Plane charge

Since the interface behaves like a capacitor, Helmholtz described it as two rigid charged planes of opposite sign [2]. For a more quantitative description Gouy and Chapman introduced a model for the electrolyte at a microscopic level [2]. In the Gouy-Chapman approach the interfacial properties are related to ionic distributions at the interface, the solvent is a dielectric medium of dielectric constant e filling the solution half-space up to the perfect charged plane—the wall. The ionic solution is considered as formed... 

Another progress in our understanding of the ideally polarizable electrode came from theoretical works showing that the metal side of the interface cannot be considered just as an ideal charged plane. A simple quantum-mechanical approach shows that the distribution of the electron gas depends both on the charge of the electrode and on the metal-solution coupling [12,13]. 

A possible reason that the problem of C < 0 did not receive much attention was the assertion [15] (BLH) that such an anomaly was forbidden. The proof was based on the statistical mechanical analysis of the primitive model of electrolytes between two oppositely charged planes, cr and —a. It was noticed in Ref. 10 that the BLH analysis missed a very simple contribution to the Hamiltonian, direct interaction between the charged walls, ItzLct (L is the distance between the walls). With proper choice of the Hamiltonian the condition on the capacitance would be C > 27re/L. It simply means that due to ionic shielding of the electric field, the capacitance exceeded its geometrical value corresponding to the electrolyte-free dielectric gap. 

Here, n(z z) is the charge density induced at z by a charged plane, of unit surface charge density, located at z, and pb is the bulk jellium density. The Hellmann-Feynman theorem was used. The full moment of n(z z) obeys... 

This plane of the center of mass of the excess ionic charge o,(x) is the effective excess charge plane on the solution side, which may be compared with the effective image plane on the metal side. In simple cases, the effective excess charge plane coincides with the outer Helmholtz plane (the plane of closest approach of hydrated ions) as shown in Fig. 5-21. 

Fig. 6-21. Charge distribution profile across a metal/aqueous solution interface (M/S) (a) the hard sphere model of aqueous solution and the jellium model of metal (the jellium-sphere model), (b) the effective image plane (IMP) and the effective excess charge plane x, (c) reduction in distance /lxd,p to the closest approach of water molecules due to electrostatic pressure, o, = differential excess charge on the solution side og = total excess charge on the solution side Oy = total excess charge on the metal side.

In a circle of unit surface area on the charged plane A-A, the negative charges acquired by the adsorbed organic ions (amphiphiles) within this area represent the surface charge density o ... 

S. Levine and C. W. Outwaite, Comparison of theories of the aqueous electric double layers at a charged plane interface, J. Chem. Soc., Faraday Trans. II, 74 (1978), pp. 1670-1689. 

Proof of boundedness of the force of interaction between two charged particles of an arbitrary shape in H3, held at a given distance from each other in an electrolyte solution, upon an infinite increase of the particle s charge. (It was shown in 2.2 that the repulsion force between parallel symmetrically charged cylinders saturates upon an infinite increase of the particle s charge. This is also true for infinite parallel charged plane interaction [9]. The appropriate result is expected to be true for particles of an arbitrary shape.)... 

The capacitor model simply assumes that all ions in the surrounding cloud are located in a thin plane or shell located at a specific distance 8 from the interface. The combination of charged interface and charged plane or shell resembles the basic design of a conductor with a difference in the electrical potential of A j/ between the two plates. Coulomb s law, which was first mentioned in 1785, quantifies the force (F) that acts between two charges qx and q2) separated by a distance (x) ... 

Figure 17. Estimated d0 between charged planes of adjacent RBCs aggregated by Dx 70 at various concentrations.

In the years 1910-1917 Gouy2 and Chapman3 went a step further. They took into account a thermal motion of the ions. Thermal fluctuations tend to drive the counterions away form the surface. They lead to the formation of a diffuse layer, which is more extended than a molecular layer. For the simple case of a planar, negatively charged plane this is illustrated in Fig. 4.1. Gouy and Chapman applied their theory on the electric double layer to planar surfaces [54-56], Later, Debye and Hiickel calculated the potential and ion distribution around spherical surfaces [57],... 

The minus sign arises because the charge planes are to the right instead of to the left of the test charge see the discussion leading to eqn. (10)]. If, in addition, we have the special situation in which the total charge of all the planes adds to zero... 

Added to these fields, of course, would be any externally applied field, E0, due to charge distributions other than the planes of space charge presently considered. Using the convention of labeling the total electric field existing in the region between charge planes — 1 and as Ejt then we can write... 

As the test charge is moved between charge planes j — 1 and j located at the positions Xj-t and xjt the electric potential V changes by Ej(xj —xj-l) where Xj —xJ-l = dj the separation distance between the two charge planes in question. Thus... 

Because Ej itself is increasing with position xh Vk increases as a higher power in xk than is the case for a single charge plane. This can be noted also from the result obtained by substituting eqn. (37) for Ej in terms of the charge density into the last equation above to obtain... 

Analytical expression for the electrophoretic velocity of a sphere can be obtained for a thin but distorted double layer. Dukhin [6] first examined the effect of distortion of thin ion cloud on the electrophoresis of a sphere in a symmetric two-species electrolyte. Dukhin s approach was later simplified and extended by O Brien [7] for the case of a general electrolyte and a particle of arbitrary shape. Since 0(k 1) double layer thickness is much smaller than the characteristic particle size L, the ion cloud can be approximated as a structure composed of a charged plane interface and an adjacent diffuse cloud of ions. Within the double layer, the length scales for variation of quantities along the normal and tangential directions are k ] and L, respectively. 

Electrostatic Surface Forces 5.4.3.1 Two Identically Charged Planes... 

First we consider the electrostatic (double layer) interaction between two identical charged plane parallel surfaces across a solution of symmetric Z Z electrolyte. The charge of a counterion (i.e., ion with charge opposite to that of the surface) is -Ze, whereas the charge of a coion is +Ze (Z = +1, +2,. ..) with e the elementary charge. If the separation between the two planes is very large, the number concentration of both counterions and coions would be equal to its bulk value, n, in the middle of the film. However, at finite separation, h, between the surfaces the two EDL overlap and the counterion and coion concentrations in the middle of the film, io and 2o> longer equal. Because the solution inside the film is supposed to be in electrochemical (Donnan) equilibrium with the bulk electrolyte solution of concentration q, we can write 20 0 or, alternatively,... 

We summarize recent work showing that condensation can be derived as a natural consequence of the Poisson-Boltzmann equation applied to an infinitely long cylindrical polyelectrolyte in the following sense Nearly all of the condensed population of counter-ions is trapped within a finite distance of the polyelectrolyte even when the system is infinitely diluted. Such behavior is familiar in the case of charged plane surfaces where the trapped ions form the Gouy double layer. The difference between the plane and the cylinder is that with the former all of the charge of the double layer is trapped, while with the latter only the condensed population is trapped. 

Such two-dimensional pair correlation functions for mobile ions adsorbed on charged planes have recently been investigated theoretically in Ref. 42. The mean separation d, = ( < 0/s )l/2 of w-valent ions on the completely neutralized plane of surface charge density s is identified as the important scaling length. The main predictions for the strongly coupled regime =... 

Sk and Si are vectors defining the position of tesserae k and I, respectively, while Rk is the radius of the sphere which tessera k belongs to. The diagonal term of D, which collects the contribution of the reaction field induced by the charge placed on tessera k on itself, is derived by the Gauss formula for an infinite charged plane with a correction term iccounting for the curvature of the convex tessera. 

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