Characterization of soft stripe-domain deformations in Sm-C and Sm-C* liquid-crystal elastomers

The neoclassical model of Sm-C (and Sm-C*) elastomers developed by Warner and Adams predicts a class of “soft” (zero energy) deformations. We find and describe the full set of stripe domains—laminate structures in which the laminates alternate between two different deformations—that can form between pairs of these soft deformations. All the stripe domains fall into two classes, one in which the smectic layers are not bent at the interfaces, but for which—in the Sm-C* case—the interfaces are charged, and one in which the smectic layers are bent but the interfaces are never charged. Striped deformations significantly enhance the softness of the macroscopic elastic response

Growth and Shape of a Chain Fountain

If a long chain is held in a pot elevated a distance h_1 above the floor, and the end of the chain is then dragged over the rim of the pot and released, the chain flows under gravity down into a pile on the floor. Not only does the chain flow out of the pot, it also leaps above the pot in a "chain-fountain". I predict and observe that if the pot is held at an angle to the vertical the steady state shape of the fountain is an inverted catenary, and discuss how to apply boundary conditions to this solution. In the case of a level pot, the fountain shape is completely vertical. In this case I predict and observe both how fast the fountain grows to its steady state hight, and how it grows quadratically in time if there is no floor. The fountain is driven by an anomalous push force from the pot that acts on the link of chain about to come into motion. I confirm this by designing two new chains, one consisting of hollow cylinders threaded on a string and one consisting of heavy beads separated by long flexible threads. The former is predicted to produce a pot-push and hence a fountain, while the latter will not. I confirm these predictions experimentally. Finally I directly observe the anomalous push in a horizontal chain-pick up experiment.Comment: 6 pages 7 figures and one movi

Spreading speeds in reducible multitype branching random walk

This paper gives conditions for the rightmost particle in the $n$th generation of a multitype branching random walk to have a speed, in the sense that its location divided by n converges to a constant as n goes to infinity. Furthermore, a formula for the speed is obtained in terms of the reproduction laws. The case where the collection of types is irreducible was treated long ago. In addition, the asymptotic behavior of the number in the nth generation to the right of na is obtained. The initial motive for considering the reducible case was results for a deterministic spatial population model with several types of individual discussed by Weinberger, Lewis and Li [J. Math. Biol. 55 (2007) 207-222]: the speed identified here for the branching random walk corresponds to an upper bound for the speed identified there for the deterministic model.Comment: Published in at http://dx.doi.org/10.1214/11-AAP813 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Supersoft elasticity in polydomain nematic elastomers

We consider the equilibrium stress-strain behavior of polydomain liquid crystal elastomers (PLCEs). We show that there is a fundamental difference between PLCEs cross-linked in the high temperature isotropic and low temperature aligned states. PLCEs cross-linked in the isotropic state then cooled to an aligned state will exhibit extremely soft elasticity (confirmed by recent experiments) and ordered director patterns characteristic of textured deformations. PLCEs cross-linked in the aligned state will be mechanically much harder and characterized by disclination textures

Elasticity of Polydomain Liquid Crystal Elastomers

We model polydomain liquid-crystal elastomers by extending the neo-classical soft and semi-soft free energies used successfully to describe monodomain samples. We show that there is a significant difference between polydomains cross-linked in homogeneous high symmetry states then cooled to low symmetry polydomain states and those cross-linked directly in the low symmetry polydomain state. For example, elastomers cross-linked in the isotropic state then cooled to a nematic polydomain will, in the ideal limit, be perfectly soft, and with the introduction of non-ideality, will deform at very low stress until they are macroscopically aligned. The director patterns observed in them will be disordered, characteristic of combinations of random deformations, and not disclination patterns. We expect these samples to exhibit elasticity significantly softer than monodomain samples. Polydomains cross-linked in the nematic polydomain state will be mechanically harder and contain characteristic schlieren director patterns. The models we use for polydomain elastomers are spatially heterogeneous, so rather than solving them exactly we elucidate this behavior by bounding the energies using Taylor-like (compatible test strain fields) and Sachs (constant stress) limits extended to non-linear elasticity. Good agreement is found with experiments that reveal the supersoft response of some polydomains. We also analyze smectic polydomain elastomers and propose that polydomain SmC* elastomers cross-linked in the SmA monodomain state are promising candidates for low field electrical actuation.Comment: 13 pages, 11 figure

Large Deviations in randomly coloured random graphs

Models of random graphs are considered where the presence or absence of an edge depends on the random types (colours) of its vertices, so that whether or not edges are present can be dependent. The principal objective is to study large deviations in the number of edges. These graphs provide a natural example with two different non-degenerate large deviation regimes, one arising from large deviations in the colourings followed by typical edge placement and the other from large deviation in edge placement. A secondary objective is to illustrate the use of a general result on large deviations for mixtures