Folding Transitions of Self-Avoiding Membranes.

Abstract: The phase structure of self-avoiding polymerized membranes is studied by extensive Hybrid Monte Carlo simulations. Several folding transitions from the flat to a collapsed state are found. Using a suitable order parameter and finite size scaling theory, these transitions are shown to be of first order. The phase diagram in the temperature-field plane is given. PACS numbers: 87.22.Bt – 05.70.Jk The phase structure of self-avoiding polymerized membranes is studied by extensive Hybrid Monte Carlo simulations. Several folding transitions from the flat to a collapsed state are found. Using a suitable order parameter and finite size scaling theory, these transitions are shown to be of first order. The phase diagram in the temperature-field plane is given. PACS numbers: 87.22.Bt – 05.70.Jk Typeset using REVTEX 1 Fluctuating membranes and surfaces are basic structural elements of biological systems and complex fluids. Recent theoretical work [1,2] and experimental studies [3,4] indicate, that these sheetlike macromolecules should have dramatically different properties than linear polymers. Polymerized membranes which contain a permanently cross-linked network of constituent molecules have a shear elasticity, giving them a large entropic bending rigidity. In the absence of self-avoidance, polymerized [5] and fluid [6] membranes adopt a crumpled random structure. Theoretical predictions by Flory mean-field approximation and Monte Carlo simulations [5] and renormalization group studies [7] supported the existence of a high temperature crumpled phase for self-avoiding polymerized membranes also, suggesting a possible finite temperature crumpling transition in the presence of an explicit bending rigidity [8]. However, more extensive computer simulations [2,9–11] found no crumpling of self-avoiding tethered membranes in a good solvent. This prediction was confirmed recently by experimental studies of graphitic oxide [3]. On the other hand, polymerized vesicles undergo a wrinkling transition [4], and upon addition of 10 vol % acetone, Spector and co-workers [3] found small compact objects, which appeared to be folded. A poor solvent leads to (short-ranged) attractive interactions, and a single membrane was found to be flat for high temperatures [9], but in a collapsed state for sufficiently low temperatures [2]. The transition between the flat and the collapsed states of the membrane proceeds through a sequence of folding transitions, which were first found by cooling of a single membrane from the flat phase [12]. Because no hysteresis was found, it was ruled out that the folded configurations are metastable states. However, this method does not give sufficient evidence of the order or even the existence of a transition. For instance, hysteresis can also be found at second order phase transitions of finite systems and the results for one system size may be misleading. In addition, the experimentally observed wrinkling transition is first order [4]. On that account, we present a systematic finite size scaling analysis of the folding transitions. The Hybrid Monte Carlo algorithm was used, which provides for simulations of the canonical ensemble with constant temperature. Moreover, ensemble averages are inde2 pendent of the discretization step size δt, i.e. systematic errors [13,14]. A suitable order parameter is defined and a phase diagram is presented. Besides the nearest neighbour interactions, the membranes were modeled similar to those of Abraham and Kardar [12]. The N particles of the polymerized membrane form the sites of a hexagonal shaped triangular lattice. The bond potential between nearest neighbour particles is