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Workshop Programme

for period 24 - 28 February 2014

Foams and Minimal Surfaces - 12 years on

24 - 28 February 2014


Monday 24 February
Session: Surface Evolver and its Applications
08:30-09:20 Registration
09:20-09:30 Welcome from Christie Marr (INI Deputy Director)
09:30-10:15 Collicott, SH (Purdue University)
  Topics in Capillary Fluid Statics Sem 1

Twenty three years ago a young assistant professor in aerospace engineering sought a research niche at his university when the field he pursued initially was found to be crowded with successful elders. Zero-gravity fluid dynamics was the new attraction. This plus Earth-bound application of capillary-dominated fluids continue to be active research and engineering topics for the speaker. Presented are highlights of a variety of capillary fluid topics from both spaceflight and terrestrial needs. Existence of static equilibrium solutions in regions of the relevant parameter space and stability of such solutions are common concepts in the topics presented. The work leans heavily on applications of the novel Surface Evolver code. Recent extensions of Concus-Finn critical wetting analysis are also presented. Highlights of experimentation are also discussed, including involvement with the emerging commercial sub-orbital rocket industry in the USA and the NASA-funded undergraduate Fluids Education experiment payload to be launched to the International Space Station on SpaceX-6 or Oribtial-4 flight in 2014 or 2015.

10:15-11:00 Neethling, SJ (Imperial College London)
  Particles and droplets in Foams - A simulation approach Sem 1

Co-author: Gareth Morris (Imperial College London)

Particles and droplets play an important role in the stability of many foam systems. For instance, particles play an important role in stabilising mineral flotation froths, while oil droplets can have a destabilising effect on the stability of detergent foams. This paper will present a number of studies in which the effect of particles, both individually and en-mass, on the stability of thin films and foams are investigated using simulations. The main simulation tool employed is Surface Evolver, which has been programmed to allow the constraints used to model the particle location to change in response to the forces exerted on the particle, both from the fluid interfaces and other particles. These simulations are coupled to small scale experiments in which the behaviour of thin films containing particles is studied using high speed video analysis. As well as particles, the influence of oil droplets on the behaviour of foams is investigated by means of simulation, with mechani sms by which these droplets can both stabilise and destabilise foams and thin films being demonstrated.

11:00-11:45 Morning Coffee
11:45-12:05 Grassia, P (University of Manchester)
  Analysis of a Model for Foam Improved Oil Recovery Sem 1

Co-authors: Elizabeth Mas Hernandez (University of Manchester), Nima Shokri (University of Manchester), Simon Cox (Aberystwyth University), Gennady Mishuris (Aberystwyth University), William Rossen (Delft University of Technology)

A model (originally developed by Shan and Rossen (2004) and de Velde Harsenhorst et al. (2013)) is considered that describes foam motion into a porous reservoir filled with surfactant solution. The model for evolution of the foam front that results is called `pressure-driven growth', and it describes processes that occur during improved oil recovery (IOR) using foam. The mathematical structure of the model is found to correspond to a special case of a more general situation called the `viscous froth model' (Glazier and Weaire 1992, Weaire and McMurry 1996). However `pressure-driven growth' turns out to be a singular limit of the viscous froth system, owing to the fact that a surface tension term has been discarded. This permits (in principle) sharp corners and kinks in the shape of the foam front. Sharp corners however tend to develop from concave regions of the front shape, whereas the main solution of interest here has a convex front. Whilst the solution of interest appears to have no sharp corners (except for some kinks that might develop spuriously owing e.g. to errors arising in a numerical scheme), it does nevertheless exhibit milder singularities in front curvature: a long-time asymptotic analytical solution for the shape of the front makes this point clear. Numerical schemes which perform robustly (avoiding the development of any spurious kinks in the above mentioned solution) are considered. Moreover some simple generalizations of this solution, all of engineering relevance, can exhibit concavities and/or sharp corner singularities as an inherent part of their evolution: propagation of such `inherent' singularities can be readily incorporated into numerical schemes.

12:05-12:30 Vincent-Bonnieu, S (Shell Global Solution International)
  Foam flow in porous media Sem 1

Co-author: Rouhi Farajzadeh (Shell Global Solution International)

The flow of foam in porous media is a complex process depending on the media properties (permeability, porosity and surface chemistry), oil saturation, gas and liquid fraction. Foam appears to be very viscous in a porous media which increases the sweep efficiency. This displacement property is used for oil recovery processes. Several works have studied the effect of oil on foam. Depending on the oil and the surfactant chemistry, oil destabilizes foam. The oil effect has been linked to various mechanisms like disjoining pressure, liquid fraction, entering and spreading coefficient, emulsification, etc. However the oil effect in porous media is still not well quantifiable. In addition, coarsening, coalescence and drainage of foam in porous media have been poorly studied. On large scale, relevant for oil recovery applications, foam flow is simulated by a mobility reduction model, which scales down the gas mobility based on empirical parameters. Effect of the physical parameters on the model will be discussed. We will present numerical and experimental works on the foam flooding and report the current challenges of the research.

12:30-13:30 Lunch at Wolfson Court
13:30-14:15 Private Discussion
14:15-14:35 Dollet, B (CNRS)
  Flow of an aqueous foam through a two-dimensional porous medium: structure-dynamics couplings Sem 1

Co-authors: Siân A. Jones (Université Paris 11), Baudouin Géraud (Université Rennes 1), Simon J. Cox (Aberystwyth University), Yves Méheust (Université Rennes 1), Isabelle Cantat (Université Rennes 1)

Flowing foams are used in many engineering and technical applications. They have peculiar flow properties that might be beneficial in applications involving porous media. In particular, viscous dissipation arises mostly from the contact zones between the soap films and the walls, which results in peculiar friction laws allowing the foam to invade narrow pores more efficiently than Newtonian fluids would. We investigate experimentally the flow of a two-dimensional foam in three geometrical configurations. We first consider a medium consisting of two parallel channels with different widths. The flow behavior is highly dependent on the foam structure within the narrowest of the two channels [Jones et al., Phys. Fluids 25, 063101 (2013)]; consequently, the flux ratio between the two channels exhibits a non-monotonic dependence on the ratio of their widths. We then consider two parallel channels that are respectively convergent and divergent. The resulting flow kinematics imposes asymmetric bubble deformations in the two channels; these deformations strongly impact the foam/wall friction, and consequently the flux distribution between the two channels. We quantitatively predict the flux ratio as a function of the channel widths by modeling pressure drops of both viscous and capillary origins. This study reveals the crucial importance of boundary-induced bubble deformation on th e mobility of a flowing foam. We then study the flow of a foam in a two-dimensional porous medium consisting of randomly-positioned cylindrical grains. Irreversibility, intermittency and non-stationarity characterize the velocity field under permanently imposed inlet flow. In this grain geometry, flow channeling appears to be different from what would be expected for a Newtonian fluid, which allows a different part of the pore population to be visited. The influence of the ratio of the typical pore size to the bubble size is also addressed.

14:35-14:55 Sen, S (Trinity College Dublin)
  A QED Coherence Model for Surface Nanobubbles Sem 1

Remarkably stable surface nanobubbles under water have been observed whose existence cannot be understood. Widely dffering explanations have been put forward. However non of them explain the size of the nanobubbles or explain why they are formed. We show that a QED coherence model of water, due to Preparata et al, can explain the reason for the existence, the size and the stability of surface nanobubbles.

14:55-15:15 Hjelt, T (VTT)
  Improved fibre product properties using foam-based forming process Sem 1

In the traditional paper making, foam is found typically to cause problems. However, in the 1970’s foam-laid technology has been demonstrated in a pilot scale based on Radfoam process [1]. Aqueous foam containing small spherical air bubbles is excellent material to transport particles in the dispersed state and this kind of foam is used instead of water as a process fluid and flowing medium in foam-laid technology.

Foam forming is found to improve many product properties including more homogenous mass distribution, reduced density, improved retention of polymers and small particles, and enhanced water removal [2]. In this presentation I will go through these improved properties and using our current knowledge to try to explain why these improvements are obtained in foam forming process.

[1].Kidner T.L.W., The Radfoam process for fine papers, Paper Technology, December 1974. [2].Lehmonen J., Jetsu P, Kinnunen K. and Hjelt T. (2013): Potential of foam-laid forming technology for paper applications, NPPRJ 28(3), 392-398.

15:15-15:45 Afternoon Tea
15:45-16:15 Private Discussion
16:15-16:35 Hutzler, S (Trinity College Dublin)
  Z-Cone Model for the Energy of a Foam Sem 1

Co-authors: Robert Murtagh (Trinity College Dublin), David Whyte (Trinity College Dublin), Steve Tobin (Trinity College Dublin), Denis Weaire (Trinity College Dublin)

We develop the Z-Cone Model, in terms of which the energy of a foam may be estimated. It is directly applicable to a structure in which every bubble has Z equivalent neighbours, but can be applied more approximately to other cases.The energy (i.e. surface area) may be analytically related to liquid fraction. It has the correct asymptotic form in the limits of dry and wet foam, with pre-factors dependent on Z. In particular, the variation of energy with compression in the wet limit is consistent with the anomalous behaviour found by Morse and Witten (1993) and Lacasse et al.(1996), with a prefactor proportional to Z.

16:35-17:00 Dennin, M (UC Irvine)
  Failure Mode of Bubble Rafts under Tension Sem 1

Co-author: Chin Chang Kuo (UC Irvine)

One of the interesting questions regarding the dynamics of foam is the characterization of their flow behavior. One often refers to distinct regimes in which the foam exhibits a primarily elastic response (solid-like) or it exhibits a primarily viscous response (fluid-like). In addition, one finds plastic flow regimes and various combinations of all three. Any attempt to provide a unified theoretical description of foam rheology must account for a wide-range of flow geometries and behavior. Most of the focus in flowing foam has been on closed geometries in which the system is driven either by a moving boundary or pressure gradient. We have recently completed a series of studies on the dynamics of foam in an open-flow situation in which the system is under tension. We use a quasi-two dimensional system of bubbles on the surface of water (bubble rafts). Two of the opposite sides of the system are free boundaries, and the other two sides are pulled apart at a constant rate. We f ocus on the failure mechanism by which the system separates in distinct sections of foam. We report on the different regimes that were observed, including fluid-like pinch-off and plastic-like fracture. We propose simple scaling arguments to account for the speed and aspect ratio dependence of the different failure modes.

17:00-18:00 Welcome Wine Reception sponsored by Oxford University Press
Tuesday 25 February
Session: Tiling space, sphere packings and wet foams, and conformal geometry
09:30-10:15 Kusner, R (University of Massachusetts at Amherst)
  Materials and Meromorphic Differentials Sem 1

We'll discuss how meromorphic differentials on Riemann surface give shorter, more conceptual proofs of old and new results on materials ranging from smectics to soap bubbles [a more extensive abstract will be posted later].

10:15-11:00 Delaney, G (CSIRO)
  Generating and Quantifying Structure in Granular and Cellular Structures Sem 1

We explore methods for the generation and analysis of simulated packings of granular and cellular structures. We consider a range of systems from simple random and ordered sphere packings, to packings of complex non-spherical shapes in 2D and 3D. We show how a simple bubble model for wet foams comprised of near spherical bubbles can be used to model the coarsening of wet foams, recreating several of the structural features seen when wet bubble crystals evolve due to bubble shrinkage at the surface of the foam. By utilising a range of order parameters to quantify structure, we are able to relate how features at the individual grain level affect the macroscopic properties of the system. Anisotropy and broken rotational symmetry at the local level are demonstrated to be of key importance in determining the macroscopic properties of granular systems in the packed stated. We are also able to decompose the contribution from the geometric shape of the grain, the inter-grain interac tion properties and the preparation method. Applications of these techniques in understanding initiation and evolution of landslides and the evolution of shape and size distribution due to particle breakage will be discussed.

11:00-11:45 Morning Coffee
11:45-12:05 Durand, M (Université Paris Diderot)
  Statistical mechanics of two-dimensional shuffled foams: prediction of the correlation between geometry and topology Sem 1

Co-authors: S. Ataei Talebi (Université Grenoble 1), S. Cox (Aberystwyth University), F. Graner (Université Paris Diderot), J. Käfer (Université Lyon 1), C. Quilliet (Université Grenoble 1)

Two-dimensional foams are characterised by their number of bubbles, $N_{}$, area distribution, $p(A)$, and number-of-sides distribution, $p(n)$. When the liquid fraction is very low (``dry'' foams), their bubbles are polygonal, with shapes that are locally governed by the laws of Laplace and Plateau. Bubble size distribution and packing (or ``topology") are crucial in determining \textit{e.g.} rheological properties or coarsening rate. When a foam is shuffled (either mechanically or thermally), $N_{}$ and $p(A)$ remain fixed, but bubbles undergo ``T1'' neighbour changes, which induce a random exploration of the foam configurations.

We explore the relation between the distributions of bubble number-of-sides (topology) and bubble areas (geometry). We develop a statistical model which takes into account physical ingredients and space-filling constraintes. The model predicts that the mean number of sides of a bubble with area $A$ within a foam sample with moderate size dispersity is given by: $$\bar{n}(A) = 3\left(1+\dfrac{\sqrt{A}}{\langle \sqrt{A} \rangle} \right),$$ where $\langle . \rangle$ denotes the average over all bubbles in the foam. The model also relates the \textit{topological disorder} $ \Delta n / \langle n \rangle =\sqrt{\langle n^2 \rangle - \langle n \rangle^2}/\langle n \rangle$ to the (known) moments of the size distribution: $$\left(\dfrac{\Delta n}{\langle n \rangle}\right)^2=\frac{ 1 }{4}\left(\langle A^{1/2} \rangle \langle A^{-1/2} \rangle+\langle A \rangle \langle A^{1/2} \rangle^{-2} -2 \right).$$ Extensive data sets arising from experiments and simulations all collapse surprisingly well on a straight line, even at extremely high values of geometrical disorder.

At the other extreme, when approaching the perfectly regular honeycomb pattern, we identify and quantitatively discuss a crystallisation mechanism whereby topological disorder vanishes.

12:05-12:30 Blumenfeld, R (Imperial College London/Cambridge University)
  Characterisation and statistical mechanics of disordered foam structures Sem 1

Co-authors: Joseph F. Jordan (Imperial College London), Rebecca Hihinashvili (Imperial College London)

The pore-scale structure of foams and cellular (open or closed) materials impacts significantly their large-scale transport and mechanical properties. A systematic several-stage method is described to derive relations between cell-scale structural characteristics and macro-scale properties in two- and three-dimensions.

The first stage involves a quantitative description of the local structure, using a tensor that captures the features most relevant to a number of physical mechanisms. The locality of the tensorial description is achieved by specialized volume elements, called quadrons. Advantages over traditional Voronoi-based descriptions are pointed out.

In the second stage, the description is used in an entropy-based statistical mechanical formalism, making possible the derivation of global structural properties as expectation values over a certain partition function.

In the third stage, we propose to use relations between structural and physical properties (e.g. permeability and heat transfer in solidified open cell foams) in order to translate the structural expectation values into expected distribution of local constitutive properties of equivalent networks.

In the fourth stage the network properties are computed on the scale of a large number of nodes, allowing us to predict upscaled properties at this scale. Further homogenisation and coarse-graining to the continuum is then possible, using conventional methods, such as effective-medium method.

The development of this programme is ongoing and initial tests of some aspects of it are presented in two- and three-dimensional systems.

12:30-13:30 Lunch at Wolfson Court
13:30-14:15 Private Discussion
14:15-14:35 Streinu, I (Smith College, USA)
  Periodic auxetics: a geometric approach Sem 1

Co-author: Ciprian S. Borcea (Rider University, USA)

We propose a purely geometric criterion for characterizing auxetic one-parameter deformations of periodic bar-and-joint frameworks. This concept is valid in arbitrary dimension. Auxetic mechanisms are then compared with expansive mechanisms defined by the stronger property that the distance between any pair of vertices increases or stays the same. For two-dimensional periodic frameworks, expansive behaviour can be explained and explored in terms of a remarkable new class of structures. We show how to generate this type of planar periodic frameworks and discuss their geometry. Since expansive implies auxetic, we obtain an endless resource for auxetic designs.

14:35-14:55 Borcea, CS (Rider University)
  Geometric flexibility in sodalite frameworks Sem 1

Co-author: Ileana Streinu (Smith College)

O'Keeffe proposed generalizing the graph of the sodalite tetrahedral structure to arbitrary dimension by taking as vertices the holes of the A*d lattice sphere packing and connecting nearest neighbours. A second d-periodic graph is obtained by replacing vertices with d-simplices which share one apex when corresponding vertices are connected by an edge. We investigate the geometric deformations of these two related periodic structures.

14:55-15:15 Mughal, A (Aberystwyth University & Friedrich-Alexander-Universität Erlangen-Nürnberg)
  Foam morphology, frustration and topological defects in a Negatively curved Hele-Shaw geometry Sem 1

We present preliminary simulations of foams and single bubbles confined in a narrow gap between parallel surfaces. Unlike previous work, in which the bounding surfaces are flat (the so called Hele-Shaw geometry), we consider surfaces with non-vanishing Gaussian curvature.

We demonstrate that the curvature of the bounding surfaces induce a geometric frustration in the preferred order of the foam. This frustration can be relieved by the introduction of topological defects (disclinations, dislocations and complex scar arrangements). We give an analysis of these defects for foams confined in curved Hele-Shaw cells and compare our results with exotic honeycombs, built by bees on surfaces of varying Gaussian curvature.

Our simulations, while encompassing surfaces of constant Gaussian curvature (such as the sphere and the cylinder), focus on surfaces with negative Gaussian curvature and in particular triply periodic minimal surfaces (such as the Schwarz P-surface and the Schoen's Gyroid surface). We use the results from a sphere-packing algorithm to generate a Voronoi partition that forms the basis of a Surface Evolver simulation, which yields a realistic foam morphology.

15:15-15:45 Afternoon Tea
15:45-16:15 Private Discussion
16:15-16:35 Bi, D (Syracuse University)
  Energy barriers and cell migration in densely packed tissues Sem 1

Co-authors: J.H. Lopez (Syracuse University), J.M. Schawarz (Syracuse University), M. Lisa Manning (Syracuse University)

Recent observations demonstrate that densely packed tissues exhibit features of glassy dynamics, such as caging behavior and dynamical heterogeneities, although it has remained unclear how single-cell properties control this behavior. Here we develop numerical and theoretical models to calculate energy barriers to cell rearrangements using Surface Evolver, which help govern cell migration in cell monolayers. In contrast to work on sheared foams, we find that energy barrier heights are exponentially distributed and depend systematically on the cell's number of neighbors. Based on these results, we predict glassy two-time correlation functions for cell motion, with a timescale that increases rapidly as cell activity decreases. These correlation functions are used to construct simple random walks that reproduce the caging behavior observed for cell trajectories in experiments. This work provides a theoretical framework for predicting collective motion of cells in wound-heali ng, embryogenesis and cancer tumorigenesis.

Related Links: • - Preprint

16:35-17:00 Vitasari, D (University of Manchester)
  Surfactant transport onto a foam lamella taking into account surface viscosity Sem 1

Co-authors: Paul Grassia (The University of Manchester), Peter Martin (The University of Manchester)

The transport of surfactant onto a foam lamella in a foam fractionation column with reflux has been simulated mathematically. Insoluble surfactant is assumed since such surfactants potentially derive more benefit from a reflux system. The transport of surfactant on the surface of a lamella is governed by the film drainage towards the Plateau border, the Marangoni effect in the direction towards the centre of the film and possibly also the surface viscous effect that balances the resultant of the other forces. The desirable condition is when the Marangoni effect dominates the film drainage, where surfactant accumulates on the surface of the lamella. The surface viscous effect takes place when there is movement on the surface and it slows down the velocity on the surface. In this study, a case without film drainage is examined as a benchmark for more complicated systems. A mathematical model of the surface velocity has been developed and results in a differential equation for s urface velocity which is solved using the finite difference method. The calculated surface velocity is used to compute the evolution of surfactant surface concentration using a material point method. The model is verified using analytical solutions for the special case where the surface viscous effect is very small. The numerical model is verified using the analytical solution in the special case where the gradient of the logarithm of surface concentration is linear in space. The Green's function solution of the differential equation is also used to verify the numerical model. The result of the simulation in the presence of surface viscosity is compared with the simulation result in the absence of surface viscosity. It was found that the surface velocity slows down markedly near the Plateau border due to the effect of surface viscous stress. At any given time, the surfactant surface concentration in the presence of surface viscosity is lower than that in the absence of surface viscosity.

Wednesday 26 February
Session: Structure and dynamics of soap films and clusters of bubbles
09:30-10:15 Goldstein, RE (University of Cambridge)
  Boundary Singularities Produced by the Motion of Soap Films Sem 1

Co-authors: Adriana I. Pesci (University of Cambridge), Keith Moffatt (University of Cambridge), James McTavish (University of Cambridge), Renzo Ricca (University of Milano-Bicocca)

Recent experiments have shown that when a soap film with the topology of a Mobius strip, is rendered unstable by slow deformation of its frame past a threshold, the film changes its topology to that of a disc by means of a ``neck-pinching'' singularity at its boundary. This behaviour is unlike the more familiar catenoid minimal surface supported on two parallel circular loops, a two-sided surface which, when rendered unstable, transitions to two disks through a neck-pinching singularity in the bulk. There is at present neither an understanding of whether the type of singularity is in general a consequence of the topology of the surface, nor of how this dependence could arise from a surface equation of motion. We investigate experimentally, computationally, and theoretically the neck-pinching route to singularities of soap films with several distinct topologies, including a family of non-orientable surfaces that are sections of Klein bottles, and provide evidence that the location of singularities (bulk or boundary) may depend on the path along which the boundary is deformed. Since in the unstable regime the driving force for soap film motion is the surface's mean curvature, the narrowest part of the neck, which can be associated with the shortest nontrivial closed geodesic of the surface at each instant of time, has the highest curvature and is thus the fastest-moving. Just before the onset of the instability there exists on the stable surface also a shortest closed geodesic, which serves as an initial condition for the evolution of the geodesics of the neck, all of which have the same topological relationship to the surface boundary. We find that if the initial geodesic is linked to the boundary then the singularity will occur at the boundary, whereas if the two are unlinked initially then the singularity will occur in the bulk. Numerical study of mean curvature flows and experiments show consistency with these conjectures.

10:15-11:00 Sethian, J (University of California, Berkeley)
  A multi-scale framework to model dry foam dynamics Sem 1

Co-author: Robert I. Saye (Department of Mathematics, University of California, Berkeley)

We describe a mathematical and computational framework for modeling soap bubble dynamics. We use a scale-separation approach to split the problem into three distinct phases that cycle over and over. During the rearrangement phase, a cluster of bubbles readjusts itself as surface tension in the membranes pushes on air in the pockets according to multi-phase incompressible flow, until a new macroscopic equilibrium is reached. Once this large-scale equilibrium is reached, we then invoke a drainage stage in which liquid in the lamellae drains into the Plateau borders according to thin film equations. Once one of the membranes becomes too thin, the model ruptures the equilibrium by removing that membrane, moving the cluster far from equilibrium, and leading back to the rearrangement stage.

This approach relies on several different computational methodologies, including (1) a Voronoi Implicit Interface Method (VIIM) to track the moving interface as a single PDE defined on a fixed mesh; (2) a second order projection method to solve the incompressible Navier-Stokes equations during the macroscopic rearrangement phase, describing the transport of fluid in the membranes; (3) a finite element formulation of a set of thin film equations for the fluid in the interfaces themselves, defined on the lamellae and Plateau borders, and linked together through coupled boundary conditions to describe drainage and (4) a rupture mechanism which includes topological rearrangement. We present results from a series of computations, including bubble cascades and thin film interference from a cluster of collapsing bubbles.

11:00-11:45 Morning Coffee
11:45-12:05 Durian, D (UPenn Physics)
  The role of bubble shape in the coarsening of wet 2d foams Sem 1

Co-authors: Adam Roth (UPenn Physics), Chris Jones (UPenn Physics), Anthony Chieco (UPenn Physics), Jennifer Rieser (UPenn Physics)

We report on the statistics of bubble size, topology, and shape and on their role in the coarsening dynamics for foams consisting of bubbles compressed between two parallel plates. The design of the sample cell permits control of the liquid content, through a constant pressure condition set by the height of the foam above a liquid reservoir. We find that in the scaling regime, all bubble distributions are independent not only of time but also of liquid content. For coarsening, the average rate decreases with liquid content due to the blocking of gas diffusion by Plateau borders inflated with liquid; we achieve a factor of four reduction from the dry limit. By observing the growth rate of individual bubbles, we find that vonNeumann's law becomes progressively violated with increasing wetness and with decreasing bubble size. We successfully model this behavior by explicitly incorporating the border blocking effect into the vonNeumann argument. Two dimensionless bubbl e shape parameters naturally arise, one of which is primarily responsible for the violation of von~Neumann's law for foams that are not perfectly dry.

12:05-12:30 Cox, S (Abersytwyth)
  Instability of stretched and twisted soap films in a cylinder Sem 1

Co-author: Sian Jones (Aberystwyth/Paris Sud)

A soap film, or a flexible area-minimizing membrane without bending and torsional stiffness, that is confined in a cylinder is shown to be susceptible to a surface-tension-driven instability when it is stretched or twisted. This leads to its breakdown and places an upper limit on the aspect ratio of such structures. A simple analysis confirms the values for the critical aspect ratio of the stretched film found in both simulations and experiments on soap films, and this threshold decreases with increasing twist of the film.

12:30-13:30 Lunch at Wolfson Court
13:30-14:15 Private Discussion
14:15-14:35 Oguey, C (LPTM, university of Cergy-Pontoise)
  Topological transformations in foams and liquid crystalline mesophases Sem 1

Topological transformations such as neighbour switches or bubble extinction play an important role in foams. They contribute to the onset of disorder during coarsening or flow; they provide fast elementary mechanisms for dissipation; they explain, to a large extent, the particular visco-plastic response of foams and similar mesophases. Geometrically, topological moves provide elementary steps to pass from one structure to another, and so offer the possibility to design new models from know templates.

The 3-arm star shaped molecules --sometimes called mikto arm polymers or linactant-- with mutually non-miscible branches self-assemble in a variety of morphologies, giving rise to a rich phase diagram. The models for these mesophases can be interpreted as partitions of space into three coloured domains satisfying particular rules. The molecular cores aggregate along triple lines at the junction of the three coloured domains.

The topological transformations occur when two or more triple line segments come into contact, generating an unstable vertex. We will see the physical reason for this instability. We will also analyse the various ways into which a vertex can relax. The simplest cases can be explored systematically, but the complexity increases rapidly. We will consider the ingredients of a possible classification. The vertex type is characterised by a sphere pattern, induced by the 3-coloured partition on a small sphere surrounding the vertex. The class of sphere patterns can be spanned by operating 2D elementary topological transformations compatible with the colouring.

14:35-14:55 Biance, A-L (ILM, CNRS and University Lyon 1)
  T1s in soap film architecture: effect of surfactants and assembly geometry Sem 1

Co-authors: Cantat Isabelle (Institut de Physique de Rennes), Seiwert Jacopo (Institut de Physique de Rennes), Petit Pauline (ILM, CNRS and University Lyon 1)

In this study, we use particular soap film architecture (cubic or parallelepipedic) to study topological rearrangement dynamics and especially the freshly film structure. In the cubic frame architecture, by measuring the thickness profile of the film and the velocity profile within the liquid, we identify two mechanisms of film formation, which are a pure elongation of a liquid meniscus or a film at rest pulled out the Plateau borders. Experiments in a parallelepipedic architecture, much more realistic for 3D foams, have underlined a different mechanism consisting in a Plateau border unzipping mechanism. For certain types of surfactants, this process is associated to oscillations (due to inertia and capillary recoil) and rupture of adjacent films. Moreover, front propagations to redistribute the liquid after the T1 in the soap film assembly have also been observed.

14:55-15:15 Rivier, NY (University of Strasbourg)
  Phyllotaxis: Crystallography under rotation-dilation, mode of growth or detachment. A foam ruled by T1 Sem 1

Co-authors: Jean-François Sadoc (LPS Orsay), Jean Charvolin (LPS Orsay)

Phyllotaxis describes the arrangement of florets, scales or leaves in composite flowers or plants (daisy, aster, sunflower, pinecone, pineapple). Mathematically, it is a foam, the most homogeneous and densest covering of a large disk by Voronoi cells (the florets). Points placed regularly on a generative spiral constitute a spiral lattice, and phyllotaxis is the tiling by the Voronoi cells of the spiral lattice. The azimuthal angle between two successive points on the spiral is 2p/ t, where t = (1+v5)/2 is the golden ratio.

If the generative spiral is equiangular (Bernoulli), the phyllotaxis is a conformal (single) crystal, with only hexagonal florets (outside a central core) and zero shear strain. Florets of equal size but not all hexagonal are generated by points on a Fermat spiral. There are annular crystalline grains of hexagonal florets (traversed by three visible reticular lines in the form of spirals, called parastichies) separated by grain boundaries.

Grain boundaries are circles of dislocations (d: dipole pentagon/heptagon) and square-shaped topological hexagons (t: squares with two truncated adjacent vertices). The sequence d t d d t d t is quasiperiodic, and Fibonacci numbers are pervasive. The two main parastichies cross at right angle through the grain boundaries and the vertices of the foam have degree 4 (critical point of a T1) . A shear strain develops between two successive grain boundaries. It is actually a Poisson shear, associated with radial compression between two circles of fixed, but different length. Thus, elastic and plastic shear can be readily absorbed by a polycrystalline phyllotactic structure described by several successive Fibonacci numbers. The packing efficiency problem is thereby solved: One grain boundary constitutes a perfect circular boundary for the disk into which objects are to be packed.

An application of phyllotaxis to growth can be seen in Agave Parryi. Structurally, it spends almost its entire life (25 years, approx.) as a single grain (13,8,5) spherical phyllotaxis, a conventional cactus of radius 0.3 m. During the last six month of its life, it sprouts (through three grain boundaries) a huge (2.5 m) mast terminating as seeds-loaded branches arranged in the (3,2,1) phyllotaxis, the final topological state before physical death.

15:15-15:45 Afternoon Tea
15:45-16:15 Private Discussion
16:15-16:35 Hilgenfeldt, S (University of Illinois at Urbana-Champaign)
  Local models for Size-Topology Correlations Sem 1

Empirical studies have long shown complex statistics in polygonal tilings of the plane or the corresponding packings of objects. Of particular interest has been the relation between domain size (area) and topology (number of neighbors) of objects. Using a simple, strictly local model of neighbor relations, we provide an analytical explanation for correlations of size and topology. We explore polydisperse, bidisperse, and anisotropic domains found in a large variety of living and inanimate systems. The model offers an explanation for long-standing empirical findings such as Lewis' law and the value of terminal polydispersity at 2D order/disorder transitions.

19:30-23:00 Conference Dinner at Emmanuel College
16:30-17:30 Informal discussion
Thursday 27 February
Session: How can physics and computer science help solve the Kelvin problem?
09:30-10:15 Evans, M (University of Erlangen-Nuremberg)
  Hyperbolic Striping on Triply-Periodic Minimal Surfaces Sem 1

Co-authors: Stephen Hyde (Australian National Univeristy), Jacob Kirkensgaard (Univeristy of Copenhagen), Liliana de Campo (Australian National University)

High symmetry dense packings of trees and lines in the two-dimensional hyperbolic plane can be projected to triply-periodic minimal surfaces. The resulting three-dimensional structures are complex and entangled. A particular family of these patterns on the Gyroid can be used to describe the self assembly of a set of hierarchical and chiral multicontinuous network structures from Y-shaped ABC and ABD three-miktoarm star terpolymers, formed during numerical simulations.

10:15-11:00 Morgan, F (Williams College)
  The Least-Perimeter Tile with n Faces Sem 1

Co-authors: Paul Gallagher (University of Pennsylvania), Whan Ghang (MIT), David Hu (Georgetown University), Zane Martin (Williams College), Maggie Miller (University of Texas at Austin), Byron Perpetua (Williams College), Steven Waruhiu (University of Chicago)

The truncated octahedron is after Kelvin conjectured to be the least-surface-area unit-volume polyhedral tile of space. Work with undergraduates seeks the least-surface-area unit-volume n-hedral tile (with n faces). The solution is proved for only two values of n.

Related Links: - "Surface-area-minimizing n-hedral Tiles" preprint

11:00-11:45 Morning Coffee
11:45-12:05 Tarnai, T (Budapest University of Technology and Economics)
  Spherical circle coverings and bubbles in foam Sem 1

One of the classical problems of discrete geometry is the following. How must a sphere be covered by n equal circles (spherical caps) so that the angular radius of the circles will be as small as possible? In the 1980’s when we started to work on this problem, proven solutions were known only for n = 2, 3, 4, 5, 6, 7, 10, 12, 14, and conjectured solutions for n = 8, 9, 32. The first gaps appeared at n = 11 and 13, for which even suggestions did not exist. We thought that the shapes of bubbles in foam might help. We considered the results of Matzke’s experimental observations, and found that the edge graph of the only bubble with 11 faces and one of the 4 bubbles with 13 faces lead to the best coverings where the Dirichlet cells of the circle system provided the same edge graphs as those of the respective bubbles.

Additionally, we could show that for n = 2 to 12, except 11, the edge network of the Dirichlet cells of the best circle covering is topologically identical to the minimal net formed by the intersection of n soap-film-like cones by a sphere (determined by A. Heppes, F.J. Almgren and J.E. Taylor).

In the range of n = 14 to 20, we considered the possible shapes of coated vesicles. These are certain kinds of “bubbles” where a part of the cellular membrane is surrounded by a clathrin basket – a polyhedron. With their help, for these values of n, except n = 19, we could construct the best so far circle coverings of a sphere.

In the lecture, we want to survey the results for n = 2 to 20, making comparison with soap-film-like cones, bubbles in foam, coated vesicles, and to compare the best circle coverings with the numerical solutions to the isoperimetric problem for polyhedra with n faces. This research was supported by OTKA grant no. K801146.

12:05-12:30 Gabbrielli, R (Università degli Studi di Trento)
  Periodic simple tilings as models for monodisperse foams Sem 1

Co-authors: Olaf Delgado-Friedrichs (ANU), Michael O'Keeffe (ASU), Davide M. Proserpio (University of Milan)

We ran a combinatorial search based on the computational tiling theory developed by Delaney, Delgado-Friedrichs, Dress and Huson aimed at the construction of periodic simple tilings of increasing complexity. Periodic tilings containing only tiles with 12 to 16 faces and 4, 5 and 6-sided faces have been considered. All Euclidean tilings with up to 11 crystallographically distinct kinds of vertices have been enumerated.

Related Links: • - Classification of periodic foams

12:30-13:30 Lunch at Wolfson Court
13:30-14:15 Private Discussion
14:15-14:35 Cañete, A (Universidad de Sevilla)
  Multiple planar bubbles Sem 1

The planar soap bubble problem looks for the least-perimeter configuration of planar curves enclosing regions Ri of prescribed areas ai, with i=1,…,n. It is known that these minimizing configurations satisfy Plateau’s laws. However, in the most general setting, the regions in a minimizing configuration might not be connected, and so some area ai could be enclosed by a region Ri with several components. This is the main difficulty of this problem, since a lot of configurations have to be taken into account. In this talk, we will review the main results obtained for this problem, and discuss new directions for attacking it.

14:35-14:55 Flikkema, E (Aberystwyth University)
  Exploring analogies between bubble clusters and particle clusters Sem 1

Co-author: Simon Cox (Aberystwyth University)

This presentation is about the minimal perimeter problem for two-dimensional clusters of equal-area bubbles. The challenge is to find the global minimum of the perimeter with respect to different bubble cluster topologies with a fixed number of bubbles N (up to N=42). The free bubble cluster is being considered, as well as bubble clusters confined to a circle or to various polygonal shapes.

In this project, an analogy between the above-mentioned bubble cluster problem and the problem of the energetic optimisation of clusters of interacting particles is explored. For clusters of atoms, finding the global minimum of the energy with respect to the positions of the atoms is important, as it is thermodynamically most likely to be formed. Many global optimisation methods have been developed for this problem.

More specifically, in this project, two dimensional clusters of particles are considered, interacting via a repulsive pair-wise potential, combined with a confining external field.

The central idea is to use a particle cluster geometry to generate a bubble cluster topology. The procedure consists of the construction of Voronoi cells on the basis of the particle positions. A local optimisation of the perimeter is then carried out using Surface Evolver. In this way, minima of the particle cluster problem are used to generate candidates for the global minimal perimeter problem for two dimensional bubble clusters. Various pair-wise potentials have been tried in combination with different confining potentials (depending on whether the bubble cluster is free or confined to a circle or polygon).

The success of this procedure in generating good candidates for the global minima of the bubble cluster problem was monitored. The various pair potentials have various success rates. Since the different potentials succeed or fail in different circumstances, the use of multiple potentials is recommended.

14:55-15:15 Plenary Discussion: Open problems in the geometry of bubble clusters
15:15-15:45 Afternoon Tea
15:45-16:15 Private Discussion
15:45-17:00 Informal discussion
Friday 28 February
Session: The effect of foam structure on dynamics and the effect of dynamics on structure
09:30-10:15 Höhler, R (Pierre et Marie Curie, Institut des Nanosciences de Paris, Paris-Est)
  Can bubble interactions in foams accurately be described by a potential? Sem 1

Elastic interactions between neighboring bubbles in a foam are of fundamental importance for the rheological behavior and osmotic pressure of this material (1,2). In many recent simulations these interactions have been modeled by a potential, depending on the distance between neighboring bubble centers. Previous surface evolver studies and experiments strongly suggest that this potential is anharmonic, depending on local packing topology (2,3). In contrast, a recent confocal microscopy study of emulsions concludes that the potential should to a good approximation be harmonic (4). We compare new 3D Surface Evolver simulations of wet foams to these various previous results and discuss to what extent bubble interactions are accurately represented in terms of a potential. We also discuss on this basis how osmotic pressure and capillary pressure in foams scale with liquid fraction in the vicinity of the jamming transition.

(1) M. Le Merrer, S. Cohen-Addad, R. Hohler, PRL 2012. Cohen-Addad, Hohler, Pitois, Annual Review of Fluid Mechanics 2013. (2) A. Maestro, W. Drenckhan, E. Rio, R. Hohler, Soft Matter, 2013. Hohler,Cheung Sang, Lorenceau, Cohen-Addad, Langmuir 2008 (3) Mason et al PRE 1997 (4) Jorjadze, Pontani, Brujic PRL 2013

10:15-11:00 Cantat, I (Institut de physique de Rennes)
  Gibbs elasticity effect in foam shear flows: a non quasi-static 2D numerical simulation Sem 1

The origin of the dissipation in liquid foams is not fully understood, especially in the large deformation, large velocity regime. Numerical simulations, now very accurate in the quasi static regime, are still sparse in the dissipative regime, and are all based on restrictive assumptions or very small bubbles numbers. Here we present the results obtained with 2D numerical simulations involving 500 bubbles under simple shear, in a non-quasi static regime. The bubble description is kept as simple as possible and the dissipation is assumed to arise from surface tension variations induced by film area variations. This model leads to a steady state stress under simple shear that is well fitted by a Herschel-Bulkley law with an exponent 0.6. We show that small tension dynamical inhomogeneities induce foam structure modifications responsible for the largest part of the stress increase.

11:00-11:45 Morning Coffee
11:45-12:05 Ketoja, JA (VTT)
  Effect of fibres on wet foams Sem 1

Co-authors: Ahmad M. Al-Qararah (VTT), Tuomo Hjelt (VTT), Antti Koponen (VTT), Ali Harlin (VTT)

Wet foam is an excellent carrier phase to produce particle or fibre structures. Thus, it is important to know how particles or fibres affect foam properties. We have studied these effects for foams generated by axially agitated mixing with 60-80% air content [1]. Onset of a flow macro-instability is observed at high rotation speeds both with and without cellulose fibres. The air content of the foam is strongly affected by this instability. By describing the effect of the air content on the stability of bubble interfaces, we explain the bubble size for broad ranges of rotation speed and surface tension. Added fibres usually reduce the bubble size for a fixed rotation speed. After the flow macro-instability, not only the air content but also the bubble size increases with reduced surface tension. Beyond the instability, the effect of fibres on foam properties becomes minor.

1) A. M. Al-Qararah, T. Hjelt, A. Koponen, A. Harlin, J. A. Ketoja, Bubble size and air content of wet fibre foams in axial mixing with macro-instabilities, Colloids and Surfaces A: Physicochem. Eng. Aspects 436 (2013) 1130-1139.

12:05-12:30 Monloubou, M (Institut de Physique de Rennes)
  Shock wave absorption in liquid foams Sem 1

Liquid foams are, amongst other usage, used to mitigate shock waves during mine-cleaning operations due to their excellent ability to absorb energy. However, the origins of this important property are still not well understood. We use a shock tube to send a shock wave on a foam sample with controlled liquid fraction and bubble size. We record images with a high speed camera and pressure signals through the sample, giving information about the wave propagation and attenuation, the foam displacement and the destroyed area.

12:30-13:30 Lunch at Wolfson Court
13:30-15:15 Private Discussion
15:15-15:45 Afternoon Tea
15:45-17:00 Private Discussion

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