# Counterion Condensation as an Exact Limiting Property of Solutions of the Poisson-Boltzmann Equation Academic Article

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### abstract

• This paper studies the transition occurring in the type of singularity, induced by a line charge to solutions of the Poisson–Boltzmann equations in $R^2$, at some critical value of the linear charge density. The appropriate problem arises in the theory of polyelectrolytes. The b.v.p. under consideration is: $\begin{gathered} \frac{1}{r} \frac{\partial }{\partial r}\left( r\frac{\partial \varphi }{\partial r} \right) = \sigma \frac{e^{\zeta \varphi } }{\int_a^1 {e^{\xi \varphi } rdr} } + N\frac{e^{z\varphi } }{\int_a^1 {e^{z\varphi } rdr} } - N\frac{e^{ - z\varphi } }{\int_a^1 {e^{ - z\varphi } rdr} },\quad a < r < 1 \\ \left( r\frac{\partial \varphi }{\partial r} \right)\bigg|_{r = a} = - \sigma ,\varphi (1) = 0,\quad a,\sigma ,N,\zeta ,z = {\text{const}} > 0 . \end{gathered}$ The central fact proved in this paper is that for $N = O(1),a \to 0$$(\text{C})\qquad \varphi (a) = \min \left( \alpha ,\frac{2}{\zeta },\frac{2}{z} \right)\ln \frac{1}{a}\, + o\left( \ln\frac{1}{a} \right).$. The case $N \to 0$ is studied and physical implications of (C) are discussed. It is pointed out that the described phenomenon represents an example of a nonbifurcational second order phase transition.

### publication date

• December 1, 1986