Lifshitz-van der Waals energy of spherical particles in cylindrical pores

Abstract

The Lifshitz-van der Waals component of the interaction energy between a spherical particle and the wall of a cylindrical pore is obtained in Hamaker's microscopic approach. In contrast to the earlier numerical studies, the present scheme results in a much simpler semianalytical result for the energy of interaction. Asymptotic analytical results are derived for particles close to the pore wall and for particles close to the pore centerline. For all feasible combinations of particle size, radial position, and pore size, the maximum deviation of analytical results from the numerical solutions is less than 20% and the mean deviation is about 5%. The energy at contact is usually several orders of magnitude larger than the energy near the centerline, and the discrepancy increases with decreasing particle size and increasing pore size. A commonly used simple model of a sphere interacting with a semi-infinite flat plate is found to be reasonable only for very small particles close to the wall, but in all other cases, it seriously underestimates the energy. Calculations of the energy for semi-infinite and finite pores show that the entry length is limited to about twice the pore radius.