Moving boundaries in ordered cylindrical foam structures

Flexural rigidity of pressurized model notochords in regular packing patterns

The biomechanics of embryonic notochords are studied using an elastic membrane model. An initial study varying internal pressure and stiffness ratio determines tension and geometric ratios as a function of internal pressure, membrane stiffness ratio, and cell packing pattern. A subsequent three-point bending study determines flexural rigidity as a function of internal pressure, configuration, and orientation. Flexural rigidity is found to be independent of membrane stiffness ratio. Controlling for number and volume of cells and their internal pressure, the eccentric staircase pattern of cell packing has more than double the flexural rigidity of the radially symmetric bamboo pattern. Moreover, the eccentric staircase pattern is found to be more than twice as stiff in lateral bending than in dorsoventral bending. This suggests a mechanical advantage to the eccentric WT staircase pattern of the embryonic notochord, over patterns with round cross-section.

Chiral photonic crystals from sphere packing

Inspired by recent developments in self-assembled chiral nanostructures, we have explored the possibility of using spherical particles packed in cylinders as building blocks for chiral photonic crystals. In particular, we focused on an array of parallel cylinders arranged in a perfect triangular lattice, each containing an identical densest sphere packing structure. Despite the non-chirality of both the spheres and cylinders, the self-assembled system can exhibit chirality due to spontaneous symmetry breaking during the assembly process. We have investigated the circular dichroism effects of the system and have found that, for both perfect electric conductor and dielectric spheres, the system can display dual-polarization photonic band gaps for circularly polarized light at normal incidence along the axis of the helix. We have also examined how the polarization band gap size depends on the dielectric constant of the spheres and the packing fraction of the cylinders. Furthermore, we have explored the effects of non-ideality and found that the polarization gap persists even in the presence of imperfections and heterogeneity. Our study suggests that a cluster formed by spheres self-assembling inside parallel cylinders with appropriate material parameters can be a promising approach to creating chiral photonic crystals.

Physical models of notochord cell packing reveal how tension ratios determine morphometry

The physical and geometric aspects of notochords are investigated using a model of finite-length notochords, with interior vacuolated cells arranged in two common packing configurations, and sheath modeled as homogeneous and thin. The key ratios governing packing patterns and eccentricity are number of cells per unit length λ and cell tension ratio Γ. By analyzing simulations that vary Γ and total number of cells N, we find that eccentricity, λ, and internal pressure approach consistent asymptotic values away from the tapering ends, as N increases. The length of the tapering ends is quantified as a function of Γ and pattern. Formulas are derived for geometric ratios, pressure, and energy as functions of Γ and pattern. These observations on the relationship between mechanics, geometry, and pattern provide a framework for further work which may provide insight into the roles of mechanosensing and pressure-volume regulation in the notochord.

Bubble packing, eccentricity, and notochord development

This paper develops a theoretical basis for the observed relationship between cell arrangements in notochords and analog physical models, and the eccentricity of their cross sections. Three models are developed and analyzed, of the mechanics of cell packing in sheaths. The key ratios governing the packing patterns and eccentricity are cells per unit length λ, tension ratio Γ, and eccentricity e. For flexible and semi-flexible sheaths, the optimal packing pattern shifts from "bamboo", with a symmetric cross section, to "staircase", with an eccentric cross section, at a critical value λ = 1.13. In rigid tubes, this threshold is lowered as imposed eccentricity is increased. Patterns can be observed which are not optimal; pattern transitions may occur below or above the critical λ values. The eccentricity of staircase patterns in flexible and semi-flexible tubes is found to be dependent on the tension ratio Γ, increasing as sheath tension decreases relative to interior cell tension. A novel "serpentine" packing pattern appears for low Γ near the critical λ. The developmental utility of enforcing notochord eccentricity is discussed, as well as potential mechanisms for such control.

Tissue self-organization underlies morphogenesis of the notochord

The notochord is a conserved axial structure that in vertebrates serves as a hydrostatic scaffold for embryonic axis elongation and, later on, for proper spine assembly. It consists of a core of large fluid-filled vacuolated cells surrounded by an epithelial sheath that is encased in extracellular matrix. During morphogenesis, the vacuolated cells inflate their vacuole and arrange in a stereotypical staircase pattern. We investigated the origin of this pattern and found that it can be achieved purely by simple physical principles. We are able to model the arrangement of vacuolated cells within the zebrafish notochord using a physical model composed of silicone tubes and water-absorbing polymer beads. The biological structure and the physical model can be accurately described by the theory developed for the packing of spheres and foams in cylinders. Our experiments with physical models and numerical simulations generated several predictions on key features of notochord organization that we documented and tested experimentally in zebrafish. Altogether, our data reveal that the organization of the vertebrate notochord is governed by the density of the osmotically swelling vacuolated cells and the aspect ratio of the notochord rod. We therefore conclude that self-organization underlies morphogenesis of the vertebrate notochord.This article is part of the Theo Murphy meeting issue on 'Mechanics of development'.

Simulation and observation of line-slip structures in columnar structures of soft spheres

We present the computed phase diagram of columnar structures of soft spheres under pressure, of which the main feature is the appearance and disappearance of line slips, the shearing of adjacent spirals, as pressure is increased. A comparable experimental observation is made on a column of bubbles under forced drainage, clearly exhibiting the expected line slip.

Corrected Article: Simulation and observation of line-slip structures in columnar structures of soft spheres [Phys. Rev. E 96, 012610 (2017)]

This corrects the article DOI: 10.1103/PhysRevE.96.012610.

Structure and dynamics of confined foams: a review of recent progress

Basic research on confined foams now points to an interesting application, a kind of microfluidics which deals with the manipulation of closely packed droplets or bubbles flowing in channels. In such systems, the minimisation of interfacial energy leads to self-organised ordering which is tightly coupled to the channel geometry, hence providing efficient means of performing controlled topological operations on droplet and bubbles structures. We have called this discrete microfluidics, and have begun to explore its possibilities and principles. Apart from the fact that such systems provide powerful tools to study the flow of foams and emulsions on the scale of a few bubbles or droplets, they also carry the promise of versatile applications for Lab-on-a-Chip technologies. In these, discrete gas or liquid samples can be generated, processed, stored and analysed within a single handheld chip. Previous work on foams and emulsions in confined geometries provides a basis for this, and is being extended progressively by new experiments and appropriate dynamic models, such as the 2d Viscous Froth Model. The result should be a practical "design kit" for more complex networks to efficiently process discrete gas and fluid samples.

Structure of liquid films of an ordered foam confined in a narrow channel

A bamboo foam is the simplest case of an ordered foam confined in a narrow channel. It is made of a regular film distribution, arranged perpendicularly to the channel. Our work consists of studying the structural properties of several films taken in a drained foam. X-ray experiments highlighted the equality of the equilibrium thickness for each film within a foam. The same thickness was found as by measurements of disjoining pressure isotherms, proving as well that films of a bamboo foam behave like isolated ones. The refinement of X-ray data by a simple model of specular reflectivity showed a significant variation of the electronic distribution of the surfactant layer for a common black film forwarding from one equilibrium state to another. A discussion on the organization of the surfactant molecules to the gas/liquid interface and film is proposed.

Instabilities in liquid foams

Instabilities play a central role in the physics of foams. Some that change the topology of a dry foam are indicated by the laws promulgated by Plateau in his 1873 book. Their occurrence is less clearly predictable in wet foams. Various other instabilities are related to gravitational loading and gas compressibility. We gather up many examples as a guide to future research and identify problems that remain, including what we call instabilities, which occur before they are expected on the basis of Plateau's laws.

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.

The foam/emulsion analogy in structure and drainage

The often quoted analogy between foams and emulsions is experimentally tested by studying properties after settling and under forced drainage of oil-in-water emulsions of drop size similar as for bubbles generally used in foam experiments. Observations with regard to structure, water fraction and drainage wave properties confirm the expected similarity in the low flow rate range. However, while for foams a convective circulation on the scale of the container sets in for values of water fraction exceeding about 0.2, no such convection is found in emulsions. Here instabilities are only encountered at water fractions of about 0.4, close to the void fraction of random packings of spheres. These take on the form of descending pulses of increased water fraction and lead to the transition from a frozen to a locally agitated structure.