Rheologically tuned modes of collective transport in active viscoelastic films

Abstract

While many living biological media combine both viscous and elastic properties, most theoretical studies employ either purely fluid- or solid-like descriptions. We here use a unified framework for active films on substrates capable of describing a broad range of viscoelastic behavior to explore the interplay between activity and rheology. The core of the study is a comprehensive state diagram showing a rich world of spatiotemporal dynamic states. Our results demonstrate the potential of tunable rheology to realize modes of controlled active transport on the microscale.

Figures (4)

• Figure 1: Dynamical state diagram as a function of the displacement relaxation time $\tau_\mathrm{d}$ and strength of active forcing $\nu_\mathrm{p}$. From left to right, the influence of elasticity increases. Colors indicate regions of different stable dynamical states. Insets in the corners illustrate idealized tracer trajectories. Blue and red dots refer to the dynamical states explored in more detail in Figs. \ref{['fig:Stripes']} and \ref{['fig:OscDomains']}, respectively. Remaining parameter values are $\gamma_\mathrm{a} = 1$, $\eta = 1$, $\mu = 1$, $\nu_\mathrm{v} = 1$, and $\nu_\mathrm{d} = 10$.
• Figure 2: Stripe pattern associated with the blue region in Fig. \ref{['fig:StateDiagram']}. (a) Snapshot of the vorticity field $\omega=(\nabla\times\mathbf{v})_z$ in a system of size $200\times 200$ at $\tau_\mathrm{d} = 0.25$, see the blue dot in Fig. \ref{['fig:StateDiagram']}. Small dark arrows indicate the velocity field $\mathbf{v}$. Transport of the patterns, given by the phase velocity $v_\mathrm{ph}=-\mathrm{Im}\{\lambda(k)\}/k$, is opposite to the averaged active flow direction $\langle\mathbf{v}\rangle$. Associated material trajectories are snake-like, see the top right inset. (b) Largest growth rates $\mathrm{Re}\{\lambda(k)\}$ of modes $\mathbf{k}\|\langle\mathbf{v}\rangle$ when starting from a uniform polar state, and phase velocity $v_\mathrm{ph}=-\mathrm{Im}\{\lambda(k)\}/k$. Remaining parameter values are $\gamma_\mathrm{a} = 1$, $\eta = 1$, $\mu = 1$, $\nu_\mathrm{v} = 1$, $\nu_\mathrm{d} = 10$, and $\nu_\mathrm{p} = 20$.
• Figure 3: Heterogeneous state of imperfect local circling motion. (a) At each spot, the vectors of polar orientational order $\mathbf{P}$ and collective flow $\mathbf{v}$ (dark arrows) rotate over time. Domains are identified by the field of phase shift $\Delta \phi_{Pv}$ (color bar) between $\mathbf{P}$ and $\mathbf{v}$. Positive and negative values are connected to clockwise and counterclockwise rotation, respectively. A sample trajectory is shown in the top right corner. The size of the snapshot is $100\times 100$. (b) Averaged spatial correlation function $C_{\Delta \phi_{Pv}}(\Delta x)$ of the phase shift $\Delta \phi_{Pv}$ between any two points of evaluation with distance $\Delta x$ from each other. The associated characteristic length scale $\ell$ of the dynamic patterns is identified with the first minimum of $C_{\Delta \phi_{Pv}}(\Delta x)$. (c) Typical domain size $\ell$ as a function of active force strength $\nu_\mathrm{p}$ for different values of the relaxation time $\tau_\mathrm{d}$. (d) Enlarged view of the marked area in (a) showing $+1$ and $-1$ defects in the velocity field as red and blue dots. (e) Time-averaged number of defect pairs $\langle N_v\rangle$ in the velocity field of a system of size $200\times 200$ as a function of $\nu_\mathrm{p}$. Remaining parameter values are $\gamma_\mathrm{a} = 1$, $\eta = 1$, $\mu = 1$, $\nu_\mathrm{v} = 1$, $\nu_\mathrm{d} = 10$, and $\nu_\mathrm{p} = 20$.
• Figure 4: Rheologically controlled transport of an array of cargo objects. (a) We periodically vary the relaxation time $\tau_\mathrm{d}$ of elastic displacements between $0.25$ and $1$. Thus, we tune the degree of elasticity. (b) Resulting trajectories for an array of four transported cargo objects. Tuning rheology allows in a controlled way to switch between persistent collective transport and intermediate storage times, roughly maintaining the spatial order within the transported cargo array. Blue and red parts in (a) and (b) correspond to each other and to the blue and red regions in the state diagram in Fig. \ref{['fig:StateDiagram']}. The remaining parameters are $\gamma_\mathrm{a} = 1$, $\eta = 1$, $\mu = 1$, $\nu_\mathrm{v} = 1$, $\nu_\mathrm{d} = 10$, and $\nu_\mathrm{p} = 20$. The initial edges in the cargo array are of length $1$.