The transport of adsorbate mixtures in porous materials: Basic equations for pores with simple geometry

The basic equations governing the transport of single and binary adsorbate mixtures through single pores are considered. An irreversible thermodynamic formulation is adopted and both viscous and diffusive terms are incorporated following the earlier work of Mason and co-workers. The links between phenomenological coefficients and molecular properties are demonstrated. For single components, the gas phase and high density limits are considered. By using simple hydrodynamic models it is shown that the phenomenological coefficients in the mixture equations can all be expressed as functions of the coefficients for the individual components in the same pore, and the properties of the component adsorption isotherms. Whilst it is appreciated that the hydrodynamic approach will be of limited value in very small pores, it is argued that useful insights can be gained into the feasibility of membrane separation processes from this method. The general equations can be used in future development of network models for porous materials.     

Solute molecular transport through polyimide asymmetric organic solvent nanofiltration (OSN) membranes and the effect of membrane-formation parameters on mass transfer

This work studies the effect of two membrane-formation parameters, evaporation time and casting thickness, on the diffusive mass transport of organic solutes through an organic solvent nanofiltration (OSN) membrane. These parameters showed a coupled effect on the final membrane thickness, which was explained in terms of top-layer formation. In a concentration-driven dialysis, both parameters, as well as the resulting membrane thickness, had a significant effect on mass transport. Casting thickness had the greatest effect on membrane mass transport rates. Multivariate regression was used to model the dialysis process with acceptable fit. A representation of the membrane morphology was obtained from SEM pictures and used to formulate an alternative mechanistic mass-transport model. A resistance-circuit analogy was used to describe transport through the top and microporous layers, which also considered diffusion through the pores and the polymer for each layer. From the analyses of the models and considering that no differences in top-layer thickness were observed by SEM, it is concluded that membrane asymmetry, determined by the formation parameters, controls mass transport, rather than top-layer thickness.     

Theoretical effects of macromolecule concentration and charge on membrane rejection coefficients

A hydrodynamic model for the convection of rigid, spherical solutes through cylindrical pores, which includes both steric and electrostatic interactions between pairs of solute particles and between solutes and the pore wall, has been developed to examine the effects of solute concentration and charge on solute rejection by membrane pores during ultrafiltration. Calculations have been performed for a wide range of charge conditions and the results are presented in terms of the membrane rejection coefficient at infinite dilution and a correction factor which accounts for the first-order effects of concentration. For pores and solutes of like charge, the rejection coefficient is predicted to decrease with increasing feed concentration or ionic strength.     

A semi-empirical approach for predicting the performance of mixed matrix membranes containing selective flakes

A successful model for mixed matrix membrane performance must address the complex geometry of the problem and accurately treat the diffusion behavior of the host–guest systems being considered. Detailed calculations based on the Maxwell–Stefan equations provide a widely accepted means of treating the diffusion of gases within zeolites. However, a full numerical solution of these equations for a complex mixed matrix membrane geometry does not offer the convenience and transparency that comes with an analytical treatment. At the same time, existing analytical equations which were formulated specifically to address mixed matrix geometry do so under the assumption of very simplistic models for diffusion. Here, an approach is presented for predicting the permeability and selectivity of mixed matrix membranes containing zeolite flakes that combines well-known analytical expressions for mixed matrix membrane performance with Maxwell–Stefan modeling for zeolite diffusion. The constant permeabilities required by the analytical models are calculated by the Maxwell–Stefan equations as a function of operating conditions, and these calculated effective permeabilities are used to predict mixed matrix membrane performance at corresponding operating conditions. The method is illustrated through two case studies: normal- and iso-butane separation by a membrane containing silicalite-1 flakes and carbon dioxide/methane separation by membranes containing CHA-type zeolites. Predictions are compared to experimental results found in the literature for both cases. Also, the applicability of the Maxwell and Cussler analytical models for mixed matrix membrane performance is explored as a function of flake loading and aspect ratio.     

Translational hydrodynamics for binary mixtures: Memory from internal molecular relaxation

We consider a binary fluid mixture made up of a monatomic species and a two level molecular species and investigate the decay of correlations in the hydrodynamic variables. By restricting our considerations to the translationally hydrodynamic situations, we deduce a set of generalised hydrodynamic equations in which the frequency, ω, and wavenumber, k, dependent memory arises solely from the internal molecular relaxation. From these generalised hydrodynamic equations, we identify various (k, ω) dependent transport coefficients and the relaxation time which depends on k. We also compute the spectrum of (polarised) light scattered from such a binary mixture. We illustrate the dependence on k, ω and concentration in the problem by a numerical calculation of the generalised transport coefficients, the Rayleigh-Brillouin spectrum and the dispersion of various modes in this spectrum for the case of thermal relaxation in a parahydrogen-helium mixture.     

Membrane potential in charged porous membranes

For charged porous membranes, the separation efficiency to charged particles and ions is affected by the electrical properties of the membrane surface. Such properties are most commonly quantified in terms of zeta-potential. In this paper, it is shown that the zeta-potential can be calculated numerically from the membrane potential. The membrane potential expression for charged capillary membranes in contact with electrolyte solutions at different concentrations is established by applying the theory of non-equilibrium thermodynamic to the membrane process and considering the space-charge model. This model uses the Nernst–Planck and Navier–Stokes equations for transport through pores, and the non-linear Poisson–Boltzmann equation, which is numerically solved, for the electrostatic condition of the fluid inside pores. The integral expressions of the phenomenological coefficients coupling the differential flow (solute relative to solvent) and the electrical current with the osmotic pressure and the electrical potential gradients are established and calculated numerically. The mobilities of anions and cations are individually specified. The variations of the membrane potential (or the apparent transport number of ions in the membrane pores) are studied as a function of different parameters: zeta-potential, pore radius, mean concentration in the membrane, ratio of external concentrations and type of ions.     

Drug transport in responding lipid membranes can be regulated by an external osmotic gradient

In this paper, we demonstrate, for the first time, how an external osmotic gradient can be used to regulate diffusion of solutes across a lipid membrane. We present experimental and theoretical studies of the transport of different solutes across a monoolein membrane in the presence of an external osmotic gradient. The osmotic gradient introduces phase transformations in the membrane, and it causes nonlinear transport behavior. The external gradient can thus act as a kind of switch for diffusive transport in the skin and in controlled release drug formulations.     

Electrohydrodynamic transport of non‐symmetric electrolyte through porous wall of a microtube

Transport of salt through the wall of porous microtube is relevant in various physiological microcirculation systems. Transport phenomena based modeling of such system is undertaken in the present study considering a combined driving force consisting of pressure gradient and external electric field. Transport of salt is modeled in two domains, in the flow conduit and in the pores of porous wall of the microtube. The solute transport in the microtube is presented by convective‐diffusive mass balance and it is solved using integral method under the framework of boundary layer analysis. The wall of the microtube is considered to be consisting of series of straight parallel cylindrical pores with charged inner surface. The solute transport through the pores is considered to be composed of diffusive, convective and electric potential gradient governed by Nernst‐Planck equation. Transport in the microtube and pores is coupled through the osmotic pressure model for the solvent and Donnan equilibrium distribution for the solute. The simulated results agree remarkably well with the experimental data conducted by in‐house experimental set up. The charge density of the porous wall is estimated through the minimization of errors involved between the experimental and simulated data for different operating conditions.     

The special features of molecular transport in nanosized channels

The molecular theory of the transport of pure substances and mixtures of molecules of different shapes in narrow slit-like pores, in which the potential of surface forces creates a strongly anisotropic distribution of molecules across pores and thereby makes the hydrodynamics equation inapplicable, is considered. The new microhydrodynamic approach is based on the lattice gas model, which takes into account the intrinsic volume of molecules and intermolecular interactions in the quasi-chemical approximation. Self-consistent calculations of dissipative coefficients taking nonlocal fluid properties into account were performed on the basis of the transition state model including information about equilibrium adsorbate distribution. Changes in fluid concentrations from the gaseous to liquid state and a broad temperature range, including the critical region, are analyzed. This allows vapor, liquid, and vapor-liquid fluid flows to be considered in the presence of capillary condensation. An increase in the size of pores transforms the equations of the theory into hydrodynamic transfer equations for gas or liquid flows, while preserving the relation of transfer coefficients to intermolecular potentials. The use of microhydrodynamic approach equations in numerical calculations and the possibility of applying this approach are discussed.     

Diffusive flows in ultrafiltration and their effect on membrane retention properties

In modelling the retention of ultrafiltration membranes, diffusive fluxes across the membrane have usually been neglected, mainly due to evidence derived from using symmetric track-etched membranes. The present paper reexamines this matter specifically for the case of “real” asymmetric membranes. A critical literature review on the use of irreversible thermodynamic (IT), hydrodynamic and Stefan-Maxwell (S-M) models is presented. It is shown that all three approaches yield the same basic retention equation for the case of non-negligible diffusive solute-flux. It is also shown that, for membranes with a coefficient close to one, a much simpler equation gives results almost identical to the more rigorous basic equation just mentioned. Furthermore an overview of available literature data indicates that diffusive fluxes do play a non-negligible role in the functioning of asymmetric ultrafiltration membranes. Further work is needed in order to predict solute transport properties in hydrodynamic terms. A simplified Stefan-Maxwell approach seems the appropriate tool for future work in studying multicomponent solutions.     

Poly(styrene sulfonic acid)–poly(vinylidene fluoride) interpolymer ion-exchange membranes. II. Ultrafiltration properties

Highly porous interpolymer ion-exchange membranes of poly(styrene sulfonic acid) and poly(vinylidene fluoride) have been investigated under pressure filtration with KCI, Na₂SO₄, erythrosin, and bovine serum albumin as solutes in the feed solution. The rejection of the ionic solutes is governed by a Donnan exclusion of electrolyte from the membrane phase. A model for the transport behavior is proposed that includes both diffusive and convective salt transport. The calculated rejections agree adequately with the observed data.     

Factors influencing the transport of short-chain alcohols through mesoporous gamma-alumina membranes

The pressure-driven transport of water, ethanol, and 1-propanol through supported gamma-alumina membranes with different pore diameters is reported. Water and alcohols had similar permeabilities when they were transported through gamma-alumina membranes with average pore diameters of 4.4 and 6.0 nm, and the permeability coefficient was found to be proportional to the square of pore size, in accordance with a viscous flow mechanism. For transport through membranes with an average diameter of 3.2 nm, the behavior of water was in accordance with the viscous flow mechanism, but the permeability of the membrane for ethanol and 1-propanol was much smaller than expected and could not be explained in terms of viscous flow. Although the low permeability of the membrane with 3.2 nm pores for ethanol and 1-propanol was partly due to the presence of small amounts of water in the alcohols, the permeability coefficients were still substantially smaller when water was absent. This intrinsic difference between water and alcohol may be due to differences in molecular size, chemisorption of alcohols on the oxide pore wall, which would lead to a reduction of the effective pore size, and/or a certain degree of translational ordering of the alcohol molecules inside the membrane pores, which leads to an effectively higher viscosity and, therefore, to a higher transport resistance.     

Pervaporation of benzene/cyclohexane mixtures using ion-exchange membrane containing copper ions

The sorption, diffusion, and pervaporation (PV) properties of benzene/cyclohexane (Bz/Cx) mixtures on cation-exchange membranes containing copper ions (Cu(II)) were investigated. The equilibrium sorption isotherms of pure vapors in the membranes and the partial solubility of binary solutions in the membranes were described using the UNIQUAC model. The τ_iM and τ_Mi values were 0.978 and 0.591 for Bz, and 0.922 and 0.475 for Cx. The transient regimes of vapor sorption were employed to calculate the concentration-dependent diffusion coefficients. Long’s model sufficiently explained the diffusivity of Bz and Cx in the membranes. The pre-exponential factors were 3×10⁻¹³ m²/s and the plasticization factors were 3.0 and 3.6 for Bz and Cx, respectively. Excellent agreement was found with the experimental results applying the solubility and diffusivity data to simulate the pervaporation performance (flux and selectivity) using the modified Maxwell–Stefan equation. The membrane containing Cu(II) demonstrates better facilitating capability for Bz transport than that with Na(I), mainly due to its preferential sorption property toward Bz. Replacing Na(I) with Cu(II) into a Neosepta membrane resulted in better separation efficiency and higher Bz flux throughout the entire Bz concentration range.     

Computer modelling of adsorbate transport in nanospaces

The passage of adsorbed molecules, through pores of nanometre dimensions, under a concentration gradient, is important in several processes. The basic equations for single-component flow in pores identify a diffusive and cooperative (viscous) component to the flux. Three simulation techniques, including non-equilibrium molecular dynamics (NEMD), have been used to investigate (spherical) methane and ethane adsorbates in model graphite pores at ambient temperatures. The NEMD method measures flux directly and shows an interesting behaviour in the total diffusion coefficients, including transitions and values substantially in excess of the Darken diffusion coefficient, calculated from self-diffusion. A simple Stokes–Einstein type of model can account for some results. The instances where this model fails can be rationalised in terms of confinement effects, and the relative contribution of kinetic energy to the Hamiltonian.     

Polylactide microspheres prepared by premix membrane emulsification—Effects of solvent removal rate

Polylactide microspheres were prepared by pre-mix membrane emulsification and subsequent extraction of solvent in a coagulation bath, and ultimately to the gas phase. The polymer was dissolved in dichloromethane and emulsified with water or water–methanol mixtures by repeated passage through a glass membrane. During and after emulsification, the droplets are exposed to a bath consisting of a mixture of water and methanol. Transfer of dichloromethane takes place into the bath and (subsequently) to the gas phase. Compared to water, the solubility of dichloromethane is increased when using water–methanol mixtures; the continuous phase can quickly dissolve a significant amount of the solvent, while transfer to the gas phase is strongly enhanced as well. This was observed experimentally and by computer simulation, using a combined model based on the Maxwell–Stefan theory for non-ideal, multi-component mass transfer.

With increasing methanol concentration, the size and span of the microspheres became smaller, and was approximately 1 μm at 30% methanol. The surface morphology of these particles was solid and smooth, whereas holes were observed in those prepared in pure water. At methanol concentrations higher than 30%, the size of the microspheres increased again. This is probably due to the swelling of the particles because of the high in-diffusion of methanol which increases the porosity of the particles. Our main conclusion is that particles of defined size and size distribution can be produced by simply adjusting the non-solvent composition of the pre-mix.     

The effect of membrane potential on the development of chemical osmotic pressure in compacted clay

When clay soils are subjected to salt concentration gradients, various interrelated processes come into play. It is known that chemical osmosis induces a water flow and that a membrane potential difference develops that counteracts diffusive flow of solutes and osmotic flow of water. In this paper, we present the results of experiments on the influence of membrane potential on chemical osmotic flow and diffusion of solutes and we show how we are able to derive the membrane potential value from theory. Moreover, the simultaneous development of water pressure, salt concentration and membrane potential difference are simulated using a model for combined chemico-electroosmosis in clays. A new method for short-circuiting the clay sample is employed to assess the influence of electrical effects on flow of water and transport of solutes.