T. C. Lubensky

Nonlinear elasticity in biological gels

The mechanical properties of soft biological tissues are essential to their physiological function and cannot easily be duplicated by synthetic materials. Unlike simple polymer gels, many biological materials—including blood vessels, mesentery tissue, lung parenchyma, cornea and blood clots—stiffen as they are strained, thereby preventing large deformations that could threaten tissue integrity. The molecular structures and design principles responsible for this nonlinear elasticity are unknown. Here we report a molecular theory that accounts for strain-stiffening in a range of molecularly distinct gels formed from cytoskeletal and extracellular proteins and that reveals universal stress–strain relations at low to intermediate strains. The input to this theory is the force–extension curve for individual semi-flexible filaments and the assumptions that biological networks composed of these filaments are homogeneous, isotropic, and that they strain uniformly. This theory shows that systems of filamentous proteins arranged in an open crosslinked mesh invariably stiffen at low strains without requiring a specific architecture or multiple elements with different intrinsic stiffness.

TOPOLOGICAL DEFECTS AND INTERACTIONS IN NEMATIC EMULSIONS

Inverse nematic emulsions, in which surfactant-coated water droplets are dispersed in a nematic host fluid, have distinctive properties that set them apart from dispersions of two isotropic fluids or of nematic droplets in an isotropic fluid. We present a comprehensive theoretical study of the distortions produced in the nematic host by the dispersed droplets and of solvent-mediated dipolar interactions between droplets that lead to their experimentally observed chaining. A single droplet in a nematic host acts like a macroscopic hedgehog defect. Global boundary conditions force the nucleation of compensating topological defects in the nematic host. Using variational techniques, we show that in the lowest energy configuration, a single water droplet draws a single hedgehog out of the nematic host to form a tightly bound dipole. Configurations in which the water droplet is encircled by a disclination ring have higher energy. The droplet dipole induces distortions in the nematic host that lead to an effective dipole-dipole interaction between droplets, and hence to chaining.

One- and two-particle microrheology

We study the dynamics of rigid spheres embedded in viscoelastic media and address two questions of importance to microrheology. First, we calculate the complete response to an external force of a single bead in a homogeneous elastic network viscously coupled to an incompressible fluid. From this response function we find the frequency range where the standard assumptions of microrheology are valid. Second, we study fluctuations when embedded spheres perturb the media around them and show that mutual fluctuations of two separated spheres provide a more accurate determination of the complex shear modulus than do the fluctuations of a single sphere.

Topological boundary modes in isostatic lattices

The mathematical connection between isostatic lattices—which are relevant for granular matter, glasses and other ‘soft’ systems—and topological quantum matter is as deep as it is unexpected.

Molecular Chirality and Chiral Parameter

The fundamental issues of symmetry related to chirality are discussed and applied to simple situations relevant to liquid crystals. We show that any chiral measure of a geometric object is a pseudoscalar (invariant under proper rotations but changing sign under improper rotations) and must involve three-point correlations which only come into play when the molecule has at least four atoms. In general, a molecule is characterized by an infinite set of chiral parameters. We illustrate the fact that these parameters can have differing signs and can vanish at different points as a molecule is continuously deformed into its mirror image. From this it is concluded that handedness is not an absolute concept but depends on the property being observed. Within a simplified model of classical interactions, we identify the chiral parameter of the constituent molecules which determines the macroscopic pitch of cholesterics.

Particle dynamics in sheared granular matter

The particle dynamics and shear forces of granular matter in a Couette geometry are determined experimentally. The normalized tangential velocity V(y) declines strongly with distance y from the moving wall, independent of the shear rate and of the shear dynamics. Local rms velocity fluctuations deltaV(y) scale with the local velocity gradient to the power 0.4+/-0.05. These results agree with a locally Newtonian, continuum model, where the granular medium is assumed to behave as a liquid with a local temperature [deltaV(y)](2) and density dependent viscosity.

Criticality and isostaticity in fibre networks

In fibre networks, mechanical stability relies on the fibres’ bending resistance—in contrast to rubbers, where entropic stretching is the key. The extent to which the mechanics of fibre networks is controlled by bending is, however, an open question. The study of a general lattice-based model of fibrous networks now reveals two rigidity critical points, one of which controls a rich crossover from stretching-dominated to bending-dominated behaviour.

Response function of a sphere in a viscoelastic two-fluid medium.

In order to address basic questions of importance to microrheology, we study the dynamics of a rigid sphere embedded in a model viscoelastic medium consisting of an elastic network permeated by a viscous fluid. We calculate the complete response of a single bead in this medium to an external force, and compare the result to the commonly-accepted, generalized Stokes-Einstein relation (GSER). We find that our response function is well approximated by the GSER only within a particular frequency range determined by the material parameters of both the bead and the network. We then discuss the relevance of this result to recent experiments. Finally we discuss the approximations made in our solution of the response function by comparing our results to the exact solution for the response function of a bead in a viscous (Newtonian) fluid.

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 that the requirement that a particles 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 of examples with spatially varying diffusion coefficients. We also derive a path integral representation for arbitrary interpretation of the noise, and use it in a perturbative study of correlations in a simple system.

Theory of bent-core liquid-crystal phases and phase transitions.

We study phases and phase transitions that can take place in the recently discovered bow-shaped or bent-core liquid-crystal molecules. We show that to completely characterize phases exhibited by such bent-core molecules a third-rank tensor T(ijk) order parameter is necessary in addition to the vector and the nematic (second-rank) tensor order parameters. We present an exhaustive list of possible liquid phases, characterizing them by their space-symmetry group and order parameters, and catalog the universality classes of the corresponding phase transitions that we expect to take place in such bent-core molecular liquid crystals. In addition to the conventional liquid-crystal phases such as the nematic phase, we predict the existence of other liquid phases, including the spontaneously chiral nematic (N(T)+2)(*) and chiral polar (V(T)+2)(*) phases, the orientationally ordered but optically isotropic tetrahedratic T phase, and a nematic N(T) phase with D(2d) symmetry that is neither uniaxial nor biaxial. Interestingly, the isotropic-tetrahedratic transition is continuous in mean-field theory, but is likely driven first order by thermal fluctuations. We conclude with a discussion of smectic analogs of these phases and their experimental signatures.