The mechanics of cells and tissues is largely governed by scaffolds of filamentous proteins such as collagen. Evidence is emerging that such structures can exhibit rich mechanical phase behavior. A classic example of a mechanical phase transition was identified by Maxwell for macroscopic engineering structures: networks of struts or springs exhibit a continuous, second-order phase transition at a critical network connectivity. This isostatic point threshold is where the number of constraints imposed by connectivity just balances the number of mechanical degrees of freedom. Importantly, most structural networks in biology are characterized by connectivity well below the Maxwell isostatic threshold. In such cases, the mechanics can be controlled by other effects, including internal stress generation and strain. We will present theoretical predictions for mechanical phase transitions and discuss experimental evidence for such behavior in both extracellular matrices and synthetic hydrogels.

Biological chiral fibrous composites, known as biological plywoods, found throughout Nature including the exoskeletons of insects and plant cell walls and collagen based structures in vertebrates have optimized structural and functional properties, such as the iridescent colors observed beetle cuticles. In many cases the micron-range chirality of the fibrous ordering can be uniform, spatially graded, multi-periodic or layered. The challenge to discover structure-property relations in biological plywoods relies on the accuracy of determining the usually space-dependent chiral pitch of the plywoods. Here we use a recently developed geometric model and computational visualization tool to determine the complex spatial gradients present in beetle cuticle which is a canonical example of graded biological plywoods, extensively studied using optical methods. The proposed computational structural characterization procedure offers a complementary tool to optical and other experimental measurements. The new procedure has wide application in biological materials characterization and in biomimetic engineering of structural and functional materials.

Various bio-minerals exhibit unique morphologies where inorganic building blocks are linked together with organic matter. The complex and sophisticated shapes of bio-minerals are influenced by organic molecules [1]. The close interplay between organic and inorganic components endows bio-minerals with unique architectures and extraordinary properties [2,3]. The calcium phosphate system is of special interest due to its widespread appearance in bio-mineralization. However several fundamental aspects are still not clearly understood. Due to unstable nature of amorphous calcium phosphate phases no unambiguous proof of its presence has been shown until now. Therefore, detecting early stages of bio-mineralization still remains a serious challenge.
In present study, different organic molecules were used to stabilize very unique calcium phosphate neuron-like structures [4]. Electron energy-loss spectroscopy (EELS) and energy-dispersive X-ray spectroscopy (EDX) combined with scanning transmission electron microscopy (STEM) imaging at high spatial and high energy resolution, as well as energy-filtered TEM (EFTEM) were used to characterize these structures using Zeiss SESAM and Jeol ARM200F microscopes at different accelerating voltages.
Typical calcium phosphate neuron-like structures consist of a dense core and thin filaments stretching out from the center in a circular form. Based on our high-resolution TEM and electron diffraction data, the structures are amorphous. EFTEM experiments using the low-loss EELS region were conducted on several neuron-like structures in order to show the chemical distribution. Energy-loss near edge fine structures (ELNES) of Ca-L_2,3 and O-K were acquired from the central regions of different neuron-like structures and were compared to the spectra acquired from standard calcium phosphate compounds. Described neuron-like structures are extremely sensitive to the electron beam and therefore special conditions should be used for their investigations and will be discussed.
References:
[1] S Weiner and PM Dove in “An Overview of Biomineralisation Processes and the Problem of the Vital Effects” (2003), Rev Mineral Geochem 54, 1-29.
[2] UGK Wegst and MF Ashby, Philos Mag 84 (2004), 2167.
[3] AP Jackson and JFV Vincent, J Mater Sci 25 (1990), 3173.
[4] M Espanol, ZT Zhao, J Almunia and M-P Ginebra, J Mater Chem B 2 (2014), 2020.

Deoxyribonucleic acid (DNA) melting, the process of double-stranded DNA separating into single-stranded DNA, is not only an essential step for the replication and transcription of genetic information, but it is also a key step for applications based on DNA such as nanodevices and sensing. Therefore, developing a controllable DNA melting process is important for DNA application and optimization.
The DNA melting process can be facilitated by increasing temperature and decreasing ionic concentration. Recently, azobenzene-modified DNA has provided a reversible pathway to control the DNA melting process via light illumination due to the wavelength-dependent switching action of the azobenzene. Azobenzene isomerizes from cis form to trans form under UV illumination (365 nm) and reverses under blue illumination (470 nm), leading to significant destabilization of double-stranded DNA. However, the influence of azobenzene on the DNA melting process is not comprehensively understood yet.
Here, we employ dynamic force spectroscopy (DFS) method to study the force-induced melting of photoswitch-modified double-stranded DNA in the unzipping and shearing geometry. In order to address the challenge of distinguishing the specific and non-specific pulls, a custom classification algorithm derived from machine learning methods is used to classify and subsequently analyze the DNA pulling process. By fitting the peak rupture force versus loading rate with Friddle model, the off rate at zero force of the photoswitch-modified DNA is extracted. Our data show that DNA is destabilized in both unzipping and shearing geometry; however, the shearing geometry is more stable than unzipping. In addition, the off rate of photoswitch- modified DNA in the unzipping geometry is more sensitive to the position of azobenzene in the DNA sequence. These results provide important insight into the kinetics of photoswitch-modified DNA melting process,, and should be helpful for the applications based on the photoswitch-modified DNA.

Polymer-peptide conjugate microcapsules that can release their contents under precise stimuli such as temperature, pH, or specific enzymatic cleavage have great potential as drug delivery vehicles. For these capsules, an atomic-level understanding is necessary in order to tailor their drug release properties. Using a cascaded all-atom simulation approach, we have modeled the interactions of the pH-responsive coiled-coils of polypeptides JR2KC and JR2EC conjugated to N-(2-hydroxypropyl)methacrylamide (HPMA) copolymers which form stable yet stimuli-responsive structures. The polypeptides JR2KC and JR2EC heterodimerize into a four-helix bundle at pH 7, but dissociate due to electrostatic repulsion at both higher and lower pH values [1,2]. However, the polymer-peptide conjugate capsules are observed experimentally to exhibit swelling but not full dissociation at low pH. Molecular dynamics simulations reveal that interactions between adjacent dimers within the film stabilize the structure, and the peptides are too tightly packed along the axis of the polymer to allow for the modeled dissociation mechanism. These results suggest that increasing the ratio of HPMA to peptide may facilitate the dissociation and improve the sensitivity of the conjugates to pH and temperature. In addition, replacing the surplus lysines in JR2KC with small neutral amino acids such as glycine or alanine could provide a less positively charged environment that will enable a more efficient response of the conjugate systems to changes in pH.
[1] D. Aili, et al. “Polypeptide Folding-Mediated Tuning of the Optical and Structural Properties of Gold Nanoparticle Assemblies” Nano Lett. 11:5564 (2011)
[2] J. Rydberg, L. Baltzer, V. Vijayalekshmi “Intrinsically unstructured proteins by design—electrostatic interactions can control binding, folding, and function of a helix-loop-helix heterodimer” J. Pept. Sci. 19:461 (2013)

Giant unilamellar vesicles or GUVs serve as excellent model cell systems. The composition of the lipid bilayer as well as the interior/exterior can be precisely controlled to mimic the in vivo situation. They are typically in the size range of micrometers which not only makes them ideal for mimicking cell morphologies, but also enables them to be observed using light microscopy. This talk will discuss membrane morphological changes caused by the following factors: i) electric fields applied to vesicles, ii) mechanical forces inducing membrane curvatures, and iii) curvature response of the membrane in the presence of various molecules.
Electric fields applied to biological membranes have some interesting applications such as cancer treatment, cell hybridization, or a means to introduce molecules into cells. Here we report GUVs subjected to AC and DC fields to investigate electrodeformation and electroporation (Dimova et al., Soft Matter, 53201, 2009). A microfluidic device will also be presented that is able to spatially confine two or more GUVs adjacently. A DC pulse is then applied via micro-electrodes to initiate so called electrofusion. In this way, GUVs can be used as micro-reactors (Robinson et al., Lab Chip, 14, 2852, 2014).
Membrane tubes within biological cells are involved in cell migration, trafficking, and signaling. Tubes can be artificially created in vitro by mechanical force using a micro-pipette aspiration and an optically trapped bead. Here we show a novel method that produces tubes inside vesicles with the advantage being that they can be used for negative spontaneous curvature studies (Dasgupta and Dimova, J. Phys. D Appl. Phys., 47:282001, 2014). Living cells are subjected to dynamic mechanical stresses and membrane proteins may be required to initiate a particular pathway in response. The role of membrane domains in this context is of great interest. Here, we present a microfluidic device that is able to controllably deform GUVs exhibiting phase-separated domains in order to study how the natural membrane can respond to such mechanical forces.
The interior of living cells is crowded with macromolecules. In such a concentrated environment, local bulk phase separation may occur, involving local composition differences and microcompartmentation. We use GUVs loaded with polymer solutions that exhibit phase separated aqueous compartments to study various phenomena related to molecular crowding in cells. Upon osmotic deflation of vesicles enclosing two aqueous phases that partially wet the membrane, one can observe morphological changes such as vesicle budding (Li et al., J. Phys. Chem. B 116:1819, 2012) and/or formation of membrane tubes (Li et al., Proc. Natl. Acad. Sci. USA 108:4731, 2011).

We identify new entangled motifs in proteins that we call tadpoles. Tadpoles arise in proteins with disulphide bridges (or in proteins with amide linkages), when termini of a protein backbone pierces through an auxiliary surface of minimal area, spanned on a covalent loop. We find that as much as 18% of all proteins with disulphide bridges in a non-redunant subset of PDB form tadpoles, and classify them into five distinct geometric classes. Based on biological classification of proteins we find that tadpoles are much more common in viruses, plants and fungi than in other kingdoms of life. During the talk I will also discuss possible functions of tadpoles. Tadpoles and associated surfaces of minimal area provide new, interesting geometric characteristics not only of proteins, but also of other biomolecules, with many potential applications.

It is well established that protein backbone chains may contain knots, and that since the functioning of the protein depends vitally on its structure this may have significant ramifications for its behaviour [1]; although once though improbable due to the time and energy necessary for the protein to form such a shape [2, 3], knots are now well recognised in over 1000 different protein chains. We refine such analysis by demonstrating that proteins may be distinguished as virtual knots, an extension of knot theory to a wider set of topological classes [4].
Proteins are open curves, and so can only be considered as topologically distinct knots by defining a closure path for their termini. Different algorithms have been developed to this purpose, but in general require the addition of extra geometric structure [5]. We present an alternative mapping of open protein chains to virtual knots [4], providing a wider set of topological types with which to distinguish the effect of knotting on protein function. A significant fraction of known knotted proteins are shown to exhibit virtually knotted structure. We further investigate whether this occurs even in proteins not currently thought to be knotted, and whether these distinct classes relate to particular families of protein curves.
[1] M Jamroz, W Niemyska, E J Rawdon, A Stasiak, K C Millett, P Su#322;kowski and J L Sulkowska. KnotProt: a database of proteins with knots and slipknots. Nucleic Acids Research 43, D306-14 (2014).
[2] F Takusagawa and S Kamitori. A real knot in protein. Journal of the American Chemical Society 118, 8945-6
(1996).
[3] M L Mansfield. Are there knots in proteins? Nature Structural Biology 1, 213-4 (1994).
[4] L H Kauffman. Introduction to virtual knot theory. JKTR 21, 1240007 (2012).
[5] J I Sulkowska, E J Rawdon, H C Millett, J N Onuchic and A Stasiak. Conservation of complex knotting and slipknotting patterns in proteins. PNAS 109, E1715-23 (2012).

Topological features of implant surface play an important role in implant-tissue reaction and osseointegration while new bone formed on the implant surface. In recent years a new electrospun nanofiber technology, along with implant surface spray coating method has been developed by co-authors Ren and Chen, in which a nanofiber coating layer can be placed on the implant surface, and can effectively stimulate osseointegration. The thickness and the microstrucutres of nanofiber networks are found to have important effects on the bone attachment, and they can be controlled through coating process parameters, mainly the coating time and target moving speeds. In order to understand the nanofiber coating effect on bone growth, and to optimize the coating process, the surface topology of the coated implant needs to be characterized.
Commonly used methods of surface characterization include the use of various light or electron microscopies, and the measurement is obtained on the projected 2D images. AFM measures 3D surface geometry, but has the limitation of stylus tip penetration depth within the coated nanofiber network, besides its slow scan speed. In this paper high-resolution laser confocal microscopy is used for obtaining 3D nanofiber surface coordinates. An automated data processing procedure has been developed to analyze fiber occupation and remaining pore geometries in 3D measurement space.
The measurement was performed on nanofiber coating layers processed under different coating times and target moving speeds. The nanofiber network and porosity are characterized by the statistic distribution of nanofiber density and the distribution of pore size over the measurement space, and the possible effect of nanofiber coating topology on new bone formation is discussed.

We use computer simulations to examine the influence of different random substrate structures on diffusion rates of polymer molecules of different sizes and with different knot types, in order to understand the physical behaviour of knotted biological polymers in agarose gel electrophoresis.
Agarose gel electrophoresis is a widely-used method for separating charged biomolecules, like DNA, according to size or conformation. In particular, two-dimensional electrophoresis yields increased sensitivity, by performing this technique with different parameters in two orthogonal directions. This method has proven to be particularly useful in the recognition of different kinds of knotted DNA molecules; the size of a molecule is determined by how compact its 3D structure is and, specifically, decreases with the average crossing number (ACN) of the knot [1]. Contrary to expectations, after 2D electrophoresis linked and knotted DNA molecules do not show a linear distribution of spots, but a characteristic arc pattern [2]. The reasons for this behaviour are not yet fully understood, though some explanations have been proposed [3]. In this context, modelling a possible structure for the gel matrix and testing specific assumptions in silico, through Brownian dynamics simulations, is proving very effective in helping understand the physical problem [4].
In our simulations we model the biological polymers as bead-spring chains, which can be parametrised and adapted to represent the physical features of various molecules, including the persistence length, molecular weight and charge. The stochastic behaviour of the polymers is modelled using Brownian dynamics simulations [5]. Different models can be used for the medium, e.g. random networks of spheres with different size distributions, channels of different geometries, and molecular networks with different connectivities. The effect of the disordered structure of the matrix on molecular diffusion is shown, and we assess the impact of different topological models of the gel on a range of different types of knotted molecules.
[1] V. Katritch, J. Bednar, D. Michoud, R. G. Scharein, J. Dubochet and A. Stasiak, Nature, 384, 142-145 (1996).
[2] S. Trigueros, J. Arsuaga, M. E. Vazquez, J. Roca et al., Nucleic acids research, 29 13, e67 (2001).
[3] J. Cebrián et al., Nucleic Acids Research, gku1255 (2014).
[4] D. Michieletto, D. Marenduzzo and E. Orlandini, arXiv preprint arXiv:1504.02327 (2015).
[5] G. O. Ibáñez-García and S. Hanna, Soft Matter, 5 22, 4464-76 (2009).

First predicted in 2005 by Kane and Mele [1] as a new class of matter, topological insulators (TIs) are of main interest in condensed matter physics thanks to their unique electronic and spin properties that arise on their interfaces. TIs gather the graphene-like transport properties with massless Dirac fermions together with the topological protection that prevents backscattering phenomena. These interfaces exhibit spin-momentum locking which polarizes the spin perpendicular to the momentum. Control of the spin and coherent spin transport are then easily achievable in TIs making them very attractive for spintronic applications [2].
Due to its strong spin-orbit interaction, HgTe is a semi-metal with a band inversion around the Γ point and has been identified as a strong TI assuming the opening of a bandgap. Quantum confinement for 2D structures [3] or application of a tensile strain for 3D bulk ones [4] allow to generate this bandgap.
With constant improvement of the growth process by molecular beam epitaxy, HgTe/CdTe structures are now characterized with sharp interfaces having interdiffusion limited to the nanometer scale range [5]. The influence of surface roughness and interface sharpness is clearly seen through low temperature quantum Hall effect measurements. Carrier mobility up to 440.000 cmsup2;.V^-1.s^-1 and density in the range of 10¹¹ cm^-2 are measured. Moreover, the topological insulator nature is evidenced as well as quantized conductance in both 2D and 3D structures.
Based on these results, we realized HgTe topological insulator nanostructures. By reducing the dimensions toward 1D and coupling them to superconducting contacts, we aim at studying some of the peculiar spin properties of the TIs. We will present the quantum transport behavior of such nanostructures.
[1] C.L. Kane and E.J. Mele, Phys. Rev. Lett. 95, 146802 (2005)
[2] V. Krueckl and K. Richter, Phys. Rev. Lett. 107, 086803, (2011)
[3] M. Konig, S. Wiedmann, C. Brune, A. Roth, H. Buhmann, L.W. Molenkamp, X.L. Qi, and S.C. Zhang, Science 318, 766 (2007)
[4] P.Ballet, C.Thomas, X.Baudry, C.Bouvier, O.Crauste, T.Meunier, G.Badano, M.Veillerot, J.P.Barnes, P.H.Jouneau and L.P.Lévy, J. Elec. Mat., 43, 2955-2962, (2014)
[5] C.Thomas, X.Baudry, J.P.Barnes, M.Veillerot, P.H.Jouneau, S.Pouget, O.Crauste, T.Meunier, L.P. Lévy, P.Ballet, to be published in J. Cryst. Growth, (2015)

The self-assembly of natural and synthetic polymers into nanoscale fibers has recently been exploited for a range of applications from tissue engineering to photonics and catalysis. However, tuning the microstructural properties of fibrous nanostructures remains a challenge. The state of the art in nanofiber assembly remains limited by external parameters such as solution polarity, applied electric field, or temperature. Regulating polymer nanofiber structure necessitates the capacity to tune polymer chain conformation, network composition, and crystal phase. Here we present a modular nanofiber fabrication technique, implemented with the rotary jet spinning system to rapidly produce nanofibers with controllable crystallinity and morphology. The rotary jet spinning system uses centrifugal forces developed in a rotating reservoir perforated with a single or multiple micron-scale orifice to circumferentially extrude polymer and protein nanofibers from solution. By exchanging modular reservoirs of varying thickness and orifice geometry, we can modify nanofiber cross-sectional shape and crystal structure. Additionally, we investigate the dependence of structural properties of nylon-6 nanofibers on system parameters, namely the shape of and flow dynamics through the force extrusion nozzle. Our results indicate that nanofiber extrusion through a non-circular nozzle (three linearly connected circular apertures) increases interchain hydrogen bonding and formation frequency of non-cylindrical nanofiber geometries. Furthermore, accelerating fiber extrusion through the non-circular nozzle shifts the crystal phase of nylon-6 fibers from kinetically favored γ-phase to thermodynamically stable #9082;-phase. We hypothesize that non-circular nozzle geometries enhance local shear forces on polymer solutions during extrusion, thus elongating polymer chains, promoting interchain interactions, and increasing stability of the resulting crystal conformation. Increasing nanofiber extrusion speed also produces a more uniform distribution of polymer microstructures across nylon-6 fibrous scaffolds. Unlike other nanofiber fabrication technologies, such as electrospinning or melt spinning, rotary jet spinning using modular extrusion nozzles also serves as a point-of-use technology; its ability to tune fiber microstructure and extrude non-cylindrical nanofiber networks is not dependent on external system parameters, such as electrical charge, high temperature, or post- processing alignment. Future applications for nanofibers with tunable shape and crystal phase could extend to functionalized filters, catalytic converters, efficient liquid transport, and networks with locally varying mechanical properties, expanding the capabilities of customizable nanotextiles.

The structural integrity of red blood cells and drug delivery carriers through blood vessels is dependent upon their ability to adapt their shape during their flow. Our goal is to examine the role of the composition of cell-mimetic and hairy vesicles on their shape during their follow through in a channel. Via the Dissipative Particle Dynamics simulation technique, we apply laminar flow in a cylindrical channel and investigate the in-channel shape transition of a bio-inspired four-component lipid vesicle encompassing DPPC, DMPC, glycolipid and cholesterol, and a vesicle with functionalized lipids with pegylated head group. We vary the channel dimensions, flow rate and composition to study their influence on the shape of the cell-mimetic and hairy vesicles. We will also characterize the critical flow rate at which the vesicles are ruptured. Our results could be potentially used to enhance the design of drug delivery and tissue engineering systems.

Herein, we report site-selective deposition of nano-features by using a nanoporous platform of poly(vinyl alcohol) (PVA) fabricated by block copolymer(BCP)-based lithography. To produce nanoporous platforms, a thin film of polystyrene-block-poly(2-vinyl pyridine) copolymer (PS-b-P2VP) is firstly spin-coated on a PVA thin layer and solvent-annealed in tetrahydrofuran (THF) vapor to induce its microphase separation. So, well-defined P2VP cylinders oriented normal to the surface are formed in PS matrix. By immersing the BCP thin film into ethanol, the surface is reconstructed to generate nanopores on the whole surface area since P2VP blocks are dissolved and diffused into ethanol but PS blocks are not mobile. This nanoporous BCP thin film is utilized as an etching mask for reactive ion etching (RIE). After RIE, the nanoporous structures of the BCP thin films are transferred to the underlying PVA layer while BCP thin films are completely removed. Then spherical micelles of polystyrene-block-poly(4-vinyl pyridine) copolymers (PS-b-P4VP) are deposited on this nanoporous PVA platform by a spin-coating method from their toluene solution. The micelles immediately tend to avoid the surface of PVA platform and spontaneously self-assemble into the nanopores selectively due to synergetic effect of both surface energy difference between PVA surface and PS shells of the micelles and height contrast of the nanopores. Changing BCP molecular weights also regulates the number of micelles self-assembled in a pore. As a result, the periodic length and separation distance of BCP micelles can be effectively controlled by nanoporous PVA platforms. Moreover, the array of BCP micelles is used as templates to synthesize inorganic nanoparticles as the micelles contain metal precursors in their PVP cores. By oxygen plasma treatment, the array of nanoparticles is achieved while organic components are completely removed. Consequently, the size and periodic length of nanoparticles are strongly dependent on both types of BCP micelles and geometry of nanoporous PVA platforms. To confirm our observation, the samples are characterized by using atomic force microscopy (AFM), scanning electron microscopy (SEM), UV-Vis spectroscopy, and grazing-incidence small angle X-ray scattering (GI-SAXS).

Topology of nanomaterials is not well studied or understood. Nanomaterials are newcomers to the scientific stage and the necessity to better understand it is an imperative to elucidate their properties and applications. Because of this we have undertaken a study of the topology of nanomaterials computationally generated using ab initio DFT techniques. In what follows we report a study of the binary metallic alloy AgAu for several concentrations.
We have generated amorphous cubic cells of Ag₇₅Au_25, Ag₅₀Au_50, Ag₂₅Au_75, using a DFT approach to generate the simulated alloy. We then remove one of the constituents at a time in order to computationally simulate a de-alloying process thereby creating nanoporous materials. This method was applied to a 32-atom crystalline face-centered cubic silver-gold supercells with their densities adjusted to experiment: The 50-50 concentration sample has a density of 14.92 g/cm³, the Ag75Au25 sample has a density of 12.69 g/cm³ and the Ag25Au75 sample has a density of 17.14 g/cm³. After the formation of each alloy cell, a molecular dynamics process at a temperature 10 K below the corresponding melting temperature T_m was performed and the structure obtained was studied. Then one of the species is removed at a time and the resulting sample is subjected to a molecular dynamics process at temperatures asymp; T_m/2. The final nanoporous structures are characterized by determining their pair distribution functions and their bond angle distribution functions. We shall present these results and carry out an analysis by comparison among themselves and to some experimental results reported in the literature.

Topology of nanomaterials is not well studied or understood. Nanomaterials are newcomers to the scientific stage and the necessity to better understand it is an imperative to elucidate their properties and applications. Because of this we have undertaken a study of the topology of nanomaterials computationally generated using ab initio DFT techniques. In what follows we report a study of the metallic-based PdH alloy for several concentrations of hydrogen.
We used an ab initio approach to generate amorphous porous Pd₅₅H₄₅, Pd₅₀H₅₀, Pd₄₅H₅₅ and Pd₄₀H₆₀, using a method which maintains the interatomic distances as in the pure crystalline palladium, swapping palladium by hydrogen in a substitutional way, thus reducing the density and making the initial supercell metastable. This method was applied to a 108-atom crystalline face-centered cubic palladium supercell, with an initial density of 12.02 g/cm³. After the hydrogen substitution we generated four supercells: a crystalline fcc supercell: Pd₅₅H₄₅, with a density of 6.60 g/cm³; an fcc supercell: Pd₅₀H₅₀, with a density of 6.06 g/cm³; an fcc supercell: Pd₄₅H₅₅, with a density of 5.50 g/cm³; and an fcc supercell: Pd₄₀H₆₀, with a density of 4.84 g/cm³. After the hydrogen insertion a molecular dynamics process at 300 K was applied, and the resulting structures were relaxed keeping the volume constant. Then a cell optimization process was performed allowing the cell parameters to change, and thus the volume of the cell. The topology of the structures was characterized by means of the pair distribution functions and the bond-angle distributions. Comparison between the results of the relaxing process and the cell optimization process are presented.

Self-assembly of block copolymers (BCPs) has been attracting a lot of attention in various fields due to their well-defined nano-scale features including sphere, cylinder, and lamellae induced by immiscibility among the chemically distinct polymers that are linked by covalent bonds at their ends. To use the BCP self-assembly directly to potential applications, it is necessary to achieve the lateral order of their nano-scale features in large area. However, existing studies still undergo several drawbacks associated with complex multi-step process and restricted experimental condition even though several methods enhancing remarkably their long-range order have been reported. In this study, we present a unique and simple method to generate long-range ordered nanostructures based on BCP self-assembly on patterned surface. By rubbing poly(tetrafluoro ethylene) (PTFE) having low friction coefficient and high wear rate, the unique array of PTFE nano-stripes with ~ 20 nm in amplitude and ~ 200 nm in pitch distance is readily generated on flat surface and highly aligned along the rubbing direction. Then, asymmetric polystyrene-block-poly(2-vinylpyridine) copolymers (PS-b-P2VP) having around 40 nm of thickness are deposited by using spin-coating method on the patterned surface. To induce BCP self-assembly on the patterned surface, the BCP thin films are solvent-annealed in tetrahydrofuran (THF) vapor that is good solvent for both PS and P2VP blocks. The underlying PTFE nano-stripes can affect the self-assembly of block copolymers and enhance dramatically their lateral ordering behavior. In addition, the long-range ordered BCP thin films are utilized as templates to synthesize functionalized inorganic nanoparticles for further applications of optoelectronics. The ordering behavior of BCP thin films on the patterned surface is also characterized by using atomic force microscopy (AFM), scanning electron microscopy (SEM), and grazing-incidence small angle X-ray scattering (GI-SAXS).

Although mammalian cells are highly complex entities composed of hundreds of unique molecules that participate in numerous interconnected signaling pathways, straightforward principles of soft matter physics can be adopted to help understand their pattern forming behaviors. We use basic principles to understand how cells sense and respond to two classes of geometric cues: curvature fields and edges. We use microfabrication techniques to study how individual, motile cells and cells in confluent monolayers align in response to non-zero Gaussian curvature fields and find that the alignment patterns resemble those observed in various physical systems. We also use concepts and approaches from the field of liquid crystals to understand how edges drive the alignment of cells in monolayers over hundreds of microns. Understanding these pattern formation phenomena may provide insight into tissue development and give tissue engineers an additional toolset for controlling cell behaviors.

Carbon nanostructures possess great potential for various practical applications and novel technologies. To realize these potentials, we need to develop synthesis methods, which enable the control of the size and the alignment of these nanostructures. It has been reported that the size of the carbon nanostructures produced on Si via CVD can be controlled by the size and the morphology of Fe catalyst nanoislands. In this study, we modeled the formation kinetics of Fe nanoislands from the initial Fe catalyst films deposited on SiC single crystalline wafers. These Fe nanoislands are used to produce 1D aligned C nanostructures through low temperature SiC vacuum decomposition. Here, we present a systematic dynamical simulation study for the spontaneous evolution of the Fe nanoislands through the mass accumulation at randomly selected regions of the film via surface drift-diffusion (induced by the capillary and mismatch stresses) on the stochastically rough initial films. The initial thickness and roughness of the film, diffusion/surface stiffness anisotropy, wetting contact angle and mismatch/residual stresses are among the several parameters that affect the final morphology of the nanoislands formed. The investigation of these material properties and process parameters will enable us to control the size and the morphology of Fe catalyst nanoislands and thus the produced 1D carbon nanostructures. Supported by TUBITAK grant no 213M481 and TUBA GEBIP award to Oren EE.

Crystalline porous materials, such as zeolites and metal-organic frameworks, contain complex networks of void channels that are exploited in many industrial applications. A key requirement for the success of any nanoporous material is that the chemical composition and pore topology must be optimal for a given application. However, this is a difficult task, since the number of possible pore topologies is extremely large: thousands of materials have been already been synthesized, and databases of millions of hypothetical structures are available. This talk will describe the development of tools for rapid screening of these large databases, to automatically select materials whose pore topology may make them most appropriate for a given application. The methods are based on computing the Voronoi tessellation, which provides a map of void channels in a given structure. Algorithms to characterize and screen the databases will be described.

We discover varioustopological Chern-Simon(CS) Structure in Nature. Topological CS structure has universal phenomena in many physical world and Biological worlds, as well as Topological crystals (Mobius, Hopflink, Cycloid Crystals and so on), Topological Quantum Fluid, Topological Insulators in condensed matter. The same structure is also found in fluid dynamics as Helicity[1]. This integral can be written as a surface integral and is regarded as a topological invariants. Vorticity is correspond to winding Number and Helicity to Chern Number as topological invariants of Mathematical term.
We discuss why the Topological Helicity (CS) Structures emerges in various systems ranging from 10^-15 m to 10²⁵m scale. [1-6].
[1] J.J.Moreau (1961) , H.K.Moffatt (1969) : Helicity(CS) in Fluid
[2] S. Tanda, et al., Nature (2002): Helicity(CS) in Crystal, Mobius Crystals
[3] J. Ishioka et al., PRL (2010), PRB(2013): Helicity(CS) in Chiral Charge Density Waves and Wave-Function
[4] H. Nobukane et al PRB(2011): Helicity(CS) in Chiral Superconductors.
[5] K. Konno et al., PRD (2008): Helicity(CS) in Space-Time
[6] T.Matsuyama PTP(1987): Helicity(CS) in Gauge fields
[2]-[6] have been done by Center of Topological Science and Technology , Hokkaido Univ.

Dislocations are line-like topological defects in crystals which dominate the plastic behaviour of metals. To describe the dislocation density distribution in terms of continuum variables remains one of the big challenges for closing the gap between the characteristics of single dislocations and the macroscopic plastic behaviour of crystals. Recently, the Kröner-Nye dislocation density tensor [1,2] (K-N-tensor) was realized as an element of a hierarchy of so-called dislocation alignment tensors which are combined with a further hierarchy of curvature tensors to provide a more complete continuum description of the dislocation state [3]. In the current talk I will give an overview of the geometrical and topological aspects of the classical and recently introduced dislocation density tensors and discuss how these relate to work-hardening of crystals.
To give some examples: it is well known that the K-N-tensor captures the topology of the crystal lattice, in that it is a continuous measure of the closure failure for (nominally closed) cycles in dislocated crystals. On the other hand, the K-N-tensor also contains information on the topology of the dislocation network, as its solenoidality reflects that dislocations do not end inside a crystal. Further topological information on the dislocation network is found in the recently introduced scalar curvature density [4] which recovers upon integration the total number of dislocation loops (for planar curves). Apparently so far overlooked was a measure for the entanglement of the dislocation state which may be defined from contractions of slip system specific K-T-tensors and similar plastic distortion tensors. The definition is closely connected to Gauss&’ linking number, and the result may be interpreted as jog-density in dislocation distributions. Jogs are out of plane dislocation segments which result from cutting processes between dislocations on different glide planes. Jogs are of importance for work-hardening in crystals because they obstruct dislocation motion and may lead to dislocation multiplication. We believe that the entanglement is essential in realizing the transition from stage I to stage II in the work-hardening of single crystals.
References
[1] J.F. Nye, Acta Metall. 1 (1953) p.153.
[2] E. Kröner and G. Rieder, Z. Phys. 145 (1956) p.424.
[3] T. Hochrainer, Philos. Mag. 95:12 (2015) p.132.
[4] T. Hochrainer, S. Sandfeld, M. Zaiser and P. Gumbsch, J. Mech. Phys. Solids 63 (2014) p.167

The mechanical rigidity of frameworks -- nodes connected by springs or rigid bars -- underlies the structural integrity of bridges, the response of granular materials, and the design of metamaterials with unusual mechanical properties. A fundamental question governing rigidity is the existence of soft modes: motions that do not significantly stretch or compress the constituent elements of the structure. We demonstrate a novel way to introduce soft modes at desired locations in a metamaterial, by exploiting the properties of a recently introduced class of topological metamaterials. These are special periodic frameworks which exhibit localized edge modes, analogous to the electronic edge states of topological insulators. We show that dislocations in such metamaterials are associated with soft modes of topological origin. The existence of the modes is determined by the interplay between two topological invariants -- the Burgers vector of the dislocation and a "polarization" characterizing the underlying lattice. Simple prototypes built out of triangular plates joined by hinges provide a visual demonstration of these modes.

Polyelectrolytes of variable molecular topology, namely linear, star, ring, and trefoil knot, are investigated with molecular dynamics simulations of a bead spring model to resolve one of the long-standings challenges in polyelectrolyte solutions, namely the interdependency between the distribution of the counter-ions and the polyelectrolytes' conformational properties. At high charge densities, the counter-ions form a ``condensed'' layer around the polyelectrolyte. These condensed counter-ions remain ``bound'' to the macroion, albeit diffusely in an extended counter-ion cloud, for long times, thus altering both the macromolecular structure and dynamics. To quantify this many-body phenomenon, we quantify the counter-ion distribution about the polymer backbone, as well as, the overall shape of the charged macromolecule-counter-ion complexes as function of the molecular topology(knotted and unknotted rings, stars, linear chains). We find that the molecular topology greatly influences the saturation in the net ionic charge around our model polyelectrolyte and the chain conformation is likewise affected in a qualitative fashion. The spatial extent of the counter-ion cloud is found to generally extend over a distance comparable to the chain radius of gyration rather than a scale on the order of the monomer size.

I will discuss a few problems with topological connections that arise in the mechanics of filaments (in such contexts as entangled hair), ordered and disordered networks of rods (in such contexts as fabrics and random networks) and membranes (in such contexts as reconstructive surgery). In each case, we will see how attempts to quantify some seemingly simple notions links topology, geometry and mechanics in unusual ways. In particular, I will show how one can get jamming via entanglement in the absence of friction, how knots form and behave, how fabric deformation is constrained by weave topology, and how topology switches allow for optimal plastic surgery. Joint work with M. Gazzola, H. King, H. Liang, A. McCormick, E. Matsumoto.

Topological data analysis is a rapidly maturing field that provides mathematically rigorous computational tools for quantifying topological structure in data. The primary mathematical theory is called persistent homology, it measures topological quantities such as connected components and loops as a function of a geometric parameter allowing us to track all topological features individually and the length-scales over which they exist.
Recent applications of persistent homology in materials science include the characterisation of configurations of spheres in large packings, atomic arrangements in fluids, the robust generation of network models for porous materials and identification of their percolating length scales. This talk will be an introductory overview of topological data analysis covering the elementary concepts, a brief outline of computational techniques and available packages, and the very latest developments in statistical methods for analysing persistent homology signatures from many samples. The potential for physical rather than just topological and geometric characterisation of materials will also be discussed.

Monopoles emerge in the ice manifold of natural and artificial materials as topological violations of the emergent gauge field that defines the manifold [1]. In magnetic materials they are endowed with a magnetic charge for pairwise Coulomb interaction. Entropic interactions from the underlying spin manifold are however a different matter, and prove to be dimensionality and topology dependent [2]. We briefly summarize the emergence of magnetic charges and monopoles in natural and artificial spin ice materials and report on how recently their low-energy collective behavior can be designed in artificial, frustrated magnetic materials of complex behavior, also known as artificial spin ices [3,4], to obtain magnetic charge order [5], magnetic charge screening [6] and in general exotic couplings between monopoles and geometry.
[1] Castelnovo, Claudio, Roderich Moessner, and Shivaji L. Sondhi. "Magnetic monopoles in spin ice." Nature 451.7174 (2008): 42-45.
[2] Chern, Gia-Wei, Muir J. Morrison, and Cristiano Nisoli. "Degeneracy and criticality from emergent frustration in artificial spin ice." Physical review letters 111.17 (2013): 177201.
[3] Nisoli, Cristiano, Roderich Moessner, and Peter Schiffer. "Colloquium: Artificial spin ice: Designing and imaging magnetic frustration." Reviews of Modern Physics 85.4 (2013): 1473.
[4] Wang, . R., Nisoli, C., Freitas, R. S., Li, J., McConville, W., Cooley, B. J., ... & Schiffer, P. (2006). Artificial ‘spin ice&’ in a geometrically frustrated lattice of nanoscale ferromagnetic islands. Nature, 439(7074), 303-306.
[5] Zhang, S., Gilbert, I., Nisoli, C., Chern, G. W., Erickson, M. J., O&’Brien, L., ... & Schiffer, P. (2013). Crystallites of magnetic charges in artificial spin ice. Nature,500(7464), 553-557.
[6] Gilbert, I., Chern, G. W., Zhang, S., O&’Brien, L., Fore, B., Nisoli, C., & Schiffer, P. (2014). Emergent ice rule and magnetic charge screening from vertex frustration in artificial spin ice. Nature Physics, 10(9), 670-675.

We consider the Kondo-lattice model on the kagome lattice and study its weak-coupling instabilities at band filling fractions for which the Fermi surface has singularities. These singularites include Dirac points, quadratic Fermi points in contact with a flat band, and Van Hove saddle points. By combining a controlled analytical approach with large-scale numerical simulations, we demonstrate that the weak-coupling instabilities of the Kondo-lattice model lead to exotic magnetic orderings. In particular, some of these magnetic orderings produce a spontaneous quantum anomalous Hall state.

Low-dimensional nanocarbon materials are tiny platforms on which beautiful interplay between geometric curvature and physical properties can be appreciated. One typical example of those materials is an axially corrugated nanocarbon cylinder, so-called peanut-shaped fullerene polymers. It was experimentally found that the axial corrugation induces drastic changes in electronic and optical properties of the materials, distinct from the case of straight, non-corrugated cylinders. Details of the experimental conditions as well as theoretical interpretation of the results are the main topics in the first half of my talk. In the second half, I will focus on the physics of carbon nanocoils, spiral-shaped cylindrical nanocarbons that hold promise for realizing novel applications in nanoelectronics, nanomechanics, and NEMS. I will talk about the latest measurements on mechanical and electric properties of carbon nanocoils with tens or hundreds nanometers in diameter, followed by a short review on the theoretical attempts to understand quantum mechanics of single-walled nanocoils.

Abstract not available.

Fundamental understanding of topology-property relationships in complex, hierarchical carbon nanomaterials is crucial for the discovery and development of new functionalities in these types of materials. However, quantifying their topological/geometrical characteristics across orders-of-magnitude differences in length scale is challenging. In this work, we use nondestructive x-ray scattering to quantitatively map the size, curvature, and order in hierarchically self-organized carbon nanotubes (CNT) from the atomic, to nano/meso, to micrometer scale. CNTs serve as a model nanocarbon for our study having a genus that ranges from g = 0 (closed single-wall), to g = 1 (open single-wall), to more complex geometries (double- and multi-wall).
Using a suite of novel x-ray beamlines at the Advanced Light Source (ALS) and Linac Coherent Light Source (LCLS), we quantitatively map structural characteristics in aligned CNT forests both spatially within the CNT material and across several length scales (~10⁰-10⁴Å) - ranging from the sp²-hybridized (graphitic) carbon to large micron-scale corrugations in the forest structure. Notably, we recently demonstrated simultaneous small-/wide-angle X-ray scattering (SAXS/WAXS) spatial mapping of CNT forests with a 1.5-µm beamspot for unprecedented spatial resolution, which enables simultaneous capture of atomic/nanoscales.
Using these novel capabilities, we find distinct hierarchical relationships between different length scales, such as how atomic spacing of graphitic shells in the CNT wall is perturbed by the presence of severe curvature in small-diameter CNTs. We also draw quantitative relationships between order at different length scales - e.g., corrugations in the CNT walls at the atomic scale or tortuosity of the CNT at the mesoscale - and we reveal a cascading decrease in order as characteristic feature size gets smaller. We use both analytical and phenomenological modeling to extract quantitative information from our x-ray experiments in order to unify for the first time the multiscale topological picture in nanocarbons.
Finally, the development of these advanced metrology tools is aimed toward understanding and connecting multiscale topology in CNT materials with their functional characteristics, such as those pertaining to fluid flow. For example, curvature (nanoconfinement) from the CNT or topological defects in the graphitic lattice can be exploited to significantly influence/control mass transport through the inside of CNT nanochannels.
This work is supported by the Defense Threat Reduction Agency (DTRA) D[MS]2 project under Contract No. BA12PHM123 and was performed under the auspices of U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344.

The exciting properties of carbon nanotubes (CNTs) and related one-dimensional systems, e.g. nanowires and nanofibers, encourage their use in multifunctional material architectures. However, the CNT morphology strongly influences their behavior, where non-idealities can lead to properties that are altered by orders of magnitude. Previous attempts to quantify and simulate the CNT morphology modeled the local curvature of the CNTs, commonly known as their waviness, using sinusoidal or helical functional forms. But such deterministic descriptions of the CNT waviness cannot adequately reproduce the stochastic nature of the CNT morphology in three dimensions. Thus a statistical description of the CNT waviness is required instead. Using a theoretical framework in conjunction with newly available information about the CNT morphology in three dimensions acquired via quantitative electron tomography, we model a CNT array comprised of 10⁵ CNTs with representative three-dimensional morphologies. The model CNT array is used to extract key quantities that are required to accurately simulate the morphology of CNTs in three-dimensions. Using the more representative stochastic description of waviness presented here, the morphology of one-dimensional nanoscale system can be simulated in three-dimensions, and further work that could enable more accurate property prediction is proposed in the context of experimental observations.

Understanding deformations of macroscopic thin plates and shells has a long and rich history, culminating with the Foeppl-von Karman equations in 1904, characterized by a dimensionless coupling constant (the "Foeppl-von Karman number") that can easily reach vK = 10⁷ in an ordinary sheet of writing paper. However, thermal fluctuations in thin elastic membranes fundamentally alter the long wavelength physics, as exemplified by experiments from the McEuen group at Cornell that twist and bend individual atomically-thin free-standing graphene sheets (with vK = 10¹³!) We review here recent results for the bending and pulling of thermalized graphene ribbons, and then move on to discuss thin amorphous spherical shells with a uniform nonzero Gaussian curvature, accessible for example with soft matter experiments on diblock copolymers. This curvature couples the in-plane stretching modes with the out-of-plane undulation modes, giving rise to qualitative differences in the fluctuations of thermal spherical shells compared to flat membranes. Interesting effects arise because a shell can support a pressure difference between its interior and exterior. Thermal corrections to the predictions of classical shell theory for microscale shells diverge as the shell radius tends to infinity.

Topological defects are critical to the structure and thermodyanics of condensed matter systems. For example, when incorporated into crystalline, 2D membranes like graphene, 5- and 7-fold disclinations drive buckling to conical- and saddle-like geometries, respectively. Motivated by a recently uncovered mapping between the metric properties of curved membranes and the geometry of inter-filament spacing, we explore influence of topological defects in the cross-section alpacking of multi-filament bundles and fibers on their equilibrium shapes. Cohesive assemblies of long, flexible nanotubes and macromolecular filaments are a common structural motifs in a diverse range of biological and synthetic materials, and understanding the relationship between inter-filament packing and the emergent shapes of the assemblies remains an outstanding challenge.
To uncover the filamentous analogs to the conical and saddles shapes found in defective membranes, we investigate the interplay between defects in the cross section of a bundle and its equilibrium shape using a combination of continuum elasticity theory and numerical simulation of cohesive bundles with a fixed packing topology. Focusing primarily on the shape instability of bundles possessing isolated disclinations, we derive the critical geometric and mechanical parameters that regulate optimal shapes of defective bundles and predict a host of new equilibria structures, some of which are without direct parallel to the analogous membrane, including torsional wrinkling, splay undulating, and helical winding. In planar crystals, isolated 5- and 7-fold disclinations are degenerate, while the 3D buckling of flexible 2D crystalline membranes weakly breaks that degeneracy. Signficantly, we find that response of filament bundle shape to 5- and 7-fold disclinations is dramatically asymmetric: 5-fold defects destabilize straight bundles of any stiffness and radius, whereas 7-fold defects drive periodic buckling only for sufficiently thick bundles. Understanding the complex relationship between bundle shapes and elementary lattice defects (e.g. disclinations) sets the stage for establishing the broader design principles needed to control 3D configurations of bundles through the 2D patterning of the filament position in cross section.

Tubular crystals are 2D lattices wrapped into cylinders and are seen in a variety of biological and materials science contexts. In the presence of external forces, one low-energy way of allowing the tube to deform is the unbinding and separation of dislocation pairs. As they travel through the system, these topological defects propagate stepwise changes in the topology of the tubular crystal, which in turn alter both the tube's geometry and its mechanical properties. Through theory and simulation, we seek to understand the resulting dynamical interplay of helicity, tube radius, and mechanical response.

Data acquired from synchrotron-based X-ray computed tomography provide complete descriptions of the stochastic positions of each fiber in large bundles within composite samples. The data can be accumulated for distances along the nominal fiber direction that are long enough to reveal meandering or misalignment. Data are analyzed for a single fiber bundle consolidated as a mini-composite specimen and a block of fibers embedded within a single ply in a tape laminate specimen. The fibers in these materials differ markedly in their departure from alignment and the patterns formed by fiber deviations. The tape laminate specimen exhibits evidence of fibers that have slipped laterally through the bundle in narrow shear bands, which may be a mechanism of bundle deformation under transverse compression and shear. This pattern is absent in the single-tow specimen, which was not subject to transverse loads in processing. We propose a combination of topological and Euclidean metrics to quantify these and other stochastic bundle characteristics. Topological metrics are based on the neighbor map of fibers, which is constructed on cross-sections of the bundle by Delaunay triangulation (or Voronoi tessellation). Variations of the neighbor map along the fiber direction describe fiber meandering, twist, etc. Euclidean metrics include factors such as local fiber density and fiber orientation. The metrics distinguish bundle types, enable quantification of the effects of the manufacturing history of bundles, and provide target statistics to be matched by virtual specimens that might be generated for use in fiber-scale virtual tests.

The striking appearance of many animals is not obtained by pigments but rather by structuring transparent materials on the order of a few hundreds of nanometers [1]. The biological world has optimized such photonic structures in fish scales, bird feathers and insect wings, since the Cambrian explosion over 500 million years ago, with which an enormous diversification of insect coloration as well as visual systems started. By changing the dimensions of such nanostructures or the amount of order, quasi-order or disorder in these systems, these diverse nanostructures allow manipulation of incident electromagnetic radiation so to achieve colors that extend over the entire visible wavelength range and that are employed in courtship, to find prey or to escape predation risk by camouflage.
A wealth of photonic systems with the most complicated and thoughtful structures can be found in animals, i.e. butterfly wing scales [2], beetle and weevil scales or bird feathers [3,4], ranging from boomerang-like shapes in bird-of-paradise feathers [5] and complex amorphous, quasi-ordered nanostructures in parrot and king fisher feathers [6,7] to completely disordered chitin networks in beetle scales [8]. Surprisingly, most pigmentary colored feathers carry disordered keratin networks.
In this talk I will present the optical properties of different topologies of biological photonics structures, from ordered to quasi-ordered to disordered ones. I will also discuss how such structures and topologies can be optically understood and serve as an inspiration to find novel optical materials, e.g. optical metamaterials [2].
References
[1] Kinoshita, S. (2008) Structural colors in the realm of nature. World Scientific.
[2] Dolan, J.A, et al. (2015) Adv. Opt. Mater.3, 12-32.
[3] Saranathan, V., et al. (2012) J. R. Soc. Interface 9, 2563-2580.
[4] Saranathan, V., et al. (2015) Nano. Lett.15, 3735-3742.
[5] Wilts, B. D., et al. (2014) Proc. Natl. Acad. Sci. U. S. A. 111, 4363-4368.
[6] Stavenga, D. G., et al. (2011) J. Exp. Biol. 214, 3960-3967.
[7] Tinbergen, J., et al. (2013) J. Exp. Biol. 216, 4358-4364.
[8] Burresi, M., et al. (2014) Sci. Rep., 4, 6075.

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 space-filling, symmetric and entangled structures composed of multiple networks or filaments, which challenge current characterisation techniques particularly from the perspective of entanglement. In this talk, I will discuss the construction and characterisation of these complex entangled structures alongside applications from star terpolymer self-assembly to skin swelling.

The promise of improved toughness and shock resistance has driven the development of materials with topologically interlocked substructures. Here we present itacolumite, a naturally occurring material of this type, which has random microstructure and exhibits mechanical behavior which is unusual for such a brittle material. These properties appear to emerge from the interlocking of quartz grains with random geometry and sizes. Mathematical modeling of such random and densely fractured structures indicates that a behavior similar to that observed experimentally occurs only in a narrow range of crack densities larger than the crack percolation threshold. We conclude that interlocking leads to a tensile stiffness percolation threshold larger than the percolation threshold associated with transport. We show that this result, which is expected in 3D, also holds in 2D structures.

Bicontinuous cubic structures, formed by biological amphiphiles such as lipids, are the subject of widespread fundamental research in both the physical and life sciences. These structures are facile to produce, and their phase behaviour is well known. As such, lipid systems represent attractive candidates for the fundamental understanding of these topologies and their in particular, their interconversion. The interconversion of bicontinuous cubic materials and their underlying mathematical surfaces have long been studied using theoretical methods. It is known that the underlying triply periodic minimal surfaces that lie at the centre of the bilayer in, for example, the double diamond (Q_II^D) and gyroid (Q_II^G) phases are topologically related by a mathematical process known as the Bonnet transformation, which preserves the Gaussian curvature, and angles, distances, and areas on the surface.¹ Although this mechanism may in theory interconvert the TPMSs, this mechanism is unlikely to occur in reality, as it require regions of the bilayer to pass through one another.²
In order to experimentally demonstrate a pathway that allows for the interconversion of the Q_II^D and Q_II^G phases, we show that studies of phase transformations using small angle x-ray scattering, between two macroscopically oriented phases allow the determination of their relative orientation, which provides evidence for a specific mechanism of the transformation. A macroscopically oriented³ Q_II^D inverse bicontinuous cubic phase of the lipid glycerol monooleate is deposited under flow as a hydrated film⁴, leading to a uniaxially oriented sample with a <110> axis parallel to the symmetry axis of the sample. Replacement of the water in the sample with a solution of poly(ethylene glycol), which draws water out of the Q_II^D sample by osmotic stress. This converts the Q_II^D phase into a Q_II^G phase with two coexisting orientations, with the <100> and <111> axes parallel to the symmetry axis. This process is reversible and the initial orientation of Q_II^D phase can be recovered. The epitaxial relationship between the two oriented mesophases is shown to be consistent with topology preserving geometric pathways that have previously been hypothesized for the transformation.⁵
(1) Schroeder-Turk, G. E.; Fogden, A.; Hyde, S. T. Euro. Phys. J. B2006, 54, 509.
(2) Seddon, J. M.; Templer, R. H. Phil. Trans. Roy. Soc. A: 1993, 344, 377.
(3) Seddon, A. M.; Lotze, G.; Plivelic, T. s. S.; Squires, A. M. J. Am. Chem. Soc.2011, 133, 13860.
(4) Squires, A. M.; Hallett, J. E.; Beddoes, C. M.; Plivelic, T. S.; Seddon, A. M. Langmuir2013, 29, 1726.
(5) Squires, A. M.; Templer, R. H.; Seddon, J. M.; Woenkhaus, J.; Winter, R.; Narayanan, T.; Finet, S. Phys. Rev. E2005, 72, 011502.

Direct writing by a laser beam focused by a high numerical-aperture objective has become a powerful tool toward the development of ultimate three-dimensional (3D) photonic devices. Biomimetic photonics is inspired by nature&’s ability to self-assemble complex nanostructured materials with superior properties to that of conventional materials. Here we demonstrate the fabrication of a novel class of 3D gyroid photonic microstructures inspired by a recent finding in butterfly wing-scales and show that these 3D gyroid structures have the ability to redirect circularly polarized light as a chiral beamsplitter. Because of the increasing demand for realising biomimetic nano-geometries, the diffraction-limited resolution associated with the two-photon-assisted fabrication should be overcome. We will show how this diffraction limit can be broken in the development of superresolution photoinduction-inhibited nanolithography (SPIN). These nano-engineered gyroid structures can support the topologic state of light if the parity symmetry is broken. More intriguingly, we will present the first experimental evidence that 8 srs single gyroid nets possess the strong optical activity without circular dichroism due to their four-fold rotational symmetry.

We propose a construction of a cholesteric pitch axis for an arbitrary nematic director field as an eigenvalue problem. Our definition leads to a Frenet-Serret description of an orthonormal triad determined by this axis, the director, and the mutually perpendicular direction. With this tool, we are able to compare defect structures in cholesterics, biaxial nematics, and smectics. Though they all have similar ground state manifolds, the defect structures are different and cannot, in general, be translated from one phase to the other.

We present a new model, numerical solutions, and
experiments of electric field driven flows in nematic suspensions.
Nematic order in the fluid allows novel ways of inducing spatial charge
separation that result in unusual response to imposed fields. This
response can be used, for example, for flow control and stirring at very
small scales. The novel properties of these suspensions generally result
from the existence, motion, and effect on transport of topological defects in
the nematic fluid that originate from topological constraint introduced by the
suspended particles. We have built a thin cell in which a nematic orientation is
imposed via polarizing nano-slits fabricated on the surface on the cell.
In order to test the model, we describe our results for periodic patterns
of nematic orientation as well as ensembles of two and three disclinations
(with zero net topological charge). Under an applied AC electric field, the
anisotropic conductivity of the nematic together with the imposed pattern
of director distortion create an oscillatory and nonuniform charge density
within the cell and streaming flows. A finite element method is used to find
the corresponding model solutions which we compare with the experiments. We
finally discuss how Liquid Crystal Enabled Osmosis can be used to manipulate
particles in ways that are not possible in conventional electro-osmosis in
isotropic systems.

Supramolecular self-assembly provides a powerful and low-cost way to achieve nanoscale materials synthesis and pattern formation for applications in advanced lithography, nanophotonics, photovoltaics, therapeutics, and related areas. The chemical and structural morphology of these materials ranges in size from 5 - 100 nm, which is not easily interrogated in real space via existing instruments based on optics (due to insufficient spatial resolution arising from the diffraction limit) or electrons (due to limited contrast between materials and the possibility of sample damage).
Infrared Photo-induced Force Microscopy (IR PiFM) is based on an atomic force microscopy (AFM) platform that is coupled to a widely tunable mid-IR laser. PiFM measures the dipole induced at or near the surface of a sample by an excitation light source by detecting the dipole-dipole force that exists between the induced dipole in the sample and the mirror image dipole in the metallic AFM tip. This interaction is strongly affected by the optical absorption spectrum of the sample, thereby providing a significant spectral contrast mechanism which can be used to differentiate between chemical species. Due to its AFM heritage, PiFM acquires both the topography and spectral images concurrently and naturally provides information on the relationship between local chemistry and topology. Due to the steep dipole-dipole force dependence on the tip-sample gap distance, PiFM spectral images surpass topographical spatial resolution, showcasing sub 10 nm spatial resolution despite the increase tip radius of metal coated tips.
PiFM studies on various self-assembled block copolymer systems will be presented. The results consist of PiFM spectral images associated with several absorption bands of different blocks along with broad spectra associated with nano-spots on sample surfaces. Images of fingerprint patterns and parallel lamellae (prepared via directed self-assembly) for both poly (styrene-b-methyl methacrylate) and poly (styrene-b-2-vinylpyridine) showed clear spectral contrast between the two blocks of each material system. For poly (styrene-b-2-vinylpyridine), PiFM contrast between blocks was far greater than is generally available by scanning electron microscopy without staining. By enabling imaging at the nm-scale with chemical specificity, PiFM provides a powerful new analytical method for deepening our understanding of nanomaterials and facilitating technological applications of such materials.

Periodic surface nano-wrinkling is found throughout biological liquid crystalline materials, such as collagen films, spider silk gland ducts, exoskeleton of beetles, and flower petals. These surface ultrastructures are responsible for structural colors observed in some beetles and plants that can dynamically respond to external conditions, such as humidity and temperature. In this paper, the formation of the surface undulations is investigated through the interaction of anisotropic interfacial tension, swelling through hydration, and capillarity at free surfaces. Focusing on the cellulosic cholesteric liquid crystals (CCLCs) material model, the generalized shape equation for anisotropic interfaces using the Cahn-Hoffman capillarity vector and the well-known Rapini-Papoular anchoring energy is applied to analyze periodic nano-wrinkling in plant-based plywood free surfaces with water-induced pitch gradients. Scaling is used to derive the explicit relations between the undulations&’ amplitude expressed as a function of the anchoring strength and the spatially varying pitch. The optical responses of the periodic nano-structured surfaces are studied through finite difference time domain simulations (FDTD) indicating that CCLCs surfaces with spatially varying pitch reflect light in a wavelength higher than that of a CCLCs surface with constant pitch. This structural color change is controlled by the pitch gradient through hydration. All these findings provide a foundation to understand structural color phenomena in Nature and for the design of optical sensor devices.

While there are four commonly observed states of matter (solid crystal, liquid, gas, and plasma), we have known for some time now that there exist many other forms of matter. Disordered hyperuniform many-particle systems [1] can be regarded to be new states of disordered matter in that they behave more like crystals or quasicrystals in the manner in which they suppress large-scale density fluctuations, and yet are also like liquids and glasses because they are statistically isotropic structures with no Bragg peaks. Thus, disordered hyperuniform systems can be regarded to possess a "hidden order" that is not apparent on short length scales, while being structurally rotationally invariant. I will describe a variety of different examples of such disordered states of matter that arise in physics, materials science and biology. Among other results, I will describe classical ground states that are disordered, hyperuniform and highly degenerate over a wide range of densities up to some critical density, below which the system undergoes a phase transition to ordered states [2]. Disordered hyperuniform systems appear to be endowed with novel physical properties, including complete photonic band gaps comparable in size to those in photonic crystals [3] and improved electronic band-gap properties. Moreover, we have recently shown that photoreceptor cell patterns (responsible for detecting light) in avian retina have evolved to be disordered and hyperuniform [4].
1. S. Torquato and F. H. Stillinger, "Local Density Fluctuations,
Hyperuniform Systems, and Order Metrics," Phys. Rev. E, 68, 041113
(2003); 2. S. Torquato, G. Zhang, and F. H. Stillinger, Ensemble Theory for
Stealthy Hyperuniform Disordered Ground States, Phys. Rev. X,
5,021020 (2015); 3. M. Florescu, S. Torquato and P. J. Steinhardt, "Designer Disordered
Materials with Large, Complete Photonic Band Gaps," Proc. Nat. Acad.
Sci., 106, 20658 (2009); 4. Y. Jiao, T. Lau, H. Haztzikirou, M. Meyer-Hermann, J. C. Corbo, and
S. Torquato, Avian Photoreceptor Patterns Represent a Disordered
Hyperuniform Solution to a Multiscale Packing Problem, Phys. Rev.
E, 89, 022721 (2014).

Magnetic skyrmions are topologically stable whirlpool-like spin textures that offer great promise as information carriers for future ultra-dense memory and logic devices. To enable such applications, particular attention has been focused on the skyrmions in highly confined geometry such as nanodisks or one dimensional nanostripes or wires. Here, we report the visualization of the skyrmion chains in FeGe nanostripes and skyrmion clusters in nanodisks by high resolution Lorentz TEM, and the electrical probing of individual skyrmions in MnSi nanowires when the wire diameter is comparable to that of a skyrmion. Specifically, we found that the highly stable skyrmion chain originated from the termination of the spin helix at the edges of the nanostripes under the action of applied field, and the field-driven transition of skyrmion cluster states in nanodisks. Complete T-H or T-W (width or diameter) phase diagram of skyrmions were outlined in nanostripes or disks. These findings demonstrate that the geometry defects can be used to control the formation of topologically nontrivial magnetic objects. Finally, we present the first electrical probing of such magnetic field-driven skyrmion cluster (SC) states in ultra-narrow single-crystal MnSi nanowires (NWs) with diameters (40 - 60 nm). In contrast to the skyrmion lattice in bulk samples, the creation or deletion of an individual skyrmion in the cluster states leads to quantized jumps in magnetoresistance (MR), which is supported by the Monte Carlo simulations.

The endoplasmic reticulum (ER) has long been considered an exceedingly important and complex intracellular organelle in eukaryotes. It is a membrane structure, part folded lamellae, part tubular network, that both envelopes the nucleus and threads its way outward, all the way to the cell&’s periphery. Despite the elegant mechanics of bilayer membranes offered by the work of Helfrich and Canham, as far as the ER is concerned, theory has mostly sat on the sidelines. However, refined imaging of the ER has recently revealed beautiful and subtle geometrical forms - suggestive of simple geometries, from the mathematical point of view - which some have called a “parking garage for ribosomes.” I&’ll review the discovery and physics of Terasaki ramps and discuss their relation to cell-biological questions, such as ER and nuclear-membrane re-organization during mitosis, and the connection between ER structures and functions.

Magnetic skyrmion is a topological spin texture observed in several helimagnets. Inspired by the novel physics tracing back to its nontrivial topology and promising applications in next generation memory device and ultra-dense data storage, a full understanding of these materials is an urgent subject. We have analyzed the symmetry of a typical helimagnet family, the B20 compounds, and developed a systematic way of writing effective spin Hamiltonian for arbitrary magnets, based on which a full classification and search strategy of helimagnets is derived. Following this guideline, we discovered a new helimagnet family, the molybdenum nitride, harboring skyrmions, which is confirmed by real space magnetic imaging. The magnetic origin of this new material is revealed by first principle calculation, and its interplay between strong correlation will be discussed.

Skyrmion crystal was discovered in chiral magnets without inversion symmetry recently. The spin anisotropy is very important for skyrmion stabilization. It was shown that an easy-axis anisotropy along the magnetic field direction is helpful to stabilize skyrmion phase. We study the equilibrium phase diagram of ultrathin chiral magnets with easy-plane anisotropy A. The vast triangular skyrmion lattice phase that is stabilized by an external magnetic field evolves continuously as a function of increasing A into a regime in which nearest-neighbor skyrmions start overlapping with each other. This overlap leads to a continuous reduction of the skyrmion number from its quantized value Q = 1 and to the emergence of antivortices at the center of the triangles formed by nearest-neighbor skyrmions. The antivortices also carry a small “skyrmion number” that grows as a function of increasing A. The system undergoes a first order phase transition into a square vortex-antivortex lattice at a critical value of A. Finally, a canted ferromagnetic state becomes stable through another first order transition for a large enough anisotropy A. Interestingly enough, this first order transition is accompanied by metastable meron solutions.

I will discuss how topological phases arise in quantum matter through spin-orbit coupling effects in the presence of protections provided by time-reversal, crystalline and particle-hole symmetries, and highlight our recent work aimed at predicting new classes of topological insulators (TIs), topological crystalline insulators, Weyl semi-metals, and quantum spin Hall insulators. [1-8] Surfaces of three-dimensional (3D) topological materials and edges of two-dimensional (2D) topological materials support novel electronic states. For example, the surface of a 3D TI supports gapless or metallic states, which are robust against disorder and non-magnetic impurities, and in which the directions of momentum and spin are locked with each other. Similarly, in 2D TIs, also called quantum spin Hall insulators, the 1D topological edge states are not allowed to scatter since the only available backscattering channel is forbidden by constraints of time-reversal symmetry. The special symmetry protected electronic states in topological materials hold the exciting promise of providing revolutionary new platforms for exploring fundamental science questions, including novel spin textures and exotic superconductors, and for the realization of multifunctional topological devices for thermoelectric, spintronics, information processing and other applications. Work supported by the U. S. Department of Energy.
[1] I. Zeljkovic et al., Nature Materials 14, 318 (2015).
[2] J. He et al., Nature Materials 14, 577 (2015).
[3] M. Neupane et al., Physical Review Letters 114, 016403 (2015).
[4] S.-Y. Xu, et al., Nature Communications 6, 6870 (2015).
[5] F.-C. Chuang, et al., Nano Letters 14, 2505 (2014).
[6] Yi Zhang et al., Nature Nanotechnology 9, 111 (2014).
[7] Y. He et al., Science 9, 608 (2014).
[8] T. Saari et al., Applied Physics Letters 104, 173104 (2014).

We will review recent developments in topological band theory, including the theory of topological insulators, topological crystalline insulators, as well as topological semimetals that feature point and line nodes in their electronic structure that are protected by symmorphic and/or non-symmorphic space group symmetries.

Band insulators appear in crystalline systems only when the filling - the number of electrons per unit cell per spin polarization - is an integer. At fractional filling, an insulating phase which preserves all symmetries is a Mott insulator, i.e. it is either gapless, or, if gapped, exhibits fractionalized excitations and topological order. I will discuss work done in collaboration with S. Parameswaran, A. Turner, and A. Vishwanath in which we raise the inverse question: at an integer filling, is a band insulator always possible? We show that lattice symmetries necessarily forbid a band insulator at certain integer fillings if the crystalline space group is non-symmorphic, a property shared by a majority of three-dimensional crystal structures. In these cases, one may infer the existence of topological order if the ground state is gapped and fully symmetric. This is demonstrated using a non-perturbative flux threading argument which generalizes the Lieb-Shultz-Mattis-Oshikawa-Hastings theorem, and has applications to quantum spin systems, bosonic insulators, and electronic band structures in the absence of spin-orbit interaction. Reference: S. Parameswaran et al., Nature Physics 9, 299 (2013).

Topological superconductors are predicted to host the long-sought-after Majorana fermions, which can form the basis for fault-tolerant quantum computing applications. Tin telluride (SnTe) is a recently discovered topological crystalline insulator whose surface states are protected by the crystal symmetry instead of time reversal symmetry. Doping SnTe with indium (In) leads to superconductivity, making In-doped SnTe a strong candidate as a topological superconductor. SnTe and In-doped SnTe nanostructures offer well-defined nanoscale morphology and high surface-to-volume ratios to enhance surface effects.
Here, we show In-doped SnTe nanoplates, In_xSn_1-xTe with x ranging from 0 to 0.1, and show that they superconduct [1]. More importantly, we observe transport signatures from the surface state of In-doped SnTe despite the high bulk carrier density. This is manifested by two-dimensional linear magnetoresistance in high magnetic fields, which is independent of temperature up to 10 K [1]. This was made possible by deliberately decreasing the bulk carrier mobility by In doping. The bulk carrier mobility decreases by more than an order of magnitude compared to SnTe nanoplates we studied previously [2]. Aging experiments show that the linear magnetoresistance is sensitive to ambient conditions while bulk transport properties such as the carrier density and the mobiliy remain unchanged, further supporting its surface origin.
Our finding suggests that, at the In doping level enough to induce superconductivity, the surface state of SnTe remains intact, strongly supporting In-doped SnTe as a candidate for a topological superconductor.
[1] J. Shen, Y. Xie, J. J. Cha, Revealing surface states in In-doped SnTe nanoplates with low bulk mobility, Nano Lett. 15, 3827-3832 (2015)
[2] J. Shen, Y. Jung, A. S. Disa, F. J. Walker, C. H. Ahn, J. J. Cha, Synthesis of SnTe nanoplates with {100} and {111} surfaces, Nano Lett. 14, 4183-4188 (2014)

The discovery of topological order without breaking time-reversal symmetry, such as that in Quantum Spin Hall (QSH) effect and Topological Insulators, is one of the most groundbreaking advancements of recent years in condensed matters physics. The approach to topological order without breaking time-reversal symmetry is particularly important in elastics because no natural elastic materials show linear nonreciprocal response. Here we illustrate the first elastic-wave system emulating QSH effect and demonstrate existence of topologically protected elastic edge states. The system represents an elastic metamaterial-based phononic crystal. In this crystal, we achieved degenerate linear dispersions for two sets of modes, classified by one of the system's symmetries. Then, by relaxing that symmetry by deliberately engineering a gauge field emulating a strong spin-orbit coupling of QSH, we observe opening a complete topological bandgap. Finally, the hallmark of the topological order, namely the presence of one-way chiral edge waves insensitive to nonmagnetic defects and disorders such as bends with arbitrary angles, is demonstrated in these elastic metacrystals. We illustrate the unique property of the elastic edge waves to flow around sharp corners without back-reflection or localization. We also show that the topologically protected edge modes retain their topological nature even if the crystal is submerged in viscous air or water causing radiative and dissipative losses.

Topological semi-metal (TSM) is a new type of quantum phases in condensed matter, which includes Dirac semi-metal (DSM) and Weyl semi-metal (WSM)
phases. The appearance of DSM phase requires additional crystal symmetry to generate Dirac points along some special directions. And the WSM phase
requires breaking of either time reversal or inversion symmetry to remove the spin degeneracy. In the present talk, I will summarize the TSM materials found
recently in our group by first principle methods. Besides the exotic physical properties of these TSMs, I will also introduce from the symmetry point of view
where and how to find these materials.

A topological quantum state of matter is characterized by a nontrivial topological structure of its Hilbert space. Intriguingly, a topological state is always accompanied by a peculiar gapless edge/surface state that characterizes the nature of the bulk state. In 3D topological insulators, a nontrivial Z₂ topology of the bulk state leads to the emergence of spin-momentum-locked Dirac fermions on the surface. Similarly, 3D topological superconductors are accompanied by surface Andreev bound states that often consist of Majorana fermions. In this talk, I will present our experimental efforts to address those exotic surface states in topological quantum materials. One may find comprehensive reviews on these materials in Refs. [1] and [2].
References
[1] Y. Ando, Topological Insulator Materials, J. Phys. Soc. Jpn. 81, 102001 (2013).
[2] Y. Ando and L. Fu, Topological Crystalline Insulators and Topological Superconductors: From Concepts to Materials, Annu. Rev. Condens. Mater Phys. 6, 361 (2015).

The geometric structure of single-particle energy bands in solids is fundamental for a wide range of many-body phenomena such as Quantum Hall insulators and topological insulators. In Graphene, many of the fascinating electronic properties are directly connected to the geometric nature of the two Dirac cones connecting the two lowest bands. The geometric structure of a single energy band is uniquely characterized by the distribution of Berry curvatures over the Brillouin zone. In real solids, Berry curvature can typically only be probed indirectly by observing e.g. transport properties. Here, we could in contrast realize an atomic interferometer to directly measure Berry flux in momentum space. This is in direct analogy to an Aharonov-Bohm interferometer that measures magnetic flux in real space. We demonstrate the interferometer for a graphene-type hexagonal optical lattice loaded with bosonic atoms and directly detect the topologically protected Berry flux associated with a Dirac point.
Extending this interferometry to beyond single bands, the resulting Berry phases generalize to matrix-valued Wilson loops, which give rise to even richer physics, including the potential for holonomic quantum computing. By combining several interferometric techniques with Bloch oscillations in the strong-gradient limit, we are able to fully characterize topological band structures and to determine the geometric tensor of Bloch bands. This generic technique enables the reconstruction of all topological invariants, including Chern numbers and Z2 invariants, and can in principle be applied to all band structures.

The quantum anomalous Hall (QAH) effect is a quantum Hall effect induced by spontaneous magnetization instead of an external magnetic field. It is the foundation for applications of dissipationless quantum Hall edge states in low-energy-consuming devices and for realization of other novel quantum phenomena such as chiral topological superconductivity and axion electrodynamics. The QAH phase can be realized a ferromagnetic topological insulator (TI) film as a result of magnetically induced gap at the Dirac surface states. With molecular beam epitaxy techniques, we have prepared thin films of Cr-doped (Bi,Sb)₂Te₃ TI with well-controlled composition, thickness and chemical potential and obtained ferromagnetic insulator phase in them. In such magnetic TI films, we have experimentally observed the quantization of the Hall resistance at h/e² at zero field, accompanied by a considerable drop in the longitudinal resistance, unambiguously demonstrating the occurrence of the QAH effect. The temperature, thickness and magnetic-doping-level dependence of the QAH effect have been systematically studied, which clarifies the roles of the band structure, electron localization and magnetic order in the effect and provides clues for obtaining high temperature QAH materials. These results pave the ways for further studies and practical applications of the QAH-related quantum phenomena.

Physical systems are frequently modeled as sets of points in space, each representing the position of an atom, molecule, or mesoscale particle. As many properties of such systems depend on the underlying ordering of their constituent particles, understanding that structure is a primary objective of condensed matter research. Although perfect crystals are fully described by a set of translation and basis vectors, real-world materials are never perfect, as thermal vibrations and defects introduce significant deviation from ideal order. Meanwhile, liquids and glasses present yet more complexity. A complete understanding of structure thus remains a central, open problem. Here we propose a unified mathematical framework, based on the topology of the Voronoi cell of a particle, for classifying local structure in ordered and disordered systems that is powerful and practical. We explain the underlying reason why this topological description of local structure is better suited for structural analysis than continuous descriptions. We demonstrate the connection of this approach to the behavior of physical systems and explore how crystalline structure is compromised at elevated temperatures. We also illustrate potential applications to identifying defects in plastically deformed polycrystals at high temperatures, automating analysis of complex structures, and characterizing general disordered systems.

This contribution is rooted in quantum chemistry and presents a novel approach to ultimately understand and predict the behaviour of condensed matter.
The language of dynamical systems (e.g. critical point, separatrix, basin, Poincaré-Hopf relationship,...) has been successfully applied to quantum mechanical density functions. When this function is the electron density then one recovers the so-called topological atom, which is a highly transferable building block of matter. A topological atom has a finite volume, a particular shape determined by its environment. Topological atoms do not overlap, leave no gaps between them and have well-defined quantum mechanical energies.
In this talk I will explain the main features of this novel approach towards the modelling of molecules and increasingly also molecular assemblies. I will show how the machine learning technique kriging can predict the properties of these topological atoms, in particular, their multipole moments and four types of fundamental energy contributions: the atomic self-energy, the Coulomb interaction, the exchange interaction and the (dynamic) correlation interaction. These energies correspond to stereo-electronic effects, charge transfer/polarity/ionicity, bond order/(hyper)conjugation and dispersion, respectively. Condensed matter can be re-interpreted and studied in terms of this minimal and rigorous energy partitioning scheme.

We discuss the classical and non-commutative C* geometry of wire systems which are the complement of triply periodic surfaces.
For the gyroid surface, these can be fabricated on a nanoscale, and the C* geometry models the electronic properties. In the presence of an ambient magnetic field, the relevant algebras become non-commutative and we classify the resulting algebras. The associated geometry in the commutative case is also very interesting. Here we have developed several other methods, such as re-gauging symmetries, the application of singularity theory and topological invariants.
We also compare the gyroid wire network to networks from other CMC surfaces. In this setting the gyroid geometry can be seen as the 3d generalization of graphene.
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3D topological insulators (TI) have surface states with Dirac fermions and spin-locked perpendicularly to momentum [1]. Magnetoresistance (MR) and surface conductivity measurements on such surface states yield useful information about the electronic behaviour including weak anti-localization, the non-trivial Berry phase, and in-plane spin polarization that are the hallmarks of TIs. In this work, using a warped Dirac Hamiltonian for Bi₂Se₃, we evaluate the non-topological component [2] of the Berry phase in presence of a time reversal symmetry (TRS) breaking perturbation [3] and the deviation from the standard value of π. The TRS breaking perturbation, due to localized ferromagnetic impurities on the surface gaps the Dirac cone at the #1043; point. In presence of such impurities, characterized by their scattering potential in a Fermi golden rule (FGR) calculation, we determine the increase in MR compared to a pristine TI. Analytic calculations show that the MR without the warping term in a gapped TI is roughly four-fold higher than in a zero-gap TI. The inclusion of warping parameter enhances the MR at points in k-space far away from the Dirac cone. Further, to accurately compute the surface conductivity for a zero-gap TI, we define a spin suppression factor cos²#1060;/2 (#1060;: scattering angle) that forbids back-scattering. The addition of this factor in the expression for scattering time in FGR yields a higher surface conductivity compared to the case where spin-protection is neglected. For a band gap open TI in a magnetic field, this factor is calculated by noting the angle between the spin-polarization vectors of the initial and final states. The scattering times are evaluated by summing over the density of states (DOS). The DOS in presence of the warping factor which is dominant at high k values is significantly reduced and lowers the scattering rate. Finally, we consider a thin TI where a scattered surface carrier tunnels through the bulk and occupies a state of opposite helicity. Scattering times are computed for a free-standing thin Bi₂Se₃ slab and one grown on a substrate which gives rise to a dipole moment. In conclusion, it is found that perturbations such as band gaps and dipole moments are important for scattering events close to the #1043; point. At points far away, the warping factor swamps these contributions. The mean free path for surface carriers is evaluated by inserting the relaxation time in a Boltzmann equation. The mean free path lambda;_fp in TIs, compared to a standard semiconductor such as GaAs, is found to be larger. A larger lambda;_fp means TIs could be employed as ballistic interconnects in a fast logic environment. Finally, we correlate the deviation from the Berry phase of π to surface conductivity; a greater deviation is reflected in decreased conductivity.
[1] M. Z. Hasan and C. L. Kane, Rev of Mod Phys, 82, 3045, (2010)
[2] P. Sengupta et al., arXiv: 1503.06224, (2015).
[3] P. Sengupta et al., Semiconductor Science and Tech, 30, 045004, (2015).

We use topology optimization to design microstructures for a range of properties, including elastic moduli and diffusivity. To avoid "weak" directions in the structures we have contrained the materials to have isotropic elastic and diffusive properties. We have have built the structures from titanium using selective laser melting, a 3D additive manufacturing technique. The materials have very high stiffness and strength compared to other lightweight titanium materials. We have found excellent agreement between the theoretical predictions and measured modulus. We discuss how the designs can be incorporated into a hierarchical design framework for designing macroscopic parts - such as bone implants - using a variable density interpolation scheme. We also show how finite element modelling of stress and strain throuhout the stucture can in some cases be used to estimate the strength.

Systems displaying wave chaos in three dimensions typically consist of dense, complicated tangles of vortex filaments (nodal lines, phase singularities). These tangles share many morphological features and statistical similarities with other physical systems of lines, such as polymer melts [1], superfluid turbulence, optical vortices in laser speckle patterns [2] or models of cosmic strings (although the dynamics may be different in these cases).
Vortex filaments in wave chaos are difficult to describe analytically using wave methods, even in the simplest cases where the wave field solves the time-dependent Schrödinger equation; this case would describe, for instance, the vortex lines of probability current for a quantum particle in a 3D quantum dot. Despite the linear nature of such eigenfunctions, the local geometry of their vortices leads to random conformations for which certain properties appear universal on large scales, such as appearing like random walks on the large scale [3].
Numerical simulations of these systems are analyzed using a variety of topological methods which identify properties of the tangled space curves, primarily identifying their knotting and linking properties. Different topological classes of knot type are discriminated by routines which combine geometric relaxation of the curves [3], together with a combination of topological knot invariants including standard polynomial invariants and invariants from 3-manifold geometry. These topological discrimination techniques are very general, and can be applied to other large topological data sets, including (for instance) conformations of biological macromolecules.
We describe the topology of the vortex tangles [4] in three model systems with degenerate energy eigenfunctions: eigenfunctions of the laplacian in the 3-torus (i.e. modes of cubes with periodic boundary conditions), hyperspherical harmonics in the 3-sphere, and the 3D harmonic oscillator. Random combinations of degenerate eigenfunctions have tangled nodal vortices, whose knotting probability, we find, increases rapidly with energy. Knots of different complexity are found to arise in different scaling regimes of curve length to system size (determined by energy), which has both similarities and differences from knotting studied in polymers. The knot conformation which occur at low energies are strongly limited by the parity properties of the eigenfunctions.
[1] E Orlandini & S G Whittington, Rev Mod Phys79, 611-42 (2007)
[2] K O&’Holleran, M R Dennis & M J Padgett, Phys Rev Lett102, 143902 (2009)
[3] A J Taylor & M R Dennis, J Phys A47, 465101 (2014)
[4] M V Berry & M R Dennis, Proc R Soc A456, 2059-79 (2000)

Fermions are closely related with topology. The Dirac string trick is a classical example of this phenomenon.The orientation-entanglement relationship of a belt attached to a macroscopic object models the fact that a process of 2Pi-rotation will change the wave-function of an electron by a sign, while a 4PI rotation leaves the wave-function unchanged. A deeper relationship occurs in understanding the annihilation/creation algebra of a fermion generated by F and F* with FF=0=F*F* and FF* + F*F = 1. One can write the fermion operators in terms of Majorana Fermion operators a and b with aa = bb = 1, ab + ba = 0 and a* = a, b* = b as follows: F = (a + bi)/sqrt(2) and F* = (a-bi)/sqrt(2). This suggests that electrons can be seen as composites of Majorana fermions. There is evidence for such composition in the behavior of shared electrons, and in the behaviour of rows of electrons in nano-wires. Remarkably, there are natural representations of the Artin braid group based on the behavior of Majorana Fermions. We shall discuss these braid group representations and how they can be potentially used for topological quantum computing. We will discuss also the Fibonacci model for topological quantum computing in relation to the Fractional Quantum Hall Effect and how the Fibonacci model can be based on the Kauffman bracket state sum for the Jones polynomial. Time permitting, we will explain a new way to think about the appearance of Clifford algebra in the Dirac equation, fermions and their topological properties.