Electrostatic and Van der Waals interactions of nanoparticles in electrolytes

The electrostatic interaction between nanoparticles caused by the overlapping of double electric layers and the van der Waals interaction caused by quantum and thermodynamic fluctuations of electromagnetic fields are considered. The linearized Poisson–Boltzmann equation for particles with a fixed electric potential on their surface is used in the case of the electrostatic interaction. An exact solution of the problem has been obtained both for identical particles and for particles with strongly different sizes. The screening of static fluctuations and the retardation of electromagnetic fields for the dispersion part of the van der Waals interaction have been taken into account. The total interaction energy of two particles has been calculated for ion concentrations in an electrolyte from 10$^{-6}$ to 10$^{-2}$ mol/L and sizes of nanoparticles from 1 to 10$^3$ nm. It has been found that the van der Waals force exceeds the screened electrostatic repulsive force at high concentrations of the electrolyte from 10$^{-3}$ to 10$^{-2}$ mol/L at both small and large interparticle distances.