Abhijeet
Joshi
,
Elias
Putzig
,
Aparna
Baskaran
* and
Michael F.
Hagan
*

Martin Fisher School of Physics, Brandeis University, Waltham, MA 02453, USA. E-mail: aparna@brandeis.edu; hagan@brandeis.edu

Received
29th October 2018
, Accepted 27th November 2018

First published on 29th November 2018

Active nematics are microscopically driven liquid crystals that exhibit dynamical steady states characterized by the creation and annihilation of topological defects. Motivated by differences between previous simulations of active nematics based on rigid rods and experimental realizations based on semiflexible biopolymer filaments, we describe a large-scale simulation study of a particle-based computational model that explicitly incorporates filament semiflexibility. We find that energy injected into the system at the particle scale preferentially excites bend deformations, reducing the apparent filament bend modulus. The emergent characteristics of the active nematic depend on activity and flexibility only through this activity-renormalized bend ‘modulus’, demonstrating that apparent values of material parameters, such as the Frank ‘constants’, depend on activity. Thus, phenomenological parameters within continuum hydrodynamic descriptions of active nematics must account for this dependence. Further, we present a systematic way to estimate these parameters from observations of deformation fields and defect shapes in experimental or simulation data.

The challenge in this task can be illustrated by considering the shape of a defect. Defects have been the subject of intense research in equilibrium nematics, since they must be eliminated for display technologies, and controlled for applications such as directed assembly and biosensing.^{8–10} Theory and experiments^{10–12} have shown that defect morphologies depend on the relative values of the bend and splay elastic moduli, k_{33} and k_{11} (Fig. 1a), defined in the Frank free energy density for a 2D nematic as f_{F} = k_{11}(∇·)^{2} + k_{33}( × (∇ × ))^{2} with the director field.^{13,14} Different experimental realizations of active nematics also exhibit variations in defect morphologies (e.g.Fig. 1c and d). However, relating these defect morphologies to material constants is much less straightforward than in equilibrium systems. Since moduli are an equilibrium concept, they cannot be rigorously defined in a non-equilibrium system such as an active nematic. If it is possible to define ‘effective moduli’ in active systems, there is currently no systematic way to measure them, and it is unclear whether and how they may depend on activity.

Fig. 1 (a and b) The equilibrium director field configuration near a +½ defect for (a) bend modulus larger than splay modulus k_{33}/k_{11} = 3.4, leading to pointed defects and (b) splay modulus larger than bend k_{33}/k_{11} = 0.3 leading to rounded defects. In each case the director field was calculated by numerically minimizing the Frank free energy around a separated ±½ defect pair. (c and d) Typical defect morphologies observed in (c) a vibrated granular nematic^{5} and (d) a microtubule-based active nematic.^{1,40} |

Computational and theoretical modeling could overcome these limitations. Symmetry-based hydrodynamic theories^{15–17} have led to numerous insights about active nematics, including describing defect dynamics (e.g.ref. 18–23), induced flows in the suspending fluid (e.g.ref. 24–26), and how confinement in planar^{27–30} and curved geometries^{31–34} controls defect proliferation and dynamics. However, hydrodynamic theories cannot predict how material constants or emergent behaviors depend on the microscale properties of individual nematogens. Further, while simulations of active nematics composed of rigid rods^{2,35–37} have elucidated emergent morphologies, these models do not account for internal degrees of freedom available to flexible nematogens.^{1,38,39}

In this work, we perform large-scale simulations on a model active nematic composed of semiflexible filaments, and determine how its emergent morphology depends on filament stiffness and activity. Fig. 2 shows representative simulation outcomes. We find that the intrinsic bending modulus of nematogens κ undergoes an apparent softening in an active nematic, according κ_{eff} ∼ κ/(f^{a})^{2}, where f^{a} is a measure of its activity. Furthermore, the characteristics of the active nematic texture – the defect density, magnitudes and characteristic scales of bend and splay deformations, the shape of motile defects, and number fluctuations – depend on activity and rigidity only through the apparent nematogen rigidity κ_{eff}. Moreover, we show that one can define effective bend and splay moduli analogous to their equilibrium counterparts, but that the apparent bend modulus is proportional to κ_{eff}. This observation demonstrates that activity preferentially dissipates into particular modes within an active material, depending on the internal degrees of freedom of its constituent units. Further, these results suggest revisiting assumptions underlying existing hydrodynamic theories of active matter, since a microscopic model for an active fluid results in macroscopic properties that depend on activity in nontrivial ways, which cannot be simply described by an active stress. Finally, we present a method of parameterizing defect shapes, which enables estimating the relative magnitudes of apparent bend and splay constants from experimental observations of defects, and thus allows direct experimental testing of our prediction.

Fig. 2 A visual summary of how active nematic emergent properties depend on the activity parameter f^{a} and the filament modulus κ. The left panel in each row shows 1/16th of the simulation box for indicated parameter values, with white arrows indicating positions and orientations of +½ defects and white dots indicating positions of -½ defects. Filament beads are colored according to the orientations of the local tangent vector. The right 4 panels are each zoomed in on a +½ defect, with indicated parameter values. The white lines are drawn by eye to highlight defect shapes. Animations of corresponding simulation trajectories are in the ESI.†^{41} The box size is (840 × 840σ^{2}) for all simulations in this work. Other parameters are τ_{1} = 0.2τ and k_{b} = 300k_{B}T_{ref}/σ^{2}. |

m_{α,i} = f^{a}_{α,i} − γṙ_{α,i} − ∇_{rα,i}U + R_{α,i}(t). | (1) |

The interaction potential includes three contributions which respectively account for non-bonded interactions between all bead pairs, stretching of each bond, and the angle potential between each pair of neighboring bonds:

(2) |

U_{nb}(r) = 4ε((σ/r)^{12} − (σ/r)^{6} + 1/4)Θ(2^{1/6}σ − r) | (3) |

U_{s}(r) = −1/2k_{b}R_{0}^{2}ln(1 − (r/R_{0})^{2}) | (4) |

U_{angle}(θ) = κ(θ − π)^{2} | (5) |

Finally, activity is modeled as a propulsive force on every bead directed along the filament tangent toward its head. To render the activity nematic, the head and tail of each filament are switched at stochastic intervals so that the force direction on each bead rotates by 180 degrees. In particular, the active force has the form, f^{a}_{α,i}(t) = η_{α}(t)f^{a}t_{α,i}, where f^{a} parameterizes the activity strength, and η_{α}(t) is a stochastic variable associated with filament α that changes its values between 1 and −1 on Poisson distributed intervals with mean τ_{1}, so that τ_{1} controls the reversal frequency.

In the FENE bond potential, R_{0} is set to 1.5σ and k_{b} = 300k_{B}T_{ref}/σ^{2} for the simulations reported in the main text (except in Fig. 3). These parameters lead to a mean bond length of b ≈ 0.84σ, which ensures that filaments are non-penetrable for the parameter space explored in this work. We have fixed the filament length at M = 20 beads, so L = (M − 1)b is the mean filament length. Varying M does not change the scaling of observables, although it shifts properties such as the defect density since the total active force per filament goes as f^{a}M. Also, the maximum persistence length above which scaling laws fail is proportional to M (see Results). In the semiflexible limit, the stiffness κ in the discrete model is related to the continuum bending modulus as , and thus the persistence length is given by l_{p} = 2bκ/k_{B}T.

We performed two sets of simulations. Initially, we set the FENE bond strength k_{b} = 30k_{B}T_{ref}/σ^{2} and τ_{1} = τ. However, for these parameters we discovered that interpenetration of filaments becomes possible at large propulsion velocities, thus limiting the simulations to f^{a} ≤ 10. To enable investigating higher activity values, we therefore performed a second set of simulations (those reported in Fig. 4–7 of the main text) with higher FENE bond strength, k_{b} = 300k_{B}T_{ref}/σ^{2} and shorter reversal timescale τ_{1} = 0.2, with κ ∈ [100, 10000]k_{B}T_{ref}. This enables simulating activity values up to f^{a} ≤ 30.

Fig. 5 Defect shape depends on the effective bending rigidity. (A) Schematic showing the defect-centered coordinate system defined in the text, with azimuthal angle ϕ and director angle θ(r, ϕ). Note that the x-axis is chosen along the orientation vector of the +1/2 defect given by the angle θ_{0}′ defined in the ESI.†^{}^{41} (B) Mode of defect shape parameter b_{1} for +½ defects as a function of κ_{eff} for f^{a} ∈ [7, 30] and κ ∈ [100, 10000], with b_{1}(r) values evaluated at a radial distance r = 12.6σ from the defect core. The dashed line and blue asterisk symbols show values of b_{1} calculated for isolated defects from equilibrium continuum elastic theory, with the ratio of bend and splay moduli k_{33}/k_{11} for each κ_{eff} set according to the measured ratio R of bend and splay deformations in Fig. 4. Images of defects at two indicated parameter sets are shown. Other parameters are τ_{1} = 0.2τ and k_{b} = 300k_{B}T_{ref}/σ^{2}. |

The shorter reversal timescale was needed to keep the system in the nematic regime at large f^{a}. When the product f^{a}τ exceeds a characteristic collision length scale the self-propulsion becomes polar in nature and model is no longer a good description of an active nematic. In particular, the filaments behave as polar self-propelled rods, as evidenced by the formation of polar clusters.^{46–50} Interestingly, increasing k_{b} increases the rate of defect formation and decreases the threshold value of f^{a}τ above which phase separation occurs. We monitored the existence of phase separation by measuring local densities within simulation boxes (see ref. 41). We found no significant phase separation (except on short scales) or formation of polar clusters for any of the simulations described here, indicating the systems were in the nematic regime. Within the nematic regime, increasing k_{b} does not qualitatively change results or scaling relations, although it does quantitatively shift properties such as the defect density. Additional plots for the simulation set with k_{b} = 30 are shown in the ESI.†^{}^{41}

Before reporting these results, let us first consider the basic physics governing the emergent phenomenology in our system. Deforming a nematogen causes an elastic stress that scales ∼κ, the filament bending modulus. This elastic stress must be balanced by stresses arising from interparticle collisions. These collisional stresses have two contributions. First there are the passive collisional stresses, which are (1) in our units (as we have chosen the thermal energy and the particle size σ as the energy and length scales). Then there are active collisional stresses that arise due to the self-replenishing active force of magnitude f^{a} acting on each monomer. The active collisional stress will scale as ν_{coll}Δp_{coll}, i.e., the collision frequency times the momentum transfer at each collision. Both the collision frequency and momentum transfer will scale with the self-propulsion velocity,^{51} which in our model is set by f^{a}. Thus, the collisional stresses will scale ∼(1 + (f^{a})^{2}). So, the system properties will be controlled by a parameter which measures the ratio of elastic to collisional stresses, κ_{eff} ≡ κ/(1 + (f^{a})^{2}).

Note that we restrict the analysis to f^{a} ≥ 5 because the system loses nematic order for smaller activity values (see Fig. S5C (ESI†)^{41} and ‘Comparison with Equilibrium’ below). The results in Fig. 3 were performed on our initial data set with τ_{1} = τ and k_{b} = 30k_{B}T_{ref}/σ^{2} and thus are limited to f^{a} ≤ 10 (see Methods).

We now test whether the arguments proposed above can describe the collective behaviors of an active nematic, using several metrics for deformations in the nematic order: the distribution of energy in splay and bend deformations, defect shape, defect density, and number fluctuations. As shown in Fig. 4–7, all of these quantities depend on bending rigidity and activity only through the combination κ_{eff}. We performed the analysis of active nematics textures (described next) on the second parameter set with τ_{1} = 0.2τ and k_{b} = 300k_{B}T_{ref}/σ^{2}, which allowed f^{a} ≤ 30.

In a 2D nematic, any arbitrary deformation of the director field can be decomposed into a bend deformation d_{bend} = ((r) × (∇ × (r))) and a splay deformation d_{splay} = (∇·(r)). We define associated strain energy densities D_{bend} = ρS^{2}|d_{bend}|^{2} and D_{splay} = ρS^{2}d_{splay}^{2}, which measure how the system's elastic energy distributes into the two linearly independent deformation modes. The prefactors of density ρ and order S allow the definitions to be valid in regions such as defect cores and voids that occur in the particle simulations.

To compare the splay and bend deformations to their form in an equilibrium nematic, let us consider the ratio of total strain energy in splay deformations to those in bend, . In equilibrium, equipartition requires that this ratio satisfy R_{eq} = k_{33}/k_{11}. In our simulations, we observe that R depends strongly on activity; however, the data for all parameter sets collapses when plotted as a function of the effective filament bend modulus κ_{eff}. This result is consistent with the concept of effective moduli, but shows that their values depend on activity.

To understand why the apparent modulus values depend on activity, we analyzed the power spectra associated with these strain energies (Fig. S4 in ref. 41). The power spectra exhibit two features: (1) Fig. 4B shows that for the smallest simulated effective rigidity, κ_{eff} = 2, the peak of the power spectrum k_{peak} is on the filament scale, indicating that most deformation energy arises due to bending of individual filaments. But for all higher values of κ_{eff}, a peak arises at the defect spacing scale, indicating that the defect density controls the texture in this active nematic. (2) Fig. 4C shows the magnitude of the power spectrum at the characteristic scale k_{peak}. The splay energy density is nearly independent of the effective filament stiffness, while the amount of bend decreases linearly with κ_{eff} for κ_{eff} ≲ κ^{max}_{eff}, where κ^{max}_{eff} ≈ 30 is the point at which the effective persistence length of the filaments is equal to their contour length. This observation demonstrates that energy injected into the system at the microscale due to activity preferentially dissipates into bend modes.

(6) |

Fig. 5B shows that the mode of b_{1} calculated from defects within our simulations depends only on κ_{eff}, thus demonstrating that defect shape is controlled by the apparent bending rigidity. Moreover, b_{1} increases logarithmically with κ_{eff}; i.e., defects become more pointed as the bending rigidity increases, consistent with the previous observations of highly pointed defects in rigid rod simulations.^{2,35,52}

To determine the dependence on κ_{eff}, we fit the data for each simulation in the range N ≤ 10^{4} to the form aN^{g}, so that g = 0 indicates equilibrium-like fluctuations and g = 0.5 would indicate linear scaling of fluctuations with system size. As shown in Fig. 7, g increases with κ_{eff}, with g = 0 for small κ_{eff} and g ≈ 0.3 for the largest effective bending rigidity values investigated; i.e. ΔN ∼ N^{0.8}. The fact that g < 0.5 for the parameters we consider may reflect suppression of fluctuations even for N < 10^{4}. Importantly, estimated values of g at different f^{a} and κ collapse onto a single function of κ_{eff}, consistent with the observations of the other characteristics of an active nematic shown above.

Returning to the defect shape measurements, the dashed line in Fig. 5B shows the defect shape parameter values b_{1} calculated by minimizing the continuum Frank free energy for an equilibrium system containing a separated pair of and defects. Here we have fixed the continuum modulus k_{11} and varied k_{33} so that matches the value obtained from our microscopic simulations at the corresponding value of effective stiffness: R_{eq}(k_{33}) = R(κ_{eff}). Although the equilibrium b_{1} values are shifted above the simulation results, consistent with the discrepancy in R between the active and passive systems described above, we observe the same scaling with bending rigidity in the continuum and simulation results. Thus, the defect shape measurement provides a straightforward means to estimate effective moduli in simulations or experiments on active nematics.

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## Footnote |

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c8sm02202j |

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