The conformations of cyclic polymers in melt are still not fully understood. The most intriguing aspect of this problem is to answer the question how mutual topological exclusion of molecules triggers compression of the molecules upon overlap. As an example we show the crumpled conformations of seven rings in a monodisperse melt of rings (see picture rhs).

To contribute to a better understanding of this problem, we simulated both non-concatenated entangled and freely interpenetrating melts of ring polymers. We found [1] that the fraction of non-concatenated rings displays an exponential decay as function of the average number of penetrations per ring. This identified the average number of penetrations f_n as the entropic effort for avoiding concatenated conformations.

On the basis of this result, a sequence of regimes for the conformations of rings in melts was proposed, starting from nearly ideal, weak and strong compression of rings and eventually leading to an overlap dominated regime for very high overlap of rings, if the problem remains dominated by pairwise entanglements in this last regime. For the weak and strong compression regime, two slightly different power laws were derived with R ~ N^2/5 and R ~ N^3/8 respectively. Recent data in literature is close to this prediction and it is not yet settled whether there might be even stronger compression or whether the overlap dominated regime is possible or not.

Since the differences between published models are rather small (powers of 2/5, 3/8 and 1/3 have been predicted for the collapsed state), the challenge for the future is to search for alternative tests of the models available in literature. One step in this direction was our recent work [2] on bi-disperse blends of rings that supports qualitatively our model.

The conformations of grafted chains are key to understand the properties of polymer brushes. One intriguing problem of an advanced description of the chain conformations is that the standard self-consistent field approaches for brushes in good solvent lead to a wrong scaling of the Free Energy. Besides, these models predict a concave shape of the end monomer distribution at small distances from the wall while a convex end monomer distribution is typically found in simulations.

Both problems could be removed simultaneously by considering the elasticity of a self-avoiding walk as basis for deriving theoretical expressions for the conformations of the brush chains [4]. On top of these qualitative changes, a significantly improved approximation for the brush density profile became available as shown in the Figure below (continuous line is model prediction, dotted line prediction based upon Gaussian chain elasticity, data points are from simulations with the Bond Fluctuation Model).

Polymer brushes are often used as protective layers in a rather harsh environment where the brush chains may degraft from the substrate. One way to reduce degrafting is to cross-link the chains and the main question is to which extent the properties of the brush are altered upon cross-linking.

We have addressed this point by computer simulations and analytical computations for brushes that are cross-linked in the swollen state [3]. Our results demonstrate that the freezing in of the monomer motion in the direction perpendicular to the grafting plane introduces only a slight height reduction of the brush but no collapse transition or a different scaling of brush height. The reason for this observation is that the driving force (monomer motion) is being reduced upon increasing degree of cross-linking. Qualitatively similar fluctuation corrections were used recently to explain corrections to rubber elasticity.

Thin film polymer melts have modified properties as compared to bulk systems, since the confining surfaces restrict the conformations of the chains and reduce the overlap of the polymers. This leads to a gradual loss of chain entanglement with a transition to non-entangled dynamics when approaching a polymer mono-layer [5]. Quite unexpectedly, the relaxation dynamics remains Rouse-like for up to one decade in time even beyond the relaxation time of individual chains. The detailed explanation of this observation will be the subject of future work.

1. Lang, M; Fischer, J.; Sommer, J.-U.
   The effect of topology on the conformations of ring polymers
   Macromolecules 45 (2012) 7642-7648.
2. Lang, M.
   Ring conformations in bidisperse blends of ring polymers
   Macromolecules 46 (2013) 1158-1166.
3. Lang, M. ; Hoffmann, M. ; Dockhorn, R. ; Werner, M. ; Sommer, J.-U.
   Fluctuation driven height reduction of crosslinked polymer brushes: A Monte Carlo study
   Journal of Chemical Physics 139 (2013) 164903.
4. Romeis, D. ; Lang, M.
   Excluded volume effects in polymer brushes at moderate chain stretching
   Journal of Chemical Physics 141 (2014) 104901.
5. Galuschko, A. ; Lang, M. ; Kreer, T. ; Sommer, J.-U.
   Monte Carlo simulation of thin film polymer melts
   Soft Materials 12 (2014) S49-S55.