The Θ points of interacting self-avoiding walks and rings on a 2D square lattice

Abstract

We propose an order parameter to locate the Θ points of interacting self-avoiding walks (ISAWs) and self-avoiding rings (SARs) on a two dimensional square lattice. Using exact enumeration results for ISAWs of finite size we find that the order parameter as a function of temperature shows a discontinuous jump at the transition temperature. The computed transition temperature fluctuates with the size of the walk and appears to converge well to the Θ point as the size increases. The value for the Θ point of the linear homopolymer phase transition obtained from our method agrees with recent high precision Monte Carlo estimates. We tested the reliability of our method for longer walk lengths by using the density of states obtained from the recently proposed flat energy histogram method. The Θ point is also computed for SARs on a square lattice using exact enumeration results available in the literature. Using two other independent methods, namely the partition function zeros on the complex temperature plane and the inflection point of microcanonical entropy, the Θ point of an SAR is computed and is found to be in agreement with the result obtained with the proposed order parameter.

Critical behavior of magnetic polymers in two and three dimensions

“…Another useful tool to use for estimating critical temperatures and, in the case of polymers, the exponent φ is to look at the behaviour of the dominant complex zero of the partition function [9,[32][33][34]. The dominant zero is the one which is closest to the real axis in the complex temperature plane, and will pinch the real axis at the transition temperature in the thermodynamic limit.…”

Polymer collapse transition: a view from the complex fugacity plane

“…Although these quantities can be easily calculated through a numerical study of partition functions, it would be interesting to explore the scaling behavior (10) directly-using numerical analysis of Z N data. We expect that such a study can reveal the way the asymptotic form (10) sets in as N increases gradually, and thus can provide more details about the asymptotic form of Z N .…”

“…In figure 7 we present some results of our numerical examination of Z N for w w θ . In view of the supposed asymptotic law (10), in the main panel of this figure we present log(µ −N Z N ) as a function of log(N) for w = 1 and w = w θ . As one can notice, on a large scale log(µ −N Z N ) seems to have essentially linear behavior modulated by the presence of some oscillations.…”

“…To take into account effective interaction between monomers in this model, one usually associates an attractive energy to each pair of neighbouring sites (monomers) that are not adjacent along the walk. The model has been studied by a variety of techniques, including computer simulations [3][4][5][6][7][8][9][10], exact enumerations on lattices [11][12][13], transfer-matrix and lattice renormalization group aproaches [14][15][16][17].…”

Interacting semi-flexible self-avoiding walks studied on a fractal lattice

“…Such an interaction represents monomer-monomer interaction mediated by the solvent and thus mimics different solvent qualities. Many aspects of the interacting model have been intensively studied concerning the polymer collapse transition [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23]. Another step that improves the model is the addition of bending energy to each bend in the walk so that various degrees of stiffness of natural polymers could be accounted for.…”