Phase Transition of a Model of Crystalline Membrane

Abstract

We study two-dimensional triangulated surfaces of sphere topology by the canonical Monte Carlo simulation. The coordination number of surfaces is made as uniform as possible. The triangulation is fixed in MC so that only the positions X of vertices may be considered as the dynamical variable. The well-known Helfrich energy function S = S 1 + bS 2 is used for the definition of the model where S 1 and S 2 are the area and bending energy functions respectively and b is the bending rigidity. The discretizations of S 1 and S 2 are identical with that of our previous MC study for a model of fluid membranes. We find that the specific heats have peaks at finite bending rigidities and obtain the critical exponents of the phase transition by the finite-size scaling technique. It is found that our model of crystalline membranes undergoes an expected second order phase transition.

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

“…In fact, it was reported recently that the Nambu-Goto model with a deficit angle term, which is an intrinsic curvature, is well-defined and undergoes a discontinuous transition between the smooth phase and a tubular phase [21]. The second model in this paper is also well-defined [31,19], because the Hamiltonian includes a bending energy, which is an extrinsic curvature defined according to the dual lattice formulation of the discrete mechanics by Lee [37].…”

“…Tethered surface models are defined on triangulated fixed connectivity surfaces representing polymerized biological membranes or membranes in the gel phase [7], and they are classified into a major class of the HPK model [15,16,17,18,19,20,21,22,23,24,25,26]. Fluid surface models are considered a different class of the HPK model defined on dynamically triangulated surfaces representing these biological membranes in the fluid phase, however, we will not discuss the fluid surface model in this paper [27,28,29,30,31,32,33,34].…”

Phase transitions of a tethered membrane model with intrinsic curvature on spherical surfaces with point boundaries

“…The surface models can be classified into two groups, which are characterized by the curvature energy in the Hamiltonian; one is an extrinsic curvature model [15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32], and the other is an intrinsic curvature model [33,34,35,36]. The extrinsic curvature model is known to undergo a first-order transition between the smooth phase and the crumpled phase on tethered spherical surfaces [22,23,24].…”

First-order phase transition of fixed connectivity surfaces

“…However, little attention has been given to the dependence of the phase transition on the Hamiltonian of tethered surfaces both for models that have surface tension [12,13,14,15,16,17,18,19,20,21] and for tensionless models [22,23,24,25,26,27]. Almost all numerical studies done so far utilize the bending energy of the ordinary form 1−n i · n j , where n i is the normal vector of the triangle i.…”