Triangular Lattice Foldings - a Transfer Matrix Study

Connection between discrete and continuum models of polymerized (tethered) surfaces has been investigated by applying a transfer matrix method to a discrete rigid-bond triangular lattice, which is allowed to fold on itself along its bonds in a two-dimensional embedding space. As its continuum counterpart, the model has an extensive entropy and the mean squared distance between two sites of a folded lattice increases logarithmically with the linear distance between the sites in the unfolded state. The model lattice with bending rigidity remains unfolded at any finite temperature, unlike real polymerized surfaces. Properties of polymerized or tethered membranes (surfaces) have been an object of numerous recent studies [l-61. We can view the polymerized surfaces as a generalization [l] of linear polymers [7]. This analogy has been used [3] to investigate the properties of self-avoiding tethered surfaces. However, unlike in the linear polymers, the long length-scale behavior of tethered surfaces strongly depends on the details of the Hamiltonian. In particular, very rigid surfaces exhibit a nontrivial [6] flat one. As the rigidity of a surface changes it undergoes a second-order phase transition [l, 4,5] from a crumpled (linear-polymerlike) phase to a flat phase. Neither the flat phase nor the crumpling transition has an analogy in linear polymers. The theoretical treatment of long length-scale properties of linear polymers rests on a fkm foundation, since in the absence of self-avoiding (excluded volume, steric) interactions their properties can be calculated exactly. In particular, it can be shown that the end-to-end distance (both in continuum and on a discrete lattice) of a long polymer described by any local Hamiltonian obeys the Gaussian probability distribution. Thus on sufficiently long length-scales the polymer can be described by an effective Hamiltonian H I = = KkB T$dx(dr/dx)2, where r is the position of a monomer in the d-dimensional embedding space, while x is the i n t e m l coordinate (label) of a monomer. A straightforward generalization of H I assumes that at long length-scale two-dimensional (2d) polymerized membranes without self-avoiding interactions will be described by H2 = KkB T$d2x(Vr)2, where x is the internal coordinate of a monomer, i.e. its position in the 2d network, while (Vr)2 (W&1)2 + (W ~ X ~) ~ .