Researchers in the US have created a new mathematical model to describe the complex evolution of foamy bubbles – something that has proved fiendishly difficult to model thanks to the hugely varying length and time scales involved. Their computed results closely match theoretical models as well as lab-based observations of foamy bubbles. The team hopes the underlying equations could have a variety of applications, including helping to make better metal and plastic foams, developing lightweight crash-absorbent materials and also to model a number of biological processes such as the growth of cell clusters.

Heady maths

Foams are all around us: from the froth on a cappuccino or beer to the soapy suds in a bubble bath. However, scientists have found it difficult to describe exactly how such clusters of bubbles coalesce, grow and change shape over time – before they ultimately go pop. An early attempt at understanding the structure of soapy foams is encapsulated in "Plateau's laws" – formulated by 19th-century Belgian physicist Joseph Plateau. Then Lord Kelvin developed his theory of an "ideal foam" of equal-sized bubbles in 1887, an accurate version of which was finally made in the lab in 2012 by a team at Trinity College, Dublin. But a more general set of equations describing bubbles on varying length and time scales remained elusive, until now. The challenge is to create mathematical models that describe how interfaces between bubbles move and how they "meet" in complicated phases.

Key phases

Now, James Sethian and Robert Saye of the University of California, Berkeley have separated the various processes that determine a foam's evolution according to the different length and time scales at which they occur – and have created a model for bulk foam dynamics. The researchers say that the model accurately describes how fluid moves within a bubble and how the individual cells form and how their junctions (or borders) are rearranged as individual bubbles within the foam burst.

To do this, Sethian and Saye identified three distinct regimes or phases of foam evolution. "We identified and separated the three phases – the drainage of liquid from a bubble's membrane, the rupture of the drained bubble and the macroscopic rearrangement of the bubbles within the foam – to simulate the system," explains Sethian.

The first set of equations describes how the liquid drains from a bubble wall, thanks to gravity, so that the wall eventually becomes so thin that it ruptures. The next set of equations explains the liquid flow at the junctions between bubble membranes; while the third set considers how the entire foam rearranges to move closer to equilibrium, a motion that happens on a macroscopic scale.

Beach bubbles

Sethian and Saye tested their formulae on bubble clusters of different sizes and found that they could accurately predict the interactions of gases and liquids in these foamy materials. They also developed a fourth set of equations that allowed them to simulate a movie that shows how light would reflect off a small foam sample as its bubbles rearrange. The researchers picked a beach scene for the simulation, so that they could "visualize and see how well the model captures what you would see in real life, while still accurately showing how the light would reflect", as Sethian explains.

These processes are all influenced by a variety of factors, including viscosity, surface tension, gravity and other terms of fluid dynamics. Some of these factors can be modified in the current model, but others, such as evaporation, that are currently not included can be added quite easily, according to the researchers.

Sethian points out that it took the team five days to solve the full set of equations of motion using a supercomputer to get the most refined solution of the algorithms. He says that the entire mathematical formulation and codes will be available to anyone who is interested in running similar simulations at whatever scales they wish, for any applications, including industrial ones.

While a large part of the aim of this work was to develop a fundamental model, the researchers claim that it could have other applications. When it comes to biological modelling, Sethian says the equations could help to understand highly complex systems, such as cell cluster growth, that may go from being organized to unorganized systems. According to him, the models might help "to better understand how cells group together and aggregate...and to study the kind of physical forces involved – such as adhesion between cell boundaries, fluid dynamics, etc – as well as the mechanisms involved in how cell cluster grow from clusters of 5 to 10 cells to those of hundreds to thousands of cells".

Take a look at the video below of a collapsing soap-bubble cluster, shown with thin-film interference and computed using Sethian and Saye's multiscale model.

The research is published in Science.