Surface integration approach: A new technique for evaluating geometry dependent forces between objects of various geometry and a plate

We present a new approach, which can be considered as a generalization of the Derjaguin approximation, that provides exact means to determine the force acting between a three-dimensional body of any shape and a half-space mutually interacting via pairwise potentials. Using it, in the cases of the Lennard-Jones, standard and the retarded (Casimir) van der Waals interactions we derive exact expressions for the forces between a half-space or a slab of finite thickness and an ellipsoid in a general orientation, which in the simplest case reduces to a sphere, a tilted fully elliptic torus, and a body obtained via rotation of a single loop generalized Cassini oval, a particular example of which mimics the shape of a red blood cell. The results are obtained for the case when the object is separated from the plane via a non-polar continuous medium that can be gas, liquid or vacuum. Specific examples of biological objects of various shapes interacting with a plate like substrates are also considered.

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► We developed a new approach, which we called surface integration approach.
► Using it we derived exact results in the case of an ellipsoid-plate interaction.
► The toroidal shape was generalized and analytical results were obtained.
► Result for the van der Waals force between red blood cell and a plate were obtained.