Molecular Imaging. Markus Rudin

is also determined by P_u.

      In the detailed following calculations, the electrostatic potential in the solvent medium at every point r is denoted by ψ(r).

      Step 1

      At any stage ν of the charging process, the values of e_i and ψ(r) are denoted by and ψ^ν(r).

      They are given by:

      r_i is the distance from the field point r to the center of ion i. ν varies between 0 at the beginning of the charging process an 1 at the end, ψ^ν(r) can then be written as in Eq. (3.50). The potential at the surface of ion 1, due to the medium and to ion 2, is obtained replacing r₁ by a₁ in Eq. (3.50). the potential there minus the self-potential, is obtained by subtracting (typo in the original) from Eq. (3.50)

      For the potential at the surface of a spherical conducting ion of charge e in a solvent medium of static dielectric constant D_s, compare, the Born equation in Ref. [21].

      The average of over the surface of ion 1 is denoted by and is found to be:

      where R is the distance between the centers of the two ions.

The average leading from Eqs. (3.51) to (3.52) is the electrostatic result that the average value of a 1/r₂ potential from a uniform distribution over a sphere is 1/R [39, 40].

      On multiplying by an increment of charge where:

      integrating over ν from 0 to 1, performing the same integration for ion 2 and summing both terms, we obtain the work term W_I required in charging step I:

      Eqs. (3.52, 3.53) yield:

      where

      When the initial charges e₁ and e₂ are both zero, the 1/R term becomes the usual Coulomb repulsion the 1/a₁ term becomes the well-known Born charging term for ion 1, and the 1/a₂ term the Born charging term for ion 2.

      Step 2

      The charges are given by Eq. (3.49b), where ν goes from 0 to 1.

      For ν = 0, (starting point of step 2), for ν = 1, (the initial charge to which one goes back in step 2).

      Let ψ_I (r) denotes the potential at the end of step 1 and ψ^ν(r) the potential at any state ν of step 2. The change of potential during step 2 is, for any ν, ψ^ν(r) − ψ_I (r). Since the medium responds now to the change of charge only via the optical dielectric constant D_op (remember that P_u is supposed fixed while P_e is following the charge) during step 2 we have:

      compare Eq. (3.50). Writing and δψ^ν = ψ^ν − ψ_I we have:

      Where

      Keeping now in mind the process that brought from Eqs. 3.50 to 3.52, we see that the average potential on the surface of ion 1 minus the self-potential, is obtained by subtracting from Eq. (3.58) and from the last term but one in Eq. (3.57), then replacing r₁ in those equations by a₁ and averaging the 1/r₂ in those equations over the surface of ion 1, thereby replacing 1/r₂ by 1/R” we thus have:

      The charging work of both ions done during this step is W_II

      The total work ΔG^r done is the sum of W_I and W_II and is the free energy of this fluctuation. It is: