Phase field modelling of the detachment of bubbles from a solid substrate

TL;DR

This work addresses bubble detachment from solid substrates in viscous liquids using a phase-field model coupled to Navier–Stokes with a diffuse interface and a dynamic contact-angle boundary condition. It derives and validates a simple quadratic scaling $Bo_c \approx l_{th} \theta^2$ with $l_{th} = \frac{3}{2^{1/3}}$, and extends it to inclined substrates and external flows with corrections that depend on $\alpha$ and $w_{in}$. The model captures topological changes such as pinch-off and satellite formation, enabling predictions of detachment thresholds under diverse conditions. The results have practical implications for optimizing electrode area and gas release in electrochemical hydrogen production.

Abstract

We develop and implement numerically a phase field model for the evolution and detachment of a gas bubble resting on a solid substrate and surrounded by a viscous liquid. The bubble has a static contact angle $θ$ and will be subject to gravitational forces. We compute, as a function of the static contact angle, the cricital Bond number over which bubbles detach from the substrate. Then, we perform similar studies for bubble resting on inclined substrates and bubbles under the action of an external flow. We provide approximate formulas for the critical Bond number under all these circumstances. Our method is also able to resolve the pinchoff of the bubble and the possible appearence of satellites.

Figures (9)

• Figure 1: Sketch of physical settings: bubble under gravity (left), over a inclined substrabe (center) and under external flow and gravity (right). $\theta$ is the contact angle and $\alpha$ the inclination angle.
• Figure 2: Bond critical number versus contact angle. Squares are numerical simulations, blue is theory and discontinuos line is the parabolic fitting of the numerical simulations $Bo=l\theta^2$, being $l_{fit}=2.25$ and $l_{th}=2.38$.
• Figure 3: Numerical simulations of the evolution of a bubble with initial contact angle $3\pi/5$ and Bond number 4 (time in dimensionless units), below criticality.
• Figure 4: Numerical simulations of the evolution of a bubble with initial contact angle $3\pi/5$ and Bond number 15 (time in dimensionless units).
• Figure 5: Bond critical number versus contact angle for an inclined substrate $\alpha=30^\circ$. Squares are numerical simulations, blue is theory and discontinuos line is the parabolic fitting of the numerical simulations $Bo=l\theta^2$, being $l_{fit}=2.71$ and $l_{th}=2.38$.