Interaction at Constant Surface Potential

In the second case, the surface potentials of the interacting particles remain constant during interaction. Consider two interacting particles 1 and 2 whose surface charges are due to adsorption of Nt ions (potential-determining ions) of valence Z adsorb onto the surface of particle i (/= 1, 2). If the configurational entropy Sc of the adsorbed ions does not depend on Ni, then the surface potential i/ oi of particle i is given by Eq. (5.10), namely. 

The free energy F of two interacting particles carrying constant surface potentials i/ oi and J/o2 can thus be expressed by 

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During interactions at constant surface potential Vs, the surface charge decreases when two surfaces approach each other and ion adsorption or molecular group dissociation is involved in the surface charging. The system (liquid and surfaces) is in thermodynamic equilibrium, and a transfer of charges from the liquid to the surfaces occurs to ensure the equality of the electrochemical potentials of the ions in the bulk and near the surface. 

For interactions at constant surface potential, y>s(0 = ips(°°)=Vs and using the corresponding Verwey—Overbeek choice C = —cKHVsO00), one obtains, integrating by parts the first, right-hand side term of eq 10, the expression... 

The form of the Helmholtz free energy F depends on the type of the origin of surface charges on the interacting particles. The following two types of interaction, that is (i) interaction at constant surface charge density and (ii) interaction at constants surface potential are most frequently considered. We denote the free energy F for the constant surface potential case by F and that for the constant surface... 

FIGURE 9.6 Schematic representation of the double-layer interaction at constant surface potential between two parallel plates 1 and 2 separated by h. j/o is the potential at the midpoint between the plates. 

AH rate calculations presented above have used the double-layer interaction given by Eq. [p3 as applied to approach at constant surface potential. Suppose the charge density had been held constant instead. How would this change affect the rate of deposition The answer is illustrated in Fig. 5. When the sphere and plate are many Debye lengths apart, the... 

At constant surface potential, the free energy per unit area of a system of two identical charged surfaces at distances rf from the middle distance can be written as the sum between a surface term (the chemical energy) and an integral over the density of the entropy of the mobile ions and all the electrostatic interactions between charges and dipoles... 

The free energy of the system is always defined up to an arbitrary constant the choice C = 0 in the Verwey— Overbeek approach leads to positive values for the free energy of the system at constant surface charge and negative at constant surface potential. However, the interaction free energy (the difference between the free energy of the system at the final separation distance and the free energy of the system at infinite separation) is not affected by the choice of the constant C. 

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. 

For interaction at constant surface charge, = 0, since the (fixed) surface charge is not in thermodynamic equilibrium with the medium, while, at constant surface potential, Fch = —2oy>s, where Vs is the surface potential.1 When neither o nor Vs is constant, the change in chemical free energy, per unit area, from infinite separation to a distance D = 2d between plates is given by... 

The van der Waals interactions between the polyelectrolyte segments and plates are included into the mean-field approach together with the excluded volume and electrostatic interactions. A repulsive contribution of the force between plates is generated by the van der Waals interactions between the segments and the plates. With increasing van der Waals interactions between the segments and the plates, the force between the plates at constant surface potential becomes more repulsive at small distances between the plates and more attractive at large... 

Figure 9.3 gives the potential distribution j/(x) across two interacting plates 1 and 2 at constant surface potential i/ o calculated with Eqs. (9.54)-(9.56) for Kh=l, 2, and oo, showing how j/(x) depends on the plate separation h. In this case, j/(x) does not depend on the plate thickness d, since there is no electric field inside the plates and j/(x) in the region x<0 remain unchanged during interaction. 

Honig and Mul [7] suggested that better approximations can be obtained if the interaction energy is expressed as a series of tanh(zei/ c/fcT) instead of i/ o and derived the interaction energy correct to tsDSi ze j/ JkT). The interaction energy V (h) per unit area correct to tanh zeij/o/kT) for the interaction between two parallel similar plates at constant surface potential separated by a distance in a symmetrical... 

ARBITRARY POTENTIAL INTERACTION AT CONSTANT SURFACE CHARGE DENSITY... 

The method for obtaining an analytic expression for the interaction energy at constant surface potential given in Chapter 9 (see Section 9.6) can be applied to the interaction between two parallel dissimilar plates. The results are given below [9]. 

We apply the Derjaguin s approximation (Eq. (12.3)) to the low-potential approximate expression for the plate-plate interaction energy, that is, Eqs. (9.53) and (9.65), obtaining the following two formulas for the interaction between two similar spheres 1 and 2 of radius a carrying unperturbed surface potential ij/f, at separation H at constant surface potential, V (H), and that for the constants surface charged density case, V (//) ... 

The next-order correction terms to Derjaguin s formula and HHF formula can be derived as follows [13] Consider two spherical particles 1 and 2 in an electrolyte solution, having radii oi and 02 and surface potentials i/ oi and 1/ 02, respectively, at a closest distance, H, between their surfaces (Fig. 12.2). We assume that i/ oi and i//q2 are constant, independent of H, and are small enough to apply the linear Debye-Hiickel linearization approximation. The electrostatic interaction free energy (H) of two spheres at constant surface potential in the Debye-Hlickel approximation is given by... 

For the interaction between two hard spheres 1 and 2 having radii ai and a2 for the case where sphere 1 is maintained at constant surface potential J/i and sphere 2 at constant surface charge density [Pg.330]

The interaction energy between spheres at constant surface potential involves only the function G (0 (which depends only on the sphere radius a,), while the interaction at constant surface charge density is characterized by the function H (i) (which depends on both sphere radius u, and relative permittivity Ep,). The interaction energy in the mixed case involves both G (i) and // (/) ... 

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