Poisson-Boltzmann equation linearized

Davis, M. E., McCammon, J. A. Solving the finite difference linearized Poisson-Boltzmann equation A comparison of relaxation and conjugate gradients methods.. J. Comp. Chem. 10 (1989) 386-394. 

If there are ions in the solution, they will try to change their location according to the electrostatic potential in the system. Their distribution can be described according to Boltzmarm. Including these effects and applying some mathematics leads to the final linearized Poisson-Boltzmann equation (Eq. (43)). 

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 ... 

Derive and solve the appropriate linear Poisson-Boltzmann equation for the interface between two immiscible solutions. 

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... 

A more detailed view of the dynamies of a ehromatin chain was achieved in a recent Brownian dynamics simulation by Beard and Schlick [65]. Like in previous work, the DNA is treated as a segmented elastic chain however, the nueleosomes are modeled as flat cylinders with the DNA attached to the cylinder surface at the positions known from the crystallographic structure of the nucleosome. Moreover, the electrostatic interactions are treated in a very detailed manner the charge distribution on the nucleosome core particle is obtained from a solution to the non-linear Poisson-Boltzmann equation in the surrounding solvent, and the total electrostatic energy is computed through the Debye-Hiickel approximation over all charges on the nucleosome and the linker DNA. 

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...

At high electrolyte concentrations, the linear approximation fails and Eqs. (31a), (31b), (32a), (32b) and (33) are no longer valid. A simple solution, even approximate, of the non-linear Poisson-Boltzmann equation is more difficult to obtain however, the general behavior of the system can be understood from the following semi-quantitative analysis. 

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... 

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