Dynamical Slowdown of Polymers in Laminar and Random Flows

The influence of an external flow on the relaxation dynamics of a single polymer is investigated theoretically and numerically. We show that a pronounced dynamical slowdown occurs in the vicinity of the coil-stretch transition, especially when the dependence on polymer conformation of the drag is accounted for. For the elongational flow, relaxation times are exceedingly larger than the Zimm relaxation time, resulting in the observation of conformation hysteresis. For random smooth flows, hysteresis is not present. Yet, relaxation dynamics is significantly slowed down because of the large variety of accessible polymer configurations. The implications of these results for the modeling of dilute polymer solutions in turbulent flows are addressed.

Figure 1

Elongational flow: (a) rescaled relaxation time vs Wi for $b=400$. The entropic force is given by the Warner law. For small Wi, $t_{\mathrm{rel}}/\tau$ grows as $1/(1–2\mathrm{Wi})$, as can be seen by replacing $\widehat{f}(r)$ with 1 in Eq. (2). For $\mathrm{Wi}\gg {\mathrm{Wi}}_{\mathrm{crit}}$, the time needed to reach the asymptotic regime is set by the time scale of the flow, ${\gamma }^{-1}$, and $t_{\mathrm{rel}}/\tau$ decreases as ${\mathrm{Wi}}^{-1}$. (b) Rescaled maximum relaxation time $t_{\max }/\tau$ vs ${\zeta }_s/{\zeta }_c$ ($b=400$).

Figure 2

(a) Effective energy at the coil-stretch transition for a polyacrylamide (PAM) molecule ($b=3953$, ${\zeta }_s/{\zeta }_c=6.87$) in the elongational flow (dashed line) and the 3D Batchelor-Kraichnan flow (solid line). The left vertical axis refers to the dashed line; the right vertical axis refers to the solid line. (b) Effective energy in the random flow for a PAM molecule with constant drag (${\zeta }_s={\zeta }_c$) (from top to bottom $\mathrm{Wi}=0.4$, 0.7, 1.0, and 1.3); (c) the same as (b), but with ${\zeta }_s=6.87{\zeta }_c$ (from top to bottom $\mathrm{Wi}=0.28$, 0.33, 0.38, and 0.43). When ${\zeta }_s>{\zeta }_c$ the transition occurs in a much narrower range of Wi.

Figure 3

Three-dimensional Batchelor-Kraichnan flow: (a) $t_{\mathrm{rel}}/\tau$ vs Wi for a PAM molecule ($b=3953$); (b) $t_{\max }/\tau$ for the following polymers: DNA (●, $b=191.5$; ○, $b=260$; □, $b=565$; +, $b=2250$), polystyrene (×, $b=673$), polyethyleneoxide (PEO) (▴, $b=1666$), Escherichia Coli DNA (▵, $b=9250$), and PAM (▪). Measures of $b$ and ${\zeta }_s/{\zeta }_c$ can be found in 18, 19, 20, 25. Synthetic polymers are modeled by the Warner law, whereas biological molecules are described by the Marko-Siggia law. Relaxation times were computed by means of the variation-iteration method [28]. For ${\zeta }_s={\zeta }_c$ they can be obtained by solving an eigenvalue problem for a Heun equation [15].

Figure 4

Brunk-Koch-Lion flow: $t_{\mathrm{rel}}/\tau$ vs $\mathrm{Wi}=\lambda \tau$ from Brownian dynamics simulations; (a) PEO; (b) PAM.