Hydrodynamics Meets Topology
Filtration and Shear Flow of Ring Polymers in Microfluidic Devices
- Fig. 1. A picture of macroscopically large, 3d-printer models having the topologies of an unknotted ring (left) as well as of a ring carrying a trefoil knot (middle) and a figure-of-eight knot (right).
- Fig. 2. A ring tank-treading on attractive spots when adsorbed gets transported through a microfluidic channel whereas the linear chain in the background gets immobilized by adsorption on the same.
- Fig. 3. (a) Sketch of a shear cell with the three major directions: the solvent velocity points in the flow (f) direction and its magnitude is proportional to the vertical distance from the box center along the gradient (g) direction. The vorticity (v) direction is perpendicular to the other two. A knotted ring polymer under strong shear aligns almost perfectly with the f-v plane and the knot can be either fully localized, as in panel (b), or delocalized, as in panel (c). The ring is swollen along the vorticity direction.
Everybody knows polymers: chain-like, primarily carbon-based macromolecules featuring an enormous richness of intriguing physical properties, both structural to rheological, stemming from the connectivity of the chains and the entropy hidden in the vast number of conformations of the thermally fluctuating macromolecules in solution.
The simplest operation that can be performed in a polymer chain, without any effect on its chemistry or interactions with its environment, is to glue together its two ends, to form thus a ring polymer. The consequences of this seemingly trivial modification have proven to be far-reaching, and they cover a broad spectrum from questions of fundamental importance to applications in material science to issues of biological relevance. The absence of free ends brings about a topological repulsion between the ring polymers, which stems from the exclusion of concatenated states between two neighboring rings . This amounts to a restriction of mutual conformations at close distances between the rings’ centers of mass, which gives rise to an additional entropic repulsion on top of the monomer steric interactions.
The presence of this topological constraint drastically alters the physics of highly concentrated ring polymer solutions or melts in comparison with their linear polymer counterparts. Whereas in linear chain melts chains are known to assume ideal random-walk conformations, in ring melts things are much more complicated. Currently, popular models for the rings’ conformations in melts are the crumpled globule model , the lattice animal model  and the threaded-ring models [4,5], the answer still being a topic of debate. The stress relaxation of ring melts is very different from that of linear chains, featuring a power-law domain due to the completely different nature of entanglements between rings and chains . Moreover, the rheological response of ring/linear mixtures is very sensitive to the amount of chains present in the melt. In Biology, both DNA and RNA can occur in linear and in circular forms and, in addition, knots of various complexities can appear on sufficiently long macromolecules (Fig.
1). The topological modifications of the molecular state carry important ramifications on their biological function. For example, circular RNA has a higher resilience against degradation compared to its linear counterpart . Moreover, proteins and DNA display a rich variety of topological effects. For example, the DNA of bacteria is present in the traditional double-helix form, but in contrast to what happens for eukaryotes, they have circular chromosomes contained in a DNA helix, which is closed into a ring . Formation of knots along the backbone of DNA and their location depending on the varying rigidity along the backbone of the macromolecule are other manifestations of the importance of topological concepts for biologically relevant processes [8,9].
Isolation of Ring-shaped Polymers
Given the enormous importance of topology for equilibrium polymer properties, the question arose of how the same affects the flow properties of ring polymers under non-equilibrium conditions and whether it was possible to take advantage of these to do something useful, i.e., to fish ring polymers out of a mixture with linear ones. The endeavor has been very rewarding, and it has been hiding quite a few surprises as well.
A first attempt to separate ring-shaped polymers from their linear counterparts focused on their transport and migration behavior in nano- and microfluidic slit channels with smooth, non-interactive walls in dilute mixtures. Here, the distance between the two channel walls corresponds to approximately five times the gyration radius and polymers are dissolved in a good solvent. Due to an intricate interplay between friction, diffusion and hydrodynamics, rings tend to stay closer to the channel center than the chains, and thus they are transported faster. However, the difference in speed is too small to effectively separate the two species. Rings, on the other hand, feature a tank-treading motion pattern, which is very much akin to the rolling of the chain-wheels of an excavator .
To take advantage of the tank-treading mode, which is unique to ring polymers, the channel walls were decorated with attractive spots. These spots, separated by approximately a Kuhn length, attract equally strongly chain- and ring-monomers and are aligned on a track in flow direction, which matches the linear topology. The monomer-spot interaction is short-ranged and based on the van-der-Waals force created by induced dipole-dipole interactions, thus applicable to all polymers, regardless the detailed chemical composition of their monomers. As soon as linear polymers adsorb onto these spots, they will get immobilized with an average transport velocity of zero as shown in Fig 2. Rings will adsorb as well, but due to their distinct topology, a part of their contour line will always remain exposed to the flow. As the exposed part constantly experiences a shear gradient, they will ‘roll’ along the attractive spots with a finite velocity (Fig. 2). Thus, the spots stabilize the tank-treading mode of ring polymers, enabling their transport.
Rolling rings will encounter at some point an already adsorbed chain on the same track. Simulations show that ring polymers will desorb in that case. Consequently, the average transport velocity of ring polymers will even increase compared to a ring that tank-treads continuously on a track without desorption. Nevertheless, purification of the spots is essential from time to time. By flushing the filter with a poor solvent for the respective polymer, adsorbed linear chains will collapse and desorb consequently.
Impact of Knots on Rings
Knowing how to separate linear chains from rings, the next goal was to investigate the impact of knots on rings on their dynamics under shear more closely . The shear geometry is shown in figure 3a. Single knots have been simulated on such long rings, with the knots being allowed to propagate freely around the chain and swell or shrink in size, but never to form a new knot or break.
A stretching of knotted polymers in flow direction has been observed under shear, as well as an alignment in the flow-vorticity plane. Whereas in the case of microfluidic transport the spots on the wall enhance the tank-treading motion of the rings, under bulk shear it is tumbling that dominates the motion as a dynamical motif, and tank-treading is strongly suppressed when full hydrodynamics is developed. The absence of rigidity on the backbone, in combination with strong local fluctuations in the solvent makes it increasingly difficult for a ring to align with the flow-gradient plane for extended periods of time to enable tank-treading to occur. Knotted and unknotted rings do not substantially differ as far as their shape under shear is concerned.
In the absence of hydrodynamics, tank-treading occurs more frequently, and the knotted section is more localized. A well-localized knotted section means that the knotted ring behaves very similarly to an unknotted one, with the distinction of a knotted section acting very much like a large bead, as far as the model is concerned. Tank-treading has been linked to a decrease of the average size of the knotted section. On the other hand, when hydrodynamics is enabled, it was possible to distinguish between two distinct states which were both more or less stable over extended periods of time – a well-localized configuration, behaving very much like an unknotted ring, and a delocalized configuration with a tight braiding section near the center of the ring, where the knot is so swollen that the polymer behaves like two elongated, concatenated rings. This delocalized configuration loses its ability to tank-tread due to the braiding section - both loops on such a polymer want to turn with the same orientation, but the braiding section connecting them causes either motion to cancel out (Figs. 3b&c).
Surprisingly, the inclusion of hydrodynamics makes a significant difference for any ring topologies in terms of shape. Any polymer under shear will tend to stretch in the shear flow direction and align in the flow-vorticity plane proportional to shear strength. However, including hydrodynamics limits the stretching and introduces a swelling in vorticity direction unique to closed polymer shapes, caused by a deflection of solvent particles from the horseshoe regions of the ring. In the absence of hydrodynamics, no such swelling is observed.
More challenges in manipulating topologically modified polymers with microfluidics lie ahead. Although it was possible to deliberately separate unknotted rings from linear chains, the same method was not able to reliably separate knotted from unknotted rings. Further investigations will be necessary to make progress towards this goal. The pronounced differences between knotted and unknotted rings in the presence of hydrodynamics, especially the strict suppression of tank-treading for the knotted ring, might also lead to some interesting behavior for mixtures of knotted and unknotted rings under shear.
Maximilian Liebetreu1, Christos N. Likos1, Lisa B. Weiss1
1Faculty of Physics, University of Vienna, Vienna, Austria
 M. D. Frank-Kamenetskii, A. V. Lukashin, and A. V. Vologodskii, Nature 258, 398 (1975). doi:10.1038/258398a0
 A. Grosberg, Y. Rabin, S. Havlin, and A. Neer, Europhys. Lett. 23, 373 (1993). doi: 10.1209/0295-5075/23/5/012
 M. Kapnistos, M. Lang, D. Vlassopoulos, W. Pyckhout-Hintzen, D. Richter, D. Cho, T. Chang, and M. Rubinstein, Nat. Mater. 7, 997 (2008). doi:10.1038/nmat2292
 R. M. Robertson and D. E. Smith, Proc. Nat. Acad. Sci. U.S.A. 104, 4824 (2007). doi:10.1073/pnas.0700137104
 D. Michieletto and M. S. Turner, Proc. Nat. Acad. Sci. U.S.A. 113, 5195 (2016). doi:10.1073/pnas.1520665113
 E. Lasda and R. Parker, RNA 20, 1829 (2014). doi:10.1261/rna.047126.114
 L. Postow, C. D. Hardy, J. Arsuaga, and N. R. Cozzarelli, Gene Dev. 18, 1766 (2004). doi:10.1101/gad.1207504
 R. Matthews, A. A. Louis and C. N. Likos, ACS Macro Lett. 1, 1352 (2012). doi:10.1021/mz300493d
 P. Poier, C. N. Likos and R. Matthews, Macromolecules 47, 3394 (2014). doi:10.1021/ma5006414
 L. B. Weiss, A. Nikoubashman and C. N. Likos, ACS Macro Lett. 6, 1426 (2017). doi:10.1021/acsmacrolett.7b00768
 M. Liebetreu, M. Ripoll and C. N. Likos, ACS Macro Lett. 7, 447 (2018). doi:10.1021/acsmacrolett.8b00059