  We report a detailed study, using state-of-the-art simulation and theoretical
methods, of the depletion potential between a pair of big hard spheres immersed
in a reservoir of much smaller hard spheres, the size disparity being measured
by the ratio of diameters q=\sigma_s/\sigma_b. Small particles are treated
grand canonically, their influence being parameterized in terms of their
packing fraction in the reservoir, \eta_s^r. Two specialized Monte Carlo
simulation schemes --the geometrical cluster algorithm, and staged particle
insertion-- are deployed to obtain accurate depletion potentials for a number
of combinations of q\leq 0.1 and \eta_s^r. After applying corrections for
simulation finite-size effects, the depletion potentials are compared with the
prediction of new density functional theory (DFT) calculations based on the
insertion trick using the Rosenfeld functional and several subsequent
modifications. While agreement between the DFT and simulation is generally
good, significant discrepancies are evident at the largest reservoir packing
fraction accessible to our simulation methods, namely \eta_s^r=0.35. These
discrepancies are, however, small compared to those between simulation and the
much poorer predictions of the Derjaguin approximation at this \eta_s^r. The
recently proposed morphometric approximation performs better than Derjaguin but
is somewhat poorer than DFT for the size ratios and small sphere packing
fractions that we consider. The effective potentials from simulation, DFT and
the morphometric approximation were used to compute the second virial
coefficient B_2 as a function of \eta_s^r. Comparison of the results enables an
assessment of the extent to which DFT can be expected to correctly predict the
propensity towards fluid fluid phase separation in additive binary hard sphere
mixtures with q\leq 0.1.
