arXiv subjects

T Prellberg

Publications and source records attributed to T Prellberg.

12 recordsLinked to original sources

arXiv2019

Adsorption of interacting self-avoiding trails in two dimensions

We investigate the surface adsorption transition of interacting self-avoiding square lattice trails onto a straight boundary line. The character of this adsorption transition depends on the strength of the bulk interaction, which induces a collapse transition of the trails from a swollen to a collapsed phase, separated by a critical state. If the trail is in the critical state, the universality class of the adsorption transition changes; this is known as the special adsorption point. Using flatPERM, a stochastic growth Monte Carlo algorithm, we simulate the adsorption of self-avoiding interacting trails on the square lattice using three different boundary scenarios which differ with respect to the orientation of the boundary and the type of surface interaction. We confirm the expected phase diagram, showing swollen, collapsed, and adsorbed phases in all three scenarios, and confirm universality of the normal adsorption transition at low values of the bulk interaction strength. Intriguingly, we cannot confirm universality of the special adsorption transition. We find different values for the exponents; the most likely explanation is that this is due to the presence of strong corrections to scaling at this point.

cond-mat.stat-mech↗

arXiv2019

Phase transitions in solvent dependent polymer adsorption in three dimensions

We consider the phase diagram of self-avoiding walks (SAW) on the simple cubic lattice subject to surface and bulk interactions, modeling an adsorbing surface and variable solvent quality for a polymer in dilute solution, respectively. We simulate SAWs at specific interaction strengths to focus on locating certain transitions and their critical behavior. By collating these new results with previous results we sketch the complete phase diagram and show how the adsorption transition is affected by changing the bulk interaction strength. This expands on recent work considering how adsorption is affected by solvent quality. We demonstrate that changes in the adsorption crossover exponent coincide with phase boundaries.

cond-mat.stat-mech↗

arXiv2017

Anomalous polymer collapse winding angle distributions

In two dimensions polymer collapse has been shown to be complex with multiple low temperature states and multi-critical points. Recently, strong numerical evidence has been provided for a long-standing prediction of universal scaling of winding angle distributions, where simulations of interacting self-avoiding walks show that the winding angle distribution for N-step walks is compatible with the theoretical prediction of a Gaussian with a variance growing asymptotically as C log N . Here we extend this work by considering interacting self-avoiding trails which are believed to be a model representative of some of the more complex behaviour. We provide robust evidence that, while the high temperature swollen state of this model has a winding angle distribution that is also Gaussian, this breaks down at the polymer collapse point and at low temperatures. Moreover, we provide some evidence that the distributions are well modelled by stretched/compressed exponentials, in contradistinction to the behaviour found in interacting self-avoiding walks.

cond-mat.stat-mech↗

arXiv2015

The role of three-body interactions in two-dimensional polymer collapse

Various interacting lattice path models of polymer collapse in two dimensions demonstrate different critical behaviours. This difference has been without a clear explanation. The collapse transition has been variously seen to be in the Duplantier-Saleur $\theta$-point university class (specific heat cusp), the interacting trail class (specific heat divergence) or even first-order. Here we study via Monte Carlo simulation a generalisation of the Duplantier-Saleur model on the honeycomb lattice and also a generalisation of the so-called vertex-interacting self-avoiding walk model (configurations are actually restricted trails known as grooves) on the triangular lattice. Crucially for both models we have three and two body interactions explicitly and differentially weighted. We show that both models have similar phase diagrams when considered in these larger two-parameter spaces. They demonstrate regions for which the collapse transition is first-order for high three body interactions and regions where the collapse is in the Duplantier-Saleur $\theta$-point university class. We conjecture a higher order multiple critical point separating these two types of collapse.

cond-mat.stat-mech↗

arXiv2015

Self-attracting polymers in two dimensions with three low-temperature phases

We study via Monte Carlo simulation a generalisation of the so-called vertex interacting self-avoiding walk (VISAW) model on the square lattice. The configurations are actually not self-avoiding walks but rather restricted self-avoiding trails (bond avoiding paths) which may visit a site of the lattice twice provided the path does not cross itself: to distinguish this subset of trails we shall call these configurations \emph{grooves}. Three distinct interactions are added to the configurations: firstly the VISAW interaction, which is associated with doubly visited sites, secondly a nearest neighbour interaction in the same fashion as the canonical interacting self-avoiding walk (ISAW) and thirdly, a stiffness energy to enhance or decrease the probability of bends in the configuration. In addition to the normal high temperature phase we find three low temperature phases: (i) the usual amorphous liquid drop-like "globular" phase, (ii) an anisotropic "$\beta$-sheet" phase with dominant configurations consisting of aligned long straight segments, which has been found in semi-flexible nearest neighbour ISAW models, and (iii) a maximally dense phase, where the all sites of the path are associated with doubly visited sites (except those of the boundary of the configuration), previously observed in interacting self-avoiding trails. We construct a phase diagram using the fluctuations of the energy parameters and three order parameters. The $\beta$-sheet and maximally dense phases do not seem to meet in the phase space and are always separated by either the extended or globular phases. We focus attention on the transition between the extended and maximally dense phases, as that is the transition in the original VISAW model. We find that for the path lengths considered there is a range of parameters where the transition is first order and it is otherwise continuous.

cond-mat.stat-mech↗

arXiv2014

Writhe-induced knotting in a lattice polymer

We consider a simple lattice model of a topological phase transition in open polymers. To be precise, we study a model of self-avoiding walks on the simple cubic lattice tethered to a surface and weighted by an appropriately defined writhe. We also consider the effect of pulling the untethered end of the polymer from the surface. Regardless of the force we find a first-order phase transition which we argue is a consequence of increased knotting in the lattice polymer, rather than due to other effects such as the formation of plectonemes.

cond-mat.soft↗

arXiv2012

Semi-flexible interacting self-avoiding trails on the square lattice

Self-avoiding walks self-interacting via nearest neighbours (ISAW) and self-avoiding trails interacting via multiply-visited sites (ISAT) are two models of the polymer collapse transition of a polymer in dilute solution. On the square lattice it has been established numerically that the collapse transition of each model lies in a different universality class. It has been shown that by adding stiffness to the ISAW model a second low temperature phase eventuates and a more complicated phase diagram ensues with three types of transition that meet at a multi-critical point. For large enough stiffness the collapse transition becomes first-order. Interestingly, a phase diagram of a similar structure has been seen to occur in an extended ISAT model on the triangular lattice without stiffness. It is therefore of interest to see the effect of adding stiffness to the ISAT model. We have studied by computer simulation a generalised model of self-interacting self-avoiding trails on the square lattice with a stiffness parameter added. Intriguingly, we find that stiffness does not change the order of the collapse transition for ISAT on the square lattice for a very wide range of stiffness weights. While at the lengths considered there are clear bimodal distributions for very large stiffness, our numerical evidence strongly suggests that these are simply finite-size effects associated with a crossover to a first-order phase transition at infinite stiffness.

cond-mat.stat-mech↗

arXiv2009

Exact Solution of the Discrete (1+1)-dimensional RSOS Model with Field and Surface Interactions

We present the solution of a linear Restricted Solid--on--Solid (RSOS) model in a field. Aside from the origins of this model in the context of describing the phase boundary in a magnet, interest also comes from more recent work on the steady state of non-equilibrium models of molecular motors. While similar to a previously solved (non-restricted) SOS model in its physical behaviour, mathematically the solution is more complex. Involving basic hypergeometric functions ${}_3\phi_2$, it introduces a new form of solution to the lexicon of directed lattice path generating functions.

cond-mat.stat-mech↗

arXiv2008

Semi-flexible hydrogen-bonded and non-hydrogen bonded lattice polymers

We investigate the addition of stiffness to the lattice model of hydrogen-bonded polymers in two and three dimensions. We find that, in contrast to polymers that interact via a homogeneous short-range interaction, the collapse transition is unchanged by any amount of stiffness: this supports the physical argument that hydrogen bonding already introduces an effective stiffness. Contrary to possible physical arguments, favouring bends in the polymer does not return the model's behaviour to that comparable to the semi-flexible homogeneous interaction model, where the canonical $θ$-point occurs for a range of parameter values. In fact, for sufficiently large bending energies the crystal phase disappears altogether, and no phase transition of any type occurs. We also compare the order-disorder transition from the globule phase to crystalline phase in the semi-flexible homogeneous interaction model to that for the fully-flexible hybrid model with both hydrogen and non-hydrogen like interactions. We show that these phase transitions are of the same type and are a novel polymer critical phenomena in two dimensions. That is, it is confirmed that in two dimensions this transition is second-order, unlike in three dimensions where it is known to be first order. We also estimate the crossover exponent and show that there is a divergent specific heat, finding $ϕ=0.7(1)$ or equivalently $α=0.6(2)$. This is therefore different from the $θ$ transition, for which $α=-1/3$.

cond-mat.soft↗

arXiv2007

Directed Paths in a Wedge

Directed paths have been used extensively in the scientific literature as a model of a linear polymer. Such paths models in particular the conformational entropy of a linear polymer and the effects it has on the free energy. These directed models are simplified versions of the self-avoiding walk, but they do nevertheless give insight into the phase behaviour of a polymer, and also serve as a tool to study the effects of conformational degrees of freedom in the behaviour of a linear polymer. In this paper we examine a directed path model of a linear polymer in a confining geometry (a wedge). The main focus of our attention is $c_n$, the number of directed lattice paths of length $n$ steps which takes steps in the North-East and South-East directions and which is confined to the wedge $Y=\pm X/p$, where $p$ is an integer. In this paper we examine the case $p=2$ in detail, and we determine the generating function using the iterated kernel method. We also examine the asymtotics of $c_n$. In particular, we show that $$ c_n = [0.67874...]\times 2^{n-1}(1+(-1)^n) + O((4/3^{3/4})^{n+o(n)}) + o((4/3^{3/4})^n) $$ where we can determine the constant $0.67874...$ to arbitrary accuracy with little effort.

cond-mat.soft↗

arXiv2007

The competition of hydrogen-like and isotropic interactions on polymer collapse

We investigate a lattice model of polymers where the nearest-neighbour monomer-monomer interaction strengths differ according to whether the local configurations have so-called ``hydrogen-like'' formations or not. If the interaction strengths are all the same then the classical $θ$-point collapse transition occurs on lowering the temperature, and the polymer enters the isotropic liquid-drop phase known as the collapsed globule. On the other hand, strongly favouring the hydrogen-like interactions give rise to an anisotropic folded (solid-like) phase on lowering the temperature. We use Monte Carlo simulations up to a length of 256 to map out the phase diagram in the plane of parameters and determine the order of the associated phase transitions. We discuss the connections to semi-flexible polymers and other polymer models. Importantly, we demonstrate that for a range of energy parameters two phase transitions occur on lowering the temperature, the second being a transition from the globule state to the crystal state. We argue from our data that this globule-to-crystal transition is continuous in two dimensions in accord with field-theory arguments concerning Hamiltonian walks, but is first order in three dimensions.

cond-mat.soft↗

arXiv2006

On a Type of Self-Avoiding Random Walk with Multiple Site Weightings and Restrictions

We introduce a new class of models for polymer collapse, given by random walks on regular lattices which are weighted according to multiple site visits. A Boltzmann weight $ω_l$ is assigned to each $(l+1)$-fold visited lattice site, and self-avoidance is incorporated by restricting to a maximal number $K$ of visits to any site via setting $ω_l=0$ for $l\geq K$. In this paper we study this model on the square and simple cubic lattices for the case K=3. Moreover, we consider a variant of this model, in which we forbid immediate self-reversal of the random walk. We perform simulations for random walks up to $n=1024$ steps using FlatPERM, a flat histogram stochastic growth algorithm. Unexpectedly, we find evidence that the existence of a collapse transition depends sensitively on the details of the model.

cond-mat.stat-mech↗