Investigation about validity of the Derjaguin approximation for electrostatic interactions for a sphere-sphere system

Validity of the Derjaguin approximation for sphere-sphere electrostatic interactions is investigated by explicitly determining the interactions between two spherical colloids using classical density functional theory (CDFT) solved in bispherical coordinates. The validity rules are summarized as follows. (i) For 1:1 type electrolyte, the Derjaguin approximation is effective for colloid sphere having a diameter down to three times the ion diameter only if the bulk concentration is higher than 0.1 M. (ii) With presence of higher-valence counter-ion, the threshold value bulk concentration rises, and increasing the colloid sphere diameter can lower greatly the threshold value bulk concentration. Encouragingly, over the valid parameter region of the Derjaguin approximation a like-charge attraction can be reproduced accurately. (iii) Both too low and too high surface charge strengths contribute to lower the quality of the Derjaguin approximation; increasing the medium permittivity or system temperature improves the accuracy of the Derjaguin approximation. (iv) Based on the mechanism analysis on the above observations, it is concluded that what matters in determining the validity of the Derjaguin approximation is the potential range of the pure inter-surface electrostatic interactions and the local Debye length rather than the bulk Debye length. Besides, the different expressivity of the influencing factors causing the effective inter-surface electrostatic interactions at different conditions determines the behavior the Derjaguin approximation deviates from the full CDFT calculations.