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Abstract We focus on a novel concept of photosynthetic soft membranes, possibly able to allow the conversion of solar energy and carbon dioxide (CO $$_2$$ 2 ) into green fuels. The considered membranes rely on self-assembled functional molecules in the form of soap films. We elaborate a multi-scale and multi-physics model to describe the relevant phenomena, investigating the expected performance of a single soft photosynthetic membrane. First, we present a macroscale continuum model, which accounts for the transport of gaseous and ionic species within the soap film, the chemical equilibria and the two involved photocatalytic half reactions of the CO $$_2$$ 2 reduction and water oxidation at the two gas–surfactant–water interfaces of the soap film. Second, we introduce a mesoscale discrete Monte Carlo model, to deepen the investigation of the structure of the functional monolayers. Finally, the morphological information obtained at the mesoscale is integrated into the continuum model in a multi-scale framework. The developed tools are then used to perform sensitivity studies in a wide range of possible experimental conditions, to provide scenarios on fuel production by such a novel approach.

We focus on a novel concept of photosynthetic soft membranes, possibly able to allow the conversion of solar energy and carbon dioxide (CO 2 ) into green fuels. The considered membranes rely on self-assembled functional molecules in the form of soap films. We elaborate a multi-scale and multi-physics model to describe the relevant phenomena, investigating the expected performance of a single soft photosynthetic membrane. First, we present a macroscale continuum model, which accounts for the transport of gaseous and ionic species within the soap film, the chemical equilibria and the two involved photocatalytic half reactions of the CO 2 reduction and water oxidation at the two gas–surfactant–water interfaces of the soap film. Second, we introduce a mesoscale discrete Monte Carlo model, to deepen the investigation of the structure of the functional monolayers. Finally, the morphological information obtained at the mesoscale is integrated into the continuum model in a multi-scale framework. The developed tools are then used to perform sensitivity studies in a wide range of possible experimental conditions, to provide scenarios on fuel production by such a novel approach. [ABSTRACT FROM AUTHOR]

Better understanding of frictional behaviour of powdered soaps used in dry wire drawing can lead to improvements in wire quality and large savings in energy consumption. The latter is particularly significant given the colossal amounts of energy expended in wire drawing processes. However, there is currently a very limited understanding of the subject, including a lack of quantitative data on soap friction. This is at least in part likely to be due to complex tribological behaviour of the powdered soaps, which is very different from that of lubricating oils. This paper studies frictional behaviour of sodium and calcium powdered soaps, using a ball-on-disc tribometer (ETM rig) incorporating a powder 'scoop', and under contact conditions pertinent to die-wire contact in wire drawing. Tests employed a WC ball and C15E steel disc to mimic die and wire materials respectively. Results show that under high sliding conditions, friction coefficient with powered soaps is remarkably low, at 0.03 and 0.04 for sodium and calcium soaps respectively. Friction generally decreased with increasing contact pressure and sliding speed. Post-test analysis of ETM specimens showed the presence of thick soap layers on the rubbed surfaces. However, in ETM tests friction was observed to sharply increase leading to the onset of scuffing, soon after the supply of soap to the contact was interrupted, suggesting that any solid deposited layers are easily removed and cannot exclusively explain the low friction. The observed trends suggest that low soap friction may be due to the ability of the soaps to form low shear stress lubricating films at high contact temperatures which are associated with the harsh conditions of high-speed and high-pressure. The results provide new insights into the frictional behaviour of powdered soaps which can be used to optimise the wire drawing processes as well as a direct input to wire deformation models. • Friction in wire drawing has major impact on wire quality and energy consumption. • Ball-on-disc ETM machine is used to study friction of typical wire drawing soaps. • Soaps have remarkably low friction (0.03-0.04) at high sliding speeds and pressures. • Continuous supply of soap is needed to ensure low friction and avoid rapid scuffing. • It is likely that soaps can form low shear stress lubricating films at high contact temperature. [ABSTRACT FROM AUTHOR]

This article contains flow visualization data from experiments conducted in an inclined gravity-driven soap film intersected by a circular cylinder undergoing controlled transverse oscillations at a Reynolds number of Re≈235. The dimensionless frequency and amplitude of cylinder oscillation were varied systematically over the ranges 0.2

In this article, I briefly discuss the remarkable scientific accomplishments of Yves Couder, their impact in the fluid dynamics community worldwide and his unique personality. It is a personal tribute to a friend and a colleague I have known for the past 35 years.

Through the simulations performed with the Surface Evolver, we explored the T1s and the eight-fold vertex related topological changes in the square cuboid frame by changing the axial ratio λ and the liquid fraction ϕ. A stability diagram is provided to illustrate the instabilities encountered in any quasi-static process. Metastable film configurations are always accompanied by pre-emptive instabilities, while the instabilities of stable film configurations are not triggered until the central quadrilateral film steadily vanishes. [ABSTRACT FROM AUTHOR]

The stability of a Plateau border between three soap films is considered, taking into account the effects of line tension and bending stiffness in the border. A simple geometry is considered, in which the border initially lies in equilibrium along the axis of a circular cylinder, with three equally spaced films radiating outwards to meet the inside wall of the cylinder. The films are pinned at the two ends of the cylinder with a fixed relative twist, so the initial film surfaces are helicoids. The stability of this system to small perturbations, involving both the films and the border, is investigated as a function of the cylinder aspect ratio, twist angle, film surface tension, border line tension and border bending stiffness. Analytically, the stability problem is reduced to finding the first occurrence of a zero eigenvalue of an infinite matrix, which is then estimated numerically. The results from this calculation are in good agreement with full numerical simulations. [ABSTRACT FROM AUTHOR]

By using a suitable triple cover we show how to possibly model the construction of a minimal surface with positive genus spanning all six edges of a tetrahedron, working in the space of BV functions and interpreting the film as the boundary of a Caccioppoli set in the covering space. After a question raised by R. Hardt in the late 1980's, it seems common opinion that an area-minimizing surface of this sort does not exist for a regular tetrahedron, although a proof of this fact is still missing. In this paper we show that there exists a surface of positive genus spanning the boundary of an elongated tetrahedron and having area strictly less than the area of the conic surface. [ABSTRACT FROM AUTHOR]

We discuss how mathematics and secondary mathematics education majors developed an understanding of Fermat points for the triangle as well as Steiner points for the square and regular pentagon, and also of soap film configurations between parallel plates where forces are in equilibrium. The activities included the use of soap films and the interactive geometry program GeoGebra. Students worked in small groups using these tools to investigate the properties of Fermat and Steiner points and then justified the results of their investigations using geometrical arguments. These activities are specific approaches of how to encourage prospective teachers to use physical experiments to support students’ development of mathematical curiosity and mathematical justifications. [ABSTRACT FROM AUTHOR]

We present experiments and Surface Evolver simulations concerning the interaction of fibres with simple arrangements of soap films, which constitute model systems for dry foams. For a fibre inserted into a soap film which spans two Plateau borders, our simulation accurately predicts the variation in the length of the film as the fibre width is varied. For a fibre introduced into a Plateau border, simulations accurately predict the variation in length of the Plateau border as the fibre diameter is varied, and as it is moved. We suggest that the force necessary to move the fibre from its equilibrium position may act to inhibit foam coarsening, in line with previous observations from experiments on fibre-laden foams. [ABSTRACT FROM AUTHOR]

In this section of Resonance, we invite readers to pose questions likely to be raised in a classroom situation. We may suggest strategies for dealing with them, or invite responses, or both. 'Classroom' is equally a forum for raising broader issues and sharing personal experiences and viewpoints on matters related to teaching and learning science. [ABSTRACT FROM AUTHOR]

Plateau's problem is to find a surface with minimal area spanning a given boundary. Our paper presents a theorem for codimension one surfaces in in which the usual homological definition of span is replaced with a novel algebraic-topological notion. In particular, our new definition offers a significant improvement over existing homological definitions in the case that the boundary has multiple connected components. Let M be a connected, oriented compact manifold of dimension and the collection of compact sets spanning M. Using Hausdorff spherical measure as a notion of 'size,' we prove: There exists an in with smallest size. Any such contains a 'core' with the following properties: It is a subset of the convex hull of M and is a.e. (in the sense of -dimensional Hausdorff measure) a real analytic -dimensional minimal submanifold. If , then has the local structure of a soap film. Furthermore, set theoretic solutions are elevated to current solutions in a space with a rich continuous operator algebra. [ABSTRACT FROM AUTHOR]

The history of the scientific research on foam films, traditionally known as soap films , dates back to as early as the late 17th century when Boyle and Hooke paid special attention to the colours of soap bubbles . Their inspiration was transferred to Newton, who began systematic study of the science of foam films. Over the next centuries, a number of scientists dealt with the open questions of the drainage, stability and thickness of foam films. The significant contributions of Plateau and Gibbs in the middle/late 19th century are particularly recognized. After the “colours” method of Newton, Reinold and Rücker as well as Johhonnot developed optical methods for measuring the thickness of the thinner “non-colour” films (first order black) that are still in use today. At the beginning of the 20th century, various aspects of the foam film science were elucidated by the works of Dewar and Perrin and later by Mysels. Undoubtedly, the introduction of the disjoining pressure by Derjaguin and the manifestation of the DLVO theory in describing the film stability are considered as milestones in the theoretical development of foam films. The study of foam films gained momentum with the introduction of the microscopic foam film methodology by Scheludko and Exerowa, which is widely used today. This historical perspective serves as a guide through the chronological development of knowledge on foam films achieved over several centuries. [ABSTRACT FROM AUTHOR]

Physical modeling as a developmental iterative process was one of the primary legacies of Frei Otto in his quest for a formfinding design tool. Exploring first hand knowledge of his sketching, and then incremental physical modeling techniques, we look at Otto's global approach to problem solving through design. Numerous actual projects of Otto including the BP Dyce Tent in Scotland and the KOCOMMAS project in Saudi Arabia are described in terms of this process as a better way of understanding his approach. A comparison of the formfinding and shape making approach to design is examined. In contrast to Otto's physical modeling, digital modeling & parametric approaches are also investigated to show Otto`s influence and evolution in contemporary modeling of complex surface geometries, but also to point out the seismic change that is presently occurring in the architectural profession. [ABSTRACT FROM AUTHOR]

The Fermat point of a triangle is the point such that minimizes the sum of the distances from that point to the three vertices. Five approaches to study the Fermat point of a triangle are presented in this article. First, students use a mechanical device using masses, strings and pulleys to study the Fermat point as the one that minimizes the potential energy of the system. Second, students use soap films between parallel planes connecting three pegs. The tension on the film will be minimal when the sum of distances is minimal. Third, students use an empirical approach, measuring distances in an interactive GeoGebra page. Fourth, students use Euclidean geometry arguments for two proofs based on the Torricelli configuration, and one using Viviani's Theorem. And fifth, the kinematic method is used to gain additional insight on the size of the angles between the segments joining the Fermat point with the vertices. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Plateau's problem is to show the existence of an area-minimizing surface with a given boundary, a problem posed by Lagrange in 1760. Experiments conducted by Plateau showed that an area-minimizing surface can be obtained in the form of a film of oil stretched on a wire frame, and the problem came to be called Plateau's problem. Special cases have been solved by Douglas, Rado, Besicovitch, Federer and Fleming, and others. Federer and Fleming used the chain complex of integral currents with its continuous boundary operator, a Poincaré Lemma, and good compactness properties to solve Plateau's problem for orientable, embedded surfaces. But integral currents cannot represent surfaces such as the Möbius strip or surfaces with triple junctions. In the class of varifolds, there are no existence theorems for a general Plateau problem. We use the chain complex of differential chains, a geometric Poincaré Lemma, and good compactness properties of the complex to solve Plateau's problem in such generality as to find the first solution which minimizes area taken from a collection of surfaces that includes all previous special cases, as well as all smoothly immersed surfaces of any genus type, orientable or nonorientable, and surfaces with multiple junctions. Our result holds for all dimensions and codimension-one surfaces in ℝ. [ABSTRACT FROM AUTHOR]

Michele Emmer discusses the relations between soap films, arts, mathematics, visual arts, architecture with very recent examples. [ABSTRACT FROM AUTHOR]

We derive the leading-order equations that govern the dynamics of the flow in a falling, free-standing soap film. Starting with the incompressible Navier-Stokes equations, we carry out an asymptotic analysis using parameters that correspond to a common experimental setup. We account for the effects of inertia, surface elasticity, pressure, viscous stresses, gravity, and air drag. We find that the dynamics of the flow is dominated by the effects of inertia, surface elasticity, gravity, and air drag. We solve the leading-order equations to compute the steady-state profiles of velocity, thickness, and pressure in an experiment in which the film is in the Marangoni elasticity regime. The computational results, which include a Marangoni shock, are in good accord with the experimental measurements. [ABSTRACT FROM AUTHOR]

Abstract: The introduction of equal soap sized bubbles into a cylindrical tube of diameter roughly equal to or smaller than the bubble diameter results in equally spaced films, arranged perpendicularly to the tube. This particular type of foam, when made with a typical surfactant solution, shows unusual drainage properties. For tube radii below 0.918l 0, where l 0 is the capillary length of the system, no measurable flow through the foam can be observed: if solution is poured onto the foam, it will accumulate at the top without descending through it. However, for tube radii above 0.918l 0, flow occurs, whatever the flow rate. This flow is highly non-uniform, unlike what is observed in disordered bulk foams: it is confined to a channel along the tube wall, resulting in localized swelling of the vertical wetting films and breaking of the symmetry of the system. This channel disappears for a critical Plateau border cross-section. A qualitative analysis attributes the inception of flow to an instability arising in the Plateau border. A semi-quantitative analysis explains why no flow occurs for tube radii less than 0.918l 0, and gives values of the critical Plateau border cross-section as a function of tube radius, in good agreement with experimental data. [Copyright &y& Elsevier]

Newton black films (NBF) can be obtained from solutions containing ionic surfactants. These thin structures have typical thickness of the order of 3 nm and are similar to biological membranes. An understanding of the NBF structure and stability is of relevance to problems concerned with the biophysics of the cell. From a practical point of view, the stability of NBFs determines the stability of dispersions and foams, which are relevant in a number of industrial areas such as foodstuffs, cosmetics or distillation. In this paper, we investigate the temperature dependence of the structure and electrostatics of sodium dodecyl sulfate (SDS) NBFs using large-scale computer simulations. We show that high temperature black films exibit a dramatic change of the film roughness. Notably, the structural changes in the film occur without a significant modification of the total thickness. The degree of polarization of the water molecules and the electrostatic fields in the NBF also show a very weak dependence on temperature, suggesting that water remains strongly ordered in the NBF even at high temperatures ( ≈ 350 K). We conclude that the anomalous dielectric response observed in NBF at ambient temperature should also be present at high temperatures. [ABSTRACT FROM AUTHOR]

In (the surface of) a convex polytope P 3 in R 4, an areaminimizing surface avoids the vertices of P and crosses the edges orthogonally. In a smooth Riemannian manifold M with a group of isometries G, an areaminimizing G-invariant oriented hypersurface is smooth (except for a very small singular set in high dimensions). Already in 3D, area-minimizing G-invariant unoriented surfaces can have certain singularities, such as three orthogonal sheets meeting at a point. We also treat other categories of surfaces such as rectifiable currents modulo v and soap films. [ABSTRACT FROM AUTHOR]

We study a toy model for electrolytic soap films, the two-dimensional two-component plasma. This model is exactly solvable for a special value of the coulombic coupling constant βq2=2. This allows us to compute the disjoining pressure of a film and to study its stability. We found that the Coulomb interaction plays an important role in this stability. Also the adhesivity that measures the attraction of soap anions to the boundaries is very important. For large adhesivity the film is stable, whereas for small adhesivity a collapse could occur. We also study the density and correlations in the film. The charge density near the boundary shows a double layered profile. We show that the charge correlations verify a certain number of sum rules. [ABSTRACT FROM AUTHOR]

We have developed an original viscometer that allows the determination of the viscoelastic properties of freely suspended soap films. We have found that soap films have a very high shear viscosity and a small shear modulus that might be associated with a glassy (gel-like) behavior. The incorporation of polymer chains inside the soap films does not change very significantly these viscoelastic properties that are governed by the dense packing of the surfactant in the interfacial films; nevertheless, the role of the polymer can be qualitatively described by scaling laws. [ABSTRACT FROM AUTHOR]

This article contains flow visualization data from experiments conducted in an inclined gravity-driven soap film intersected by a circular cylinder undergoing controlled transverse oscillations at a Reynolds number of Re ≈ 235 . The dimensionless frequency and amplitude of cylinder oscillation were varied systematically over the ranges 0.2 < f * < 1.8 and 0.1 < A * < 1.3 . A high-speed camera was used to capture the interference fringe patterns reflected from the soap film. These videos show the structure of the wake behind the cylinder, including the initial formation of vortices and the extended 'vortex street'. Several wake patterns were identified, including the classic 2S, P+S, 2P, and 2T patterns, which are discussed in detail in the accompanying research article titled "The wake of a transversely oscillating circular cylinder in a flowing soap film at low Reynolds number" [1]. The videos presented in this article can be accessed through the Virginia Tech University Libraries' Repository at https://doi.org/10.7294/14448027.v5[2]., Competing Interests: The authors declare that they have no known competing financial interests or personal relationships which have, or could be perceived to have, influenced the work reported in this article., (© 2022 Published by Elsevier Inc.)

A new mathematical model of soap films is proposed, called the 'covering space model.' The two sides of a film are modeled as currents on different sheets of a covering space branching along the film boundary. Hence a film may be seen as the minimal cut separating one sheet of the covering space from the others. The film is thus the oriented boundary of one sheet, which represents the exterior of the film. As oriented boundaries, films may be calibrated with differential forms on the covering space, a version of the min-cut, max-flow duality of network theory. This model applies to unoriented films, films with singularities, films touching only part of a knotted curve, films that deformation retract to their boundaries, and other examples that have proved troublesome for previous soap film models. [ABSTRACT FROM AUTHOR]

One of the many interests of the late Dietrich Stauffer was the modelling of mortality. Here we review the features of mortality data for various biological species. Age specific mortality (death rate) leads to a discussion of possible models of the death rate, including that of Gompertz, raising the interesting question: is our lifetime finite or could we contemplate living for ever? The answer judging from many different data sources is that without radical changes in our biology it seems death above age 120 is extremely unlikely. We then show how a toy model, linking mortality to the immune system, can predict the general variation of the death rate with time, spanning both infant and adult phases. The outcome provides underpinning support for the many nutritionists and medical experts who increasingly advocate the benefits to mortality of a healthy lifestyle. Age specific mortality within social networks is also shown to be significantly affected by both psychological and physical shocks. The review concludes with the description of novel experiments using soap films for the study of failure. These allow for a reproduction of high infant mortality, the so-called bath-tub curve of mortality, and the Gompertz law. • For humans, infant mortality extends into late teenage years decreasing parabolically. • Psychological shocks create mortality spikes and give insight into social links. • A toy model links mortality to immune strength. • Soap film data shows how mortality is linked to system defects. [ABSTRACT FROM AUTHOR]

Previous research on animations of soap bubbles, films, and foams largely focuses on the motion and geometric shape of the bubble surface. These works neglect the evolution of the bubble's thickness, which is normally responsible for visual phenomena like surface vortices, Newton's interference patterns, capillary waves, and deformation-dependent rupturing of films in a foam. In this paper, we model these natural phenomena by introducing the film thickness as a reduced degree of freedom in the Navier-Stokes equations and deriving their equations of motion. We discretize the equations on a non-manifold triangle mesh surface and couple it to an existing bubble solver. In doing so, we also introduce an incompressible fluid solver for 2.5D films and a novel advection algorithm for convecting fields across non-manifold surface junctions. Our simulations enhance state-of-the-art bubble solvers with additional effects caused by convection, rippling, draining, and evaporation of the thin film. [ABSTRACT FROM AUTHOR]

Soap bubbles are widely appreciated for their fragile nature and their colorful appearance. The natural sciences and, in extension, computer graphics, have comprehensively studied the mechanical behavior of films and foams, as well as the optical properties of thin liquid layers. In this paper, we focus on the dynamics of material flow within the soap film, which results in fascinating, extremely detailed patterns. This flow is characterized by a complex coupling between surfactant concentration and Marangoni surface tension. We propose a novel chemomechanical simulation framework rooted in lubrication theory, which makes use of a custom semi-Lagrangian advection solver to enable the simulation of soap film dynamics on spherical bubbles both in free flow as well as under body forces such as gravity or external air flow. By comparing our simulated outcomes to videos of real-world soap bubbles recorded in a studio environment, we show that our framework, for the first time, closely recreates a wide range of dynamic effects that are also observed in experiment. [ABSTRACT FROM AUTHOR]