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

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

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

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

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

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

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 $M$ and solid angle deficit, $\Delta\Omega/4\pi = 8\pi G\eta^2$, where $\eta$ is the symmetry breaking scale. The deficit angle equals $4\pi$ when $\eta=1/\sqrt{8\pi G} \equiv M_p$. We find that: (1) if $\eta <M_p$, there exists a unique globally static non-singular solution with a well defined mass, $M_0<0$. $M_0$ provides a lower bound on $M$. If $M_0<M<0$, the solution oscillates. There are no inflating solutions in this symmetry breaking regime. (2) if $\eta \ge M_p$, non-singular solutions with an inflating core and an asymptotically cosmological exterior will exist for all $M<0$. (3) if $\eta$ is not too large, there exists a finite range of values of $M$ where a non-inflating monopole will also exist. These solutions appear to be metastable towards inflation. If $M$ 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, $(\eta, M)$ 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