On the interaction of colloidal particles IV. general mathematical theory for two identical particles

Abstract

A general theory of the interaction of two charged identical colloidal particles of arbitrary shape is developed. An expression for the Helmholtz free energy of the electric double layers is obtained by the methods of statistical mechanics. The condition that there is equilibrium between the ions adsorbed on the surfaces of the colloidal particles and those dissolved in the dispersion medium is accounted for by requiring that the free energy of the whole system be a minimum with respect to variation of the ionic density on the surfaces.The theory presented here is a further development of the work of Verwey and Overbeek. The conclusions of this paper are that in dilute sols, to which the present investigations are restricted, the results of these authors require extension in two directions. First, there is a correction to the mutual energy of two particles, due to the Coulomb interaction of the ions in the bulk of the solution. Secondly, no special assumption concerning the relation between the surface potential (or charge) and interparticle separation need be introduced. The equations set up to determine the free energy of interaction at the same time yield the ‘adsorption isotherm’ for the ion type which is common to the solution and the surface of the particle.

Statistical mechanics and lifshitz theory for electrolytes. Part 1

“…First we note that our assumption that the surfaces are mathematically sharp boundaries means that Ap(r) is a delta function and the corresponding integral a surface integral. Second, on using (22) and the fact that the surface charge density is equal to e times the surface density of potential determining ions, we may rewrite the contribution from a(Ap)/aA in (14) as…”

On the interaction of colloidal particles V. nature of repulsion between particles in dilute sols

“…where A = I, Q o = ne 1 and R is large. For small r, the approximation exp (2T) = (1 + T) 2 is introduced and e lt K 0 and T are each multiplied by A. Thus…”

“…f Present address: Department of Mathematics, The University, Manchester, 13. 2. General form of Debye-Huckel energy of ions for a single particle.…”

“…General form of Debye-Huckel energy of ions for a single particle. We shall first consider the case of a single particle, and assume the volume of the dispersion medium is so large (infinite) that the density of i ion n° (A, n) in the interior of the solution is constant, independent of n and equal to NJ V. The average potential at position r in the diffuse layer of the particle is denoted by ifr, which is assumed to satisfy the equationŝ (2) where p is the average charge density in the diffuse layer and 3The surface density of ions v will have the form (IV, 10), and it is convenient to designate partial derivatives of the quantities rjr and v with respect to A or n simply by a subscript. v K = 0, since the charging process is carried out at constant surface distribution of ions.…”

“…v K = 0, since the charging process is carried out at constant surface distribution of ions. Then from (2) where/' is the first derivative of the function/(A^r) with respect to Xijr. Let D 1 be the dielectric constant of the colloidal particle and i/r 1 the potential function inside the particle.…”