47 research outputs found
Interactions and Correlations of Particulate Inclusions in a Columnar Phase
We calculate the elastic field mediated interaction between macroscopic
particles in a columnar hexagonal phase. The interaction is found to be
long-ranged and non-central, with both attractive and repulsive parts. We show
how the interaction modifies the particle correlations and the column
fluctuations. We also calculate the interaction of particles with the
topological defects of the columnar phase. The particle-defect interaction
reduces the mobility of the defects.Comment: RevTeX4 8 pages, 4 eps figures, submitted to Euro. Phys. J.
A geometric formulation of Schaefer's theory of Cosserat solids
The Cosserat solid is a theoretical model of a continuum whose elementary
constituents are notional rigid bodies. Here we present a formulation of the
mechanics of a Cosserat solid in the language of modern differential geometry
and exterior calculus, motivated by Schaefer's "motor field" theory. The solid
is modelled as a principal fibre bundle and configurations are related by
translations and rotations of each constituent. This kinematic property is
described in a coordinate-independent manner by a bundle map. Configurations
are equivalent if this bundle map is a global Euclidean isometry. Inequivalent
configurations, representing deformations of the solid, are characterised by
the local structure of the bundle map. Using Cartan's magic formula we show
that the strain associated with infinitesimal deformations is the Lie
derivative of a connection one-form on the bundle, revealing it to be a Lie
algebra-valued one-form. Extending Schaefer's theory, we derive the finite
strain by integrating the infinitesimal strain along a prescribed path. This is
path independent when the curvature of the connection one-form is zero. Path
dependence signals the presence of topological defects and the non-zero
curvature is then recognised as the density of topological defects. Mechanical
stresses are defined by a virtual work principle in which the Lie
algebra-valued strain one-form is paired with a dual Lie algebra-valued stress
two-form to yield a scalar work volume form. The d'Alembert principle for the
work form provides the balance laws, which is shown to be integrable for a
hyperelastic Cosserat solid. The breakdown of integrability, relevant to active
oriented solids, is briefly examined. Our work elucidates the geometric
structure of Cosserat solids, aids in constitutive modelling of active oriented
materials, and suggests structure-preserving integration schemes.Comment: 15 pages, 7 figure
Diffusivity dependence of the transition path ensemble
Transition pathways of stochastic dynamical systems are typically
approximated by instantons. Here we show, using a dynamical system containing
two competing pathways, that at low-to-intermediate temperatures, instantons
can fail to capture the most likely transition pathways. We construct an
approximation which includes fluctuations around the instanton and, by
comparing with the results of an accurate and efficient path-space Monte Carlo
sampling method, find this approximation to hold for a wide range of
temperatures. Our work delimits the applicability of large deviation theory and
provides methods to probe these limits numerically.Comment: 5 pages, 4 figure
Internal friction controls active ciliary oscillations near the instability threshold.
Ciliary oscillations driven by molecular motors cause fluid motion at micron scale. Stable oscillations require a substantial source of dissipation to balance the energy input of motors. Conventionally, it stems from external fluid. We show, in contrast, that external fluid friction is negligible compared to internal elastic stress through a simultaneous measurement of motion and flow field of an isolated and active Chlamydomonas cilium beating near the instability threshold. Consequently, internal friction emerges as the sole source of dissipation for ciliary oscillations. We combine these experimental insights with theoretical modeling of active filaments to show that an instability to oscillations takes place when active stresses are strain softening and shear thinning. Together, our results reveal a counterintuitive mechanism of ciliary beating and provide a general experimental and theoretical methodology to analyze other active filaments, both biological and synthetic ones