Linearized PB equation

Another attempt to go beyond the cell model proceeds with the Debye-Hiickel-Bjerrum theory [38]. The linearized PB equation is used as a starting point, however ion association is inserted by hand to correct for the non-linear couplings. This approach incorporates rod-rod interactions and should thus account for full solution properties. For the case of added salt the theory predicts an osmotic coefficient below the Manning limiting value, which is much too low. The same is true for a simplified version of the salt free case. 

Two theoretical techniques worthy of serious review here, perturbation and Green function methods, can be considered complementary. Perturbation methods can be employed in systems which deviate only slightly from regular shape (mostly from planar geometry, but also from other geometries). However, they can be used to treat both linear and nonlinear PB problems. Green function methods on the other hand are applicable to systems of arbitrary irregularity but are limited to low surface potential surfaces for which the use of the linear PB equation is permitted. Both methods are discussed here with reference to surfactant solutions which are a potentially rich source of nonideal surfaces whether these be solid-liquid interfaces with adsorbed surfactants or whether surfactant self-assembly itself creates the interface. 

Since the coefficients of each member in the set of differential equations are spatially dependent and the equations themselves inhomogeneous, higher order terms are obtained with increasing difficulty. This is in contrast to the case with solving the linear PB equation with the same undulating surface. In this case the set member differential equations are simply homogeneous and have constant coefficients,... 

Before discussing the BEM, let us first set to record the fundamental Green functions to the linear PB equation. These are the free space Green functions which satisfy Eq. (106) together with the boundary condition at infinity ... 

The problem considered by Ohshima and Kondo [28] starts with a charged plate and a charged ion-penetrable sphere with radius a,separated by a distance h. Thus we first approximate short oligonucleotides as an ion penetrable sphere which for oligos 7-8 bases long is reasonable but short by experimental standards. The potential field, ip is assumed to obey a linearized PB equation,... 

The above comparison against experiment was based on calculating the electrostatic interaction energy from the linearized PB equation. It has been found that, when the full PB equation was used, agreement with experiment improved, albeit modestly [19]. This underscores the point that a rigorous treatment of elecfrostatic interactions is essential for the accuracy of calculated ko-... 

The full or nonlinear form of the problem given in Eq. [5] is often simplified to the linearized PB equation by replacing the sinhtp(x) term with its first-order approximation, sinh[Pg.355]

Solution of the PB equation in cylindrical coordinates is relevant for the case of pores with charged walls filled with an electrolyte solution. This case has been treated analytically by several authors (Morrison and Osterle 1965 Rice and Whitehead 1965 Philip and Wooding 1970 Rice and Home 1981 Sigal and Ginsburg 1981 Olivares and McQuarrie 1985 Rice 1985 Rice and Home 1985). For the linearized PB Equation 3.11 applied to a pore fiUed with a uniform solution (Rice and Whitehead 1965), the potential is given by... 

Several models that depart partially or totally from DLVO have been proposed. One such model is the Spitzer dissociative electrical double layer theory (Spitzer 1984, 2003 and references therein). It essentially uses the linearized PB equation (which is consistent with Maxwellian electromagnetism) along with a coion exclusion boundary, which prevents ions of the same charge as the surface becomes too close in practice, this avoids negative concentrations, which will be predicted by the linear PB theory. It also includes a double layer association parameter a, which gives the fractions of counterions that are associated to the surface forming the Stern layer (Spitzer 1992). This theory, however, has not been further developed. 

Delphi Code to solve the linearized PB equation. Available http //wiki. c2b2.columbia.edu/honiglab public/index.php/Main Page. 

Another frequently used form of the PB equation can be derived by applying the linearization procedure, which is justified if the maximum term z e if/kT < 1. By expanding the exponential terms and exploiting the electroneutrality condition, one obtains the linear PB equation... 

In the case of the double layer around a spherical particle, an exact solution can only be derived for the linear PB equation. The potential distribution has the form... 

However, exact analytical formulas can be derived for two plates bearing different surface charges cij and ct immersed in an electrolyte solution of arbitrary composition. The solution of the linear PB equation with the constant charge (c.c.) boundary condition [Eq. (17)] gives the following expression for the electric potential distribution in the gap between the plates ... 

The difference between both models appearing at short separations seems highly unphysical. It is caused by the violation of the low-potential assumption. Indeed, in order to observe the c.c. boundary conditions, the surface potential of the plates should tend to infinity when they closely approach each other, even if these potentials were very low at large separations. As a consequence, / 1 for A 0 and the linear PB equation is not valid. Hence, Eqs. (33) and (34) are incoherent for the c.c. model and should not be used for short separations. 

To overcome some of the shortcomings of the above simple model, one can make use of the linearized PB equation ... 

The solution to the linearized PB equation results in an electrostatic potential map generated by the set of permanent point charges embeded in a low dielectric cavity, which is surrounded by a higher dielectric solvent. The total linearized PB energy results from a product of the partial charge at the site i qi) with the computed electrostatic potential V ... 

The solution of the linearized PB Equation 8.5 with the boundary conditions in Equation 8.4 reads as... 

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