Unified Entanglement Scaling for Flexible, Semiflexible, and Stiff Polymer Melts and Solutions

Unified understanding of the impact of semiflexibility, concentration, and molecular weight on macromolecular-scale ring diffusion

Conformationally fluctuating, globally compact macromolecules such as polymeric rings, single-chain nanoparticles, microgels, and many-arm stars display complex dynamic behaviors due to their rich topological structure and intermolecular organization. Synthetic rings are hybrid objects with conformations that display both ideal random walk and compact globular features, which can serve as models of genomic DNA. To date, emphasis has been placed on the effect of ring molecular weight on their unusual behaviors. Here, we combine simulations and a microscopic force-level theory to build a unified understanding for how key aspects of ring dynamics depend on different tunable molecular properties including backbone rigidity, monomer concentration, degree of traditional entanglement, and molecular weight. Our large-scale molecular dynamics simulations of ring melts with very different backbone stiffnesses reveal unanticipated behaviors which agree well with our generalized theory. This includes a universal master curve for center-of-mass diffusion constants as a function of molecular weight scaled by a chemistry and thermodynamic state-dependent critical molecular weight that generalizes the concept of an entanglement cross-over for linear chains. The key physics is how backbone rigidity and monomer concentration induced changes of the entanglement length, interring packing, degree of interpenetration, and liquid compressibility slow down space-time dynamic-force correlations on macromolecular scales. A power law decay of the center-of-mass diffusion constant with inverse molecular weight squared is the first consequence, followed by an ultraslow activated hopping transport regime. Our results set the stage to address slow dynamics and kinetic arrest in different families of compact synthetic and biological polymeric systems.

Entanglement Kinetics in Polymer Melts Are Chemically Specific

We investigate the universality of entanglement kinetics in polymer melts. We compare predictions of a recently developed constitutive equation for disentanglement to molecular dynamics simulations of both united-atom polyethylene and Kremer-Grest models for polymers in shear and extensional flow. We confirm that entanglements recover on the retraction time scale, rather than the reptation time scale. We find that the convective constraint release parameter β is independent of molecular weight, but that it increases with the ratio of Kuhn length bK to packing length p as β ∼ (bK/p)α, with an exponent α = 1.9, which may suggest that disentanglement rate correlates with an increase in the tube diameter. These results may help shed light on which polymers are more likely to undergo shear banding.

Activity-induced stiffness, entanglement network and dynamic slowdown in unentangled semidilute polymer solutions

Active polymers possess numerous unique properties that are quite different from those observed in the system of small active molecules due to the intricate interplay between their activity and topological constraints. This study focuses on the conformational changes induced by activity, impacting effective stiffness and crucially influencing entanglement and dynamics. When the two terminals of a linear chain undergo active modification through coupling to a high-temperature thermal bath, there is a substantial increase in chain size, indicating a notable enhancement in effective stiffness. Unlike in passive semiflexible chains where stiffness predominantly affects local bond angles, activity-induced stiffness manifests at the scale of tens of monomers. While activity raises the ambient temperature, it significantly decreases diffusion by over an order of magnitude. The slowdown of the dynamics observed can be attributed to increased entanglement due to chain elongation.

Cooperative Rheological State-Switching of Enzymatically-Driven Composites of Circular DNA And Dextran

Polymer topology, which plays a principal role in the rheology of polymeric fluids, and non-equilibrium materials, which exhibit time-varying rheological properties, are topics of intense investigation. Here, composites of circular DNA and dextran are pushed out-of-equilibrium via enzymatic digestion of DNA rings to linear fragments. These time-resolved rheology measurements reveal discrete state-switching, with composites undergoing abrupt transitions between dissipative and elastic-like states. The gating time and lifetime of the elastic-like states, and the magnitude and sharpness of the transitions, are surprisingly decorrelated from digestion rates and non-monotonically depend on the DNA fraction. These results are modeled using sigmoidal two-state functions to show that bulk state-switching can arise from continuous molecular-level activity due to the necessity for cooperative percolation of entanglements to support macroscopic stresses. This platform, coupling the tunability of topological composites with the power of enzymatic reactions, may be leveraged for diverse material applications from wound-healing to self-repairing infrastructure.

Molecular Dynamics Simulation of Entangled Melts at High Rates: Identifying Entanglement Lockup Mechanism Leading to True Strain Hardening

In the present work, molecular dynamics simulations are carried out based on the bead-spring model to indicate how the entanglement lockup manifests in the late stage of fast Rouse-Weissnberg number (Wi_R >>1) uniaxial melt stretching of entangled polymer melts. At high strains, distinct features show up to reveal the emergence of an increasingly tightened entanglement network. Chain tension can build up, peaking at the middle of the chain, to a level for chain scission, through accumulated interchain interactions, as if there is a tug-of-war ongoing for each load-bearing chain. Thanks to the interchain uncrossability, network junctions form by the pairing of two or more hairpins. It is hypothesized that the interchain entanglement at junctions can lockup through prevailing twist-like interchain couplings as long as Wi_R > 9. In this limit, a significant fraction of chains act like cyclic chains to form a network held by interchain uncrossability, and appreciable chain tension emerges.

How Softness Matters in Soft Nanogels and Nanogel Assemblies

Softness plays a key role in determining the macroscopic properties of colloidal systems, from synthetic nanogels to biological macromolecules, from viruses to star polymers. However, we are missing a way to quantify what the term "softness" means in nanoscience. Having quantitative parameters is fundamental to compare different systems and understand what the consequences of softness on the macroscopic properties are. Here, we propose different quantities that can be measured using scattering methods and microscopy experiments. On the basis of these quantities, we review the recent literature on micro- and nanogels, i.e. cross-linked polymer networks swollen in water, a widely used model system for soft colloids. Applying our criteria, we address the question what makes a nanomaterial soft? We discuss and introduce general criteria to quantify the different definitions of softness for an individual compressible colloid. This is done in terms of the energetic cost associated with the deformation and the capability of the colloid to isotropically deswell. Then, concentrated solutions of soft colloids are considered. New definitions of softness and new parameters, which depend on the particle-to-particle interactions, are introduced in terms of faceting and interpenetration. The influence of the different synthetic routes on the softness of nanogels is discussed. Concentrated solutions of nanogels are considered and we review the recent results in the literature concerning the phase behavior and flow properties of nanogels both in three and two dimensions, in the light of the different parameters we defined. The aim of this review is to look at the results on micro- and nanogels in a more quantitative way that allow us to explain the reported properties in terms of differences in colloidal softness. Furthermore, this review can give researchers dealing with soft colloids quantitative methods to define unambiguously which softness matters in their compound.

Validation and Refinement of Unified Analytic Model for Flexible and Semiflexible Polymer Melt Entanglement

We combine molecular dynamics simulations and topological analyses (TA) to validate and refine a recently proposed unified analytic model [Hoy, R. S.; Kröger, M. Phys. Rev. Lett. 2020, 124, 147801] for the reduced entanglement length, tube diameter, and plateau modulus of polymer melts. While the functional forms of the previously published expressions are insensitive to the choice of the TA method and N _e -estimator, obtaining better statistics and eliminating all known sources of systematic error in the N _e -estimation alters their numerical coefficients. Our revised expressions quantitatively match bead-spring simulation data over the entire range of chain stiffnesses for which systems remain isotropic, semiquantitatively match all available experimental data for flexible, semiflexible, and stiff polymer melts (including new data for conjugated polymers that lie in a previously unpopulated stiffness regime), and outperform previously developed unified scaling theories.

Predicting the Plateau Modulus from Molecular Parameters of Conjugated Polymers

The relationship between Kuhn length l k , Kuhn monomer volume v 0, and plateau modulus G N 0, initially proposed by Graessley and Edwards for flexible polymers, and extended by Everaers, has a large gap in experimental data between the flexible and stiff regimes. This gap prevents the prediction of mechanical properties from the chain structure for any polymer in this region. Given the chain architecture, including a semiflexible backbone and side chains, conjugated polymers are an ideal class of material to study this crossover region. Using small angle neutron scattering, oscillatory shear rheology, and the freely rotating chain model, we have shown that 12 polymers with aromatic backbones populate a large part of this gap. We also have shown that a few of these polymers exhibit nematic ordering, which lowers G N 0. When fully isotropic, these polymers follow a relationship between l k , v 0, and G N 0, with a simple crossover proposed in terms of the number of Kuhn segments in an entanglement strand N e.

Facile equilibration of well-entangled semiflexible bead-spring polymer melts

The widely used double-bridging hybrid (DBH) method for equilibrating simulated entangled polymer melts [Auhl et al., J. Chem. Phys. 119, 12718-12728 (2003)] loses its effectiveness as chain stiffness increases into the semiflexible regime because the energy barriers associated with double-bridging Monte Carlo moves become prohibitively high. Here we overcome this issue by combining DBH with the use of core-softened pair potentials. This reduces the energy barriers substantially, allowing us to equilibrate melts with N ≃ 40N_e and chain stiffnesses all the way up to the isotropic-nematic transition using simulations of no more than 100 × 10⁶ time steps. For semiflexible chains, our method is several times faster than the standard DBH; we exploit this speedup to develop improved expressions for Kremer-Grest melts' chain-stiffness-dependent Kuhn length ℓ_K and entanglement length N_e.

Topological tuning of DNA mobility in entangled solutions of supercoiled plasmids

Ring polymers in dense solutions are among the most intriguing problems in polymer physics. Because of its natural occurrence in circular form, DNA has been extensively used as a proxy to study the fundamental physics of ring polymers in different topological states. Yet, torsionally constrained-such as supercoiled-topologies have been largely neglected so far. The applicability of existing theoretical models to dense supercoiled DNA is thus unknown. Here, we address this gap by coupling large-scale molecular dynamics simulations with differential dynamic microscopy of entangled supercoiled DNA plasmids. We find that, unexpectedly, larger supercoiling increases the size of entangled plasmids and concomitantly induces an enhancement in DNA mobility. These findings are reconciled as due to supercoiling-driven asymmetric and double-folded plasmid conformations that reduce interplasmid entanglements and threadings. Our results suggest a way to topologically tune DNA mobility via supercoiling, thus enabling topological control over the (micro)rheology of DNA-based complex fluids.

Crossover from Rouse to Reptation Dynamics in Salt-Free Polyelectrolyte Complex Coacervates

Considerable interest in the dynamics and rheology of polyelectrolyte complex coacervates has been motivated by their industrial application as viscosity modifiers. A central question is the extent to which classical Rouse and reptation models can be applied to systems where electrostatic interactions play a critical role on the thermodynamics. By relying on molecular simulations, we present a direct analysis of the crossover from Rouse to reptation dynamics in salt-free complex coacervates as a function of chain length. This crossover shifts to shorter chain lengths as electrostatic interactions become stronger, which corresponds to the formation of denser coacervates. To distinguish the roles of Coulomb interactions and density, we compare the dynamics of coacervates to those of neutral, semidilute solutions at the same density. Both systems exhibit a universal dynamical behavior in the connectivity-dominated (subdiffusion and normal diffusion) regimes, but the monomer relaxation time in coacervates is much longer and increases with increasing Bjerrum length. This is similar to the cage effect observed in glass-forming polymers, but the local dynamical slowdown is caused here by strong Coulomb attractions (ion pairing) between oppositely charged monomers. Our findings provide a microscopic framework for the quantitative understanding of coacervate dynamics and rheology.

Unified Analytic Expressions for the Entanglement Length, Tube Diameter, and Plateau Modulus of Polymer Melts

By combining molecular dynamics simulations and topological analyses with scaling arguments, we obtain analytic expressions that quantitatively predict the entanglement length N_{e}, the plateau modulus G, and the tube diameter a in melts that span the entire range of chain stiffnesses for which systems remain isotropic. Our expressions resolve conflicts between previous scaling predictions for the loosely entangled [Lin-Noolandi, Gℓ_{K}^{3}/k_{B}T∼(ℓ_{K}/p)^{3}], semiflexible [Edwards-de Gennes: Gℓ_{K}^{3}/k_{B}T∼(ℓ_{K}/p)^{2}], and tightly entangled [Morse, Gℓ_{K}^{3}/k_{B}T∼(ℓ_{K}/p)^{1+ϵ}] regimes, where ℓ_{K} and p are, respectively, the Kuhn and packing lengths. We also find that maximal entanglement (minimal N_{e}) coincides with the onset of local nematic order.