Spontaneous rotation of active droplets in two and three dimensions

We use numerical simulations and linear stability analysis to study active nematic droplets, in the regime where the passive phase is isotropic. We show that activity leads to the emergence of nematic order and of spontaneous rotation in both two and three dimensions. In 2D the rotation is caused by the formation of a chiral $+1$ defect at the center of the drop. With increasing activity the droplet deforms to an ellipse, and then to a rotating annulus. Growing droplets form extended active arms which loop around to produce holes. In 3D the rotation is due to a disclination which loops away from and back to the surface, defining the rotation axis. In the bulk the disclination loop ends at a skyrmion. Active extensile flows deform the droplet to an oblate ellipsoid, contractile flows elongate it along the rotation axis. We compare our results on rotation in two-dimensional droplets with experiments on microtubule and motor protein suspensions and find a critical radius $\sim 700 \mu m$ above which the spontaneous rotation gives way to active turbulence. Comparing the simulation parameters with experiments on epithelial cell colonies shows that the crossover radius for cell colonies could be as large as $2 mm$, in agreement with experiments.


INTRODUCTION
There are many examples where small aggregates of living particles spontaneously rotate.In vivo, persistent angular motion of Xenopus and Drosophila eggs [1] has been shown to be essential for proper embryonic development [2].Multicellular human mammary cell (MCF10A) spheroids embedded in an alginate and Matrigel-based extracellular matrix rotate [3] and there is evidence that the correct formation of spheres of epithelial cells surrounding a hollow lumen relies on rotation [4][5][6].Rotation has been observed in vitro in experiments on confined epithelial colonies [7], active microtubule networks [8], and bacterial systems [9,10] and also in unconfined layers of cells [11][12][13] and bacteria [14].
There is increasing interest in interpreting biological phenomena in terms of the theories of active matter [15][16][17][18][19]. Active matter models in 2D that lead to collective rotation include generalised Potts models [13], simulations of rod-like swimmers [20], agent-based simulations of self-propelling particles with intrinsic angular velocity [21,22], or in the presence of randomly distributed obstacles [23], annular confinements [24,25], and activity gradients [26].Continuum simulations have also been able to predict collective rotation in particular geometries.These include circular confinements with anchoring conditions that impose net topological charges [27], colonies with an imposed motility difference between the center and the rim [12], and polar systems [28].
In 3D, continuum simulations have identified rotation in confined 3D nematic droplets with strong passive anchoring [29], in chiral active nematics * mehrana.raeisiannejad@physics.ox.ac.uk [30], and in active polar systems moving on curved surfaces [31,32].Agent-based simulations of selfpropelling particles moving on a fixed sphere have also reported the formation of collective rotation [33,34].
Continuum theories of active nematics have been particularly successful in describing the properties of microtubule motor mixtures and bacterial and epithelial cell monolayers [15,[35][36][37][38].Here we describe how this approach can predict spontaneous rotation in cell aggregates in both two-and three-dimensions.We work in the limit, relevant to systems of isotropic cells, where there is no nematic ordering in the absence of activity [39], and the boundaries at the edge of the colonies have no imposed confinement or anchoring.
In 2D we find a spontaneous transition to a rotating state above an activity threshold.The rotation is driven by active shear flow due to director misalignments which result from the formation of a +1 or two +1/2 topological defects at the centre of the colony.Increasing activity elongates the droplets and can lead to lumen (hole) formation at the droplet centre.
We also observe rotation in 3D droplets, with a disclination loop forming along the axis of rotation.The resultant active flows extend the droplet along (perpendicular to) the rotation axis for extensile (contractile) active forcing.
We first introduce the equations of motion.We then perform a linear stability analysis to find the onset of collective motion, and explore the phase diagram of 2D active isotropic droplets in terms of surface tension, droplet radius, and activity.We also discuss the role of growth in the emergence of the different dynamical steady states.In the next section of the paper we study the behaviour of active isotropic droplets in 3D in the absence and presence of growth.Finally, we discuss our results and compare them to experiments.

EQUATIONS OF MOTION
To investigate the dynamical behaviour of colonies in two and three dimensions we solve the continuum equations of motion for an active nematic droplet.The relevant hydrodynamic variables are an orientational order parameter Q = Sd (nn − I/d) /(d − 1), which describes the magnitude S and direction of the nematic order n in d dimensions, the concentration of the active material φ, and the velocity u [40].
The dynamics of the nematic tensor is governed by [41] where γ is the rotational diffusivity and S is the co-rotational advection term that accounts for the impact of the strain rate E = (∇u T + ∇u)/2 and vorticity Ω = (∇u T − ∇u)/2 on the director field.
The co-rotational advection has the form where the flow-aligning parameter λ controls the coupling between the orientation field and the flow, determining whether the nematogens align or tumble in a shear flow.The relaxation of the orientational order is described by a free energy F = f Q dV through the molecular field, The nematic free energy density is We choose A, B, C, so that the passive system is in the isotropic phase in equilibrium.The final term encodes the elastic free energy density due to spatial inhomogeneities in the nematic tensor.We assume a single Frank elastic constant K Q [40].
The concentration field φ defines the position of the nematic droplet, with φ = 1 corresponding to the active phase and φ = 0 to the passive phase.It evolves according to [42] where the mobility Γ φ quantifies how fast the concentration field responds to gradients in the chemical potential µ = ∂∇φ ), and the last term is a source term which describes droplet growth.The concentration free energy density is where K φ and A control the surface tension and interface width.The dynamics of the velocity field is governed by the incompressible Navier-Stokes equations, which read: Here, ρ is the density and Π is a generalized stress tensor that has both passive and active contributions.The passive part of the stress includes the viscous stress, Π visc = 2ηE, elastic stress, and capillary stress due to the inhomogeneous concentration field, In the equations for the passive stress, P is the isotropic pressure and η is the viscosity [41].
The active stress drives changes in the flow field caused by continuous energy injection at the microscopic scale.The activity generates flows for nonzero divergence of the nematic tensor, and the active stress takes the form [43] The parameter ζ determines the strength of the activity.Extensile (contractile) activity is represented by The equations of active nematohydrodynamics are solved using a hybrid lattice Boltzmann and finite difference method [44,45], with the discrete space and time steps defining the simulation units.A list of parameters and initial conditions is introduced in SM.For all the simulations, we use the same set of parameters unless otherwise stated.

Linear stability analysis:
As a first step towards understanding the behaviour of the isotropic colony, we perform a linear stability analysis around an inert isotropic phase, Q = 0 + δQ and u = 0 + δu, and study the growthrate of perturbations in δQ and δu.Introducing the Fourier transform for any fluctuating field f as f (r, t) = dω dq f (q, t) e iq•r+ωt , Fourier transforming Eqs.(1,(7)(8) gives where the wave-vector q = q(cos θ, sin θ), B 1 = γ(A + K Q q 2 ), and B 2 = (ζλ sin 4θ)/(4η).From Eq. ( 12) the growth rate of the perturbations is These equations show that nematic perturbations are suppressed by elasticity K Q and the free energy term A that favor an isotropic phase.Conversely, (ω 2 ) also includes a term, depending on activity ζ, that destabilizes the isotropic phase.For an active system confined to a droplet, the largest wave vector corresponds to the inverse of the droplet size q ∼ 1/ 0 .Fig. 1 displays simulation results for the onset of nematic ordering and rotation in active droplets showing the expected collapse of the data to a line predicted by Eq. ( 14) when plotted in terms of K Q / 2 0 for fixed A. At the linear level, the droplet velocity is given by showing that once nematic order is created the velocity inside the droplet becomes non-zero.Eq. ( 14) shows that perturbations in the nematic tensor and the rotational flows grow due to shear, through the tumbling parameter λ.In Fig. S1 in the SM we confirmed this by performing simulations where we ignored the vorticity, and only considered shear flow.
The graphs show that without vorticity the angle between the director and the radius of the droplet is equal to π/4.This means that there is a spiral defect at the center of the droplet.The presence of vorticity decreases (increases) the angle in contractile (extensile) systems as contractile (extensile) activity produces splay (bend) when nematic order is formed.

Dynamical steady states:
Beyond the linear regime, surface tension becomes important in controlling droplet behaviour, and the drop shape becomes more complex.The possible steady-state droplet configurations are illustrated in Fig. 2 for varying surface tension K φ for fixed activity ζ = 0.01 and an active area equivalent to 0 ≈ 70 for the case of a single circular droplet.
As expected, for large values of the surface tension droplets remain circular and active flows form a chiral +1 defect at the center of the droplets and lead to spontaneous rotation (Movie 1, and Fig. 2(a), green diamonds).As the surface tension is decreased the droplet deforms to a rotating ellipse and the chiral +1 defect splits into two +1/2 defects (Movie 2, and Fig. 2

(a), red circles).
A further decrease in surface tension shifts the steady state to a droplet with a hole at its center (Fig. 2(a), yellow stars).Within the active annulus, the nematic director still has a chiral orientation and, as a result, the ring of active material rotates (Movie 3).Fig. S2 in the SM compares contributions to the droplet free energy as the activity is increased.The figure shows that droplet elongation and the separation of the two +1/2 defects as the activity increases leads to a decrease in the elastic energy.The surface tension energy remains almost unchanged, but shows a sudden jump when the hole forms.The decrease in the elastic free energy density is not large enough to compete with the increase in the surface free energy density; therefore the hole formation can be identified as an active process.

Hysteresis:
To quantify the change in the shape of the droplet, in Fig. 2 (b) we define 1 and 2 (r 1 and r 2 ) as the smaller and larger outer (inner) radius of the droplet.In Fig. 2 (c) we start the simulations from a circular droplet for each value of the surface tension, and measure 1 and 2 .Decreasing the surface tension, the droplet first elongates, and then forms a hole.By contrast, in Fig. 2 (d) we start the simulations from an annulus droplet and measure r 1 and r 2 as a function of surface tension.As expected, for large values of surface tension, the annulus collapses back to a configuration with no hole.Decreasing the surface tension, the radius of the active annulus increases.The same sequence of transitions, from circle to ellipse, to annulus, is seen as the activity or the drop radius is increased (Figs.S2 and S3 in the SM).
Figs. 2 (c) and (d) show that the system has considerable hysteresis, with the steady state depending strongly on the initial conditions.In particular the elliptical configuration can be reached from an initial circle, but not from an annulus as the surface tension is increased.
In Fig. 3 we show similar hysteresis as a function of activity.We measured 1 and 2 as a function of activity, starting from two different initial conditions, a circular droplet and an annulus.The graph shows that there is a range of activity, 0.008 < ζ < 0.01, for which the steady state in the simulations preserves the topology of the initial condition, forming a rotating elongating droplet without, or with, a hole.For very large activity ζ ≥ 0.01, the circular droplets also create a hole.

Growing droplets:
The theories of active nematohydrodynamics have proved successful in modelling the dynamics of bacteria and epithelial cell layers.These are systems where cell division is relevant and therefore it is interesting to study the configurations of growing active droplets.We consider parameters where the time scale for the growth is much larger than the active time scale ).Without any temporal perturbations, the drop grows symmetrically and the growth suppresses rotation and hole formation.
However, this behaviour changes when we add a uniform noise in the concentration field of magnitude |δφ| < 0.1 every ∆T = 30000 time-steps.Fig. 4 shows snapshots from simulations of a growing droplet (see also Movie 4).The initial radius of the droplet is sufficiently small that the droplet is isotropic and inert at the beginning of the simula-tions (t 1 ).As the droplet grows, it forms a vortex at the center and starts rotating (t 2 ).As it becomes larger, it accommodates more than one internal vortex and the interface of the droplet starts deviating from a circle (t 4 ).When the droplet becomes larger, still active flows become strong enough to initiate the formation of arms that fold and form holes (t 5 - In the opposite regime where the growth timescale is smaller than the active time-scale, we do not see rotation and hole formation as the growth suppresses the rotational flows created by the activity.

THREE DIMENSIONS
We next investigate whether spontaneous rotation can also be observed in 3D isotropic droplets.Fig. 5(a) shows how the the steady states of the droplet vary for different values of activity.Firstly we observe, as in 2D, that for very small values of activity, the isotropic phase is stable and the droplet does not show any collective behavior.Increasing activity, nematic order emerges and the droplet starts rotating (orange stars in Fig. 5(a)) for both extensile and contractile driving.
The rotation is caused by a disclination, which loops from the surface into the bulk and back to the surface, and which defines the axis of rotation (Fig. 5(b)), which we take to be the z-axis, with the positive z direction from the centre of the drop towards the ends of the defect line.Measuring the director field in cross sections perpendicular to the rotation axis shows two +1/2 defects close to the droplet surface which merge into a +1 defect further from the surface (Fig. 5(c) and (d) top and middle row).Beyond the point where the loop terminates in the bulk, the director configuration is a skyrmion (Fig. 5(c) and (d) bottom row).The difference between a +1 defect and a skyrmion is that the magnitude of the order goes to zero at the center of the former, whereas it remains large and points out of the cross-sectional plane for the latter.
The details of the director field measured on planes perpendicular to dislocation lines have been shown to vary between extensile and contractile systems; twist-type configurations, where the director points out of the cross-sectional plane are more common in extensile systems [46].We quantify this in terms of the the twist angle β, defined as the angle between the rotation vector ∆, and the vector t tangent to the disclination line (Fig. 5(e)).For +1/2 (-1/2) defects, ∆ is parallel (antiparallel) to t, and cos β = 1 (cos β = −1).The out-of-plane twist component is maximal for cos β = 0.In Fig. 5(f) we plot Stronger active flows alter the shape of the droplet.An initially spherical extensile droplet deforms to an oblate spheroid, contracting along the z-axis (red stars in Fig. 5(a)).This makes it easier for the two twist defects in the two arms of the dislocation line to separate.A further increase in extensile activity then leads to droplet break-up (blue circles in Fig. 5(a)).
By contrast, for contractile activity, the +1/2 defects are more stable [46] and the disclination line pushes the active fluid towards the skyrmion, and as a result, the droplet forms an arm that grows towards the skyrmion (green triangles in Fig. 5(a)).
Similar shape changes are seen when we consider growing drops for which the growth time-scale t g = 1/k d = 2 × 10 5 is much larger than activity time-scale t a = η/ζ ∼ 10 2 .The droplet growth is illustrated in Fig. 6, and Movies 5-7.As before, extensile droplets grow to oblates.Contractile droplets grow to prolates or, for lower surface tension, form an arm in the direction of the disclination line.The rotation and growth of the 3D contractile droplets is reminiscent of the unidirectional egg chamber elongation observed in Drosophila [1].For growth which is very rapid compared to the active time scale no rotation or droplet shape changes are observed.
We now comment briefly on droplets that have a smaller value of the elastic constant (K Q = 0.02, K φ = 0.02, 0.02 < |ζ| < 0.04).For these parameters, the disclination remains as a single line with cross sections that look like +1 defects in the plane perpendicular to the disclination line.The droplet shapes are more stable, and the rotation is accompanied by a self-propulsion due to the active flows along z.As expected extensile and contractile droplets move in opposite directions with respect to the orientation of the central disclination line.
Moreover, in the contractile case for larger activities (−0.05 < ζ ≤ −0.04) we observed a wave that propagates along the disclination line (Fig. 6(d) and Movie 8).This is caused by the rotation of the two +1/2 defects in the cross section around each other as also observed in 2D.In extensile systems, +1/2 defects are not stable, and twist regions in the defect loops suppress the wave.
Finally, in this regime of small elasticity, for specific values of the parameters (K φ = 0.06, −0.06 < ζ ≤ −0.04), we observed an unusual phase in which defect loops form and move toward the rotation axis.This is shown in Fig. 6(e).

DISCUSSION
We have solved the equations of active nematohydrodynamics to investigate the spontaneous rotation of unconfined droplets in both two and three di-mensions.We work in the regime where the passive material has an isotropic director field, and nematic ordering is a consequence of active flows.
The activity leads to spontaneous rotation in 2D droplets.Using linear stability analysis, we show that the critical activity needed for the formation of order and collective motion scales linearly with 0 , where K Q and 0 are the elastic constant and the radius of the droplet, respectively.The rotation is caused by a chiral +1 defect, which can split into a chiral configuration of two +1/2 defects.Decreasing surface tension or increasing activity leads to droplet elongation, and then hole formation and a rotating annulus.The parameters for which the hole forms depend strongly initial conditions.
There are many experiments that have observed the rotation of small cell colonies.At but low densities, confined C2C12 myoblasts selforganized into configurations with persistent rotation [7].Heinrich et al. [12] observed rotation in circular also of epithelial cells with a spiral +1 defect at the centre; in this paper, a polar model was used to explain the experimental data.Formation of spiral +1 defects and rotation has also been observed in sea urchin eggs where microtubules order to form spiral +1 defects which drive cytoplasmic flows 47].Rotation has also been observed in confined confluent MDCK epithelial cells [48], and in growing colonies of T84 cells, as long as the colonies were small and approximately circular [11].In the latter case the results were interpreted in terms of an active nematic model.
Our simulations predict that the activity also leads to the formation of nematic order and unidirectional rotation in three dimensions.In 3D, a disclination line is formed which loops from the surface into the bulk and back to the surface, and defines the axis of rotation.Beyond the point where the disclination loops back, the rotation axis is marked by a skyrmion.
In extensile systems, the disclination line has large twist sections, and cross sections with +1 defect configurations are less favored.As a result, the two arms of the disclination line increase their distance apart and the droplet elongates perpendicular to the disclination line.In contractile systems, on the other hand, +1/2 defects in disclination line cross-sections are more stable and the disclination line has stable sections with charge +1 in the planes perpendicular to the disclination line.We showed that activity leads to formation of flows along the disclination line.These tend to lengthen the disclination line in contractile systems and shrink it in extensile systems.As a result of this, and the different director configurations, extensile droplets tend to form oblate ellipsoids whereas contractile droplets tend to elongate along the rotation axis.
In 3D, spontaneous persistent rotation has been observed in several biological contexts.Epithelial gland morphogenesis, where cylindrical branches transform into spherical alveoli during growth, is a morphogenetic transition which is accompanied by rotation of the emerging alveoli [49].The epithelial cell layer that lines the Drosophila egg chamber [1] rotates during development, possibly to align fibrilin the basement membrane surrounding the egg and to promote egg elongation.similar to the geometry we here, small spherical tissues rotate, an effect often termed coherent angular motion in the literature.Palamidessi al. [50] observed rotation in cancer spheroids of radius ∼ 100 µm and Brandstatter et al. [3] in spheroids of similar dimensions consisting of human mammary cells.
There are still many questions about both the physical description of the rotations, and about possible biological advantages they confer.It would be of interest to look experimentally the changes in shape and hole formation in 2D colonies, predicted here, which are most likely to be observed in highly active cellular systems.It would also be interesting to study the nematohydrodynamics of growing active shells in 3D to gain insight into the rotation of epithelial layers that surround a lumen.
Supplemental Material: Spontaneous rotation of active droplets in two and three dimensions

I. MOVIE DESCRIPTION
Movie 1: Coarsening of droplets starting from a homogeneous concentration φ = 0.4.When the drops become large enough, active flows induce nematic order, the director forms a +1 chiral defect and the drops start rotating.The direction of the rotation is random.On the left, the background color shows the magnitude of the nematic order, the director is shown in black and the boundary of the drop is shown in light green.+1/2 (−1/2) defects are shown in yellow (magenta).On the right, the background color shows the concentration of the active material, arrows show the velocity field and the boundary of the drop is shown in cyan.
Movie 2: Rotation of an elongated drop for intermediate values of surface tension K φ = 0.085.By decreasing surface tension, elastic interactions become more important and the central +1 defect splits into two +1/2 defects.On the left, the background color shows the magnitude of the nematic order, the director is shown in black and the boundary of the drop is shown in light green.+1/2 (−1/2) defects are shown in yellow (magenta).On the right, the background color shows the concentration of the active material, arrows show the velocity field and the boundary of the drop is shown in cyan.
Movie 3: Formation of a hole at the center of a droplet.Due to activity the initial circle becomes unstable.The resulting arms fold and form an annulus with no central defect which starts rotating.On the left, the background color shows the magnitude of the nematic order, the director is shown in black and the boundary of the drop is shown by light green.In the right, the background color shows the concentration of the active material, arrows show the velocity field and the boundary of the drop is shown in cyan.Movie 7: Shows the growth of a contractile isotropic droplet.The surface tension is smaller than in Movie 6, and that leads to the formation of a protrusion, instead of leading to elongation of the whole drop.This shows that the unidirectional growth is indeed caused by the disclination line.Note that, similar to Movie 6, the growth is unidirectional (along the hemisphere without the disclination).Left shows the 3D view of the drop with the central disclination line.The middle shows the trajectory of a tracer inside the drop that moves with the average flow, and the right shows the three elongation axis of the drop.The larger axis is shown in red and the two shorter axes are shown in black.
For the results presented in the phase diagram in Fig. 1, the system was initialised with a circle in a 200 × 200 box with periodic boundary condition and a noise with a uniform distribution in the interval 0 < δθ < 2π in the orientation field.The magnitude of the initial order was equal to 0.01, and the noise in the initial concentration was in the interval 0 < δφ < 0.1.
For the results presented in the phase diagram in Fig. 2, the system was initialised with the homogeneous concentration φ = 0.4, and simulations were run for t = 8 × 10 7 time steps to ensure that a dynamical steady state was reached.
For the results presented in Figs.For the growing droplet in Fig. 4, the initial droplet radius was 0 = 38.
For the 3D growing drops in Fig. 6 we used a growth rate k d = 5 × 10 −6 , and initial radius 0 = 15.

FIG. 2. (a)
Variation in elasticity free energy density with activity.The scale is chosen so that this is measured relative to the free energy density of a circular droplet with a spiral defect at its center (green shading).Increasing activity, the +1 defect separates to two +1/2 defects and the droplet elongates (pink shading), then a hole is formed eliminating the defects (yellow shading).This leads to a steady small decrease in the elasticity free energy density.(b) Variation in surface tension free energy density with activity, showing that the formation of the hole increases the free energy density by a large amount.(LB units) FIG. 3. Starting from a circular droplet, we find the steady state of the system for different values of the radius 0. Increasing the radius leads to the same sequence of phases (rotating circular drop, rotating elongated drop, rotating annulus) that we observed by decreasing the surface tension or increasing the activity.(e) (f) angular average of velocity u = u 2 x + u 2 y + u 2 z as a function of r3D = x 2 + y 2 + z 2 e) and f) show that the magnitude of the velocity decreases from the center to the boundary of the drop.This is in contrast to 2D drops and is due to the presence of a disclination line that ends at a skyrmion structure at the center of the 3d drops.

FIG. 1 .
FIG.1.Emergence of spontaneous rotation in active isotropic droplets in the plane of activity ζ and scaled elastic constant KQ/ 2 0 where 0 is the droplet radius.The black line is a linear fit to the data.

FIG. 2
FIG.2.a) Dynamical steady states of a 2D droplet as a function of the surface tension parameter K φ .For large K φ a circular droplet is stable and undergoes spontaneous rotation (green diamonds).Upon decreasing K φ , the central +1 defect splits into two +1/2 defects, and the circular droplet deforms to an ellipse (red circles).Decreasing K φ further, the droplet forms a hole and the +1 defect disappears (yellow stars).The simulations are started from a homogeneous concentration which leads to the formation of a droplet with radius 0 = 70.b) 1 and 2 (r1 and r2) are defined as the smaller and the larger outer (inner) radius of the droplet, respectively.The remaining panels demonstrate the hysteretic nature of the transitions: in (c) the initial condition (for each data point) is a droplet of radius 70; in (d) the initial condition is an annulus.

FIG. 3 .
FIG. 3. Hysteretic nature of the transitions as activity is varied: in (a) the initial condition is a droplet of radius 70 (for each data point); in (b) the initial condition is an annulus with the same amount of active material and inner radius 40. 1 and 2 are the smaller and the larger outer radius of the droplet, respectively.In both (a) and (b) K φ = 0.05.All the other parameters are as listed in the SM.

FIG. 4 .
FIG. 4. Dynamics of an isotropic growing droplet.The blue background color shows the concentration of the active material and the arrows show the velocity field.The holes (e.g.yellow outline in t8) result from the formation of isotropic arms (e.g.red outline in t8) that fold.The green arrows show the direction of time.The snapshots are zoom-in views.Here, we used K φ = 0.1.

FIG. 5 .
FIG. 5. (a) Dynamical steady states of a 3D isotropic droplet.The pink background color indicates regions in which the droplet spontaneously rotates.Increasing the magnitude of the activity from zero leads to spontaneous rotation.A further increase in the magnitude of the activity in extensile droplets leads to the formation of an oblate ellipsoid.In contractile droplets, increasing activity leads to the formation of arms.Both extensile and contractile droplets break up at high activities (only shown for the extensile case here).(b) Snapshot of a 3D rotating droplet with a stable disclination line.The blue dashed lines with arrows show the direction of rotation of the droplet.Disclination is colour coded by cos β where β is the twist angle.(c) Director field in different cross-sections along the disclination line.The color shows the magnitude of the nematic order.(d) Schematic of the director field around the disclination line in the different cross-sections.The cyan points show the center of defects.(e) The vector around which the director rotates looping around a defect is shown by ∆.The twist angle β is defined as the angle between ∆ and the vector t along the disclination line.(f) Distribution of cos β in extensile (top) and contractile (bottom) droplets.Twist defects are more common in extensile systems.(g) Breakdown of apolar symmetry due to the positioning of the disclination line on one side of the droplet leads to flows along the disclination line.The flows are along +z in the extensile system and −z in the contractile case.(h) Distribution of the lengths of the disclination line in extensile and contractile systems.In extensile systems, the disclination lines are shorter, as the active flows are in the +z direction.Here, we used KQ = 0.05, K φ = 0.02, and 0 = 20.

FIG. 6 .
FIG. 6.(a) A growing extensile droplet forms a rotating oblate ellipsoid.(b) A growing contractile droplet forms a rotating prolate ellipsoid.(c) For smaller values of surface tension (K φ = 0.01), a contractile growing droplet forms an arm.(d) Snapshot of a droplet where a wave is moving along the disclination line, and corresponding director conflguration in the yellow plane perpendicular to the disclination line showing two +1/2 defects which rotate around each other.The background color shows the magnitude of the nematic order.(e) Snapshot showing disclination loop creation.Disclination lines located at two sides of the droplet shoot disclination loops along its symmetry axis.Disclinations are colour coded by cos β where β is the twist angle.The blue dashed lines with arrows show the direction of rotation of the droplets.

Movie 4 :
Growth of a 2D drop.On the left, the background color shows the magnitude of the nematic order, the director is shown in black and the boundary of the drop is shown by light green.+1/2 (−1/2) defects are shown in yellow (magenta).On the right, the background color shows the concentration of the active material, arrows show the velocity field and the boundary of the drop is shown in cyan.Movie 5: Growth of an extensile droplet in 3D.The drop grows in the plane perpendicular to the disclination line.Left: 3D view of the droplet showing the disclination.The colour bar indicates cos β where β is the twist angle.Middle: the trajectory of a tracer inside the drop that moves with the average flow.Right: lengths of the three axes of the drop.Two larger axes are shown in red and the shorter axis is shown in black.Movie 6: Growth of a contractile droplet in 3D.The drop grows along the rotation axis defined by the disclination line.Note that the growth is unidirectional (towards −z).Left: 3D view of the droplet showing the disclination.The colour bar indicates cos β where β is the twist angle.Middle: trajectory of a tracer inside the drop that moves with the average flow.Right: lengths of the three axes of the drop.The larger axis is shown in red and the two shorter axes are shown in black.

Movie 8 :
Propagation of a wave along the disclination line in a 3D contractile droplet for large activity and small elastic constant (K Q = 0.02, K φ = 0.02).In the planes perpendicular to the disclination line, +1/2 defects rotate around each other.Top left: 3D view of the drop showing the central disclination line.The black line shows the rotation axis.The colour bar indicates cos β where β is the twist angle showing that the disclination line cross-section comprises mostly wedge-type +1/2 defects (β = +1).Top centre: trajectory of a tracer inside the drop that moves with the average flow.Top right: Three axes of the drop are shown in black.Bottom left: director field in a cross-section perpendicular to the z-axis.The background color shows the magnitude of the velocity along z.Bottom right: velocity field in a cross-section perpendicular to the z-axis.The background color shows the magnitude of the arXiv:2305.04018v1[cond-mat.soft]6 May 2023 nematic order.
2(c) and 3(a) we started the simulations with a circle with radius 0 = 70.For the results presented in Figs.2(d) and 3(b) we started the simulations with a circular annulus with inner radius r 0 = 40, and outer radius 0 = 80.

FIG. 1 .
FIG. 1.Average orientation for different values of activity as a function of distance from the center of the 2D droplet.Data in black shows the result of simulations with simplified equations that only include shear flow.Without vorticity a spiral +1 defect with θ = 45 o forms inside the droplet and bend and splay distortions do not form.Solving the full equations for extensile systems gives θ > 45 o indicating bend distortions which increase with activity.For contractile systems θ < 45 o indicating splay distortions.This confirms that the rotation is caused by the shear and that vorticity increases (decreases) the value of θ in the simulations of contractile (extensile) systems.

FIG. 4 .
FIG. 4. (a), (b) Rotational velocity u θ as a function of r2D = x 2 + y 2 , averaged over z (c), (d) angular average of the z component of the velocity uz as a function of r3D = x 2 + y 2 + z 2(e) (f) angular average of velocity u = u 2 x + u 2 y + u 2 z as a function of r3D = x 2 + y 2 + z 2 e) and f) show that the magnitude of the velocity decreases from the center to the boundary of the drop.This is in contrast to 2D drops and is due to the presence of a disclination line that ends at a skyrmion structure at the center of the 3d drops.