Validity of the Derjaguin approximation on electrostatic effect in the Frumkin-Derjaguin approach

Abstract

The Frumkin-Derjaguin theory relates the macroscopic contact angle of a droplet with the disjoining pressure of the thin film in equilibrium with the droplet. To obtain the analytic expressions of the disjoining pressure, the Derjaguin approximation has been used, combined with the Debye-Huckel theory on the electrical double layer. As a result, the disjoining pressure can be determined without consideration of the overall geometry of the liquid surface. The validity of the Derjaguin approximation has been regarded to be limited to small contact angles, and its validity for large contact angles has rarely been assessed analytically. In this paper, the electrostatic force acting on the meniscus of a droplet (which enforces the droplet to spread) is obtained by integrating the Maxwell stress and the osmotic pressure acting on the liquid surface. The Debye-Huckel theory is employed for direct comparison with the results of the Derjaguin approximation in the two cases of the constant surface potential and the constant surface charge boundary conditions. The present electromechanical approach provides an exact result for the electrostatic contribution to the contact angle for a given model of the electrical double layer. It is shown (for the constant potential case) that if the modification of the interfacial energy at the liquid-surrounding fluid interface by the electrocapillary effect is separately considered, the Derjaguin approximation gives an exact interaction free energy, regardless of the magnitude of the contact angle. For the constant charge case, on the contrary, the interaction free energy derived based on the Derjaguin approximation for prediction of the contact angle is shown to have evident deficiency except for the case of vanishing surface charge density at the liquid surface. Such deficiency originates from the neglect of the contribution of the tangential-stress component.