Conformational dynamics and topological analysis for polymer rings via atomistic molecular-dynamics simulations and comparison with experimental data
Dimitrios G. Tsalikis,1 Vlasis G. Mavrantzas1,2
1 Department of Chemical Engineering, University of Patras&FORTH/ICE-HT, Patras, GR 26504, Greece
2 Particle Technology Laboratory, Department of Mechanical and Process Engineering, ETH-Z, CH-8092 Zurich, Switzerland
Due to their chain-like structure and uncrossability, a number of microscopic topological constraints are generated in high-molar mass polymers known as entanglements which dominate their dynamical and rheological properties. For linear or branched polymer architectures, these topological interactions are well understood today thanks to the tube model, an effective medium theory proposed independently by de Gennes1 and Doi-Edwards2 built on the concept of primitive path (PP). According to the tube model, entanglements constrain the lateral motion of the chains which is thus restricted within a curvilinear tube-like region encompassing the chain. The primitive path is the shortest disconnected path along the chain contour which avoids crossings with other chains and has the same topology as the chain itself. For a linear chain with two free ends, the PP diffuses (i.e., reptates) backward and forward along the tube axis. An arm of a branched polymer has one free end which fluctuates attempting to reach the other, immobile end.
There exists, however, a class of polymers, the so called ring polymers, which lack chain ends and whose dynamics cannot be described by the conceptual framework of the tube model. Due to the absence of chain ends and their particular loop topology, polymer rings exhibit dynamic and viscoelastic properties that cannot be explained by the reptation theory. Even for the simplest case of unlinked (i.e., non-concatenated) ring polymers, issues related with chain configuration, molecular shape, local and terminal dynamics and stress relaxation are today only partially understood, although several advances have been made over the years. The key idea is that entangled rings contract into a folded form in order to satisfy the constraint that they should never link with neighboring rings (the so called lattice animal picture). Several extensions and improvements of this picture have been proposed recently, all however agreeing on the contraction and presence of local double folds or loops.6-9
Polymer rings serve, however, as a model system for understanding dynamics of fundamental ring structures in nature such as mitochondrial and plasmic DNA, and this explains the big interest in their properties (static, dynamic, and viscoelastic) in the last few years. From an experimental point of view, a major difficulty to overcome in the measurements is the production of highly purified and monodisperse polymer rings in sufficient quantities, since small contamination of the melt by linear chains can have a dramatic effect on the dynamics and molecular rheology of the melt.
To help in the direction of understanding the nature and role of microscopic topological interactions in polymer rings we have thus resorted to computer simulations and to the design of a simulation strategy involving three steps: a) execution of very long molecular dynamics simulations with model ring polymer structures using a well-studied system, polyethylene oxide (PEO), b) topological analysis of the accumulate trajectories to define local contact points, and c) detailed geometric analysis using vector calculus to identify threading events, classify them into weak and strong, and compute their dynamics from birth to death.
Key to the success of the entire methodology is the simulation of model PEO structures of molecular weights spanning the regime of MWs addressed experimentally. This means very large simulation cells and thus the use of supercomputing time. For example, we run simulations with systems containing up to 200,000 atomistic units. We also need to study the flow properties of these systems, which due to Lees Edwards boundary conditions necessitate simulation cells that on the flow direction are as long as the fully extended size of the chains, i.e., on the order of 100 nm. This increases even further the number of particles in the simulation cell, up to 500,000.
So far, we have already carried out test simulations with short-to-moderately long PEO melts (MW = 5k, 10k, and 20k) and the results are impressive. We have found outstanding agreement with experimental data for the ring center-of-mass self-diffusion coefficient and the normalized single-chain dynamic structure factor from small angle neutron scattering (SANS), neutron spin echo (NSE), and pulse-field gradient NMR (PFG-NMR) measurements).3 Furthermore, we have quantified ring-ring threading in the simulated ring PEO melts which reveals a variety of topological interactions corresponding to single and multiple penetrations, that can last up to several times the average ring polymer orientational relaxation time.4,5 And we have shown that these interactions can explain, at least in part, the appearance of slow relaxation modes observed experimentally in entangled rings.6
 P.G. De Gennes, J. Chem. Phys.55, 572 (1971).
 M. Doi and S.F. Edwards, The Theory of Polymer Dynamics (Clarendon Press, 1986).
 D G. Tsalikis, T. Koukoulas, V.G. Mavrantzas, D. Vlassopoulos, W. Pyckhout-Hintzen, and D. Richter, Macromolecules, in preparation (2015).
 D.G. Tsalikis and V. G. Mavrantzas, ACS Macro Lett. 3, 763 (2014).
 D.G. Tsalikis, V.G. Mavrantzas, D. Vlassopoulos, Phys. Rev. Lett., submitted (2015).
 M. Kapnistos, M. Lang, D. Vlassopoulos, W. Pyckhout-Hintzen, D. Richter, D. Cho, T. Chang, and M. Rubinstein, Nature Mater. 7, 997 (2008).