Department of Chemistry, University of Louisville, Louisville, KY 40292, USA.
Biopolymers. 2010 Jul;93(7):619-39. doi: 10.1002/bip.21421.
An analytical approximation to the nonlinear Poisson-Boltzmann (PB) equation is applied to charged macromolecules that possess one-dimensional symmetry and can be modeled by a plane, infinite cylinder, or sphere. A functional substitution allows the nonlinear PB equation subject to linear boundary conditions to be transformed into an approximate linear (Debye-Hückel-type) equation subject to nonlinear boundary conditions. A simple analytical result for the surface potential of such polyelectrolytes follows, leading to expressions for the amount of condensed (or renormalized) charge and the electrostatic Helmholtz energy for polyelectrolytes. Analytical high-charge/low-salt and low-charge/high-salt limits are shown to be similar to results obtained by others based on PB or counterion condensation theory. Several important general observations concerning polyelectrolytes treated within the context of PB theory can be made including: (1) all charged surfaces display some counterion condensation for finite electrolyte concentration, (2) the effect of surface geometry is described primarily by the sum of the Debye constant and the mean curvature of the surface, (3) two surfaces with the same surface charge density and mean curvature condense approximately identical fractions of counterions, (4) the amount of condensation is not determined by a predefined "condensation distance" although such a distance can be determined uniquely from it, and (5) substantial condensation occurs if the Debye constant of the electrolyte is much less than the mean curvature of a highly charged polyelectrolyte.
一种非线性泊松-玻尔兹曼（PB）方程的解析逼近方法被应用于具有一维对称性的带电大分子，可以用平面、无限圆柱或球体来建模。通过函数替换，可以将具有线性边界条件的非线性 PB 方程转换为具有非线性边界条件的近似线性（德拜-休克尔型）方程。这种聚电解质的表面电势的简单解析结果随之而来，导致聚电解质的凝聚（或归一化）电荷和静电亥姆霍兹自由能的表达式。高电荷/低盐和低电荷/高盐的解析极限与基于 PB 或抗衡离子凝聚理论的其他人的结果相似。在 PB 理论的背景下处理聚电解质时，可以得出几个重要的一般性观察结果，包括：（1）对于有限电解质浓度，所有带电表面都显示出一定程度的抗衡离子凝聚；（2）表面几何形状的影响主要由德拜常数和表面平均曲率之和来描述；（3）具有相同表面电荷密度和平均曲率的两个表面凝聚的抗衡离子大致相同；（4）尽管可以从它唯一地确定，但凝聚的量不由预定的“凝聚距离”决定；（5）如果电解质的德拜常数远小于高度带电聚电解质的平均曲率，则会发生大量凝聚。
