Poisson-Boltzmann equation nonlinear

Sharp, K. A., Honig, B. Calculating total electrostatic energies with the nonlinear Poisson-Boltzmann equation. J. Phys. Chem. 94 (1990) 7684-7692. Zhou, H.-X. Macromolecular electrostatic energy within the nonlinear Poisson-Boltzmann equation. J. Chem. Phys. 100 (1994) 3152-3162. 

Rahaman and Hatton [152] developed a thermodynamic model for the prediction of the sizes of the protein filled and unfilled RMs as a function of system parameters such as ionic strength, protein charge, and size, Wq and protein concentration for both phase transfer and injection techniques. The important assumptions considered include (i) reverse micellar population is bidisperse, (ii) charge distribution is uniform, (iii) electrostatic interactions within a micelle and between a protein and micellar interface are represented by nonlinear Poisson-Boltzmann equation, (iv) the equilibrium micellar radii are assumed to be those that minimize the system free energy, and (v) water transferred between the two phases is too small to change chemical potential. 

The electrostatic force (F ) between two charged plates separated by an electrolyte solution can be determined from an existing imphcit solution to the nonlinear Poisson-Boltzmann equation and can be expressed in the form [187]... 

Figure 6.10 Electrostatic double-layer force between a sphere of R = 3 /um radius and a flat surface in water containing 1 mM monovalent salt. The force was calculated using the nonlinear Poisson-Boltzmann equation and the Derjaguin approximation for constant potentials (tpi = 80 mV, ip2 = 50 mV) and for constant surface charge (i/2/Ad so that at large distances both lead to the same potential.

Comparison with Predictions of the [X Nonlinear Poisson-Boltzmann Equation... 

Rocchia W, Alexov E, Honig B (2001) Extending the applicability of the nonlinear Poisson-Boltzmann equation Multiple dielectric constants and multivalent ions. J Phys Chem B 105 6507-6514... 

A second and more rigorous approach to the calculation of the double-layer contribution is to solve the nonlinear Poisson-Boltzmann equation, taking into account the following dissociation equilibria to determine the surface charge and potential at each value of x ... 

To calculate the double-layer force, the nonlinear Poisson-Boltzmann equation was solved for the case of two plane parallel plates, subject to boundary conditions which arise from consideration of the simultaneous dissociation equilibria of multiple ionizable groups on each surface. Deijaguin s approximation is then used to extend these results to calculate the force between a sphere and a plane. Details of the method can be found in Ref. (6). 

In Figure 6a, the force per unit area between surfaces with grafted polyelectrolyte brushes, plotted as a function of their separation distance 2d, calculated in the linear approximation, is compared with the numerical solution of the nonlinear Poisson—Boltzmann equations, for a system with IV = 1000, a = 1 A, ce = 0.01 M, s2 = 1000... 

Understandably, it is much more common to see analyses of problems based on Eq. (32) since for simple geometries the solution can be written down in closed form, expressed in terms of simple functions. For plane surfaces, for example, the solutions are elementary hyperbolic functions while for an isolated spherical surface the Debye-Huckel potential expression prevails. For two charged spherical surfaces the general solution can be written down as a convergent infinite series of Legendre polynomials [16-19]. The series is normally truncated for calculation purposes [16] K For an ellipsoidal body ellipsoidal harmonics are the natural choice for a series representation [20]. (The nonlinear Poisson Boltzmann equation has been solved numerically for a ellipsoidal body... 

In the paragraphs below, we first examine the simple, analytical results that can be derived from the linear Poission-Boltzmann equation for a single particle interacting with a flat surface. Next, more complicated physical situations are considered, including interactions between many particles and a wall between a particle and a deformable interface between a protein and a wall and between a moving particle and a wall. In Sec. Ill, solutions to the nonlinear Poisson-Boltzmann equation are considered, and comparisons are made between the linear and nonlinear versions and also with more... 

III. ACCURACY OF LINEAR AND NONLINEAR POISSON-BOLTZMANN EQUATIONS... 

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 ... 

A. Comparisons between the Linear and Nonlinear Poisson-Boltzmann Equations... 

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... 

Perhaps the first comparison that should be made is that between the potential near an isolated, charged plate as predicted by the linear and nonlinear Poisson-Boltzmann equations. The linear result is given by... 

In all of the discussion above, comparisons have been made between various types of approximations, with the nonlinear Poisson-Boltzmann equation providing the standard with which to judge their validity. However, as already noted, the nonlinear Poisson-Boltzmann equation itself entails numerous approximations. In the language of liquid state theory, the Poisson-Boltzmann equation is a mean-field approximation in which all correlation between point ions in solution is neglected, and indeed the Poisson-Boltzmann results for sphere-sphere [48] and plate-plate [8,49] interactions have been derived as limiting cases of more rigorous approaches. For many years, researchers have examined the accuracy of the Poisson-Boltzmann theory using statistical mechanical methods, and it is... 

The TPE-HNC/MS theory reduces to an integral form of the nonlinear Poisson-Boltzmann equation in the limit of point ions [8,44]. Hence, in that limit agreement between the two methods is exact. For a 0.1 M, 1 1 electrolyte separating plates with surface potentials of 70 mV, Lozada-Cassou and Diaz-Herrera [8] show excellent agreement between the TPE-HNC/MS theory and the Poisson-Boltzmann equation. The agreement becomes very poor, however, at a higher concentration of 1 M. In addition, like the Monte Carlo and AHNC results, the TPE-HNC/MS theory predicts attractive interactions at sufficiently high potentials and/or salt concentrations, and such effects are missed entirely by the Poisson-Boltzmann equation. 

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. 

Sharp KA, Honig B, (1990) Calculating Total Electrostatics energies with the Nonlinear Poisson-Boltzmann Equation, J. Phys. Chem. 94 7684—7692... 

This transformed the nonlinear Poisson-Boltzmann equation into the linear Helmholtz-type equation... 

Y. N. Vorobjev, J. A. Grant, and H. A. Scheraga, ]. Am. Chem. Soc., 114, 3189 (1992). A Combined Iterative and Boundary-Element Approach for Solution of the Nonlinear Poisson-Boltzmann Equation. 

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