Poisson-Boltzmann equation 250, Table

Marmur [12] has presented a guide to the appropriate choice of approximate solution to the Poisson-Boltzmann equation (Eq. V-5) for planar surfaces in an asymmetrical electrolyte. The solution to the Poisson-Boltzmann equation around a spherical charged particle is very important to colloid science. Explicit solutions cannot be obtained but there are extensive tabulations, known as the LOW tables [13]. For small values of o, an approximate equation is [9, 14]... 

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

Similar calculations, based on the same principles, were carried out for spherical particles. Since the Poisson-Boltzmann equation cannot be integrated analytically in spherical symmetry, a numerical integration was performed. The computer-generated numerical tables of reduced potential as a function of reduced distance of Loeb, Wiersema, and... 

Clint el al. (4) measured the rate of deposition of 0.43-gm-diameter polystyrene latex particles onto a rotating disk coated with a polystyrene film in Ba(N03)s solutions of three different ionic strengths. Results are reported ia Table II. Also reported in this table are the surface potentials of the disk which are needed to force agreement between predicted and observed rates. Speculations 1 and 2 again refer to approach at constant surface potential or charge, respectively, when the potential is small enough to linearize the Poisson-Boltzmann equation. However, when the... 

Loeb et al. tabulated numerical computer solutions to the nonlinear spherical Poisson-Boltzmann equation (1.63). On the basis of their numerical tables, they discovered the following empirical formula for the cT-ij/o relationship ... 

In Table 4.1, which shows the relationship between tdon and jo, we compare exact numerical results obtained by solving the nonlinear Poisson-Boltzmann equations (4.59) and (4.60) and approximate results obtained via Eqs. (4.67) and (4.74) for several values of jo, Ka, and Kb. Good agreement is seen between exact and approximate results. 

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). 

Table 5.1. Analytical solutions of the Poisson-Boltzmann equation for single electrolytes ...

Table 1 presents a list of the major software that currently solves the Poisson-Boltzmann equation for biomolecular systems. A variety of such programs exist, ranging from multipurpose computational biology packages (e.g., CHARMM, Jaguar, UHBD, and MacroDox) to specialized PB solvers (e.g., APBS, MEAD, and DelPhi). 

Ff may be estimated from experimental measurements of the zeta potential [173] using numerical solutions of the Poisson-Boltzmann equations [174]. F i is, of course, positive and its contribution to the total AGJ value is generally small (see Table 3.11). 

The experimental and theoretical values are given in Table IV and in Figure 27. The agreement is better than ever before and namely better than that obtained by the resolution of the Poisson-Boltzmann Equation [23, 19]. 

We use the Gouy-Chapman theory for the diffuse layer which is based on the Poisson-Boltzmann (P.B.) equation for the potential distribution. Although the different corrections to the P.B. equation in double-layer theory have been investigated (20, 21, 22, 23), it is difficult to state precisely the range of validity of this equation. In the present problem the P.B. equation seems a reasonable approximation at 0.1M of a 1-1 electrolyte to 50mV for the mean electrostatic potential pd at the ohp (24) this upper limit for pd increases with a decrease in electrolyte concentration. All the values for pd calculated in Tables I-IV are less than 50 mV— most of them are well below. If n is the volume density of each ion type of the 1-1 electrolyte in the substrate, c the dielectric constant of the electrolyte medium, and... 

Region 1 is controlled by a hard-sphere potential. The potentials of mean force of regions 2 and 3, and are split into two parts according to Eq. (73). The electrostatic parts W j are obtained as solutions of Poisson-Boltzmann differential equations and appropriate boundary conditions the nonelectrostatic parts can be chosen as step potentials, that is, TP = const, TP = 0. The total mean force potentials are compiled in Table II. The parameter k is the reciprocal Debye length as defined by the relationship... 

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