Poisson-Boltzmann electrostatic forces

Beyond the IHP is a layer of charge bound at the surface by electrostatic forces only. This layer is known as the diffuse layer, or the Gouy-Chapman layer. The innermost plane of the diffuse layer is known as the outer Helmholtz plane (OHP). The relationship between the charge in the diffuse layer, o2, the electrolyte concentration in the bulk of solution, c, and potential at the OHP, 2> can be found from solving the Poisson-Boltzmann equation with appropriate boundary conditions (for 1 1 electrolytes (13))... 

The electrostatic force (F ) between two charged plates separated by an electrolyte solution can be determined from an existing imphcit solution to the nonlinear Poisson-Boltzmann equation and can be expressed in the form [187]... 

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. 

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.

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... 

This model is based on the Gouy-Chapman theory (diffuse double-layer theory). The theory states that in the area of the boundary layer between solid and aqueous phase, independently of the surface charge, increased concentrations of cations and anions within a diffuse layer exists because of electrostatic forces. In contrast to the constant-capacitance model, the electrical potential does not change up to a certain distance from the phase boundaries and is not immediately declining in a linear manner (Fig. 14 a). Diffusion counteracts these forces, leading to dilution with increasing distance from the boundary. This relation can be described physically by the Poisson-Boltzmann equation. 

A thorough discussion of the basic theory describing electrostatic interactions can be found in [7] the pertinent points are discussed below. Electrostatic forces arise from the osmotic pressure difference between two charged surfaces as a result of the local increase in the ionic distribution around each charged surface. For a single electrified interface, the local ion distribution is coupled to the potential distribution near that surface and can be described using the Poisson-Boltzmann equation. The solution of this equation shows that for low surface potentials the potential follows an exponential function with distance from the interface, D, given by... 

M. K. Gilson, M. E. Davis, B. A. Luty, and J. A. McCammon, /. Phys. Chem., 97, 3591 (1993). Computation of Electrostatic Forces on Solvated Molecules Using the Poisson-Boltzmann Equation. 

Gilson MK, Davis ME, Luty BA, McCammon JA. 1993. Computation of electrostatic forces on solvated molecules using the Poisson-Boltzmann equation. J. Phys. Chem. 97 3591-600... 

Im W, Beglov D, Roux B. 1998. Continuum solvation model computation of electrostatic forces from numerical solutions to the Poisson-Boltzmann equation. Comput. Phys. Commun. 111 59-75... 

Numerous approaches to handling molecular solute-continuum solvent electrostatic interactions, are described in detail in several recent reviews. - The methods most widely used and most often applied to Brownian dynamics simulations, however, fall in the category of finite difference solutions to the Poisson-Boltzmann equation. So, here we concentrate on that approach, providing a review of the basic theory along with the state-of-the-art methods in calculating potentials, energies, and forces. 

Computation of Electrostatic Forces on Solvated Molecules Using the Poisson-Boltzmann Equation. 

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