Small water droplets dispersed in a nematic liquid crystal exhibit a novel class of colloidal interactions, arising from the orientational elastic energy of the anisotropic host fluid. These interactions include a short-range repulsion and a long-range dipolar attraction, and they lead to the formation of anisotropic chainlike structures by the colloidal particles. The repulsive interaction can lead to novel mechanisms for colloid stabilization.
Frames, or lattices consisting of mass points connected by rigid bonds or central-force springs, are important model constructs that have applications in such diverse fields as structural engineering, architecture, and materials science. The difference between the number of bonds and the number of degrees of freedom of these lattices determines the number of their zero-frequency "floppy modes". When these are balanced, the system is on the verge of mechanical instability and is termed isostatic. It has recently been shown that certain extended isostatic lattices exhibit floppy modes localized at their boundary. These boundary modes are insensitive to local perturbations, and appear to have a topological origin, reminiscent of the protected electronic boundary modes that occur in the quantum Hall effect and in topological insulators. In this paper we establish the connection between the topological mechanical modes and the topological band theory of electronic systems, and we predict the existence of new topological bulk mechanical phases with distinct boundary modes. We introduce model systems in one and two dimensions that exemplify this phenomenon.Isostatic lattices provide a useful reference point for understanding the properties of a wide range of systems on the verge of mechanical instability, including network glasses 1,2 , randomly diluted lattices near the rigidity percolation threshold 3,4 , randomly packed particles near their jamming threshold 5-10 , and biopolymer networks [11][12][13][14] . There are many periodic lattices, including the square and kagome lattices in d = 2 dimensions and the cubic and pyrochlore lattices in d = 3, that are locally isostatic with coordination number z = 2d for every site under periodic boundary conditions. These lattices, which are the subject of this paper, have a surprisingly rich range of elastic responses and phonon structures [15][16][17][18][19] that exhibit different behaviors as bending forces or additional bonds are added.The analysis of such systems dates to an 1864 paper by James Clerk Maxwell 20 that argued that a lattice with N s mass points and N b bonds has N 0 = dN s − N b zero modes. Maxwell's count is incomplete, though, because N 0 can exceed dN s − N b if there are N ss states of self-stress, where springs can be under tension or compression with no net forces on the masses. This occurs, for example, when masses are connected by straight lines of bonds under periodic boundary conditions. A more general Maxwell relation 21 ,is valid for infinitesimal distortions. In a locally isostatic system with periodic boundary conditions, N 0 = N ss . The square and kagome lattices have one state of self-stress per straight line of bonds and associated zero modes along lines in momentum space. Cutting a section of N sites from these lattices removes states of selfstress and O( √ N ) bonds and necessarily leads to O( √ N ) zero modes, which are essentially identical to the bulk zero modes. Recently Sun et al. 22 studied a twisted kagome lattice in which states ...
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.
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 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 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. 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 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 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. Disciplines Physical Sciences and Mathematics | Physics
We studied the Brownian motion of isolated ellipsoidal particles in water confined to two dimensions and elucidated the effects of coupling between rotational and translational motion. By using digital video microscopy, we quantified the crossover from short-time anisotropic to long-time isotropic diffusion and directly measured probability distributions functions for displacements. We confirmed and interpreted our measurements by using Langevin theory and numerical simulations. Our theory and observations provide insights into fundamental diffusive processes, which are potentially useful for understanding transport in membranes and for understanding the motions of anisotropic macromolecules.
Disordered fibre networks are the basis of many man-made and natural materials, including structural components of living cells and tissue. The mechanical stability of such networks relies on the bending resistance of the fibres, in contrast to rubbers, which are governed by entropic stretching of polymer segments. Although it is known that fibre networks exhibit collective bending deformations, a fundamental understanding of such deformations and their effects on network mechanics has remained elusive. Here we introduce a lattice-based model of fibrous networks with variable connectivity to elucidate the roles of single-fibre elasticity and network structure. These networks exhibit both a low-connectivity rigidity threshold governed by fibre-bending elasticity and a high-connectivity threshold governed by fibre-stretching elasticity. Whereas the former determines the true onset of network rigidity, we show that the latter exhibits rich zero-temperature critical behaviour, including a crossover between various mechanical regimes along with diverging strain fluctuations and a concomitant diverging correlation length.
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