Session S40: Physics of Ring Polymers

Rheology modification with ring polymers

It is now established that experimental unconcatenated ring polymers can be purified effectively by means of fractionation at the critical condition. For molecular weights well above the entanglement threshold, purified rings relax stress via power-law (with an exponent of about -0.4), sharply departing from their linear counterparts. Experimental results are in harmony with modeling predictions and simulations. Here, we present results from recent interdisciplinary efforts and discuss two challenges: (i) the nonlinear shear rheology of purified ring melts is also very different from that of unlinked chains. Whereas the latter exhibit features that can be explained, to a first approach, in the framework in the tube model, the former behave akin to unentangled chains with finite extensibility and exhibit much small deformation at steady state. (ii) blends of rings and linear polymers exhibit unique features in different regimes: The addition of minute amounts of linear chains drastically affects ring dynamics. This relates to ring purity and the ability of unlinked linear chains to thread rings. With the help of simulations, it is possible to rationalize the observed surprisingly slow viscoelastic relaxation, which is attributed to ring-linear and ring-ring penetrations. On the other hand, adding small amounts of rings to linear polymers of different molecular weights influences their linear and nonlinear rheology in an unprecedented way. The blend viscosity exceeds that of the slower component (linear) in this non-interacting mixture, and its dependencies on composition and molecular weight ratio are examined, whereas the role of molecular architecture is also addressed. Consequently, closing the ends of a linear chain can serve as a powerful means for molecular manipulation of its rheology. This presentation reflects collaborative efforts with S. Costanzo, Z-C. Yan, R. Pasquino, M. Kaliva, S. Kamble, Y. Jeong, P. Lutz, J. Allgaier, T. Chang, D. Talikis, V. Mavrantzas and M. Rubinstein.   

Analysis of the morphology, stability, and folding pathways of ring polymers with supramolecular topological constraints using molecular simulation and nonlinear manifold learning

Ring polymers offer a wide range of natural and engineered functions and applications, including as circular bacterial DNA, crown ethers for cation chelation, and ``molecular machines'' such as mechanical nanoswitches. The morphology and dynamics of ring polymers are governed by the chemistry and degree of polymerization of the ring, and intramolecular and supramolecular topological constraints such as knots or mechanically-interlocked rings. We perform molecular dynamics simulations of polyethylene ring polymers as a function of degree of polymerization and in different topological states, including a knotted state, catenane state (two interlocked rings), and borromean state (three interlocked rings). Applying nonlinear manifold learning to our all-atom simulation trajectories, we extract low-dimensional free energy surfaces governing the accessible conformational states and their relative thermodynamic stability. The free energy surfaces reveal how degree of polymerization and topological constraints affect the thermally accessible conformations, chiral symmetry breaking, and folding and collapse pathways of the rings, and present a means to rationally engineer ring size and topology to preferentially stabilize particular conformational states.   

Scale-Dependent Viscosity in Ring and Linear Polymer Fluids

Traditional polymer rheology considers the response of the polymer fluids caused by external forces applied on a macroscopic scale. By contrast, in live cells, and especially in the nucleus, one encounters the situation in which forces are applied on a molecular scale, for instance, by molecular motors or other active ATP-consuming molecules. The influence of these locally applied forces can cascade upwards and lead to large-scale motion of the surrounding fluids which can not be captured by the traditional notion of viscosity. In this work, we examine the response of polymer fluids to forces applied at finite wave vector and frequency. Using simple physical arguments we construct a ``phase diagram'' of various frequency and wave vector-dependent regimes of effective viscosity for polymer fluids, including non-entangled and entangled melts, semidilute solutions with and without hydrodynamic interactions, as well as the more exotic case of a melt of unconcatenated ring polymers.   

Determining the Structure of Ring Polymers and Catenanes in Solution

Small-angle neutron scattering (SANS) is used to investigate the conformation of ring polymers and catenanes (mechanically interlocked rings). The conformations and interaction parameters of cyclic polystyrene and its exact linear analog are determined from a random phase approximation (RPA) analysis of the scattering data, taken in deuterated cyclohexane as a function of temperature, using new scattering form factors. Scattering data show distinct differences between linear and ring polymers. Finally, SANS measurements from mechanically interlocked rings are presented, and compared to measurements of linear and cyclic polystyrene. Scattering from catenanes finds a conformation that is very similar to linear polymers.   

Nanoparticle Motion in Entangled Melts of Linear and Non-Concatenated Ring Polymers

The motion of nanoparticles (NPs) in entangled melts of linear polymers and non-concatenated ring polymers are compared by large-scale molecular dynamics simulations. The comparison provides a paradigm for the effects of polymer architecture on the dynamical coupling between NPs and polymers in nanocomposites. Strongly suppressed motion of NPs with diameter $d$ larger than the entanglement spacing $a$ is observed in a melt of linear polymers before the onset of Fickian NP diffusion. This strong suppression of NP motion occurs progressively as $d$ exceeds $a$, and is related to the hopping diffusion of NPs in the entanglement network. In contrast to the NP motion in linear polymers, the motion of NPs with $d>a$ in ring polymers is not as strongly suppressed prior to Fickian diffusion. The diffusion coefficient $D$ decreases with increasing $d$ much slower in entangled rings than in entangled linear chains. NP motion in entangled non-concatenated ring polymers is understood through a scaling analysis of the coupling between NP motion and the self-similar entangled dynamics of ring polymers.   

Force-Based Theory for the Center of Mass Dynamics of Dense Ring Polymer Liquids

Recent simulations of concentrated ring polymer melts liquids have found they form collapsed globules on global scales but still significantly interpenetrate. In contrast to entangled chains, their internal modes relax relatively quickly compared to the center-of-mass (CM) motion time scale. Curiously, the ring diffusion constant scales roughly as the inverse square of the degree of polymerization ($N$), but with an onset well above that of the chain analog. The suggestion of a type of macromolecular glass transition at very large values of $N$ has been made. We formulate a statistical mechanical theory to describe the CM dynamics of rings based on assuming the internal modes are relaxed, the effective time-dependent forces on the ring CM are related to the inter-ring packing structure, and the ring liquid equation-of-state is qualitatively the same as for chains. Solving the theory at the level of dynamic second order perturbation theory results in a near $N^{-2}$ scaling of the long time diffusion constant, which is not predicted for non-interpenetrating strictly compact globules. A macromolecular glass-like localization transition is predicted as $N$ becomes very large which signals a crossover to activated hopping transport controlled by an entropic barrier.   

Confinement in Melts of Chains with Junction Points, but No Ends

Measurements of surface fluctuations of 4-arm star and "8-shaped" analogs of the same polystyrene (PS) chain show that elimination of chain ends is much more important in dictating the fragility in a thin film than is the introduction of a branch point in the molecule. Both the viscosities derived from surface fluctuations and rheological measurements for the 8-shaped PS manifest a lower value than the 4-arm star PS analog, with the discrepancy increasing as the temperature approaches the glass transition temperature, $T_{\mathrm{g,bulk}}$. Comparison among different chain topologies shows the effect of the number of chain ends and junction point on the viscosity. The viscosity behavior of the 8-shaped PS is quite different from that of the star analog, but similar to that of the simple cycle analog. The fragility of the 8-shaped molecule in the thin film is reduced relative to that in the bulk, manifesting a nanoconfinement effect.   

Topological order of a Polymer Chain in a Time-Dependent Potential

We study the effect of a polymer chain under a time-dependent external potential which presents broken time-reversal symmetry and find that a polymer chain can exhibit the classical counterpart of the Thouless quantum pump in the limit of strong potentials, signaling the existence of topological order in a polymer system. The topological order is quantized in terms of Chern numbers which is observed after explicit calculations. Since this quantization is topological protected against a small amount of disorder and other factors, our model can be used to the design of topological polymer's materials that illustrates different configurations of a polymer for practical uses   

Tuning polymer dynamics by chain-end association

Functional end groups in polymers are a molecular tool to reversibly connect and disconnect chains to combine properties of both short polymer and large supramolecular structures. The topology of the latter – and in turn the respective properties – can be tuned by the choice of the functional group. This is shown in a series of polydimethyl siloxanes (PDMS) of different molecular weight (MW) which are terminated by amino and carboxylic (-COOH) groups, respectively. Differential scanning calorimetry and dielectric spectroscopy measurements reveal that segmental dynamics are identical for the chains with two different end groups. In contrast, rheology unravels a mechanical reinforcement for PDMS-COOH and a rise in viscosity by ~2 decades. This is accompanied by a 2nd Tg and a corresponding dielectric relaxation process which indicates phase separation of the end groups in clusters forming a physically crosslinked network. As a consequence, the viscoelastic properties can be tuned from common short polymer chains at high temperatures to highly entangled or even crosslinked systems at T close and below the 2nd Tg. This suggests a promising route to combine easy processibility of low MW polymers with the desired mechanical performance of high MW polymers or even crosslinked networks.   

Polymer brushes in weakly interpenetrating regimes

We employ Molecular Dynamics (MD) simulations and develop new scaling laws to probe the behavior of semi-dilute polymer brushes in the weakly interpenetrating regime. This particular regime is characterized by the condition d$_{\mathrm{g}}$ being more than d$_{\mathrm{0}}$ but less than 2d$_{\mathrm{0}}$, where d$_{\mathrm{g}}$ is the gap between two opposing surfaces with grafted polymer brushes and d$_{\mathrm{0}}$ is the unperturbed brush height. Our results, showing excellent match between the MD simulation and scaling theory predictions, establish (a) unlike the classically studied case of strongly interpenetrating polymer brushes with d$_{\mathrm{g}}$ less than d$_{\mathrm{0}}$, here the brush height ($d)$, instead of being solely dictated by the interpenetration length, can be expressed in a power law form where $d$ scales as $N^{\chi }$ (where $N$ is the polymer size), (b) the exponent $\chi $ shows a monotonic increase with a decrease in the degree of interpenetration, (c) the interpenetration length shows a different scaling behavior as compared to the strongly interpenetrated case, and (d) the scaling behavior of the experimentally-witnessed variation of the compressive energy between the brushes can be reproduced.   

Comparison of Critical Adsorption Points of Ring Polymers with Linear Polymers

The Critical Adsorption Points (CAP) for ring and linear polymers are determined and compared using Monte Carlo simulations and liquid chromatography experiments. The CAP is defined as the co-elution point of ring or linear polymers with different molecular weights (MW). Computational studies show that the temperature at the CAP, TCAP, for rings is higher than TCAP for linear polymers regardless of whether the chains are modelled as random walks or self-avoiding walks. The difference in the CAP can be attributed to the architectural difference. Experimentally, four pairs of linear and ring polystyrenes (PS) of different MW were synthesized and purified. Elution of these polymers using a C18 bonded silica stationary phase and a CH2Cl2/CH3CN mixed eluent were studied. The temperature at the co-elution point, TCAP, and the co-elution time at the CAP, tE,CAP were determined for both ring and linear polymers. Experimentally it was found that T\textunderscore CAP of linear PS is lower than TCAP of cyclic PS and tE,CAP of linear PS is shorter than tE,CAP of ring PS. Therefore, at the CAP of linear polymers, ring polymers elute later in order of increasing MW while, at the CAP of ring polymers, linear polymers elute earlier in order of decreasing MW. This is in excellent agreement with the Monte Carlo computer simulation results.   

Progress in the synthesis and purification of cyclic polymers: towards accurate physical characterization

The physical characterization of ring polymers has been the subject of substantial experimental efforts since the first reports of high molecular weight cyclic polymers nearly 40 years ago. Despite numerous investigations, a number of fundamental questions remain unresolved in large part because of the difficulty in preparing cyclic polymers with high purity. A number of recent synthetic improvements offer promise to address this limitation. One such method, the ``click'' cyclization approach, exhibits unprecedented versatility in backbone chemistry and provides ring polymers and their exact linear analogs for direct physical comparison. Recent efforts have confirmed the ability to prepare cyclic polymers using this route with greater than 99.5{\%} purity. Using this approach the thermal properties of cyclic polycaprolactone and their linear analogs have been probed, including trends in glass transition and crystallization temperatures. Furthermore, blends of linear and cyclic polycaprolactone show a deviation in their observed crystallization temperature relative to a simple mixing law, suggesting that threading events are affecting their thermal properties.