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ARTICLE Received 8 Jun 2016 | Accepted 30 Sep 2016 | Published 21 Nov 2016 DOI: 10.1038/ncomms13483 OPEN 1 1 1 1 Rui Zhang , Ye Zhou , Mohammad Rahimi & Juan J. de Pablo When a thin ﬁlm of active, nematic microtubules and kinesin motor clusters is conﬁned on the surface of a vesicle, four þ 1/2 topological defects oscillate in a periodic manner between tetrahedral and planar arrangements. Here a theoretical description of nematics, coupled to the relevant hydrodynamic equations, is presented here to explain the dynamics of active nematic shells. In extensile microtubule systems, the defects repel each other due to elasticity, and their collective motion leads to closed trajectories along the edges of a cube. That motion is accompanied by oscillations of their velocities, and the emergence and annihilation of vortices. When the activity increases, the system enters a chaotic regime. In contrast, for contractile systems, which are representative of some bacterial suspensions, a hitherto unknown static structure is predicted, where pairs of defects attract each other and ﬂows arise spontaneously. Institute for Molecular Engineering, University of Chicago, Chicago, Illinois 60637, USA. Correspondence and requests for materials should be addressed to J.J.d.P. (email: depablo@uchicago.edu). NATURE COMMUNICATIONS | 7:13483 | DOI: 10.1038/ncomms13483 | www.nature.com/naturecommunications 1 ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms13483 ctive systems consume and transform energy into local deformation modes. For extensile systems at relatively low 1,2 mechanical work at microscopic length scales . activity, defects move in closed trajectories. As activity ASuch systems arise in living cells, as is the case of the increases, the system becomes unstable and enters a previously cytoskeleton , or can be realized in vitro, for example, in mixtures unreported chaotic regime that is characterized by open 4–7 of bioﬁlaments and their associated motor proteins . At longer trajectories. In contrast, contractile systems, which are 8,9 length scales, microswimmers and even ﬂocks of birds and representative of some bacteria and for which experimental schools of ﬁsh have been examined within the context of active data are not yet available, yield a static defect structure at low and 10,11 systems . Active materials can also be prepared from intermediate activities. The defects are attracted to each other in 12,13 inorganic components, by relying on active colloids or pairs, and the corresponding ﬂow patterns produce intriguing 14,15 vibrating granular systems . Of particular relevance to this stagnation points that could potentially be used for applications. work are recent experiments that have sought to elucidate the interplay between activity and geometric conﬁnement. Examples Results include studies of cytoplasmic streaming , hydrodynamic Model system. The active nematic shells considered here 16,17 4,18 instabilities , pattern formation and counter rotating are conﬁned between two concentric spherical surfaces that boundary layers . Insightful qualitative arguments have been exhibit strong degenerate planar anchoring. A no-slip boundary advanced to unravel the physics that underlie the above condition is enforced on both surfaces. Note that the nematic 20,21 phenomena . coherence length x , which is usually comparable to the Many active systems consist of elongated molecules or constituent’s size, ranges from 1 (in microtubules) to 10mm assemblies; the dynamics of such systems are much more 9,31 (in bacteria) . Here we choose x ¼ 2mm to describe the complex and potentially useful. To interpret their behaviour, behaviour of a typical microtubule system. The shell is centred at one can rely on theoretical descriptions of nematic liquid the origin O, and has inner radius R ¼ 14 and outer radius in crystals , which have been particularly helpful in elucidating a R ¼ 18 in lattice units, which can be mapped onto a vesicle out 23,24 number of phenomena associated with active suspensions . of radius 32mm, comparable to that used in experiments. Despite the above successes, however, the ability to control The timescale of the model is determined by choosing ordered dynamics in active nematics has been limited and the characteristic length scale and the viscosity. By setting the 25,26 remains in its infancy . rotational viscosity of the nematic material to g ¼ 0.1 Pa s, the Conﬁnement has a profound effect on the collective dynamics unit of time becomes t ¼ 4 ms. The choice of elastic constant of active systems. On the one hand, conﬁnement dictates the K ¼ 10 pN is consistent with that adopted in numerous past structure of a nematic phase via topology and anchoring. For the 22,31 studies of liquid crystalline systems . The equilibrium scalar particular case of a static nematic ‘shell’ conﬁned by two order parameter of the bulk, static nematic material is set to concentric spherical surfaces (with degenerate planar anchoring), q ’ 0:62, and all our calculations are restricted to the ﬂow the ground-state conﬁguration has four þ 1/2 disclination lines aligning regime by choosing a material constant x ¼ 0.8, to reﬂect 27–29 arranged into a tetrahedral conﬁguration . On the other the fact that the aspect ratio of the typical biopolymer ﬁlaments in hand, conﬁnement can shape and stabilize spontaneous ﬂows . experiments is deep in the prolate regime. For instance, the When taken together, these two factors lead to non-trivial in vitro microtubules used in refs 6,25 have a length-to-width phenomena. Simulations under appropriate boundary conditions, ratio of ’ 60. An initial static system is prepared via a Ginsburg– for example, suggest that active nematics in a capillary can Landau relaxation . A baseball-like director ﬁeld is formed as the develop bidirectional ﬂows or helical vortices . And recent four defects, with topological charge þ 1/2, repel each other and calculations show that an active nematic drop on a surface can 27–29 adopt a tetrahedral arrangement . To quantify the relative exhibit self-propulsion along well-deﬁned directions .In positions of the defects, an angular distance a is introduced, ij experiments, droplets of bacterial suspensions squeezed between given by the angle between radii Oi and Oj, where i and j refer two plates exhibit spontaneous circulation only for certain aspect to the ith and jth defects, respectively. The average angular ratios . And a microtubule emulsion droplet under similar distance hai¼ a represents the mean of the angles of the six ij conﬁnement may become motile . defect pairs. At equilibrium, for a tetrahedral arrangement When microtubules and kinesin are encapsulated within a a a 109:47 , and therefore hai¼ 109:47 . ij 0 shell, a new type of dynamic ordering emerges in which four The activity of the material is controlled by parameter z. þ 1/2 topological defects move in a well-deﬁned pattern . That Hybrid lattice Boltzmann simulations are evolved for a duration ordering is particularly relevant for fundamental studies of active 6 4 of t ¼ 4 10 t, which corresponds toB1.6 10 s. We ﬁrst focus systems in that it provides a perfectly well-bounded, periodic on an extensile system with z40. In Fig. 1, our simulation results system in which to interpret emerging views of active materials. are contrasted with the experimental images reported in ref. 25. The original experiments of Keber et al. were analysed within the Four different conﬁgurations are compared. Two snapshots are framework of four coupled points (the defects) constrained to close to the ground state at which the four defects adopt a move on the surface of a sphere. That approach did not consider tetrahedral conﬁguration (Fig. 1a,b,e,f). The other two snapshots the molecular structure of the material explicitly, and in this work are close to the excited state (Fig. 1c,d,g,h). As can be appreciated we rely on a continuum model of active nematics to describe in the ﬁgure, our simulated images capture quantitatively the and understand such a system. More speciﬁcally, we couple a defect structure observed in experiments. Landau–de Gennes representation of a nematic liquid crystal to a hydrodynamic framework that accounts for activity to probe the effect of spherical-shell conﬁnement on active nematics. The Low-activity behaviour. The system remains passive until resulting patterns predicted by the model compare favourably the activity reaches a value of zZ0.0007, at which point a with those observed experimentally , serving to validate the spontaneous ﬂow is generated. Below the onset activity, the defect underlying theoretical treatment. The model is then used to conﬁguration is deformed but remains static, as the elasticity elucidate the spatiotemporal details of the velocity and director balances the activity. Figure 2 shows a time sequence of ﬁelds, as well as the system’s free energy. Recent experimental representative images, separated by B80 s, which reveal the observations are interpreted in terms of distinct contributions position of the defects, along with the corresponding streamlines, to the free energy arising from enthalpy, ‘bend’ and ‘splay’ for z ¼ 0.001. The highest velocities, which reach values as high as 2 NATURE COMMUNICATIONS | 7:13483 | DOI: 10.1038/ncomms13483 | www.nature.com/naturecommunications NATURE COMMUNICATIONS | DOI: 10.1038/ncomms13483 ARTICLE 0.15mms , are always associated with the defects, implying that symmetry axis, which is consistent with experimental the spatial gradient of the nematic order parameter induces the observations . It is convenient to deﬁne this direction as the ﬂows. The mean ﬂow direction at the þ 1/2 defect is along its orientation of the defect. As can be appreciated in the ﬁgure, the Experiment Simulation Experiment Simulation a bc d e f h Figure 1 | Representative conﬁgurations of active nematic shells at four different times. They are labelled by a,c,e,g (experimental images from ref. 25, reprinted with permission from AAAS) and b,d,f,h (simulated structures). The two images within each labelled panel are the projections of opposite hemispheres. In all images, the coloured dots indicate the defect positions. In simulations, the blue lines correspond to the director ﬁeld. The inset panels illustrate the defect conﬁguration. a bc g de f h Figure 2 | Vortex formation during evolution. Time sequence (a–f) illustrate the velocity ﬁeld of an active nematic shell for z ¼ 0.001. Defects are shown by yellow cylinders, velocity ﬁelds are shown by arrows and streamlines are shown by the white curved lines. The conﬁgurations shown here 6 6 6 6 6 6 correspond to t/t ¼ 1.956 10 (a), 2 10 (b), 2.192 10 (c), 2.246 10 (d), 2.43 10 (e) and 2.482 10 (f). The positions of two of the defects are encircled. (g,h) The corresponding optical images and the director ﬁelds of c,f, respectively. NATURE COMMUNICATIONS | 7:13483 | DOI: 10.1038/ncomms13483 | www.nature.com/naturecommunications 3 ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms13483 system develops vortices that are separated by defects. resultant average angular distance becomes hai 120 . In this Occasionally (in Fig. 2a,c,e), four equally sized parallel eddies mode, the defect–defect distance is on average smaller than that appear, with two extensional ﬂows located at the two poles. in the tetrahedral conﬁguration. The system’s free energy Neighbouring vortices counter-rotate and, as shown later, this reaches a maximum, as shown in Fig. 3d, indicating that this feature corresponds to an excited state. arrangement corresponds to an excited state. For z ¼ 0.004, Defect dynamics are controlled by two competing effects: one the system oscillates between these two modes with period is the repulsive ‘force’ between the four, positively charged þ 1/2 T ¼ 3:1410 t ¼ 125 s. This estimate is consistent with the defects, which has its origins in the elasticity of the material and experimental value reported in ref. 25, and it is established by the drives the system towards its ground state. The other is the system size and the average ﬂow velocity. When the system is in activity. The singularity of the director ﬁeld at the defect serves to the tetrahedral mode, the free energy is close to that of the static excite the system above its ground state. When one takes a system. There are, however, two differences between the ground derivative of equation (11), at rest, the last term of that equation state of the extensile active system and the static state of a does not vanish, leading to the spontaneous emergence of ﬂow. nematic. One is that the ground state of the extensile active It is found that for the range of activity between system cannot be a short-arc state (if one draws a geodesic line 0:0007 z 0:005, the four defects move in closed trajectories connecting the two defects following the director ﬁeld, the curve along the edges of a deformed cube. One can observe that each is a short arc ). As the defects try to move along their pair of defects is symmetric about the symmetry axis of the cube, orientations (symmetry axis) they necessarily form a long-arc and therefore a ¼ a , a ¼ a and a ¼ a (Supplementary state (if one draws a geodesic line connecting the two defects 12 34 13 24 14 23 Note 1), implying that only three independent angular distances following the director ﬁeld, the curve is a long arc). As shown are necessary to describe the conﬁguration of the defects. For later, for a contractile system the conﬁguration always consists of conciseness, we use a , a and a to denote these and thus a short-arc state. The other difference is that the orientations of 1 2 3 hai¼ða þ a þ a Þ=3. the defect pairs are not directly ‘against’ each other. As the system 1 2 3 The results in Fig. 3, for z ¼ 0.001, serve to explain the system’s passes the tetrahedral state, two approaching defects deﬂect of entire motion mechanism. As the four defects move collectively, each other at an angle due to elastic repulsions. As shown in their conﬁgurations oscillate between a tetrahedral mode and a Fig. 2g, this deﬂection requires that the orientations of the defects planar mode. When they move to the four corners of the do not point towards each other on the sphere. deformed cube, they form a tetrahedron. The defect–defect To elucidate the precise origin of the oscillatory nature of the distance is at its maximum, and a ¼ a ¼ a ¼hai free energy, in Fig. 3e we plot the different contributions to 1 2 3 a ¼ 109:47 . As shown in Fig. 3d, the system is in its free- the free energy of the system. Note that for our quasi-two- energy ground state. When the defects move to the midpoints of dimensional geometry, the twist is 0 and the saddle-splay is time the four parallel sides of the cube, they form a square within a invariant. We therefore only show splay, bend and defect (phase) single plane. At that point a ¼ 180 and a ¼ a ¼ 90, and the energies. The defect (phase) energy is the Landau–de Gennes 1 2 3 112.4° A B C D Splay Bend 2 Defect –2 112.4° 108 110 112 114 116 118 120 0 0.5 1 1.5 2 <> (°) t (10 ) Figure 3 | Defect conﬁgurations and trajectories for low activity. (a) Defect positions for (ﬁlled circles) z ¼ 0.001 at three consecutive times, marked by A, B and C. The defects’ orientations are illustrated by arrows, and the dashed lines connect the defects and the origin. (b) Correspondence between the free energy F and hai.(c) Time evolution of the mean angular distance; (d) free energy of the system; (e) splay, bend and defect energies, referenced by initial value; (f) mean velocity of the four defects. The static system’s free energy F is used as a reference. The three vertical lines mark the times of the three snapshots. The fourth vertical line d marks a secondary maximum in the velocity plot of f. 4 NATURE COMMUNICATIONS | 7:13483 | DOI: 10.1038/ncomms13483 | www.nature.com/naturecommunications F–F (10 k T) 0 B < > (°) Energies –4 F-F (10 k T) 0 B v (10 /) N NATURE COMMUNICATIONS | DOI: 10.1038/ncomms13483 ARTICLE short-range free-energy term f deﬁned in equation (5). When the time points B and C in Fig. 3), the defect velocity reaches a system is in its ground state, the splay and bend energies are maximum, and the temporal curves of both hai and E show el relatively low. In contrast, when the system is in its excited state, sharp declines. If t , t and t are used to denote the times A B C the splay energy reaches its peak value, and the bend energy is corresponding to the three time points shown in Fig. 3, the relatively high. There is a phase shift between the splay and bend percent of time spent by the system climbing the free-energy contributions. We also note that the defect energy exhibits a barrier b ¼ðt t Þ=ðt t Þ can be used to characterize the B A C A sharp decline right after the system passes the energy barrier, asymmetry of the curves; for z ¼ 0.001, we have b ¼ 0.78. The as the release of defect core energy drives defect motion towards asymmetry in our dynamics could potentially be useful for the ground state. Interestingly, when the velocity reaches a engineering purposes; for instance, during self propulsion, as in secondary peak (shown in Fig. 3f), the phase energy goes to a the case of an active nematic vesicle, the asymmetric oscillation of minimum but the splay contributions reach a secondary peak. the defects could be used to induce directional motion. It is also of interest to examine the evolution of the spatial A secondary peak in the velocity plot can also be appreciated, distribution of different contributions to the free energy. The labelled as time point D in Fig. 3. If one plots the system’s free relevant distributions are shown and discussed in Supplementary energy F and its associated hai at different times on a single Note 2 (Supplementary Fig. 1), where one can appreciate the ﬁgure, those data collapse onto a single curve that describes the formation of a low-splay band between pairs of defects. free-energy landscape in terms of hai (Fig. 3b). The slope of such It is worth pointing out here that the effect of biopolymer a curve shows a secondary minimum at a ¼ 112.4, which is the ﬂexibility can be included implicitly in our model through a exact angle when the system reaches a secondary defect velocity reduced bend elastic constant. When the activity is sufﬁciently maximum. strong, however, the ﬁlaments may buckle and lose their rod-like Our simulation results at low activity agree with the shape. In such cases, a more elaborate treatment of ﬁlament phenomenological model proposed in ref. 25 in the following ﬂexibility is required. We leave this issue and the possible effects respects: (1) both models predict a ratchet-like shape when of disparate elastic constants for subsequent studies. represented in a a-plot. As explained above, that shape is The temporal evolution curves of hai and the system’s free manifestation of the interplay between activity and elasticity. energy F are ratchet like. The defect velocity is plotted in Fig. 3f. (2) Both models exhibit a threshold/onset activity, below which When the system moves from the ground (tetrahedral) state the system cannot overcome the elasticity to enter the oscillatory towards the excited, planar state (between time points A and B in dynamic state. However, the defect trajectories predicted by the Fig. 3), the dynamics becomes slow as the system gains potential two models are different. In the treatment presented in ref. 25, energy. However, when the system is moving from an the defects form pairs, and the paired defects revolve around energetically unfavourable state towards a ground state (between the pair’s centre of mass . In our simulations and in the a c –2 –4 –6 –8 012340123 40 1 2 3 4 –4 –1 –4 –1 –4 –1 1/T (10 )1/T (10 )1/T (10 ) d f 0.5 –0.5 –1 –1 –0.5 00.5 –1 –0.5 0 0.5 –1 –0.5 0 0.5 1 x /L x /L x /L x x x n n n z z z gh i n n n y n y n x x x Figure 4 | Spectral analysis of defect conﬁgurations. System evolution from periodic (a,d,g, with z ¼ 0.0042) to quasiperiodic (b,e,h, with z ¼ 0.0052) and chaotic (c,f,i, with z ¼ 0.01). (a–c) Power spectrum of the time series of hai.(d–f) Projection of one defect trajectory onto the xy plane. (g–i) The four defect trajectories in three dimensions. Trajectories in g,h are made transparent and that in i are not to assist eyes. NATURE COMMUNICATIONS | 7:13483 | DOI: 10.1038/ncomms13483 | www.nature.com/naturecommunications 5 y/L y I A ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms13483 experiments, defect trajectories exhibit a more complicated 0:006 z 0:01, one can still measure a ‘period’ T by deﬁning behaviour: the defects do not form pairs, and their trajectories it as the average time spent between two consecutive planar are not simple circles. Instead, the defects can occasionally turn modes (illustrated in Supplementary Fig. 2d,e). For that measure, by B90 during motion. 1/T decreases slightly but is still linear with respect to z. The inset The vesicle in the experiments can be made more ﬂexible by of Fig. 5c shows that the underlying characteristic ﬂow velocity is changing the hypertonic stress. Experiments show that the þ 1/2 approximately proportional to z, which is consistent with 25 6,24 defects on a ﬂexible vesicle can grow protrusions . We think this literature reports . The transition of the period plot from one phenomenon arises from the interplay between the dynamics of linear regime to another is due to the transition of the system the þ 1/2 defect, the curvature of the vesicle and the excess from periodic to chaotic dynamics. surface area provided by the hypertonic stress. As the comet-like þ 1/2 defect moves, the microtubules at the tail of the defect Contractile system. We conclude with a discussion of a con- move along the tangential plane of the vesicle, but the curvature tractile system, for which zo0. When z 0:05, the system forces the motion to bend, and follow the sphere’s surface. This comes to rest (in terms of its defect structure) after a few yields an outward stress that is able to protrude the vesicle. oscillations. The steady state corresponds to four defects forming two pairs; within each pair, the defects are attracted to each other. Intermediate- and high-activity behaviour. We now consider As shown in Fig. 6b, the two pairs appear at the two poles how the system evolves under high activity. In Supplementary (one pair is not shown for clarity). They are oriented back to Figs 2 and 3, we show the defect trajectories for intermediate back, forming a short arc. Although the director ﬁeld and defect activities. As explained below, a power spectrum analysis reveals a conﬁguration are stationary, a velocity ﬁeld still exists (Fig. 6a). transition from periodic to quasi-periodic and to chaotic It is primarily localized near the defect pairs. The mean ﬂow dynamics as z increases . When z is gradually raised to 0.005, the direction at the defect is opposite to its orientation, in contrast to cube deforms and the asymmetry in the hai plot fades away. extensile systems, where the mean ﬂow direction is along the The above trend is discussed in Supplementary Note 3. When defect orientation. Extensional ﬂows emerge between the defect z40.005, the hai plot no longer consists of a single periodic pair, a feature that could be useful for design of microﬂuidic oscillation (Supplementary Fig. 3 for details). When zZ0.006, applications. In this case, the activity drives the system to a the defect trajectories become open (Supplementary Fig. 4). state from which it cannot escape. The activity tries to attract two Figure 4a–c shows the power spectrum of the time series of hai þ 1/2 defects, but this is prevented by elasticity. As shown in for low (z ¼ 0.0042), medium (z ¼ 0.0052) and high activity Fig. 6d, as activity increases in a contractile state, the angular (z ¼ 0.01). At z ¼ 0.0042, the defect trajectories in the distance between attracted defects at steady state becomes two-dimensional and three-dimensional plots (Fig. 4d,g) are smaller, and the system’s free energy increases. For even more closed, and the power spectrum shows sharp peaks corresponding negative values of z, the four-defect structure becomes unstable. to their oscillation period and its harmonics. When z ¼ 0.0052, Occasionally, it fuses into two þ 1 defects located at the poles, the defect trajectories are still closed but exhibit a more intricate forming a bipolar structure. But this state is fragile, the system geometry (Fig. 4e,h). The corresponding power spectrum exhibits evolves into a multi-defect conﬁguration, and momentum is no equally spaced frequencies, but with signiﬁcant noise in between. longer conserved. We are not aware of contractile biopolymer For z ¼ 0.01, the defect trajectories are chaotic outside a depletion systems, despite the fact that some materials are contractile-like . region (Fig. 4f,i, see Supplementary Note 4 for discussion). However, non-nematic micro-swimmer systems, such as algae, The power spectrum shows no evidence of periodicity, and in can be contractile. We therefore propose that future experiments Supplementary Fig. 4 we further show that the chaotic system is on bacterial swimmers in nematic shells could be used to assess ergodic. the merits of our predictions and, if correct, could be used as the Importantly, the oscillation period T is highly sensitive to basis for creation of self-propelled microﬂuidic devices capable of activity z. As shown in Fig. 5c, 1/T is linear in z for producing both controlled shear and elongational forces. 0:0007 z 0:005, the closed-trajectory regime, where T varies We conclude by noting that, to address any possible size 6 4 from 2.16 10 t to 2.5 10 t. This dependence could in fact effects, we have performed simulations for system with R ¼ 34 in be used to either measure the macroscopic quantity z or to and R ¼ 38. The results for large systems are similar to those out quantify the concentration of ATP in experiments. When the for their smaller counterparts, but, as expected, the transition system transitions to the less-ordered state, in the range value of the activity z is different. 1 0 a 10 b 10 c 1 –3 ×10 0.8 10 0.6 –1 0.4 Open trajactories 0 0.005 0.01 –1 10 0.2 Closed trajactories –2 10 0 –2 –1 0 24 6 8 10 10 10 10 01 2 3 4 –3 –3 <> – (°) (10 ) (10 ) min 0 Figure 5 | Activity dependence plots. (a) Log–log scatter plot of the minimum of hai a and the minimum of F F for 0:0007 z 0:004; (b) Shape 0 0 asymmetry b ¼ðt t Þ=ðt t Þ of the angular distance curve on activity z; t , t and t are the times of three consecutive tetrahedral, planar and B A C A A B C tetrahedral modes, respectively, as illustrated in Fig. 3; (c) Oscillation period T as a function of z. The inset shows the maximum ﬂow velocity (x =t) at the defect versus z. 6 NATURE COMMUNICATIONS | 7:13483 | DOI: 10.1038/ncomms13483 | www.nature.com/naturecommunications (F –F ) (100 KT) 0 min – 0.5 –4 –1 1/T(10 ) NATURE COMMUNICATIONS | DOI: 10.1038/ncomms13483 ARTICLE ab c d 50 –2 –4 –10 –10 –0.06 –0.04 –0.02 0 Figure 6 | Contractile system structure and dynamics. Streamlines (a), defects (cylinders) and director ﬁeld (b) of contractile system for z¼ 0.007 at steady state. In b, the curve connecting the two defects along the director ﬁeld is a short arc. The angular distance between these defects is a minimum. Activity dependence of the minimum angular distance (a ) (c) and free energy F F at steady state (d). ij min 0 Discussion cube. In contrast, for contractile systems, which are representative Recent experiments with microtubule ﬁlaments and kinesin of bacteria or algae, the mean ﬂow direction at a defect is opposite motor clusters encapsulated in shells have revealed the existence to its orientation. Such ﬂows correspond to an intriguing static of intriguing structures and closed-loop trajectories . In this state of the director ﬁeld, where defects are attracted to each other work, a theory for active nematics, coupled to the relevant in pairs, forming a short-arc state. momentum conservation equations, has been used to explain their temporal evolution. In agreement with experiment, the Methods theory predicts the formation of a four-defect structure that Governing equations. A hybrid lattice-Boltzmann method is used to simultaneously solve a modiﬁed Beris–Edwards equation and a momentum oscillates periodically between a tetrahedral conﬁguration (ground equation, which account for the activity of the nematic material. The nematic state) and a planar conﬁguration (excited state), thereby lending phase is described by a tensorial order parameter Q ¼ nn I , where support to the proposed model. It is then shown that the unit vector n describes the director orientation, I the identity tensor and underlying physics leading to the observed periodic oscillations of hi denotes an ensemble average. By introducing a velocity gradient W ¼ @ u , ij j i T T A ¼ W þ W 2, O ¼ W W 2, and a generalized advection term structure is the competition between elasticity and activity. More speciﬁcally, a complex interplay between splay, bend and defect SWðÞ ; Q¼ ðxA þOÞðQ þ I=3ÞþðQ þ I=3ÞðxA OÞ ð1Þ energies is identiﬁed in which, in the ground state, the splay and 2xðQ þ I=3ÞTrðQWÞ: 35,36 bend energies have a near-minimal value but the defect adopts one can write a modiﬁed Beris–Edwards equations according to an intermediate size. In contrast, in the excited state, splay ð@ þ u rÞQ SðW; QÞ¼ GH þ lQ: ð2Þ deformations, as well as the size of the defect cores, reach a The constant x is related to the material’s aspect ratio, and G is related to the maximum. The corresponding velocity of the defects is also near its rotational viscosity g of the system by G ¼ 2q =g (ref. 37), where q is the scalar 1 0 0 1 maximum value at that point. order parameter of the nematic phase. In equation (2), l represents the ﬁrst activity For low to intermediate activity, the four-defect structure is parameter, which is equivalent to varying the static nematic order parameter . stable. In extensile systems, a moving defect induces a ﬂow, and The molecular ﬁeld H, which drives the system towards thermodynamic equilibrium, is given by the mean ﬂow direction follows the defect’s orientation. For low st activity, defect trajectories form a closed loop that can be mapped dF H ¼ ; ð3Þ onto the edges of a deformed cube. This fact can be used to dQ rationalize the oscillating behaviour of the defect conﬁguration. st where½ ::: is a symmetric and traceless operator, and F is the total free energy of As the activity increases, the closed trajectories deform and the the system, deﬁned by oscillation frequency increases linearly. This observation suggests Z Z F ¼ dVf þ dSf : ð4Þ that one may in fact use such a dependence to measure the bulk surf macroscopic activity. By increasing the activity even further, the bulk surface defect trajectories open up and they enter an ergodic state outside The terms f and f represent the bulk and surface contributions to the free bulk surf the depletion region, which is deﬁned by the symmetry axes of the energy, respectively. Here f ¼ f þ f , where f is the short-range or ‘phase’ bulk p e p NATURE COMMUNICATIONS | 7:13483 | DOI: 10.1038/ncomms13483 | www.nature.com/naturecommunications 7 (α ) (°) ij min F –F (10 k T) 0 B ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms13483 energy and f is the long-range elastic energy. The phase energy f is given by a we calculate the elastic energies on bulk points with order parameter q40.45 e p Landau–de Gennes expression of the form refs 38,39 (q ¼ q ’ 0:62 for undistorted nematics and q ’ 0:2 at defect cores). A U A U A U 0 0 0 2 2 3 2 Data availability. Data and analysis codes are available from the authors upon f ¼ 1 Tr Q Tr Q þ Tr Q : ð5Þ 2 3 3 4 request. 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We solve the evolution equations, equations (2) and (9), using a ﬁnite- Lett. 110, 268102 (2013). difference method. The momentum equation, equation (10), is solved 20. Marchetti, M. C. et al. Hydrodynamics of soft active matter. Rev. Mod. Phys. 85, simultaneously via a lattice Boltzmann method over a D3Q15 grid . The 1143 (2013). implementation of stress follows the approach proposed by Guo et al. . Our model 21. Khoromskaia, D. & Alexander, G. P. Motility of active ﬂuid drops on surfaces. and implementation were validated by comparing our simulation results to Phys. Rev. E 92, 062311 (2015). 22,47–49 predictions using the Ericksen-Leslie-Parodi (ELP) theory in the absence of 22. Kleman, M. & Lavrentovich, O. D. Soft Matter Physics (Springer, 2001). activity. We refer the reader to ref. 41 for additional details on the numerical 23. DeCamp, S. J., Redner, G. S., Baskaran, A., Hagan, M. F. & Dogic, Z. methods used here. Orientational order of motile defects in active nematics. Nat. 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Nano Lett. 2, 1125 through (2002). 29. Lopez-Leon, T., Koning, V., Devaiah, K. B. S., Vitelli, V. & Fernandez-Nieves, A. 1 1 L ¼ K þ ðK K Þ ; 1 2 22 33 11 2q 3 0 Frustrate nematic order in spherical geometries. Nat. Phys. 7, 391 (2011). L ¼ ðK K Þ; 2 2 11 24 30. Ravnik, M. & Yeomans, J. M. Conﬁned active nematic ﬂow in cylindrical ð14Þ L ¼ ðK K Þ; capillaries. Phys. Rev. Lett. 110, 026001 (2013). 3 3 33 11 2q 1 31. Wolgemuth, C. W. Collective swimming and the dynamics of bacterial L ¼ ðK K Þ: 4 24 22 turbulence. Biophys. J. 95, 1564 (2008). 32. Ravnik, M. & Zumer, S. Landau-de gennes modelling of nematic liquid crystal By adopting a one-elastic-constant approximation, K ¼ K ¼ K ¼ K K, 11 22 33 24 one has L ¼ L K=2q and L ¼ L ¼ L ¼ 0. Point wise, n is the eigenvector colloids. Liq. Cryst. 36, 1201–1214 (2009). 1 2 2 2 associated with the greatest eigenvalue of the Q-tensor at each lattice point. The 33. Sec,D. et al. Defect trajectories in nematic shells: role of elastic anisotropy and derivatives of n are obtained via a ﬁnite-difference method. To avoid singularities, thickness heterogeneity. Phys. Rev. E 86, 020705 (R) (2012). 8 NATURE COMMUNICATIONS | 7:13483 | DOI: 10.1038/ncomms13483 | www.nature.com/naturecommunications NATURE COMMUNICATIONS | DOI: 10.1038/ncomms13483 ARTICLE materials was supported by the University of Chicago Materials Research Science and 34. Strogatz, S. H. Nonliner Dynamics and Chaos: With Applications to Physics, Engineering Center (NSF DMR-1420709). We are grateful for the support of the Biology, Chemistry, and Engineering. 1st edn (Westview, 1994). University of Chicago Research Computing Center for assistance with the calculations 35. Beris, A. N. & Edwards, B. J. Thermodynamics of Flowing Systems with Internal carried out in this work. We thank Prof Julia Yeomans, Prof M. Cristina Marchetti, Microstructure (Oxford Univ. Press, 1994). Prof Miha Ravnik, Prof Daniel J. Needleman, Prof Margaret Gardel, Dr Stephen J. 36. Marenduzzo, D., Orlandini, E., Cates, M. E. & Yeomans, J. M. Steady-state DeCamp, Dr Arnout Boelens, Dr Abelardo Ramirez-Hernandez and Zhihong You for hydrodynmaic instabilities of active liquid crystals: hybrid lattice Boltzmann helpful discussions. simulations. Phys. Rev. E 76, 031921 (2007). 37. Denniston, C., Orlandini, E. & Yeomans, J. M. Lattice Boltzmann simulations of liquid crystal hydrodynamics. Phys. Rev. E 63, 056702 (2001). Author contributions 38. de Gennes, P. & Prost, J. The Physics of Liquid Crystals (Oxford Univ. Press, R.Z. and J.J.d.P designed the research; R.Z. and J.J.d.P. performed the research; R.Z., 1995). Y.Z., M.R. and J.J.d.P. analysed the data; R.Z. and J.J.d.P. wrote the paper; J.J.d.P. 39. Landau, L. & Lifshitz, E. Statistical Physics 3rd edn (Pergamon, 1980). supervised the research. All authors discussed the progress of research and reviewed the 40. Fournier, J. & Galatola, P. Modeling planar degenerate wetting and anchoring manuscript. in nematic liquid crystals. Europhys. Lett. 72, 403 (2005). 41. Zhang, R., Roberts, T., Aranson, I. & de Pablo, J. J. Lattice Boltzmann simulation of asymmetric ow in nematic liquid crystals with ﬁnite anchoring. Additional information J. Chem. Phys. 14, 084905 (2016). Supplementary Information accompanies this paper at http://www.nature.com/ 42. Batista, V. M. O., Blow, M. L. & Telo da Gama, M. M. The effect of anchoring naturecommunications on the nematic ﬂow in channels. Soft Matter 11, 4674–4685 (2015). 43. ichi Fukuda, J., Yokoyama, H., Yoneya, M. & Stark, H. Interaction between Competing ﬁnancial interests: The authors declare no competing ﬁnancial interests. particles in a nematic liquid crystal: numerical study using the Landau-de Reprints and permission information is available online at http://npg.nature.com/ Gennes continuum theory. Mol. Cryst. Liq. Cryst. 435, 63–74 (2005). reprintsandpermissions/ 44. Voituriez, R., Joanny, J. F. & Prost, J. Spontaneous ﬂow transition in active polar gels. Europhys. Lett. 70, 404 (2005). How to cite this article: Zhang, R. et al. Dynamic structure of active nematic shells. 45. Guo, Z. & Shu, C. Lattice Boltzmann Method and Its Applications in Nat. Commun. 7, 13483 doi: 10.1038/ncomms13483 (2016). Engineering 1st edn (World Scientiﬁc, 2013). 46. Guo, Z., Zheng, C. & Shi, B. Discrete lattice effects on the forcing term in the Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in lattice Boltzmann method. Phys. Rev. E 65, 046308 (2002). published maps and institutional afﬁliations. 47. Ericksen, J. L. Continuum theory of liquid crystals of nematic type. Mol. Cryst. Liq. Cryst. 7, 153 (1969). This work is licensed under a Creative Commons Attribution 4.0 48. Leslie, F. M. Some constitutive equations for anisotropic ﬂuids. Q. J. Mech. International License. The images or other third party material in this Appl. Math. 19, 357 (1966). article are included in the article’s Creative Commons license, unless indicated otherwise 49. Parodi, O. Stress tensor for a nematic liquid crystal. J. Phys. France 31, 581–584 in the credit line; if the material is not included under the Creative Commons license, (1970). users will need to obtain permission from the license holder to reproduce the material. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/ Acknowledgements The development of the non-equilibrium lattice Boltzmann method presented here for active nematics was supported by NSF DMR-1410674. The analysis of active nematic r The Author(s) 2016 NATURE COMMUNICATIONS | 7:13483 | DOI: 10.1038/ncomms13483 | www.nature.com/naturecommunications 9
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Published: Nov 21, 2016
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