The capillary bridge between two spheres: New closed-form equations in a two century old problem

Area of Science:

• Physics
• Materials Science
• Surface Science

Background:

• Capillary forces between solid spheres are crucial in various scientific and engineering fields.
• Previous models, like the toroidal and Derjaguin approximations, had limitations regarding accuracy and applicability to different geometries and liquid volumes.
• Understanding these forces is essential for predicting phenomena from self-assembly to microfluidics.

Purpose of the Study:

• To derive more accurate and general closed-form expressions for capillary forces between solid spheres.
• To overcome the limitations of existing approximations for capillary bridges.
• To provide a versatile equation applicable to spheres of equal and unequal sizes with varying contact angles.

Main Methods:

• Utilizing numerical solutions to analyze capillary bridges between spheres.
• Developing and validating analytical expressions based on numerical data.
• Comparing the new closed-form solution with established approximations like the Derjaguin equation.

Main Results:

• A new, simple, explicit algebraic equation accurately fits numerical results for capillary bridges.
• The derived closed-form solution is applicable to spheres of equal and unequal sizes.
• The new equation demonstrates higher accuracy and broader applicability than the Derjaguin approximation, especially for larger liquid volumes and separation distances.

Conclusions:

• The newly derived closed-form equation offers a more accurate and general description of capillary forces between spheres.
• This advancement provides a valuable tool for researchers and engineers working with capillary phenomena.
• The improved model extends the understanding of capillary bridge stability and rupture points.