An Analysis of Bubble Deformation by a Sphere Relevant to the Measurements of Bubble-Particle Contact Interaction and Detachment Forces

Atomic force microscopy makes it possible to measure the interacting forces between individual colloidal particles and air bubbles, which can provide a measure of the particle hydrophobicity. To indicate the level of hydrophobicity of the particle, the contact angle can be calculated, assuming that no interfacial deformation occurs with the bubble retaining a spherical profile. Our experimental results obtained using a modified sphere tensiometry apparatus to detach submillimeter spherical particles show that deformation of the bubble interface does occur during particle detachment. We also develop a theoretical model to describe the equilibrium shape of the bubble meniscus at any given particle position, based on the minimization of the free energy of the system. The developed model allows us to analyze high-speed video captured during detachment. In the system model deformation of the bubble profile is accounted for by the incorporation of a Lagrange multiplier into both the Young-Laplace equation and the force balance. The solution of the bubble profile matched to the high-speed video allows us to accurately calculate the contact angle and determine the total force balance as a function of the contact point of the bubble on the particle surface.