91 research outputs found
Fixed-Connectivity Membranes
The statistical mechanics of flexible surfaces with internal elasticity and
shape fluctuations is summarized. Phantom and self-avoiding isotropic and
anisotropic membranes are discussed, with emphasis on the universal negative
Poisson ratio common to the low-temperature phase of phantom membranes and all
strictly self-avoiding membranes in the absence of attractive interactions. The
study of crystalline order on the frozen surface of spherical membranes is also
treated.Comment: Chapter 11 in "Statistical mechanics of Membranes and Surfaces", ed.
by D.R. Nelson, T. Piran and S. Weinberg (World Scientific, Singapore, 2004);
25 pages with 26 figures (high resolution figures available from author
Delaunay Surfaces
We derive parametrizations of the Delaunay constant mean curvature surfaces
of revolution that follow directly from parametrizations of the conics that
generate these surfaces via the corresponding roulette. This uniform treatment
exploits the natural geometry of the conic (parabolic, elliptic or hyperbolic)
and leads to simple expressions for the mean and Gaussian curvatures of the
surfaces as well as the construction of new surfaces.Comment: 16 pages, 11 figure
Effects of scars on crystalline shell stability under external pressure
We study how the stability of spherical crystalline shells under external
pressure is influenced by the defect structure. In particular, we compare
stability for shells with a minimal set of topologically-required defects to
shells with extended defect arrays (grain boundary "scars" with non-vanishing
net disclination charge). We perform Monte Carlo simulations to compare how
shells with and without scars deform quasi-statically under external
hydrostatic pressure. We find that the critical pressure at which shells
collapse is lowered for scarred configurations that break icosahedral symmetry
and raised for scars that preserve icosahedral symmetry. The particular shapes
which arise from breaking of an initial icosahedrally-symmetric shell depend on
the F\"oppl-von K\'arm\'an number.Comment: 8 pages, 6 figure
Shapes and singularities in triatic liquid crystal vesicles
Determining the equilibrium configuration and shape of curved two-dimensional
films with (generalized) liquid crystalline (LC) order is a difficult infinite
dimensional problem of direct relevance to the study of generalized
polymersomes, soft matter and the fascinating problem of understanding the
origin and formation of shape (morphogenesis). The symmetry of the free energy
of the LC film being considered and the topology of the surface to be
determined often requires that the equilibrium configuration possesses singular
structures in the form of topological defects such as disclinations for nematic
films. The precise number and type of defect plays a fundamental role in
restricting the space of possible equilibrium shapes. Flexible closed vesicles
with spherical topology and nematic or smectic order, for example, inevitably
possess four elementary strength disclination defects positioned at the
four vertices of a tetrahedral shell. Here we address the problem of
determining the equilibrium shape of flexible vesicles with generalized LC
order. The order parameter in these cases is an element of , for any
positive integer . We will focus on the case , known as triatic LCs.
We construct the appropriate order parameter for triatics and find the
associated free energy. We then describe the structure of the elementary
defects of strength in flat space. Finally, we prove that sufficiently
floppy triatic vesicles with the topology of the 2-sphere equilibrate to
octahedral shells with strength defects at each of the six vertices,
independently of scale.Comment: New results and new sections added, 4 new figures and updated
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Topological Sound and Flocking on Curved Surfaces
Active systems on curved geometries are ubiquitous in the living world. In
the presence of curvature orientationally ordered polar flocks are forced to be
inhomogeneous, often requiring the presence of topological defects even in the
steady state due to the constraints imposed by the topology of the underlying
surface. In the presence of spontaneous flow the system additionally supports
long-wavelength propagating sound modes which get gapped by the curvature of
the underlying substrate. We analytically compute the steady state profile of
an active polar flock on a two-sphere and a catenoid, and show that curvature
and active flow together result in symmetry protected topological modes that
get localized to special geodesics on the surface (the equator or the neck
respectively). These modes are the analogue of edge states in electronic
quantum Hall systems and provide unidirectional channels for information
transport in the flock, robust against disorder and backscattering.Comment: 15 pages, 6 figure
Defect unbinding in active nematics
We formulate the statistical dynamics of topological defects in the active
nematic phase, formed in two dimensions by a collection of self-driven
particles on a substrate. An important consequence of the non-equilibrium drive
is the spontaneous motility of strength +1/2 disclinations. Starting from the
hydrodynamic equations of active nematics, we derive an interacting particle
description of defects that includes active torques. We show that activity,
within perturbation theory, lowers the defect-unbinding transition temperature,
determining a critical line in the temperature-activity plane that separates
the quasi-long-range ordered (nematic) and disordered (isotropic) phases. Below
a critical activity, defects remain bound as rotational noise decorrelates the
directed dynamics of +1/2 defects, stabilizing the quasi-long-range ordered
nematic state. This activity threshold vanishes at low temperature, leading to
a re-entrant transition. At large enough activity, active forces always exceed
thermal ones and the perturbative result fails, suggesting that in this regime
activity will always disorder the system. Crucially, rotational diffusion being
a two-dimensional phenomenon, defect unbinding cannot be described by a
simplified one-dimensional model.Comment: 15 pages (including SI), 4 figures. Significant technical
improvements without changing the result
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