Poisson-Boltzmann equation, calculation

Fig. 5.1. Reaction rate for the complexation of Ni2 + and PADA (Pyridine-2-azo-p-dimethyl-aniline) in sodium decylsulfate (NaDeS) solutions. — calculated rate from the Poisson-Boltzmann equation. - calculated rate using the assumption that all Ni2+ ions are bound to the micellar surface. (From Ref.285 )...

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. ... 

Sharp, K. A., Honig, B. Calculating total electrostatic energies with the nonlinear Poisson-Boltzmann equation. J. Phys. Chem. 94 (1990) 7684-7692. Zhou, H.-X. Macromolecular electrostatic energy within the nonlinear Poisson-Boltzmann equation. J. Chem. Phys. 100 (1994) 3152-3162. 

Another way of calculating the electrostatic component of solvation uses the Poisson-Boltzmann equations [22, 23]. This formalism, which is also frequently applied to biological macromolecules, treats the solvent as a high-dielectric continuum, whereas the solute is considered as an array of point charges in a constant, low-dielectric medium. Changes of the potential within a medium with the dielectric constant e can be related to the charge density p according to the Poisson equation (Eq. (41)). 

Calculation of Electrostatic Potential by the Poisson-Boltzmann Equation... 

In addition to the nearest-neighbor interaction, each ion experiences the electrostatic potential generated by the other ions. In the literature this has generally been equated with the macroscopic potential 0 calculated from the Poisson-Boltzmann equation. This corresponds to a mean-field approximation (vide infra), in which correlations between the ions are neglected. This approximation should be the better the low the concentrations of the ions. 

Two remaining problems relating to the treatment of solvation include the slowness of Poisson-Boltzmann calculations, when these are used to treat electrostatic effects, and the difficulty of keeping buried, explicit solvent in equilibrium with the external solvent when, e.g., there are changes in nearby solute groups in an alchemical simulation. Faster methods for solving the Poisson-Boltzmann equation by means of parallel finite element techniques are becoming available, however.22 24... 

A solvated MD simulation is performed to determine an ensemble of conformations for the molecule of interest. This ensemble is then used to calculate the terms in this equation. Vm is the standard molecular mechanics energy for each member of the ensemble (calculated after removing the solvent water). G PB is the solvation free energy calculated by numerical integration of the Poisson-Boltzmann equation plus a simple surface energy term to estimate the nonpolar free energy contribution. T is the absolute temperature. S mm is the entropy, which is estimated using... 

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

The presence of the diffuse layer determines the shape of the capacitance-potential curves. For a majority of systems, models describing the double-layer structure are oversimplified because of taking into account only the charge of ions and neglecting their specific nature. Recently, these problems have been analyzed using new theories such as the modified Poisson-Boltzmann equation, later developed by Lamper-ski. The double-layer capacitanties calculated from these equations are... 

For our work, expressions of Ohshlma et. al. (37) obtained from an approximate form of the Poisson-Boltzmann equation were used. These analytical expressions agree with the exact solution for xRp 2. (All of our calculations meet this criterion.) The relation between the surface potential and the surface charge density Is (37)... 

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. 

The potential distribution ( ) in the space-charge region is described, with due account for Eq. (13), by the self-consistent Poisson-Boltzmann equation. Its first integral can be calculated analytically, so that the electric field at the semiconductor surface Ssc = — /dx is expressed as (Garrett and Brattain, 1955 see also Frankl, 1967)... 

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. 

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... 

In order to determine the force in a specific situation, the potential must first be calculated. This is done by solving the Poisson-Boltzmann equation. In a second step, the force per unit area is calculated. It does not matter for which point we calculate II, the value must be the same for every . 

In order to calculate the potential distribution in the gap we need not only the Poisson-Boltzmann equation, but in addition, boundary conditions must be specified. Two common types of boundary conditions are ... 

Figure 6.10 Electrostatic double-layer force between a sphere of R = 3 /um radius and a flat surface in water containing 1 mM monovalent salt. The force was calculated using the nonlinear Poisson-Boltzmann equation and the Derjaguin approximation for constant potentials (tpi = 80 mV, ip2 = 50 mV) and for constant surface charge (i/2/Ad so that at large distances both lead to the same potential.

From the solution of the Poisson-Boltzmann equation one can calculate the electrostatic contribution to the free energy. It is illustrative to divide G into two parts302. The first is concerned with the free energy of the electric field and is given by ... 

Nielsen, J. E. and Vriend, G. (2001) Optimizing the hydrogen-bond network in Poisson-Boltzmann equation-based pKa calculations. Proteins 43,403-412. 

An alternative theoretical approach is the application of the Poisson-Boltzmann equation on the so-called cell model, assuming a parallel and equally spaced packing of rod-like polyions [62, 63]. This allows one to calculate at finite concentration according to ... 

Two approaches for the calculation of the double-layer contribution are explored. Hogg el of. (16) linearized the Poisson- -Boltzmann equation to compute the double-layer force between two dissimilar plane surfaces, then used Derjaguin s approximation to extend this result to the interaction of two spheres of different radii. When the radius of one sphere is infinite, their result becomes... 

A second and more rigorous approach to the calculation of the double-layer contribution is to solve the nonlinear Poisson-Boltzmann equation, taking into account the following dissociation equilibria to determine the surface charge and potential at each value of x ... 

To calculate the double-layer force, the nonlinear Poisson-Boltzmann equation was solved for the case of two plane parallel plates, subject to boundary conditions which arise from consideration of the simultaneous dissociation equilibria of multiple ionizable groups on each surface. Deijaguin s approximation is then used to extend these results to calculate the force between a sphere and a plane. Details of the method can be found in Ref. (6). 

Ruckenstein and Schiby derived4 an expression for the electrochemical potential, which accounted for the hydration of ions and their finite volume. The modified Poisson-Boltzmann equation thus obtained was used to calculate the force between charged surfaces immersed in an electrolyte. It was shown that at low separation distances and high surface charges, the modified equation predicts an additional repulsion in excess to the traditional double layer theory of Deijaguin—Landau—Verwey—Overbeek. 

Let us first calculate the free energy of interactions between two planar, nonundulating interfeces. The potential obeys the Poisson—Boltzmann equation... 

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