Deformation of Platonic foam cells: Effect on growth rate

The diffusive growth rate of a polyhedral cell in dry three-dimensional foams depends on details of shape beyond cell topology, in contrast to the situation in two dimensions, where, by von Neumann's law, the growth rate depends only on the number of cell edges. We analyze the dependence of the instantaneous growth rate on the shape of single foam cells surrounded by uniform pressure; this is accomplished by supporting the cell with films connected to a wire frame and inducing cell distortions by deforming the wire frame. We consider three foam cells with a very simple topology; these are the Platonic foam cells, which satisfy Plateau's laws and are based on the trivalent Platonic solids (tetrahedron, cube, and dodecahedron). The Surface Evolver is used to model cell deformations induced through extension, compression, shear, and torsion of the wire frames. The growth rate depends on the deformation mode and frame size and can increase or decrease with increasing cell distortion. The cells have negative growth rates, in general, but dodecahedral cells subjected to torsion in small wire frames can have positive growth rates. The deformation of cubic cells is demonstrated experimentally

Foam Micromechanics

Foam evokes many different images: waves breaking at the seashore, the head on a pint of Guinness, an elegant dessert, shaving, the comfortable cushion on which you may be seated... From the mundane to the high tech, foams, emulsions, and cellular solids encompass a broad range of materials and applications. Soap suds, mayonnaise, and foamed polymers provide practical motivation and only hint at the variety of materials at issue. Typical of mukiphase materiaIs, the rheoIogy or mechanical behavior of foams is more complicated than that of the constituent phases alone, which may be gas, liquid, or solid. For example, a soap froth exhibits a static shear modulus-a hallmark of an elastic solid-even though it is composed primarily of two Newtonian fluids (water and air), which have no shear modulus. This apparent paradox is easily resolved. Soap froth contains a small amount of surfactant that stabilizes the delicate network of thin liq- uid films against rupture. The soap-film network deforms in response to a macroscopic strain; this increases interracial area and the corresponding sur- face energy, and provides the strain energy of classical elasticity theory [1]. This physical mechanism is easily imagined but very challenging to quantify for a realistic three-dimensional soap froth in view of its complex geome- try. Foam micromechanics addresses the connection between constituent properties, cell-level structure, and macroscopic mechanical behavior. This article is a survey of micromechanics applied to gas-liquid foams, liquid-liquid emulsions, and cellular solids. We will focus on static response where the foam deformation is very slow and rate-dependent phenomena such as viscous flow can be neglected. This includes nonlinear elasticity when deformations are large but reversible. We will also discuss elastic- plastic behavior, which involves yield phenomena. Foam structures based on polyhedra packed to fill space provide a unify- ing geometrical theme. Because a two-dimensional situation is always easier to visualize and usually easier to analyze, the roots of foam micromechanics lie in the plane packed with polygons. There are striking similarities as well as obvious differences between 2D and 3D

Cell shape analysis of random tessellations based on Minkowski tensors

To which degree are shape indices of individual cells of a tessellation characteristic for the stochastic process that generates them? Within the context of stochastic geometry and the physics of disordered materials, this corresponds to the question of relationships between different stochastic models. In the context of image analysis of synthetic and biological materials, this question is central to the problem of inferring information about formation processes from spatial measurements of resulting random structures. We address this question by a theory-based simulation study of shape indices derived from Minkowski tensors for a variety of tessellation models. We focus on the relationship between two indices: an isoperimetric ratio of the empirical averages of cell volume and area and the cell elongation quantified by eigenvalue ratios of interfacial Minkowski tensors. Simulation data for these quantities, as well as for distributions thereof and for correlations of cell shape and volume, are presented for Voronoi mosaics of the Poisson point process, determinantal and permanental point processes, and Gibbs hard-core and random sequential absorption processes as well as for Laguerre tessellations of polydisperse spheres and STIT- and Poisson hyperplane tessellations. These data are complemented by mechanically stable crystalline sphere and disordered ellipsoid packings and area-minimising foam models. We find that shape indices of individual cells are not sufficient to unambiguously identify the generating process even amongst this limited set of processes. However, we identify significant differences of the shape indices between many of these tessellation models. Given a realization of a tessellation, these shape indices can narrow the choice of possible generating processes, providing a powerful tool which can be further strengthened by density-resolved volume-shape correlations.Comment: Chapter of the forthcoming book "Tensor Valuations and their Applications in Stochastic Geometry and Imaging" in Lecture Notes in Mathematics edited by Markus Kiderlen and Eva B. Vedel Jense

A Model for the Elasticity of Compressed Emulsions

We present a new model to describe the unusual elastic properties of compressed emulsions. The response of a single droplet under compression is investigated numerically for different Wigner-Seitz cells. The response is softer than harmonic, and depends on the coordination number of the droplet. Using these results, we propose a new effective inter-droplet potential which is used to determine the elastic response of a monodisperse collection of disordered droplets as a function of volume fraction. Our results are in excellent agreement with recent experiments. This suggests that anharmonicity, together with disorder, are responsible for the quasi-linear increase of $G$ and $\Pi$ observed at $\varphi_c$.Comment: RevTeX with psfig-included figures and a galley macr

Nonlinear Stress Fluctuation Dynamics of Sheared Disordered Wet Foam

Sheared wet foam, which stores elastic energy in bubble deformations, relaxes stress through bubble rearrangements. The intermittency of bubble rearrangements in foam leads to effectively stochastic drops in stress that are followed by periods of elastic increase. We investigate global characteristics of highly disordered foams over three decades of strain rate and almost two decades of system size. We characterize the behavior using a range of measures: average stress, distribution of stress drops, rate of stress drops, and a normalized fluctuation intensity. There is essentially no dependence on system size. As a function of strain rate, there is a change in behavior around shear rates of $0.07 {\rm s^{-1}}$.Comment: accepted to Physical Review

Deformation of Small Compressed Droplets

We investigate the elastic properties of small droplets under compression. The compression of a bubble by two parallel plates is solved exactly and it is shown that a lowest-order expansion of the solution reduces to a form similar to that obtained by Morse and Witten. Other systems are studied numerically and results for configurations involving between 2 and 20 compressing planes are presented. It is found that the response to compression depends on the number of planes. The shear modulus is also calculated for common lattices and the stability crossover between f.c.c.\ and b.c.c.\ is discussed.Comment: RevTeX with psfig-included figures and a galley macr

Theoretical study of the tube flow of one component sticky foam solutions. [As access deterrents in Safeguard program]

A theoretical model describing the flow of a solution that foams during tube flow was developed. The larger viscosity and smaller density of the foam phase relative to the original solution are responsible for the large resistance to flow that occurs when foaming takes place. The experimental observations of P.B. Rand and K.C. Goettsche that describe a maximum in flow rate as the temperature is increased are explained by the model. The temperature-dependence of the Freon 12 vapor pressure causes the decrease in flow rate at higher temperatures. Though the viscosities of the foam and solution may decrease with increasing temperature, the higher foaming pressure causes a larger proportion of the tube to be filled with the more viscous foam. The result is a reduced mass flow rate. The practical implications of this result on the high-temperature performance of a foam delivery system must be considered by the design engineer since foaming has the greatest detrimental effect under those conditions. (These polymer solution systems serve as access deterrents in the Safeguard program)

Foam Micromechanics

Foam evokes many different images: waves breaking at the seashore, the head on a pint of Guinness, an elegant dessert, shaving, the comfortable cushion on which you may be seated... From the mundane to the high tech, foams, emulsions, and cellular solids encompass a broad range of materials and applications. Soap suds, mayonnaise, and foamed polymers provide practical motivation and only hint at the variety of materials at issue. Typical of mukiphase materiaIs, the rheoIogy or mechanical behavior of foams is more complicated than that of the constituent phases alone, which may be gas, liquid, or solid. For example, a soap froth exhibits a static shear modulus-a hallmark of an elastic solid-even though it is composed primarily of two Newtonian fluids (water and air), which have no shear modulus. This apparent paradox is easily resolved. Soap froth contains a small amount of surfactant that stabilizes the delicate network of thin liq- uid films against rupture. The soap-film network deforms in response to a macroscopic strain; this increases interracial area and the corresponding sur- face energy, and provides the strain energy of classical elasticity theory [1]. This physical mechanism is easily imagined but very challenging to quantify for a realistic three-dimensional soap froth in view of its complex geome- try. Foam micromechanics addresses the connection between constituent properties, cell-level structure, and macroscopic mechanical behavior. This article is a survey of micromechanics applied to gas-liquid foams, liquid-liquid emulsions, and cellular solids. We will focus on static response where the foam deformation is very slow and rate-dependent phenomena such as viscous flow can be neglected. This includes nonlinear elasticity when deformations are large but reversible. We will also discuss elastic- plastic behavior, which involves yield phenomena. Foam structures based on polyhedra packed to fill space provide a unify- ing geometrical theme. Because a two-dimensional situation is always easier to visualize and usually easier to analyze, the roots of foam micromechanics lie in the plane packed with polygons. There are striking similarities as well as obvious differences between 2D and 3D

Foam drainage

Transient drainage from a column of persistent foam has been analyzed theoretically. Gravity-driven flow was assumed to occur through an interconnected network of Plateau borders that define the edges of foam cells taken to be regular pentagonal dodecahedrons. A small liquid volume fraction and monodisperse cell size distribution were assumed. In the basic model, it is assumed that all liquid is contained in Plateau borders that are bounded by rigid gas-liquid interfaces. The predicted half life, the time required for one half of the liquid to drain from the foam, is inversely proportional to the square of the cell diameter, illustrating the importance of foam structure in drainage. Liquid hold up in the films separating adjacent cells, nonuniform initial liquid volume fraction distribution and interfacial mobility are explored. Border suction due to reduced pressure in the Plateau borders provides a mechanism for film drainage. Simultaneous film drainage and flow through the Plateau borders are analyzed. Sufficient conditions for neglecting film drainage kinetics are obtained. The results indicate that improved foam stability is related to small cells, liquid hold up in the films and slow film drainage kinetics