Two-dimensional Dry Foam Research Articles

Average evolution and size-topology relations for coarsening 2d dry foams

Two-dimensional dry foams coarsen according to the von Neumann law as dA/dt ∝ (n − 6) where n is the number of sides of a bubble with area A. Such foams reach a self-similar scaling state where area and side-number distributions are stationary. Combining self-similarity with the von Neumann law, we derive time derivatives of moments of the bubble area distribution and a relation connecting area moments with averages of the side-number distribution that are weighted by powers of bubble area. To test these predictions, we collect and analyze high precision image data for a large number of bubbles squashed between parallel acrylic plates and allowed to coarsen into the self-similar scaling state. We find good agreement for moments ranging from 2–20.

Quasistatic rheology of soft cellular systems using the cellular Potts model.

Soft cellular systems, such as foams or biological tissues, exhibit highly complex rheological properties, even in the quasistatic regime, that numerical modeling can help to apprehend. We present a numerical implementation of quasistatic strain within the widely used cellular Potts model (CPM). The accuracy of the method is tested by simulating the quasistatic strain of two-dimensional dry foams, both ordered and disordered. The implementation of quasistatic strain in CPM allows the investigation of sophisticated interplays between stress-strain relationship and structural changes that take place in cellular systems.

Numerical study on rheology of two-dimensional dry foam

We study the mechanical response of two-dimensional aqueous foams subjected to an oscillating shear strain using numerical simulations based on the immersed boundary method. Foams have unique rheological properties ranging from solid-like to fluid-like. They have an elastic property under a small strain with a linear stress-strain relationship. As the strain increases, the foams undergo topological rearrangements with a sudden release of energy and stress. Then the energy-strain and stress-strain curves exhibit hysteresis behaviors under the oscillating shear strain, and the macroscopic response of foams changes from a viscoelastic solid to a viscoelastic fluid. These wide-ranging dynamical responses of foams result from avalanches of topological rearrangements which are concentrated in a single localized region. This is called a shear localization, and we investigate the effect of the topological rearrangements and the shear localization on the rheology of foams.

Numerical Study on Statistical Behaviors of Two-Dimensional Dry Foam

Numerical Study on Statistical Behaviors of Two-Dimensional Dry Foam

Simulation of surfactant transport during the rheological relaxation of two-dimensional dry foams.

We describe a numerical model to predict the rheology of two-dimensional dry foams. The model accurately describes soap film curvature and viscous friction with the walls, and includes the transport of surfactant within the films and across the vertices where films meet. It accommodates the changes in foam topology that occur when a foam flows and, in particular, accurately represents the relaxation of the foam following a topological change. The model is validated against experimental data, allowing the prediction of elastic and viscous parameters associated with different surfactant solutions.

Geometry and Topology of Two-Dimensional Dry Foams: Computer Simulation and Experimental Characterization.

Pseudo-two-dimensional (2D) foams are commonly used in foam studies as it is experimentally easier to measure the bubble size distribution and other geometric and topological properties of these foams than it is for a 3D foam. Despite the widespread use of 2D foams in both simulation and experimental studies, many important geometric and topological relationships are still not well understood. Film size, for example, is a key parameter in the stability of bubbles and the overall structure of foams. The relationship between the size distribution of the films in a foam and that of the bubbles themselves is thus a key relationship in the modeling and simulation of unstable foams. This work uses structural simulation from Surface Evolver to statistically analyze this relationship and to ultimately formulate a relationship for the film size in 2D foams that is shown to be valid across a wide range of different bubble polydispersities. These results and other topological features are then validated using digital image analysis of experimental pseudo-2D foams produced in a vertical Hele-Shaw cell, which contains a monolayer of bubbles between two plates. From both the experimental and computational results, it is shown that there is a distribution of sizes that a film can adopt and that this distribution is very strongly dependent on the sizes of the two bubbles to which the film is attached, especially the smaller one, but that it is virtually independent of the underlying polydispersity of the foam.

The Immersed Boundary Method for Two-Dimensional Foam with Topological Changes

AbstractWe extend the immersed boundary (IB) method to simulate the dynamics of a 2D dry foam by including the topological changes of the bubble network. In the article [Y. Kim, M.-C. Lai, and C. S. Peskin, J. Comput. Phys. 229:5194-5207,2010], we implemented an IB method for the foam problem in the two-dimensional case, and tested it by verifying the von Neumann relation which governs the coarsening of a two-dimensional dry foam. However, the method implemented in that article had an important limitation; we did not allow for the resolution of quadruple or higher order junctions into triple junctions. A total shrinkage of a bubble with more than four edges generates a quadruple or higher order junction. In reality, a higher order junction is unstable and resolves itself into triple junctions. We here extend the methodology previously introduced by allowing topological changes, and we illustrate the significance of such topological changes by comparing the behaviors of foams in which topological changes are allowed to those in which they are not.

Growth of Aqueous Foam on Flexible Membranes

I consider the coarsening dynamics of two-dimensional dry foam sandwiched by deformable membranes. The time-varying deformation of the confining membranes gives rise to a significant alteration in the evolution of polygonal cells of bubbles when compared to the case of rigid membranes. This alteration is attributed to the correlation between the rate of inter-cell gas transfer and temporal fluctuation in surface curvature within a cell domain. Existing material constants are referred to show that the correlation can yield sizeable consequences in actual coarsening experiments.

Emergence of foams from the breakdown of the phase field crystal model

The phase field crystal (PFC) model captures the elastic and topological properties of crystals with a single scalar field at small undercooling. At large undercooling, new foamlike behavior emerges. We characterize this foam phase of the PFC equation and propose a modified PFC equation that may be used for the simulation of foam dynamics. This minimal model reproduces von Neumann's rule for two-dimensional dry foams and Lifshitz-Slyozov coarsening for wet foams. We also measure the coordination number distribution and find that its second moment is larger than previously reported experimental and theoretical studies of soap froths, a finding that we attribute to the wetness of the foam increasing with time.

Numerical simulations of two-dimensional foam by the immersed boundary method

In this paper, we present an immersed boundary (IB) method to simulate a dry foam, i.e., a foam in which most of the volume is attributed to its gas phase. Dry foam dynamics involves the interaction between a gas and a collection of thin liquid-film internal boundaries that partition the gas into discrete cells or bubbles. The liquid-film boundaries are flexible, contract under the influence of surface tension, and are permeable to the gas, which moves across them by diffusion at a rate proportional to the local pressure difference across the boundary. Such problems are conventionally studied by assuming that the pressure is uniform within each bubble. Here, we introduce instead an IB method that takes into account the non-equilibrium fluid mechanics of the gas. To model gas diffusion across the internal liquid-film boundaries, we allow normal slip between the boundary and the gas at a velocity proportional to the (normal) force generated by the boundary surface tension. We implement this method in the two-dimensional case, and test it by verifying the von Neumann relation, which governs the coarsening of a two-dimensional dry foam. The method is further validated by a convergence study, which confirms its first-order accuracy.

Modelling light transport in dry foams by a coarse-grained persistent random walk

A two-dimensional persistence random walk model with two states is considered. In the immobile state, the particle chooses a new direction according to some probability function, while in the mobile state it persists on its direction and performs a ballistic motion. Starting from a master equation, we are able to calculate the diffusion constant for this model. The model was developed to explain Monte Carlo simulations of photon diffusion in two-dimensional dry foams. While it gives the leading term in the diffusion constant D, it fails to reproduce the details in D which depend on the amount of disorder in foams. This might show that diffusing photons probe local structural correlations in foams.

SELF-ORGANIZED CRITICAL BEHAVIOR OF TWO-DIMENSIONAL FOAMS

A simple dynamical model for two-dimensional dry foam rheology is constructed for which surface tension effects and viscous dissipation at Plateau borders (intersections of three cell boundaries) are taken into account and is studied by computer simulation. Under externally applied shear strain increasing at small rates, the system exhibits avalanche-like release of stress that has been accumulating under increasing strain. There is a close similarity with earthquake models that show self-organized criticality (SOC). We discuss related simulation of two-dimensional wet foams under statically applied strain by Hutzler et al. [Phil. Mag. B71, 277 (1995)] showing critical behavior.