Van der Waals interaction between spherical metallic nanoparticles with mutually induced polarization

An influence of multipole plasmonic resonances, excited on the surface of the spherical metallic nanoparticles, on van der Waals interaction between nanoparticles is investigated in this work. The relations for the size dependences of the interaction free energy, Hamaker parameter and van der Waals force are obtained. It is shown that the improper integral, included in the obtained expressions, is convergent, and the sum is easily calculated due to the fast convergence of the series at increasing multipolarity. The calculations were performed for the case of interaction between spherical nanoparticles of the different radii and different metals in air, on the surface of which the localized plasmonic resonances are excited. It is found that the increase in the distance between the nanoparticles results in the decrease in free energy and van der Waals force and in the increase in Hamaker parameter. In turn, the free energy practically does not change with the change of nanoparticle material and increases sharply with the increase in radius of nanoparticles. In contrast to the free energy, the increase in the nanoparticle radius results in the decrease in Hamaker parameter. The decrease in Hamaker parameter at the same distance between particles takes place when changing their composition (using metals with decreasing plasma frequency). The distance between nanoparticles, at which the sharp decrease in van der Waals force changes to the smooth one, has been determined. The comparison of the calculation results with the case of van der Waals interaction between spherical nanoparticles, caused by electromagnetic fluctuations with the continuous spectrum, is carried out. It is shown that the qualitative character of the size dependences of the free energy and Hamaker parameter remains the same: the free energy decreases and Hamaker parameter increases with increasing distance between interacting nanoparticles. At the same time in the case of the particles with the localized plasmons, excited on their surfaces, the free energy is greater and Hamaker parameter is less than in the case of electromagnetic fluctuations with the continuous spectrum.