The Poisson-Boltzmann Equation.

Consider now a collection of fixed charges (usually embedded in macromolecules of some type) surrounded by a ``gas" of simple ions, where the quotation marks emphasize that in chemical applications the simple ions are electrolytes which are dissolved in a (polar) liquid solvent. For example, a biological macroion such as DNA (negatively charged due to phosphate groups) immersed in water will be surrounded also by positive counterions (K<sup>+</sup>, Na<sup>+</sup>, Ca<sup>+2</sup>, Mg<sup>+2</sup>) which ionize when the (originally charge neutral) DNA molecule is placed in the solvent [34], plus any other ions that may be present in the solution (including H<sup>+</sup> and OH<sup>-</sup> ions due to the natural ionization of water, and additional impurity ions such as can be introduced by dissolving salt in the solution.) On a more macroscopic scale, highly charged colloidal spheres made of tightly entangled polystrene chains can be prepared in various solvents by well-developed synthetic methods. Such ``polyballs" can range in diameter from less than 50 nm to microns [3,35]. Just as in the case of biological macroions, overall electroneutrality requires that the solution contain counterions which were originally bound to the polyballs. In addition, salts can be added to the solution in order to increase its ionic strength.

To describe the equilibrium distribution of the electrolytic ions from first principles is not easy. These ions interact with each other and with the fixed charge distribution embedded in the macroion(s). The Coulomb force is very long ranged, so we have to study the thermal motion of a strongly interacting many-body system. The difficulty of doing this, even via numerical simulation carried out on modern computers, has led to the development of approximate treatments of the equilibrium properties of ``ionic atmospheres" surrounding static charge distributions. One of the most widely utilized approximate descriptions is the Poisson-Boltzmann (PB) Equation, which can be motivated heuristically as a modification of Poisson's Equation, namely,

$$\vec \nabla \cdot (\epsilon (\vec r) \vec \nabla \phi (\vec... ...r)-v(\vec r)} - q \bar n_- e^{\beta q\phi(\vec r)-v(\vec r)} ]$$ (31)

Note that there are three contributions to the electric charge distribution that appears on the r.h.s. of Poisson's Eq. The first term accounts for the fixed source charges (again, usually embedded in macroions). The other two terms count, respectively, the contributions of the mobile positive and negative electrolyte ions. These are assumed to be in thermal equilibrium at temperature T ($\beta \equiv 1/k_B T$). The potential energy of charge q located at position $\vec r$ is given by $q\phi (\vec r)$, where $\phi$ is the electric potential (which remains to be determined!). The quantities $\bar n_{\pm}$ are the bulk number densities of $\pm$ ions far away from the macroion sources (taking the asymptotic value of $\phi$ as zero). Furthermore, q is the charge on each of the positive ions, usually a low multiple of the magnitude of the electronic charge reflecting the valency of the mobile cations; for simplicity, we consider the case that the negative ions have the opposite charge as the positive ones, namely -q. Finally v is a dimensionless excluded volume potential, i.e. v=0 in regions of space that are accessible to the mobile ions, and $v=\infty$ in regions that are inaccessible to them.

The introduction of this equation raises several questions. Where does the PB equation come from? (We have only given an empirical motivation above.) What is its range of validity? How can we solve it for desired charge distributions and ionic strengths (it is nonlinear due to the exponential Boltzmann factors)? Once it has been solved, how can knowledge of the electric potential field $\phi$ be used to calculate thermodynamic properties of the macroion/electrolyte system?