Exact Solution of the Discrete (1+1)-dimensional RSOS Model in a Slit with Field and Wall Interactions

We present the solution of a linear Restricted Solid--on--Solid (RSOS) model confined to a slit. We include a field-like energy, which equivalently weights the area under the interface, and also include independent interaction terms with both walls. This model can also be mapped to a lattice polymer model of Motzkin paths in a slit interacting with both walls and including an osmotic pressure. This work generalises previous work on the RSOS model in the half-plane which has a solution that was shown recently to exhibit a novel mathematical structure involving basic hypergeometric functions ${}_3\phi_2$. Because of the mathematical relationship between half-plane and slit this work hence effectively explores the underlying $q$-orthogonal polynomial structure to that solution. It also generalises two other recent works: one on Dyck paths weighted with an osmotic pressure in a slit and another concerning Motzkin paths without an osmotic pressure term in a slit

Exact Solution of Semi-Flexible and Super-Flexible Interacting Partially Directed Walks

We provide the exact generating function for semi-flexible and super-flexible interacting partially directed walks and also analyse the solution in detail. We demonstrate that while fully flexible walks have a collapse transition that is second order and obeys tricritical scaling, once positive stiffness is introduced the collapse transition becomes first order. This confirms a recent conjecture based on numerical results. We note that the addition of an horizontal force in either case does not affect the order of the transition. In the opposite case where stiffness is discouraged by the energy potential introduced, which we denote the super-flexible case, the transition also changes, though more subtly, with the crossover exponent remaining unmoved from the neutral case but the entropic exponents changing

First-order scaling near a second-order phase transition: Tricritical polymer collapse

The coil-globule transition of an isolated polymer has been well established to be a second-order phase transition described by a standard tricritical O(0) field theory. We provide compelling evidence from Monte Carlo simulations in four dimensions, where mean-field theory should apply, that the approach to this (tri)critical point is dominated by the build-up of first-order-like singularities masking the second-order nature of the coil-globule transition: the distribution of the internal energy having two clear peaks that become more distinct and sharp as the tricritical point is approached. However, the distance between the peaks slowly decays to zero. The evidence shows that the position of this (pseudo) first-order transition is shifted by an amount from the tricritical point that is asymptotically much larger than the width of the transition region. We suggest an explanation for the apparently contradictory scaling predictions in the literature.Comment: 4 pages, 2 figures included in tex

A self-interacting partially directed walk subject to a force

We consider a directed walk model of a homopolymer (in two dimensions) which is self-interacting and can undergo a collapse transition, subject to an applied tensile force. We review and interpret all the results already in the literature concerning the case where this force is in the preferred direction of the walk. We consider the force extension curves at different temperatures as well as the critical-force temperature curve. We demonstrate that this model can be analysed rigorously for all key quantities of interest even when there may not be explicit expressions for these quantities available. We show which of the techniques available can be extended to the full model, where the force has components in the preferred direction and the direction perpendicular to this. Whilst the solution of the generating function is available, its analysis is far more complicated and not all the rigorous techniques are available. However, many results can be extracted including the location of the critical point which gives the general critical-force temperature curve. Lastly, we generalise the model to a three-dimensional analogue and show that several key properties can be analysed if the force is restricted to the plane of preferred directions.Comment: 35 pages, 14 figure

Forcing Adsorption of a Tethered Polymer by Pulling

We present an analysis of a partially directed walk model of a polymer which at one end is tethered to a sticky surface and at the other end is subjected to a pulling force at fixed angle away from the point of tethering. Using the kernel method, we derive the full generating function for this model in two and three dimensions and obtain the respective phase diagrams. We observe adsorbed and desorbed phases with a thermodynamic phase transition in between. In the absence of a pulling force this model has a second-order thermal desorption transition which merely gets shifted by the presence of a lateral pulling force. On the other hand, if the pulling force contains a non-zero vertical component this transition becomes first-order. Strikingly, we find that if the angle between the pulling force and the surface is beneath a critical value, a sufficiently strong force will induce polymer adsorption, no matter how large the temperature of the system. Our findings are similar in two and three dimensions, an additional feature in three dimensions being the occurrence of a reentrance transition at constant pulling force for small temperature, which has been observed previously for this model in the presence of pure vertical pulling. Interestingly, the reentrance phenomenon vanishes under certain pulling angles, with details depending on how the three-dimensional polymer is modeled

Identification of a polymer growth process with an equilibrium multi-critical collapse phase transition: the meeting point of swollen, collapsed and crystalline polymers

We have investigated a polymer growth process on the triangular lattice where the configurations produced are self-avoiding trails. We show that the scaling behaviour of this process is similar to the analogous process on the square lattice. However, while the square lattice process maps to the collapse transition of the canonical interacting self-avoiding trail model (ISAT) on that lattice, the process on the triangular lattice model does not map to the canonical equilibrium model. On the other hand, we show that the collapse transition of the canonical ISAT model on the triangular lattice behaves in a way reminiscent of the $\theta$-point of the interacting self-avoiding walk model (ISAW), which is the standard model of polymer collapse. This implies an unusual lattice dependency of the ISAT collapse transition in two dimensions. By studying an extended ISAT model, we demonstrate that the growth process maps to a multi-critical point in a larger parameter space. In this extended parameter space the collapse phase transition may be either $\theta$-point-like (second-order) or first-order, and these two are separated by a multi-critical point. It is this multi-critical point to which the growth process maps. Furthermore, we provide evidence that in addition to the high-temperature gas-like swollen polymer phase (coil) and the low-temperature liquid drop-like collapse phase (globule) there is also a maximally dense crystal-like phase (crystal) at low temperatures dependent on the parameter values. The multi-critical point is the meeting point of these three phases. Our hypothesised phase diagram resolves the mystery of the seemingly differing behaviours of the ISAW and ISAT models in two dimensions as well as the behaviour of the trail growth process

Four-dimensional polymer collapse II: Pseudo-First-Order Transition in Interacting Self-avoiding Walks

In earlier work we provided the first evidence that the collapse, or coil-globule, transition of an isolated polymer in solution can be seen in a four-dimensional model. Here we investigate, via Monte Carlo simulations, the canonical lattice model of polymer collapse, namely interacting self-avoiding walks, to show that it not only has a distinct collapse transition at finite temperature but that for any finite polymer length this collapse has many characteristics of a rounded first-order phase transition. However, we also show that there exists a `$\theta$-point' where the polymer behaves in a simple Gaussian manner (which is a critical state), to which these finite-size transition temperatures approach as the polymer length is increased. The resolution of these seemingly incompatible conclusions involves the argument that the first-order-like rounded transition is scaled away in the thermodynamic limit to leave a mean-field second-order transition. Essentially this happens because the finite-size \emph{shift} of the transition is asymptotically much larger than the \emph{width} of the pseudo-transition and the latent heat decays to zero (algebraically) with polymer length. This scenario can be inferred from the application of the theory of Lifshitz, Grosberg and Khokhlov (based upon the framework of Lifshitz) to four dimensions: the conclusions of which were written down some time ago by Khokhlov. In fact it is precisely above the upper critical dimension, which is 3 for this problem, that the theory of Lifshitz may be quantitatively applicable to polymer collapse.Comment: 30 pages, 14 figures included in tex

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.Comment: 10 pages, 2 figure