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Let us approximate the free energy of a solution of macro-ions, counterions, and added salt by the following simple functional of the ion concentrations:

 

(1)

The second term is the mean-field entropic free energy of the ions, with n_i being the concentration of the ith ion species carrying charge z_ie (n₀ sets the zero of the potential--see the following). The first term is the electrostatic energy, with the charge density r(r) being the sum of charge densities of the macro-ions and the mobile ions:

 

(2)

The local electrostatic potential is Y(r). The charge density and the potential are related by Poisson's equation, ѲY = (4p/e)r(r), where e is the dielectric constant of the continuum--the aqueous medium in which the ions are dissolved. Minimization of equation 1 with respect to the ion concentrations leads to the condition that they obey the Boltzmann distribution. More explicitly, using Poisson's equation, we obtain the relation

 

(3)

for the potential outside the surface of the macro-ions. This nonlinear differential equation, which is known as the Poisson­Boltzmann (PB) equation, must be solved under the boundary condition (Gauss's law) that the electric field E = -ÑY at the surface of a macro-ion be consistent with its fixed charge density s. That is, -ÑYp/e)s.

The electrostatic self-energy of a macro-ion is computed by inserting the solution of the PB equation into equation 1 for an isolated macro-ion, and then subtracting the free energy with all charges set equal to zero. When this calculation is carried out for a charged rod, the self-energy is found to be positive; the increase in entropic free energy induced by the confinement of the ions near the rod exceeds the lowering of their electrostatic energy.

The force acting between macro-ions is found by integrating the stress tensor

across a surface surrounding each macro-ion. For large distances r, the dimensionless electrostatic potential eY(r)/kT in between the macro-ions is small compared to one and the PB equation reduces to the well-known Debye­Hückel (DH) equation:

 

(4)

The Debye screening length is k^-1 (k² 8pl_Bn₀ ), l_B e ²/ekT is the Bjerrum length, and z is the magnitude of the z_i. It is straightforward to solve equation 4 for two parallel line charges each with a charge per unit length l and separated by a distance r. We can then use this solution to compute the force on a rod provided we integrate the stress tensor over a cylindrical surface located outside the rod with a radius big enough compared to k^-1 for the DH approximation to be valid. The effective interaction computed in this way is V(r) (2l^*/e)(p/2kr)^1/2exp(-kr), kr >> 1. Here, l^* is an effective or renormalized charge per unit length whose relation to the bare charge density l must come from a complete solution of the PB equation. For b <l_B, l^*/l is found to equal the Manning­Oosawa parameter x l_B/b and to equal one otherwise (b is the distance between fixed charges on the rod). A similar calculation for spherical macro-ions shows that the effective charge z^*e of an isolated sphere equals the bare charge ze, consistent with there being no counterion condensation in this case. For any nonzero concentration of spheres, however, z^*e is of order R/l_B, with R being the sphere radius. For charged planar surfaces, on the other hand, the renormalized charge per unit area is effectively zero.

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