292 research outputs found
Hamilton's equations for a fluid membrane: axial symmetry
Consider a homogenous fluid membrane, or vesicle, described by the
Helfrich-Canham energy, quadratic in the mean curvature. When the membrane is
axially symmetric, this energy can be viewed as an `action' describing the
motion of a particle; the contours of equilibrium geometries are identified
with particle trajectories. A novel Hamiltonian formulation of the problem is
presented which exhibits the following two features: {\it (i)} the second
derivatives appearing in the action through the mean curvature are accommodated
in a natural phase space; {\it (ii)} the intrinsic freedom associated with the
choice of evolution parameter along the contour is preserved. As a result, the
phase space involves momenta conjugate not only to the particle position but
also to its velocity, and there are constraints on the phase space variables.
This formulation provides the groundwork for a field theoretical generalization
to arbitrary configurations, with the particle replaced by a loop in space.Comment: 11 page
Untethered micro-robotic coding of three-dimensional material composition
Complex functional materials with three-dimensional micro- or nano-scale dynamic compositional features are prevalent in nature. However, the generation of three-dimensional functional materials composed of both soft and rigid microstructures, each programmed by shape and composition, is still an unsolved challenge. Herein, we describe a method to code complex materials in three-dimensions with tunable structural, morphological, and chemical features using an untethered magnetic micro-robot remotely controlled by magnetic fields. This strategy allows the micro-robot to be introduced to arbitrary microfluidic environments for remote two- and three-dimensional manipulation. We demonstrate the coding of soft hydrogels, rigid copper bars, polystyrene beads, and silicon chiplets into three-dimensional heterogeneous structures. We also use coded microstructures for bottom-up tissue engineering by generating cell-encapsulating constructs
Axially symmetric membranes with polar tethers
Axially symmetric equilibrium configurations of the conformally invariant
Willmore energy are shown to satisfy an equation that is two orders lower in
derivatives of the embedding functions than the equilibrium shape equation, not
one as would be expected on the basis of axial symmetry. Modulo a translation
along the axis, this equation involves a single free parameter c.If c\ne 0, a
geometry with spherical topology will possess curvature singularities at its
poles. The physical origin of the singularity is identified by examining the
Noether charge associated with the translational invariance of the energy; it
is consistent with an external axial force acting at the poles. A one-parameter
family of exact solutions displaying a discocyte to stomatocyte transition is
described.Comment: 13 pages, extended and revised version of Non-local sine-Gordon
equation for the shape of axi-symmetric membrane
Helfrich-Canham bending energy as a constrained non-linear sigma model
The Helfrich-Canham bending energy is identified with a non-linear sigma
model for a unit vector. The identification, however, is dependent on one
additional constraint: that the unit vector be constrained to lie orthogonal to
the surface. The presence of this constraint adds a source to the divergence of
the stress tensor for this vector so that it is not conserved. The stress
tensor which is conserved is identified and its conservation shown to reproduce
the correct shape equation.Comment: 5 page
Guided and magnetic self-assembly of tunable magnetoceptive gels
Self-assembly of components into complex functional patterns at microscale is common in nature, and used increasingly in numerous disciplines such as optoelectronics, microfabrication, sensors, tissue engineering and computation. Here, we describe the use of stable radicals to guide the self-assembly of magnetically tunable gels, which we call ‘magnetoceptive’ materials at the scale of hundreds of microns to a millimeter, each can be programmed by shape and composition, into heterogeneous complex structures. Using paramagnetism of free radicals as a driving mechanism, complex heterogeneous structures are built in the magnetic field generated by permanent magnets. The overall magnetic signature of final structure is erased via an antioxidant vitamin E, subsequent to guided self-assembly. We demonstrate unique capabilities of radicals and antioxidants in fabrication of soft systems with heterogeneity in material properties, such as porosity, elastic modulus and mass density; then in bottom-up tissue engineering and finally, levitational and selective assembly of microcomponents
Recapitulating cranial osteogenesis with neural crest cells in 3-D microenvironments
The experimental systems that recapitulate the complexity of native tissues and enable precise control over the microenvironment are becoming essential for the pre-clinical tests of therapeutics and tissue engineering. Here, we described a strategy to develop an in vitro platform to study the developmental biology of craniofacial osteogenesis. In this study, we directly osteo-differentiated cranial neural crest cells (CNCCs) in a 3-D in vitro bioengineered microenvironment. Cells were encapsulated in the gelatin-based photo-crosslinkable hydrogel and cultured up to three weeks. We demonstrated that this platform allows efficient differentiation of p75 positive CNCCs to cells expressing osteogenic markers corresponding to the sequential developmental phases of intramembranous ossification. During the course of culture, we observed a decrease in the expression of early osteogenic marker Runx2, while the other mature osteoblast and osteocyte markers such as Osterix, Osteocalcin, Osteopontin and Bone sialoprotein increased. We analyzed the ossification of the secreted matrix with alkaline phosphatase and quantified the newly secreted hydroxyapatite. The Field Emission Scanning Electron Microscope (FESEM) images of the bioengineered hydrogel constructs revealed the native-like osteocytes, mature osteoblasts, and cranial bone tissue morphologies with canaliculus-like intercellular connections. This platform provides a broadly applicable model system to potentially study diseases involving primarily embryonic craniofacial bone disorders, where direct diagnosis and adequate animal disease models are limited
Lipid membranes with an edge
Consider a lipid membrane with a free exposed edge. The energy describing
this membrane is quadratic in the extrinsic curvature of its geometry; that
describing the edge is proportional to its length. In this note we determine
the boundary conditions satisfied by the equilibria of the membrane on this
edge, exploiting variational principles. The derivation is free of any
assumptions on the symmetry of the membrane geometry. With respect to earlier
work for axially symmetric configurations, we discover the existence of an
additional boundary condition which is identically satisfied in that limit. By
considering the balance of the forces operating at the edge, we provide a
physical interpretation for the boundary conditions. We end with a discussion
of the effect of the addition of a Gaussian rigidity term for the membrane.Comment: 8 page
Stresses in lipid membranes
The stresses in a closed lipid membrane described by the Helfrich
hamiltonian, quadratic in the extrinsic curvature, are identified using
Noether's theorem. Three equations describe the conservation of the stress
tensor: the normal projection is identified as the shape equation describing
equilibrium configurations; the tangential projections are consistency
conditions on the stresses which capture the fluid character of such membranes.
The corresponding torque tensor is also identified. The use of the stress
tensor as a basis for perturbation theory is discussed. The conservation laws
are cast in terms of the forces and torques on closed curves. As an
application, the first integral of the shape equation for axially symmetric
configurations is derived by examining the forces which are balanced along
circles of constant latitude.Comment: 16 pages, introduction rewritten, other minor changes, new references
added, version to appear in Journal of Physics
Geometry of lipid vesicle adhesion
The adhesion of a lipid membrane vesicle to a fixed substrate is examined
from a geometrical point of view. This vesicle is described by the Helfrich
hamiltonian quadratic in mean curvature; it interacts by contact with the
substrate, with an interaction energy proportional to the area of contact. We
identify the constraints on the geometry at the boundary of the shared surface.
The result is interpreted in terms of the balance of the force normal to this
boundary. No assumptions are made either on the symmetry of the vesicle or on
that of the substrate. The strong bonding limit as well as the effect of
curvature asymmetry on the boundary are discussed.Comment: 7 pages, some major changes in sections III and IV, version published
in Physical Review
Modelling the dynamics of global monopoles
A thin wall approximation is exploited to describe a global monopole coupled
to gravity. The core is modelled by de Sitter space; its boundary by a thin
wall with a constant energy density; its exterior by the asymptotic
Schwarzschild solution with negative gravitational mass and solid angle
deficit, , where is the symmetry
breaking scale. The deficit angle equals when . We find that: (1) if , there exists a unique globally
static non-singular solution with a well defined mass, . provides
a lower bound on . If , the solution oscillates. There are no
inflating solutions in this symmetry breaking regime. (2) if ,
non-singular solutions with an inflating core and an asymptotically
cosmological exterior will exist for all . (3) if is not too large,
there exists a finite range of values of where a non-inflating monopole
will also exist. These solutions appear to be metastable towards inflation. If
is positive all solutions are singular. We provide a detailed description
of the configuration space of the model for each point in the space of
parameters, and trace the wall trajectories on both the interior
and the exterior spacetimes. Our results support the proposal that topological
defects can undergo inflation.Comment: 44 pages, REVTeX, 11 PostScript figures, submitted to the Physical
Review D. Abstract's correcte
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