Size effects and dislocation patterning in two-dimensional bending

We perform atomistic Monte Carlo simulations of bending a Lennard-Jones single crystal in two dimensions. Dislocations nucleate only at the free surface as there are no sources in the interior of the sample. When dislocations reach sufficient density, they spontaneously coalesce to nucleate grain boundaries, and the resulting microstructure depends strongly on the initial crystal orientation of the sample. In initial yield, we find a reverse size effect, in which larger samples show a higher scaled bending moment than smaller samples for a given strain and strain rate. This effect is associated with source-limited plasticity and high strain rate relative to dislocation mobility, and the size effect in initial yield disappears when we scale the data to account for strain rate effects. Once dislocations coalesce to form grain boundaries, the size effect reverses and we find that smaller crystals support a higher scaled bending moment than larger crystals. This finding is in qualitative agreement with experimental results. Finally, we observe an instability at the compressed crystal surface that suggests a novel mechanism for the formation of a hillock structure. The hillock is formed when a high angle grain boundary, after absorbing additional dislocations, becomes unstable and folds to form a new crystal grain that protrudes from the free surface.Comment: 15 pages, 8 figure

Cooperative Chiral Order in Copolymers of Chiral and Achiral Units

Polyisocyanates can be synthesized with chiral and achiral pendant groups distributed randomly along the chains. The overall chiral order, measured by optical activity, is strongly cooperative and depends sensitively on the concentration of chiral pendant groups. To explain this cooperative chiral order theoretically, we map the random copolymer onto the one-dimensional random-field Ising model. We show that the optical activity as a function of composition is well-described by the predictions of this theory.Comment: 13 pages, including 3 postscript figures, uses REVTeX 3.0 and epsf.st

Order and Frustration in Chiral Liquid Crystals

This paper reviews the complex ordered structures induced by chirality in liquid crystals. In general, chirality favors a twist in the orientation of liquid-crystal molecules. In some cases, as in the cholesteric phase, this favored twist can be achieved without any defects. More often, the favored twist competes with applied electric or magnetic fields or with geometric constraints, leading to frustration. In response to this frustration, the system develops ordered structures with periodic arrays of defects. The simplest example of such a structure is the lattice of domains and domain walls in a cholesteric phase under a magnetic field. More complex examples include defect structures formed in two-dimensional films of chiral liquid crystals. The same considerations of chirality and defects apply to three-dimensional structures, such as the twist-grain-boundary and moire phases.Comment: 39 pages, RevTeX, 14 included eps figure

An extremal model for amorphous media plasticity

An extremal model for the plasticity of amorphous materials is studied in a simple two-dimensional anti-plane geometry. The steady-state is analyzed through numerical simulations. Long-range spatial and temporal correlations in local slip events are shown to develop, leading to non-trivial and highly anisotropic scaling laws. In particular, the plastic strain is shown to statistically concentrate over a region which tends to align perpendicular to the displacement gradient. By construction, the model can be seen as giving rise to a depinning transition, the threshold of which (i.e. the macroscopic yield stress) also reveal scaling properties reflecting the localization of the activity.Comment: 4 pages, 5 figure

Emergence of hexatic and long-range herringbone order in two-dimensional smectic liquid crystals : A Monte Carlo study

Using a high resolution Monte Carlo simulation technique based on multi-histogram method and cluster-algorithm, we have investigated critical properties of a coupled XY model, consists of a six-fold symmetric hexatic and a three-fold symmetric herringbone field, in two dimensions. The simulation results demonstrate a series of novel continues transitions, in which both long-range hexatic and herringbone orderings are established simultaneously. It is found that the specific-heat anomaly exponents for some regions in coupling constants space are in excellent agreement with the experimentally measured exponents extracted from heat-capacity data near the smecticA-hexaticB transition of two-layer free standing film

Thurston's pullback map on the augmented Teichm\"uller space and applications

Let $f$ be a postcritically finite branched self-cover of a 2-dimensional topological sphere. Such a map induces an analytic self-map $\sigma_f$ of a finite-dimensional Teichm\"uller space. We prove that this map extends continuously to the augmented Teichm\"uller space and give an explicit construction for this extension. This allows us to characterize the dynamics of Thurston's pullback map near invariant strata of the boundary of the augmented Teichm\"uller space. The resulting classification of invariant boundary strata is used to prove a conjecture by Pilgrim and to infer further properties of Thurston's pullback map. Our approach also yields new proofs of Thurston's theorem and Pilgrim's Canonical Obstruction theorem.Comment: revised version, 28 page

Quantum resource estimates for computing elliptic curve discrete logarithms

We give precise quantum resource estimates for Shor's algorithm to compute discrete logarithms on elliptic curves over prime fields. The estimates are derived from a simulation of a Toffoli gate network for controlled elliptic curve point addition, implemented within the framework of the quantum computing software tool suite LIQ$Ui|\rangle$. We determine circuit implementations for reversible modular arithmetic, including modular addition, multiplication and inversion, as well as reversible elliptic curve point addition. We conclude that elliptic curve discrete logarithms on an elliptic curve defined over an $n$-bit prime field can be computed on a quantum computer with at most $9n + 2\lceil\log_2(n)\rceil+10$ qubits using a quantum circuit of at most $448 n^3 \log_2(n) + 4090 n^3$ Toffoli gates. We are able to classically simulate the Toffoli networks corresponding to the controlled elliptic curve point addition as the core piece of Shor's algorithm for the NIST standard curves P-192, P-224, P-256, P-384 and P-521. Our approach allows gate-level comparisons to recent resource estimates for Shor's factoring algorithm. The results also support estimates given earlier by Proos and Zalka and indicate that, for current parameters at comparable classical security levels, the number of qubits required to tackle elliptic curves is less than for attacking RSA, suggesting that indeed ECC is an easier target than RSA.Comment: 24 pages, 2 tables, 11 figures. v2: typos fixed and reference added. ASIACRYPT 201

Role of disclinations in determining the morphology of deformable fluid interfaces

We study the equilibrium shapes of vesicles, with an in-plane nematic order, using a Monte-Carlo scheme and show that highly curved shapes, like tubes and discs, with a striking similarity to the structures engendered by certain curvature sensing peripheral membrane proteins, can be spontaneously generated by anisotropic directional curvature with nematic disclinations playing and important role. We show that the coupling between nematic order and local curvature could lead to like defects moving towards each other and unlike defects moving away, in turn leading to tube formation. Thermally induced defect pair production lead to branched tubular structures. It is also shown that helical arrangement of the membrane tubes, with nematic field spiraling around it, is a dominant soft mode of the system.Comment: 6 Figures; Soft Matter, Advance Article 201

Travelling colourful patterns in self-organized cellulose-based liquid crystalline structures

Publisher Copyright: © 2021, The Author(s)Cellulose-based systems are useful for many applications. However, the issue of self-organization under non-equilibrium conditions, which is ubiquitous in living matter, has scarcely been addressed in cellulose-based materials. Here, we show that quasi-2D preparations of a lyotropic cellulose-based cholesteric mesophase display travelling colourful patterns, which are generated by a chemical reaction-diffusion mechanism being simultaneous with the evaporation of solvents at the boundaries. These patterns involve spatial and temporal variation in the amplitude and sign of the helix´s pitch. We propose a simple model, based on a reaction-diffusion mechanism, which simulates the observed spatiotemporal colour behaviour.publishersversionpublishe