An improvement on Derjaguin's expression at small potentials for the double layer interaction energy of two spherical colloidal particles

Abstract

We describe a new method of solving the linear Debye-Huckel (D.H.) equation for the electric potential in the overlapping diffuse layers of two, identical, spherical, colloidal particles at separation R. The mathematical approach involves expressing the potential in terms of a distribution of electric dipoles on the two spherical surfaces and constructing an integral equation which governs this dipole distribution. On the assumption of a uniform potential Ψ on each particle, a simple, approximate solution of the integral equation is obtained. This is suitable for values of κa ≳ 5, where a is the radius of a particle and 1 κ is the D.H. thickness of the diffuse layer. The Derjaguin-Landau-Verwey-Overbeek formula at small potentials for the free energy of the electric double layers is expressed as a surface integral of the dipole distribution. This integral is evaluated approximately for κa ≳ 5 to yield a generalization of Derjaguin's formula for the double layer free energy of interaction, namely, δF= ϵaZ 2(r−a) R log e 1+ a (R−a) exp[− k(R−2a)] , where ϵ is the dielectric constant of the dispersion medium. This simplifies to Derjaguin's formula if R − 2 a ⪡ 2 a. Comparison with numerical calculations show that the new formula is superior to the corresponding Derjaguin expression, having an error of less than 3% for all separations when κa ≳ 5. The earlier numerical solutions by Hoskin and Levine of the Poisson-Boltzmann equation and their values for the interaction free energy of two spherical particles are recalculated with greater accuracy.