Publications

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Surface Evoluer models of soap froth with a wide range of cell-size distributions are used to investigate random cellular morphology. Geometric properties of foams and foam cells are analyzed. A simple, accurate theory relates the total suface area of foam to the cell-size distribution. The total surface area is approximately equal to the total edge length when both quantities are scaled by average cell volume. Voronoi structures are significantly different from foams, which raises questions over their use for predicting structure-property relationships. © 2006 WILEY-VCH Verlag GmbH & Co. KGaA.

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The properties of solid foams depend on their structure, which usually evolves in the fluid state as gas bubbles expand to form polyhedral cells. The characteristic feature of foam structure-randomly packed cells of different sizes and shapes-is examined in this article by considering soap froth. This material can be modeled as a network of minimal surfaces that divide space into polyhedral cells. The cell-level geometry of random soap froth is calculated with Brakke's Surface Evolver software. The distribution of cell volumes ranges from monodisperse to highly polydisperse. Topological and geometric properties, such as surface area and edge length, of the entire foam and individual cells, are discussed. The shape of struts in solid foams is related to Plateau borders in liquid foams and calculated for different volume fractions of material. The models of soap froth are used as templates to produce finite element models of open-cell foams. Three-dimensional images of open-cell foams obtained with x-ray microtomography allow virtual reconstruction of skeletal structures that compare well with the Surface Evolver simulations of soap-froth geometry.

The microrheology of dry soap foams subjected to large, quasistatic, simple shearing deformations is analyzed. Two different monodisperse foams with tetrahedrally close-packed (TCP) structure are examined: Weaire-Phelan (A15) and Friauf-Laves (C15). The elastic-plastic response is evaluated by calculating foam structures that minimize total surface area at each value of strain. The minimal surfaces are computed with the Surface Evolver program developed by Brakke. The foam geometry and macroscopic stress are piecewise continuous functions of strain. The stress scales as T/V{sup 1/3} where T is surface tension and V is cell volume. Each discontinuity corresponds to large changes in foam geometry and topology that restore equilibrium to unstable configurations that violate Plateau's laws. The instabilities occur when the length of an edge on a polyhedral foam cell vanishes. The length can tend to zero smoothly or abruptly with strain. The abrupt case occurs when a small increase in strain changes the energy profile in the neighborhood of a foam structure from a local minimum to a saddle point, which can lead to symmetry-breaking bifurcations. In general, the new foam topology associated with each stable solution branch results from a cascade of local topology changes called T1 transitions. Each T1 cascade produces different cell neighbors, reduces surface energy, and provides an irreversible, film-level mechanism for plastic yield behavior. Stress-strain curves and average stresses are evaluated by examining foam orientations that admit strain-periodic behavior. For some orientations, the deformation cycle includes Kelvin cells instead of the original TCP structure; but the foam does not remain perfectly ordered. Bifurcations during subsequent T1 cascades lead to disorder and can even cause strain localization.

The microrheology of liquid foams is discussed for two different regimes: static equilibrium where the capillary number Ca is zero, and the viscous regime where viscosity and surface tension are important and Ca is finite. The Surface Evolver is used to calculate the equilibrium structure of wet Kelvin foams and dry soap froths with random structure, i.e., topological disorder. The distributions of polyhedra and faces are compared with the experimental data of Matzke. Simple shearing flow of a random foam under quasistatic conditions is also described. Viscous phenomena are explored in the context of uniform expansion of 2D and 3D foams at low Reynolds number. Boundary integral methods are used to calculate the influence of Ca on the evolution of foam microstructure, which includes bubble shape and the distribution of liquid between films, Plateau borders, and (in 3D) the nodes where Plateau borders meet. The micromechanical point of view guides the development of structure-property-processing relationships for foams.

Foams, like most highly structured fluids, exhibiting rheological behavior that is both fascinating and complex. We have developed microrheological models for uniaxial extension and simple shearing flow of a `dry`, perfectly ordered, three-dimensional foam composed of thin films with uniform surface tension T and negligible liquid content. We neglect viscous flow in the thin films and examine large elastic-plastic deformations of the foam. The primitive undeformed foam structure is composed of regular space-filling tetrakaidecahedra, which have six square and eight hexagonal surfaces. This structure possesses the film-network topology that is necessary to satisfy equilibrium: three films meet at each edge, which corresponds to a Plateau border, and four edges meet at vertex. However, to minimize surface energy, the films must meet at equal angles of 120{degrees} and the edges must join at equal tetrahedral angles of cos{sup {minus}1}({minus}1/3) {approx} 10.947{degree}. No film in an equilibrium foam structure can be a planar polygon because no planar polygon has all angles equal to the tetrahedral edge. In the equilibrium foam structure known as Kelvin`s minimal tetrakaidecahedron, the `squares` are planar quadrilateral surfaces with curved edges and the `hexagons` are non-planar saddle surfaces with zero mean curvature. As the foam structure evolves with the macroscopic flow, each film maintains zero mean curvature because the pressure is the same in every bubble. In general, the shape of each thin film, defined by z = h(x,y), satisfies R{sub 1}/1 + R{sub 2}/1 = {del}{center_dot} (1 + {vert_bar}{del}h{vert_bar}){sup {1/2}} = O where R{sub 1}{sup {minus}1} and A{sub 2}{sup {minus}1} are the principal curvatures. The appropriate boundary conditions correspond to three films meeting at equal angles. For the homogeneous deformations under consideration, the center of each film moves affinely with the flow. 5 refs

A boundary layer theory for the flow of power-law fluids in a converging planar channel has been developed. This theory suggests a Reynolds number for such flows, and following numerical integration, a boundary layer thickness. This boundary layer thickness has been used in the generation of a finite element mesh for the finite element code FIDAP. FIDAP was then used to simulate the flow of power-law fluids through a converging channel. Comparison of the analytic and finite element results shows the two to be in very good agreement in regions where entrance and exit effects (not considered in the boundary layer theory) can be neglected. 6 refs., 8 figs., 1 tab.