Finite-size scaling functions for directed polymers confined between attracting walls

The exact solution of directed self-avoiding walks confined to a slit of finite width and interacting with the walls of the slit via an attractive potential has been calculated recently. The walks can be considered to model the polymer-induced steric stabilisation and sensitised floculation of colloidal dispersions. The large width asymptotics led to a phase diagram different to that of a polymer attached to, and attracted to, a single wall. The question that arises is: can one interpolate between the single wall and two wall cases? In this paper we calculate the exact scaling functions for the partition function by considering the two variable asymptotics of the partition function for simultaneous large length and large width. Consequently, we find the scaling functions for the force induced by the polymer on the walls. We find that these scaling functions are given by elliptic theta-functions. In some parts of the phase diagram there is more a complex crossover between the single wall and two wall cases and we elucidate how this happens.

On the location of the surface-attached globule phase in collapsing polymers

We investigate the existence and location of the surface phase known as the Surface-Attached Globule (SAG) conjectured previously to exist in lattice models of three-dimensional polymers when they are attached to a wall that has a short range potential. The bulk phase, where the attractive intra-polymer interactions are strong enough to cause a collapse of the polymer into a liquid-like globule and the wall either has weak attractive or repulsive interactions, is usually denoted Desorbed-Collapsed or DC. Recently this DC phase was conjectured to harbour two surface phases separated by a boundary where the bulk free energy is analytic while the surface free energy is singular. The surface phase for more attractive values of the wall interaction is the SAG phase. We discuss more fully the properties of this proposed surface phase and provide Monte Carlo evidence for self-avoiding walks up to length 256 that this surface phase most likely does exist. Importantly, we discuss alternatives for the surface phase boundary. In particular, we conclude that this boundary may lie along the zero wall interaction line and the bulk phase boundaries rather than any new phase boundary curve.

Controlling Secondary Structures of Bio-Polymers with Hydrogen-Like Bonding

We present results for a lattice model of bio-polymers where the type of $beta$-sheet formation can be controlled by different types of hydrogen bonds depending on the relative orientation of close segments of the polymer. Tuning these different interaction strengths leads to low-temperature structures with different types of orientational order. We perform simulations of this model and so present the phase diagram, ascertaining the nature of the phases and the order of the transitions between these phases.