Phase Transition of Meshwork Models for Spherical Membranes

Abstract

We have studied two types of meshwork models by using the canonical Monte Carlo simulation technique. The first meshwork model has elastic junctions, which are composed of vertices, bonds, and triangles, while the second model has rigid junctions, which are hexagonal (or pentagonal) rigid plates. Two-dimensional elasticity is assumed only at the elastic junctions in the first model, and no two-dimensional bending elasticity is assumed in the second model. Both of the meshworks are of spherical topology. We find that both models undergo a first-order collapsing transition between the smooth spherical phase and the collapsed phase. The Hausdorff dimension of the smooth phase is H ≃ 2 in both models as expected. It is also found that H ≃ 2 in the collapsed phase of the second model, and that H is relatively larger than 2 in the collapsed phase of the first model, but it remains in the physical bound, i.e., H < 3. Moreover, the first model undergoes a discontinuous surface fluctuation transition at the same transition point as that of the collapsing transition, while the second model undergoes a continuous transition of surface fluctuation. This indicates that the phase structure of the meshwork model is weakly dependent on the elasticity at the junctions.

Phase structure of a spherical surface model on fixed connectivity meshes

“…Meshwork models in [22,23] have no vertex inside the compartments, which have finite size n. The phase structure of such meshwork model of finite n is considered to be dependent on the elasticity of junctions [22,23]. Therefore, it is interesting to study the dependence of the surface fluctuation transition on n in the meshwork model, where the elasticity of junctions is identical to that in the model of this Letter.…”

Flat Histogram Monte Carlo Simulations of Triangulated Fixed-Connectivity Surface Models

“…In those inhomogeneous surface models in R 3 , the transitions separating two neighboring phases are discontinuous [21,22,23]. Thus, the first-order nature of transitions seems to be a common feature of the shape transformation transitions in the triangulated surface models.…”

“…By including certain inhomogeneous components such as cytoskeletal structure or holes in the abovementioned homogeneous models, we obtain a variety of surface models for numerical studies [21,22,23]. Lateral diffusion of lipids can also be implemented in the models by the so-called dynamical triangulation technique, which introduces non-uniform coordination numbers q to the triangulated surfaces [24].…”

“…Thus, we understand that the surface collapses and wrinkles in the limit of b → 0 while it swells and becomes smooth in the limit of b → ∞ [8,9,10]. Theoretical studies utilizing the renormalization group technique predict that the crumpling transition is continuous [11,12,13,14,15], while density-matrix renormalization group studies on the folding of triangular lattice [16] and recent numerical simulations of the model on triangulated surfaces [17,18,19,20] indicate that the crumpling transition is of the first-order and accompanies the collapsing transition.By including certain inhomogeneous components such as cytoskeletal structure or holes in the abovementioned homogeneous models, we obtain a variety of surface models for numerical studies [21,22,23]. Lateral diffusion of lipids can also be implemented in the models by the so-called dynamical triangulation technique, which introduces non-uniform coordination numbers q to the triangulated surfaces [24].…”

Spherical Surface Models with Directors

“…A well-known two-dimensional curvature energy is the so-called Helfrich and Polyakov Hamiltonian, which is rigorously defined by using the notions of two-dimensional differential geometry and plays a role for maintaining the surface shape [8,9,10]. A linear bending energy in the compartmentalized surfaces can also maintain the surface shape [11,12,13,14,15]. A unit tangential vector along the compartment defines the linear bending energy in those compartmentalized models [11,12,13,14,15], while a unit normal vector of the triangles defines the twodimensional bending energy of Helfrich and Polyakov in the conventional models [16,17,18].…”