Relaxation of chemical potential and a generalized diffusion equation

The effect of slow structural relaxation in a solvent of high viscosity on the chemical potential driving the diffusion of penetrant molecules is described by a generalized diffusion equation with a memory term. The linearized version of this equation is solved for some special cases, and the correlation function of concentration fluctuations in thermodynamic equilibrium is calculated. As a result of the memory term, for very slow relaxation two different stages of the diffusion process can be distinguished.     

Diffusion competition between solvent and nonsolvent during the coagulation process of cellulose / ammonia / ammonium thiocynate fiber spinning system

An investigation of the diffusion competition between solvent and nonsolvent in a coagulation bath is presented for the formation of a new cellulosic fiber by wet-spinning. The system consisted of the spinnable cellulose solution with a mixture of liquid ammonia/ammonia thiocynate as the solvent and low-molecular-weight alcohols as the nonsolvents. The diffusion competition between solvent and nonsolvent was quantitatively characterized in terms of their mass transfer rate differences. The measurements of this rate difference were performed on the model filament shaped from gelled cellulose solutions. Results revealed that an increase in molecular size of coagulant, bath temperature, and coagulant concentration in the bath enhanced preferential diffusion of solvent from cellulose solution. Fiber spinning experiments showed that a higher value of the initial modulus of the fiber was attained with a coagulation condition providing a lower value of mass transfer rate difference. The importance of mass transfer rate difference was also shown in the influence of the fiber cross-sectional shapes.     

Disentanglement and reptation during dissolution of rubbery polymers

The dissolution mechanism of rubbery polymers was analyzed by dividing the penetrant concentration field into three regimes that delineate three distinctly different transport processes. The solvent penetration into the rubbery polymer was assumed to be Fickian. The mode of mobility of the polymer chains was shown to undergo a change at a critical penetrant concentration expressed as a change in the diffusion coefficient of the polymer. It was assumed that beyond the critical penetrant concentration, reptation was the dominant mode of diffusion. Molecular arguments were invoked to derive expressions for the radius of gyration, the plateau modulus, and the reptation time, thus leading to an expression for the reptation diffusivity. The disentanglement rate was defined as the ratio between the radius of gyration of the polymer and the reptation time. Transport in the second penetrant concentration regime was modeled to occur in a diffusion boundary layer adjacent to the polymer-solvent interface, where a Smoluchowski type diffusion equation was obtained. The model equations were numerically solved using a fully implicit finite difference technique. The results of the simulation were analyzed to ascertain the effect of the polymer molecular weight and its diffusivity on the dissolution process. The results show that the dissolution can be either disentanglement or diffusion controlled depending on the polymer molecular weight and the thickness of the diffusion boundary layer. © 1996 John Wiley & Sons, Inc.     

GAS SORPTION,DIFFUSION AND PERMEATION OF ORDERED POLYMERIC MEMBRANES

He, CO_2, O_2, N_2, CH_4, C_3H_8, and t-C_4H_(10) gas permeability coefficients and diffusion coefficients of poly(4-methylpentene-1) (PMP) with various degrees of crystallization were plotted against the degree of crystallization. The plotdemonstrated a linear relationship. The gas permeability coefficient and diffusion coefficient of pure amorphous and purecrystalline PMPs were evaluated by a linear extrapolation to zero and 100% crystallinity, respectively. The relationshipbetween the diffusion coefficient of crystalline parts of PMP and the kinetic diameter of penetrant gases was discussed.Syndiotactic polystyrene (SPS) could exist as δ form crystals complexed with organic solvents such as benzene, toluene,xylene, and ethylbenzene. The mesophase of SPS is prepared by annealing the δ form of crystalline complexes at a certaintemperature for 1 h. The desorption of solvent during annealing almost does not result in changes of both the conformation ofbackbone chains and the crystal lattice. We could prepare the mesophase containing molecular cavities with the size andshape of the organic solvent molecules. The mesophase could sorb the same solvent after the manner of Langmuir sorption atlow vapor pressure range while this would not be the case for solvents of different size and shape. This suggests a molecularrecognition of organic solvent, and mesophase SPS might be useful for separation membrane and adsorptive material.     

Diffusion phenomena in ternary systems polymer-nonsolvent-solvent

Diffusion in a boundary between a polymer+solvent solution and non-solvent was treated by accounting for the presence of the four diffusion coefficients that describe the isothermal transport process in a three component system. Diffusion equations were integrated assuming a concentration dependence of diffusion coefficients that account for the thermodynamic conditions on the cross diffusion terms of Eq. (1). The presence of non-zero cross terms promotes an incongruent diffusion of polymer whose concentration increases at the boundary between the polymer+solvent solution and the non-solvent. Although our model describes diffusion in the range of homogeneous solution, this incongruent polymer diffusion is a process similar to that promoted by the solvent evaporation from the polymer+solvent film that some authors suggested as an intermediate step before the film immersion into the coagulation bath to obtain good asymmetric membranes.     

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

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

Determination of penetrant solubility from transient permeation measurements

The permeability coefficient for the transport of a gas, vapor, or liquid through a polymer film is the product of the penetrant solubility and a diffusion coefficient. A transient permeation experiment known as the time-lag technique can be used to separate this product, provided the diffusion coefficient is independent of penetrant concentration. In this well-known experiment the polymer is initially free of penetrant. A new transient permeation experiment where the polymer is initially saturated with penetrant is suggested here. A general mathematical proof is given to show that by using the results form these two transient experiments which have different initial conditions one can determine the penetrant solubility no matter how the diffusion coefficient depends on penetrant concentration. Also one can determine two different concentration averaged diffusion coefficients from the results.     

Modeling of penetrant diffusion in glassy polymers with an integral sorption deborah number

A mathematical model was developed to explain the anomalous penetrant diffusion behavior in glassy polymers. The model equations were derived by using the linear irreversible thermodynamics theory and the kinematic relations in continuum mechanics, showing the coupling between the polymer mechanical behavior and penetrant transport. The Maxwell model was used as the stress–strain constitutive equation, from which the polymer relaxation time was defined. An integral sorption Deborah number was proposed as the ratio of the characteristic relaxation time in the glassy region to the characteristic diffusion time in the swollen region. With this definition, an integral sorption process was characterized by a single Deborah number and the controlling mechanism was identified in terms of the value of the Deborah number. The model equations were two coupled nonlinear differential equations. A finite difference method was developed for solving the model equations. Numerical simulation of integral sorption of penetrants in glassy polymers was performed. The simulation results show that (1) the present model can predict Case II transport behavior as well as the transition from Case II to Fickian diffusion and (2) the integral sorption Deborah number is a major parameter affecting the transition. © 1993 John Wiley & Sons, Inc.     

Polymer structure and the compensation effect of the diffusion pre-exponential factor and activation energy of a permeating solute

In the present work, the relation between the pre-exponential factor and the apparent activation energy of diffusion, ln D(0) = alpha + betaE(D), so-called compensation effect, is re-examined and critically discussed for diffusion of gases in rubbery and glassy polymers. In principle, the above equation could be derived from the enthalpy-entropy compensation in the framework of the transition state theory. However, one should consider the influence of the jump length term contained in the pre-exponential factor, which may be affected by permeating species and polymer properties. We found that parameter alpha depends on penetrant size and polymer properties, such as local chain mobility and free volume. This can be interpreted by the fact that the jump length is affected by both penetrant and polymer properties. Finally, methods for estimating the jump length are discussed.     

Diffusion in glassy polymers. II

The Fickian diffusion coefficient of methylene chloride in a glassy epoxy polymer is calculated with the use of Crank's model of discontinuous change of D with concentration C. The diffusion constant is obtained as 1.93 × 10^?6 cm²/sec. The swollen layer behind the advancing solvent front is essentially in the rubbery state of the same polymer. The case II swelling by benzene is discussed in terms of a convective transport arising from the partial stress (internal) tensor of the penetrant. The superposition of Fickian and case II diffusion found with mixtures of methylene chloride and benzene is also discussed briefly.     

Diffusion dynamics of 1-Butyl-3-methylimidazolium chloride from cellulose filament during coagulation process

The diffusion dynamics of 1-Butyl-3-methylimidazolium chloride ([BMIM]Cl) during coagulation process of cellulose filaments with H₂O as non-solvent were investigated in detail. The diffusion coefficients of [BMIM]Cl was calculated based on the Fick’s second law of diffusion according to the experimental data. Several factors which affect the coagulation process including polymer concentration, concentration and temperature of coagulation bath were discussed respectively. It is found that the diffusion rate of [BMIM]Cl decreased with the increasing polymer content in the spinning solutions and the initial concentration of [BMIM]Cl in the coagulation bath, while the diffusion coefficients increased largely with the coagulation temperature becoming higher. The diffusion coefficients of [BMIM]Cl is relatively lower, in contrast with the conventional solvent in the solution spinning process, which is coordinate with the result of polyacrylonitrile [BMIM]Cl system by Zhang et al. (Polym Eng Sci 48(1):184–190, 2008). Compared with the diffusion process of N-methylmorpholine-N-oxide (NMMO) from cellulose filament, the diffusion coefficients of [BMIM]Cl is lower, which suggested a stronger coagulation and washing conditions should be taken to produce regenerated cellulose fiber with [BMIM]Cl as solvent.     

Diffusion in atactic polystyrene above the glass transition point

Diffusivities and solubilities in the system n-pentane-polystyrene were measured for low penetrant concentrations at several temperatures above the glass transition temperature. The relatively sharp changes in the activation energy of diffusion and the heat of solution near 150°C. are interpreted tentatively as indicating that a second-order transition exists in atactic polystyrene above the glass transition temperature. Evidence indicating the existence of this transition has been obtained by other techniques and the effects observed in this study should also be present in the diffusion of other penetrants in polystyrene. Correlation of these and other available data for diffusion in polystyrene as a function of the molecular size of the penetrant indicates that specific thermodynamic interactions between polymer and penetrant have little influence on the diffusion process.     

Sorption of propane and propylene in polyethylene

Measurements are reported of the solubility and concentration of propane and propylene in polyethylene, at temperatures from ?30°C to +30°C and pressures from 1.68 to 3.52 atm. Solubility of both gases in polymer depends on penetrant activity. Henry's law is not obeyed at high values of the penetrant activity, i.e., in the vicinity of the condensation point of the gas. The interaction between the solvent and the polymer is independent of pressure and a function of temperature. The propylene-polyethylene interaction seems to reach a maximum at 10°C within the range investigated. A physical mechanism, based on opposite effects of temperature upon polymer and penetrant, is suggested to explain the results.     

Local solvent density augmentation around a solute in supercritical solvent bath: 1. A mechanism explanation and a new phenomenon

A recently proposed partitioned density functional (DF) approximation (Phys. Rev. E 2003, 68, 061201) and an adjustable parameter-free version of a Lagrangian theorem-based DF approximation (LTDFA: Phys. Lett. A 2003, 319, 279) are combined to propose a DF approximation for nonuniform Lennard-Jones (LJ) fluid. Predictions of the present DF approximation for local LJ solvent density inhomogeneity around a large LJ solute particle or hard core Yukawa particle are in good agreement with existing simulation data. An extensive investigation about the effect of solvent bath temperature, solvent-solute interaction range, solvent-solute interaction magnitude, and solute size on the local solvent density inhomogeneity is carried out with the present DF approximation. It is found that a plateau of solvent accumulation number as a function of solvent bath bulk density is due to a coupling between the solvent-solute interaction and solvent correlation whose mathematical expression is a convolution integral appearing in the density profile equation of the DF theory formalism. The coupling becomes stronger as the increasing of the whole solvent-solute interaction strength, solute size relative to solvent size, and the closeness to the critical density and temperature of the solvent bath. When the attractive solvent-solute interaction becomes large enough and the bulk state moves close enough to the critical temperature of the solvent bath, the maximum solvent accumulation number as a function of solvent bath bulk density appears near the solvent bath critical density; the appearance of this maximum is in contrast with a conclusion drawn by a previous investigation based on an inhomogeneous version of Ornstein-Zernike integral equation carried out only for a smaller parameter space than that in the present paper. Advantage of the DFT approach over the integral equation is discussed.     

The concentration dependence of the diffusion and permeability in a homogeneous membrane

For the sorption and diffusion coefficient dependence on the concentration of the penetrant the transport properties of a homogeneous medium are calculated. The diffusion current is assumed to be proportional to the negative gradient of the chemical potential. This is in contrast with the first Fick's law that assumes this current to be proportional to the negative gradient of the concentration of the penetrant. The difference between the two cases depends on the concentration dependence of the sorption coefficient. In a homogeneous membrane the chemical potential formulation leads to an equation which is very similar to the Fickian expression. The apparent diffusion coefficient, however, depends not onlly on the transport resistance but also on the deviation of the sorption coefficient from constancy.     

Effective molecular diffusion coefficient in a two-phase gel medium

We derive a mean-field expression for the effective diffusion coefficient of a probe molecule in a two-phase medium consisting of a hydrogel with large gel-free solvent inclusions, in terms of the homogeneous diffusion coefficients in the gel and in the solvent. Upon comparing with exact numerical lattice calculations, we find that our expression provides a remarkably accurate prediction for the effective diffusion coefficient, over a wide range of gel concentration and relative volume fraction of the two phases. Moreover, we extend our model to handle spatial variations of viscosity, thereby allowing us to treat cases where the solvent viscosity itself is inhomogeneous. This work provides robust grounds for the modeling and design of multiphase systems for specific applications, e.g., hydrogels as novel food agents or efficient drug-delivery platforms.     

Pair diffusion, hydrodynamic interactions, and available volume in dense fluids

We calculate the pair diffusion coefficient D(r) as a function of the distance r between two hard sphere particles in a dense monodisperse fluid. The distance-dependent pair diffusion coefficient describes the hydrodynamic interactions between particles in a fluid that are central to theories of polymer and colloid dynamics. We determine D(r) from the propagators (Green's functions) of particle pairs obtained from molecular dynamics simulations. At distances exceeding ~3 molecular diameters, the calculated pair diffusion coefficients are in excellent agreement with predictions from exact macroscopic hydrodynamic theory for large Brownian particles suspended in a solvent bath, as well as the Oseen approximation. However, the asymptotic 1/r distance dependence of D(r) associated with hydrodynamic effects emerges only after the pair distance dynamics has been followed for relatively long times, indicating non-negligible memory effects in the pair diffusion at short times. Deviations of the calculated D(r) from the hydrodynamic models at short distances r reflect the underlying many-body fluid structure, and are found to be correlated to differences in the local available volume. The procedure used here to determine the pair diffusion coefficients can also be used for single-particle diffusion in confinement with spherical symmetry.     

Diffusion in inhomogeneous films and membranes

In a number of different contexts one is interested in the diffusion of a penetrant into an inhomogeneous film or membrane. We review exact results in the theory of permeation through membranes containing fixed inhomogeneities. We also present the available exact results on permeation, time lag, and sorption for a sufficiently dilute penetrant in an inhomogeneous slab or film. We discuss and compare the kind of information provided by permeation and sorption studies using a variety of examples of simple types of inhomogeneities.     

Sorption and diffusion of methyl substituted benzenes through cross-linked nitrile rubber/poly(ethylene co-vinyl acetate) blend membranes

The diffusion and sorption of methyl substituted benzenes through cross-linked nitrile rubber/poly(ethylene co-vinyl acetate) (NBR/EVA) blend membranes has been studied. The influence of blend composition, cross-linking systems, temperature and size of penetrants on the transport behaviour has been analysed. It was observed that as the EVA content increases in the blends, the solvent uptake decreases. An increase in the penetrant size also decreases the solvent uptake. The diffusion experiments were carried out in the temperature range 23–75 °C. As temperature increases the equilibrium uptake also increases. The transport coefficients namely diffusion coefficient, sorption coefficient and permeation coefficient have been calculated. The sorption data has been used to estimate the activation energies for permeation and diffusion. The van’t Hoff relationship was used to determine the thermodynamic parameters. The affine and phantom models for chemical cross-links were used to predict the nature of cross-links. Models for permeability were used and the theoretical values compared with the experimental results.     

Continuous desorption rate measurement from a shallow-bed of poly(styrene-divinylbenzene) particles with correction for experimental artifacts

A 0.50 mm high bed, containing ca. 3 mg of the nominally non-porous poly(styrene-divinylbenzene) (PS-DVB) sorbent Hamilton PRP-infinity, is located in a valve. After the bed is pre-equilibrated with a (7/3) methanol/water solution of naphthalene (NA), the valve is switched and (7/3) methanol/water solvent flows continuously through the bed at a high linear velocity. This causes NA to desorb into a constantly refreshed solvent, creating a "shallow-bed" contactor with an "infinite bath" kinetic condition. The effluent from the bed passes through a UV-absorbance detector which generates the observed instantaneous desorption rate curve for NA. The same experiment is performed using the solute phloroglucinol (PG), which is not sorbed by PRP-infinity and serves as an "impulse response function marker" (IRF-Marker). The resulting peak-shaped IRF curve is used in two ways (i.e. subtraction and deconvolution) in order to correct the observed instantaneous rate curve of NA for the following experimental artifacts: hold-up volume of the bed and valve, transit-delay time between the bed and the detector and instrument bandbroadening of the NA zone. The cumulative desorption rate curve, which is a plot of moles NA desorbed versus time, is obtained by integration. It is found to be accurately described by the theoretical equation for homogeneous spherical diffusion. The diffusion coefficient of NA inside the PRP-infinity particles (5.0+/-0.6) x 10(-11) cm2/s, agrees with the literature value that was obtained from the sorption rate of NA into the same particles. This constitutes virtually conclusive evidence for diffusion control of intra-particle kinetics of NA in the PS-DVB matrix of PRP-infinity and related polymers. The influence of both sorbent and solute properties on the method is evaluated.