van der Waals interactions between bodies having inhomogeneous dielectric permittivities

We present analytical results for the contribution of electromagnetic fluctuations in the distribution of interaction energy and pressure in isotropic systems whose properties depend only on one spatial coordinate. If we neglect the continuous inhomogeneity introduced here and consider the simplest case of two macroscopic homogeneous bodies separated by a homogeneous film our result reduces to the well-known Lifshitz formula. As a first application of theory, a one-dimensional modulated system with a homogeneous layer embedded in it is considered and a suitable perturbation theory for this system is developed. In the main part of this paper we limit the calculations to the non-retarded case, that is only the calculation of van der Waals interaction energy is given. As a second application of theory we consider the van der Waals interaction between two semi-infinite media across a planar region within which there is a thin film having an arbitrary variation of the dielectric permittivity. The importance of the precise evaluation of the transverse magnetic surface mode dispersion relation in inhomogeneous media is elucidated. For concreteness the influence of the transition layer between water and lipid in a symmetrical configuration is considered in some detail. The zero-frequency term and the dispersion-only contribution to the Hamaker coefficient are given analytically using some approximations and modelling of the dielectric constants reasonable for these dielectrics at room temperature. As a whole the results indicate the necessity of performing Lifshitz-type calculations on realistic inhomogeneous layered models, as are the models described here, for accurate interaction energy modelling. Of course further work is needed for real justification of the continuous variation of dielectric permittivities across phase interfaces.