A continuum description of the buckling of a line of spheres in a transverse harmonic confining potential

A line of contacting hard spheres, placed in a transverse confining potential, buckles under compression or when tilted away from the horizontal, once a critical tilt angle is exceeded. This interesting nonlinear problem is enriched by the combined application of both compression and tilt. In a continuous formulation, the profile of transverse sphere displacement is well described by numerical solutions of a second-order differential equation (provided that buckling is not of large amplitude). Here we provide a detailed discussion of these solutions, which are approximated by analytic expressions in terms of Jacobi, Whittaker and Airy functions. The analysis in terms of Whittaker functions yields an exact result for the critical tilt for buckling without compression.

The energy of fcc and hcp foams

We present Surface Evolver evaluations of the difference in energy between face-centred cubic (fcc) and hexagonal close-packed (hcp) foams in the usual idealized model, for liquid fractions ranging from the dry to the wet limit. The difference vanishes in both limits, and favours hcp for all intermediate liquid fractions, as has been proven. The maximum relative energy difference is very small, of the order of 10-5. The asymptotic dependence on liquid fraction is non-analytic in both limits: we present explicit expressions in both cases, derived from first principles. They have been obtained from identifying node interactions (dry limit) and contact interactions (wet limit) as the respective sources for energy differences between fcc and hcp. The wet limit is well described by Morse-Witten theory which has proven to be very powerful for the analytic computation of the surface energy of slightly deformed bubbles.

Theory of rotational columnar structures of soft spheres

There is a growing interest in cylindrical structures of hard and soft particles. A promising new method to assemble such structures has recently been introduced by Lee et al. [Lee, Gizynski, and Grzybowski, Adv. Mater. 29, 1704274 (2017)ADVMEW0935-964810.1002/adma.201704274]. They used rapid rotations around a central axis to drive spheres of lower density than the surrounding fluid towards the axis. This resulted in different structures as the number of spheres is varied. Here, we present comprehensive analytic energy calculations for such self-assembled structures, based on a generic soft sphere model, from which we obtain a phase diagram. It displays interesting features, including peritectoid points. These analytic calculations are complemented by preliminary numerical simulations for finite sample sizes with soft spheres. A similar analytic approach could be used to study packings of spheres inside cylinders of fixed dimensions, but with a variation in the number of spheres.

Ideal wet two-dimensional foams and emulsions with finite contact angle

We present simulations that show that the equilibrium structure of an ideal two-dimensional foam with a finite contact angle develops an inhomogeneity for high liquid fraction φ. In liquid-liquid emulsions this inhomogeneity is known as flocculation. In the case of an ordered foam this requires a perturbation, but in a disordered foam inhomogeneity grows steadily and spontaneously with φ, as demonstrated in our simulations performed with the Surface Evolver.

Bubble-bubble interactions in a 2d foam, close to the wet limit

Following the general approach of Morse and Witten for the deformation of a bubble in contact with neighbouring bubbles, we develop a model for contacting bubbles in two dimensions which can be solved analytically. The force-displacement relations are derived by elementary methods; unlike the case of 3d, no logarithmic factors arise in two dimensions. We also discuss the case of a uniform compression of a symmetric foam structure; the (osmotic) compressibility depends on the number of contacts, as was shown in earlier work by Lacasse et al. Our model, which is based on first principles, without any free parameters, may be extended to simulate 2d foams.

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.

Phyllotaxis, disk packing, and Fibonacci numbers

We consider the evolution of the packing of disks (representing the position of buds) that are introduced at the top of a surface which has the form of a growing stem. They migrate downwards, while conforming to three principles, applied locally: dense packing, homogeneity, and continuity. We show that spiral structures characterized by the widely observed Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, ...), as well as related structures, occur naturally under such rules. Typical results are presented in an animation.

Theoretical analysis of the performance of a foam fractionation column

A model system for theory and experiment which is relevant to foam fractionation consists of a column of foam moving through an inverted U-tube between two pools of surfactant solution. The foam drainage equation is used for a detailed theoretical analysis of this process. In a previous paper, we focused on the case where the lengths of the two legs are large. In this work, we examine the approach to the limiting case (i.e. the effects of finite leg lengths) and how it affects the performance of the fractionation column. We also briefly discuss some alternative set-ups that are of interest in industry and experiment, with numerical and analytical results to support them. Our analysis is shown to be generally applicable to a range of fractionation columns.

Theory of cylindrical dense packings of disks

We have previously explored cylindrical packings of disks and their relation to sphere packings. Here we extend the analytical treatment of disk packings, analyzing the rules for phyllotactic indices of related structures and the variation of the density for line-slip structures, close to the symmetric ones. We show that rhombic structures, which are of a lower density, are always unstable, i.e., can be increased in density by small perturbations.

A model system for foam fractionation

A model system for theory and experiment that is relevant to foam fractionation consists of a column of foam moving through an inverted U-tube between two pools of surfactant solution. The foam drainage equation and its variants are used for a theoretical analysis of this process. In the limit in which the lengths of the two legs is large , exact analytic formulae may be derived for the key properties of the system. Numerical computations and experiments support these results.

Dense packings of spheres in cylinders: simulations

We study the optimal packing of hard spheres in an infinitely long cylinder, using simulated annealing, and compare our results with the analogous problem of packing disks on the unrolled surface of a cylinder. The densest structures are described and tabulated in detail up to D/d=2.873 (ratio of cylinder and sphere diameters). This extends previous computations into the range of structures which include internal spheres that are not in contact with the cylinder.

Phyllotactic description of hard sphere packing in cylindrical channels

We develop a simple analytical theory that relates dense sphere packings in a cylinder to corresponding disk packings on its surface. It applies for ratios R=D/d (where d and D are the diameters of the hard spheres and the bounding cylinder, respectively) up to R=1+1/sin(π/5). Within this range the densest packings are such that all spheres are in contact with the cylindrical boundary. The detailed results elucidate extensive numerical simulations by ourselves and others by identifying the nature of various competing phases.

Foam as a complex system

What is a 'complex system'? The two-dimensional foam, as originally popularized by Cyril Stanley Smith, provides an ideal context in which to explore this question.

The viscous froth model: steady states and the high-velocity limit

The steady-state solutions of the viscous froth model for foam dynamics are analysed and shown to be of finite extent or to asymptote to straight lines. In the high-velocity limit, the solutions consist of straight lines with isolated points of infinite curvature. This analysis is helpful in the interpretation of observations of anomalous features of mobile two-dimensional foams in channels. Further physical effects need to be adduced in order to fully account for these.

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.

Steady drainage in emulsions: corrections for surface Plateau borders and a model for high aqueous volume fraction

We compare extensive experimental results for the gravity-driven steady drainage of oil-in-water emulsions with two theoretical predictions, both based on the assumption of Poiseuille flow. The first is from standard foam drainage theory, applicable at low aqueous volume fractions, for which a correction is derived to account for the effects of the confinement of the emulsion. The second arises from considering the permeability of a model porous medium consisting of solid sphere packings, applicable at higher aqueous volume fractions. We find quantitative agreement between experiment and the foam drainage theory at low aqueous volume fractions. At higher aqueous volume fractions, the reduced flow rate calculated from the permeability theory approaches the master curve of the experimental data. Our experimental data demonstrates the analogy between the problem of electrical flow and liquid flow through foams and emulsions.

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 response of 2D foams to continuous applied shear in a Couette rheometer

The continuum model that has reproduced the experimental observation of exponential shear localisation for straight-edge boundary conditions is adapted to the case of circular geometry. Essentially the same effect is found. However, the scenario of possible velocity profiles is much richer. Our calculations elucidate many recent experiments qualitatively and suggest further extensions of them. Various limits are analysed. In particular, the localisation length vanishes as the inner-boundary velocity tends to zero.

Two-dimensional foam rheology with viscous drag

We formulate and apply a continuum model that incorporates elasticity, yield stress, plasticity, and viscous drag. It is motivated by the two-dimensional foam rheology experiments of Debregeas et al. [Phys. Rev. Lett. 87, 178305 (2001)10.1103/PhysRevLett.87.178305] and Wang et al. [Phys. Rev. E 73, 031401 (2006)10.1103/PhysRevE.73.031401], and is successful in exhibiting their principal features, which are an exponentially decaying velocity profile and strain localization. Transient effects are also identified.

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.

Drainage induced convection rolls in foams I. Convective bubble motion in a tilted tube

When liquid is added to a foam at sufficiently large flow rates, convective bubble motion will occur. Experiments are described in which the foam is confined in a tube which is tilted from the vertical. The theory of foam drainage is applied to this problem to show that the critical angle of tilt theta(c) at which convection occurs is related to the liquid flow-rate Q by theta(c) proportional to Q(-3/4).

Two-dimensional viscous froth model for foam dynamics

The two-dimensional viscous froth model is a simple tractable model for foam rheology and coarsening. It includes, but is not confined to, the quasistatic regime. Here we present a detailed analysis and implementation of the model, illustrated with various examples. With certain simplifying assumptions, it provides significant insight into strain-rate-dependent effects in foam rheology and elsewhere, particularly in relation to recent experiments.

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.

Topological changes in a two-dimensional foam cluster

Experiments on a small cluster of bubbles in a nominally two-dimensional foam show an instability in which a topological change forces one of the bubbles to be ejected to the outside of the cluster at a point where this is not predicted by a two-dimensional model of a foam. This is interpreted in terms of the energy of the initial and ejected states and of the finite liquid content of the experimental system. A description of the distribution of liquid in various experimental set-ups suggests that the exact response may depend critically upon the type of system used. This is demonstrated experimentally with reference to small clusters of bubbles undergoing a single topological change.

The transition from two-dimensional to three-dimensional foam structures

Small cells in an experimental sample of two-dimensional foam, such as that which is contained between two glass plates, may undergo a transition to a three-dimensional form, becoming detached from one boundary. We present the first detailed observations of this phenomenon, together with computer simulations. The transition is attributed to an instability of the Rayleigh-Plateau type. A theoretical analysis is given which shows that an individual cell is susceptible to this instability only if it has less than six sides.