Theoretical study of van der Waals dispersion pressures considering one-dimensional material distributions in the in-plane direction

The van der Waals dispersion pressures between a half-space consisting of a uniform material and a half-space with a one-dimensional material distribution in the in-plane direction have been theoretically derived. Two patterns of material distribution were considered: a periodic distribution of materials (Pattern 1) and a distribution of two materials with a single interface (Pattern 2). The van der Waals pressure for Pattern 1 was derived based on a Fourier series, while the van der Waals pressure for Pattern 2 was derived as elementary functions. Both of the van der Waals pressures derived consist of two terms: a conventional term between half-spaces made of uniform materials and a spatial fluctuation term due to the material distribution. The basic characteristics of these van der Waals pressures were quantitatively clarified. Furthermore, an approximate method for obtaining the van der Waals pressure of Pattern 1 from Pattern 2 was proposed.