhibit a mesophase/crystalline state is greatly reduced. This paper is dedicated to Prof. Harry Allcock for his scientific contributions to inorganic and organometallic polymers.     

Oxygen transport in crosslinked,silicon-containing copolymers of methyl methacrylate

Copolymers of methyl methacrylate and 1,3-bis(methacryloxy methyl)-1,1,3,3-tetramethyl disiloxane were prepared by chemically induced copolymerization/crosslinking at 60°C and 49 mm Hg. Crosslinked, glassy copolymers were obtained with copolymer mole fraction of the silicon-containing monomer varying from 0.09 to 0.55. Oxygen transport studies were performed with thin films as prepared and after sub-T_g annealing. The results of this study indicated that an enhancement of both the steady state oxygen permeation rate and the oxygen diffusion coefficient was achieved through copolymerization. The oxygen diffusion coefficients through the copolymers were found, within experimental error, to be independent of silicon content and ranged from 0.80 × 10^?7 to 1.90 × 10^?7 cm²/s vs. oxygen diffusion coefficient for pure poly(methyl methacrylate) of 1.5 × 10^?8 cm²/s. Sub-T_g annealing effected a reduction of approximately equal magnitude in both the oxygen diffusion coefficient and the steady state oxygen flux for the copolymers. In addition, the normalized oxygen flux data were predicted with Fick's law, assuming constant boundry conditions and diffusion coefficient. These results were explained in terms of the free volume theory and the combined effects of increased crosslinking density, chain mobility, and oxygen solubility with increased copolymer silicon content.     

The atomistic simulation of the gas permeability of poly(organophosphazenes). Part 2. Poly[bis(2,2,2-trifluoroethoxy)phosphazene]

Self-diffusion and sorption of seven gases (He, H₂, O₂, N₂, CH₄, CO₂, and Xe) in poly[bis(2,2,2-trifluoroethoxy)phosphazene] (PTFEP) have been investigated by molecular dynamics and Grand Canonical Monte Carlo (GCMC) simulations of two amorphous cells and an α-orthorhombic crystalline supercell. In the case of MD simulation of diffusion coefficients, values obtained for both amorphous and crystalline PTFEP are similar and comparable to experimental values reported for semicrystalline samples. These results indicate that gas diffusion is unrestricted in the crystalline state of PTFEP as has been reported for poly(4-methyl-1-pentene) (PMP) and, more recently, for a crystalline form of syndiotactic polystyrene (sPS).In contrast to both PMP and sPS that have low-density crystalline forms, only He exhibits any solubility in the α-orthorhombic crystalline cells of PTFEP during simulation. In addition, values of the solubility coefficients obtained from simulation of the amorphous cells are three to five times larger than would be expected by extrapolating values reported for semicrystalline samples to 100% amorphous content. These results suggest that while the crystalline domains do not restrict gas diffusivity in PTFEP, they significantly reduce gas solubility in semicrystalline PTFEP through the reduction of amorphous content and through some additional effect of the crystallites on amorphous-phase solubility, possibly through chain immobilization of the amorphous phase. Similar solubility behavior has been suggested for polyethylene on the basis of recent simulation studies.As reported in a prior communication, the solubility of CO₂ in PTFEP is very high compared to other gases due to a weak quadrupole-dipole interaction between CO₂ and the trifluoroethoxy group of PTFEP. As a result, the solubility coefficients of CO₂ obtained from GCMC simulation of the amorphous cells and from permeability measurements of semicrystalline samples are both larger than predicted by a simple correlation of gas solubility coefficients with the Lennard-Jones potential well parameter, ε/k, of other gases as proposed by Teplyakov and others. A modified form of this correlation that includes a Flory interaction term is shown to fit all gas solubility data for this polymer including that of CO₂.     

Gas separation properties of aromatic polyetherimides from 1,4-bis(3,4-dicarboxyphenoxy)benzene dianhydride and 3,5-diaminobenzic acid or its esters

The gas transport properties of a series polyetherimides, which were prepared from 1,4-bis(3,4-dicarboxyphenoxy)benzene dianhydride (HQDPA) with 1,3-phenylenediamine or 3,5-diaminobenzic acid (DBA) or its esters are reported. The effects of carboxylic group (—COOH) and carboxylic ether groups (—COOR), at five positions of 1,3-phenylenediamine moiety, on H₂ CO₂, O₂, and N₂ permeability, diffusivity, and solubility of the polyetherimides were investigated. The gas permeability, diffusion, and solubility coefficients of the polyetherimides containing COOR are bigger than those of HQDPA-PDA, but the ideal separation factors and ideal diffusivity selectivity factors are much smaller than that of HQDPA-PDA because COOR decreases chain segmental packing efficiency and increases chain segmental mobility. The permeability coefficients of HQDPA-DBA to H₂, CO₂, and O₂ are bigger than those of HQDPA-PDA; the ideal separation factors for gas pairs H₂/N₂, CO₂/N₂, and O₂/N₂ are also much bigger than those of HQDPA-PDA. Both the diffusion coefficients of CO₂ and O₂ and the ideal diffusivity selectivity factors for CO₂/N₂ and O₂/N₂ are bigger than those of HQDPA-PDA because COOH decreases both chain segmental packing efficiency and chain segmental mobility. The copolyimides, which were prepared from 3,5-diaminobenzic acid and 3,5-diaminobenzic esters, have both high permeability and high permselectivity. © 1997 John Wiley & Sons, Inc.     

Influence of flexible oligo(tetrafluoroethene) segment on the sorption and diffusion of carbon dioxide in poly(amide‐imide) membranes

Molecular simulations have been used to study the sorption and diffusion properties of carbon dioxide in a series of poly (amide‐imide) (PAI) membranes containing oligo(tetrafluoroethene) segment with various numbers (n = 0, 1, 2, 3, and 4) of tetrafluoroethene units. The solubility and self‐diffusion coefficients were computed by the Grand Canonical Monte Carlo (GCMC) method and molecular dynamics (MD) simulations respectively. It was found that increasing the fluorine content of the polymer membrane reduced the associated glass transition temperature (T_g) and led to an increase in diffusion coefficient of carbon dioxide. Results indicate that penetrant molecule's diffusion coefficient is strongly dependent on chain mobility. It is also noticed that the radial distribution functions (RDFs) are inconsistent with the d‐spacings of PAIs calculated form X‐ray data. This is also thought to be tied to the number of degrees of freedom of the chain. Finally, this study gives a useful insight into how PAIs with high fluorine content can be tailored with a high permeability to carbon dioxide. © 2010 Wiley Periodicals, Inc. J Appl Polym Sci, 2011     

Atomistic molecular dynamics simulations of gas diffusion and solubility in rubbery amorphous hydrocarbon polymers

Diffusion coefficients D of H₂, He, O₂, N₂, and CO₂ in different rubbery amorphous polymeric matrices were estimated by atomistic molecular dynamics simulations at 298 K using the Einstein relationship, and compared with the relevant experimental values, where available. The simulated diffusion coefficients D of all the gases in all polymers considered almost regularly decreased with increasing molecular gas volumes and increasing polymer glass transition temperature. Further, solubility coefficients and heats of solution were obtained for all gases from Grand Canonical Monte Carlo simulations, which were also used to calculate sorption isotherms. In general, there is a good agreement between experimental and simulated values of diffusion and solubility coefficients for all gases considered.     

Diffusion of gases and vapors through methacrylates with oxyethylene units in the pendent chain

The diffusion of He, Ar, O₂, N₂, CO₂, CH₄, CH₃CH₃ and CH₂CH₂ has been determined in three polymethacrylates with one, two and three oxyethylene units as side chains. For these three rubbery polymers we have used the acronyms PEEMA, PDEMA and PTEMA, respectively. The results have been compared to those of poly(ethyl methacrylate) on one hand, and on the other with previous results on methacrylates of varying length alkyl side chains. It has been found that the oxyethylene units side chains methacrylates show diffusion coefficients which are slightly lower than those methacrylates bearing alkyl side chain, when comparing polymers with the same number of atoms in the side chain. These diffusion data are put in relation to relevant structural features such as the glass transition temperature and the fractional free volume of both families of rubbery methacrylates.     

Study of transport of pure and mixed CO2/N2 gases through polymeric membranes

The permeations of pure CO₂ and N₂ gases and a binary gas mixture of CO₂/N₂ (20/80) through poly(dimethylsiloxane) (PDMS) membrane were carried out by the new permeation apparatus. The permeation and separation behaviors were characterized in terms of transport parameters, namely, permeability, diffusion, and solubility coefficients which were precisely determined by the continuous‐flow technique. In the permeation of the pure gases, feed pressure and temperature affected the solubility coefficients of CO₂ and N₂ in opposite ways, respectively; increasing feed pressure positively affects CO₂ solubility coefficient and negatively affects N₂ solubility coefficient, whereas increasing temperature favors only N₂ sorption. In the permeation of the mixed gas, mass transport was observed to be affected mainly by the coupling in sorption, and the coupling was analyzed by a newly defined parameter permeation ratio. The coupling effects have been investigated on the permeation and separation behaviors in the permeation of the mixed gas varying temperature and feed pressure. © 2000 John Wiley & Sons, Inc. J Appl Polym Sci 78: 179–189, 2000     

Estimation of diffusion and solubility coefficients for water and CO2 in reaction injection molded parts

The evolution of gases from Reaction Injection Molded (RIM) parts during painting causes “pinhole” surface defects, which result in scrap. The transport of gases plays a major role in this outgassing. The diffusion rates of CO₂ and water through reaction injection molded parts is measured in this work. Despite the presence of glassy hard domains and dispersed microbubbles of size <25 μm (nucleated parts), the diffusion showed Fickian characteristics, and diffusion coefficients were independent of the concentration of water and CO₂ in nucleated and non-nucleated RIM parts. The presence of microbubbles enhances diffusion, which could be predicted from the diffusion rate through the non-nucleated material by using a simplistic one-dimensional model. The diffusion coefficients of water and CO₂ follow an Arrhenius relationship. The solubility coefficients follow a van't Hoff relationship over a wide range of 20 to 150°C. Apart from predicting diffusion and solubility coefficients, for various RIM materials and processing conditions, the estimated parameters will be used to interpret the outgassing phenomenon in a subsequent paper.     

The desorption of ethane–butane mixtures from polyethylene

The desorption of mixtures of ethane and butane at atmospheric pressure from low-density polyethylene was investigated over the temperature range from 20 to 60°C. Desorbed penetrants were continuously trapped in glass tubes immersed in liquid nitrogen, and composition was determined as a function of time by means of gas chromatography. The ratio of the quantity of desorbed gas at any time t, q_t, to the quantity at complete desorption, q_∞, was used to determine diffusion coefficients and solubility constants. The diffusion coefficients for both ethane and butane increase with increasing butane concentration in the temperature interval investigated. The solubility of both penetrants can be correlated by Henry's law at 40, 50, and 60°C. However, at 20 and 30°C. the solubility constant for both penetrants increases with increasing butane concentration. This trend is consistent with experimental observations for single-component diffusion and solubility of several hydrocarbons in polyethylene, where increasing concentration of penetrant plasticizes the polymer, resulting in increasing diffusion coefficients and solubility constants.     

Measurement and prediction of diffusion coefficients of supercritical CO2 in molten polymers

The solubility and diffusivity of supercritical carbon dioxide (sc‐CO₂) in low‐density polyethylene (LDPE), high‐density polyethylene (HDPE), polypropylene (PP), ethylene‐ethylacrylate copolymer (EEA) and polystyrene (PS) were measured at temperatures from 150°C to 200°C and pressures up to 12 MPa by using the Magnetic Suspension Balance (MSB), a gravimetric technique for gas sorption measurements. The solubility of CO₂ in each polymer was expressed by Henry's constant. The interaction parameter between CO₂ and polymer could be obtained from the solubility data, and it was used to estimate the Pressure‐Volume‐Temperature relationship and the specific free volume of polymer/CO₂ mixtures. The diffusion coefficients were also measured by the MSB for each polymer. The resulting diffusion coefficients were correlated with the estimated free volume of polymer/CO₂ mixture. Combining Fujita's and Maeda and Paul's diffusion models, a model was newly developed in order to predict diffusion coefficients for the polymers studied. Polym. Eng. Sci. 44:1915–1924, 2004. © 2004 Society of Plastics Engineers.     

采用泰勒分散法测量蜡分子扩散系数

蜡分子扩散系数是蜡沉积预测模型中非常重要的物性参数。为测量蜡分子（高碳数正构烷烃）在液相体系中的分子扩散系数,建立了基于泰勒分散法的扩散系数测量装置,并对装置操作注意事项进行了详细探讨。采用该装置测量了丙酮水溶液、n-C₆+n-C₇溶液以及甲醇水溶液中溶质在溶剂中的分子扩散系数,以文献值对测量值进行校验,验证了装置的可靠性。分别以n-C₁₈、n-C₂₀、n-C₂₂、n-C₂₄、n-C₂₆为溶质,以n-C₇为溶剂,在不同温度、浓度下测量蜡分子在液相体系中的分子扩散系数,并将实验测量值与Hayduck-Minhas关系式计算值进行比较。结果表明：蜡分子扩散系数随温度升高而线性增大,随溶液中蜡分子摩尔分数的增大而以指数形式减小;在相同条件下,高碳数正构烷烃的分子扩散系数低于低碳数正构烷烃的分子扩散系数。采用Hayduck-Minhas关系式的计算结果比实验测量结果平均小50%,应用于蜡沉积预测时,将低估蜡分子扩散质量流。     

Molecular simulations of small gas diffusion and solubility in copolymers of styrene

The objective of this study is to investigate the relationship between gas permeability and the chemical structure and conformational properties for copolymers of styrene and its homopolymer. The diffusion and the solubility coefficients of small gas molecules (He, H₂, Ne, O₂, N₂, CH₄, Ar, CO₂) in amorphous structures of poly (styrene-alt-maleic anhydride) copolymer (SMA), poly (styrene-stat-butadiene) rubber (SBR), and atactic polystyrene (PS) are investigated by the transition state approach. Simulation results are found to be in good agreement with the experimentally measured values. The transport behavior of H₂O molecules is also studied in the same bulk structures by fully atomistic molecular dynamics simulations. In general, the diffusion coefficients of the gases in these matrices decrease in the following order: SBR>PS>SMA, whereas the solubility coefficients follow the reverse order. The differences in the mobility of the matrices seem to be the dominant determining factor for diffusion. And the solubility coefficients depend on the free volume distribution of the matrices.     

Preparation and C3H8/Gas Separation Properties of a Synthesized Single Layer PDMS Membrane

In this paper, a new polydimethylsiloxane (PDMS) membrane was synthesized and its ability for separation of heavier gases from lighter ones was examined. Sorption, diffusion, and permeation of H₂, N₂, O₂, CH₄, CO_2, and C₃H₈ in the synthesized membrane were investigated as a function of pressure at 35°C. PDMS was confirmed to be more permeable to more condensable gases such as C₃H₈. This result was attributed to very high solubility of larger gas molecules in hydrocarbon?based PDMS in spite of their low diffusion coefficients relative to small molecules. The synthesized membrane showed much better gas permeation performance than others reported in the literature. Increasing upstream pressure increased solubility, permeability and diffusion coefficients of C₃H₈, while these values decreased slightly or stayed constant for other gases. Local effective diffusion coefficient of C₃H₈ and CO₂ increased with increasing penetrant concentration which indicated plasticization effect of these gases over the range of penetrant concentration studied. C₃H₈/gas solubility, diffusivity and overall selectivities also increased with increasing feed pressure. Ideal selectivity values of 4, 13, 18, 20, and 36 for C₃H₈ over CO₂, CH₄, H₂, O_2, and N₂, respectively, at upstream pressure of 7 atm, confirmed the outstanding separation performance of the synthesized mebrane.     

Solubility and diffusivity of CO2 in polypropylene/micro-calcium carbonate composites

This work is aimed at studying the effects of the fillers and interface bonding condition between the fillers and polymer matrix on the solubility and diffusivity of CO₂ in polypropylene (PP)/Micro-calcium carbonate (MicroCaCO₃) composites. The solubility of CO₂ in PP and its composites containing 5% and 10% MicroCaCO₃ was determined precisely by using magnetic suspension balance (MSB) combined with experimental swelling correction at 200 and 220 °C and CO₂ pressures up to 22 MPa. It was found that the solubility of CO₂ in the PP/MicroCaCO₃ composites without the interface compatibilizer increased with increasing the filler content, while the CO₂ solubility remained almost unchanged in PP composites with compatibilizer. The Henry's law and a modified Henry's law were used to well correlate the solubility of CO₂ in the PP composites with and without the interface compatibilizer, respectively. The diffusion coefficient of CO₂ in the PP composites was found to decrease with increasing the filler content. The mutual diffusion coefficients of CO₂ in the PP composites can be correlated within an average relative deviation of 10% by the free volume model proposed by Kulkarni and Stern with a parameter accounting for the barrier effect of the filler.     

Gas transport properties of polyaniline membranes

The transport properties of He, H₂, CO₂, O₂, N₂, and CH₄ gases in solvent cast, HCl doped, and undoped polyaniline (PANi) membranes were determined. Measurements were carried out at 40 psi pressure from 19°C to 60°C. An excellent correlation was found between the diffusion coefficients and the molecular diameters of gases. The solubility coefficients of gases were found to correlate with their boiling points or critical temperatures. The sepa-ration factors for CO₂/N₂ and CO₂/CH₄ are dominated by the high solubility of CO₂. These correlations enable us to predict the permeability, diffusion, and solubility coefficients of other gases. After the doping-undoping process, the fluxes of gases with kinetic diameters smaller than 3.5 Å increased but those of larger gases decreased. This results in a higher separation factor for a gas pair involving a small gas molecule and a larger one. © 1996 John Wiley & Sons, Inc.     

Diffusion coefficient of hydrogen in a Pd-Ag membrane: Effect of hydrogen solubility

The permeabilities of hydrogen through a Pd₇₅-Ag₂₅ membrane have been measured at temperatures ranging from 423 to 573 K and under hydrogen pressure differences ranging from 69 to 256 kPa. From the available literature solubility data a neural network model has been developed in order to simulate the variation of the hydrogen content in the alloy as a function of pressure and temperature. Then, from steady state permeability measurements and calculated solubilities, the diffusion coefficients of hydrogen have been computed. At a given temperature, the diffusion coefficient has been found to decrease with the hydrogen content (0.1 ≤ n = H/M ≤ 0.37). A simple relation is then suggested to predict the variation of the diffusion coefficient on both temperature and hydrogen content. To account for the variation of the diffusion coefficient with n, the integration of Fick's law of diffusion has been performed numerically, resulting in non-linear profile of hydrogen content through the membrane under steady state permeation.