Molecular Imaging. Markus Rudin

There will also be a surface charge distribution on the electrode surface.” As in Chapter 2 (see Ref. [4]), “the potential can be expressed in terms of contributions from the charges and from the polarized volume elements.” The potential arises then from the following contributions:

      (i)Volume charges of the ions in ionic atmospheres

      (ii)Surface charges on the central ions

      (iii)Surface charges on the surface of the M electrode

      (iv)Polarized volume elements

      (v)Potential difference at the interface

This potential “is the same as the ‘inner potential’ φ(r) described by Parsons [5] and Lange [6].”

      “We have:

      where ∇_r is the gradient operator and where

      The volume integrals are over the entire volume of the dielectric (the solution). The surface integral is over the surface of each central ion and over the surface of the electrode M:

      It is convenient to define a function ψ(r) which is identical with the potential employed in Marcus’ Part I (Ref. [4]):

      For the electric field strength E(r), we have

      E(r) = −∇φ = ∇ψ” [1, p. 212].

      Equation (4.1) is like Eq. (2.11a) in Chapter 2 with the addition of the and β terms, characteristic of the electrode surface and of the electrode–solution interface.

       4.4.Electrode Charge Distribution and the Method of Electrical Images

      The term is expressed by M. using the method of electrical images.

“In systems having equilibrium polarization the method of images [7] can be used to determine the charge distribution” [1, p. 217].

M. briefly discusses the quantum limitations to the image force theory in reference 9 of Ref. [1] deducing from Ref. [8] that the percent error in the energy change computed from the image force theory is small.

      “The same method is also suitable for systems having nonequilibrium polarization. For simplicity, consider a planar electrode, that is, any electrode whose radius of curvature is appreciably greater than the thickness of the double layer. Let the electrode–dielectric interface be situated in the plane x = 0 and let the dielectric occupy the semi-infinite region x > 0. As before, the coordinates of any point in the dielectric will be specified by the vector r drawn from any arbitrary origin to the point. The coordinates of the ‘image point,’ having the same y and z and differing only in the sign of x, will be specified by a vector r_im. Mirror image functions and will then be defined in the region of negative x, according to Eq. (4.4). They are undefined in the region of positive x, i.e., in the dielectric.

      The last three expressions relate the components of the vector to those of P. In Eq. (4.4), σ refers only to the surface charge density on the central ions and not to that on the electrode.”

      Figure 1 shows schematically the electrical image of a central ion, represented by a small dashed circle. The dashed shell around it represents the image of its ionic cloud. The region between the solid line representing the electrode–solution interface and the dashed line parallel to it is the region of the electrode where the images of most of the ions of the electrical double layer are to be found.

      Note that because both charge and volume in the image charge density are opposite to those in the charge density, that is, dV = dxdydz in the dielectric but dV = (−dx)dydz in the metal. Notice that not only the charge but also the polarization vector function has an image.

      Fig. 1*.

      “Applying the method of images, the electrode charge distribution which satisfies the condition of constant potential on the electrode M and in the body of the solution obeys the following equation:

      In Eq. (4.5), r and r′ denote any points on the electrode’s Msurface and in the dielectric medium, respectively. The first surface integral is over M and the second over the mirror images of each central ion. The volume integrals are over the entire mirror image of the volume of the dielectric.”

      It appears in this way very clearly that the charge induced on the surface of the electrode can be expressed in terms of the dielectric images: “The total charge in the dielectric is equal and opposite to the total image charge, which in turns equals the total electrode charge.”⁽¹⁾

      The integrals extended to the ions can be evaluated treating the ions as spheres with surfaces of uniform charge densities. “It can be readily shown (see Chapter 2) that:

      Where r is the distance from the ion to the field point, see Fig. 2, and q is the ionic charge.

      Fig. 2*.

      Similarly,