Constant surface charge density

Cjnn is the capacitance due to the inner layer, which can be experimentally obtained from the plot of 1/Ld (with Cd being the capacitance measured at a given charge density) for several electrolyte concentrations versus the calculated l/LG-ch at a constant surface charge density (Parsons and Zobel plot) [2]. If this plot is not linear, this is an indication that specific adsorption occurs. 

The chemical contribution FChe is 0 at the constant surface charge density of the large particles and —constant surface potential //o of the large particles, where a is the surface charge per large particle. 

For constant surface charge density o, the condition of overall electroneutrality 2o = — fdd p(z) dz leads to... 

The first term is the Verwey—Overbeek expression for the interactions at constant surface charge density and the second accounts for the variation of the surface charge with the separation distance. 

The present approach reduces to the traditional ones within their range of application (imaginary charging processes for double layer interactions between systems of arbitrary shape and interactions either at constant surface potential or at constant surface charge density, and the procedure based on Langmuir equation for interactions between planar, parallel plates and arbitrary surface conditions). It can be, however, employed to calculate the interaction free energy between systems of arbitrary shape and any surface conditions, for which the traditional approaches cannot be used. 

At constant surface potential, the force provided by the new model is smaller than predicted by the Gouy-Chapman theory. In contrast, at constant surface charge density, the dielectric constant and the thickness of region I have effects opposite to those at constant surface potential. 

The chemical contribution AFchem is 0 at constant surface charge density. 

H. Ohshima, Diffuse double layer interaction between two spherical particles with constant surface charge density in an electrolyte solution, Colloid Polymer Sci. 263, 158-163 (1975). 

It is instructive also to consider the case where the sphere and wall surfaces have constant surface charge densities boundary conditions (33) and (34) are replaced by... 

Dungan and Hatton [12] solved Eq. (6) together with Eq. (48) for the problem depicted in Fig. 6, where a spherical particle is interacting with an oppositely charged deformable interface. To obtain their solution they used a boundary-integral method, in which the surfaces of the interface and sphere are discretized and assigned constant surface charge density boundary con-... 

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. 

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