Poisson-Boltzmann equation, for

The inner potentials have to be calculated by solving the Poisson-Boltzmann equations for the potentials this is done in Appendix A. 

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

The purpose of the present chapter is to introduce some of the basic concepts essential for understanding electrostatic and electrical double-layer pheneomena that are important in problems such as the protein/ion-exchange surface pictured above. The scope of the chapter is of course considerably limited, and we restrict it to concepts such as the nature of surface charges in simple systems, the structure of the resulting electrical double layer, the derivation of the Poisson-Boltzmann equation for electrostatic potential distribution in the double layer and some of its approximate solutions, and the electrostatic interaction forces for simple geometric situations. Nonetheless, these concepts lay the foundation on which the edifice needed for more complicated problems is built. 

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),... 

Solve the Poisson-Boltzmann equation for a spherically symmetric double layer surrounding a particle of radius Rs to obtain Equation (38) for the potential distribution in the double layer. Note that the required boundary conditions in this case are at r = Rs, and p - 0 as r -> oo. (Hint Transform p(r) to a new function y(r) = r J/(r) before solving the LPB equation.)... 

We return to the solution of the Poisson-Boltzmann equation for a spherical particle, Equation (19), with B = 0 ... 

Equation (1.23b) is the equilibrium Boltzmann distribution in a potential field. Substitution of (1.23b) into (1.9c) yields the Poisson-Boltzmann equation for equilibrium electric potential of the form... 

A plot of equilibrium r/e as a function of the external parameter a is schematically presented in Fig. 2.3.5a. The plot in Fig. 2.3.5a is markedly different from that in Fig. 2.3.4a by its lack of bifurcation. (Uniqueness of the appropriate solutions of the Poisson-Boltzmann equation for any values of a is proved in [18].) In the (F, ri) or (F, aeS) plane this corresponds to the existence of solutions of the Poisson-Boltzmann equations with finite F (bounded norm of the appropriate solution with a subtracted singular part due to the effective line charge) only for aeS < with adetermined by Conjecture 2.1. This is schematically illustrated in Fig. 2.3.5b. Note that F as a function of creS is constructed in a single counterion case by solving (2.3.3a) with a = of J e rdr and with the boundary conditions tp(a) = -aeS lna, = 0, and by going to the limit a- 0. 

Study of counterion condensation as a limiting property of the solutions of the Poisson-Boltzmann equation for arbitrary, charged cylindrical manifolds in H3 (see 2.3). 

T. Odijk, On the limiting solution of the cylindrical Poisson-Boltzmann equation for polyelectrolytes, Chem. Phys. Lett. 100 (1983), p. 145. 

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. 

The corresponding modified Poisson-Boltzmann equations for the planar interface located at x=0 are ... 

The modified Poisson-Boltzmann equation for a uni-univalent electrolyte confined between parallel plates is ... 

In a second method, the free energy was calculated by adding the electric (eq 1), entropic (eq 3), and chemical contributions, and the interaction free energy was obtained by subtracting the free energy for infinite separation. The integrals were calculated using the numerical solution of the Poisson—Boltzmann equation for y>(z,Z) and o(Z). 

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... 

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.)... 

Taking the Laplacian of Eq. (25), we write the Poisson-Boltzmann equations for the unary and binary potentials in the form... 

The Poisson-Boltzmann equations for the two first iterations can be rewritten as... 

Fogolari F, Brigo A, Molinari H (2002) The Poisson-Boltzmann equation for biomolecular electrostatics a tool for structural biology, J Mol Recognit, 15 377—392... 

The potential distribution, and hence the extent of the band bending, within the space charge layer of a planar macroscopic electrode may be obtained by solution of the one-dimensional Poisson-Boltzmann equation [95]. However, since the particles may be assumed to have spherical geometry, the Poisson-Boltzmann for a sphere must be solved. This has been done by Albery and Bartlett [131] in a treatment that was recently extended by Liver and Nitzan [125]. For an n-type semiconductor particle of radius r0, the Poisson-Boltzmann equation for the case of spherical symmetry takes the form ... 

Electrostatic. In many practical situations, both membrane and solute have net negative charges. Hence, as the solute approaches a pore in the membrane it experiences an electrostatic repulsion. A quantitative theoretical description of this interaction requires solution of the non-linear Poisson-Boltzmann equation for the interacting solute and membrane followed by calculation of the resulting force by integrating the electric stress tensor on the solute surface. Due to the complexity of the geometry... 

The exact solution for the Poisson-Boltzmann equation for flat plate with constant surface potential geometry with a S3nmnetrical electro-... 

Numerical Solution to Nonlinear Poisson-Boltzmann Equation for Bispherical Coordinates... 

Using a linearized flat plate solution to the Poisson—Boltzmann equation for the potential distribution of a sphere, what is the osmotic pressure of a 0.01 volume fraction suspension of 0.1 /an spheres immersed in a 1 1 salt solution at 0.1 M. The surface potential of the particle is 25 mV. Compare this value with that for the salt solution only. 

Fig. 3.37. Steps in the solution of the linearized Poisson-Boltzmann equation for point-charge ions.

This is the Poisson-Boltzmann equation for the potential distribution i/ (r). The surface charge density a of the particle is related to the potential derivative normal to the particle surface as... 

Combining Eqs. (1.17) and (1.19) gives the following Poisson-Boltzmann equation for the potential distribution ij/(x) ... 

We obtain the potential distribution around a sphere of radius a having a surface potential i/ o immersed in a solution of general electrolytes [9]. The Poisson-Boltzmann equation for the electric potential i//(r) is given by Eq. (1.94), which, in terms of/(r), is rewritten as... 

THE POISSON-BOLTZMANN EQUATION FOR A SURFACE WITH AN ARBITRARY FIXED SURFACE CHARGE DISTRIBUTION... 

On the basis of the theory of Camie and McLaughhn [5], the Poisson-Boltzmann equation for j/(x) can be modified in the present case as follows. Consider the end of a rod that carries the +e charge. As in the problem of rod-like divalent cations,... 

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