Nature of project: **software**, theory

Available to full-time physicists or joint students.

This project is about simulating a lattice model for polymers. Polymers are (long) chains of repeated units called monomers. Lattice models, where the monomers are assumed to lie on the sites of a regular lattice, are very popular in polymer simulation. Although a seemingly crude approximation, valuable information can be gained from lattice models. A simple model for a flexible polymer in solution is the Self-Avoiding-Walk (SAW) model. A SAW is a walk on a lattice, consisting of steps from lattice sites to neighbouring lattice sites, where the walker cannot revisit lattice sites it has visited before. A SAW can be interpreted as a polymer chain, where the lattice sites visited by the walker correspond to monomers and the steps correspond to chemical bonds.

A naive way of generating SAW's would be to perform the walk step by step, each time selecting a neighbour that has not yet been visited before. However, this creates a bias that needs to be corrected for, using a non-trivial set of weights. This algorithm is known as Rosenbluth sampling [1], and it becomes increasingly inefficient for long chains.

In order to overcome the difficulties mentioned above, various algorithms have been proposed. Many of them are based on the Metropolis Monte Carlo scheme. The Metropolis scheme consists of repeating a simple step many times. At each step the SAW of the previous step is altered (e.g. by moving some of the monomers). Then a decision is made whether to accept the new SAW or to revert back to the SAW from the previous step. In the context of SAW's this decision process is rather simple: if the new SAW is indeed self-avoiding (i.e. no revisits), the new SAW is accepted. If the new walk is not self-avoiding, it is rejected.

Various versions of the Metropolis Monte Carlo scheme exist for SAW's. They mainly differ in the choice of 'moves', i.e. the ways in which the SAW's are altered at each step. Possible moves include so-called kink-jump and crankshaft motions where little groups of monomers are moved around. A very effective move for SAW's is the reptation move, also known as the 'slithering snake' move [2]. Here a monomer is removed from one end of the chain, while a new monomer is grown at the other end of the chain.

The SAW model can be made more realistic by adding nearest neighbour interaction: for each monomer-monomer contact, the energy is reduced by a fixed amount. This attractive interaction favours compact conformations, leading to a 'collapse' of the polymer chain as the temperature is reduced. In order to study this collapse transition, the histogram of monomer-monomer contacts, also known as the 'density of states' must be determined accurately.

Although the Metropolis scheme, in combination with the reptation move, is a very efficient way of sampling SAW's in general, it is not very accurate when it comes to calculating the density of states. Especially the tails of the distribution (low-energy, high-energy) are not accurately sampled. A more recently developed scheme for accepting or rejecting moves, known as Wang-Landau sampling [3], is expected to be more accurate. Here a system of adaptive weights is introduced, which is tuned to produce a flat histogram for the energy, leading to an estimate of the density of states that is equally accurate all across the spectrum.

It is the purpose of this project to use Wang-Landau sampling, in combination with the reptation move, to accurately determine the density of states for a SAW with nearest-neighbour interaction. The student(s) will compare the results with ordinary Monte Carlo sampling (and exact enumeration for small chains).

*A successful project will develop beyond the above in one/some of the following directions:*

This project can be developed further by using the density of states that has been obtained to study the phase diagram for polymers in solution [4] and the collapse transition.

When considering where to take your project, please bear in mind the time available. It is preferable to do fewer things well than to try many and not get conclusive results on any of them. However, sometimes it is useful to have a couple of strands of investigation in parallel to work on in case delays occur.

*Additional scope or challenge if taken as a Year-4 project:* For an MPhys project, the student could also look at ring-polymers (closed SAW's) using the kink-jump and crankshaft motions.

*Initial literature for students:*

- Rosenbluth, M.N.; Rosenbluth, A.W.; J. Chem. Phys, 1955, 23, 356.
- Wall, F.T.; Mandel, F.; J. Chem. Phys., 1975, 63, 4592.
- Wang, F.; Landau, D.P.; Physical Review Letters, 2001, 86, 2050.
- Szleifer, I.; J. Chem. Phys., 1990, 92, 6940.

Wang-Landau sampling of Self-avoiding walks has probably been done before. Indeed, a quick literature search using the keywords "Wang-Landau" and "polymer" leads to a handful of hits. However, variations on the theme are possible, e.g. looking at ring-polymers and/or considering size ranges that have not been studied before. Furthermore, the density of states can be used to study phase separation using Szleifer's method [4].The supervisor has already implemented (in Fortran) a first version of a Wang-Landau sampling code for SAW's using reptation moves. The student(s) can use this as a starting point. This project will involve coding, and this will probably be the most challenging part of the project. Once a working code is produced, production runs will be performed. This should be rather straightforward. The data-analysis can be challenging, especially if Szleifer's method [4] is going to be used.

milestone | to be completed by |
---|---|

Familiarisation with existing Wang-Landau code. | Christmas |

Implementation of new features in code. | mid-March |

Production runs. | Easter |

Analysis of data. | Easter |