Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-17T20:11:46.775Z Has data issue: false hasContentIssue false
  • English
  • Français

An analytical theory for the capillary bridge force between spheres

Published online by Cambridge University Press:  22 December 2016

N. P. Kruyt  and
O. Millet
N. P. Kruyt*
Affiliation:
Department of Mechanical Engineering, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands
O. Millet
Affiliation:
LaSIE-UMR CNRS 7356, La Rochelle University, France
*
†Email address for correspondence: n.p.kruyt@utwente.nl
Get access Rights & Permissions [Opens in a new window]

Abstract

An analytical theory has been developed for properties of a steady, axisymmetric liquid–gas capillary bridge that is present between two identical, perfectly wettable, rigid spheres. In this theory the meridional profile of the capillary bridge surface is represented by a part of an ellipse. Parameters in this geometrical description are determined from the boundary conditions at the three-phase contact circle at the sphere and at the neck (i.e. in the middle between the two spheres) and by the condition that the mean curvature be equal at the three-phase contact circle and at the neck. Thus, the current theory takes into account properties of the governing Young–Laplace equation, contrary to the often-used toroidal approximation. Expressions have been developed analytically that give the geometrical parameters of the elliptical meridional profile as a function of the capillary bridge volume and the separation between the spheres. A rupture criterion has been obtained analytically that provides the maximum separation between the spheres as a function of the capillary bridge volume. This rupture criterion agrees well with a rupture criterion from the literature that is based on many numerical solutions of the Young–Laplace equation. An expression has been formulated analytically for the capillary force as a function of the capillary bridge volume and the separation between the spheres. The theoretical predictions for the capillary force agree well with the capillary forces obtained from the numerical solutions of the Young–Laplace equation and with those according to a comprehensive fit from the literature (that is based on many numerical solutions of the Young–Laplace equation), especially for smaller capillary bridge volumes.

JFM classification

Interfacial Flows (free surface): Interfacial Flows (free surface) Interfacial Flows (free surface): Liquid bridges
Type
Papers
Information
Journal of Fluid Mechanics , Volume 812 , 10 February 2017 , pp. 129 - 151
Check for updates
Copyright
© 2016 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ardito, R., Corigliano, A., Frangi, A. & Rizzini, F. 2014 Advanced models for the calculation of capillary attraction in axisymmetric configurations. Eur. J. Mech. A 47, 298–308.Google Scholar
Bostwick, J. B. & Steen, P. H. 2015 Stability of constrained capillary surfaces. Annu. Rev. Fluid Mech. 47, 539–568.Google Scholar
Cundall, P. A. & Strack, O. D. L. 1979 A discrete numerical model for granular assemblies. Géotechnique 9, 47–65.Google Scholar
Eggers, J. & Villermaux, E. 2008 Physics of liquid jets. Rep. Prog. Phys. 71, 1–79.Google Scholar
Erle, M. A., Dyson, D. C. & Morrow, N. R. 1971 Liquid bridges between cylinders, in a torus, and between spheres. AIChE J. 17, 115–121.Google Scholar
Farmer, T. P. & Bird, J. C. 2015 Asymmetric capillary bridges between contacting spheres. J. Colloid Interface Sci. 454, 192–199.Google Scholar
Fisher, R. A. 1926 On the capillary forces in an ideal soil; correction of formulae given by W. B. Haines. J. Agricultural Sci. 16, 492–505.Google Scholar
Gagneux, G. & Millet, O. 2014 Analytical calculation of capillary bridge properties deduced as an inverse problem from experimental data. Trans. Porous Med. 105, 117–139.CrossRefGoogle Scholar
Gladkyy, A. & Schwarze, R. 2014 Comparison of different capillary bridge models for application in the Discrete Element Method. Granul. Matt. 16, 911–920.Google Scholar
Harireche, O., Faramarzi, A. & Alani, A. M. 2013 A toroidal approximation of capillary forces in polydisperse granular assemblies. Granul. Matt. 15, 573–581.Google Scholar
Harireche, O., Faramarzi, A. & Alani, A. M. 2015 Prediction of inter-particle capillary forces for non-perfectly wettable granular assemblies. Granul. Matt. 17, 537–543.CrossRefGoogle Scholar
Israelachvili, J. N. 2011 Intermolecular and Surface Forces, 3rd edn. Elsevier.Google Scholar
Korn, G. A. & Korn, T. M. 1968 Mathematical Handbook for Scientists and Engineers. McGraw-Hill.Google Scholar
Kralchevsky, P. A. & Nagayama, K. 2001 Particles at Fluid Interfaces and Membranes. Elsevier.Google Scholar
Li, Y. & Sprittles, J. E. 2016 Capillary breakup of a liquid bridge: identifying regimes and transitions. J. Fluid Mech. 797, 29–59.Google Scholar
Lian, G. & Seville, J. P. K. 2016 The capillary bridge between two spheres: new closed-form equations in a two century old problem. Adv. Colloid Interface Sci. 227, 53–62.Google Scholar
Lian, G., Thornton, C. & Adams, M. J. 1993 A theoretical study of the liquid bridge forces between two rigid spherical bodies. J. Colloid Interface Sci. 161, 138–147.CrossRefGoogle Scholar
Mason, G. & Clark, W. C. 1965 Liquid bridges between spheres. Chem. Engng Sci. 20, 859–866.Google Scholar
Megias-Alguacil, D. & Gauckler, L. J. 2011 Accuracy of the toroidal approximation for the calculus of concave and convex liquid bridges between particles. Granul. Matt. 13, 487–492.Google Scholar
Mielniczuk, B., Hueckel, T. & El Youssoufi, M. S. 2015 Laplace pressure evolution and four instabilities in evaporating two-grain liquid bridges. Powder Technol. 283, 137–151.Google Scholar
Orr, F. M., Scriven, L. E. & Rivas, A. P. 1975 Pendular rings between solids: meniscus properties and capillary force. J. Fluid Mech. 67, 723–742.Google Scholar
Rabinovich, Y. I., Esayanur, M. S. & Moudgil, B. M. 2005 Capillary forces between two spheres with a fixed volume liquid bridge: theory and experiment. Langmuir 21, 10992–10997.Google Scholar
Reyssat, E. 2015 Capillary bridges between a plane and a cylindrical wall. J. Fluid Mech. 773, R1.CrossRefGoogle Scholar
Richefeu, V., El Youssoufi, M. S., Peyroux, R. & Radjaï, F. 2008 A model of capillary cohesion for numerical simulations of three-dimensional polydisperse granular media. Intl J. Numer. Anal. Meth. Geomech. 32, 1365–1383.Google Scholar
Soulié, F., Cherblanc, F., El Youssoufi, M. S. & Saix, C. 2006 Influence of liquid bridges on the mechanical behaviour of polydisperse granular materials. Intl J. Numer. Anal. Meth. Geomech. 30, 213–228.CrossRefGoogle Scholar
Than, V. D., Khamseh, S., Tang, A. M., Pereira, J. M., Chevoir, F. & Roux, J. N. 2016 Basic mechanical properties of wet granular materials: a DEM study. J. Engng Mech. ASCE C4016001.Google Scholar
Wexler, J. S., Heard, T. M. & Stone, H. A. 2014 Capillary bridges between soft substrates. Phys. Rev. Lett. 112, 066102.Google Scholar
Willett, C. D., Adams, M. J., Johnson, S. A. & Seville, J. P. K. 2000 Capillary bridges between two spherical bodies. Langmuir 16, 9396–9405.CrossRefGoogle Scholar