Full Poisson-Boltzmann equation

More sophisticated approaches to describe double layer interactions have been developed more recently. Using cell models, the full Poisson-Boltzmann equation can be solved for ordered stmctures. The approach by Alexander et al shows how the effective colloidal particle charge saturates when the bare particle charge is increased [4o]. Using integral equation methods, the behaviour of the primitive model has been studied, in which all the interactions between the colloidal macro-ions and the small ions are addressed (see, for instance, [44, 45]). 

Plot the potential versus distance for surface potentials of 60 mV, 100 mV, and 140 mV using the solution of the linearized and the full Poisson-Boltzmann equation for an aqueous solution with 2 mM KC1. 

Exact numerical results are used to validate the available approximate models described by Eqs. (16)-(19). The comparison is shown in Fig. 4 for particles with scaled radii Rk = 0.1 and Rk = 15. The interaction energy was determined for two identical spheres in a z z electrolyte solution. The approximate solutions are given by Eqs. (16)-(19) and the equation for the HHF model given in Table 3. For the exact numerical solution, the full Poisson-Boltzmann equation was discretized and solved by the finite volume method. The results have been plotted for two particle sizes kR = 0.1 (Fig. 4A) and k/ = 15 (Fig. 4B). 

Fig. 4 Comparison of the model improvements on the Derjaguin approximation to the exact numerical computational results of the full Poisson-Boltzmann equation for two spheres with the scaled radius Rk — 0.1 and Rk — 15 and constant surface potential [j/ ez/(k-gT) = 1. The scaled energy, G h), on the vertical axis is defined by G(h) = (/,)/jsM...

Qui et al. have compared experimental and calculated hydration free energies for a set of 35 small organic molecules with diverse functional groups by using the OPLS force field and the GB/SA hydration model [57], These calculations resulted in a mean absolute error of 0.9 kcal/mol. It is of interest to note that the results obtained with the GB/SA model were very similar to those obtained by the corresponding calculations using the full Poisson-Boltzmann equation. 

The integral in the denominator of the second term is taken over the system volume Vi accessible to species i and ensures the proper normalization of the number of ions for a system with a finite volume. When this integral is included, Eqs. [3] and [4] constitute what we refer to as the full Poisson-Boltzmann equation. For systems with a bulk electrolyte, the number of ions is considered infinite so the Boltzmann expression for mobile ions as written in the third line of Eq. [4] is used and the concentration at the outer boundary cf is replaced by a bulk concentration cf. Also, since ions of species 0 are finite in number (compared to bulk species), they are then neglected from the summation. 

Solving Eq. [382] for < j gives the discretized version of the full Poisson-Boltzmann equation (Eqs. [378] and [379]) applicable to non-Cartesian grids ... 

Let us finally turn to the more general case of two dissimilar surfaces. We start with two parallel, planar surfaces having constant and dissimilar surface potentials / tjjj. It is not possible to find a simple analytical expression using the full Poisson-Boltzmann equation. Depending on the approximations made, different expressions have been proposed [416-418.]. 

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... 

Holst, M., R. Kozack, F. Saied and S. Subramaniam. (1994a). Protein electrostatics-Rapid multigrid-based Newton algorithm for solution of the full nonlinear Poisson-Boltzmann equation. J. Biomol. Struct. Dynam. 11 1437-1445. 

Here we briefly discuss the calculation of the electrostatic energy of a molecular system from a finite difference solution of the linearized Poisson-Boltzmann equation. Calculations of the molecular electrostatic energy from grid solutions of the full nonlinear Poisson-Boltzmann equation are more involved and are discussed in detail elsewhere. ... 

In order to determine the values of the constants ci and C2, these equations must be matched to a solution valid in the outer region (i.e., f < 5). We followed Russel and Sdgter s approach in calculating f in the outer region numerically and matched f to the Fuoss solution for y > 5 to obtain a complete solution of the full nonlinear Poisson-Boltzmann equation for a arbitrary surface charge density... 

The interaction between two charged particles in a polar media is related to the osmotic pressure created by the increase in ion concentration between the particles where the electrical double-layers overlap. The repulsion can be calculated by solving the Poisson-Boltzmann equation, which describes the potential, or ion concentration, between two overlapping double-layers. The full theory is quite complicated, although a simplified expression for the double-layer interaction energy, V dl( ) between two spheres, can be written as follows ... 

Coimterion condensation has detractors (28-34), who point to flaws in the concept s derivation, such as artificial subdivision of the counterions into two populations, inappropriate extrapolation of the Debye-Hiickel approximation to regions of high electrostatic potential, and inconsistent treatment of counterions. The full nonlinear Poisson-Boltzmann equation offers a more rigorous way to interpret electrostatic phenomena in electrolyte solutions, but the physical picture obtained through this equation is different in some ways from the one suggested by condensation (21,34,35). In particular, a Poisson-Boltzmann analysis does not readily identify distinct populations of condensed and free counterions but rather a smoothly varying Gouy-Chapman layer. Nevertheless, Poisson-Boltzmann-based... 

Protein Electrostatics—Rapid Multigrid-Based Newton Algorithm for Solution of the Full Nonlinear Poisson-Boltzmann Equation. 

Full evaluation of equation (2.4) thus requires knowledge of the charge distribution at the electrode - electrolyte interface, a problem that has been explored in various works.For example, Dickinson and Compton recently used numerical modelling to solve the Poisson - Boltzmann equation, which describes the electric field in an electrolyte solution under thermodynamic equilibrium, for hemispherical electrodes. The simulations revealed a transition between two classical limits a planar double layer as predicted by the Gouy - Chapman model and the spherical double layer associated with a point charge (Coulomb s Law). This is illustrated in Fig. 2.2, in which the dimensionless charge density, Q ( FrqjRTEQEg) is plotted as a function of the dimensionless hemispherical electrode radius,... 

Let us compare results obtained with the linearized Poisson-Boltzmann equation (4.9) with the full solution equation (4.24). Figure 4.3 (left) shows the potential calculated for a monovalent salt at a concentration of 20 mM in water. The Debye length is 6.8 nm. For a low surface potential of 50 mV, both results agree well. When the surface potential is increased to 100,150, or even 200 mV, the full solution leads to lower potentials. At distances below kn /2, the decay is, therefore, steeper than just the exponential decay. This steep decay at small distances becomes progressively more effective at higher and higher surface potentials, which leads to saturation behavior. For example, the potential at a distance of one Debye length can never exceed 40 mV irrespective of the surface potential. 

Figure 4.3 Potential versus distance for surface potentials of 50, 100, 150, and 200mV (from bottom to top) with 2 m M monovalent salt. Left Planar surface. Results calculated with the full solution equation (4.24) and the solution of the linearized Poisson-Boltzmann equation (4.9)...

Simultaneous measurements of d and osmotic pressure provide a relation between the separation of bilayers and their mutual repulsive pressure. Measurement of the electrostatic repulsion is, in fact, a determination of the electrostatic potential midway between bilayers relative to the zero of potential in the dextran reservoir. The full nonlinear Poisson-Boltzmann differential equation governing this potential has been integrated (I) from the midpoint to the bilayer surface to let us infer the surface potential. The slope of this potential at the surface gives a measure of the charge bound. 

It should also be noted that the full complexity of the PB equation is not always necessary. In many cases mobile ion densities are small enough that the exponential factors in the mobile ion distribution can be approximated by keeping only terms linear in the potential, leading to the linearized Poisson— Boltzmann (LPB) equation. In this case the ionic terms can be added to the diagonal of the coefficient matrix A, and extremization along any component direction is analytic because the surface is parabolic. Another advantage of LPB is the additivity of terms in the electrostatic potential, which enables contributions made by any part of a molecule to be determined. 

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