Image interactions

If the simulated system uses periodic boundary conditions, the logical long-range interaction includes a lattice sum over all particles with all their images. Apart from some obvious and resolvable corrections for self-energy and for image interaction between excluded pairs, the question has been raised if one really wishes to enhance the effect of the artificial boundary conditions by including lattice sums. The effect of the periodic conditions should at least be evaluated by simulation with different box sizes or by continuum corrections, if applicable (see below). 

The common example of real potential is the electronic work ftmction of the condensed phase, which is a negative value of af. This term, which is usually used for electrons in metals and semiconductors, is defined as the work of electron transfer from the condensed phase x to a point in a vacuum in close proximity to the surface of the phase, hut heyond the action range of purely surface forces, including image interactions. This point just outside of the phase is about 1 pm in a vacuum. In other dielectric media, it is nearer to the phase by e times, where e is the dielectric constant. 

Models for the compact layer of the metal-electrolyte interface have become more and more elaborate, providing better and better representations of observed electrocapillary data for different metals, solvents, and temperatures, but almost always leaving the metal itself out of consideration, except for consideration of image interactions of the solvent dipoles. For reviews of these models, see Parsons,13 Reeves,14 Fawcett et a/.,15 and Guidelli,16 who gives detailed discussion of the mathematical as well as the physical assumptions used. 

Electrochemical interfaces are sometimes referred to as electrified interfaces, meaning that potential differences, charge densities, dipole moments, and electric currents occur. It is obviously important to have a precise definition of the electrostatic potential of a phase. There are two different concepts. The outer or Volta potential ij)a of the phase a is the work required to bring a unit point charge from infinity to a point just outside the surface of the phase. By just outside we mean a position very close to the surface, but so fax away that the image interaction with the phase can be ignored in practice, that means a distance of about 10 5 — 10 3 cm from the surface. Obviously, the outer potential i/ a U a measurable quantity. 

The dielectric displacement must be calculated from electrostatics for a reactant in front of a metal surface the image force has to be considered. For the simple case of a spherical ion in front of a metal electrode experiencing the full image interaction, a straightforward calculation gives ... 

The energy involves electronic energy s°, solvent interaction energy Ep, and the image interaction energy e, . The distance-dependent effective solvent dielectric constant k(x), which appears in the image term, was taken as... 

The situation is quite different for physisorbed molecules. In that case, there is no transfer of charge, the mechanical renormalization is weaker due to a much weaker metal-molecule bond and also the image interaction is smaller as the molecule probably is adsorbed further out from the surface. In a recent IRS investigation of CO physisorbed on Al(100) the measured frequency is only shifted down a few cm from the gasphase value. However, there is for this system also a short range intermolecular interaction that certainly will affect the vibrational frequency. As yet there exist no theoretical calculations for the van der Waals interaction between a CO molecule and a metal. 

For the monomers (d) the image interaction of one water molecule is given by... 

Again, in a very diluted electrolyte solution, the major contribution will make the self-image interaction. The self-image interaction potential can be defined on the basis of the fluctuation potential as [7,18]... 

Numerical calculations can be carried out on the basis of Eqs. (70)-(74), and to show the influence of the image forces on the disjoining pressure between two plates, we will compare these results with the frequently used expression in Eq. (75). We have still to adhere to the approximations accepted in the text, and, first of all, the bulk electrolyte concentration c0 = 1 mmol/L will be chosen. For this concentration the Bierrum (or plasma) parameter is n — ky q = 0.04, and, as a consequence, for characteristic interparticle distances of a colloid domain the relationship a = Kr /x is much smaller than one. This allows to simplify our equations sufficiently leading to the final expression for the disjoining pressure, where the dominant role plays the self-image interaction... 

FIG. 6 The disjoining pressure with (ncorrected) and without (nDerjaguin) image interaction. 

We have also to say a few words about a smeared out analogue utilized in our consideration to represent the surface charge distribution. There was an indication in the literature that the image interaction within this adsorption layer may influence the interaction on a macroscopic scale, i.e., which is being commensurate with the range of colloidal forces [29]. However, later results... 

In this case the author constructed the total interaction potential on the basis of an additivity of the particle-wall and particle-image interactions. The electrostatic contribution is given then by... 

Combining these two contributions and accounting for reactant-electrode image interactions within the transition state [31b] yields eqn. (19). 

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