Nonlinear Poisson-Boltzmann

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. 

We have used the procedure of Marcelja et al (22) to compute r for spherical molecules with the same charge and volume as 200 bp rodlike DNA. Each molecule is at the center of a Wigner-Seitz cell of volume (4/3)with bulk salt concentrations spanning the experimental range (2). The nonlinear Poisson-Boltzmann e( uation is solved numerically with appropriate boundary conditions at the particle surface and the cell boundary. The results are that F = 155 at about 3 mg/mL DNA (twice the experimental concentration) with no added salt, but F is always < 155 for added salt in the experimental range. For NaPSS, with dp 3800 at 1-4 X 10 mg/mL, F > 300, consistent with the observation of a structure factor maximum. 

Rahaman and Hatton [152] developed a thermodynamic model for the prediction of the sizes of the protein filled and unfilled RMs as a function of system parameters such as ionic strength, protein charge, and size, Wq and protein concentration for both phase transfer and injection techniques. The important assumptions considered include (i) reverse micellar population is bidisperse, (ii) charge distribution is uniform, (iii) electrostatic interactions within a micelle and between a protein and micellar interface are represented by nonlinear Poisson-Boltzmann equation, (iv) the equilibrium micellar radii are assumed to be those that minimize the system free energy, and (v) water transferred between the two phases is too small to change chemical potential. 

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]... 

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. 

The interaction between two charged surfaces in liquid depends on the surface charge. Here, we only consider the linear case and assume that the surface potentials are low. If we had to use the nonlinear Poisson-Boltzmann theory, the calculations would become substantially more complex. In addition, only monovalent salts are considered. An extension to other salts can easily be made. 

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.

Comparison with Predictions of the Nonlinear Poisson-Boltzmann... 

Analysis of ion atmospheres around highly charged macromolecules has traditionally been performed using numerical solutions to the nonlinear Poisson-Boltzmann (P-B) equation (Anderson and Record, 1980 Bai et al, 2007 Baker, 2004), in which the macromolecule is approximated as a collection of point charges embedded in a low dielectric cavity surrounded by a high-dielectric solvent. This approach utilizes the precise three-dimensional structure of the macromolecule (albeit in a static sense). We would not expect such a framework to capture subtleties, which are dependent on the partial dehydration of ions. 

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

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). 

In Figure 6a, the force per unit area between surfaces with grafted polyelectrolyte brushes, plotted as a function of their separation distance 2d, calculated in the linear approximation, is compared with the numerical solution of the nonlinear Poisson—Boltzmann equations, for a system with IV = 1000, a = 1 A, ce = 0.01 M, s2 = 1000... 

Understandably, it is much more common to see analyses of problems based on Eq. (32) since for simple geometries the solution can be written down in closed form, expressed in terms of simple functions. For plane surfaces, for example, the solutions are elementary hyperbolic functions while for an isolated spherical surface the Debye-Huckel potential expression prevails. For two charged spherical surfaces the general solution can be written down as a convergent infinite series of Legendre polynomials [16-19]. The series is normally truncated for calculation purposes [16] K For an ellipsoidal body ellipsoidal harmonics are the natural choice for a series representation [20]. (The nonlinear Poisson Boltzmann equation has been solved numerically for a ellipsoidal body... 

In the paragraphs below, we first examine the simple, analytical results that can be derived from the linear Poission-Boltzmann equation for a single particle interacting with a flat surface. Next, more complicated physical situations are considered, including interactions between many particles and a wall between a particle and a deformable interface between a protein and a wall and between a moving particle and a wall. In Sec. Ill, solutions to the nonlinear Poisson-Boltzmann equation are considered, and comparisons are made between the linear and nonlinear versions and also with more... 

III. ACCURACY OF LINEAR AND NONLINEAR POISSON-BOLTZMANN EQUATIONS... 

In Secs. II.A and II.B above, we examined some common, approximate solutions to the linear Poisson-Boltzmann equation, and commented on the level of their agreement with exact solutions of that same equation. However, these approximations are no more accurate than the exact solutions, and the accuracy of the latter can only be ascertained by comparison with solutions to the complete, nonlinear Poisson-Boltzmann equation. From the... 

How good an approximation is the linear Poisson-Boltzmann equation to the full, nonlinear Poisson-Boltzmann equation ... 

A. Comparisons between the Linear and Nonlinear Poisson-Boltzmann Equations... 

The ability of the linear Poisson-Boltzmann equation to yield accurate results (i.e., results close to those for the full, nonlinear Poisson-Boltzmann equation) can reasonably be expected to depend on both geometry and boundary conditions. Comparisons for different geometries seem to yield... 

FIG. 8 Predictions of the linear and nonlinear Poisson- Boltzmann solutions for the potential near a flat plate. 

Perhaps the first comparison that should be made is that between the potential near an isolated, charged plate as predicted by the linear and nonlinear Poisson-Boltzmann equations. The linear result is given by... 

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