Dirt Softens Soap: Anomalous Elasticity of Disordered Smectics

Abstract

We show that a smectic in a disordered medium (e.g., aerogel) exhibits anomalous elasticity, with the compression modulus B(k) vanishing and the bend modulus K(k) diverging as k → 0. In addition, the effective disorder develops long ranged correlations. These divergences are much stronger than those driven by thermal fluctuations in pure smectics, and are controlled by a zero temperature glassy fixed point, which we study in an ǫ = 5 − d expansion. We discuss the experimental implications of these theoretical predictions. 64.60Fr,05.40,82.65Dp The effects of quenched disorder on the properties of condensed matter systems continues to be a fascinating area of active research, which includes the study of disordered superconductors [1], charge density waves [2], Josephson junction arrays [3], and Helium in aerogel [4], to name a few. Some of this attention has focussed [5,6] on liquid crystals in the random environment of an aerogel. While a complete picture of aerogel-confined liquid crystals is still being developed [6], in this Letter we show that a smectic phase of these systems possesses strong anomalous elasticity, when subjected to an arbitrarily weak amount of quenched disorder. This result has important experimental consequences.The anomalous elasticity of pure smectics was predicted sometime ago [7], and is characterized by bulk compressional and tilt moduli B(k) and K(k) which, respectively, vanish and diverge at long wavelengths (k → 0). This is a general property of all one-dimensional crystals in which the direction of the 1d ordering wavevector is chosen spontaneously. As a consequence of this spontaneous breaking of rotational symmetry, (a property of smectics but not of charge density waves) in such systems, compression can be relieved by smoothing out fluctuations (wrinkles in the smectic layers), so the effective layer compressional modulus B(k) vanishes at long wavelengths. Similarly, in the presence of fluctuations, a bending of smectic layers necessarily leads to a compression, which implies that the effective tilt modulus K(k) diverges at long wavelengths. Unfortunately, this thermally driven behavior in pure smectics is difficult to observe experimentally, because the effect is very weak (logarithmic) in 3d.The main ingredients necessary for anomalous elasticity, namely spontaneously broken rotational invariance, and fluctuations, both exist even at zero temperature in quenched disordered smectics. In this Letter we demonstrate the existence of anomalous elasticity in quencheddisordered smectics which is significantly stronger than the marginal anomalous elasticity of thermal smectics, and is described by a zero-temperature fixed point that is perturbatively accessible in d = 5 − ǫ dimensions. The elastic anomaly is much stronger in quenched disordered smectics because layer fluctuations are much larger, even at T = 0, than in a pure smectic at T > 0.One experimental signature of these divergences is in the smectic correlation length for smectics in aerogel, which has a different, univers...

Dirty, Skewed, and Backwards: The SmecticA−CPhase Transition in Aerogel

“…To include the effects of the quenched disorder of the aerogel, we add to the pure Hamiltonian (9) random fields coupling to u and its gradients: giving us (10) where h( r) is a quenched random field that for simplicity we take to be Gaussian zero distributed mean, and characterized by short-ranged anisotropic correlations:…”

Universal Elasticity and Fluctuations of Nematic Gels

“…Although such quenched disorder and thermally-driven elastic anomalies, controlled by a nontrivial low temperature fixed point have been previously predicted in a variety of other systems [12,14,15], to our knowledge, nematic elastomers are a first example of a threedimensional solid, where these exotic theoretical predictions can be directly experimentally tested.…”

“…The remaining longitudinal mode is encoded through in-plane density fluctuations w ii , that can be integrated out, leading to a finite shift in B z , with the model reducing to that of a randomly strained smectic [12]. It is reassuring that this fixed point S, is precisely the same as that found in the study of smectics in aerogel [12]. We are not aware of any physical system that is described by the fixed point X, characterized by g L = 0.…”

“…In fact, a renormalization group analysis and expansion about five dimensions suggests that orientational order of nematic elastomers is stable to weak network heterogeneity [16], in strong contrast to nematics confined in rigid gels [11,12,13].…”

“…As discussed in closely related contexts [12,14,15], at long length scales, the elastomer network heterogeneities lead to quenched local random stress σ αβ (x) and anisotropy g(x) fields [17] …”