Non-linear Poisson Boltzmann equation

A more detailed view of the dynamies of a ehromatin chain was achieved in a recent Brownian dynamics simulation by Beard and Schlick [65]. Like in previous work, the DNA is treated as a segmented elastic chain however, the nueleosomes are modeled as flat cylinders with the DNA attached to the cylinder surface at the positions known from the crystallographic structure of the nucleosome. Moreover, the electrostatic interactions are treated in a very detailed manner the charge distribution on the nucleosome core particle is obtained from a solution to the non-linear Poisson-Boltzmann equation in the surrounding solvent, and the total electrostatic energy is computed through the Debye-Hiickel approximation over all charges on the nucleosome and the linker DNA. 

At high electrolyte concentrations, the linear approximation fails and Eqs. (31a), (31b), (32a), (32b) and (33) are no longer valid. A simple solution, even approximate, of the non-linear Poisson-Boltzmann equation is more difficult to obtain however, the general behavior of the system can be understood from the following semi-quantitative analysis. 

Electrostatic. Virtually all colloids in solution acquire a surface charge and hence an electrical double layer. When particles interact in a concentrated region their double layers overlap resulting in a repulsive force which opposes further approach. Any theory of filtration of colloids needs to take into account the multi-particle nature of such interactions. This is best achieved by using a Wigner-Seitz cell approach combined with a numerical solution of the non-linear Poisson-Boltzmann equation, which allows calculation of a configurational force that implicitly includes the multi-body effects of a concentrated dispersion or filter cake. 

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... 

W.R. Bowen and A.O. Sharif, Adaptive finite element solution of the non-linear Poisson-Boltzmann equation—a charged spherical particle at various distances from a charged cylindrical pore in a charged planar surface, J. Colloid Interface Sci. 187 (1997)... 

B. A. Luty, M. E. Davis, and J. A. McCammon,/. Comput. Chem., 13, 1114 (1992). Solving the Finite-Difference Non-Linear Poisson—Boltzmann Equation. 

Sharp, K. and B. Honig. (1990a). Calculating total electrostatic energies with the non-linear Poisson-Boltzmann Equation. J. Phys. Chem. 94 7684-7692. 

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

Suppose that the system is in the steady state and that there is no fluid flow or imposed electric fields. Further suppose that the geometry is such that the electrolyte-substrate interface is an iso-surface of xjrk- Then it readily follows from Eqs. (5) and (10) and the boundary condition of no flux into the wall that V t/fjt = 0 everywhere. Therefore, Hk = exp —Zke

charge density pe and substituting in Eq. (4), we get the non-linear Poisson-Boltzmann equation for determining the potential... 

M. A. Lampert and R. U. Martinelli, Chem. Phys., 88, 399 (1984). Solution of the Non-Linear Poisson-Boltzmann Equation in the Interior of Charged, Spherical and Cylindrical Vesicles. I. The High-Charge Limit. 

Simulation results were compared with the predictions of the Ornstein-Zernike (OZ) equation with the hypernetted chain (HNC) closure approximation and the non-linear Poisson-Boltzmann equation, both augmented by pertinent Lifshitz NES potentials. We show in Fig. 1 that there is very good agreement between modified Poisson-Boltzmann theory, MC simulations, and HNC calculations when the counterions and co-ions are monovalent. There is also good agreement between the different approaches with divalent co-ions (not shown here). However, the results from MPBE cannot account for ion correlation effects that occur in Fig. 2 when the counterions are divalent. The reason is simply that the... 

Fig. 1. Concentration profiles near a macroion (gm = 30 A and —20 o) in a monovalent electrolyte solution of ionic strength 1.0 M for (a) NaCl and (b) Nal. Open circles represent counterion concentrations and dark circles the co lons. Solid lines are numerical solutions of the non-linear Poisson-Boltzmann equation and dashed lines are for the OZ-HNC integral equation. ...

The polyelectrolyte chain is often assumed to be a rigid cylinder (at least locally) with a uniform surface charge distribution [33-36], On the basis of this assumption the non-linearized Poisson-Boltzmann (PB) equation can be used to calculate how the electrostatic potential

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A new theory of electrolyte solutions is described. This theory is based on a Debye-Hiickel model and modified to allow for the mutual polarization of ions. From a general solution of the linearized Poisson-Boltzmann equation, an expression is derived for the activity coefficient of a central polarized ion in an ionic atmosphere of non-spherical symmetry that reduces to the Debye-Hiickel limiting laws at infinite dilution. A method for the simultaneous charging of an ion and its ionic cloud is developed to allow for ionic polarization. Comparison of the calculated activity coefficients with experimental values shows that the characteristic shapes of the log y vs. concentration curves are well represented by the theory up to moderately high concentrations. Some consequences in relation to the structure of electrolyte solutions are discussed. 

This series arises naturally, when expressing the Coulomb potential of a charge separated by a distance s from the origin in terms of spherical coordinates. The positive powers result when r < s, while for r > s the potential is described by the negative powers. Similarly the solutions of the linearized Poisson-Boltzmann equation are generated by the analogous expansion of the shielded Coulomb potential exp[fix]/r of a non-centered point charge. Now the expansion for r > s involves the modified spherical Bessel-functions fo (x), while lor r < s the functions are the same as for the unshielded Coulomb potential,... 

The final charge on the surface is balanced by counterions from the solution, establishing the so-called diffuse double layer. The solution of the non-linear Poisson-Boltzmann (PB) equation yields the density of counterions, the potential and the electric field at any point [19]. The characteristic length or thickness of the diffuse double layer is the so-called Debye length, which determines the exponential decay of the counterion density away from the surface. It can also be viewed as a screening length and depends solely on the properties of the liquid such as the concentration of the electrolyte, and not on any property of the surface. For a 1 mM solution of a 1 1 electrolyte it is approximately 10 nm [19]. 

One of the simplest equations is obtained using the Debye-Hiickel approximation (for low potentials) and the superposition principle. The latter assumes that the unperturbed potential near a charged surface can be simply added to that potential due to the other (unperturbed) surface. Thus, for the example shown in the Figure 6.12, it follows that /m = 2 /d/2- This is precisely valid for Coulomb-type interactions, where the potential at any point can be calculated from the potentials produced by each fixed charge, individually. However, the Poisson-Boltzmann equation is non-linear (this has to do with the fact that in the diffuse double-layer the ions are not fixed but move because of their kinetic energy) and so this is formally not correct although it still offers a useful approximation. 

In the literature sometimes the statement is made that the Poisson-Boltzmann equation is only compatible with electrostatics if linearized, which is not correct. The argument refers to the superposition principle which relies on the presupposed linearity of Poisson s equation. Note, however, that Poisson s equation is not linear if the charge density depends on f itself in a non-linear way as it is the case here. 

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. 

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 ... 

Hogg, Healy, and Fuerstenau [7] developed their HHF theory to describe the interactions of two particles of different size. In 1985, Matijevi and Barouch [8] evaluated the validity of the HHF theory for the electrostatic interaction between two surfaces of different sizes for both unlike particles with potentials opposite in sign, and for particles with same sign potentials. The computational calculations overcame the problem of the accuracy in the evaluation of incomplete elliptic integrals of the first kind, which is a direct consequence of a non-linearity of the Poisson-Boltzmann equation. They concluded that for systems with dissimilar particles with either opposite signs or the same sign, the approximation of the HHF theory achieved good results. However, when potential differences increased, marked deviations from the HHF theory were found. 

The combination of the Boltzmann distribution to the Poisson equation for net excess charge density leads to a non-linear second-order differential equation for the electric potential 0. Therefore, the Poisson-Boltzmann equation is written as ... 

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