Acoustic tokamak with strongly coupled toroidal bubbles.

Gas bubbles stabilized in toroidal 3D-printed cages are good acoustic resonators with an unusual topology. We arrange them in a circular array to obtain what we call an "acoustic tokamak" because of the torus shape of the whole array. We demonstrate experimentally and theoretically that the system features several acoustic modes resulting from the acoustic interaction between tori. The fundamental acoustic mode has a much lower frequency than that of the individual bubbles. The acoustic field along the circle inside the acoustic tokamak is remarkably homogeneous, as shown by our 3D simulations.

Acoustic Resonance Frequencies of Underwater Toroidal Bubbles.

Underwater bubbles display an acoustic resonance frequency close to spherical ones. In order to obtain a resonance significantly deviating from the spherical case, we stabilize bubbles in toroidal frames, resulting in bubbles which can be slender while still compact. For thin tori the resonance frequency increases greatly. Between a pair of bubble rings, we can achieve a flat acoustic pressure field for a critical distance between rings, a condition reminiscent of Helmholtz coils in magnetostatics. This opens the possibility to shape the acoustic field using long tunnels of rings.

Effect of Gaussian curvature modulus on the shape of deformed hollow spherical objects.

A popular description of soft membranes uses the surface curvature energy introduced by Helfrich, which includes a spontaneous curvature parameter. In this paper we show how the Helfrich formula can also be of interest for a wider class of spherical elastic surfaces, namely with shear elasticity, and likely to model other deformable hollow objects. The key point is that when a stress-free state with spherical symmetry exists before subsequent deformation, its straightforwardly determined curvature ("geometrical spontaneous curvature") differs most of the time from the Helfrich spontaneous curvature parameter that should be considered in order to have the model being correctly used. Using the geometrical curvature in a set of independent parameters unveils the role of the Gaussian curvature modulus, which appears to play on the shape of an elastic surface even though this latter is closed, contrary to what happens for surfaces without spontaneous curvature. In appendices, clues are given to apply this alternative and convenient formulation of the elastic surface model to the particular case of thin spherical shells of isotropic material (TSSIMs).

Numerical deflation of beach balls with various Poisson's ratios: from sphere to bowl's shape.

We present a numerical study of the shape taken by a spherical elastic surface when the volume it encloses is decreased. For the range of 2D parameters where such a surface may model a thin shell of an isotropic elastic material, the mode of deformation that develops a single depression is investigated in detail. It occurs via buckling from sphere toward an axisymmetric dimple, followed by a second buckling where the depression loses its axisymmetry through folding along portions of meridians. For the thinnest shells, a direct transition from the spherical conformation to the folded one can be observed. We could exhibit unifying master curves for the relative volume variation at which first and second buckling occur, and clarify the role of Poisson's ratio. In the folded conformation, the number of folds and inner pressure are investigated, allowing us to infer shell features from mere observation and/or knowledge of external constraints.

Elastic properties of hollow colloidal particles.

The elastic properties of micrometer-sized hollow colloidal particles obtained by emulsion templating are probed by nanoindentation measurements in which point forces are applied to solvent-filled particles supported on a flat substrate. We show that the shells respond linearly up to forces of 7-21 nN, where the indentation becomes of the order of the shell thickness (20-40 nm). In the linear region, the particle deformation is reversible. The measured Young's modulus (approximately 200 MPa) is comparable to values for stiff rubbers or soft polymers. At larger applied force, we observe a crossover into a nonlinear regime, where the shells assume a buckled shape. Here, the force increases approximately as the square root of the indentation, in agreement with the theory of elasticity of thin shells. We also observe permanent deformation of the shells after probing them repetitively beyond the linear regime. Finally, the measured elastic properties of the shells nicely explain their spontaneous buckling in solution and due to drying.

Anisotropic colloids through non-trivial buckling.

We present a study on buckling of colloidal particles, including experimental, theoretical and numerical developments. Oil-filled thin shells prepared by emulsion templating show buckling in mixtures of water and ethanol, due to dissolution of the core in the external medium. This leads to conformations with a single depression, either axisymmetric or polygonal depending on the geometrical features of the shells. These conformations could be theoretically and/or numerically reproduced in a model of homogeneous spherical thin shells with bending and stretching elasticity, submitted to an isotropic external pressure.

Depressions at the surface of an elastic spherical shell submitted to external pressure.

Elasticity theory calculations predict the number N of depressions that appear at the surface of a spherical thin shell submitted to an external isotropic pressure. Using a model that mainly considers curvature deformations, we show that N depends on the relative volume variation and on an adimensional parameter that takes into account both the relative spontaneous curvature and the relative thickness of the shell. Equilibrium configurations show single depression (N = 1) for small volume variations, then N increases, at maximum up to 6, before decreasing more abruptly due to steric constraints, down to N = 1 again for maximal volume variations. These static predictions are consistent with previously published experimental observations.