Journal Club for October 2017: Multiscale modeling and simulation of active matter

Tong Gao

Department of Mechanical Engineering and Department of Computational Mathematics, Science, and Engineering, Michigan State University

Introduction

Active matter, a novel class of non-equilibrium materials made up of constituents that are self-driven, present new challenges to design, control, and analysis of materials. Examples of active matter include swarms of swimming bacteria [1], self-propelled colloidal particles [2-5], and mixtures of cytoskeletal filaments and molecular motors [6-9]; see Fig. 1. Despite their differences in composition and length-scale, these diverse systems show common features absent in equilibrium systems, including collective motion, non-equilibrium ordering transitions, and anomalous fluctuations and mechanical properties, which cannot be explained by the conventional equilibrium physics. The intriguing physical properties of active matter have generated considerable excitement over the last decade in disciplines as diverse as applied mathematics, biophysics, developmental biology, materials science, and mechanical/chemical engineering.

FIG. 1: Examples of active matter systems: (a) Bacterial (Bacillus subtilis) swarm [1]; (b) Self-driven bimetallic particles [3]; (c) Flow field of a bacterial “turbulence” (Bacillus subtilis) [14]; (d) Active liquid crystalline phase of a microtubule (MT)-motor suspension featured by motile disclination defects [9].

Self-assembly

To understand the multiscale nature of the complex dynamics in active matter through a bottom-up approach, we first examine the microscale particle motions. Here we focus on the active matter system reported by Sanchez et al. [9] where the complex dynamics is driven by concentrated suspensions comprising microtubule (MT) and molecular motors. At the microscopic level, the local dynamical behaviors therein can be captured through discrete particle simulations without incorporating the long-range hydrodynamic interactions (e.g., Brownian Dynamics and Monte-Carlo method [15]). Figure 2 shows an example of an immersed suspension of polar MTs.  We consider adjacent MTs that are coupled by plus-end directed crosslinking motors consisting of one motor head on each MT connected by a tether that responds as a spring to stretching (Fig. 2a). The motor on each crosslink endpoint moves with a linear force-velocity relation (Fig. 2b) [16].  For a nematically aligned suspension there are two basic types of MT pair interaction. For polar anti-aligned MTs (Fig. 2c) the motors on each end of an active crosslink move in opposite directions, stretching the tether. This creates forces on each MT that, acting against fluid drag, slide the MTs relative to each other towards their minus-ends. This process is termed “polarity sorting” [17]. Conversely, for polar-aligned MTs the motors on each end of the crosslink move in the same direction, there is little or no net sliding, and the tether pulling on the leading motor causes stretched tethers to relax (Fig. 2d). For the long-time behavior, Fig. 2e shows a nematic state of MTs interacting only through thermal fluctuations and steric interactions (without motors); Fig. 2f suggests that the system now exhibits active MT flows driven by polarity sorting, leading to the formation of polar lanes (domains of MTs with similar polarity). These polar lanes are highly dynamic and show large fluctuations.

 FIG. 2: (a) Schematic of a cluster of polar-aligned and anti-aligned MTs, with plus ends marked by red rings. Motors walk on neighboring MTs, and (b) exert spring-like forces with a piecewise linear force-velocity relation. (c) An anti-aligned MT pair. (d) A polar-aligned MT pair. Grey arrows characterize the magnitude of the extensile stress. (e) System with no crosslinks, illustrating the 2D nematic state. (f) An active system with mobile crosslinks exhibits active flows and formation of polar lanes [15].

Material instability

To analyze the large-scale collective dynamics, we construct continuum models (e.g., Doi-Onsager theory and macro liquid crystal models) through systematic coarse-graining, based on the measured particle stresses in the particle simulations [15]. As shown in Figure 3, we analyze the coherent structures observed in the simulations, and have identified that the complex collective dynamics is in fact developed through a concatenation of hydrodynamic instabilities as the system evolves from an initially isotropic state where the rod-like particles are randomly oriented. For concentrated active suspensions like MT/motor assemblies, the system can spontaneously evolve towards a nematic state due to the enhanced steric interactions when the particles are densely packed. Immediately after the isotropic-nematic transition, a bending instability develops from the nematic state to induced jet-like active flows that are perpendicular to the alignment direction, as shown in Fig. 3a, and eventually lead to an active liquid-crystalline phase featured by motile disclination defects and chaotic flows (Fig. 3b and 3c).

Fig 3: (a) Bending of nematic field lines of a concentrated MT/motor suspension of rod-like particles that are initially aligned in the horizontal direction. Inset: the fluid velocity vector field (blue) and the eigenmode (red line). The separation distance between the adjacent bends characterizes the length scale of the nematic structures. (b) Genesis of disclination defects at late times. (c) Large scale active nematic flows featured by motile disclination defects and incipient crack formation. In (a-c), the vector field represents the nematic director; the background colormap represents the scalar order parameter [15].

Confinement control

To effectively control the collective dynamics in various internally-driven systems, it is critical to manipulate the emergent coherent structures. One way of doing this is to tune the suspension concentration and the amount of chemical fuels. Alternatively, we can take advantage of the particle interactions, either individually or collectively, with obstacles and geometric boundaries to manipulate the system more directly. By trapping active suspensions within the straight and curved boundaries, stable flow patterns, such as unidirectional circulations, traveling waves, density shocks, and rotating vortices [18-26] (also see Fig. 4a-c), have already been constructed. More interestingly, active nematics under soft confinement by surface tension are able to generate internal flows to break symmetry and drive the whole-body movement, which is highlighted in Fig. 4d for sequential simulation snapshots of a moving droplet whose motion is driven by internal collective dynamics of active nematic flows [27-30].

Fig 4: Snapshots of the motile declination defects in a circular disk (a), an annulus (b), and biconcave chamber (c) [26]. (d) Sequential simulation snapshots of a moving droplet driven by collective dynamics of active particles. Top: evolution of particle alignment within the drop. Bottom: force generation [27].

Conclusion

People so far have investigated various aspects of these active-matter systems at different scales, from the dynamics and mechanical properties of filament bundles to macroscopic behavior and stability of active suspensions. From the modeling perspective, one biggest challenge is how to construct coarse-grained models that have precise connections with some unique microscale dynamics for various different active systems, such as self-swimming or rotation of active particles. As shown by the above example of active nematics, bottom-up multiscale approaches are preferred compared to some rather general top-down models where phenomenological models are often used without justification. Furthermore, constructing coarse-grained models may assume separation of time and length scales in the system, which needs to be carefully examined by comparing with the experimental observations, or through the direct particle simulations with both short- and long-range interactions being appropriately resolved. In general, it is desired to develop hybrid algorithms that combine computational fluid dynamics  techniques and stochastic methods to perform large-scale simulations, together with coarse-grained modeling, to reveal how to hierarchically design, analyze, and control novel active materials.

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