Shape of a particle-laden bubble at a liquid surface

Particle-laden bubbles are widely observed in nature and industrial applications such as foaming, emulsification, froth flotation, and gas evolution reactions, with the shape of bubbles at the liquid surface being important for understanding these processes. Although the static shape of bare bubbles has been well studied, how an attached particle affects the bubble shape remains elusive. Here we establish a theoretical model based on force equilibrium to describe the profile of a particle-laden bubble at the liquid surface. The accuracy of the theoretical model is validated to well describe the experimental observations. We find that increasing the particle size and density pulls the bubble down and stretches the bubble vertically, yielding a smaller bubble cap angle and aspect ratio. However, the increase in particle contact angle barely changes the bubble cap angle and causes the aspect ratio to first increase and then decrease. Analysis of the acting forces indicates that the bubble cap is pulled down mainly by particle gravity, while the bubble aspect ratio is under the combination effect of particle gravity and the curvature at the three-phase contact line that is influenced by the contact angle. Our findings can provide guidance for further studies on the dynamics of particle-contaminated bubbles.