Olaf Stenull

Semisoft nematic elastomers and nematics in crossed electric and magnetic fields.

Nematic elastomers with a locked-in anisotropy direction exhibit semisoft elastic response characterized by a plateau in the stress-strain curve in which stress does not change with strain. We calculate the global phase diagram for a minimal model, which is equivalent to one describing a nematic in crossed electric and magnetic fields, and show that semisoft behavior is associated with a broken symmetry biaxial phase and that it persists well into the supercritical regime. We also consider generalizations beyond the minimal model and find similar results.

Phase Transitions and Soft Elasticity of Smectic Elastomers

Smectic-C elastomers can be prepared by cross-linking, e.g., liquid crystal polymers, in the smectic-A phase followed by a cooling through the smectic-A to smectic-C phase transition. This transition from D(infinityh) to C(2h) symmetry spontaneously breaks rotational symmetry in the smectic plane as does the transition from a smectic-A to a biaxial smectic phase with D(2h) symmetry. We study these transitions and the emergent elasticity of the smectic-C and biaxial phases in three related models and show that these phases exhibit soft elasticity analogous to that of nematic elastomers.

Topological Phonons and Weyl Lines in Three Dimensions.

Topological mechanics and phononics have recently emerged as an exciting field of study. Here we introduce and study generalizations of the three-dimensional pyrochlore lattice that have topologically protected edge states and Weyl lines in their bulk phonon spectra, which lead to zero surface modes that flip from one edge to the opposite as a function of surface wave number.

Anomalous elasticity of nematic elastomers

We study the anomalous elasticity of nematic elastomers by employing the powers of renormalized field theory. Using general arguments of symmetry and relevance, we introduce a minimal Landau-Ginzburg-Wilson elastic energy for nematic elastomers. Performing a diagrammatic low-temperature expansion, we analyze the fluctuations of the displacement fields at and below the upper critical dimension 3. Our analysis reveals an anomaly of certain elastic moduli in the sense that they depend on the length scale. In d = 3 this dependence is logarithmic and below d = 3 it is of power law type with anomalous scaling exponents. One of the 4 relevant shear moduli vanishes at long length scales whereas the only relevant bending modulus diverges.

Dynamics of nematic elastomers

We study the low-frequency, long-wavelength dynamics of soft and semisoft nematic elastomers using two different but related dynamic theories. Our first formulation describes the pure hydrodynamic behavior of nematic elastomers in which the nematic director has relaxed to its equilibrium value in the presence of strain. We find that the sound-mode structure for soft elastomers is identical to that of columnar liquid crystals. Our second formulation generalizes the derivation of the equations of nematohydrodynamics by Forster et al. to nematic elastomers. It treats the director explicitly and describes slow modes beyond the hydrodynamic limit.

Topological boundary modes in jammed matter

Granular matter at the jamming transition is poised on the brink of mechanical stability, and hence it is possible that these random systems have topologically protected surface phonons. Studying two model systems for jammed matter, we find states that exhibit distinct mechanical topological classes, protected surface modes, and ubiquitous Weyl points. The detailed statistics of the boundary modes shed surprising light on the properties of the jamming critical point and help inform a common theoretical description of the detailed features of the transition.

Elasticity of a filamentous kagome lattice.

The diluted kagome lattice, in which bonds are randomly removed with probability 1-p, consists of straight lines that intersect at points with a maximum coordination number of 4. If lines are treated as semiflexible polymers and crossing points are treated as cross-links, this lattice provides a simple model for two-dimensional filamentous networks. Lattice-based effective-medium theories and numerical simulations for filaments modeled as elastic rods, with stretching modulus μ and bending modulus κ, are used to study the elasticity of this lattice as functions of p and κ. At p=1, elastic response is purely affine, and the macroscopic elastic modulus G is independent of κ. When κ=0, the lattice undergoes a first-order rigidity-percolation transition at p=1. When κ>0, G decreases continuously as p decreases below one, reaching zero at a continuous rigidity-percolation transition at p=p(b)≈0.605 that is the same for all nonzero values of κ. The effective-medium theories predict scaling forms for G, which exhibit crossover from bending-dominated response at small κ/μ to stretching-dominated response at large κ/μ near both p=1 and p(b), that match simulations with no adjustable parameters near p=1. The affine response as p→1 is identified with the approach to a state with sample-crossing straight filaments treated as elastic rods.

Effective-medium theory of a filamentous triangular lattice

We present an effective-medium theory that includes bending as well as stretching forces, and we use it to calculate the mechanical response of a diluted filamentous triangular lattice. In this lattice, bonds are central-force springs, and there are bending forces between neighboring bonds on the same filament. We investigate the diluted lattice in which each bond is present with a probability p. We find a rigidity threshold p(b) which has the same value for all positive bending rigidity and a crossover characterizing bending, stretching, and bend-stretch coupled elastic regimes controlled by the central-force rigidity percolation point at p(CF)=/~2/3 of the lattice when fiber bending rigidity vanishes.

Anomalous elasticity of nematic and critically soft elastomers

Uniaxial elastomers are characterized by five elastic constants. If their elastic modulus C5 describing the energy of shear strains in planes containing the anisotropy axis vanishes, they are said to be soft. In spatial dimensions d less than or equal to 3, soft elastomers exhibit anomalous elasticity with certain length-scale-dependent bending moduli that diverge and shear moduli that vanish at large length scales. Using renormalized field theory at d=3 and to first order in epsilon=3-d, we calculate critical exponents and other properties characterizing the anomalous elasticity of two soft systems: (i) nematic elastomers in which softness is a manifestation of a Goldstone mode induced by the spontaneous symmetry breaking associated with a transition from an isotropic state to a nematic state, and (ii) a particular version of what we call a critically soft elastomer in which C(5)=0 corresponds to a critical point terminating the stability regime of a uniaxial elastomer with C5>0.

Smectic-

Experimentally it is possible to manipulate the director in a (chiral) smectic- A elastomer using an electric field. This suggests that the director is not necessarily locked to the layer normal, as described in earlier papers that extended rubber elasticity theory to smectics. Here, we consider the case that the director is weakly anchored to the layer normal assuming that there is a free energy penalty associated with relative tilt between the two. We use a recently developed weak-anchoring generalization of rubber elastic approaches to smectic elastomers and study shearing in the plane of the layers, stretching in the plane of the layers, and compression and elongation parallel to the layer normal. We calculate, inter alia, the engineering stress and the tilt angle between director and layer normal as functions of the applied deformation. For the latter three deformations, our results predict the existence of an instability towards the development of shear accompanied by smectic- C-like order.