PB equation

The use of PB modeling by practitioners has been hmited for two reasons. First, in many cases the kinetic parameters for the models have been difficult to predict and are veiy sensitive to operating conditions. Second, the PB equations are complex and difficult to solve. However, recent advances in understanding of granulation micromechanics, as well as better numerical solution techniques and faster computers, means that the use of PB models by practitioners should expand. 

SASA), a concept introduced by Lee and Richards [9], and the electrostatic free energy contribution on the basis of the Poisson-Boltzmann (PB) equation of macroscopic electrostatics, an idea that goes back to Born [10], Debye and Htickel [11], Kirkwood [12], and Onsager [13]. The combination of these two approximations forms the SASA/PB implicit solvent model. In the next section we analyze the microscopic significance of the nonpolar and electrostatic free energy contributions and describe the SASA/PB implicit solvent model. 

In Section III we described an approximation to the nonpolar free energy contribution based on the concept of the solvent-accessible surface area (SASA) [see Eq. (15)]. In the SASA/PB implicit solvent model, the nonpolar free energy contribution is complemented by a macroscopic continuum electrostatic calculation based on the PB equation, thus yielding an approximation to the total free energy, AVP = A different implicit... 

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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Fig. 4. Relationship between the electrostatic potential calculated from the PB equation and the charge density parameter at distances 5,10, and 15 A from the axis of a cylinder with 3 A radius. Points indicate experimental data listed in Table 1 [42]...

Further simphfication of the SPM and RPM is to assume the ions are point charges with no hard-core correlations, i.e., du = 0. This is called the Debye-Huckel (DH) level of treatment, and an early Nobel prize was awarded to the theory of electrolytes in the infinite-dilution limit [31]. This model can capture the long-range electrostatic interactions and is expected to be valid only for dilute solutions. An analytical solution is available by solving the Pois-son-Boltzmann (PB) equation for the distribution of ions (charges). The PB equation is... 

Relatively high values of the experimental capacity Q have led Samec et al. [16] to propose that ions can penetrate into the irmer layer over some distance. The effect of the ion penetration was taken into account by solving the linearized Poisson-Boltzmann (PB) equation in all three regions of the MVN model with the result [17],... 

Considering that the ions of the aqueous supporting electrolyte can be present in the region 0 < x < d, the PB equation takes the form... 

A theoretical approach based on the electrical double layer correction has been proposed to explain the observed enhancement of the rate of ion transfer across zwitter-ionic phospholipid monolayers at ITIES [17]. If the orientation of the headgroups is such that the phosphonic group remains closer to the ITIES than the ammonium groups, the local concentration of cations is increased at the ITIES and hence the current observed due to cation transfer is larger than in the absence of phospholipids at the interface. This enhancement is evaluated from the solution of the PB equation, and calculations have been carried out for the conditions of the experiments presented in the literature. The theoretical results turn out to be in good agreement with those experimental studies, thus showing the importance of the electrostatic correction on the rate of ion transfer across an ITIES with adsorbed phospholipids. 

Another advantage of PB based pKa calculations is that effects of electrolytes are readily accounted for in the PB equation. The Coulombic contribution in conjunction with salt dependence to the abnormally depressed pAVs of histidine in staphylococcal nuclease has been experimentally tested [56], Recently, the methodology used in the PB calculations (Eqs. 10-11 and 10-12) has been combined with the generalized Born (GB) implicit solvent model [94] to offer pKa predictions at a reduced computational cost [52],... 

Electro-osmotic drag phenomena are closely related to the distribution and mobility of protons in pores. The molecular contribution can be obtained by direct molecular d5mamics simulations of protons and water in single iono-mer pores, as reviewed in Section 6.7.2. The hydrod5mamic contribution to n can be studied, at least qualitatively, using continuum approaches. Solution of the Poisson-Boltzmann (PB) equation. 

This is the important Poisson-Boltzmann (PB) equation and the model used to derive it is usually called the Gouy-Chapman (GC) theory. It is the basic equation for calculating all electrical double-layer problems, for flat surfaces. In deriving it we have, however, assumed that all ions are point charges and that the potentials at each plane x are uniformly smeared out along that plane. These are usually reasonable assumptions. 

If we can now determine V /ni as a function of the separation distance d between the surfaces, we can calculate the total double-layer (pressure) interaction between the planar surfaces. Unfortunately, the PB equation cannot be solved analytically to give this result and instead numerical methods have to be used. Several approximate analytical equations can, however, be derived and these can be quite useful when the particular limitations chosen can be applied to the real situation. 

The Poisson equation is valid under conditions of zero ionic strength. If dissolved, mobile electrolytes are present in the solvent, the Poisson-Boltzmann (PB) equation applies instead... 

Note that it is fairly common in the literature for continuum solvation calculations to be reported as having been carried out using Poisson-Boltzmann electrostatics even when no electrolyte concentration is being considered, i.e., the Poisson equation is considered a special case of the PB equation and not named separately. 

For certain ideal cavity shapes, the relevant PB equations have particularly simple analytic solutions. While such ideal cavities are not typically to be expected for arbitrary solute molecules, consideration of some examples is instructive in illustrating how more sophisticated modeling may be undertaken by generalization therefrom. 

Determine the electrostatic potential at each grid point by numerical solution of the PB equation this process is typically iterative. 

The primary area where classical PB equations find application is to biomolecules, whose size for the most part precludes application of quantum chemical methods. The dynamics of such macromolecules in solution is often of particular interest, and considerable work has gone into including PB solvation effects in the dynamics equations (see, for instance, Lu and Luo 2003). Typically, force-field atomic partial charges are used for the primary solute charge distribution. 

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