Unified description of viscous, viscoelastic, or elastic thin active films on substrates

Abstract

It is frequent for active or living entities to find themselves embedded in a surrounding medium. Resulting composite systems are usually classified as either active fluids or active solids. Yet, in reality, particularly in the biological context, a broad spectrum of viscoelasticity exists in between these two limits. There, both viscous and elastic properties are combined. To bridge the gap between active fluids and active solids, we here systematically derive a unified continuum-theoretical framework. It covers viscous, viscoelastic, and elastic active materials. Our continuum equations are obtained by coarse-graining a discrete, agent-based microscopic dynamic description. In our subsequent analysis, we mainly focus on thin active films on supporting substrates. Strength of activity and degree of elasticity are used as control parameters that control the overall behavior. We concentrate on the analysis of transitions between spatially uniform analytical solutions of collective migration. These include isotropic and polar, orientationally ordered states. A stationary polar solution of persistent directed collective motion is observed for rather fluid-like systems. It corresponds to the ubiquitous swarming state observed in various kinds of dry and wet active matter. With increasing elasticity, persistent motion in one direction is prevented by elastic anchoring and restoring forces. As a consequence, rotations of the spatially uniform migration direction and associated flow occur. Our unified description allows to continuously tune the material behavior from viscous, via viscoelastic, to elastic active behavior by variation of a single parameter. Therefore, it allows in the future to investigate the time evolution of complex systems and biomaterials such as biofilms within one framework.

Figures (4)

• Figure 1: Sketch of the microscopic picture that we use for coarse-graining to derive our continuum theory. $N$ active agents, labeled by $n$, are embedded in a surrounding continuous viscous, viscoelastic, or elastic medium, indicated by the yellow background. Every active agent $n$ exerts an active force $f_\mathrm{a}\mathbf{N}_n$ (black arrows) on the surrounding medium, see the inset. This force results from self-propulsion by interaction with an underlying substrate or due to external forcing in artificial systems. $f_\mathrm{a}$ is the magnitude of the active force and $\mathbf{N}_n$ its direction, here identified with the orientation of the $n$th object. Flows $\mathbf{v}$ (brown arrows) in the medium affect the orientations $\mathbf{N}_n$ of the active objects and lead to their displacements, resulting in velocities $d\mathbf{X}_n/dt$ (blue arrows).
• Figure 2: Spatially uniform dynamic states of rotating polar orientational order and flow. (a) Snapshots of the uniform velocity field during one period $T$ for a genuinely elastic solid ($\tau_\mathrm{d}\rightarrow\infty$). The arrows in the main plots display the current state of the overall flow field $\mathbf{v}$. Circular insets visualize the current states of the polar orientational order parameter $\mathbf{P}$, overall flow velocity $\mathbf{v}$, and field of memorized elastic displacements $\mathbf{u}$. Phase shifts between these quantities are obvious. These results agree with numerical solutions in a small periodic box. (b) Corresponding time evolution of the spatially averaged $x$-components of $\mathbf{P}$, $\mathbf{v}$, and $\mathbf{u}$. We rescale $\langle v_x \rangle$ and $\langle u_x \rangle$ by the amplitudes $v_0$ and $u_0$, respectively. We set the remaining parameters to $\gamma_\mathrm{a} = 1$, $\kappa = 0$, $\eta = 1$, $\mu = 1$, $\nu_\mathrm{v} = 1$, $\nu_\mathrm{d} = 10$, and $\nu_\mathrm{p} = 20$.
• Figure 3: Analysis of the transition between the different dynamic states. (a) Amplitude $v_0$ of the velocity, see Eqs. (\ref{['eq:PolarForm']}), as a function of the strength $\nu_\mathrm{p}$ of active forcing for elastic solid-like systems ($\tau_\mathrm{d}\rightarrow\infty$). Analytical calculations (black lines) reveal a subcritical transition from the isotropic to the rotating state of polar orientational order. Here, solid lines denote stable and dashed lines unstable solutions. Numerical results (red and blue data points) confirm our analytical results and the associated hysteretic behavior. (b) Rotational frequency $\omega_0$ of the direction of spatially uniform motion as a function of $\nu_\mathrm{p}$ for the rotating solution, see Eqs. (\ref{['eq:PolarForm']}). Results are displayed for decreasing relaxation time $\tau_\mathrm{d}$ of the memory of elastic displacements. This corresponds to declining elastic and increasing fluid-like behavior. Along this trend, we observe that the transition from quiescent to rotational dynamic states turns from sub- to supercritical. (c) Magnitude of polar orientational order $|\mathbf{P}|$ as a function of activity $\nu_\mathrm{p}$ for various values of the relaxation time $\tau_\mathrm{d}$ of the elastic memory. Isotropic and polar stationary solutions, see Eqs. (\ref{['eq:PolarSolution']}), as well as stable and unstable rotational solutions are shown. The remaining parameters in all cases are set to $\gamma_\mathrm{a} = 1$, $\kappa = 0$, $\eta = 1$, $\mu = 1$, $\nu_\mathrm{v} = 1$, and $\nu_\mathrm{d} = 10$.
• Figure 4: State diagram for small, idealized systems that combine activity and viscoelasticity. We here impose spatial uniformity of the solutions to enable analytical considerations for illustration of the theory. Dropping this idealization leads to a reconsideration of the state diagram and novel dynamic states as described in a companion paper reinken2025rheologically. We hatch the areas of significant deviation when the idealization of spatial uniformity is dropped. Such deviations mainly occur when both activity and (visco)elastic effects become substantial, which corresponds to the major focus of our considerations. The diagram is displayed as a function of the relaxation time of elastic memory $\tau_\mathrm{d}$, which measures the importance of elastic effects, and the strength of active forcing $\nu_\mathrm{p}$. Green and yellow areas denote the regions of uniform stationary polar and isotropic solutions, respectively. The spatially homogeneous rotational solution exists above the red curve. A hysteretic region is indicated by the dashed black line. We set the remaining parameters to $\gamma_\mathrm{a} = 1$, $\kappa = 0$, $\eta = 1$, $\mu = 1$, $\nu_\mathrm{v} = 1$, and $\nu_\mathrm{d} = 10$.