Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userOsmotic Pressure of Confined Square Lattice Self-Avoiding Walks
296
0
0.0
(
0
)
Added by
Esaias J Janse van Rensburg
Publication date
2018
fields
Physics
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
Flory-Huggins theory is a mean field theory for modelling the free energy of dense polymer solutions and polymer melts. In this paper we use Flory-Huggins theory as a model of a dense two dimensional self-avoiding walk confined to a square in the square lattice. The theory describes the free energy of the walk well, and we estimate the Flory interaction parameter of the walk to be $chi_{saw} = 0.32(1)$.
rate research
Read More
We study a restricted class of self-avoiding walks (SAW) which start at the origin (0, 0), end at $(L, L)$, and are entirely contained in the square $[0, L] times [0, L]$ on the square lattice ${mathbb Z}^2$. The number of distinct walks is known to grow as $lambda^{L^2+o(L^2)}$. We estimate $lambda = 1.744550 pm 0.000005$ as well as obtaining strict upper and lower bounds, $1.628 < lambda < 1.782.$ We give exact results for the number of SAW of length $2L + 2K$ for $K = 0, 1, 2$ and asymptotic results for $K = o(L^{1/3})$. We also consider the model in which a weight or {em fugacity} $x$ is associated with each step of the walk. This gives rise to a canonical model of a phase transition. For $x < 1/mu$ the average length of a SAW grows as $L$, while for $x > 1/mu$ it grows as $L^2$. Here $mu$ is the growth constant of unconstrained SAW in ${mathbb Z}^2$. For $x = 1/mu$ we provide numerical evidence, but no proof, that the average walk length grows as $L^{4/3}$. We also consider Hamiltonian walks under the same restriction. They are known to grow as $tau^{L^2+o(L^2)}$ on the same $L times L$ lattice. We give precise estimates for $tau$ as well as upper and lower bounds, and prove that $tau < lambda.$
The growth constant for two-dimensional self-avoiding walks on the honeycomb lattice was conjectured by Nienhuis in 1982, and since that time the corresponding results for the square and triangular lattices have been sought. For the square lattice, a possible conjecture was advanced by one of us (AJG) more than 20 years ago, based on the six significant digit estimate available at the time. This estimate has improved by a further six digits over the intervening decades, and the conjectured value continued to agree with the increasingly precise estimates. We discuss the three most successful methods for estimating the growth constant, including the most recently developed Topological Transfer-Matrix method, due to another of us (JLJ). We show this to be the most computationally efficient of the three methods, and by parallelising the algorithm we have estimated the growth constant significantly more precisely, incidentally ruling out the conjecture, which fails in the 12th digit. Our new estimate of the growth constant is $$mu(mathrm{square}) = 2.63815853032790, (3).$$
We consider self-avoiding walks terminally attached to a surface at which they can adsorb. A force is applied, normal to the surface, to desorb the walk and we investigate how the behaviour depends on the vertex of the walk at which the force is applied. We use rigorous arguments to map out some features of the phase diagram, including bounds on the locations of some phase boundaries, and we use Monte Carlo methods to make quantitative predictions about the locations of these boundaries and the nature of the various phase transitions.
Self-avoidance is a common mechanism to improve the efficiency of a random walker for covering a spatial domain. However, how this efficiency decreases when self-avoidance is impaired or limited by other processes has remained largely unexplored. Here we use simulations to study the case when the self-avoiding signal left by a walker both (i) saturates after successive revisits to a site, and (ii) evaporates, or dissappears, after some characteristic time. We surprisingly reveal that the mean cover time becomes minimum for intermediate values of the evaporation time, leading to the existence of a nontrivial optimum management of the self-avoiding signal. We argue that this is a consequence of complex blocking effects caused by the interplay with the signal saturation and, remarkably, we show that the optimum becomes more and more significant as the domain size increases.
A numerical simulation shows that the osmotic pressure of compressed lattice knots is a function of knot type, and so of entanglements. The osmotic pressure for the unknot goes through a negative minimum at low concentrations, but in the case of non-trivial knot types $3_1$ and $4_1$ it is negative for low concentrations. At high concentrations the osmotic pressure is divergent, as predicted by Flory-Huggins theory. The numerical results show that each knot type has an equilibrium length where the osmotic pressure for monomers to migrate into or our of the lattice knot is zero. Moreover, the lattice unknot is found to have two equilibria, one unstable, and one stable, whereas the lattice knots of type $3_1$ and $4_1$ have one stable equilibrium each.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
