Poisson-Boltzmann differential

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. 

Report of relation (19.9) in relation (19.8) gives linear Poisson-Boltzmann differential equation ... 

Poisson s equation and Eq. (10) are combined to form the Poisson-Boltzmann differential equation that, for the case of electrostatic potential variation in one dimension and one predominant defect, results in ... 

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... 

The third point implies that it is possible to develop a physical theory for ionic interactions that is relatively simple and still useful. The most frequently used is the Poisson-Boltzmann (P-B) equation, which combines the Poisson equation from classical electrostatics with the Boltzmann distribution from statistical mechanics. This is a second-order nonlinear differential equation and its solution depends on the geometry and the boundary conditions. 

A. Friedman and K. Tintarev, Boundary asymptotics for solutions of the Poisson-Boltzmann equation, J. Differential Equations, 69 (1987), pp. 15-38. 

K. Tintarev, Fundamental solution of the Poisson-Boltzmann equation, in Differential Equations and Mathematical Physics, I. W. Knowles and Y. Saito, eds., Lecture Notes in Math. 1285, Springer-Verlag, Berlin, New York, 1987. 

Often this equation is referred to as the Poisson-Boltzmann equation. It is a partial differential equation of second order, which in most cases has to be solved numerically. Only for some simple geometries can it be solved analytically. One such geometry is a planar surface. 

For the simple case of a planar, infinitely extended planar surface, the potential cannot change in the y and z direction because of the symmetry and so the differential coefficients with respect to y and z must be zero. We are left with the Poisson-Boltzmann equation which contains only the coordinate normal to the plane x ... 

The electrostatic field in the stationary state is described by the Poisson-Boltzmann equation. The PB model constitutes the fundamental equation of electrostatics and is based on the differential Poisson equation which describes the electrostatic potential 4>(r) in a medium with a charge density p(r) and a dielectric scalar field e(r) ... 

Moreira IS, Fernandes PA and Ramos MJ (2005) Accuracy of the numerical differentiation of the Poisson-Boltzmann equation, J Mol Struct (Theochem), 729 11-18... 

Poisson-Boltzmann equation — The Poisson-Boltz-mann equation is a nonlinear, elliptic, second-order, partial differential equation which plays a central role, e.g., in the Gouy-Chapman (- Gouy, - Chapman) electrical -> double layer model and in the - Debye-Huckel theory of electrolyte solutions. It is derived from the classical -> Poisson equation for the electrostatic potential... 

The three-dimensional, second-order, nonlinear, elliptic partial differential equation may be simplified in the limit of weak electrolyte solutions, where the hyperbolic sine of is well approximated by 4). This yields the linearized Poisson—Boltzmann equation... 

Numerical solution of the Poisson and Poisson-Boltzmann equations is more complicated since these are three dimensional partial differential equations, which in the latter case can be non-linear. Solutions in planar, cylindrical and spherical geometry, are... 

Note the non-tmncated form of the Poisson-Boltzmann equation has terms in powers of and would be a non-linear differential equation. 

To solve the PB equation for arbitrary geometries requires some type of discretization, to convert the partial differential equation into a set of difference equations. Finite difference methods divide space into a cubic lattice, with the potential, charge density, and ion accessibility defined at the lattice points (or grid points ) and the permittivity defined on the branches (or grid lines ). Equation [1] becomes a system of simultaneous equations referred to as the finite difference Poisson-Boltzmann (FDPB) equation ... 

If one accepts the continuum, linear response dielectric approximation for the solvent, then the Poisson equation of classical electrostatics provides an exact formalism for computing the electrostatic potential (r) produced by a molecular charge distribution p(r). The screening effects of salt can be added at this level via an approximate mean-field treatment, resulting in the so-called Poisson-Boltzmann (PB) equation [13]. In general, this is a second order non-linear partial differential equation, but its simpler linearized form is often used in biomolecular applications ... 

The substitutions translate the second order, nonlinear Poisson-Boltzmann equation into two coupled first order differential equations. The boundary conditions are ... 

In the case of electrode-electrolyte solution interfaces, the Poisson-Boltzmann equation has been modified for integrating many effects as, for example, finite ion size, concentration dependence of the solvent, ion polarizability, and so on. More often, this modification consists in the introduction of one or several supplementary terms to the energetic contribution in the distribution, which leads to modified Poisson-Boltzmann (MPB) nonlinear differential equations [52],... 

Experimentally, the fraction of free counterions in salt-free polyelectrolyte solutions is believed to give the main contribution to the system osmotic pressure. In the framework of the two-state models, the osmotic pressure is equal to the osmotic pressure of the free counterions. The more accurate analysis of the counterion effect on the solution osmotic pressure can be done in the framework of the Poisson-Boltzmann approach and its two-zone model simplification. In order to obtain an expression for the osmotic pressure in the framework of the two-zone model, one has to know the counterion concentration at the outer boundary of the spherical region. This requires knowledge of the electrostatic potential within the spherical zone. However, one can avoid solving the nonlinear Poisson-Boltzmann equation and use the relation between the local pressure P(r) and the electric field (r). To obtain this relation, one has to combine the differential form of the Gauss law ... 

The Poisson-Boltzmann equation (23.5) is a nonlinear second-order differential equation from which you can compute ip if you know the charge density on P and the bulk salt concentration, rioo- This equation can be solved numerically by a computer. However, a linear approximation, which is easy to solve without a computer, applies when the electrostatic potential is small. For small potentials, zeip/kT 1, you can use the approximation sinh(x) s [(I + x) - (I - x)] 12 = X (which is the first term of the Taylor series expansion for the two exponentials in sinh(x) (see Appendix C, Equation (C.l)). Then Equation (23.5) becomes... 

Equations (23.22) and (23.23) come from the linearized Poisson-Boltzmann Equation (23.6). It is a general property of linear differential equations that you can sum their solutions. For example, if you have two point charges in a uniform salt solution, q[ at distance r from P, and q-2 at from P, the... 

The Poisson-Boltzmann equation is a difficult nonlinear differential equation. Even if we solve it exactly, it will lead to some inconsistencies in the solution, resulting from the approximations introduced above. Fortunately, since we are interested only in the limit of low-solute densities, where the average interionic distances are large, we can further assume that... 

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