Schedule, Notes/Handouts & Videos

We discuss, in precise geometric language, the notions of expansive and auxetic deformations for

periodic bar-and-joint frameworks. Expansive implies auxetic. In dimension two, expansive periodic structures

can be understood in terms of pseudo-triangulations. From this characterization we derive an effective procedure to generate infinitely many planar periodic expansive designs. We also present a few principles to generate infinite auxetic families in higher dimensions. We illustrate these techniques with an abundance of new auxetic blueprints in dimension three, which is most important for metamaterial design.

This is joint work with Ciprian Borcea.

This talk will give a constructive approach to exploring tangled filaments in a periodic box, where hyperbolic line packings decorate triply-periodic minimal surfaces. A particular arrangement of tangled filaments will be explored in the context of keratin filaments in human skin cells, as well as the broader idea of geometric form as a consequence of the restoring force of a liquid. Finally, I will connect the idea of filament tangling with framework material design.

Beamline 8.3.2 at the Advanced Light Source is an instrument for Hard X-ray Micro-Tomography, providing its users with micron-resolution 3D images, which they analyze by measurement and modeling of the imaged structures. I will present examples of the types of challenges faced during this analysis, and the questions that can be answered in areas including including earth science (why does "flexible sandstone" bend? can shape predict CO2 flow in sandstone?), biology (what is the path of CO2 and H2O through a leaf? why did spiders evolve different head shapes?), and materials science (why do cracks form in the shape they do during processing and then loading of ceramic matrix composites?).

Advances in imaging for the design and investigation of materials have been remarkable: the growth of x-ray brilliance was 18 orders of magnitude in 5 decades, and extremely quick snapshots have enabled description of dynamic systems at the atomic scale. From industry to national laboratories, shape and structural properties of new compounds imaged through x-rays are used to measure the function and resilience of new materials. What drastically changed is the frequency in which this data modality is collected and used as a key scientific record, which is unprecedented. One of the main challenges is how to couple increasing data rate experiments to new Data Science methods in support of more automated analytical tasks for scientific discovery. Recent efforts in machine learning applied to data representation and structural fingerprints have streamlined sample sorting and ranking, including the identification of special materials configurations from million-sized image databases. Methods such as convolutional neural networks have allowed automated characterization of abstract pictures, such as scattering patterns, based on prototypes stipulated by experts, or simulated at leadership computing facilities. Such characterizations or signatures show accelerated image similarity search with real-time feedback in million-size image collections. The ability to survey samples more broadly allied to computational algorithms to compare millions of samples offers unique opportunities for deeper scientific interpretation of experiments, but also impose hurdles such as availability of storage, data transfers, large memory footprint and intensive computation. This talk discuss some strategies and software that tackles detection, segmentation and classification of materials such as carbon fibers, concrete, CMC and more.

Thanks to the Materials Genome Initiative, there is now a database of millions of different classes of nanoporous materials, in particular zeolites. In this talk I will describe a computational approach to tackle high-throughput screening of this database to find the the best nano-porous materials for a given application, using a topological data analysis-based descriptor (TD) recognizing pore shapes. For methane storage and carbon capture applications, our method enables us to predict performance properties of zeolites. When some top-performing zeolites are known, TD can be used to efficiently detect other high-performing materials with high probability. We expect that this approach could easily be extended to other applications by simply adjusting one parameter: the size of the target gas molecule

Soap froth – the quintessential foam – is composed of polyhedral gas bubbles separated by thin liquid films. How are the bubbles shaped and how are they packed? Why do foams have a shear modulus and yield stress, which we usually associate with solids? These and other questions have been explored through simulations with the Surface Evolver, a computer program developed by Ken Brakke. We will describe foam structures ranging in complexity from perfectly ordered foams based on the Kelvin cell to random polydisperse foams with 12^3 cells in which the individual cells have a wide distribution of shapes and sizes – the former is highly idealized and the latter are very realistic. The calculations are in excellent agreement with seminal experiments by Matzke (1946) on the foam structure, and shear modulus measurements by Princen & Kiss (1986). The connection between elastic-plastic rheology and foam structure involves intermittent cascades of topological transitions; this cell-neighbor switching is a fundamental mechanism of foam flow. We will also discuss diffusive coarsening, a mechanism of foam aging, and crushing low-density solid foams with open cells.

Consistency of properties is critical for materials performance, and fundamentally depends on the microstructure on the level of the grains. The evolution of the grain structure is governed by the grain boundary energy and mobility, both functions on the five-dimensional space of grain boundary parameters. Despite decades of experimental effort, the properties of these functions in most regions of the space remain unknown. The result is that existing simulations of grain structure evolution usually employ simple analytic formulas for the grain boundary energy and mobility, and cannot quantitatively reproduce experimental grain structure evolution. Microstructure information made available by recently-developed three-dimensional microscopy techniques could soon be used to infer more realistic grain boundary energy and mobility functions. However, existing front-tracking codes generally make assumptions about the grain boundary network topology that are inconsistent with the microstructures that could arise, or restrict the allowed topological transitions to a small set that could cause substantial deviations from experimental trajectories. This talk will outline our recent efforts to substantially expand the allowed sets of grain boundary topologies and topological transitions, to formulate a physical criterion for the selection of a topological transition, and to develop equations of motion suitable for arbitrary grain boundary energies and mobilities. The intention is to prepare a front-tracking code able to perform predictive simulations of grain structure evolution on the day that realistic grain boundary energy and mobility functions become available.

 Topological materials can exhibit robust properties that are protected

  against disorder even in the absence of quantum effects. In

  mechanics and 

  optics, topological protection has been primarily

  applied to linear waves and 

  non-interacting systems at zero temperature. In this talk, we demonstrate

  how to construct topologically protected states that arise from the

  combination of strong interactions and thermal fluctuations inherent to soft

  matter. Specifically, we consider fluctuating lines under tension, subject to a class of

  spatially modulated substrate potentials. At equilibrium, the lines acquire

  a collective tilt proportional to an integer topological invariant called the

  Chern number. This quantized tilt is robust against substrate disorder,

  as verified by classical Langevin dynamics simulations. We establish the

  topological underpinning of this pattern via a mapping that we develop 

  between the line fluid and Thouless pumping of an imaginary-time Mott insulator in which

  excitations are gapped by interactions. Our work points to a new class of classical 

  topological phenomena in which the topological signature manifests itself in a

  structural property rather than a transport measurement.