Abstract
We numerically study surface models defined on hexagonal disks with a free boundary. 2D surface models for planar surfaces have recently attracted interest due to the engineering applications of functional materials such as graphene and its composite with polymers. These 2D composite meta-materials are strongly influenced by external stimuli such as thermal fluctuations if they are sufficiently thin. For this reason, it is very interesting to study the shape stability/instability of thin 2D materials against thermal fluctuations. In this paper, we study three types of surface models including Landau-Ginzburg (LG) and Helfirch-Polyakov models defined on triangulated hexagonal disks using the parallel tempering Monte Carlo simulation technique. We find that the planar surfaces undergo a first-order transition between the smooth and crumpled phases in the LG model and continuous transitions in the other two models. The first-order transition is relatively weak compared to the transition on spherical surfaces already reported. The continuous nature of the transition is consistent with the reported results, although the transitions are stronger than that of the reported ones.
Highlights
The two-dimensional surface model proposed by Helfrich is a model for biological membranes composed of lipid molecules, and it shares almost the same mathematical structure with the Polyakov’s rigid string model for elementary particles in subatomic scales [1,2]
The so-called parallel tempering Monte Carlo (PTMC) technique developed for the spin glass model at low temperatures is successfully applied to the first-order crumpling transition of the canonical model on spherical lattices [28]
This nP1 is fixed to nP1 = 10, and this implies that the total number of Metropolis MC (MMC)
Summary
The two-dimensional surface model proposed by Helfrich is a model for biological membranes composed of lipid molecules, and it shares almost the same mathematical structure with the Polyakov’s rigid string model for elementary particles in subatomic scales [1,2]. For almost all of these discrete models, MC studies predict that the models undergo a first-order crumpling transition if the lattice is of spherical topology and allowed to self-intersect (⇔ self-intersecting) [20]. Intrinsic curvature models have a first-order crumpling transition even on a disk surface [21], and the intrinsic curvature models are out of consideration in this paper. We expect that the PTMC technique can be used to study the phase structure of the surface models in this paper even if these models have first-order transitions [28]. The surface condition of graphenes is altered by corrugations, and ripples, wrinkles and crumples emerge [33,34] These surface states modify or enhance the material properties such as mechanical, electrical and optical ones [35].
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