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Bending rigidity and higher order curvature terms for the hard-sphere fluid near a curved wall

In this work I derive analytic expressions for the curvature dependent fluid-substrate surface tension of a hard sphere fluid on a hard curved wall. In a first step, the curvature thermodynamic properties are found as truncated power series in the activity in terms of the known second and third order cluster integral of the hard-sphere fluid near spherical and cylindrical walls. These results are then expressed as packing fraction power series and transformed to different reference regions which is equivalent to consider different positions of the dividing surface. All the obtained series expansions are rigorous results. Based on the truncated series it is shown that the bending rigidity of the system is non-null and that higher order terms in the curvature also exist. In a second step, approximate analytic expressions for the surface tension, the Tolman length, the bending rigidity and the Gaussian rigidity as functions of the packing fraction are found by considering the known terms of the series expansion complemented with a simple fitting approach. It is found that the obtained formulas accurately describe the curvature thermodynamic properties of the system, further, they are more accurate than any of the previously published expressions.