Comparison of theory and experiment for diffusion in dilute polymer solutions

Results from a number of theories for the concentration dependence of the mutual diffusion coefficient in dilute polymer solutions are examined, and clarifications are made as to what forms of the equations for these theories should be used in comparisons with experimental diffusivity data. An evaluation of the available theories for the concentration dependence of the diffusivity under theta conditions is carried out using experimental diffusivity data taken using sharp fractions of polystyrene. It is concluded that the Pyun—Fixman theory appears to provide the most promising method for estimating the concentration dependence of the mutual diffusion coefficient in dilute polymer solutions at the present time.     

New methods for the determination of the parameters of a concentration‐dependent diffusion law for molecular penetrants from transient permeation or sorption data

New methods were developed for the determination of the plasticization parameters and the limit diffusion coefficient of a penetrant whose diffusivity varies exponentially with its concentration. No specific computer softwares is required for their use and they are user friendly. The method using differential permeation data is based on the correlations between the slope of the reduced permeation flux versus time plot at the inflexion point, and the two key parameters of the concentration‐dependent diffusivity laws. For the transient sorption method, the slope of the penetrant mass uptake versus square‐root of time curve leads to the same parameters of the diffusion law via similar correlations.     

Diffusion in heterogeneous media containing impermeable domains arranged in parallel arrays of variable orientation

Effective diffusivity through heterogeneous media containing impermeable and geometrically anisotropic domains depends on the domain orientation relative to the diffusion direction. A particular class of heterogeneous membranes in which domain orientation can be controlled in situ, side-chain liquid-crystalline polymers (SCLCPs), are being investigated as responsive, or variable permeability membranes for use in a variety of applications including controlled release drug delivery. This paper describes a geometric model for predicting the effective diffusivity through this type of membranes as a function of domain orientation, volume fraction, and aspect ratio. Estimates of effective diffusivity were also generated using random walk simulations. Results from the two methods agree well with each other. Predicted effective diffusivities in systems with domains aligned either orthogonal or parallel to the diffusion direction are compared to results from available literature models. Model predictions are compared to experimentally obtained transport properties through SCLCPs in which liquid-crystalline domain orientation is modulated by externally applied electric field.     

Diffusion of condensable vapors in single adsorbent particles measured via concentration-swing frequency response

A concentration-swing frequency response method is extended to examine mass transfer mechanisms and the concentration dependence of mass transfer rates for adsorption of condensable vapors in single adsorbent particles. The adsorption kinetics of water and hexane in BPL activated carbon and the adsorption of water in silica gel are determined at several different concentrations. The mechanism that best describes the adsorption of water in BPL activated carbon is nanopore diffusion. The diffusivity of water in BPL activated carbon has a clear minimum at approximately P/Po = 0.5, and the concentration dependence of the diffusion data are not described well by the Darken relationship. Both nanopore diffusion and the Glueckauf linear driving force models can be used to describe the diffusion of hexane in BPL activated carbon for the pressure range studied, and the dependence of the diffusivity on concentration can be described approximately using the Darken relationship. However, the diffusion of water in silica gel cannot be described by the nanopore diffusion model and is best characterized by the Glueckauf linear driving force model. The results illustrate the ability of concentration-swing frequency response to accurately determine adsorption rate mechanisms and quantify the complex adsorption kinetics of condensable vapors using small quantities of adsorbent.     

Interdiffusion of solvent into glassy polymer films: a molecular dynamics study

Large scale molecular dynamics and grand canonical Monte Carlo simulation techniques are used to study the behavior of the interdiffusion of a solvent into an entangled polymer matrix as the state of the polymer changes from a melt to a glass. The weight gain by the polymer increases with time t as t(1/2) in agreement with Fickian diffusion for all cases studied, although the diffusivity is found to be strongly concentration dependent especially as one approaches the glass transition temperature of the polymer. The diffusivity as a function of solvent concentration determined using the one-dimensional Fick's model of the diffusion equation is compared to the diffusivity calculated using the Darken equation from simulations of equilibrated solvent-polymer solutions. The diffusivity calculated using these two different approaches are in good agreement. The behavior of the diffusivity strongly depends on the state of the polymer and is related to the shape of the solvent concentration profile.     

The permeation of organophosphorus compounds in silicone rubber membranes

The permeabilities, solubilities, and diffusivities of eight organophosphorus chemicals in silicone rubber were measured at saturation concentration using two different experimental methods: permeation experiments and absorption experiments. All tests were carried out at 25°C (±3°C). The eight organophosphorus chemicals investigated are dimethyl methylphosphonate, diethyl methylphosphonate, dimethyl hydrogenphosphonate, diethyl hydrogenphosphonate, trimethylphosphate, triethylphosphate, trimethylphosphite, and triethylphosphite. These eight chemicals were selected based on their similarities to organophosphorus chemicals used as pesticides and chemical warfare agents. The experimental data were analyzed using solutions of Fick's second law of diffusion and boundary conditions representative of the experimental settings. An unsteady-state diffusion model using boundary conditions that represent uniform initial concentration in the polymer and constant but different surface concentrations was used to interpret the permeation experimental data. In this model, the effective diffusivity calculated from the steady-state permeability and equilibrium solubility of each chemical was used and was assumed to be constant.     

Film-Pore Diffusion Control for the Batch Sorption of Cadmium Ions from Effluent onto Bone Char

The sorption equilibrium and kinetics of cadmium ions from aqueous solution onto bone char have been studied. Equilibrium isotherms for the sorption system were correlated by Langmuir and bi-Langmuir equations. The application of the bi-Langmuir equation was developed because the mechanistic analysis in this research indicated that cadmium removal occurs ion exchange and physical adsorption onto different surface sites. The bi-Langmuir equation provides a better fit to the experimental data. In addition, the removal rates of cadmium ions based on the Langmuir models have been investigated. The effective diffusivity was calculated using the effects of initial metal ion concentration and bone char mass. Two mass-transport models based on film-pore diffusion control have been applied to analyze the concentration decay curves. The film and pore diffusion coefficients using an analytical equation are equal to 1.26x10(-3) cm/s and 5.06x10(-7) cm(2)/s, respectively. The pore diffusion coefficient obtained from the numerical method is 4.89x10(-7) cm(2)/s. A sensitivity analysis showed that the film-pore diffusion model and constant effective diffusivity could be used to describe the mass-transport mechanism of the sorption system with a high degree of correlation. Copyright 2001 Academic Press.     

Molecular dynamics simulation of solvent-polymer interdiffusion: Fickian diffusion

The interdiffusion of a solvent into a polymer melt has been studied using large scale molecular dynamics and Monte Carlo simulation techniques. The solvent concentration profile and weight gain by the polymer have been measured as a function of time. The weight gain is found to scale as t(1/2), which is expected for Fickian diffusion. The concentration profiles are fit very well assuming Fick's second law with a constant diffusivity. The diffusivity found from fitting Fick's second law is found to be independent of time and equal to the self-diffusion constant in the dilute solvent limit. We separately calculated the diffusivity as a function of concentration using the Darken equation and found that the diffusivity is essentially constant for the concentration range relevant for interdiffusion.     

Inverse gas chromatography study on the effect of humidity on the mass transport of alcohols in an ethylene-vinyl alcohol copolymer near the glass transition temperature

Inverse gas chromatography (IGC) was used to study the effect of moisture on transport properties of three low molecular weight alcohols (methanol, ethanol, and 1-butanol) through high barrier copolymers of ethylene-vinyl alcohol with an ethylene content of 38%mol (EVOH38) at 40 degrees C. The value of the partition coefficient (K) was obtained by using two approaches: (a) the fit of the slope of sorption isotherms obtained through the method of Kiselev and Yashin; and (b) the solution to the model of Romdhane and Danner obtained by using the law of moments. The second method also allowed the estimation of the diffusion coefficient (D(p)) at the different humidity conditions. None of these two methods were applicable at low values of relative humidity. With the first method, the diffusion of the permeants through the copolymer was not fast enough to allow them to reach the core of the EVOH particles used as stationary phase resulting in sorption values unrealistically low. The fit of the chromatograms obtained by using the second method also suggested questionable values of the mass transport parameters. Although the theoretical curve perfectly described the chromatogram, the low extent of the interaction and the slow diffusion resulted in interdependent values of the coefficients K and D(p), with infinite pairs of values providing the same curve profile. As the relative humidity of the carrier gas increased, the diffusivity and the sorption of the alcohols also increased, making both methods applicable. In the case of the partition coefficient, the sorption of the biggest molecules (ethanol and 1-butanol) was the most affected, the increment of K for methanol being moderate. As regards the D(p) value, methanol was the most influenced compound and 1-butanol the least. Finally, a sharp increment of the D(p) of the three alcohols was observed between 35 and 47% RH and attributed to the plasticization of the copolymer.     

Which surfactants reduce surface tension faster? A scaling argument for diffusion-controlled adsorption

Consider the example of surfactant adsorbing from an infinite solution to a freshly formed planar interface. There is an implicit length scale in this problem, the adsorption depth h, which is the depth depleted to supply the interface with the absorbed surfactant. From a mass balance, h can be shown to be the ratio of the equilibrium surface concentration gamma eq to the bulk concentration C infinity. The characteristic time scale for diffusion to the interface is tau D = h2/D, where D is the diffusivity of the surfactant in solution. The significance of this time scale is demonstrated by numerically integrating the equations governing diffusion-controlled adsorption to a planar interface. The surface tension equilibrates within 1-10 times tau D regardless of bulk concentration, even for surfactants with strong interactions. Dynamic surface tension data obtained by pendant bubble method are rescaled using tau D to scale time. For high enough bulk concentrations, the re-normalized surface tension evolutions nearly superpose, demonstrating that tau D is indeed the relevant time scale for this process. Surface tension evolutions for a variety of surfactants are compared. Those with the smallest values for tau D equilibrate fastest. Since diffusion coefficients vary only weakly for surfactants of similar size, the differences in the equilibration times for various surfactant solutions can be attributed to their differing adsorption depths. These depth are determined by the equilibrium adsorption isotherms, allowing tau D to be calculated a priori from equilibrium surface tension data, and surfactant solutions to be sorted in terms of which will reduce the surface tension more rapidly. Finally, trends predicted by tau D to gauge what surfactant properties are required for rapid surface tension reduction are discussed. These trends are shown to be in agreement with guiding principles that have been suggested from prior structure-property studies.     

颗粒内传质参数作为工具用于催化剂颗粒的设计(英文)

A chromatographic method and a dynamic Wicke-Kallenbach method (DMWK) were used to determine the diffusion characteristics of two industrial copper containing catalysts. The first catalyst was used in nitrobenzene hydrogenation to aniline and the second was used in a low temperature water-gas shift reaction. Experimental results show that application of these two methods leads to similar results. Experimental data obtained allow for monitoring changes in the texture of the catalyst grains and intraparticle diffusivity of gaseous reagents at different states of the catalyst activity and use, which can be used as criteria for designing optimal industrial catalyst pellets.     

Sorption Kinetics and Intracrystalline Diffusion of Methanol in Ferrierite: An Example of Disguised Kinetics

Sorption kinetics of methanol in large crystals of ferrierite have been studied in detail by interference microscopy (IFM) and infra-red microscopy (IRM). The IFM measurements yield the transient concentration profiles, thus providing a direct measurement of both the surface resistance to mass transfer and the internal diffusion resistance. It is shown that, for this system, the uptake rate is controlled by the combined effects of surface resistance and diffusion through the 8-ring channels (in the y-direction). Transport through the 10-ring channels (in the z-direction) appears to be blocked by surface resistance. Although the overall uptake curves conform well to the “root t law” the diffusivity values derived from the uptake curves vary widely depending on the assumed direction of diffusion. Even if the correct direction of diffusion is assumed, the diffusivity values derived from the uptake curves are seriously in error as a result of the intrusion of surface resistance. The existence of transport resistances at the crystal surface is clearly apparent from the transient concentration profiles but is not obvious from the uptake curves.     

Diffusion of sodium dodecyl sulfate micelles in agarose gels

The gradient diffusion of ionic sodium dodecyl sulfate micelles in agarose gel was investigated at moderate concentrations above the CMC. Of particular interest were the effects of micelle, gel, and sodium chloride concentration on the micelle diffusivity. Holographic interferometry was used to measure the gradient diffusion coefficient at three sodium chloride concentrations (0, 0.03, 0.10 M), three gel concentrations (0, 1, 2 wt%), and several surfactant concentrations. Time-resolved fluorescence quenching was used to measure aggregation numbers both in solution and gel. The micelle diffusivity increased linearly with surfactant concentration at the two larger sodium chloride concentrations and all gel concentrations. In general, the strength of this effect increased with decreasing sodium chloride concentration and increased with gel concentration. This behavior is evidence of decreasing micelle-micelle electrostatic interactions with increasing sodium chloride concentrations, and increasing excluded volume effects and hydrodynamic screening with increasing gel concentration, respectively. The only exception was at 0.1M sodium chloride and 2 wt% agarose, which showed a slight reduction in the slope compared to 1 wt% agarose. It was found that the concentration effect is quite strong for charged solutes: at a NaCl concentration of 0.03 M in a 2% agarose gel, in a solution with 3% SDS micelles by volume, the micelle diffusion coefficient is doubled relative to its value in the same gel at infinite dilution. The extrapolated, infinite-dilution diffusion coefficients and the rate at which the micelle diffusivity increased with surfactant concentration were compared with predictions of previously published theories in which the micelles are treated as charged, colloidal spheres and the gel as a Brinkman medium. The experimental data and theoretical predictions were in good agreement.     

Transport Properties of Ethane, Butanes and Their Binary Mixtures in MFI-Type Zeolite and Zeolite-Membrane Samples

Transport properties of ethane, butane and their binary mixtures in large crystals of silicalite-1, ZSM-5, and an MFI-zeolite membrane as well as agglomerates (pellets) of silicalite zeolites have been investigated by the Zero Length Column (ZLC) method. It was found that in the large crystals of silicalite-1 and ZSM-5, and in the membrane sample desorption of iso-butane was controlled by micropore diffusion, while in the case of pelleted silicalite sample it was controlled by macropore diffusion. The effective thickness of the zeolite membrane can be reasonably evaluated by comparing the diffusivity data obtained from the ZLC and gas permeation measurements. Desorption of ethane and n-butane in the large crystals of silicalite-1 and ZSM-5 and the membrane sample is attributed to both equilibrium effects and micropore diffusion. The diffusivity of ethane is significantly reduced in the presence of iso-butane giving rise to a micropore diffusion-controlled process. Furthermore, diffusion of iso-butane in the zeolite samples is affected by the counter flow of ethane.     

Diffusion of methyl ethyl ketone in polyisobutylene: Comparison of spectroscopic and gravimetric techniques

In this study, vapor sorption FTIR–ATR (Fourier Transform Infrared Attenuated Total Reflectance) spectroscopy was combined with a conventional gravimetric sorption balance to examine diffusion in polymers. Mutual diffusion coefficients of methyl ethyl ketone in polyisobutylene were measured using both methods at various penetrant activities and temperatures in the range 40–60°C. Actual penetrant concentrations were determined from the sorption balance. The diffusion coefficients from the two techniques agree very well with each other. In addition, the diffusivity data from both techniques could be correlated successfully as a function of temperature and concentration with the Vrentas and Duda free-volume model. © 1997 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 35 : 1261–1267, 1997     

Statistical theory of turbulent mass transfer in electrochemical systems

The paper presents an overview of the statistical theory of turbulent mass transfer in electrochemical systems and some new results which generalize the previously obtained relations for the flows of complex geometry. The developed theory does not use traditional semi-empirical hypotheses and analogies, but directly addresses to the solving of the critical for turbulent transfer the closure problem. The mathematical procedure for solving of the closure problem makes use of new equations for the correlations between concentration and velocity fluctuations in different space points and at different time moments; the dumping of turbulent pulsations in the viscous sublayer allows to neglect high order moments and obtain a closed equation for the turbulent mass flux. In general, the relation between the turbulent mass flux and the mean concentration gradient is non-local. Using available experimental information, the non-local equation for the turbulent mass flux is reduced to the traditional local one and the functional form of the turbulent diffusion coefficient is obtained. It is demonstrated that Reynolds analogy cannot been used for the prediction of the turbulent diffusivity. Applications of the developed theory to chemical engineering and to electrochemical flow diagnostics (prediction of flow characteristics using limiting diffusion current measurements) are discussed.     

Solubility,diffusivity, and mobility of n-pentane and ethanol in poly(1-trimethylsilyl-1-propyne)

The diffusion coefficient of ethanol and of n-pentane in PTMSP, at 27°C, was measured as a function of concentration up to a penetrant content of about 12% by weight, for polymer samples obtained through different processes; differential sorptions and desorptions with vapor phases were considered. In the case of ethanol a nonmonotonous behavior was observed for the diffusivity, while in the case of n-pentane the same property was found to monotonously decrease with increasing the penetrant content. The sorption isotherms were also reported, indicating that n-pentane exhibits a typical dual mode behavior, while ethanol follows an unusual s-shape curve. The chemical potential of the dissolved penetrants, calculated directly from the isotherms, shows the very different importance of the energetic interactions of the two penetrants with the polymer units. In spite of the remarkably different concentration dependencies observed for both solubility and diffusivity of the two penetrants, the mobility factors are in both cases monotonously decreasing with the penetrant concentration, and follow very similar trends. The significant differences observed for the concentration dependence of the diffusion coefficients are, thus, associated to the thermodynamic contributions, which are very different for n-pentane and ethanol. Different polymeric films, obtained through different solvent evaporation processes, show quite different solubility, diffusivity and mobility for both ethanol and n-pentane. On the other hand, the ratio between the mobility of the two penetrants as well as the slope of mobility as function of the concentration remains the same for all the different samples inspected. © 1997 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 35 : 2245–2258, 1997     

Kinetics and Equilibrium Swelling Properties of Hydrophilic Polymethacrylate Networks

Summary: Hydrogels were synthesized by polymerization of 2-Hydroxy ethylmethacrylate (HEMA) in the presence of ethyleneglycoldimethacrylate (EGDMA) as a crosslinking agent. Structural information and thermophysical properties of the hydrogels were analyzed using Fourier-Transform Infrared spectroscopy, thermogravimetrical analysis, and differential scanning calorimetry. The swelling behaviour of the obtained chemically crosslinked P(HEMA-EGDMA) networks in aqueous solution was investigated as a function of the pH value and concentration of crosslinking agent. Plateau values were found at equilibrium swelling for a low pH value after one day swelling, whereas increasing water uptake was obtained for pH = 6.32 even at swelling times of more than five days. For short swelling times, a linear relationship between swelling ratio and time was found. Experimental data were rationalized using Fick's second law of diffusion. For early and moderate times of diffusion, threshold values were found in all cases considered here, indicating a Fickian behaviour below and a non-Fickian diffusion mechanism above the threshold.     

Protein adsorption on novel acrylamido-based polymeric ion exchangers. II. Adsorption rates and column behavior

Uptake kinetics and breakthrough behavior were determined for bovine serum albumin (BSA) and alpha-chymotrypsinogen (alphaCHY) in new polymeric ion-exchange media based on acrylamido monomers. Two anion exchangers and a cation exchanger were investigated. As shown in Part I of this work, the two anion exchangers have different morphologies. The first one, BRX-Q, comprises a low-density gel with a matrix of denser polymeric aggregates. While this material has a very low size-exclusion limit for neutral probes, it exhibits an extremely high binding capacity for BSA. The second anion exchanger, BRX-QP, comprises large open pores but has a very low binding capacity. The cation exchanger, BRX-S, also comprises large open pores but exhibits an intermediate capacity; likely as a result of the presence of smaller pores. Dynamic protein uptake experiments showed that the highest mass transfer rates are obtained with BRX-Q. The apparent diffusivity is also highest for this material and increases substantially as the protein concentration is reduced. For these particles, the external film resistance is dominant at very low protein concentrations. Much lower rates and apparent diffusivities are obtained for BRX-QP. Finally intermediate rates and apparent diffusivities are found with BRX-S. The concentration dependence of the apparent pore diffusivity is much less pronounced in this case. The apparently paradoxical result that mass transfer rates are highest for the material with the smallest neutral-probe size-exclusion limit can be explained in terms of a general conceptual model where parallel pore and adsorbed-phase diffusion paths exist in these particles. In the first case, adsorbed phase diffusion in gel pores is dominant, while in the second transport is dominated by diffusion in a macroporous network. In the third case, both contributions are important. The conceptual model provides an accurate prediction of the breakthrough behavior of columns packed with these media using independently determined rate parameters. Dynamic binding capacities of 80-140 mg/ml were observed for BSA on BRX-Q in ca. 1.5 cm columns operated at 300-900 cm/h in agreement with theoretical predictions.