Van der Waals interactions between bodies having inhomogeneous dielectric permittivities

Abstract

We present analytical results for the contribution of electromagnetic fluctuations inthe distribution of interaction energy and pressure in isotropic systems whoseproperties depend only on one spatial coordinate. If we neglect the continuousinhomogeneity introduced here and consider the simplest case of two macroscopichomogeneous bodies separated by a homogeneous film our result reduces to thewell-known Lifshitz formula. As a first application of theory, a one-dimensionalmodulated system with a homogeneous layer embedded in it is considered and asuitable perturbation theory for this system is developed. In the main part ofthis paper we limit the calculations to the non-retarded case, that is only thecalculation of van der Waals interaction energy is given. As a second applicationof theory we consider the van der Waals interaction between two semi-infinitemedia across a planar region within which there is a thin film having an arbitraryvariation of the dielectric permittivity. The importance of the precise evaluationof the transverse magnetic surface mode dispersion relation in inhomogeneousmedia is elucidated. For concreteness the influence of the transition layer betweenwater and lipid in a symmetrical configuration is considered in some detail. Thezero-frequency term and the dispersion-only contribution to the Hamaker coefficient aregiven analytically using some approximations and modelling of the dielectricconstants reasonable for these dielectrics at room temperature. As a whole theresults indicate the necessity of performing Lifshitz-type calculations on realisticinhomogeneous layered models, as are the models described here, for accurateinteraction energy modelling. Of course further work is needed for real justificationof the continuous variation of dielectric permittivities across phase interfaces.