We analyse van der Waals interactions between a pair of dielectrically anisotropic plane-layered media interacting across a dielectrically isotropic solvent medium. We develop a general formalism based on transfer matrices to investigate the van der Waals torque and force in the limit of weak birefringence and dielectric matching between the ordinary axes of the anisotropic layers and the solvent. We apply this formalism to study the following systems: (i) a pair of single anisotropic layers, (ii) a single anisotropic layer interacting with a multilayered slab consisting of alternating anisotropic and isotropic layers, and (iii) a pair of multilayered slabs each consisting of alternating anisotropic and isotropic layers, looking at the cases where the optic axes lie parallel and/or perpendicular to the plane of the layers. For the first case, the optic axes of the oppositely facing anisotropic layers of the two interacting slabs generally possess an angular mismatch, and within each multilayered slab the optic axes may either be the same, or undergo constant angular increments across the anisotropic layers. In particular, we examine how the behaviors of the van der Waals torque and force can be tuned by adjusting the layer thicknesses, the relative angular increment within each slab, and the angular mismatch between the slabs.
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