Ring polymer dynamics and tumbling-stretch transitions in planar mixed flows

The properties of dilute polymer solutions are governed by the conformational dynamics of individual polymers which can be perturbed in the presence of an applied flow. Much of our understanding of dilute solutions comes from studying how flows manipulate the molecular features of polymer chains out of equilibrium, primarily focusing on linear polymer chains. Recently there has been an emerging interest in the dynamics of nonlinear architectures, particularly ring polymers, which exhibit surprising out-of-equilibrium dynamics in dilute solutions. In particular, it has been observed that hydrodynamics can couple to topology in planar elongational and shear flows, driving molecular expansion in the nonflow direction that is not observed for linear chains. In this paper, we extend our understanding of dilute ring polymer dynamics to mixed flows, which represent flow profiles intermediate between simple shear or planar elongation. We map the conformational behaviors at a number of flow geometries and strengths, demonstrating transitions between coiled, tumbling, and stretched regimes. Indeed, these observations are consistent with how linear chains respond to mixed flows. For both linear and ring polymers, we observe a marked first-order-like transition between tumbling and stretched polymers that we attribute to a dynamic energy barrier between the two states. This manifests as bimodal extension distributions in a narrow range of flow strengths and geometries, with the primary difference between rings and linear chains being the presence of molecular expansion in the vorticity direction.

Figure 1

Mixed flows represent linear combinations of shear flow (leftmost flow field, $\alpha =0$) and planar elongational flow (rightmost flow field, $\alpha =1$). Streamlines for various $\alpha$ demonstrate that the principal axis of extension (green arrows denoting $x_f$) varies in mixed flows $0<\alpha <1$.

Figure 2

Linear polymer fractional extension as a function of Weissenburg number and mixed flow parameter $\alpha$ (a) along the axis of principal extension $x_f$, (b) along $y_f$, anad (c) in the vorticity direction $z_f$. For the $x_f$ direction, a linear change undergoes a combination of tumbling or stretching depending on Wi and $\alpha$. Conversely, the chain contracts in both the $y_f$ and $z_f$ directions.

Figure 3

Ring polymer fractional extension as a function of Weissenburg number and mixed flow parameter $\alpha$ (a) along the axis of principal extension $x_f$, (b) along $y_f$, and (c) in the vorticity direction $z_f$. Similarly to the linear case, ring polymers stretch and/or tumble at large Wi in the $x_f$ direction. In contrast to linear chains, however, stretching is observed also in the $z_f$ direction.

Figure 4

(a) Linear and (b) ring polymer conformational phase diagrams as a function of strain rate Wi and the mixed flow parameter $\alpha$. Points correspond to simulations, which are categorized by the criteria described in the text. Coiled chains are near-equilibrium conformations, tumbling chains are rapidly oscillating between slightly stretched and un-stretched conformations, and fully stretched chains are stable near $\mathrm{\Delta }x\approx 0.8$–1.0. We find a coexistence-like region, which is in orange, where there is a bimodal distribution of chain extensions $\mathrm{\Delta }{x}_f/L$. Boxed points correspond to the probability distribution functions in Figs. 7 and 8.

Figure 5

Fractional extension as a function of shifted $\beta \mathrm{Wi}$ for ring and linear polymers with $\alpha =1$. The linear data is shifted by a factor $\beta =1.45$, whereas for the ring data $\beta =1.0$.

Figure 6

Simulation traces of ring polymer extension in $x_f$ (black squares) and $z_f$ (red circles) for $\alpha =0.001$ and (a) Wi = 630, (b) Wi = 794, and (c) Wi = 1000 [corresponding to the probability distribution function in Fig. 8]. We note the coexistence of two distinct conformational behaviors: a rapidly fluctuating extension length $\mathrm{\Delta }{x}_f/L$ and $\mathrm{\Delta }{z}_f/L$ that corresponds to molecular tumbling and a stretched conformation at high values of $\mathrm{\Delta }{x}_f/L$ and $\mathrm{\Delta }{z}_f/L$. (d) Snapshots at Wi $=794$ are indicated by the numerals in (b) and show the unextended (i) and extended (ii) tumbling rings and a fully stretched ring (iii).

Figure 7

Probability density function of linear polymer stretch along the principal axis of extension for (a) $\alpha =0.001$, (b) $\alpha =0.005$, and (c) $\alpha =0.05$ and selected flow rates Wi. These points are indicated in Fig. 4 by boxes.

Figure 8

Probability density function of ring polymer stretch along the principal axis of extension for (a) $\alpha =0.001$, (b) $\alpha =0.005$, and (c) $\alpha =0.05$ and selected flow rates. These points are indicated in Fig. 4 by boxes.

Figure 9

Characteristic lifetime of an extended state for linear and ring polymers (a) at constant $\alpha$ and varying Wi (b) at constant Wi and varying $\alpha$. Insets are the probability distribution of extended lifetimes for the linear polymer case and dashed lines are exponential fits via ${\lambda }_{\mathrm{ext}}$.

Figure 10

Power spectral density of the polymer orientation angle for a (a) linear polymer at constant $\alpha =0.001$, (b) ring polymer at constant $\alpha =0.005$, (c) linear polymer at constant Wi = 82, and (d) ring polymer at constant Wi = 81.

Figure 11

(a) Probability density function of ring and linear polymer stretch along the principal axis of extension and the vorticity ($z_f$) direction at $\alpha =0.005,{\mathrm{Wi}}_{\mathrm{ring}}=79,{\mathrm{Wi}}_{\mathrm{linear}}=33$. These points are chosen to have the same $\alpha$ and average extension $\langle \mathrm{\Delta }{x}_f/L\rangle$. Closed points correspond to the linear chain and open to the ring. The $\mathrm{\Delta }{z}_f/L$ distribution has been plotted on a second $y$ axis. We note a shoulder in the $\mathrm{\Delta }{z}_f/L$ distribution for the ring polymer. (b) Contour plot of $P\left(\{\mathrm{\Delta }{x}_f/L,\mathrm{\Delta }{z}_f/L\}\right)$ for a linear polymer at $\alpha =0.005,{\mathrm{Wi}}_{\mathrm{linear}}=33$. (c) Contour plot of $P\left(\{\mathrm{\Delta }{x}_f/L,\mathrm{\Delta }{z}_f/L\}\right)$ for a ring polymer at $\alpha =0.005,{\mathrm{Wi}}_{\mathrm{ring}}=79$.