Weaire-Phelan Structure

The Weaire-Phelan structure has 0.3 percent less surface area than the Kelvin structure, which was thought to be the solution to the Kelvin problem since 1887 until a better solution (the Weaire-Phelan structure) was discovered by Denis Weaire at Trinity College Dublin using software which was modern in the early nineties. As part of his "experimentation", Denis Weaire visited the local bar with some bubble making instruments and investigated the bubble foam at the top glasses of beer. The software he used to make the calculations is called Surface Evolver. Designs using the Weaire-Phelan structure include a sculpture at Trinity College and the Beijing Water Cube. The insulation caused by the shape of the plastic cells helps to keep heat in the building. [1]

The main goal of phoamatomics (a combination of the words “foam” and “atomics") is to induce components to self-organize. The reason that this endogenous arranging is important is that the natural framework of the Wearie-Phelan structure allows only a few specific frequencies of light to permeate while simultaneously impeding others. This phenomenon is called a photonics band gap. This is a behavior also expressed by normal semiconductors, but the use of carbon has to be limited to slow the rate of climate change in the environment due to climate change. [2]

As scientists in a variety of fields have taken to studying the Weiare-Phelan structure, such as its shear and photonic properties. In the nineteenth century, Lord Kelvin posed a three-dimensional space-filling problem regarding the shape of idealized bubbles: “What space-filling arrangement of similar cells of equal volume has minimal surface area?” The resulting structure was the Kelvin Structure, consisting of 14-sided truncated octahedrons, which have a very slight curvature on the hexagonal faces. In 1994, Dennis Weaire, a professor in the school of Physics at Trinity College Dublin, and his Phd student Robert Phelan decided to tackle the ideal foam structure problem. Dennis and Robert used computer simulations to find that there is an alternative structure available that is better than the Kelvin Structure. The Weaire-Phelan unit cell consists of six fourteen-sided polyhedra and two twelve-sided polyhedra with irregular faces, with all sides having some convex curvature except for the hexagonal faces. Researchers in this group at Trinity College Dublin and in other laboratories all over the world attempted to actually make this structure using soap bubbles. Since one can make the Kelvin Structure, one would think that Weaire-Phelan structure is also producible. Making this structure proved very difficult due to the complexity of the boundary conditions.

An Italian researcher, Gabbrielli, who was working on this problem, said that he had an idea of how to make this structure, using a template. With funding from SFI, he spent two months working together with Ken Brakke, who performed further computer simulations. Brakke worked out the ideal surface studies that were needed to trigger the formation of this structure in a bubbly fluid. First, one creates normal soap bubbles and then uses a box to fill the template’s structure with these bubbles. The Weaire-Phelan structure was only fully realized in the laboratory once they stopped using containers with flat surfaces. The detailed convex templates allowed them to replicate this structure in real life. They looked at the samples and found that they were able to produce it many times. The bubbles were only a few millimeters in diameter. The Kelvin Problem is a nice mathematical problem and the Weaire-Phelan Structure was a solution which, more accurately, met the expectation of the problem than the previously accepted Kelvin Structure. The discovery of the Weaire-Phelan structure inspired the design of the Beijing Natatorium, which is called the Water Cube. The architects were interested in the optics of the structure. In future optical studies, the Weaire-Phelan could also be used to produce a unique type of photonic crystal through which carefully chosen frequencies of light could be sent.

Kelvin’s curiosity about the even division of three-dimensional space was mostly motivated by his interest in ether, the fifth element, according to Plato, which is essentially air. The Kelvin structure is a tiling of truncated octahedra. Despite the fact that Kelvin’s belief in ether was pseudoscientific, there are modern applications in biology, physics, and mathematics. The Weaire-Phelan structure is made up of twelve and fourteen sided polyhedral bubbles. It was almost impossible to experimentally create the structure in bubble form the way that the Kelvin structure could. As it turns out, the reason for that is the fact that the containers used to create the bubble structure had flat surfaces. The containers are essentially a form that helps the bubbles form. The Weaire-Phelan structure needs curved surfaces in the template so that the structure can form. Using calculations from the program Surface Evolver, the design for the template was created. Then, it was 3D printed out of plastic. The correct bubble size was estimated. Then the bubbles were made from liquid dish soap and then released into the template. After that, the container is closed and shaken up. This was the first experimental realization of the Weaire-Phelan structure. [3]

Dimensions of the Weaire-Phelan structure from : "On Quantization with the Weaire-Phelan Structure" [4]:

The paper “Quasicrystalline three-dimensional foams” focuses on numerical analysis the of the Kelvin and Weaire-Phelan foams relating to Frank-Kasper phases. Each soap-film relies upon the surface tension to minimize the surface area and the amount of gas retained in each bubble can be held constant and congruent. In the Weaire-Phelan structure, “the bubble centroids are located at the vertices of the well-known A15 Frank-Kasper (F-K) phase. F-K phases are polytetrahedral periodic packings, which can be easily dualyzed, leading to plausible skeletons for foam structures.” The Frank-Kasper phase is related to quasicrystalline materials. Plateaus laws dictate that all minimum surface area partitions “must meet three-fold along edges (at equal angles of 120 degrees) and the edges along which the interfaces meet four-fold at vertices with the tetrahedral angle which is approximately equals 109.5 degrees which approximately equals 1.91 radians.” The balanced quasicrystalline metallic alloys inhabit a limited category in their corresponding phase diagrams. This means that their stability is reliant on the properties of the chemicals of which they are made. The Frank-Kasper phases are relevant in this case. When a metallic allow forms a compound which is partially one metallic element are partially another, it is categorized as Frank-Kasper. In this case the Weaire-Phelan structure is organized following the Frank-Kaspar schematics. This study failed to find a more efficient structure than the Weaire-Phelan structure. [5]

In order to make metamaterials the internal degrees of freedom and a periodic structure are necessary because of negative effective mass and stiffness. Basing metamaterials on closed-cell, crystalline foams make sense to use because their microstructure is more complicated, leading to internal resonance. Recently, self-assembly processes have been applied to the Kelvin structure and the Weaire-Phelan structure. Using numerical models it is possible to show that these foams are superanisotropic which means that they can behave as both a fluid and a solid. The band structure of a foam depends upon the lattice constant, the structural symmetry, the properties of the entrained fluid, and the film resonance frequencies which lead to frequency-dependent anisotropy, Bragg, and resonant scattering. The Debye regime is bounded by film resonances leading to scattering of wavelengths that are even much larger than the cell lattice constant. Closed-cell, crystalline foams exhibit pentamode characteristics depending upon the large difference in fluid compressibility and stiffness of the solid phase. Various measurements have been made concerning Kelvin and Weaire-Phelan structure relating to these quantities. [6]

Thermal transport characteristics are extremely important in foam applications and rely upon the microstructure of the foam. Analysing the thermal transport characteristics is extremely complicated because the Weaire-Phelan foam has such a complex geometry. Using Surface Evolver software, the mass, momentum, and energy equations concerning the air convection in these foams are solved using the finite-element method for values of cell size and porosity. The Weaire-Phelan structure is more computationally heavy than the Kelvin foam in terms of these calculations. Open cell foams are a type of porous material and have the possibility to be used for thermal insulation, heat exchangers, burners, filter, and also in biomedical tasks. There is a difference in the Weaire-Phelan and Kelvin foam models because of the difference in the porosity. This is perhaps because the Weaire-Phelan structure fills space in a more "disordered" way. The different type of filling space leads to differences in the velocity profile. If the porosity is smaller, the differences are greater due to the fact that the larger fraction of solid material in the open-cell foam, which affects the convective heat transfer. The lower porosity profile leads to a larger convective heat transfer coefficient and a lower friction factor. The authors conclude that the Kelvin structure is much simpler than the Weaire-Phelan structure and making the acurate measurements more complicated. [7]

The results of this experiment show that a high variability in the length of the ligaments as well as features having to do with ideal geometry which are found in the Weaire-Phelan structure. The geometry of metal foams is closely connected to the geometry of bubbles, soapy froths, and wet foams, which has led to many comparisons between various geometries made by observing real foams. The Weaire-Phelan structure is a counter-example to the Kelvin structure. The isoperimetric quotient is the way to measuring the area minimization at a fix volume, or a maximized volume with a fixed area and the Weaire-Phelan structure has .3 percent less surface area per unit volume than the Kelvin structure. Even though the Weaire-Phelan unit cell has equal-sized pores, the pressures are not the same. This is different from the Kelvin structure which has equal-pressured cells with equal volume as well. There have been various structures designed since the discovery of the Weaire-Phelan structure, but none of them have a better isoperimetric quotient than it does. Previously, the Kelvin cell had been regarded as the idealized foam for describing metal foams and can be described as a single, regular polyhedron. Since the Weaire-Phelan structure is so much more complicated as a model, the Kelvin structure continued to be used. [8]

Insulation materials are expensive and building energy consumption increased 41.1 percent since 1980. Concrete and plaster do not provide sufficient thermal resistance for heat transfer. Silica aerogels usually consist of a silicon dioxide particles and have low thermal conductivity, thus they are more ideal for using in construction. The silicon dioxide particles are called beads and usually have a diameter of about 7 nm. They are connected to other beads through a complicated, non-regular structure. The silica aerogel structure is made up of silicone and dioxide atoms. Depending on the pH, the backbones are longer or shorter. The thermal conductivity is found by adding three Debye integrals which are dependent upon the velocity of sound in the pores. The Kelvin and Weaire-Phelan structure densities were calculated for this simulation by adding their volume percentages of the tetrahedral elements and multiplying that with the intrinsic densities of the gas and the solids. The results found indicated that the Kelvin and Weaire-Phelan cells were closer to the structure of real aerogels than cubes. In this simulation, the center node was placed and then the tetrahedral elements were built around it. For very low densities, it was found that the Weaire-Phelan structure estimates the effective thermal conductivity better than the Kelvin structure. This experiment is incredibly interesting and the paper is worth reading if only just to learn more details about the topic of aerogels. [9]

Foams are promising options for self-organizing photonic networks with physical characteristics that would make them useful for various applications. The largest phoamatomic band gap found corresponded to the Weaire-Phelan structure. The Weaire-Phelan foam can be made by relaxing a weighted Voronoi tessellation of the A15 crystal. The Weaire-Phelan and the C15 networks have nearly isotropic photonic band gaps which make them useful for thermal radiation sources, adjustable bending angles in waveguides, and photonic circuits. A wet Weaire-Phelan foam is stable and its solidification could directly lead to a photonic network but the band structure for a network with Plateau borders still has to be calculated. The gap size of the Weaire-Phelan network is better than that of geometrically optimized synthetic opals. [10]

Bibliography

[1] Drenckhan, Wiebke. “Bubble Crawling in Dublin - the Mathematical Intelligencer.” SpringerLink, 7 Nov. 2008, link.springer.com/article/10.1007/BF02984702.

[2] Schultz, Steven. “Foam Offers Way to Manipulate Light.” Princeton University, www.pacm.princeton.edu/news/foam-offers-way-manipulate-light. Accessed 21 May 2023.

[3] Gabbrielli, Ruggero, et al. An Experimental Realization of the Weaire–Phelan Structure In ..., www.tandfonline.com/doi/abs/10.1080/09500839.2011.645898. Accessed 22 May 2023.

[4]Kashyap, N., and D.L. Neuhoff. On Quantization with the Weaire-Phelan Partition | IEEE Journals ..., ieeexplore.ieee.org/document/945264/. Accessed 22 May 2023.

[5] Cox, S J, et al. Quasicrystalline Three-Dimensional Foams - Arxiv.Org, arxiv.org/pdf/1610.09286.pdf. Accessed 22 May 2023.

[6]Spadoni, Alessandro. “Closed-Cell Crystalline Foams: Self-Assembling, Resonant Metamaterials.” Asa.Scitation.Org, asa.scitation.org/doi/10.1121/1.4867375. Accessed 21 May 2023.

[7] Cunsolo, Salvatore, et al. “Lord Kelvin and Weaire–Phelan Foam Models: Heat Transfer and Pressure Drop.” ASME Digital Collection, 21 Oct. 2015, asmedigitalcollection.asme.org/heattransfer/article/138/2/022601/384494/Lord-Kelvin-and-Weaire-Phelan-Foam-Models-Heat.

[8] Bock, Jessica, and Anthony Jacobi. “Geometric Classification of Open-Cell Metal Foams Using X-Ray Micro-Computed Tomography.” Materials Characterization, 12 Oct. 2012, www.sciencedirect.com/science/article/pii/S1044580312002641.

[9] Steven K. Latré a, et al. “Comparative Study of a Cubic, Kelvin and Weaire-Phelan Unit Cell for the Prediction of the Thermal Conductivity of Low Density Silica Aerogels.” Microporous and Mesoporous Materials, 8 Apr. 2020, www.sciencedirect.com/science/article/abs/pii/S1387181120302092.

[10] Klatt, Michael, et al. Phoamtonic Designs Yield Sizeable 3D Photonic Band Gaps | PNAS, www.pnas.org/doi/10.1073/pnas.1912730116. Accessed 22 May 2023.