  The leading order terms in a curvature expansion of the surface tension, the
Tolman length (first order), and rigidities (second order) have been shown to
play an important role in the description of nucleation processes. This work
presents general and rigorous expressions to compute these quantities for any
non-local density functional theory (DFT). The expressions hold for pure fluids
and mixtures, and reduce to the known expressions from density gradient theory
(DGT). The framework is applied to a Helmholtz energy functional based on the
perturbed chain polar statistical associating fluid theory (PCP-SAFT) and is
used for an extensive investigation of curvature corrections for pure fluids
and mixtures. Predictions from the full DFT are compared to two simpler
theories: predictive density gradient theory (pDGT), that has a density and
temperature dependent influence matrix derived from DFT, and DGT, where the
influence parameter reproduces the surface tension as predicted from DFT. All
models are based on the same equation of state and predict similar Tolman
lengths and spherical rigidities for small molecules, but the deviations
between DFT and DGT increase with chain length for the alkanes. For all
components except water, we find that DGT underpredicts the value of the Tolman
length, but overpredicts the value of the spherical rigidity. An important
basis for the calculation is an accurate prediction of the planar surface
tension. Therefore, further work is required to accurately extract Tolman
lengths and rigidities of alkanols, because DFT with PCP-SAFT does not
accurately predict surface tensions of these fluids.
