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Mean-field model for the bubble size distribution in coarsening wet foams

Jacob Morgan, Simon Cox

TL;DR

This work addresses predicting the scaling-state bubble-size distribution in coarsening wet foams as a function of liquid fraction $\phi$ up to the unjamming transition $\phi_c \approx 0.36$. It introduces a three-dimensional mean-field border-blocking growth law that accounts for film-area effects and bubble pressure, and derives an analytic scaling-state distribution $\rho(\hat{R})$ via a Lifshitz–Slyozov–Marqusee framework. The authors validate the growth law against bubble-scale simulations and show that the scaling state is approached from various initial conditions, with a larger population of small bubbles at $\phi>0$ and qualitative differences from prior results attributed to the absence of rattlers. The findings provide a tractable, parameter-free description of the scaling-state distribution across moderate wetness and offer a baseline for incorporating bulk diffusion and adhesion effects in future work, with potential relevance to industrial and ISS foam systems. The work also highlights how border-blocking, film-based gas transfer modifies the small-bubble regime relative to the classic dry-Lemlich/Lifshitz–Slyozov picture, informing both theory and experimental interpretation of coarsening foams.

Abstract

Aqueous foams are subject to coarsening, whereby gas from the bubbles diffuses through the liquid phase. Gas is preferentially transported from small to large bubbles, resulting in a gradual decrease of the number of bubbles and an increase in the average bubble size. Coarsening foams are expected to approach a scaling state at late times in which their statistical properties are invariant. However, a model predicting the experimentally observed bubble-size distribution in the scaling state of foams with moderate liquid content, as a function of the liquid fraction $φ$, has not yet been developed. To this end, we propose a three-dimensional mean-field bubble growth law for foams without inter-bubble adhesion, validated against bubble-scale simulations, and use it to derive a prediction of the scaling-state bubble-size distribution for any $φ$ from zero up to the unjamming transition $φ_\text{c} \approx 36\%$. We verify that the derived scaling state is approached from a variety of initial conditions using mean-field simulations implementing the proposed growth law. Comparing our predicted bubble-size distribution with previous simulations and experimental results, we likewise find a large population of small bubbles when $φ> 0$, but there are qualitative differences from prior results which we attribute to the absence of rattlers, i.e. bubbles not pressed into contact with their neighbours, in our model.

Mean-field model for the bubble size distribution in coarsening wet foams

TL;DR

This work addresses predicting the scaling-state bubble-size distribution in coarsening wet foams as a function of liquid fraction up to the unjamming transition . It introduces a three-dimensional mean-field border-blocking growth law that accounts for film-area effects and bubble pressure, and derives an analytic scaling-state distribution via a Lifshitz–Slyozov–Marqusee framework. The authors validate the growth law against bubble-scale simulations and show that the scaling state is approached from various initial conditions, with a larger population of small bubbles at and qualitative differences from prior results attributed to the absence of rattlers. The findings provide a tractable, parameter-free description of the scaling-state distribution across moderate wetness and offer a baseline for incorporating bulk diffusion and adhesion effects in future work, with potential relevance to industrial and ISS foam systems. The work also highlights how border-blocking, film-based gas transfer modifies the small-bubble regime relative to the classic dry-Lemlich/Lifshitz–Slyozov picture, informing both theory and experimental interpretation of coarsening foams.

Abstract

Aqueous foams are subject to coarsening, whereby gas from the bubbles diffuses through the liquid phase. Gas is preferentially transported from small to large bubbles, resulting in a gradual decrease of the number of bubbles and an increase in the average bubble size. Coarsening foams are expected to approach a scaling state at late times in which their statistical properties are invariant. However, a model predicting the experimentally observed bubble-size distribution in the scaling state of foams with moderate liquid content, as a function of the liquid fraction , has not yet been developed. To this end, we propose a three-dimensional mean-field bubble growth law for foams without inter-bubble adhesion, validated against bubble-scale simulations, and use it to derive a prediction of the scaling-state bubble-size distribution for any from zero up to the unjamming transition . We verify that the derived scaling state is approached from a variety of initial conditions using mean-field simulations implementing the proposed growth law. Comparing our predicted bubble-size distribution with previous simulations and experimental results, we likewise find a large population of small bubbles when , but there are qualitative differences from prior results which we attribute to the absence of rattlers, i.e. bubbles not pressed into contact with their neighbours, in our model.

Paper Structure

This paper contains 8 sections, 31 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Mean-field growth law for wet foams
  3. Derivation of scaling state
  4. Simulations
  5. Simulation methods
  6. Simulation results
  7. Discussion
  8. Conclusion

Figures (12)

  • Figure 1: Border-blocking growth rate (expressed as the rate of change in bubble area $A_\text{b} = \pi R^2$) versus effective radius for $1024$ bubbles in 2D simulated foams at (a) $\phi = 2\%$ and (b) $\phi = 10\%$ without inter-bubble adhesion, the simulation data having been taken from our previous study.2024-morgan2023-morgan-data The bubble-area distribution is a compressed exponential fitted to experimental data by 2013-roth The individual bubble data is shown alongside its binned mean ($20$ equal bins are used; the error bars give the standard deviation within each bin). Comparison is made with eqn \ref{['eqn:growth-law-bb-2d']} and the 2D version of eqn \ref{['eqn:growth-law-pasquet']}, in which $R_\text{c}$ is obtained by numerically solving $\langle d A_\text{b} / d t \rangle = 0$ (i.e. conservation of total gas area1978-lemlich) and $\bar{\Pi}$ is measured in the simulations.2024-morgan The calculation of $d A_\text{b} / d t$ in the simulations is explained in the text.
  • Figure 2: Border-blocking relative growth rate2013-cantat (where $V = 4 \pi R^3 / 3$ is the bubble volume) versus effective radius for $256$ bubbles (aggregated from four $64$-bubble simulations at equal $\phi$) in 3D foams, using finite-element simulations described in ref. 2025-morgan-thesis. There is no inter-bubble adhesion, and the liquid fraction is (a) $\phi = 10\%$ and (b) $\phi = 30\%$. The bubble-size distribution2010-lambert is lognormal with standard deviation $0.4$ with respect to $R / \langle R \rangle$. The simulation data is plotted as in Fig. \ref{['fig:growth-law-2d']}, except $5$ equal bins are used due to the smaller number of bubbles, and is compared with eqns \ref{['eqn:growth-law-bb']} and \ref{['eqn:growth-law-pasquet']};2023-pasquet-b where $\hat{\Pi}$ and $R_{3 2}$ are the mean values measured in the aggregated simulations, and $R_\text{c}$ is obtained1978-lemlich by numerically solving $\langle d V / d t \rangle = 0$. The means of calculating $d V / d t$ is stated in the text. Comparison is also made to eqn \ref{['eqn:growth-law-bb']} with $R_{3 2}$ replaced by $R_\text{c}$ as described in the text, where $R_\text{c}$ is again calculated by solving $\langle d V / d t \rangle = 0$.
  • Figure 3: Probability distribution $\rho$ of the relative bubble size $\hat{R}$ in the scaling state, predicted by eqn \ref{['eqn:probability-dist']} ($C$ is obtained by numerical integration of the distribution), for various liquid fractions. We also plot $\rho$ in the limiting cases of a dry foam and a foam at the unjamming transition, given respectively by eqns \ref{['eqn:probability-dist-dry']}1961-wagner1985-markworth and \ref{['eqn:probability-dist-wet']}.
  • Figure 4: Various properties of eqn \ref{['eqn:probability-dist']} as functions of liquid fraction. The ratios of critical radius and cutoff radius to $R_{2 1}$ (and $R_{3 2}$ in the former case) are shown, alongside the polydispersity2004-kraynik$\mathcal{P}$ and geometric disorder2013-cantat$\sigma_V$, which are defined in the text, and the standard deviation $\sigma_R$ of $R / \langle R \rangle$. These are all evaluated numerically from eqns \ref{['eqn:probability-dist']} and \ref{['eqn:r-cutoff']}.
  • Figure 5: Coarsening rate (see text) versus liquid fraction $\phi$, as predicted by the growth law \ref{['eqn:growth-law-bb']} via eqn \ref{['eqn:x-crit-and-cutoff']}, alongside the prediction of 2023-pasquet-b2023-pasquet-b The predictions agree at $\phi = 0$, where they equal unity with the chosen scaling, and they both reach zero at $\phi = \phi_\text{c}$. The variation of $\rho(0)$ with $\phi$, given by eqn \ref{['eqn:probability-dist']} (with $C$ again being obtained numerically), is also plotted ($\rho(0) = 0$ at $\phi = 0$).
  • ...and 7 more figures