Kinetics of polymer tumbling in shear flow: A coarse-grained description

We study the classic problem of dynamical evolution of a polymer in a shear flow. Interestingly, the polymer goes through several distinctly identifiable conformations during its passage from coiled to stretched states back and forth. We identify these conformations assumed by the polymer while tumbling and study the kinetics of the process in terms of the residence and recurrence times of individual conformations. The distribution of residence times exhibits exponentially decaying tails which helps us build an effective Markovian picture of the truly non-Markovian problem. We present the explicit $\mathbb{W}$ matrix for the coarse-grained evolution via a master equation and study its elements as a function of the Weissenberg number. We show that the timescales of decay of the autocorrelation function for the full Langevin dynamics compare quite well with the approximate results from the master equation approach.

Figure 1

(a) Sample conformations in coarse-grained states $i$ during tumbling: coiled (c), half-dumbbell (hd), half-folded (hf), forked (fk), half-dumbbell-half-folded (hd-hf), dumbbell (d), folded (f), kink (k), and stretched (s). The corresponding values of $n_i\in \{0,...,8\}$ are shown alongside (see the text). (b) The range of values of $X/C_L$ of the nine states are shown. (c) A hf conformation is shown as an example with its $X/C_L,d_1,d_2$, and $d_3$ and $N_1,N_2$, and $N_3$ values.

Figure 2

Probability of different conformations with varying Wi.

Figure 3

Variation of autocorrelation $C\left(\tau \right)$ with lag $\tau$ for different Wi's. The inset (log-log scale) shows that ${\omega }_0\sim {\mathrm{Wi}}^{0.67}$.

Figure 4

(a) Distribution of residence times ${\tau }_{\mathrm{res}},\mathrm{Wi}=15(\circ )$ and $\mathrm{Wi}=75\left(▵\right)$ and (b) the algebraic decay of the characteristic residence time ${\tau }_{\mathrm{res}}^c$ with the Wi. The inset in (a) shows the nonexponential behavior at short times, and the dashed line is a fit used to estimate the characteristic time of decay. The corresponding quantities for the recurrence time ${\tau }_{\mathrm{rec}}$ are shown in parts (c) and (d). The symbols represent the corresponding conformations: $▿$, c; $\blacksquare$, hd; $\circ$, hf; $⧫$, fk; $◊$, hd-hf; $▴$, d; $◐$, f; $\square$, k; $•$, s.

Figure 5

Color map of the $\mathbb{W}$ matrix for (a) $\mathrm{Wi}=15$ and (b) $\mathrm{Wi}=75$. With increasing Wi the jump rates increase, e.g., ${\mathbb{W}}_{56}$ is higher for $\mathrm{Wi}=75$ as compared to that of $\mathrm{Wi}=15$. The diagonal elements of the $\mathbb{W}$ matrix are negative. (c) The colored boxes indicate the matrix elements of $\mathbb{W}$ for $\mathrm{Wi}=75$ which are zero for $\mathrm{Wi}=15$. The elements ${\mathbb{W}}_{ij}$ are shifted for the purpose of visibility.

Figure 6

Autocorrelation $C\left(\tau \right)$ for the time series of states from the Langevin dynamics simulations and master equation for (a) $\mathrm{Wi}=15$, (b) $\mathrm{Wi}=75$, and (c) $\mathrm{Wi}=150$. It is evident that the two autocorrelations decay at nearly the same timescale [see (d)].