Asymptotic solution of the cylindrical nonlinear Poisson–Boltzmann equation at low salt concentration: Analytic expressions for surface potential and preferential interaction coefficient

Abstract

The analytic solution to the nonlinear Poisson-Boltzmann equation describing the ion distributions surrounding a nucleic acid or other cylindrical polyions as a function of polyion structural quantities and salt concentration ([salt]) has been sought for more than 80 years to predict the effect of these quantities on the thermodynamics of polyion processes. Here we report an accurate asymptotic solution of the cylindrical nonlinear Poisson-Boltzmann equation at low to moderate concentration of a symmetrical electrolyte (<0.1 M 1:1 salt). The approximate solution for the potential is derived as an asymptotic series in the small parameter ؊1 , where ϵ ؊1 ͞a, the ratio of the Debye length ( ؊1 ) to the polyion radius (a). From the potential at the polyion surface, we obtain the coulombic contribution to the salt-polyelectrolyte preferential interaction (Donnan) coefficient (⌫ u coul ) per polyion charge at any reduced axial charge density . ⌫ u coul is the sum of the previously recognized low-salt limiting value and a salt-dependent contribution, analytically derived here in the range of low-salt concentra- T he cylindrical nonlinear Poisson-Boltzmann (NLPB or PB) equation is widely used for calculating electrostatic potential around rod-like charged objects surrounded by mobile ions both in the theory of polyelectrolyte solutions (1, 2), in applications in plasma physics (3) and in colloid and surface sciences (4). In particular, the surface coulombic potential (and͞or its distance dependence) of a charged polyion in an electrolyte solution is required for calculations of such thermodynamic properties as electrostatic free energy (5, 6), polyion-ligand binding constant (7,8), and especially the fundamental thermodynamic quantity ⌫ u coul , the coulombic contribution to the preferential interaction coefficient (equivalent to the experimentally observable Donnan coefficient). This coefficient is required for interpretation of thermodynamic experiments on salt-polyion interactions and on effects of salt concentration ([salt], C b ) on polyion processes (1, 9). Numerical calculations of NLPB solution for the model of a long periodically charged polyion as an infinite charged cylinder characterized by only two structural quantities, reduced charge density § and radius a, are known to successfully describe experimentally measured thermodynamic properties of polyelectrolyte solutions (1). Monte Carlo simulations confirm the accuracy of the cylindrical PB equation in the presence of added univalent salt up to 0.1 M (10, 11). However, despite numerous studies devoted to solving the NLPB equation (2, 3), no sufficiently accurate analytic solution for the cylindrical NLPB equation was known at low-to moderate-salt concentration.Solution of cylindrical NLPB equation is more challenging at low-salt (LS) concentration than at high [salt], where several useful accurate approximations for the potential, electrostatic free energy and preferential interaction coefficient are known (4,(12)(13)(14). In the absence of added salt, the ...

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