Liquid-supported membranes in chromium(VI) optical sensing: transport modelling

Flat sheet liquid-supported membranes (FSLSM) containing Aliquat 336 as a carrier have been evaluated as sample interface in an optical sensor for Cr(VI) monitoring. A model describing the transport mechanism of Cr(VI) through the membrane is reported. The model considers a diffusion process through a feed aqueous diffusion layer, a fast interfacial chemical reaction and a diffusion of ALQHCrO₄ and (ALQ)₂CrO₄ species through the membrane (Aliquat 336, ALQ). The mathematical equations describing the transport rate are derived and they correlate the membrane permeability coefficient to diffusional and equilibrium parameters as well as to the chemical composition of the system, i.e. extractant concentration in the membrane phase and acidity in the feed phase. The experimental data are explained by the derived equations and the diffusion resistances to mass transfer are evaluated. The influence of other experimental parameters, such as stirring speed in the feed phase and nature of the diluent and stripping agent on the transport is also discussed. Experiments with optical detection demonstrate the suitability of liquid-supported membranes (LSM) containing ALQ as interfaces for optical sensing.     

Investigation of gas transport through porous membranes based on nonlinear frequency response analysis

Theoretical development of a new experimental method for investigation of mass transport in porous membranes, based on the principle of the modified Wicke-Kallenbach diffusion cell and the nonlinear frequency response analysis is presented. The method is developed to analyze the transport of a binary gas mixture in a porous membrane. The mixture is assumed to consist of one adsorbable and one inert component. Complex mass transfer mechanism in the membrane, where bulk or transition diffusion in the pore volume and surface diffusion take place in parallel, is assumed. Starting from the basic mathematical model equations and following a rather standardized procedure, the frequency response functions (FRFs) up to the second order are derived. Based on the derived FRFs, correlations between some characteristic features of these functions on one side, and the whole set of equilibrium and transport parameters of the system, on the other, are established. As the FRFs can be estimated directly from different harmonics of the measured outputs, these correlations give a complete theoretical basis for the proposed experimental method. The method is illustrated by quantifying the transport of helium (inert gas) and C₃H₈ and CO₂ (adsorbable gases) through a porous Vycor glass membrane.     

Thermodynamics of ions and water transport in porous media

The thermodynamic framework of Prigogine, de Groot, and Mazur is extended to study the transport of ions and water in thermoporoelastic materials assuming infinitesimal deformations. New expressions are developed for the first and second principles of nonequilibrium thermodynamics of multicomponent systems and a generalized power balance equation is derived. For porous materials, all the components cannot be treated on a symmetric basis. A Lagrangian framework associated with deformation of the solid phase is introduced and, in this framework, Curie's principle is used to set up the form of the linear constitutive equations describing the transport of ions, water, and heat through the pore network. The material properties entering these equations were recently obtained by Revil and Linde [J. Colloid Interface Science 302 (2006) 682-694] using a volume-averaging approach based in the Nernst-Planck and Stokes equations. This provides a way to relate the material properties entering the constitutive equations to two textural parameters characterizing the topology of the pore space of the material (namely the tortuosity of the pore space and the permeability). The generalized power balance equation is used to derive the linear poroelastic constitutive equations (including the osmotic pressure) to describe the reversible contribution of deformation of the medium in response to ions and water transport through the connected porosity.     

Design-Oriented Modelling of Different Quenching Solutions in Induction Plasma Synthesis of Copper Nanoparticles

The aim of this paper is to compare the effects of different mechanisms underlying the synthesis of copper nanoparticles using an atmospheric pressure radio-frequency induction thermal plasma. A design oriented modelling approach was used to parametrically investigate trends and impact of different parameters on the synthesis process through a thermo-fluid dynamic model coupled with electromagnetic field equations for describing the plasma behaviour and a moment method for describing nanoparticles nucleation, growth and transport. The effect of radiative losses from Cu vapour on the precursor evaporation efficiency is highlighted, with occurrence of loading effect even with low precursor feed rate due to the decrease in plasma temperature. A method to model nanoparticle deposition on a porous wall is proposed, in which a sticking coefficient is employed to model particle sticking on the porous wall used to carry a quench gas flow into the chamber. Two different reaction chamber designs combined with different quench gas injection strategies (injection through a porous wall for “active” quenching; injection of a shroud gas for “passive” quenching) are analysed in terms of process yield and size distribution of the synthetized nanoparticles. Conclusion can be drawn on the characteristics of each quenching strategy in terms of throughput and mean diameter of the synthesized nanoparticles.     

The rheology of pseudoplastic fluids in porous media using network modeling

This paper considers the rheology of pseudoplastic (shear thinning) fluids in porous media. The central problem studied is the relationship between the viscometric behavior of the polymer solution and its observed behavior in the porous matrix. In the past, a number of macroscopic approaches have been applied, usually based on capillary bundle models of the porous medium. These simplified models have been used along with constitutive equations describing the fluid behavior (usually of power law type) to establish semiempirical macroscopic equations describing the flow of non-Newtonian fluids in porous media. This procedure has been reasonably successful in correlating experimental results on the flow of polymer solutions through both consolidated and unconsolidated porous materials. However, it does not allow an interpretation of polymer flow in porous media in terms of the flows on a microscopic scale; nor does it allow us to predict changes in macroscopic behavior resulting from variations at a microscopic level in the characteristics of the porous medium such as pore size distribution. In this work, we use a network approach to the modeling of non-Newtonian rheology, in order to understand some of the more detailed features of polymjer flow in porous media. This approach provides a mathematical bridge between the behavior of the non-Newtonian fluid in a single capillary and the macroscopic behavior as deduced from the pressure drop-flow rate relation across the whole network model. It demonstrates the importance of flow redistribution within the elements of the capillary network as the overall pressure gradient varies. As an example of a pseudoplastic fluid in a porous medium, we consider the flow of xanthan biopolymer. This polymer is important as a displacing fluid viscosifier in enhanced oil recovery applications and, for that reason, a considerable amount of experimental data has been published on the flow of xanthan solutions in various porous media.     

Estimation-identification problem for diffusive transport in porous materials based on single reservoir test: results for silica hydrogel

In this paper the problems of selection of an appropriate model of diffusive transport in silica hydrogel and estimation of model parameters are considered. The analytical solutions and simulations of diffusive transport in a single reservoir test with equilibrium model of sorption are developed. The mathematical models refer to transport of solute from a porous material to water or in reverse direction assuming arbitrary initial concentrations and Dirichlet or mixed type of boundary conditions. In order to quantify the relative importance of the model parameters, logarithmic sensitivity analysis of solute concentration with respect to estimated parameters describing diffusion, sorption, and interfacial effects is performed. The application of the discussed models for estimation of transport parameters is discussed using experimental results for diffusion of sodium chloride in silica hydrogel.     

Structural and mathematical models for pressure-dependent electrodiffusion of an electrolyte through heterogeneous ion-exchange membranes: Pressure-dependent electrodiffusion of NaOH through the MA-41 anion-exchange membrane

Structural and mathematical models are proposed for describing electrolyte transport through heterogeneous anion-exchange membranes under conditions of pressure-dependent electrodiffusion. The idea that mesopores and macropores are present in the membrane provides the basis for the structural model. The Nernst-Planck equations with a convective term are used to describe ion transport in the solution filling the pores. Results of the solution to the mathematical problem and the experimental investigations demonstrate the possibility of decreasing the transport numbers of sodium ions through an anion-exchange membrane by applying a pressure gradient in the same direction as the electrolyte diffusion flux in the membrane.     

Mathematical models for estimating effective diffusion parameters of spherical drug delivery devices

Mathematical modeling of drug delivery is of increasing academic and industrial importance in many aspects. In this paper, we propose an optimization approach for the estimation of the parameters characterizing the diffusion process of a drug from a spherical porous polymer device to an external finite volume. The approach is based on a nonlinear least-squares method and a novel mathematical model which takes into consideration both boundary layer effect and initial burst phenomenon. An analytical solution to the model is derived and a formula for the ratio of the mass released in a given time interval and the total mass released in infinite time is also obtained. The approach has been tested using experimental data of the diffusion of prednisolone 21-hemisuccinate sodium salt from spherical devices made of porous poly(2-hydroxyethyl methacrylate) hydrogels. The effectiveness and accuracy of the method are well demonstrated by the numerical results. The model was used to determine the diffusion parameters including the effective diffusion coefficient of the drug from a series of devices that vary in both the porous structure and the drug loading levels. The computed diffusion parameters are discussed in relation to the physical properties of the devices.     

Hydrothermal Synthesis of Porous Al2O3/Al Metal Ceramics: II. Mechanism of Formation of a Porous Al(OH)3/Al Composite

The formation of a porous layer of aluminum hydroxide on the surface of aluminum particles and the aggregation of Al(OH)₃/Al composite particles were analyzed theoretically. It was found that the diffusion mass transfer of the hydroxo complexes of aluminum through the porous structure of a growing layer of aluminum hydroxide to the outer surface is a rate-limiting step in the synthesis of the porous composite. A model mechanism of formation of the porous composite was developed, and rate equations were derived for describing the growth of an aluminum hydroxide layer on the surface of an aluminum particle and changes in the degree of aluminum conversion and the contact radius between composite particles. Based on the developed mathematical model and experimental data, the diffusion coefficient of the hydroxo complexes of aluminum in the porous structure of aluminum hydroxide was calculated.     

Filtration in Nonuniform Capillary–Porous Media: Equations of the Statistical Theory

Relationships between the statistical distribution of the filtration coefficient in a nonuniform capillary–porous medium and its total value for a filter layer are obtained. A closed set of equations where higher-order correlation moments of statistical fields serve as transferable entities is derived. The approach developed in this study makes it possible to study detailed laws of filtration in a nonuniform capillary–porous medium using the data on the filtration coefficient statistics, which is determined by the structure of the porous space.     

Rational surface design for molecular dynamics simulations of porous polymer adsorbent media

The construction and use of nonflat agarose surfaces in a simulation box, together with the employment of criteria for the immobilization of a set of dextran polymer chains on the nonflat agarose surfaces whose mathematical physics is compatible with that of the criteria used for the immobilization of the same set of dextran polymer chains on flat agarose surfaces, are shown to generate, through the use of molecular dynamics simulations whose simulation box has linear dimensions along the lateral directions that are the same when flat and nonflat agarose surfaces are used, dextran porous polymer structures whose pore sizes at the outermost surface and in the vicinity of the outermost surface of the porous medium can be controlled by an indirect manner through the variation of the parameters that characterize the nonflat surface. The use of a nonflat surface for the generation of desired large pores requires only a small or modest increase in the number of solvent molecules in the simulation box, while the use of a flat surface for the construction of the same desired large pores requires significant increases in the size of the linear dimensions of the flat surface. This increases so substantially the number of solvent molecules that the computational loads become intractable. The results in this work show that through the use of nonflat surfaces porous dextran polymer layers having pores of desired sizes can be effectively constructed, and this approach could be used for the design and construction of polymer-based porous adsorbent media that could effectively facilitate the transport and adsorption of an adsorbate biomolecule of interest that must be separated from a mixture of components. A useful definition about the properties that a porous polymer structure must have in order to become, for an adsorbate biomolecule of interest of known molecular size, a useful adsorbent medium, is presented and is used to (1) evaluate the porous polymer structures generated through the employment of different nonflat surface models and (2) determine and select the nonflat surface model from a set of nonflat surface models that is effective in producing promising porous structures. Then a procedure is presented by which a set of porous polymer media is generated through the use of the selected nonflat surface model, and the desired porous structure from this set is determined and could be considered to be used for the transport and immobilization of the selected affinity groups/ligands and the subsequent transport and adsorption of the desired to be separated adsorbate.     

Ionic transport in porous media for high zeta potentials

Ionic transport coefficients are numerically calculated in various porous media for high zeta potentials zeta by solving the Poisson-Boltzmann, the Stokes, and the convection-diffusion equations on the pore scale. Theoretical formulae for the simple case of a plane Poiseuille flow are derived in this limit. These formulae, when expressed in terms of the characteristic length Lambda, provide an excellent estimate of the ionic transport coefficients whatever the porous medium and whatever the value of zeta.     

Molecular weight distributions in radiation-induced polymerization. IV. A comprehensive kinetic scheme for polymerization in the liquid state

A comprehensive kinetic scheme is proposed which takes into consideration γ-ray-initiated polymerization via free radical, cationic, or radical-cationic species, or their combinations, and the relative contribution of each depending upon experimental conditions. From this kinetic model, equations describing the kinetics of polymerization and the resulting molecular weight distributions have been derived. The resulting expressions are complex in nature, particularly for the case where a combination of mechanisms is involved. In view of this, a general approach to the problem of generating theoretical molecular weight distributions based on the application of high speed digital computers has been presented. The proposed technique is particularly powerful since no simplifying assumptions or mathematical approximations are required in order to obtain the molecular weight distribution.     

Emulsion polymerization: Theory of particle size distribution in copolymerization system

A mathematical model for describing the particle size distribution (PSD) in emulsion copolymerization system is developed by analogy to that in emulsion homopolymerization system as proposed by Lichti and co-workers. By use of the appropriate combinations of the kinetic parameters of the comonomers, the complicated equations for copolymerization systems can be reduced to simpler equations identical to those of homopolymerization systems. The two calculation examples, styrene–methyl methacrylate and styrene-butadiene systems, are given to demonstrate the applicability of the proposed theory. The conditions for producing bimodal PSD from a seeded emulsion polymerization are discussed.     

Simulation of suspension separation by two-stage pressure-head flotation in a cylindrical direct-flow hydrocyclone

A mathematical model of separation of suspensions with a non-Newtonian dispersion medium by two-stage pressure-head flotation in a cylindrical direct-flow hydrocyclone was developed. A system of differential equations describing convective diffusion and motion of a particle-bubble complex was solved numerically. The concentration field was modeled and integral separation parameters were determined.     

Hydrodynamics of a porous sphere molecule

A mathematical model is proposed which treats a hypothetical polymer molecule as a porous sphere. Fluid flow within the sphere obeys Darcy's law while the creeping motion equations are used outside the sphere. Equations are derived relating the permeability and radius of the sphere to hydrodynamic properties of dilute solutions of the polymer. In checking against experimental data, it is found that the model, despite its simplicity, may be useful in explaining the hydrodynamic behavior of molecules which are highly branched or crosslinked.     

Dependence of the Self-Diffusion Coefficient of Liquid Molecules in a Porous Medium on Its Geometric Parameters

Based on the most general concepts of the translational mobility and geometry of a porous medium, an expression is derived for the self-diffusion coefficient of liquid molecules in such a medium. An analytical relation between the self-diffusion coefficient and the effective geometric parameters of the pores, as well as the sizes of diffusant molecules, is proposed. The derived expression agrees well with experimental results.     

Orientational Brownian motion in a viscoelastic medium

A model has been proposed for the complete (2D) orientational Brownian motion of axially symmetric particles in a Kelvin viscoelastic medium. It has been shown that the elastic response of the carrier medium can be correctly determined by introducing an additional variable, the moment elastic forces, which, together with the particle-orientation vector, composes the complete set of phase variables of a system. A set of stochastic equations (Langevin equations) has been derived for these values. This set has been employed to construct, under the effective field approximation, a simple set of moment equations describing the magnetization dynamics of a ferrogel, i.e., a dispersion of single-domain nanoparticles in a gel matrix. The theoretical results qualitatively agree with the data of magnetic microrheology of gelatin-based ferrogels.