Lubensky, Tom C
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Publication Elasticity and Response in Nearly Isostatic Periodic Lattices(2009-11-13) Souslov, Anton; Liu, Andrea J.; Lubensky, Tom C.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 the perimeter. We analytically study the approach to this isostatic limit as the spring constant k' for the next-nearest-neighbor bond vanishes. We identify a characteristic frequency ω* ~ √(k') and length ω* ~ √(k/k' ) for both lattices. The shear modules C44 = k' of the square lattice vanishes with k', but that for the kagome lattice does not.Publication Rheological Microscopy: Local Mechanical Properties from Microrheology(2003-03-14) Chen, D. T.; Crocker, John C; Weeks, E. R.; Islam, M. F.; Verma, R.; Gruber, J.; Lubensky, Thomas C.; Levine, A. J.; Yodh, A. G.We demonstrate how tracer microrheology methods can be extended to study submicron scale variations in the viscoelastic response of soft materials; in particular, a semidilute solution of lambda-DNA. The polymer concentration is depleted near the surfaces of the tracer particles, within a distance comparable to the polymer correlation length. The rheology of this microscopic layer alters the tracers’ motion and can be precisely quantified using one- and two-point microrheology. Interestingly, we found this mechanically distinct layer to be twice as thick as the layer of depleted concentration, likely due to solvent drainage through the locally perturbed polymer structure.Publication Effect of Randomness on Critical Behavior of Spin Models(1975) Lubensky, Thomas C.; Harris, A. BrooksRenormalization 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 1Publication Smectic-C Tilt Under Shear in Smectic-A Elastomers(2008-08-08) Stenull, Olaf; Lubensky, Thomas C.; Adams, J. M.; Warner, MarkStenull and Lubensky [Phys. Rev. E 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 [e-print arXiv:0708.2024; Phys. Rev. E 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 71, 021708 (2005)] and use it to study the magnitude of SmC-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 sheetlike sample, thus inducing instabilities that we discuss.Publication Influence of Particle Shape on Bending Rigidity of Colloidal Monolayer Membranes and Particle Deposition during Droplet Evaporation in Confined Geometries(2012-06-01) Yunker, Peter J; Gratale, Matthew; Lohr, Matthew A; Lubensky, Thomas C.; Still, Tim; Yodh, Arjun G.We investigate the influence of particle shape on the bending rigidity of colloidal monolayer membranes (CMMs) and on evaporative processes associated with these membranes. Aqueous suspensions of colloidal particles are confined between glass plates and allowed to evaporate. Confinement creates ribbonlike air-water interfaces and facilitates measurement and characterization of CMM geometry during drying. Interestingly, interfacial buckling events occur during evaporation. Extension of the description of buckled elastic membranes to our quasi-2D geometry enables the determination of the ratio of CMM bending rigidity to its Young’s modulus. Bending rigidity increases with increasing particle anisotropy, and particle deposition during evaporation is strongly affected by membrane elastic properties. During drying, spheres are deposited heterogeneously, but ellipsoids are not. Apparently, increased bending rigidity reduces contact line bending and pinning and induces uniform deposition of ellipsoids. Surprisingly, suspensions of spheres doped with a small number of ellipsoids are also deposited uniformly.Publication Renormalization-Group Approach to Percolation Problems(1975-08-11) Harris, A. Brooks; Lubensky, Thomas C; Holcomb, William K; Dasgupta, ChandanThe relation between the s-state Ashkin-Teller-Potts (ATP) model and the percolation problem given by Fortuin and Kasteleyn is used to formulate a renormalization-group treatment of the percolation problem. Both an ε expansion near 6 spatial dimensions and cluster approximations for the recursion relations of a triangular lattice are used. Series results for the ATP model are adapted to the percolation problem.Publication Renormalization-Group Treatment of the Random Resistor Network in 6−ε Dimensions(1978-02-01) Harris, A. Brooks; Dasgupta, Chandan; Lubensky, Thomas CWe consider a hypercubic lattice in which neighboring points are connected by resistances which assume independently the random values σ>−1 and σ<−1 with respective probabilities p and 1−p. For σ<=0 the lattice is viewed as consisting of irreducible nodes connected by chains of path length L. This geometrical length is distinct from the characteristic length Lr which sets a scale of resistance in the random network or Lm which sets a scale of effective exchange in a dilute magnet. Near the percolation concentration pc one sets L~|p−pc|−ζ, Lr~|p−pc|−ζr and Lm~|p−pc|−ζm. Stephen and Grest (SG) have already shown that ζm=1+o(ε2) for spatial dimensionality d=6−ε. Here we show in a way similar to SG that ζr=1+o(ε2). Thus it is possible that ζm=ζr=1 for a continuous range of d below 6. However, increasing evidence suggests that this equality does not hold for d<4, and in particular a calculation in 1+ε dimensions analogous to that of SG for ζm does not seem possible.Publication Flux Phases in Two-Dimensional Tight-Binding Models(1989-08-01) Harris, A. Brooks; Lubensky, Tom C; Mele, Eugene JUsing a gauge-invariant tight-binding model on a rigid square lattice, we discuss the transition between a low-temperature flux phase in which orbital magnetic moments alternate antiferromagnetically in sign from plaquette to plaquette and a normal metallic phase. The order parameter, which may be chosen to be the magnetic flux penetrating a plaquette, goes continuously to zero at the transition. We also consider similar phases in a model with m spin colors antiferromagnetically exchange coupled.Publication Effective-medium theory of a filamentous triangular lattice(2013-04-09) Stenull, Olaf; Mao, Xiaoming; Lubensky, Thomas C.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 pb 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 pCF≃2/3 of the lattice when fiber bending rigidity vanishes.Publication ε Expansion for the Conductivity of a Random Resistor Network(1984-08-20) Harris, A. Brooks; Lubensky, Tom C.; Kim, SoobangWe present a reanalysis of the renormalization-group calculation to first order in ε=6−d, where d is the spatial dimensionality, of the exponent, t, which describes the behavior of the conductivity of a percolating network at the percolation threshold. If we set t=(d−2)νp+ζ, where νp is the correlation-length exponent, then our result is ζ=1+(ε/42). This result clarifies several previously paradoxical results concerning resistor networks and shows that the Alexander-Orbach relation breaks down at order ε.