Elastic Response and Ward Identities in Stressed Nematic Elastomers

Nematic elastomers exhibit a rich elastic response to external stresses. Of particular interest is the semisoft response of elastomers with an anisotropy direction (z) frozen in by a double cross-linking process. This response is characterized by a stress-strain curve for stresses along x perpendicular to z that rises initially, exhibits a nearly flat plateau between two critical values of strain, and then rises again. This paper explores elastic response in semisoft elastomers as a function of externally applied strain. It derives general Ward identities for elastic moduli and shows that the elastic modulus measuring response to xz shears vanishes at the boundaries of the semisoft plateau whereas moduli measuring response to shears perpendicular to the xz plane do not. It then calculates all relevant moduli in a simple model of elastomers and verifies the general Ward-identity predictions

Stripes, Zigzags, and Slow Dynamics in Buckled Hard Spheres

We study the analogy between buckled colloidal monolayers and the triangular-lattice Ising antiferromagnet. We calculate free-volume-induced Ising interactions, show how lattice deformations favor zigzag stripes that partially remove the Ising model ground-state degeneracy, and identify the martensitic mechanism prohibiting perfect stripes. Slowly inflating the spheres yields jamming as well as logarithmically slow relaxation reminiscent of the glassy dynamics observed experimentally

Soft Elasticity in Biaxial Smectic and Smectic-\u3cem\u3eC\u3c/em\u3e Elastomers

Ideal (monodomain) smectic-A elastomers cross-linked in the smectic-A phase are simply uniaxial rubbers, provided deformations are small. From these materials smectic-C elastomers are produced by a cooling through the smectic-A to smectic-C phase transition. At least in principle, biaxial smectic elastomers could also be produced via cooling from the smectic-A to a biaxial smectic phase. These phase transitions, respectively, from D∞h to C2h and from D∞h to D2h symmetry, spontaneously break the rotational symmetry in the smectic planes. We study the above transitions and the elasticity of the smectic-C and biaxial phases in three different but related models: Landau-like phenomenological models as functions of the Cauchy-Saint-Laurent strain tensor for both the biaxial and the smectic-C phases and a detailed model, including contributions from the elastic network, smectic layer compression, and smectic-C tilt for the smectic-C phase as a function of both strain and the c-director. We show that the emergent phases exhibit soft elasticity characterized by the vanishing of certain elastic moduli. We analyze in some detail the role of spontaneous symmetry breaking as the origin of soft elasticity and we discuss different manifestations of softness like the absence of restoring forces under certain shears and extensional strains

Effect of Randomness on Critical Behavior of Spin Models

Renormalization group methods are used to analyze the critical behavior of random Ising models. The Wilson‐Fischer ε‐expansion for the recursion relations for n‐component continuous spin models are developed for randomly inhomogeneous systems. In addition to the usual variables for a homogeneous system there appears a variable which in essence describes local fluctuations in T c . From the structure and stability of the fixed points we conclude that critical exponents are unaffected by randomness for n≳4 but are renormalized by randomness for 1\u3c

Randomly Diluted xy and Resistor Networks Near the Percolation Threshold

A formulation based on that of Stephen for randomly diluted systems near the percolation threshold is analyzed in detail. By careful consideration of various limiting procedures, a treatment of xy spin models and resistor networks is given which shows that previous calculations (which indicate that these systems having continuous symmetry have the same crossover exponents as the Ising model) are in error. By studying the limit wherein the energy gap goes to zero, we exhibit the mathematical mechanism which leads to qualitatively different results for xy-like as contrasted to Ising-like systems. A distinctive feature of the results is that there is an infinite sequence of crossover exponents needed to completely describe the probability distribution for R(x,x’), the resistance between sites x and x’. Because of the difference in symmetry between the xy model and the resistor network, the former has an infinite sequence of crossover exponents in addition to those of the resistor network. The first crossover exponent φ1=1+ε/42 governs the scaling behavior of R(x,x’) with ‖x-x’‖≡r: [R(x,x’)]c~xφ1/ν, where [ ]c indicates a conditional average, subject to x and x’ being in the same cluster, ν is the correlation length exponent for percolation, and ε=6-d, where d is the spatial dimensionality. We give a detailed analysis of the scaling properties of the bulk conductivity and the anomalous diffusion constant introduced by Gefen et al. Our results show conclusively that the Alexander-Orbach conjecture, while numerically quite accurate, is not exact, at least in high spatial dimension. We also evaluate various amplitude ratios associated with susceptibilities, χn involving the nth power of the resistance R(x,x’), e.g., limp→pcχ2χ0/χ12=2[1(19ε/420)]. In an appendix we outline how the calculation can be extended to treat the diluted m-component spin model for m\u3e2. As expected, the results for φ1 remain valid for m\u3e2. The techniques described here have led to several recent calculations of various infinite families of exponents

Internal Stresses, Normal Modes, and Nonaffinity in Three-Dimensional Biopolymer Networks

We numerically investigate deformations and modes of networks of semiflexible biopolymers as a function of crosslink coordination number z and strength of bending and stretching energies. In equilibrium filaments are under internal stress, and the networks exhibit shear rigidity below the Maxwell isostatic point. In contrast to two-dimensional networks, ours exhibit nonaffine bending-dominated response in all rigid states, including those near the maximum of z = 4 when bending energies are less than stretching ones

Potts-Model Formulation of the Random Resistor Network

The randomly diluted resistor network is formulated in terms of an n-replicated s-state Potts model with a spin-spin coupling constant J in the limit when first n, then s, and finally 1/J go to zero. This limit is discussed and to leading order in 1/J the generalized susceptibility is shown to reproduce the results of the accompanying paper where the resistor network is treated using the xy model. This Potts Hamiltonian is converted into a field theory by the usual Hubbard-Stratonovich transformation and thereby a renormalization-group treatment is developed to obtain the corrections to the critical exponents to first order in ε=6-d, where d is the spatial dimensionality. The recursion relations are shown to be the same as for the xy model. Their detailed analysis (given in the accompanying paper) gives the resistance crossover exponent as φ1=1+ε/42, and determines the critical exponent, t for the conductivity of the randomly diluted resistor network at concentrations, p, just above the percolation threshold: t=(d-2)ν+φ1, where ν is the critical exponent for the correlation length at the percolation threshold. These results correct previously accepted results giving φ=1 to all orders in ε. The new result for φ1 removes the paradox associated with the numerical result that t\u3e1 for d=2, and also shows that the Alexander-Orbach conjecture, while numerically quite accurate, is not exact, since it disagrees with the ε expansion

Spin-Glass and Related Orderings in Quenched Random-Spin Systems

A general model in which the nearest-neighbor exchange interactions J(x⃗ ,x⃗ ′) are of the form J(x⃗ ,x⃗ ′)=J1+J2ε(x⃗ )ε(x⃗ ′)+J3[ε(x⃗ )+ε(x⃗ ′)]+J4μ(x⃗ ,x⃗ ′) is considered. Here ε(x⃗ ) is a random site variable and μ(x⃗ ,x⃗ ′) is a random bond variable. It is argued that J4 tends to produce the Edwards-Anderson-type spin glass, whereas J2 produces a different type of phase which is like an alloy of up and down spins. That these two phases are distinct follows from the existence of a phase boundary in the J2−J4 plane when J1=J3=0. A consistent, but qualitative, discussion of this model via the renormalization group is also given

Mean-Field Theory and ε Expansion for Anderson Localization

A general field-theoretic formulation of the Anderson model for the localization of wave functions in a random potential is given in terms of n-component replicated fields in the limit n→0, and is analyzed primarily for spatial dimension d≥4. Lengths ξ1 and ξ2 associated with the spatial decay of correlations in the single-particle and two-particle Green\u27s functions, respectively, are introduced. Two different regimes, the weak coupling and strong coupling, are distinguished depending on whether ξ1−1 or ξ2−1, respectively, vanishes as the mobility energy, Ec, is approached. The weak-coupling regime vanishes as d→4+. Mean-field theory is developed from the uniform minimum of the Lagrangian for both the strong- and weak-coupling cases. For the strong-coupling case it gives the exponents va=1/4, γa=βa=1/2, η=0, and μ=1, where βa is the exponent associated with the density of extended states and μ is that associated with the conductivity. Simple heuristic arguments are used to verify the correctness of these unusual mean-field values. Infrared divergences in perturbation theory for the strong-coupling case occur for d\u3c8, and an ε expansion (ε=8−d) is developed which is found to be identical to that previously analyzed for the statistics of lattice animals and which gives βa=1/2−ε/12, η=−ε/9, va=1/4+ε/36, and μ=1−5ε/36. The results are consistent with the Ward identity, which in combination with scaling arguments requires that βa+γa=1. The treatment takes account of the fact that the average of the on-site Green\u27s function [G(x⃗ ,x⃗ ;E)]av is nonzero and is predicated on this quantity being real, i.e., on the density of states vanishing at the mobility edge. We also show that localized states emerge naturally from local minima of finite action in the Lagrangian. These instanton solutions are analyzed on a lattice where the cutoff produced by the lattice constant leads to lattice instantons which exist for all d, in contrast to the case for the continuum model where instanton solutions seem not to occur for d\u3e4. This analysis leads to a density of localized states ρloc satisfying 1nρloc~−E2 at large E and 1nρloc~−|E−Ec|−ζ at the mobility edge, where for the weak-coupling case ζ=(1/2)(d−4) and for the strong-coupling case ζ=(d−2+η)va−2βa=1/2+ε/18 for d\u3c8 and ζ=(1/4)(d−6) for d\u3e8. A brief discussion of the relationship between this work and the theories of localization below four dimensions is presented