Hydrodynamic Inflation of Ring Polymers Under Shear

Topology profoundly affects polymer properties, with ring polymers exhibiting distinctive behaviour in solutions and melts. Under steady shear, polymers commonly tumble; linear chains do not fully stretch in simple shear and require mixed flows for full extension. Prior work showed that with fully developed hydrodynamic interactions (HI), rings—but not linear chains—swell in the vorticity direction due to backflow from horseshoe regions. The present study asks whether a distinct shear-induced inflation phase exists in ring polymers under pure shear with HI, how its onset depends on contour length N, and how knots along the backbone influence this behaviour. The motivation spans fundamental nonequilibrium polymer physics and potential applications, such as using shear-induced conformational differences for microfluidic separation of rings by size or distinguishing rings from chains. The authors aim to quantify and explain the interplay between HI and ring topology, test the role of knots, and delineate regimes of tumbling, inflation, and post-inflation dynamics.

The paper situates its work within extensive literature on ring polymer topology, conformations, dynamics, and rheology in bulk and under flow. It notes: (i) rich single-molecule and melt phenomena unique to rings, including self-organization and glassy states; (ii) decreasing probability of unknotted states with contour length and the prevalence of knots in DNA/proteins; (iii) prior investigations of knot effects in equilibrium, confinement, tension, and sedimentation; (iv) universal tumbling in steady shear for various architectures, and the need for mixed flows for full stretching of linear chains; (v) earlier findings that rings swell in vorticity direction only when HI are fully accounted for; (vi) recent Brownian dynamics in mixed planar flows (Young et al.) which did not report inflation in pure shear—attributed here to lower N, different solvent quality, and hydrodynamic approximations (RPY). This work extends these by demonstrating a ring-exclusive inflation phase in pure shear with explicit solvent hydrodynamics, mapping its onset versus N and topology, and contrasting with chains and rings without HI.

Computational study using Molecular Dynamics (MD) of single ring polymers coupled to an explicit mesoscopic solvent via Multi-Particle Collision Dynamics (MPCD) to capture fully developed hydrodynamic interactions. Shear implemented with Lees–Edwards boundary conditions establishing planar Couette flow v_x(y)=γ̇ y. Polymer model: Kremer–Grest bead–spring model with Weeks–Chandler–Andersen (WCA) nonbonded interactions (σ=a=1.0, k_B T=1.0), FENE bonds with k=30 ε/σ^2 and maximum extension R_1=1.5 σ; expected equilibrium bond length ≈0.965 σ. MD integrated with Velocity-Verlet, timestep δt=h/100=0.001 (in (k_B T/m)^1/2 units). Monomers of mass M participate in MPCD collisions. Solvent (MPCD): Collision cell length a as unit of length; rotation angle χ=130°; average solvent particles per cell ⟨N⟩=10; collision time step h=0.1 [(k_B T)^{1/2} m^{1/2} a^{-1}] (and tests at h=0.05). Maxwellian cell-level thermostat to avoid viscous heating. Dynamic viscosity η≈8.70 [(k_B T)^{1/2} m^{1/2} a^{-2}] for h=0.1 and η≈16.67 for h=0.05. Simulations also performed with HI switched off for selected chains and rings for control. Systems: Single rings with degree of polymerisation N=100, 150, 200; topologies: unknot 0_1 and prime knots 3_1, 4_1, 5_1, 5_2, 6_1, 6_2, 6_3, 7_1, 8_1 (Alexander–Briggs notation). Comparison chains (linear) for reference (e.g., N=75, 100). Shear rates γ̇ scanned to reach Weissenberg numbers Wi=γ̇ τ_R up to O(10^3), well into the nonlinear regime. Longest relaxation times τ_R determined from equilibrium simulations (Supplementary Table 1) to compute Wi. Observables: Gyration tensor G_αβ=(1/N) ∑_i s_{iα} s_{iβ} (with s_i relative to center of mass) and its diagonal components G_xx (flow), G_yy (vorticity), G_zz (gradient). Shape parameters (anisotropy δ, acylindricity c) as in Supplementary Methods. Orientation angle θ (between eigenvector for largest eigenvalue of G and flow axis) and orientational resistance m_g=Wi tan(2θ). Inter-monomer forces (FENE tension components) resolved along coordinates; average bond length ⟨b⟩ vs γ̇. Flow fields around the polymer’s COM in flow–vorticity plane to visualise backflow. Knot detection and localization via Alexander polynomial and minimum-closure scheme (Locknot); knotted section size tracked. Simulations performed in elongated boxes to minimise finite-size effects: V=(100×60×60) a^3 for N=100; V=(150×80×80) a^3 for N=150,200; test at V=(200×120×120) a^3 reproduced behaviour. GPU-accelerated CUDA/C++ implementation; visualisation via matplotlib and VMD.

- Discovery of a shear-induced inflation phase unique to ring polymers with fully developed hydrodynamic interactions. This phase features uniform expansion and stretching, strong suppression of tumbling (TB), and onset of a distinct tank-treading-like rigid rotation; tumbling resumes at higher shear. - Non-monotonic swelling in vorticity and gradient directions for rings: G_yy (vorticity) shows decrease then sharp increase to a maximum and then decreases; G_zz (gradient) exhibits a swelling anomaly with a local minimum, rise to local maximum, and subsequent decrease—visible for sufficiently long rings (N≥150), nearly absent at N=100. In flow direction, G_xx increases then decreases as rings align into the flow–vorticity plane. - Chains (linear) with or without HI and rings without HI do not exhibit vorticity swelling nor the inflation anomaly; their G_αα vary monotonically with Wi, underscoring the necessity of both ring topology and HI. - Orientational resistance m_g shows a crossover in scaling with Wi, defining the onset of inflation: in the tumbling regime m_g∼Wi^{0.6}; in the inflation regime m_g∼Wi^{~1}, implying θ is approximately constant across this decade in γ̇. After full inflation, θ decreases again as the ring aligns closer to the flow axis and tumbling reappears. - Crossover shear rates and Weissenberg numbers (from Table 1) quantify onset of inflation (examples): 0_1, N=150: γ̇_c=7.2×10^{-3}, Wi_c=1.57×10^2; 0_1, N=200: γ̇_c=7.2×10^{-3}, Wi_c=1.86×10^2; 0_1, N=100: γ̇_c=1.4×10^{-2}, Wi_c=1.19×10^2 (no fully developed inflation); 3_1, N=200: γ̇_c=5.2×10^{-3}, Wi_c=0.82×10^2; 4_1, N=200: γ̇_c=3.8×10^{-3}, Wi_c=0.52×10^2; 7_1, N=200: γ̇_c=6.4×10^{-3}, Wi_c=1.05×10^2. Across knot types, qualitative behaviour and anomalies are the same. - Flow-field maps show pronounced solvent backflow from the horseshoe regions, stronger for longer rings (e.g., N=200) and maximal near γ̇≈2×10^{-2}, coinciding with G_yy anomaly peak. - Force analysis: In the inflation regime, inter-monomer forces (FENE tension) are largest near the COM and directed toward it; a broad ring segment exhibits F_{m,y}≈0 corresponding to local horizontal segments with stable equilibrium against hydrodynamic forces. Average bond length increases with γ̇, with greater stretching for larger N. - Knotted rings under strong shear: knotted sections tighten and localize, acting as stabilisation anchors typically residing in horseshoe regions; when moving, they tank-tread rapidly between horseshoe regions. Overall shapes, orientation dynamics, and inflation behaviour of knotted rings mirror unknotted rings of the same N. - Scaling theory rationalises suppression of tumbling: defining tip elevation y_c and tension blob size ξ_τ, tumbling requires ξ_τ≥y_c. With y_c∼γ̇^{-0.1} N^{0.4} and ξ_τ∼γ̇^{0.9} N^{-0.4}, increasing γ̇ quickly violates ξ_τ≥y_c, suppressing tumbling and inducing inflation. The predicted crossover shear rate scales as γ̇_x∼k_B T/(η σ^2) N^{-1}, consistent with simulations (γ̇_x≈10^3 in units used for η≈10 and N≈10^2).

The findings demonstrate that ring topology, when coupled with explicit hydrodynamic interactions, generates a unique backflow in the horseshoe regions that inflates the ring in the vorticity direction. This backflow enhances chain tension and elevates ring tips, stabilising an orientation with nearly constant angle θ over a finite γ̇ range—the inflation phase—during which tumbling is suppressed. The universal m_g scaling crossover from Wi^{0.6} to ~Wi^1 quantitatively marks this transition and explains the fixed orientation. The absence of the anomaly in chains and in rings without HI confirms that both closed topology and solvent-mediated HI are essential. A scaling argument based on tip elevation and blob tension sizes explains why increasing γ̇ suppresses tumbling (ξ_τ decreases faster than y_c grows), and yields γ̇_x∝N^{-1}. Comparison to colloidal Peclet-number scaling supports viewing the fully inflated ring as a rigid, non-Brownian object composed of N monomers connected rigidly. Knots do not qualitatively alter the inflation transition; they tighten and act as anchors, reinforcing stabilisation and exhibiting intermittent tank-treading between horseshoe regions. At higher γ̇ beyond inflation, θ decreases and tumbling reappears (re-entrant tumbling), consistent with increased Brownian susceptibility as y_c diminishes. The results clarify the mechanistic role of backflow-induced forces and elastic tensions in determining ring dynamics under shear and suggest practical implications: in Poiseuille flow, size-dependent inflation and wall lift could focus larger rings to the channel center and smaller ones toward walls, enabling size-based microfluidic separation; differences from studies using RPY hydrodynamics and lower N likely stem from weaker HI approximations and insufficient polymer length for inflation to manifest.

This work identifies and characterizes a shear-induced inflation phase unique to ring polymers in solution when fully developed hydrodynamic interactions are present. The phase is marked by vorticity swelling, a non-monotonic anomaly in G_yy and G_zz, suppressed tumbling, a fixed orientation angle, and eventual re-entrant tumbling at higher shear. The onset depends on contour length and follows a simple N^{-1} scaling for γ̇_x derived from a blob-based scaling theory consistent with a rigidified ring picture. Knotted rings of high contour length behave similarly to unknotted rings, with knots tightening and acting as stabilising anchors. The contrast with linear chains and with rings without HI underscores the critical role of hydrodynamics and topology. Potential applications include microfluidic separation of rings by size and differentiating rings from chains under flow. Future research directions include exploring how the inflation and stretching transition affect the rheology of dilute and semidilute ring solutions and examining the interplay between inflation and polymer rigidity (e.g., short DNA rings), along with experimental validation.

The study is computational and focuses on single-molecule simulations; experimental validation is pending. Only three contour lengths (N=100, 150, 200) and a selected set of knot types were examined, limiting extrapolation across all N. While polymer conformations were robust to box-size changes, solvent flow details exhibit finite-size features (e.g., distant vortices). Results depend on the explicit-solvent MPCD model parameters; alternative hydrodynamic treatments (e.g., RPY) and different solvent qualities may yield quantitative differences. Inflation does not occur without hydrodynamic interactions, so applicability is restricted to conditions with fully developed HI. At N=100 a fully developed inflation phase did not emerge, indicating a minimum N is required. The scaling arguments are asymptotic and neglect some prefactors; quantitative thresholds may vary with model details.