Nodoids and toroids: comparison of two geometries for the meniscus profile of a wetting liquid between two touching isolated spheres and extensions to the model of a collection of packed spheres

In the mid 1960s the present authors published two papers dealing with penetration of nonwetting liquids such as mercury into the interstitial void spaces using the model of uniform packed spheres. A circular arc was used to approximate the liquid-vapor interface in both papers. However, our circular arc-toroid values for the pressure-volume relationship in the pendular ring which exists between two touching spheres was criticized. The authors concluded that our approximation led to unacceptably large differences compared to the values calculated from the exact nodoid shape. This incorrect conclusion was never rebutted and has, in fact, been misinterpreted by subsequent workers to include values calculated for the shape of the access opening and the associated pressure for penetration into the void space of a collection of spheres. This leaves a cloud of uncertainty, not only over our original work on nonwetting fluids, but on the application of our procedures to the field of wetting fluids. The contrast in the geometrical shapes of the toroid and nodoid is depicted and the pressure values are compared at equal volumes. In contrast to the claim of excessive error, we show the toroid geometry, in conjunction with a pressure-volume work derivation, to have a maximum error of 0.06% as compared to the nodoid at all liquid-solid contact angles. The toroid also has the advantage of using a readily derived work versus surface free energy balance rather than requiring the use of incomplete elliptic integrals to evaluate the nodoid. Attempts to use radii of curvature to evaluate the toroid shape are shown to give extremely poor approximations of the exact values for the pressure. Values reported for access to the interior void space of a collection of spheres still need adjustment for the effect of contact angles between 0° and 180° for characterizing assemblies of real solids by computing “equivalent spherical” particle size from porosity and mercury penetration data. However, there is no anticipation that use of the circular arc will introduce large errors in the results. This gives confidence to us and others working with wetting media to test the potential applicability of the packed sphere model to various diverse fields.