Bubble Rafts Research Articles

Comparison of low-amplitude oscillatory shear in experimental and computational studies of model foams

A fundamental difference between fluids and solids is their response to applied shear. Solids possess static shear moduli, while fluids do not. Complex fluids such as foams display an intermediate response to shear with nontrivial frequency-dependent shear moduli. In this paper, we conduct coordinated experiments and numerical simulations of model foams subjected to boundary-driven oscillatory planar shear. Our studies are performed on bubble rafts (experiments) and the bubble model (simulations) in two dimensions. We focus on the low-amplitude flow regime in which T1 events, i.e., bubble rearrangement events where originally touching bubbles switch nearest neighbors, do not occur, yet the system transitions from solid- to liquidlike behavior as the driving frequency is increased. In both simulations and experiments, we observe two distinct flow regimes. At low frequencies omega, the velocity profile of the bubbles increases linearly with distance from the stationary wall, and there is a nonzero total phase shift between the moving boundary and interior bubbles. In this frequency regime, the total phase shift scales as a power law Delta approximately omegan with n approximately 3. In contrast, for frequencies above a crossover frequency omega>omegap, the total phase shift Delta scales linearly with the driving frequency. At even higher frequencies above a characteristic frequency omeganl>omegap, the velocity profile changes from linear to nonlinear. We fully characterize this transition from solid- to liquidlike flow behavior in both the simulations and experiments and find qualitative and quantitative agreements for the characteristic frequencies.

Bubble-raft model for a paraboloidal crystal

We investigate crystalline order on a two-dimensional paraboloid of revolution by assembling a single layer of millimeter-sized soap bubbles on the surface of a rotating liquid, thus extending the classic work of Bragg and Nye on planar soap bubble rafts. Topological constraints require crystalline configurations to contain a certain minimum number of topological defects such as disclinations or grain boundary scars whose structure is analyzed as a function of the aspect ratio of the paraboloid. We find the defect structure to agree with theoretical predictions and propose a mechanism for scar nucleation in the presence of large Gaussian curvature.

Impact of boundaries on velocity profiles in bubble rafts

Under conditions of sufficiently slow flow, foams, colloids, granular matter, and various pastes have been observed to exhibit shear localization, i.e., regions of flow coexisting with regions of solidlike behavior. The details of such shear localization can vary depending on the system being studied. A number of the systems of interest are confined so as to be quasi two-dimensional, and an important issue in these systems is the role of the confining boundaries. For foams, three basic systems have been studied with very different boundary conditions: Hele-Shaw cells (bubbles confined between two solid plates); bubble rafts (a single layer of bubbles freely floating on a surface of water); and confined bubble rafts (bubbles confined between the surface of water below and a glass plate on top). Often, it is assumed that the impact of the boundaries is not significant in the "quasistatic limit," i.e., when externally imposed rates of strain are sufficiently smaller than internal kinematic relaxation times. In this paper, we directly test this assumption for rates of strain ranging from 10(-3) to 10(-2) s(-1). This corresponds to the quoted rate of strain that had been used in a number of previous experiments. It is found that the top plate dramatically alters both the velocity profile and the distribution of nonlinear rearrangements, even at these slow rates of strain. When a top is present, the flow is localized to a narrow band near the wall, and without a top, there is flow throughout the system.

The crystal structure of bubbles in the wet foam limit.

We have observed a rich variety of three-dimensional crystal and defect structures spontaneously formed by small (diameter 200 µm) bubbles in a wet foam. The observations confirm and extend those made by Bragg and Nye in 1947. However, while their experiments with two-dimensional bubble rafts have stimulated many researchers, their work on assemblages does not appear to have been followed up. These ordered packings now pose intriguing questions for the physics of foams. The bubbles seem too large for conventional thermodynamics and kinetics to easily explain the high degree of ordering.

Statistics of bubble rearrangements in a slowly sheared two-dimensional foam

Many physical systems exhibit plastic flow when subjected to slow steady shear. A unified picture of plastic flow is still lacking; however, there is an emerging theoretical understanding of such flows based on irreversible motions of the constituent "particles" of the material. Depending on the specific system, various irreversible events have been studied, such as T1 events in foam and shear transformation zones (STZ's) in amorphous solids. This paper presents an experimental study of the T1 events in a model, two-dimensional foam: bubble rafts. In particular, I report on the connection between the distribution of T1 events and the behavior of the average stress and average velocity profiles during both the initial elastic response of the bubble raft and the subsequent plastic flow at sufficiently high strains.

Velocity Profiles in Slowly Sheared Bubble Rafts

Measurements of average velocity profiles in a bubble raft subjected to slow, steady shear demonstrate the coexistence between a flowing state and a jammed state similar to that observed for three-dimensional foams and emulsions [P. Coussot et al., Phys. Rev. Lett. 88, 218301 (2002)]. For sufficiently slow shear, the flow is generated by nonlinear topological rearrangements. We report on the connection between this short-time motion of the bubbles and the long-time averages. We find that velocity profiles for individual rearrangement events fluctuate, but a smooth, average velocity is reached after averaging over only a relatively few events.

How many plates?

Herein I address the current number of plates, the number there should be, and whether there is a pattern in the plate mosaic. Related issues are the optimal sizes and shapes of plates and spacings of ridges, trenches, and transform faults. Similar questions arise in studies of foams, bubble rafts, buckyballs, mudcracks, columnar jointing, the tessellation of spheres, and the planforms of convection. In sphere-covering problems, and in dynamic problems, pentagons replace the familiar hexagons. The “ground” state of plate tectonics on a homogeneous planet may involve ∼12 plates with five nearest and five next-nearest neighbors. The plate mosaic may be a self-organized network of plates and force chains, which are readily reorganized by stress changes. This paper starts with the premise that the mosaic may have simple and surficial explanations rather than convective or plutonic causes. The study of the tessellation of Earth can be called “platonics” to distinguish it from the idea that the lithosphere necessarily mirrors the planform of mantle convection.

Stresses in compressed bubble rafts

We present a simple geometric method to determine the stress field in a compressed bubble raft. We show that bubble rafts exhibit arches phenomena as do granular materials, and examine the relaxation of stresses following a T 1 transformation. To cite this article: C. Ybert, J.-M. di Meglio, C. R. Physique 3 (2002) 555–559.

Stability of two-dimensional foam

The stability of two-dimensional foam comprising bubble rafts constrained to a fixed area of liquid surface has been examined. It is found that a perfectly sixfold-coordinated foam is unstable against loss of cohesion, the foam breaking after some 20 min to leave a significant surface area free of bubbles. Foams containing a single dislocation or point defect are stable against such breakage. It is argued that it is bubble deformations induced by the elastic stress fields around the defects which are responsible for this stabilization.

Two-dimensional bubble rafts

Abstract We show that magnetic bubbles and lipid monolayer bubbles belong to the same family of cellular pattern systems, whose best studied member is the two-dimensional soap froth. We analyse the family resemblances by showing that these systems share the phase space structure of the ideal planar froth. The dynamics of these systems is studied with the goal of seeing how their topological structure might arise and unravel. Monolayer bubbles evolve in time, while magnetic bubbles do not; we show that this is due to a symmetry breaking in the interactions that form the domains. We propose the Gibbs-Thompson effect as responsible for this breaking in monolayers, and show that it induces a dynamics which is, to lowest order, von Neumann's law.

Proton implantation into GaAs: Transmission electron microscopy results

Semi-insulating liquid-encapsulated Czochralski grown GaAs wafers were implanted at room temperature with protons at energies of 2, 4, and 30 keV at doses up to 1×1018 cm−2. Without using further annealing treatments the samples were inspected, also using cross-sectional techniques, by transmission electron microscopy. Surface amorphization of the bombarded GaAs was found. Excess hydrogen precipitates in the form of large bubbles in the amorphous layer. Nearly spherical hydrogen bubbles were detected in the crystalline layer below the amorphous zone. At 30 keV, pressurized bubble rafts, where a certain number of bubbles are located in the plane of a microcrack, were detected. The recent observations of similar bubble rafts by Neethling and Snyman [J. Mater. Sci. 23, 2697 (1988)] and the present rafts are discussed in the light of the theoretical treatment by d’Olieslaeger et al. [Philos. Mag. B 63, 1321 (1991)]. The bubble rafts have presumably been produced by the collapse of pressurized hydrogen-filled microcracks.

Bubble raft model for indentation with adhesion

Indentation hardness tests are now widely used to measure the mechanical properties of solid surfaces1–3. Recent developments of this technique4,5 permit the analysis of the outermost 10 nm of materials. Experimental and theoretical questions arise regarding the physical and mechanical processes involved in such small indentations. We describe here an indentation experiment on a microscopic scale, using soap bubbles blown onto a water surface. Bubble rafts provide a simple two-dimensional model for indentation behaviour; as for other materials, their behaviour is governed by two principal attraction–repulsion forces6, and by geometrical constraints. A crystalline two-dimensional lattice is obtained by using bubbles of uniform size7–9, whereas bubbles of two sizes give an amorphous structure10,11. Indentation can be represented by the contact between a triangular crystalline raft and a rectangular crystalline raft bordered by an amorphous layer. The flow of the materials, which is dependent on both adhesion and the force between the two rafts, can be analysed during the experiment.

Epithelia as bubble rafts: A new method for analysis of cell shape and intercellular adhesion in embryonic and other epithelia

Cell-cell interactions may often be characterized by a work of adhesion or equivalent surface tension between cells. This intercellular surface tension may vary from one part of an epithelium to another. Such gradients in surface tension may be important driving factors in morphogenesis. A computational method for the measurement of parameters of cell shape involved in surface tension has been developed. It permits detailed observational analysis of cell-cell interface curvature. The information thus obtained is utilized in an algorithm for estimating gradients of surface tension in epithelia. The use of these methods should permit cell biologists and embryologists to decide whether or not developing cells behave like bubbles, and if so, to what extent their adhesive forces contribute to various aspects of embryogenesis.

Analysis of plastic flow in an amorphous soap bubble raft by the use of an inter-bubble potential

Abstract The changes in potential energy, free volume, and internal shear strain in clusters undergoing shear transformations during the shearing of amorphous soap bubble rafts have been analysed by means of a generalized inter-bubble potential initially developed by Nicolson and by Lomer. A unique inverse correlation was found between the activation free energy barriers of transformations and the initial state of free volume of the transforming sites. Two separate branches of correlation, one for concentrated shear transformations and the other for diffuse shear transformations have been identified. When proper scaling between the inter-bubble potential and the appropriate close-packed metal potential is applied to the results, the distribution of activation free energies found from the soap bubble model gives remarkably good agreement with the experimentally measured distributions in metallic glasses obtained by Argon and Kuo earlier. The inverse correlation, found between activation free energy and free volume, has important implications with respect to shear relaxations near a glass transition and the glass transition process itself.

The potential and force law between different-size bubbles in soap bubble rafts

Abstract Since the Bragg bubble model continues to be a powerful tool for modelling interatomic binding and inelastic processes in close-packed metals with particularly attractive applications to the mechanism of plastic flow in metallic glasses, the inter-bubble potential previously developed by Nicolson and Lomer has been generalized to cover interactions between bubbles of unequal size. Fitting the binding energy between pairs of bubbles to the ab initio potential of a close-packed metal such as copper gives good superposition in the trough region of equilibrium and at the inflection point as well. Measurements of Young's moduli and biaxial cohesive strain in perfect rafts and uniaxial cohesive strain between pairs of bubbles of different size are in good agreement with predictions made from the generalized bubble potential and force law.

Triangle rafts — extended Zachariasen schematics for structure modeling

A systematic method is described for generating arrays of triangles equivalent to extended Zachariasen schematics. The resulting “triangle rafts” are convenient tools for visualizing ideal random network structures and structural defects in a similar way to the utility of bubble rafts for demonstrating crystal structures and crystal defects. The ideal random network structure (especially the ring statistics) is conveniently characterized by a log-normal probability distribution function (PDF). Defect structures such as channels (continuous strings of large interstices) or non-bridging oxygens (in a sodium-modified glass) drive the ring statistics away from the simple log-normal PDF. For a multicomponent glass, the “clustering” of modifier ions can be visualized in the framework of a randomly generated ring structure.

Microwave Emissivity Measurements of Bubbles and Foam

Measurements were made in the laboratory of the emissivities of a variety of bubble and foam samples at a wavelength of 3.2 cm. The technique was an active one in which the absorptivities of the samples were determined. Both free-space and waveguide measurements were made, and emissivities were found to be near 0.6 for some single-layer bubble rafts and near 1.0 for mixed foams of 3- to 4-mm thickness. These measurements support the previously expressed belief that it should be possible to infer sea state and approximate wind speed from passive microwave measurements interpreted in percentage of foam coverage.

An optical method of studying the diffraction from imperfect crystals II. Crystals with dislocations

Lipson’s optical diffractometer has been used to determine the diffraction patterns of gratings representing crystals with dislocations. The optical method lends itself readily to the solution of the two-dimensional problem of diffraction by a single edge dislocation. The intensity distribution near the ideal reciprocal lattice points is, in general, complex, but for certain special points it is relatively simple and in agreement with that deduced by a new theory of Suzuki. An earlier theory of Wilson’s fails to explain the observed intensity distribution. Diffuse scattering, not predicted by Suzuki’s theory, occurs in the form of streaks joining the reciprocal lattice points. The diffraction has also been studied from gratings consisting of photographs of dislocated bubble rafts, and from a grating whose diffraction pattern is related to that of a screw dislocation. A brief discussion is given of the expected diffraction patterns from crystals with various arrays of dislocations.

A dynamical model of a crystal structure - IV. Grain boundaries

Four photographs of bubble rafts are used as a basis for discussion of the structure of grain boundaries in pure metals. In these photographs one can follow the gradual transition from a small-angle boundary made up of clearly separate dislocations to a large-angle boundary where the dislocation structure is hardly recognizable. As the angle is increased, a continuous shortening of the dislocations, accompanied by the widening of a crack on the tensile side, is seen, and the process culminates in a structure which is perhaps best described in terms of local fit and misfit. The fact is also illustrated that the dislocation content of the boundary depends on the angle of the boundary, as well as on the disorientation of the crystals that it separates. If a boundary turns it must therefore gain or lose dislocations. The bearing of this on the measurement of grain-boundary energies is discussed. Other points considered concern the range of validity of calculations of the energy of dislocation walls, and slip and diffusion along grain boundaries.

Optical diffraction patterns produced by bubble rafts

A method of obtaining diffraction patterns, analogous to X -ray and electron diffraction patterns, from photographs of bubble rafts is described. The results of some early experiments are shown, and the conditions for obtaining patterns of higher resolution and more uniform illumination are deduced. An experimental technique for realizing these conditions is described and the results are shown. A method is described for viewing the raft in the light of a selected region of the diffraction pattern, and some results of this technique are shown.