arXiv subjects

T. C. Lubensky

Publications and source records attributed to T. C. Lubensky.

At least 37 records · Page 2Linked to original sources

arXiv2010

Non-affine displacements in flexible polymer networks

The validity of the affine assumption in model flexible polymer networks is explored. To this end, the displacements of fluorescent tracer beads embedded in polyacrylamide gels are quantified by confocal microscopy under shear deformation, and the deviations of these displacements from affine responses are recorded. Non-affinity within the gels is quantified as a function of polymer chain density and cross-link concentration. Observations are in qualitative agreement with current theories of polymer network non-affinity. The measured degree of non-affinity in the polyacrylamide gels suggests the presence of structural inhomogeneities which likely result from heterogeneous reaction kinetics during gel preparation. In addition, the macroscopic elasticity of the polyacrylamide gels is confirmed to behave in accordance with standard models of flexible polymer network elasticity.

cond-mat.soft↗

arXiv2010

Centrifugal compression of soft particle packings - theory and experiment

An exact method is developed for computing the height of an elastic medium subjected to centrifugal compression, for arbitrary constitutive relation between stress and strain. Example solutions are obtained for power-law media and for cases where the stress diverges at a critical strain -- for example as required by packings composed of deformable but incompressible particles. Experimental data are presented for the centrifugal compression of thermo-responsive N-isopropylacrylamide (NIPA) microgel beads in water. For small radial acceleration, the results are consistent with Hertzian elasticity, and are analyzed in terms of the Young elastic modulus of the bead material. For large radial acceleration, the sample compression asymptotes to a value corresponding to a space-filling particle volume fraction of unity. Therefore we conclude that the gel beads are incompressible, and deform without deswelling. In addition, we find that the Young elastic modulus of the particulate gel material scales with cross-link density raised to the power 3.3+-0.8, somewhat larger than the Flory expectation.

cond-mat.soft↗

arXiv2010

Coherent potential approximation of random nearly isostatic kagome lattice

The kagome lattice has coordination number $4$, and it is mechanically isostatic when nearest neighbor ($NN$) sites are connected by central force springs. A lattice of $N$ sites has $O(\sqrt{N})$ zero-frequency floppy modes that convert to finite-frequency anomalous modes when next-nearest-neighbor ($NNN$) springs are added. We use the coherent potential approximation (CPA) to study the mode structure and mechanical properties of the kagome lattice in which $NNN$ springs with spring constant $κ$ are added with probability $\Prob= Δz/4$, where $Δz= z-4$ and $z$ is the average coordination number. The effective medium static $NNN$ spring constant $κ_m$ scales as $\Prob^2$ for $\Prob \ll κ$ and as $\Prob$ for $\Prob \gg κ$, yielding a frequency scale $ω^* \sim Δz$ and a length scale $l^*\sim (Δz)^{-1}$. To a very good approximation at at small nonzero frequency, $κ_m(\Prob,ω)/κ_m(\Prob,0)$ is a scaling function of $ω/ω^*$. The Ioffe-Regel limit beyond which plane-wave states becomes ill-define is reached at a frequency of order $ω^*$.

cond-mat.dis-nn↗

arXiv2010

Soft modes and elasticity of nearly isostatic lattices: randomness and dissipation

The square lattice with central-force springs on nearest-neighbor bonds is isostatic. It has a zero mode for each row and column, and it does not support shear. Using the Coherent Potential Approximation (CPA), we study how the random addition, with probability $\mathcal{P}=(z-4)/4$ ($z$ = average number of nearest neighbors), of springs on next-nearest-neighbor ($NNN$) bonds restores rigidity and affects phonon structure. We find that the CPA effective $NNN$ spring constant $\tildeκ_m(ω)$, equivalent to the complex shear modulus $G(ω)$, obeys the scaling relation, $\tildeκ_m(ω) = κ_m h(ω/ω^*)$, at small $\mathcal{P}$, where $κ_m = \tildeκ'_m(0)\sim \mathcal{P}^2$ and $ω^* \sim \mathcal{P}$, implying that elastic response is nonaffine at small $\mathcal{P}$ and that plane-wave states are ill-defined beyond the Ioffe-Regel limit at $ω\approx ω^*$. We identify a divergent length $l^* \sim \mathcal{P}^{-1}$, and we relate these results to jamming.

cond-mat.soft↗

arXiv2009

Elasticity and Response in Nearly Isostatic Periodic Lattices

The square and kagome lattices with nearest neighbor springs of spring constant $k$ are isostatic with a number of zero-frequency modes that scale with their perimeter. We analytically study the approach to this isostatic limit as the spring constant $k'$ for next-nearest-neighbor bonds vanishes. We identify a characteristic frequency $ω^* \sim \sqrt{k'}$ and length $l^* \sim \sqrt{k/k'}$ for both lattices. The shear modulus $C_{44}= k'$ of the square lattice vanishes with $k'$, but that for the kagome lattice does not.

cond-mat.soft↗

arXiv2008

Dynamics of Gas-Fluidized Granular Rods

We study a quasi-two-dimensional monolayer of granular rods fluidized by a spatially and temporally homogeneous upflow of air. By tracking the position and orientation of the particles, we characterize the dynamics of the system with sufficient resolution to observe ballistic motion at the shortest time scales. Particle anisotropy gives rise to dynamical anisotropy and superdiffusive dynamics parallel to the rod's long axis, causing the parallel and perpendicular mean squared displacements to become diffusive on different timescales. The distributions of free times and free paths between collisions deviate from exponential behavior, underscoring the non-thermal character of the particle motion. The dynamics show evidence of rotational-translational coupling similar to that of an anisotropic Brownian particle. We model rotational-translation coupling in the single-particle dynamics with a modified Langevin model using non-thermal noise sources. This suggests a phenomenological approach to thinking about collections of self-propelling particles in terms of enhanced memory effects.

cond-mat.soft↗

arXiv2008

Smectic-C tilt under shear in Smectic-A elastomers

Stenull and Lubensky [Phys. Rev. E {\bf 76}, 011706 (2007)] have argued that shear strain and tilt of the director relative to the layer normal are coupled in smectic elastomers and that the imposition of one necessarily leads to the development of the other. This means, in particular, that a Smectic-A elastomer subjected to a simple shear will develop Smectic-C-like tilt of the director. Recently, Kramer and Finkelmann [arXiv:0708.2024, Phys. Rev. E {\bf 78}, 021704 (2008)] performed shear experiments on Smectic-A elastomers using two different shear geometries. One of the experiments, which implements simple shear, produces clear evidence for the development of Smectic-C-like tilt. Here, we generalize a model for smectic elastomers introduced by Adams and Warner [Phys. Rev. E {\bf 71}, 021708 (2005)] and use it to study the magnitude of Smectic-C-like tilt under shear for the two geometries investigated by Kramer and Finkelmann. Using reasonable estimates of model parameters, we estimate the tilt angle for both geometries, and we compare our estimates to the experimental results. The other shear geometry is problematic since it introduces additional in-plane compressions in a sheet-like sample, thus inducing instabilities that we discuss.

cond-mat.soft↗

arXiv2008

Smectic-$A$ elastomers with weak director anchoring

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.

cond-mat.soft↗

arXiv2007

Fluctuations and Rheology in Active Bacterial Suspensions

We probe non-equilibrium properties of an active bacterial bath through measurements of correlations of passive tracer particles and the response function of a driven, optically trapped tracer. These measurements demonstrate violation of the fluctuation-dissipation theorem and enable us to extract the power spectrum of the active stress fluctuations. In some cases, we observe $1/\sqrtω$ scaling in the noise spectrum which we show can be derived from a theoretical model incorporating coupled stress, orientation, and concentration fluctuations of the bacteria.

cond-mat.soft↗

arXiv2007

State-dependent diffusion: thermodynamic consistency and its path integral formulation

The friction coefficient of a particle can depend on its position as it does when the particle is near a wall. We formulate the dynamics of particles with such state-dependent friction coefficients in terms of a general Langevin equation with multiplicative noise, whose evaluation requires the introduction of specific rules. Two common conventions, the Ito and the Stratonovich, provide alternative rules for evaluation of the noise, but other conventions are possible. We show the requirement that a particle's distribution function approach the Boltzmann distribution at long times dictates that a drift term must be added to the Langevin equation. This drift term is proportional to the derivative of the diffusion coefficient times a factor that depends on the convention used to define the multiplicative noise. We explore the consequences of this result in a number examples with spatially varying diffusion coefficients. We also derive path integral representations for arbitrary interpretation of the noise, and use it in a perturbative study of correlations in a simple system.

cond-mat.soft↗

arXiv2007

Unconventional elasticity in smectic-A elastomers

We study two aspects of the elasticity of smectic-$A$ elastomers that make these materials genuinely and qualitatively different from conventional uniaxial rubbers. Under strain applied parallel to the layer normal, monodomain smectic-$A$ elastomers exhibit a drastic change in Young's modulus above a threshold strain value of about 3%, as has been measured in experiments by Nishikawa and Finkelmann [Macromol. Chem. Phys. {\bf 200}, 312 (1999)]. Our theory predicts that such strains induce a transition to a smectic-$C$-like state and that it is this transition that causes the change in elastic modulus. We calculate the stress-strain behavior as well as the tilt of the smectic layers and the molecular orientation for strain along the layer normal, and we compare our findings with the experimental data. We also study the electroclinic effect in chiral smectic-$A^\ast$ elastomers. According to experiments by Lehmann {\em et al}. [Nature {\bf 410}, 447 (2001)] and Köhler {\em et al}. [Applied Physics A {\bf 80}, 381 (2003)], this effect leads in smectic-$A^\ast$ elastomers to a giant or, respectively, at least very large lateral electrostriction. Incorporating polarization into our theory, we calculate the height change of smectic-$A^\ast$ elastomer films in response to a lateral external electric field, and we compare this result to the experimental findings.

cond-mat.soft↗

arXiv2007

Semi-soft Nematic Elastomers and Nematics in Crossed Electric and Magnetic Fields

Nematic elastomers with a locked-in anisotropy direction exhibit semi-soft 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 semi-soft 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.

cond-mat.soft↗

arXiv2006

Dynamics of smectic elastomers

We study the low-frequency, long-wavelength dynamics of liquid crystal elastomers, crosslinked in the smectic-$A$ phase, in their smectic-$A$, biaxial smectic and smectic-$C$ phases. Two different yet related formulations are employed. One formulation describes the pure hydrodynamics and does not explicitly involve the Frank director, which relaxes to its local equilibrium value in a non-hydrodynamic time. The other formulation explicitly treats the director and applies beyond the hydrodynamic limit. We compare the low-frequency, long-wavelength dynamics of smectic-$A$ elastomers to that of nematics and show that the two are closely related. For the biaxial smectic and the smectic-$C$ phases, we calculate sound velocities and the mode structure in certain symmetry directions. For the smectic-$C$ elastomers, in addition, we discuss in some detail their possible behavior in rheology experiments.

cond-mat.soft↗

arXiv2006

Soft elasticity in biaxial smectic and smectic-C elastomers

Ideal (monodomain) smectic-$A$ elastomers crosslinked 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 $D_{\infty h}$ to $C_{2h}$ and from $D_{\infty h}$ to $D_{2h}$ 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.

cond-mat.soft↗

arXiv2006

Nonaffine Correlations in Random Elastic Media

Materials characterized by spatially homogeneous elastic moduli undergo affine distortions when subjected to external stress at their boundaries, i.e., their displacements $\uv (\xv)$ from a uniform reference state grow linearly with position $\xv$, and their strains are spatially constant. Many materials, including all macroscopically isotropic amorphous ones, have elastic moduli that vary randomly with position, and they necessarily undergo nonaffine distortions in response to external stress. We study general aspects of nonaffine response and correlation using analytic calculations and numerical simulations. We define nonaffine displacements $\uv' (\xv)$ as the difference between $\uv (\xv)$ and affine displacements, and we investigate the nonaffinity correlation function $\mathcal{G} = < [\uv'(\xv) - \uv' (0)]^2>$ and related functions. We introduce four model random systems with random elastic moduli induced by locally random spring constants, by random coordination number, by random stress, or by any combination of these. We show analytically and numerically that $\mathcal{G}$ scales as $A |\xv|^{-(d-2)}$ where the amplitude $A$ is proportional to the variance of local elastic moduli regardless of the origin of their randomness. We show that the driving force for nonaffine displacements is a spatial derivative of the random elastic constant tensor times the constant affine strain. Random stress by itself does not drive nonaffine response, though the randomness in elastic moduli it may generate does. We study models with both short and long-range correlations in random elastic moduli.

cond-mat.soft↗

arXiv2006

Dynamics, dynamic soft elasticity and rheology of smectic-C elastomers

We present a theory for the low-frequency, long-wavelength dynamics of soft smectic-C elastomers with locked-in smectic layers. Our theory, which goes beyond pure hydrodynamics, predicts a dynamic soft elasticity of these elastomers and allows us to calculate the storage and loss moduli relevant for rheology experiments as well as the mode structure.

cond-mat.soft↗

arXiv2005

Poisson-Bracket Approach to the Dynamics of Nematic Liquid Crystals. The Role of Spin Angular Momentum

Nematic liquid crystals are well modeled as a fluid of rigid rods. Starting from this model, we use a Poisson-bracket formalism to derive the equations governing the dynamics of nematic liquid crystals. We treat the spin angular momentum density arising from the rotation of constituent molecules about their centers of mass as an independent field and derive equations for it, the mass density, the momentum density, and the nematic director. Our equations reduce to the original Leslie-Ericksen equations, including the inertial director term that is neglected in the hydrodynamic limit, only when the moment of inertia for angular momentum parallel to the director vanishes and when a dissipative coefficient favoring locking of the angular frequencies of director rotation and spin angular momentum diverges. Our equations reduce to the equations of nematohydrodynamics in the hydrodynamic limit but with dissipative coefficients that depend on the coefficient that must diverge to produce the Leslie-Ericksen equations.

cond-mat.soft↗

arXiv2005

Phase transitions and soft elasticity of smectic elastomers

Smectic-C elastomers can be prepared by crosslinking, 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 $D_{\infty h}$ 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.

cond-mat.soft↗