The Linearized Poisson-Boltzmann Equation

The stage is now set for the calculation of the potential V r ihe charge density in terms of known parametes of the solution. 

Notice that one has obtained two expressions for the charge density in the volume element dV at a distance r from the central ion. One has the Poisson equation [Eq. (3.4)] 

If one equates these two expressions, one can obtain the linearized Poisson-Boltzmann (P-B) expression 

The constants in the right-hand parentheses can all be lumped together and called a new constant k, i.e., 

At this point, the symbol k is used only to reduce the tedium of writing. It turns out later, however, that k is not only a shorthand symbol it contains information concerning several fundamental aspects of the distribution of ions around an ion in solution. In Chapter 6 it will be shown that it also contains information concerning the distribution of charges near a metal surface in contact with an ionic solution. In terms ofK, the linearized P-B expression (3.19) is 

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For small deviations from electroneutrality, the charge density at x is proportional to -(x)/kT9 where < is the difference of the electrostatic potential from its (constant) value when there is no charge density (the density of a species of charge z is proportional to 1 - zkT on linearizing the Boltzmann exponential). Then the Poisson equation [Eq. (44)] becomes the linearized Poisson-Boltzmann equation ... 

As a final topic in this section, we briefly consider the effect of electrolyte concentration on the solvent properties. The linearized Poisson-Boltzmann equation [31,121] can be used instead of (2) and (3) when the dielectric medium... 

Combining Eqs. (2.18) and (2.19) yields the linearized Poisson-Boltzmann equation... 

The Poisson-Boltzmann equation. The slab model is based on a solution of the linearized Poisson-Boltzmann equation that is valid only for low electrostatic surface potentials. As... 

The above equation is known as the linearized Poisson-Boltzmann equation since the assumption of low potentials made in reaching this result from Equation (29) has allowed us make the right-hand side of the equation linear in p. This assumption is also made in the Debye-Hiickel theory and prompts us to call this model the Debye-Hiickel approximation. Equation (33) has an explicit solution. Since potential is the quantity of special interest in Equation (33), let us evaluate the potential at 25°C for a monovalent ion that satisfies the condition e p = kBT ... 

For studying the stability of colloidal particles in suspension (Chapter 13) or for determining the potential at the surface of particles (Chapter 12), one often needs expressions for potential distributions around small particles that have curved surfaces. Solving the Poisson-Boltzmann equation for curved geometries is not a simple matter, and one often needs elaborate numerical methods. The linearized Poisson-Boltzmann equation (i.e., the Poisson-Boltzmann equation in the Debye-Hiickel approximation) can, however, be solved for spherical electrical double layers relatively easily (see Section 12.3a), and one obtains, in place of Equation (37),... 

The solution of the linearized Poisson-Boltzmann equation around cylinders also requires numerical methods, although when cylindrical symmetry and the Debye-Hiickel approximation are assumed the equation can be solved. The solution, however, requires advanced mathematical techniques and we will not discuss it here. It is nevertheless useful to note the form of the solution. The potential for symmetrical electrolytes has been given by Dube (1943) and is written in terms of the charge density a as... 

A new theory of electrolyte solutions is described. This theory is based on a Debye-Hiickel model and modified to allow for the mutual polarization of ions. From a general solution of the linearized Poisson-Boltzmann equation, an expression is derived for the activity coefficient of a central polarized ion in an ionic atmosphere of non-spherical symmetry that reduces to the Debye-Hiickel limiting laws at infinite dilution. A method for the simultaneous charging of an ion and its ionic cloud is developed to allow for ionic polarization. Comparison of the calculated activity coefficients with experimental values shows that the characteristic shapes of the log y vs. concentration curves are well represented by the theory up to moderately high concentrations. Some consequences in relation to the structure of electrolyte solutions are discussed. 

Table I. Constants of Integration for the General Solution of the Linearized Poisson—Boltzmann Equation...

The mean field potential for this system, a solution of the linear Poisson-Boltzmann equation, Eq. (32), will appropriately have the same periodic structure as the surface boundary condition. Thus, we expect that if/ will have the Fourier series,... 

Other surface charge conditions are naturally also possible and indeed necessary to cover the range of physical situations. Zhmud and House [71], for example, have recently solved the linear Poisson-Boltzmann equation to analyze the behavior of the electrical double layer near a surface which is nonuniformly charged with regions of altering ionization ability. The calculations in this case are only slightly more involved than those above. The generalization to curved surfaces has also been undertaken [72]. 

Equation (6) is the linear Poisson-Boltzmann equation. Although generally considered to be less accurate than its nonlinear counterpart, it has the advantage of being considerably easier to solve. In addition, in several cases it has been shown to give results very close to Eq. (4), even when the surface potentials are as high as one to two times the thermal voltage kT/e (i.e., 25-5 mV). Hence, Eq. (6) can yield information relevant to real colloidal systems under certain conditions. 

Relating Cb to P by using Eq. (2) and expanding the exponential as in Eq. (5) yields a form of Eq. (11) that is appropriate for use with the linearized Poisson-Boltzmann equation ... 

We denote the stress tensor inside the curly brackets in Eq. (13) asT. Equation (13) shows that the solution for the potential and its gradient at the particle surface are all that are required to calculate the force on a particle via the linearized Poisson-Boltzmann equation. 

This simple and appealing result shows that, for H 1 /k, the sphere-wall interaction depends linearly on the charge densities of each surface, and decays exponentially with the separation distance. The result does not depend on whether the surfaces are considered to be constant charge density or constant potential, because the potentials of an isolated wall and sphere were used in its derivation. Phillips [13] has compared Eq. (24) with a numerical solution of the linear Poisson-Boltzmann equation, and shows that it errs by less than about 10% for xh>3 when 0.5 [Pg.257]

The physical problem is shown in Fig. 2, where the geometry is described by using a cylindrical coordinate system with its origin on the wall. The linear Poisson-Boltzmann equation is... 

Because of the relative simplicity of the linear Poisson-Boltzmann equation, it is usually used in the first attempts to study relatively complex situations, such as when several particles are interacting simultaneously... 

FIG. 3 Comparison of the linear Derjaguin approximation with a numerical solution of the linear Poisson-Boltzmann equation for (a) constant potential and (b) constant charge density boundary conditions. (From Ref. 13.)... 

Unlike the other examples in this section, the equation governing the electrostatics here [i.e., Eq. (53)] is not the linearized Poisson-Boltzmann equation. However, considering interactions outside of thin double layers does have the effect of linearizing the problem. In Eq. (54), n is the fluid viscosity, K is the conductivity, is the zeta potential of the z th surface, and is a bipolar coordinate that is constant on the sphere and wall surfaces. It is this last condition (54), derived by Bike and Prieve [36] as a requirement to satisfy charge conservation, that couples the fluid mechanics with the electrostatics. 

In Secs. II.A and II.B above, we examined some common, approximate solutions to the linear Poisson-Boltzmann equation, and commented on the level of their agreement with exact solutions of that same equation. However, these approximations are no more accurate than the exact solutions, and the accuracy of the latter can only be ascertained by comparison with solutions to the complete, nonlinear Poisson-Boltzmann equation. From the... 

How good an approximation is the linear Poisson-Boltzmann equation to the full, nonlinear Poisson-Boltzmann equation ... 

The ability of the linear Poisson-Boltzmann equation to yield accurate results (i.e., results close to those for the full, nonlinear Poisson-Boltzmann equation) can reasonably be expected to depend on both geometry and boundary conditions. Comparisons for different geometries seem to yield... 

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. 

A hierarchy of approximations now exists for calculating interactions between a charged particle and a charged, planar interface in electrolyte solutions. At moderate surface potentials less than approximately 2(kT/e the linear Poisson-Boltzmann equation provides a good approximation in many circumstances, provided the solution is a 1 1 electrolyte at low to moderate ionic strength. The relative simplicity of the linear equation makes it particularly useful for examining problems that are complicated in other ways, such as interactions involving many particles, interactions with deformable interfaces, and interactions where the detailed structure and properties of the particle (or macromolecule) play an important role. 

In the limit of low potential right-hand side of the Poisson-Boltzmann equation can be linearized by using the first two terms of its Taylor series to get the linearized Poisson-Boltzmann equation ... 

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