Knots, links, rings and other things: simulating topological interaction in polymer systems

(supervisor: Edwin Flikkema)

Nature of project: software, theory

Available to full-time physicists or joint students.

Project description and methodology

A polymer is a macro-molecule built from many monomers. A linear polymer is a simple chain of monomers. More complicated architectures are possible, such as branched polymers. In this study we focus on ring-polymers, which are similar to linear polymers, but form a closed loop.

Polymer systems tend to be very flexible: the position of the monomers (the so-called 'conformation') changes constantly over time. An interesting feature of a ring-polymer system is that, although the monomers move, the topological state of the system will not change over time. This is because, during their motion, the monomers are not allowed to cross. For example, if a ring-polymer is knotted, it will remain knotted and the type of knot will not change. Another example is that if initially two rings are not linked, they will remain unlinked. It is sometimes said that a ring-polymer system is 'topologically constrained'.

Topological constraints imply that the conformational space of the polymer system is partitioned into regions that are mutually inaccessible. This can lead to an interaction between the rings that is of an entropic nature, which is sometimes referred to as 'topological interaction'. An example of this is the topological repulsion between two unlinked rings: If the two rings are far apart, every combination of conformations is allowed. However, if the rings are close together, some conformations are not allowed, because they do not correspond to the unlinked state. So the entropy of the system is reduced as the rings are brought closer together. This gives rise to a repulsive entropic force: topological repulsion.

It is the aim of this project to investigate topological interaction in various ring-polymer systems using Monte Carlo simulation. In [1,2] a Monte Carlo algorithm was developed for studying such systems. This algorithm has been designed specifically to ensure that the topological state of the system does not change, even if the monomers have zero volume. In this algorithm an explicit non-crossing constraint is implemented: moves that might change the topological state of the system are rejected.

This algorithm is quite general and allows us to study various ring-polymer systems. The simplest such system is a single polymer ring. Here the polymer chain can be knotted in various topologically different ways. Knots have been tabulated by mathematicians and have been classified according to properties such as the minimum number of intersections in a 2D projection of the knot. More complicated knots will give rise to more intersections. Here the size (radius of gyration) of the polymer can be calculated as a function of the complexity of the knot. It is expected that the radius of gyration will decrease as the knot becomes more complex.

The next simplest system consists of two ring-polymers. In the case of unlinked rings, the entropic repulsion between the two can be studied as a function of their distance.

A successful project will develop beyond the above in one/some of the following directions:
To further develop the project, the student can generalize the methodology and study more complicated systems. If a pair of rings are linked, a force can be applied that tries to pull the rings apart. Here, the separation can be studied as a function of the applied force. This can be generalised to larger systems with multiple rings, e.g. chains of N=3,4,5 rings.

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: A 4th-year student can generalize the methodology by including other types of interaction, such as repulsion between monomers. Multiple unlinked rings confined to a box can also be studied.

Initial literature for students:

  1. E. Flikkema and G. ten Brinke. Osmotic pressure of ring polymer solutions: a Monte Carlo study. J. Chem. Phys., 113:11393, 2000.
  2. E Flikkema, Monte Carlo simulation of polymer systems with topological constraints, http://irs.ub.rug.nl/ppn/239363280
  3. M. E. Cates and J. M. Deutsch, Conjectures on the statistics of ring polymers. J. Physique, 47:2121, 1986.

Novelty, degree of difficulty and amount of assistance required

This project consists of novel applications of an existing algorithm. The project will involve a considerable amount of coding. This will probably be the most challenging part of the project and it is expected that the student(s) will need a lot of supervision at this stage. Once the coding is completed, production runs will be performed. This, plus the interpretation of the results, should be fairly straightforward. The student(s) should be able to do this fairly independently.

Project milestones and deliverables (including timescale)

milestoneto be completed by
Familiarisation with existing code.end of November
Implementation/coding of Monte Carlo simulation algorithm.mid-March
Production runs.Easter
Analysis of results.Easter