Ion-penetrable membrane

In this chapter, mathematical procedures for the estimation of the electrical interactions between particles covered by an ion-penetrable membrane immersed in a general electrolyte solution is introduced. The treatment is similar to that for rigid particles, except that fixed charges are distributed over a finite volume in space, rather than over a rigid surface. This introduces some complexities. Several approximate methods for the resolution of the Poisson-Boltzmann equation are discussed. The basic thermodynamic properties of an electrical double layer, including Helmholtz free energy, amount of ion adsorption, and entropy are then estimated on the basis of the results obtained, followed by the evaluation of the critical coagulation concentration of counterions and the stability ratio of the system under consideration. 

For planar particles covered by an ion-penetrable membrane an efficient numerical algorithm is available for the resolution of (1). The distribution of fixed charge in the membrane phase can be nonuniform, and the electrolyte in the liquid phase can be either a b or mixed (a b) + (c d) [35]. For two identical, planar parallel particles, this algorithm can be applied to estimate... 

TWO PARALLEL SEMI-INFINITE ION-PENETRABLE MEMBRANES (POROUS PLATES)... 

In Fig. 13.3, we plot the potential distribution i/ (x) between two parallel similar ion-penetrable membranes with i/tdoni = iAdon2 = don (or >Aoi = o2 = >Ao) for Kh = 0, 1,2, and oo. In Fig. 13.3, we have introduced the following scaled potential y, scaled unperturbed surface potential y, and scaled Donnan potential yooN-... 

Consider the double-layer interaction between two parallel porous cylinders 1 and 2 of radii and a2, respectively, separated by a distance R between their axes in an electrolyte solution (or, at separation H = R ai—a2 between their closest distances) [5]. Let the fixed-charge densities of cylinders 1 and 2 be and Pfix2. respectively. As in the case of ion-penetrable membranes and porous spheres, the potential distribution for the system of two interacting parallel porous cylinders is given by the sum of the two unperturbed potentials... 

Consider two parallel planar dissimilar ion-penetrable membranes 1 and 2 at separation h immersed in a solution containing a symmetrical electrolyte of valence z and bulk concentration n. We take an x-axis as shown in Fig. 13.2 [7-9]. We denote by Ni and Zi, respectively, the density and valence of charged groups in membrane 1 and by N2 and Z2 the corresponding quantities of membrane 2. Without loss of generality we may assume that Zj > 0 and Z2 may be either positive or negative and that Eq. (13.1) holds. The Poisson-Boltzmann equations (13.2)-(13.4) for the potential distribution j/(x) are rewritten in terms of the scaled potential y = zeif/IkT as... 

Consider two parallel planar ion-penetrable membranes 1 and 2, which may not be identical, at separation h in a symmetrical electrolyte solution of valence z and bulk concentration n (Fig. 16.1). We take an x-axis perpendicular to the membranes with its origin at the surface of membrane 1. The electric potential i/ (x) at position X between the membranes (relative to the bulk solution phase, where is set equal to zero) is assumed to be small so that the linearized Poisson-Boltzmann equation can be employed. Membranes 1 and 2, respectively, consist of N and M layers. All the layers are perpendicular to the x-axis. Let the thickness and the density of membrane-fixed charges of the ith layer of membrane j (7=1, 2) be and The linearized Poisson-Boltzmann equation for the /th layer... 

FIGURE 16.1 Interaction of two ion-penetrable membranes 1 and 2 consisting of N and M layers, respectively. 

It is interesting to note that i/ oi and 1//02, are, respectively, the unperturbed surface potentials of membranes 1 and 2 (i.e., the surface potentials at /i = 00) and that Eq. (16.9) states that the interaction force is proportional to the product of the unperturbed surface potentials of the interacting membranes. This is generally true for the Donnan potential-regulated interaction between two ion-penetrable membranes in which the distribution of the membrane-fixed charges far inside the membranes is uniform but may be arbitrary in the region near the membrane surfaces (see Eq. (8.28)). 

FIGURE 16.2 Electrostatic force P between two ion-penetrable membranes, each consisting of two layers, immersed in a monovalent symmetrical electrolyte solution as a function of the electrol3de concentration n (M). The Donnan potentials at n = 0.1 M in the respective... 

Electrostatic Interaction Between Ion-Penetrable Membranes in a Salt-Free Medium... 

Electric behaviors of colloidal particles in a salt-free medium containing counterions only are quite different from those in electrolyte solutions, as shown in Chapter 6. In this chapter, we consider the electrostatic interaction between two ion-penetrable membranes (i.e., porous plates) in a salt-free medium [1]. 

Before considering the interaction between two ion-penetrable membranes, we here treat the interaction between two similar ion-impenetrable hard plates 1 and 2 carrying surface charge density cr at separation h in a salt-free medium containing counterions only (Fig. 18.1) [2]. We take an x-axis perpendicular to the plates with its origin on the surface of plate 1. As a result of the symmetry of the system, we need consider only the region 0 < x < h 2. Let the average number density and the valence of counterions be o and z, respectively. Then we have from electroneutrality condition that... 

Now consider two parallel identical ion-penetrable membranes 1 and 2 at separation h immersed in a salt-free medium containing only counterions. Each membrane is fixed on a planar uncharged substrates (Fig. 18.2). We obtain the electric potential distribution i/ (x). We assume that fixed charges of valence Z are distributed in the membrane of thickness d with a number density of A (m ) so that the fixed-charge density pgx within the membrane is given by... 

FIGURE 18.2 Schematic representation of the electrostatic interaction between two parallel identical ion-penetrable membranes (porous plates) of thickness d separated by a distance h between their surfaces. Each membrane is fixed on an uncharged planar substrate. 

FIGURE 18.3 Distributions of counter-ion concentration n x) across two parallel identical ion-penetrable membranes of thickness d separated by a distance h between their surfaces in a salt-free medium. Distributions were calculated at /i = 0, 2, 6, and lOnm for Z—z— 1, d = 5nm, and the charge amount per unit area (j = ZeNd — 0.2Clm in water at 25 C (fir = 78.55). From Ref. [1]. 

Ohshima, H. and Kondo, T., pH dependence of electrostatic interaction between ion-penetrable membranes, Biophys. Chem., 32 (3), 161-166, 1988. 

Kuo, Y.C. and Hsu, J.P., Exact solution of the linearized Poisson-Boltzmann equation ion-penetrable membrane bearing fixed charges, J. Chem. Phys., 102 (4), 1806-1815, 1995. 

Hsu, J.P. and Kuo, Y.C., Properties of a double layer with asymmetric electrolytes cylindrical and spherical particles with an ion-penetrable membrane, J. Coll. Interf. Sci, 171 (2), 331-339, 1995. 

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