Fixed charge density

We seek to determine an effective fixed-charge density which influences ion uptake and transport. This may be different from an analytical determination of total carboxyl content because some groups may not be in swollen regions of the polymer and so may not... 

In conclusion, it can be claimed that a combination of kinetic and equilibrium conductance and membrane potential measurements provides a powerful method for investigating the permselective properties of membranes of low fixed charge density. Such methods should be applicable also to other polymers useful in hyperfiltration if they can be prepared in the form of homogeneous membranes. 

Low inversion channel mobility on a vertical sidewall due to high interface state density at the p-SiC/SiO interface and high fixed charge density in the insulator. 

Here N(P) (cul/cm3) is the fixed charge density in the membrane, assumed to be a known function of position (negative in a cation-exchange membrane and positive in an ion-exchange one). In most modern ion-exchange membranes IV is typically higher than 102 cul/cm3 ( IV /F > 10 3mol/cm3). 

To illustrate some of the notions introduced so far, let us consider the Ci and p fields in an electrodialysis cell at equilibrium. For simplicity, let us limit our consideration to a 1,1 valent electrolyte at bulk (feed) concentration Co- Assume a constant fixed charge density N(—N) for the an- (cat-) ion membrane. 

A terminological remark is due. An equilibrium between two media with different fixed charge density (e.g., an ion-exchanger in contact with an electrolyte solution) is occasionally termed the Donnan equilibrium. The corresponding potential drop between the bulks of the respective media is then termed the Donnan potential. By the same token, we speak of the local Donnan equilibrium and the local Donnan potential, referring, respectively, to the local equilibrium and the interface potential jump at the surface of discontinuity of the fixed charge density, considered in the framework of the LEN approximation. 

Below we present a well-known calculation of membrane potential based on the classical Teorell-Meyer-Sievers (TMS) membrane model [2], [3]. The essence of this model is in treating the ion-selective membrane as a homogeneous layer of electrolyte solution with constant fixed charge density and with local ionic equilibrium at the membrane/solution interfaces. In spite of the obvious idealization involved in the first assumption the TMS model often yields useful results and represents in fact the main tool for practical membrane calculations. We shall return to TMS once again in 4.4 when discussing the electric current effects upon membrane selectivity. In the case of our present interest, the simplest TMS model of membrane potential for a 1,2 valent electrolyte reads... 

Preliminaries. In the previous chapter we dealt with locally electro-neutral time-dependent electro-diffusion under the condition of no electric current in a medium with a spatially constant fixed charge density (ion-exchangers). It was observed that under these circumstances electrodiffusion is equivalent to nonlinear diffusion with concentration-dependent diffusivities. 

For similar reasons ion diffusivities will be assumed piecewise constant, and we shall mostly limit ourselves to a discussion of one-dimensional situations, for which the combination of local electro-neutrality and piecewise constant fixed charge density yields instructive explicit solutions. 

We point out that the results of locally electro-neutral studies should be extrapolated upon the nonreduced systems with a certain caution even for e very small. This is so because, to the best of the author s knowledge, no asymptotic procedure for the singularly perturbed one-dimensional system (4.1.1), (4.1.2) has been developed so far that would be uniformly valid for the entire range of the operational parameters (e.g., for arbitrary voltages and fixed charge densities). 

The piecewise constant fixed charge density N(x) is chosen of the form ... 

On this low current branch, concentration variations within the charged layers are small (of order Cq). The condition (4.3.17d) merely says that in order for the lower limiting current to exist, the total concentration gain er in the enrichment layers (i — 1,4), has to be smaller than the appropriate total concentration decrease in the depletion layers (i = 2,3). (According to (4.3.15), the total concentration variation within each layer is proportional to the passing current /, fixed charge density Ni, and the thickness A of the layer.) In particular, (4.3.17d) implies the existence of some critical iV4 (N 4r) beyond which the low current branch with saturation ceases to exist. 

Finally ( 5.5), in order to illustrate an alternative asymptotic approach, available for systems with nowhere vanishing fixed charge density, we shall treat, following Please [4], a p — n junction (bipolar membrane). Presentation of this chapter is purely heuristic, based on numerical or formal asymptotic results. For some rigorous results concerning related matters, the reader is referred to [5]—[7]. 

Consider the following simplest prototype problem for stationary electrodiffusion of a univalent symmetric electrolyte through a bipolar ion-exchange membrane with an antisymmetric piecewise constant fixed charge density XN(x). 

Soft biological structures exhibit finite strains and nonlinear anisotropic material response. The hydrated tissue can be viewed as a fluid-saturated porous medium or a continuum mixture of incompressible solid (s), mobile incompressible fluid (f), and three (or an arbitrary number) mobile charged species a, (3 = p,m, b). A mixed Electro-Mechano-Chemical-Porous-Media-Transport or EMCPMT theory (previously denoted as the LMPHETS theory) is presented with (a) primary fields (continuous at material interfaces) displacements, Ui and generalized potentials, ifi ( , r/ = /, e, to, b) and (b) secondary fields (discontinuous) pore fluid pressure, pf electrical potential, /7e and species concentration (molarity), ca = dna/dVf or apparent concentration, ca = nca and c = Jnca = dna/dVo. The porosity, n = 1 — J-1(l — no) and no = no(Xi) = dVj/dVo for a fluid-saturated solid. Fixed charge density (FCD) in the solid is defined as cF = dnF/dV , cF = ncF, and cF = cF (Xf = JncF = dnF/d o. 

For other models of flow of electrolytes through porous media the reader is referred to [2], [5], [6]. To take into account FCD (fixed charge density) one has to impose additional condition on the interface T (w) and the electroneutrality condition. A challenging problem is to use homogenisation methods for the case of finitely deformable skeleton, even hyperelastic. The permeability would then necessarily depend on strains. Such a dependence (nonlinear) is important even for small strain, cf. [7]. It is also important to include ion channels [8]. 

Gu, W.Y., Justiz, M.A. and Yao, H. (2002) Electrical conductivity of lumbar annulus fibrosis Effects of porosity and fixed charge density. Spine 27, 2390-2395... 

To obtain membranes with different ion-exchange capacity the sulfonated polyetheretherketone or polysulfone can be mixed with unsulfonated polymer in a solvent such as N-methylpyrrolidone. By changing the ratio of the sulfonated to unsulfonated polymer the fixed-charge density can easily be adjusted to a desired value. 

Finally, it was observed from the calculations that, if particle and collector surfaces bear fixed charge densities of opposite sign but sufficiently different magnitudes, then the double-layer repulsion which occurs at small separations... 

In some previous calculations [16-18] of the interactions involving polyelectrolyte chains grafted to two surfaces, the charge of the polyelectrolyte chains was assumed to be constant and this fixed charge density was introduced into a one-dimensional Poisson-Boltzmann equation to calculate the electrical potential profile. All the above treatments involved a unidimensional model. 

A q is the scaled fixed charge density defined by iVoAn/Aj and A4 is the value of N based on Nq. 

We first treat the case in which the fixed-charge density ZeN is low. Then Eqs. (4.50) and (4.51) are linearized to give... 

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