Constant surface potential boundary condition

Until we discovered the constancy of the surface potential from the uniaxial stress results, like most other people, I had been more interested in constant surface charge models. If you do not know how the valency of a macroion varies with the external conditions, it is reasonable to assume it to be constant unless given evidence to the contrary. Given the evidence that y/0 70 mV is roughly constant for the n-butylammonium vermiculite system, what other consequences follow from this In particular, what happens if we apply the coulombic attraction theory with the constant surface potential boundary condition ... 

Our experimental conclusion — that s is constant and equal to 2.6 0.4 in the range 3 mM < cex <120 mM — accords well with the prediction from the new generalized Donnan equilibrium made in Chapter 4. We recall that the coulombic attraction theory, with the constant surface potential boundary condition s = 70 mV, predicts that. v is constant and equal to 2.8. A factor of x40 in c provides a severe test of the prediction and it passes, although the quantitative agreement between the theoretical and experimental values of s in this case should be treated with caution because of the severity of the approximations used in deriving the theoretical result. The pure Donnan prediction that s = 4.0 for s = 70 mV is definitely invalidated by... 

We first discuss the case of constant surface potential, which can be directly compared to the numerical profiles. Then we note the differences with the constant surface charge boundary condition in which the interesting phenomenon of charge overcompensation is discussed in detail. 

It is important to mention here that the case of constant zeta potential boundary condition, as considered for the above analysis, may not be a very common feature for strongly interacting EDLs. Rather, the concerned boundary condition may be posed more appropriately by simultaneously invoking the pertinent equations for chemical equilibrium and surface charge density-surface potential relationship. For example, one may use [6]... 

The equations are transcendental for the general case, and their solution has been discussed in several contexts [32-35]. One important issue is the treatment of the boundary condition at the surface as d is changed. Traditionally, the constant surface potential condition is used where po is constant however, it is equally plausible that ag is constant due to the behavior of charged sites on the surface. 

In this paper it is shown that the rate of deposition of Brownian particles on the collector can be calculated by solving the convective diffusion equation subject to a virtual first order chemical reaction as a boundary condition at the surface. The boundary condition concentrates the surface-particle interaction forces. When the interaction potential between the particle and the collector experiences a sufficiently high maximum (see f ig. 2) the apparent rate constant of the boundary condition has the Arrhenius form. Equations for the apparent activation energy and the apparent frequency factor are established for this case as functions of Hamaker s constant, dielectric constant, ionic strength, surface potentials and particle radius. The rate... 

To solve the Poisson-Boltzmann equation, boundary conditions are required. For a constant surface potential tf o on the surface of the large particles,... 

The boundary condition that iJ/ must satisfy at an interface can vary between the conductive and insulating cases mentioned earlier. A conductive surface infers a constant surface potential leading to a Dirichet boundary condition,... 

Even with these useful results from statistical mechanics, it is difficult to specify straightforward criteria delineating when the Poisson-Boltzmann or linear Poisson-Boltzmann equations can be expected to yield quantitatively accurate results for particle-wall interactions. As we have seen, such criteria vary greatly with different types of boundary conditions, what type of electrolyte is present, the electrolyte concentration and the surface-to-surface gap and double layer dimensions. However, most of the evidence supports the notion that the nonlinear Poisson-Boltzmann equation is accurate for surface potentials less than 100 mV and salt concentrations less than 0.1 M, as stated in the Introduction. Of course, such a statement might not hold when, for example, the surface-to-surface separation is only a few ion diameters. We have also seen that the linear Poisson-Boltzmann equation can yield results virtually identical with the nonlinear equation, particularly for constant potential boundary conditions and with surface potentials less than about 50 mV. Even for constant surface charge density conditions the linear equation can be useful, particularly when Ka < 1 or Kh > 1, or when the particle and wall surfaces have comparable charge densities with opposite signs. 

FIGURE 103 Electrostatic repulsion force, F/njAg T versus separation, h, between two identically charged plates with surface potentials of 25 mV in a 1 1 electrolyte solution at 0.001 Hf according to the exact theory with either constant charge or constant potential boundary conditions. Calculated from F = n Jk T (A - A ) with definitions of Aj given in the text for constant charge and potential boundary conditions. 

Eq. (20) can be discretized in the computational domain bounded hy = o2,o ) and r] = (0,7t). The discretization can be carried out using either the finite volume or finite difference scheme, producing a system of nonlinear equations for the reduced potential in the computational domain, which can be solved employing the Newton-Ralphson method or other relaxation techniques. The boundary conditions of constant surface potentials are described by 0 = (p at the surface of sphere 1 with = o and r] = (0,7t), and (p = (p2 the surface of sphere 2 with — 02 and r] — 0,n). The boundary conditions of constant surface charge are described by... 

Equation (6.5.12), subject to the boundary conditions indicated, can be solved analytically only under limiting conditions on the Debye length described below. In general, it must be solved numerically. Figure 6.5.2 gives numerical solutions obtained by Gross Osterle (1968) for a constant surface potential 2-79 for various values of the Debye length ratio A It can be seen that for A " <0.1 the potential is zero over most of the capillary cross section, whereas for A " > 10 the potential is nearly constant over the cross section. 

Here A and B are constants defined by boundary conditions, which are either constant surficial electric potential (Dirichlet boxmdary conditions) or constant density of the surface charge (Neumann boundary conditions). If we assume q>(x = 0) equal and (x -> o°) equal 0, then A = q>, and B = 0, and equation (2.124) is simplified ... 

The linear approximation and an employment of the boundary condition of constant surface potentials yields ... 

The two standard cases presented so far are by no means comprehensive, and in reahty, other types of boundary conditions could be operative as well. For example, out of the two parallel plates, plate 1 might be of constant surface potential, and plate 2 might be of constant surface charge. The EDL free energy of formation, in that case, can be described as... 

To obtain accurate values for the variation of the repulsive force with separation, equation 2.52 must be solved. Constant-charge and constant-potential boundary conditions furnish bounds on the force-distance relation for surfaces that regulate their charge according to the mass action equilibria studied in conneetion with the isolate plate. 

To obtain the force per unit area, we applied/ = —dV /dx. Equation (4.60) is valid for constant surface potential. For constant charge boundary conditions, we get 417]... 

For high surface potentials and thinner films, we have to specify the electrostatic boundary condition. For constant surface potential, we apply Eq. (4.60). For constant surface charge, Eq. (4.61) is suitable. 

This function is a solution of Laplace s equation regardless of the values of constants, and our goal is to find such of them that the potential satisfies the boundary condition on the surface of the given ellipsoid of rotation and at infinity. In order to solve this problem we have to discuss some features of Legendre s functions. First of all, as was shown in Chapter 1, the Legendre s function of the first kind P (t]) has everywhere finite values and varies within the range... 

Equation (11.2) remains valid as the first boundary condition in this case. The surface concentrations, c, of the reactants will remain constant, in accordance with the Nemst equation, when the electrode potential is held constant during current flow (and activation polarization is absent). Hence, the second boundary condition can be formulated as... 

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