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Register a new userFluctuating Nematic Elastomer Membranes: a New Universality Class
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Xiangjun Xing
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We study the flat phase of nematic elastomer membranes with rotational symmetry spontaneously broken by in-plane nematic order. Such state is characterized by a vanishing elastic modulus for simple shear and soft transverse phonons. At harmonic level, in-plane orientational (nematic) order is stable to thermal fluctuations, that lead to short-range in-plane translational (phonon) correlations. To treat thermal fluctuations and relevant elastic nonlinearities, we introduce two generalizations of two-dimensional membranes in a three dimensional space to arbitrary D-dimensional membranes embedded in a d-dimensional space, and analyze their anomalous elasticities in an expansion about D=4. We find a new stable fixed point, that controls long-scale properties of nematic elastomer membranes. It is characterized by singular in-plane elastic moduli that vanish as a power-law eta_lambda=4-D of a relevant inverse length scale (e.g., wavevector) and a finite bending rigidity. Our predictions are asymptotically exact near 4 dimensions.
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Nematic elastomers dramatically change their shape in response to diverse stimuli including light and heat. In this paper, we provide a systematic framework for the design of complex three dimensional shapes through the actuation of heterogeneously patterned nematic elastomer sheets. These sheets are composed of textit{nonisometric origami} building blocks which, when appropriately linked together, can actuate into a diverse array of three dimensional faceted shapes. We demonstrate both theoretically and experimentally that: 1) the nonisometric origami building blocks actuate in the predicted manner, 2) the integration of multiple building blocks leads to complex multi-stable, yet predictable, shapes, 3) we can bias the actuation experimentally to obtain a desired complex shape amongst the multi-stable shapes. We then show that this experimentally realized functionality enables a rich possible design landscape for actuation using nematic elastomers. We highlight this landscape through theoretical examples, which utilize large arrays of these building blocks to realize a desired three dimensional origami shape. In combination, these results amount to an engineering design principle, which we hope will provide a template for the application of nematic elastomers to emerging technologies.
Instabilities in thin elastic sheets, such as wrinkles, are of broad interest both from a fundamental viewpoint and also because of their potential for engineering applications. Nematic liquid crystal elastomers offer a new form of control of these instabilities through direct coupling between microscopic degrees of freedom, resulting from orientational ordering of rod-like molecules, and macroscopic strain. By a standard method of dimensional reduction, we construct a plate theory for thin sheets of nematic elastomer. We then apply this theory to the study of the formation of wrinkles due to compression of a thin sheet of nematic liquid crystal elastomer atop an elastic or fluid substrate. We find the scaling of the wrinkle wavelength in terms of material parameters and the applied compression. The wavelength of the wrinkles is found to be non-monotonic in the compressive strain owing to the presence of the nematic. Finally, due to soft modes, the critical stress for the appearance of wrinkles can be much higher than in an isotropic elastomer and depends nontrivially on the manner in which the elastomer was prepared.
The complex interplay between the various attractive and repulsive forces that mediate between biological membranes governs an astounding array of biological functions: cell adhesion, membrane fusion, self-assembly, binding-unbinding transition among others. In this work, the entropic repulsive force between membranes---which originates due to thermally excited fluctuations---is critically reexamined both analytically and through systematic Monte Carlo simulations. A recent work by Freund cite {Freund13} has questioned the validity of a well-accepted result derived by Helfrich cite{Helfrich78}. We find that, in agreement with Freund, for small inter-membrane separations ($d$), the entropic pressure scales as $psim 1/d $, in contrast to Helfrichs result: $psim 1/d^3$. For intermediate separations, our calculations agree with that of Helfrich and finally, for large inter-membrane separations, we observe an exponentially decaying behavior.
In this paper, the two-dimensional pure bending of a hyperelastic substrate coated by a nematic liquid crystal elastomer (abbreviated as NLCE) is studied within the framework of nonlinear elasticity. The governing system, arising from the deformational momentum balance, the orientational momentum balance and the mechanical constraint, is formulated, and the corresponding exact solution is derived for a given constitutive model. It is found that there exist two different bending solutions. In order to determine which the preferred one is, we compare the total potential energy for both solutions and find that the two energy curves may have an intersection point at a critical value of the bending angle $alpha_c$ for some material parameters. In particular, the director $bm n$ abruptly rotates $dfrac{pi}{2}$ from one solution to another at $alpha_c$, which indicates a director reorientation (or jump). Furthermore, the effects of different material and geometric parameters on the bending deformation and the transition angle $alpha_c$ can be revealed using the obtained bending solutions. Meanwhile, the exact solution can offer a benchmark problem for validating the accuracy of approximated plate models for liquid crystal elastomers.
We introduce two new concepts, frictional rigidity percolation and minimal rigidity proliferation, to help identify the nature of the frictional jamming transition as well as significantly broaden the scope of rigidity percolation. For frictional rigidity percolation, we construct rigid clusters in two different lattice models using a $(3,3)$ pebble game, while taking into account contacts below and at the Coulomb threshold. The first lattice is a honeycomb lattice with next-nearest neighbors, the second, a hierarchical lattice. For both, we generally find a continuous rigidity transition. Our numerical results suggest that, for the honeycomb lattice, the exponents associated with the transition found with the frictional $(3,3)$ pebble game are distinct from those of a central-force $(2,3)$ pebble game. We propose that localized motifs, such as hinges, connecting rigid clusters that are allowed only with friction could give rise to this new frictional universality class. However, the closeness of the order parameter exponent between the two cases hints at potential superuniversality. To explore this possibility, we construct a bespoke cluster generating algorithm invoking generalized Henneberg moves, dubbed minimal rigidity proliferation. The minimally rigid clusters the algorithm generates appear to be in the same universality class as connectivity percolation, suggesting superuniversality between all three types of transitions. Finally, the hierarchical lattice is analytically tractable and we find that the exponents depend both on the type of force and on the fraction of contacts at the Coulomb threshold. These combined results allow us to compare two universality classes on the same lattice via rigid clusters for the first time to highlight unifying and distinguishing concepts within the set of all possible rigidity transitions in disordered systems.
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