On the Interaction of Two Colloidal Particles, Using the Complete Debye-Hückel Equation

Abstract

Making use of the original Debye‐Hückel equation in the theory of electrolytes, an approximate method of calculating the mutual electrical energy of two colloidal particles has been developed. Only binary electrolytes (symmetrical type) have been considered. The final expression for the energy is of the form $${\mathrm{(Q}}_0^2{\mathrm{/Da)\{f}}_1{\mathrm{(R/a,\kappa a)+(Q}}_0{\mathrm{/Da)}}^2{\mathrm{(z\epsilon /kT)}}^2\mathrm{g(R/a,\kappa a)\}}$$ or $${\zeta }^2{\text{Da(1+κa)}}^2{\mathrm{\{f}}_1{\mathrm{(R/a,\kappa a)+\zeta }}^2{\mathrm{(z\epsilon /kT)}}^2\mathrm{G(R/a,\kappa a)\},}$$ where Q₀, a and ζ are the charge, radius and electrokinetic potential, respectively, of each particle, R is the mutual separation of the two particles, κ the characteristic quantity in the Debye‐Hückel theory, being proportional to the square root of the electrolyte concentration, z—z the valency type of the binary electrolyte, D the dielectric constant of the solution, T the temperature, ε the electronic charge and k Boltzmann's constant. Contrary to the opinion held by many, it is shown that for symmetrical electrolytes, the commonly quoted restriction zεζ/kT≪1 need not be satisfied in order that the approximate Debye‐Hückel equation be applicable. Provided the particle radius is not too large, the latter equation leads to fairly satisfactory values for both ζ and the mutual energy at ordinary ζ potentials. The coagulating powers of the 1–1, 2–2 and 3–3 valency types of electrolytes are studied in detail and qualitative agreement with experiment is obtained. It is shown that the stability of a sol is sensitive to changes in the ζ potential and that in comparing the relative precipitation values of the different electrolytes one must be careful of any small change in ζ. The distinction between the monovalent ions which coagulate at high concentrations and high potentials and the other ions which precipitate at lower potentials and smaller concentrations is emphasized. The energy of the two particles at large separations is again negative and there is a minimum in the energy as a function of the particle distance.