Numerical analysis of capillarity in packed spheres: planar hexagonal-packed spheres

General solutions of the capillary pressure for liquids as a function of contact angle and volume in planar close-packed spheres have been calculated numerically using Surface Evolver software. Applied pressure differences between liquid and vapor result in undulating (puckered) menisci exhibiting anticlastic curvature in the narrower spaces near particle contacts. The corresponding capillary pressures exhibit maxima with infiltration volume (minima with drainage), corresponding to critical pressures for engulfment of the spheres by the liquid (vapor). The analysis also reveals the formation of residual pendular rings of the wetting phase around particle contacts. Pendular ring formation is explored further by analyzing hexagonally packed spheres separated by 1/10 their radius. The results are discussed relative to the wide range of approximate solutions available in the literature.