Simple Relaxation Algorithms for the Poisson-Boltzmann Equation

In Sect. 2 we outlined Jacobi, Gaus-Seidel and SOR tecgniques for solving the Poisson Eq. The same schemes can be applied applies to the PB Eq., except that here the charge on site $\vec n$ depends on the electric potential at that point. Thus, Eq. (13) can be used, with the identification

$$\rho_s (\vec r) + \bar n_+ q e^{-\beta q\phi (\vec r)-v(\ve... ...q e^{\beta q\phi (\vec r)-v(\vec r)} \rightarrow \rho (\vec r)$$ (40)

(That is, the total charge density at point $\vec r$, including both fixed source terms and the mobile electrolytes is computed.) This method of solution applied to the PB Eq. with the field-dependent charge distribution in Eq. 40 has been used extensively in the biophysics community [39,40] to study electrostatic properties of proteins, enzymes and DNA, as well as solvation properties of organic molecules [41].

There is one other issue involved in a lattice based solution procedure for the PB Equation, namely the value of $\phi$ on the boundary surface. This information is not generally prescribed as a boundary condition; rather, charges on the macroions are given as inputs. Fortunately, if there is a finite collection of macroions surrounded by a large reservoir of electrolyte molecules, the physical situation is simple: the electrolyte ion atmosphere screens the electric fields produced by the macroion source charges. Thus the net electric field decays rapidly to zero away from the macroions, or equivalently, the electric potential field tends to a constant value. The actual value of this constant is in fact arbritary (since shifting the potential field by an overall constant does not change the electric field). It is usually convenient to choose it to be zero. For example, this choice means that $\bar n_{\pm}$are the bulk $\pm$ electrolyte concentrations of the reservoir. Therefore, in the case where a finite number of macroions are surrounded by a large electrolyte reservoir we should set $\phi=0$ at the boundaries. In light of the discussion above, once convergence has been obtained with the initially chosen simulation box, it is necessary to repeat the calculation with a bigger box, to make sure that the potential field does not change in the region near the macroion. In this way it can be verified that the potential field is truly zero on the boundary surface - otherwise imposition of this boundary condition is artificial and does not provide the solution of the problem which we wish to study.

(For a system consisting of extended collections of macroions, such as a colloidal suspension or crystal, it is more sensible to use periodic boundary conditions to describe the $\phi$ field. This means that there is no boundary surface. The parameters $\bar n_{\pm}$ should then be regarded as adjustable activity parameters chosen to scale the simple ion densities so as to put the desired number of simple ions in the simulation box. This entails an extra step: these parameters must be updated after each sweep based on the current value of the $\phi$ field. See Ref. [38] for details.)