Nonlinear PB equation

In another study, the nonlinear PB equation with added salt has been approximately solved for a cylindrical polyion of radius a, by matching the near- and far-field solutions in an asymptotic expansion in terms of a small parameter, s= l/ln(l/Ka), and where k 1 is the Debye screening length [60],... 

We now discuss an asymptotic methodology of calculating the preferential interaction coefficient of nucleic acids. Assume that the electric potential of a cylindrical polyion satisfies the nonlinear PB equation. The polyion is immersed in an aqueous solution containing symmetric electrolytes. If y ze i/kBT denotes the reduced potential, where / is the actual potential, then the preferential interaction coefficient can be expressed in terms of the surface potential yfl[65-67, 77, 78]... 

The above results can be compared with low salt solutions obtained from numerical solutions of the nonlinear PB equation [77, 79, 80]. The preferential interaction coefficient is found to be of the form [77-79]... 

Taubes et al. developed rigorous bounds on preferential interaction coefficient [72], Their staring point is the reduced electrostatic potential u of a polyion of radius a and length L in an ionic solution containing monovalent and divalent counterions. The reduced potential w(p) satisfies the nonlinear PB equation [72, 81]... 

Only one of the three roots of Eq. (17) is physically realistic. Note that a solution to the nonlinear PB equation for a 2 1 1 electrolyte has been given before [12], but in a less flexible form. 

The electrostatic potential only can be determined relative to a reference point which normally is chosen to be zero at r — oo. However, this equation is still very difficult to solve and an analytical solutions are only available in special cases. Useful solutions occur at low surface potential, where the PB can be linearized (see Debye-Hiickel below). A famous analytical solution was derived by Gouy [12] and Chapman [13] independently (see below) for one flat surface in contact with an infinite salt reservoir. The interaction between two flat and charged surfaces in absence of salt, can also be solved analytically [14]. In other situations the nonlinearized PB equation has to be solved numerically. 

FIGURE IIS Osmotic pressure versus volume fraction for = 100 mV and ko = 3.3 (a) body-centered cubic DLVO, (b) faceconfigurational osmotic pressure. Reprinted firtan [28] with permission by Academic Press. 

The free energy functional in Eq. (2) must be minimized with respect to all relevant degrees of freedom in a self-consistent manner. In particular, functionally minimizing F with respect to the mobile ion concentrations leads to the familiar nonlinear PB equation [26-36] ... 

Up to now the following observation repeatedly turned up from the simulations. The nonlinear PB equation provides a fairly good description of the cell model, but it suffers from systematic deviations in strongly coupled or dense systems. It underestimates the extent of counterion condensation and at the same time overestimates the osmotic coefficient. As the common reason for both problems, the neglect of correlations has been proposed, basically for two reasons ... 

The precise value of Zeff is derived by numerically solving the full, nonlinearized PB equation (quite easy integration) and identifying the asymptote with Eq. 2. While Zeff obviously... 

UHBD Solves the linearized or nonlinear PB equation. Available http // mccammon.ucsd.edu/uhbd.html. 

Increment the coxmter p p + and repeat the nested cycle of equations (29) and (30) until convergence. This iterative procedure is related to the normal method of solving the nonlinear PB equation for screening charge densities. Indeed, the DHH analysis can be regarded as... 

The boundary conditions for the nonlinear PB equation as well as for its limiting form [Eq. (10)] and the Laplace equation (Eq. (11)] are the same as previously expressed by Eq. (8). However, in the limiting case when particle dimensions are much larger than the screening length, the field inside particle can be neglected and Eq. (8) simplifies to... 

Equations (25) and (26) describe the potential distribution in the difiuse double layer by neglecting all specific adsorptions of ions in the Stem layer adjacent to the surface [24,25]. They represent practically the only exact solution of the nonlinear PB equation known in literature. At larger separations, when xjLe 1, Eq. (25) assumes the asymptotic form... 

The deficiency of the linear c.c. model was also demonstrated in Refs. 32 and 33 by analyzing the asymptotic behavior of the nonlinear PB equation in the limit of small plate... 

In addition to always giving a positive coion concentration, and, for small potentials, at the same level of accuracy as Eq. [79], Eqs. [84] and [85] give the product of the anion/cation concentrations for z z salts as unity, as it is with the nonlinear PB equation. (Of course, use of Eq. [85] violates electroneutrality.)... 

The fact that the apparent linear charge density is lower than the actual charge density can be seen as confirmation that PB theory does predict some sort of counterion condensation. We mention that because the ADH method results from a (linear) DH interpretation of the (nonlinear) PB equation, it also has a (rather tenuous) mathematical connection to Manning s coimterion condensation theory, which also makes use of the DH solution. " From either Eq. [192] or Eq. [263], the fraction of linear charge density neutralized can be written as (ignoring a contribution from Q... 

Garrett and Poladian demonstrate the uniqueness of the nonlinear PB equation for the case of a constant dielectric coefficient. The extension for a variable (and positive) dielectric coefficient is readily shown by a suitable modification of Green s theorem based on Eq. [3]. However, care must be taken in the numerical solution of some modified PB theories as nonuniqueness has been observed. ... 

In addition to neglecting ion correlation, using the mean electrostatic potential has the undesirable consequence that the (nonlinear) PB equation no longer satisfies a reciprocity condition that use of the potential of mean force would obey. Linearization of the equation by Debye and Hiickel regained this condition. These considerations led Outhwaite and others to propose modifications of the PB equation to treat these problems. Within this modified Poisson-Boltzmaim (MPB) theory, the effect of ion correlation is expressed in terms of a fluctuation potential for which a first-order (local) expression, written as an activity coefficient, can be derived. Their result for bulk hard-sphere electrolyte ions of valence z, and common radius a gives the formula ... 

Kirkwood gave the first detailed derivation of the nonlinear PB equation, including corrections due to correlation (fluctuation) and finite ion size. These were elaborated on by Levine, culminating in a paper with Bell" ° dealing with the electric double layer near a polyelectrolyte surface. The leading terms in the Bell-Levine treatment were kept and some restrictions and approximations were applied to obtain a modified Poisson-Boltzmann (MPB) equation in the form of an integrodifference/differential equation that took into account the fluctuation potential of Kirkwood. 

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