Anisotropic Membranes

We describe the statistical behavior of anisotropic crystalline membranes. In particular we give the phase diagram and critical exponents for phantom membranes and discuss the generalization to self-avoiding membranes.Comment: LATTICE98(surfaces) 5 pages, 4 Postscript figure

Phase transition of an extrinsic curvature model on tori

We show a numerical evidence that a tethered surface model with extrinsic curvature undergoes a first-order crumpling transition between the smooth phase and a non-smooth phase on triangulated tori. The results obtained in this Letter together with the previous ones on spherical surfaces lead us to conclude that the tethered surface model undergoes a first-order transition on compact surfaces.Comment: 13 pages with 10 figure

Monte Carlo simulations of a tethered membrane model on a disk with intrinsic curvature

A first-order phase transition separating the smooth phase from the crumpled one is found in a fixed connectivity surface model defined on a disk. The Hamiltonian contains the Gaussian term and an intrinsic curvature term.Comment: 10 pages with 6 figure

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

Nonvanishing string tension of elastic membrane models

By using the grand canonical Monte Carlo simulations on spherical surfaces with two fixed vertices separated by the distance L, we find that the second-order phase transition changes to the first-order one when L is sufficiently large. We find that string tension \sigma \not= 0 in the smooth phase while \sigma \to 0 in the wrinkled phase.Comment: 10 pages with 6 figure

New Analytical Results on Anisotropic Membranes

We report on recent progress in understanding the tubular phase of self-avoiding anisotropic membranes. After an introduction to the problem, we sketch the renormalization group arguments and symmetry considerations that lead us to the most plausible fixed point structure of the model. We then employ an epsilon-expansion about the upper critical dimension to extrapolate to the physical interesting 3-dimensional case. The results are $\nu=0.62$ for the Flory exponent and $\zeta=0.80$ for the roughness exponent. Finally we comment on the importance that numerical tests may have to test these predictions.Comment: LATTICE98(surfaces), 3 pages, 2 eps figure

Pathways to faceting of vesicles

The interplay between geometry, topology and order can lead to geometric frustration that profoundly affects the shape and structure of a curved surface. In this commentary we show how frustration in this context can result in the faceting of elastic vesicles. We show that, under the right conditions, an assortment of regular and irregular polyhedral structures may be the low energy states of elastic membranes with spherical topology. In particular, we show how topological defects, necessarily present in any crystalline lattice confined to spherical topology, naturally lead to the formation of icosahedra in a homogeneous elastic vesicle. Furthermore, we show that introducing heterogeneities in the elastic properties, or allowing for non-linear bending response of a homogeneous system, opens non-trivial pathways to the formation of faceted, yet non-icosahedral, structures

Paraboloidal Crystals

The interplay between order and geometry in soft condensed matter systems is an active field with many striking results and even more open problems. Ordered structures on curved surfaces appear in multi-electron helium bubbles, viral and bacteriophage protein capsids, colloidal self-assembly at interfaces and in physical membranes. Spatial curvature can lead to novel ground state configurations featuring arrays of topological defects that would be excited states in planar systems. We illustrate this with a sequence of images showing the Voronoi lattice (in gold) and the corresponding Delaunay triangulations (in green) for ten low energy configurations of a system of classical charges constrained to lie on the surface of a paraboloid and interacting with a Coulomb potential. The parabolic geometry is considered as a specific realization of the class of crystalline structures on two-dimensional Riemannian manifolds with variable Gaussian curvature and boundary.Comment: 2 page

First-order phase transition of the tethered membrane model on spherical surfaces

We found that three types of tethered surface model undergo a first-order phase transition between the smooth and the crumpled phase. The first and the third are discrete models of Helfrich, Polyakov, and Kleinert, and the second is that of Nambu and Goto. These are curvature models for biological membranes including artificial vesicles. The results obtained in this paper indicate that the first-order phase transition is universal in the sense that the order of the transition is independent of discretization of the Hamiltonian for the tethered surface model.Comment: 22 pages with 14 figure

Phase transitions of a tethered membrane model on a torus with intrinsic curvature

A tethered surface model is investigated by using the canonical Monte Carlo simulation technique on a torus with an intrinsic curvature. We find that the model undergoes a first-order phase transition between the smooth phase and the crumpled one.Comment: 12 pages with 8 figure