Analytical Analysis of the Conformational and Rheological Properties of Flexible Active Polar Linear Polymers under Shear Flow

Abstract

The conformational and rheological properties of active polar linear polymers (APLPs) under linear shear flow are studied analytically. We describe a discrete APLP as an inextensible flexible Gaussian bead-spring chain supplemented by active forces along the bonds. The linear, non-Hermitian equations of motion are solved by an eigenfunction expansion in terms of a biorthogonal basis set. The model reveals an intimate coupling between activity and shear flow, which implies activity-enhanced polymer conformational and rheological properties. Compared to a passive polymer, we find a significantly enhanced shrinkage transverse to the flow direction with increasing shear rate, with a power-law exponent $-4/3$, compared to the passive values of $-2/3$. This conformational change is tightly linked with a strongly amplified shear-thinning behavior, where the shear viscosity exhibits the same power law. The characteristic shear rate for the onset of these effects is determined by the activity. In the asymptotic limit of large activities, the shear-induced features become independent of activity and equal to those of passive polymers.

Figures (8)

• Figure 1: A continuum representation of an active polymer under linear shear flow. The green arrows show the direction of the active force, while the blue arrows indicate the direction of the imposed shear flow.
• Figure 2: Stretching coefficient $\mu$ as function of the Weissenberg number $Wi$ (Eq. \ref{['eq:def_wi']}) for various Péclet numbers $Pe$ (legend) and the number of beads $N=200$. Inset: scaled stretching coefficient, illustrating the universal crossover from the activity-dominated to the passive shear-rate regime.
• Figure 3: (a) Relaxation times $\mu \tau_n/\tau_l$ and (b) frequencies $\omega_n \tau_l$ as a function of the mode number $n$ for the Péclet number $Pe =150$ and various Weissenberg numbers $Wi$ (legend). The relaxation time $\tau_l$ is defined as $\tau_R/N^2$. The dashed line in (a) indicates the power law $1/n^2$.
• Figure 4: Normalized polymer end-to-end vector correlation function $C_e(t)$ (Eq. \ref{['eq:end_corr']}) as a function of the scaled time $t/\tau_R$ for the polymer length $N=200$ and various Péclet numbers (legend).
• Figure 5: Scaled characteristic relaxation time $\tau_e$ of APLPs in absence of shear as a function of $Pe$ for the polymer lengths $N=100$ and $N=200$. The solid line (blue) indicates the power $\tau_e \sim Pe^{-1}$. The relaxation time $\tau_l=\tau_R/N^2 = \gamma l^2/(3k_BT)$.