Stress tensor of inhomogeneous fluids

We derive explicit expressions for the stress tensor for general inhomogeneities in a one-component simple fluid in terms of density gradients and moments of the direct correlation function. The expressions follow from the change in grand potential ΔΩV that takes place in a selected portion of fluid of volume V as an arbitrary strain is applied to it. We employ the free energy density functional approach to determine ΔΩV, and consider two model mean-field density functionals, one spatially non-local and the other a local Landau form that contains a squared-gradient and a squared-laplacian terms. In the first case the expression for the stress tensor is spatially non-local, and in the latter case the tensor can be written down in the form of a volume σ and a surface contribution τ. These two terms can be modified by the transformation of volume into surface terms, or vice versa, of the fixed change ΔΩV, and in this way symmetric forms for both σ and τ can be obtained. We observe that the surface term τ and, more generally, the non-locality of the stress tensor are features that arise as the finite range of molecular interactions is restored in a local description by incorporation of higher order terms in the derivatives of the density. Our results corroborate expressions for the elastic bending free energy terms of curved interfaces derived previously.