Estimation of thickness of concentration boundary layers by osmotic volume flux determination

The estimation method of the concentration boundary layers thicknesses (δ) in a single-membrane system containing non-electrolytic binary or ternary solutions was devised using the Kedem-Katchalsky formalism. A square equation used in this method contains membrane transport (L(p), σ, ω) and solution (D, C) parameters as well as a volume osmotic flux (J(v)). These values can be determined in a series of independent experiments. Calculated values δ are nonlinearly dependent on the concentrations of investigated solutions and the membrane system configuration. These nonlinearities are the effect of a competition between spontaneously occurring diffusion and natural convection. The mathematical model based on Kedem-Katchalsky equations and a concentration Rayleigh number (R(C)) was presented. On the basis of this model we introduce the dimensionless parameter, called by us a Katchalsky number (Ka), modifies R(C) of membrane transport. The critical value of this number well describes a moment of transition from the state of diffusion into convective diffusion membrane transport.     

Laser interferometric investigation of solute transport through membrane-concentration boundary layer system

We investigate diffusive transport in a membrane system with a horizontally mounted membrane under concentration polarization conditions performed by a laser interferometry method. The data obtained from two different theoretical models are compared to the experimental results of the substance flux. In the first model, the membrane is considered as infinitely thin, while in the second one as a wall of finite thickness. The theoretical calculations show sufficient correspondence with the experimental results. On the basis of interferometric measurements, the relative permeability coefficient (ζ_s) for the system, consisting of the membrane and concentration boundary layers, was also obtained. This coefficient reflects the concentration polarization of the membrane system. The obtained results indicate that the coefficient ζ_s of the membrane-concentration boundary layer system decreases in time and seems to be independent of the initial concentration of the solute.     

A model equation for the gravielectric effect in electrochemical cells

The gravielectric effect model equation for a single-membrane system was elaborated. This model for binary and ternary ionic solutions was verified using a cell with a horizontally mounted membrane. In this cell, the membrane and transition potentials were measured as a function of gravitational configuration. In these experiments, a 0.001 M aqueous solution of sodium chloride was placed on one side of the membrane. The opposite side of the membrane was exposed to either aqueous sodium chloride solutions, with densities greater than that of 0.001 M aqueous NaCl, or ethanol/NaCl/water solutions. On the basis of the experimental results, the influence of constrained release and the gravielectric effect were established. These experimental findings are interpreted in terms of a convective gravitational instability that reduces boundary layer dimensions and increases the permeability coefficient of the complex system: boundary layer/membrane/boundary layer. A concentration-gradient Rayleigh number is used in a mathematical model for gravitationally sensitive membrane potential.     

Mass transport at rotating disk electrodes: effects of synthetic particles and nerve endings

An unstirred layer (USL) exists at the interface of solids with solutions. Thus, the particles in brain tissue preparations possess a USL as well as at the surface of a rotating disk electrode (RDE) used to measure chemical fluxes. Time constraints for observing biological kinetics based on estimated thicknesses of USLs at the membrane surface in real samples of nerve endings were estimated. Liposomes, silica, and Sephadex were used separately to model the tissue preparation particles. Within a solution stirred by the RDE, both diffusion and hydrodynamic boundary layers are formed. It was observed that the number and size of particles decreased the following: the apparent diffusion coefficient excluding Sephadex, boundary layer thicknesses excluding silica, sensitivity excluding diluted liposomes (in agreement with results from other laboratories), limiting current potentially due to an increase in the path distance, and mixing time. They have no effect on the detection limit (6 ± 2 nM). The RDE kinetically resolves transmembrane transport with a timing of approximately 30 ms.     

The role of mechanical pressure difference in the generation of membrane voltage under conditions of concentration polarization

The mechanical pressure difference across the bacterial cellulose membrane located in a horizontal plane causes asymmetry of voltage measured between electrodes immersed in KCl solutions symmetrically on both sides of the membrane. For all measurements, KCl solution with lower concentration was above the membrane. In configuration of the analyzed membrane system, the concentration boundary layers (CBLs) are created only by molecular diffusion. The voltages measured in the membrane system in concentration polarization conditions were compared with suitable voltages obtained from the model of diffusion through CBLs and ion transport through the membrane. An increase of difference of mechanical pressure across the membrane directed as a difference of osmotic pressure always causes a decrease of voltage between the electrodes in the membrane system. In turn, for mechanical pressure difference across the membrane directed in an opposite direction to the difference of osmotic pressure, a peak in the voltage as a function of mechanical pressure difference is observed. An increase of osmotic pressure difference across the membrane at the initial moment causes an increase of the maximal value of the observed peak and a shift of this peak position in the direction of higher values of the mechanical pressure differences across the membrane.     

Current voltage curves of biomolecular lipid membranes.

The first part of this paper describes the current voltage curves of bimolecular membranes of oxidized cholesterol formed between two aqueous solutions of tetrabutylammonium chloride. These membranes are selectively permeable for cations and the membrane interfaces are electrically uncharged. The dependence of the membrane conductivity on the membrane potential can be described as the product of the conductivity at zero current ("zero conductivity") and a function called "overlinearity". The zero conductivity increases linearly with the concentration of tetrabutylammonium chloride. The overlinearity is independent of the concentration of tetrabutylammonium chloride. In the second part the Nernst-Planck and Poisson equations are integrated numerically for a three-phase system consisting of an aqueous electrolyte solution, a membrane and an aqueous electrolyte solution. Each phase is characterized by material constants. Appropriate boundary conditions cause the electric current to build up electrical double layers on both sides of the membrane. The opposing double layers with opposite electrical signs inject the soluble ions into the membrane. This ion injection accounts for the overlinearity of the current voltage curves, thus explaining the measured characteristics.     

Evolution of concentration field in a membrane system

This paper deals with the evolution of concentration field at a single membrane system. Concentration field evolution is described by concentration effect of stable boundary layers, which originate in this system. The concentration effect of boundary layers (CBLE) is studied experimentally on the basis concentration profiles obtained from computer analysis of interferometric pictures of near-membrane regions. Besides experimental results, we also report theoretical investigations and numerical calculations of this effect for two models of membranes (an infinite thin wall and the wall of thickness l). Evolution of concentration field at different distances from membrane surface describes accurately the spatio-temporal structure of the concentration boundary layers (CBLs). Results have shown that their spatial structure is fully established and these layers develop diffusively.     

Probing protein colloidal behavior in membrane‐based separation processes using spectrofluorometric Rayleigh scattering data

One of the primary problems in membrane‐based protein separation is membrane fouling. In this study we explored the feasibility of employing Rayleigh light scattering data from fluorescence studies combined with chemometric techniques to determine whether a correlation could be established with membrane fouling phenomena. Membrane flux was measured in a dead‐end UF filtration system and the effect of protein solution properties on the flux decline was systematically investigated. A variety of proteins were used as a test case in this study. In parallel, the colloidal behavior of the protein solutions was assessed by employing multiwavelength Rayleigh scattering measurements. To assess the usefulness of Rayleigh scattering measurements for probing the colloidal behavior of proteins, a protein solution of β‐lactoglobulin was used as a base‐case scenario. The colloidal behavior of different β‐lactoglobulin solutions was inferred based on published data for this protein, under identical solution conditions, where techniques other than Rayleigh scattering had been used. Using this approach, good agreement was observed between scattering data and the colloidal behavior of this protein. To test the hypothesis that a high degree of aggregation will lead to increased membrane fouling, filtration data was used to find whether the Rayleigh scattering intensity correlated with permeate flux changes. It was found that for protein solutions which were stable and did not aggregate, fouling was reduced and these solutions exhibited reduced Rayleigh scattering. When the aggregation behavior of the solution was favored, significant flux declines occurred and were highly correlated with increased Rayleigh scattering. © 2010 American Institute of Chemical Engineers Biotechnol. Prog., 2010     

Thermodynamic Model Equations for Heterogeneous Multicomponent Non-Ionic Solution Transport in a Multimembrane System

Non-equilibrium thermodynamic model equations for non-ionic and heterogeneous n-component solution transport in a m-membrane system are presented. This model is based on two equations. The first one describes the volume transport of the solution and the second the transport of the solute. Definitions of the hydraulic permeability, reflection and diffusive permeability coefficients of the m-membrane system and relations between the coefficients of the m-membrane system and the respective membranes of the system are also given. The validity of this model for binary and ternary solutions was verified, using a double-membrane cell with a horizontally mounted membrane. In the cell, volume and solute fluxes were measured as a function of concentration and gravitational configuration.     

The Investigation of Time-dependent Solute Transport through Horizontally Situated Membrane: The Effect of Configuration Membrane System

This paper presents an experimental and theoretical investigation ofsolute transport through a horizontally situated membrane. Theexperimental investigation was carried out by the laserinterferometric method in association with a computer system ofinterference image analysis. On the basis of this analysis thicknessof near-membrane layers, solution concentration drops on these layersas well as diffusion fluxes of diluted substance are determined.Different fluxes of the soluble substance are observed depending onthe configuration of the system. The results of the experimental andtheoretical investigation of diffusion fluxes are conformable inrespect of measurement error, with one adjustment parameter, i.e. thesolute partition coefficient.     

Measurement of membrane potential and estimation of effective fixed-charge density in membranes.

Electrical potentials Em arising across cross-linked phenolsulfonate membrane separating NaCl solutions of molality M1 and M2 have been measured at 25 degrees C. These values of Em have been used in the Nernst equation to calculate values for the apparent transport number ti(app) for the counterion or the co-ion in the membrane. Values of ti(app) together with the limiting value for the cation transport number in the aqueous phase have been used in the equation developed by Kobatake and co-workers to evaluate the membrane permselectivity Ps as a function of external electrolyte concentration. With the help of the equation relating Ps to phiX, the effective fixed-charge density in the membrane (where phi is a constant, 0 less than phi less than 1, and X is the membrane stochiometric charge density and can be evaluated by chemical analysis of the membrane phase), values for phiX and phi have been determined. Values of phi were low in dilute solutions and increased with increase in the concentration of the external solution. Similar behavior was noted in the case of another membrane system, cross-linked polymethacrylic acid in contact with KOH solutions. On the other hand, the membrane system, "untreated" collodion in contact with KCl solutions, exhibited a behavior in which the values of phi, low in dilute solutions, increased and then decreased following a gradual increase in the external concentration. This slight divergence in its behavior was attributed to the heterogeneity of the collodion membrane structure. The reliability of this potentiometric method to estimate effective fixed-charge density in membranes has been discussed in relation to a similar but old method due to Teorell, Meyer and Sievers. Also the significance of the values derived for phi has been pointed out.     

Steady-state electrodiffusion. Scaling, exact solution for ions of one charge, and the phase plane.

This is the first of two papers dealing with electrodiffusion theory (the Nernst-Planck equation coupled with Gauss's law) and its application to the current-voltage behavior of squid axon. New developments in the exact analysis of the steady-state electrodiffusion problem presented here include (a) a scale transformation that connects a given solution to an infinity of other solutions, suggesting the po-sibility of direct comparison of electrical data for membranes with different thicknesses and other properties; (b) a first-integral relation between the electric field and ion densities more general than analogous relations previously reported, and (c) an exact solution for the homovalent system, i.e., a membrane system permeated by various ion species of the same charge. The latter is a generalization of the known one-ion solution. The properties of the homovalent solution are investigated analytically and graphically. In particular we study the phase-plane curves, which reduce to the parabolas discussed by K. S. Cole in the special case in which the current-density parameter (a linear combination of the ionic current densities) is zero.     

Bipartite expressions for diffusional mass transport in biomembranes

Analytical expressions for solute diffusion through a membrane barrier for different initial and boundary conditions are available in the literature. The three commonest initial and boundary conditions are for a membrane without solute respectively immersed in a solution of constant concentration, immersed in such a solution for one side but with the other side isolated, and immersed in such a solution for one side and with the other side kept at zero concentration. The physical quantities for the first two initial and boundary conditions are concentration and average concentration (the total solute entering the membrane) with amperometric current (flux) and solute that permeates through the membrane (charge passed) for the third initial and boundary condition. Expressions for these methods in the literature are inconvenient for practical applications because of the infinite mathematical series required. An investigation of convergence of these expressions was therefore carried out. Simple but accurate bipartite expressions for these methods were constructed and provided theoretical support for studies on mass transport characterization of biomembranes. As a specific application, these expressions enabled a direct fit of the simulated observables to experimental values to obtain diffusion coefficients. For these initial and boundary conditions and corresponding physical quantities, simple one point methods for diffusion coefficient estimation are also suggested. These latter diffusion coefficients can be initial values for numerical fit methods.     

Transport numbers in biomembranes from emf measurements.

Initial membrane potentials of biological periderm and cuticular membranes have been measured with KCl solution using Ag/AgCI electrodes. For the emf measurements, the concentration in both compartment were first brought to equilibrium with 0.01 M KCI solution. After equilibration the concentration in one compartment (inner surface of the membrane) was kept constant as 0.01 M and the concentration in the other compartment was varied between 10(-4) M and 1 M. The procedure was reversed as the concentration in the outer surface of the membrane was fixed and the other side was varied. Transport number of K(+) ion corresponding to the each of the two surface layers (homogeneous and heterogeneous) of the membrane was evaluated from the initial time emf measurements. The results obtained were compared with charged polysulfone with polyester supported membrane.     

Study of the Solute Flows of Multicomponent and Heterogeneous Non-Ionic Solutions in Double-Membrane System

The solute flows were studied in a double-membrane osmotic-diffusive cell, in which two membranes mounted in horizontal planes separate three compartments (l,m,r) containing the non-homogeneous, non-electrolytic binary and ternary solutions. The volume of inter-membrane compartment (m), which is the infinitesimally layer of solution, and volume of external compartments (l and r) fulfill the conditions V _m 0 and V _l =V _r , respectively. In an initial moment, the solution concentrations satisfy the condition (C ^o _s ) _l < (C ^o _s ) _m >(C ^o _s ) _r. The double-membrane osmotic-diffusive cell is composed of two complexes: boundary layer/membrane/boundary layer, mounted in horizontal planes. In the cell, solute flux was measured as a function of concentration and gravitational configuration. The linear dependencies of the solute flux on concentration difference in binary solutions and nonlinear – in ternary solutions were obtained. It was shown that the double-membrane osmotic-diffusive cell has rectifying and amplifying properties of solute flows.     

Diffusion through a membrane: Approach to equilibrium

The transfer of solute through a membrane separating two aqueous solutions is studied with the time-dependent diffusion equation for composite media. By introducing new independent and dependent variables it is shown that the differential equations and boundary conditions can be transformed into a dimensionless form which does not explicitly depend on the diffusivities of the media. Laplace transforms are used to derive explicit solutions for the solute concentration as a function of position and time. It is shown that at large time the concentration approaches the equilibrium distribution exponentially. Explicit results are given for the decay time as a function of the parameters of the system. In addition, an accurate and simplified expression is derived for the decay time for the case of small membrane permeability. The accuracy of the analytic solutions for the concentration profiles is tested by comparing them with numerical results obtained by solving the diffusion equations by the method of finite differences. Excellent agreement is found. Research supported in part by a grant from the National Science Foundation.     

On the theory of diffusion of electrolytes

In the theory of diffusion of electrolytes the following assumptions are frequently made: (i) the electrolytic solution is electrically neutral everywhere, (ii) the ionic concentrations and the electric potential all depend on a single Cartesian coordinate as the only space variable. Often the electric potential of the solution is determined on the basis of the Poisson equation alone, disregarding any other relation between this potential and the ionic concentrations. Since the Poisson equation only represents a condition which the potential fulfills, the use of this equation alone may lead to error unless the explicit relation for the potential involving a space integration of ionic concentrations is also taken into account. But if this relation is used the Poisson equation becomes redundant and, more important, assumptions (i) and (ii) appear unacceptable, the former because it leads to a zero electric potential everywhere, the latter because it is mathematically incorrect. The present paper is based on general equations of diffusion of ions, excluding the Poisson equation. These equations form a system of nonlinear integrodifferential quations whose number equals the number of ionic species present in the solution. It appears that when all ions are distributed symmetrically around a point all functions related to the above system of equations can be made dependent on a single space coordinate: the distance from the center of symmetry. Two methods of successive approximations are given for the solution of the equations in the case of spherical symmetry with limitation to the steady state. These methods are then applied to the study of the distribution of ionic concentrations and electrical potentials inside a cell of spherical shape in equilibrium with its surroundings. These methods are rapidly convergent; exact theoretical values of the electric potential are calculable on the boundary of the cell. It appears that the potential at the center of the cell is not more than ∼50% higher than at its boundary and that variation of concentration inside the cell is not very large. For instance, with 100 mV on the boundary the ionic concentration there is about four times higher than at the center. Calculations show that extremely small amounts of electricity are sufficient to account for the electric potentials currently observed. In a cell of 100 micra diameter an average concentration of only 10⁻¹⁴ mole/cm³ of a monovalent ion would be sufficient to give 1 millivolt on the boundary. This concentration is directly proportional to the voltage and inversely proportional to the square of the cell diameter. Most of the numerical results given above are obtained by considering only those ions whose electrical charge is not compensated for by ions of an opposite sign. The total concentrations may be much higher than those quoted. The theory does not take into account possible effects of structural heterogeneities which may exist in the cell, particularly of various phase boundaries. An incidental result shows that the Boltzmann distribution function in the form employed in modern theory of electrolytes is fundamentally a consequence of the mathematical theory of diffusion alone. It is pointed out, however, that Boltzmann distribution is not always compatible with the definition of the electric potential.     

The development of anoxia following occlusion

Following arteriolar occlusion, tissue oxygen concentration decreases and anoxic tissue eventually develops. Although anoxia first appears in the region most distal to the capillary at the venous end, it eventually spreads throughout the entire region of supply. In this paper the changing oxygen concentration, from the time of occlusion until the tissue is entirely anoxic, is examined mathematically. The equations governing oxygen transport to tissue are solved by iterating a nonlinear integral equation. This solution is valid until anoxia first appears. After anoxia develops it is necessary to solve a moving boundary problem. This is done using the method of matched asymptotic expansions, and accurate solutions are obtained for a wide range of physiological conditions.     

Transport of macromolecules in arterial wallin vivo: A mathematical model and analytical solutions

A mathematical model has been developed to simulatein vivo transmural accumulation of an intravenously injected tracer in the aortic wall of experimental animals. Parameters have been included to represent the following processes that affect tracer distribution: permeation of the blood-tissue interface, diffusion through the layers of the artery wall,convective solute drag through the same, and degradation. Of particular interest for thein vivo situation situation is the inclusion of boundary conditions that account for the variation in the plasma concentration of injected tracer as a function of time. Two analytical solutions are presented. The first describes a system in which two boundaries must be delineated; it pertains if the tracer is allowed to circulate until it enters the avascular media of the artery wall both across its luminal boundary and from the capillaries in its outer layer. The second applies to shorter duration experiments in which entry across only the luminal boundary is considered. A limiting case of the solution for short circulation times is presented, compared with a previously published solution, and examined for its potential utility in parameter estimation. Because of its treatment of time-dependent boundary conditions, the model has unique application toin vivo experiments related to macromolecular transport in atherosclerosis that may otherwise elude proper interpretation. This work was supported by National Institutes of Health Grants HL-29582 and HL-07242.     

Temperature distribution in human skin and subdermal tissues

Temperature profiles have been computed in the skin and subdermal part of a human body for (i) various values of environmental temperature, rate of sweat evaporation and wind velocity, (ii) rate of blood mass flow, (iii) rate of metabolic heat generation and (iv) three different sets of thicknesses of skin layers. The mathematical equations have been considered for a one-dimensional steady-state case. The two important physical parameters, namely rate of blood mass flow and rate of metabolic heat generation, have been assigned position-dependent values. The latter is also taken as linearly dependent on the tissue temperature. Analytic solutions have been obtained for the three layers of the region. These forms of solution facilitate the study of parameter dependence.