Topology of three-dimensional active nematic turbulence confined to droplets

Active nematics contain topological defects which under sufficient activity move, create and annihilate in a chaotic quasi-steady state, called active turbulence. However, understanding active defects under confinement is an open challenge, especially in three-dimensions. Here, we demonstrate the topology of three-dimensional active nematic turbulence under the spherical confinement, using numerical modelling. In such spherical droplets, we show the three-dimensional structure of the topological defects, which due to closed confinement emerge in the form of closed loops or surface-to-surface spanning line segments. In the turbulent regime, the defects are shown to be strongly spatially and time varying, with ongoing transformations between positive winding, negative winding and twisted profiles, and with defect loops of zero and non-zero topological charge. The timeline of the active turbulence is characterised by four types of bulk topology-linked events --- breakup, annihilation, coalescence and cross-over of the defects --- which we discuss could be used for the analysis of the active turbulence in different three-dimensional geometries. The turbulent regime is separated by a first order structural transition from a low activity regime of a steady-state vortex structure and an offset single point defect. We also demonstrate coupling of surface and bulk topological defect dynamics by changing from strong perpendicular to inplane surface alignment. More generally, this work is aimed to provide insight into three-dimensional active turbulence, distinctly from the perspective of the topology of the emergent three-dimensional topological defects.


I. INTRODUCTION
The ability of the material to employ externally or internally stored energy to spontaneously organize, flow, move, or change shape -i.e. be active -is found in a variety of materials [32]. A major class of active materials are active nematics [9], which exhibit strong collective behaviour and self organisation, emergent as local orientational order of material building blocks in material systems such as bacteria [4] and microtubule-kinesin mixtures [37,45]. Active nematic formalism could be also applied to describe other biological systems, such as cytoskeleton [5,43] and biofilm dynamics [24,46]. A recurring behaviour in active nematics is the formation of topological defects, i.e. regions of broken orientational order, which are subject to topological invariant conservation [1]. Due to strong localized deformation of the orientational order, some active nematic defects can perform as strong effective sources of material flow, as dependent on the symmetry of the defects [7,15]. A recurring property of topological defects in different active nematic systems is that at larger activities, they undergo into a regime of irregular and chaotic motion at low Reynolds numbers -called active turbulence. In the turbulent regime, generally, the topological defects are constantly created and annihilated [4,45] which can be characterised by using statistical tools from classical turbulence [14,53]. The threshold between regular and irregular dynamics is strongly dependent on the confinement [10,21], higher order force multipoles [31] and friction, which can even lead to stabilization of ordered defect phases in active nematics [7,44].
Geometrical confinements is today seen as one of the prime mechanisms for controlling the dynamics of active nematics [29,50,58,59], notably also in the context of possible energy harvesting [49]. Active nematics interact with their confinement through hydrodynamic boundary conditions, such as slip, no-slip and friction, as well as through boundary conditions on the surface imposed orientational ordering, also called surface anchoring. Active nematic droplets based on the mixture of microtubules and molecular motors were shown to periodically morph their shape [25]. Active nematic layers organised at the surface of spherical droplets are also explored from the perspective of topological defect trajectories and their mutual interactions [21,25,26,60]. When interfacing an ordered passive nematic fluid, active nematics can also exhibit directional streaming along the passive nematic director [17], and actuate the defects in the passive nematic host [18]. Fluid flow inside a droplet can produce net propulsion of the droplet, which itself acts as an individual active swimmer particle [13,30,51].
Realising and controlling three-dimensional active nematic materials is an open emergent challenge in experiments [2,8,39,40,47,61] and numerical simulations [19,48,53,57], where one of the standing major complexities is how to characterise the complex 3D spatially and time-varying flow and orientational fields, that moreover have embedded topological defect regions [53]. Analogous systems exist that are dominated by topological defects in three-dimensions, distinctly passive (non-active) liquid crystal colloids [33,52] and chiral ferromagnets [12,41], where structures are controlled and characterised by using topological approaches. Indeed, due to the apolar nature of nematic director, the nematic (passive or active) in three dimensions can form not only point defects (as in 2D), but also defect lines and loops, which can be characterised with different topological invariants, including winding number, topological charge and self linking number [1,62]. In passive nematics, coarsening structures during relaxation of a quenched state from isotropic to nematic also show notable analogy in view of defect loop topology to 3D active turbulent defect profiles [38]. To generalize, the idea of this paper is to apply concepts of topology to characterize structural events in 3D active nematic turbulence.
In this paper, we explore the topological defects regimes of three-dimensional active nematic under the spherical confinement and no-slip surface, specifically focusing on the topology-affecting events in the active turbulence. We show that for moderate activities, the active nematic assumes an oriented state with an off-center shifted single point defect in the form of a small loop, and vortex flow with direction-set angular momentum. Upon increasing activity, we observe the onset of topological turbulence, characterised by a firstorder structural transition and hysteresis between the offset point defect and the turbulent regime, also showing the corresponding phase diagram as dependent on activity and droplet size. In the turbulent regime, we show that the active turbulent dynamics can be interpreted as a time-series of topological defect affecting events: defect splitting, annihilation, merging and crossovers. Depending on surface anchoring conditions, the defects are in the form of closed loops (perpendicular anchoring) or surface-to-surface lines that effectively wet the confining surface (inplane anchoring), resulting in interesting surface-to-bulk conditioned turbulent dynamics. More generally, the results demonstrate dynamics of strongly confined active nematic and point towards using concepts of topology for controlling the three-dimensional active nematic turbulence.

II. RESULTS
The defect phenomena of the active nematic is explored in an elementary confinement of a droplet with fixed spherical shape by using mescoscopic numerical modelling of active nematodynamics [9,15,18,20], which was shown to give good agreement with experiments on dense active nematic systems [9,14,56]. The approach relies on the dynamic coupling between the material flow and the mesocospic order parameter tensor Q ij that covers the orientational ordering of active nematic. The activity is described by dipolar-like forcing via the active stress contribution. The confining spherical surface is set to impose strong perpendicular (homeotropic; in Figs. 1-3) or inplane (degenerate planar; in Fig. 4) alignment of the active nematic at the surface, which experimentally, would correspond to different surface functionalities [34]. Using strong anchoring and fixed spherical shape allows us to discern the effects of shape from the effects of topology that are otherwise inherently intertwined. Generally, such fixed shape regime corresponds to having materials with large surface tension or a background medium that is rigid enough to support the shape of the active nematic droplet (e.g. gel-like [11]). We use no-slip boundary condition at the surfaces to simulate the host medium that resists flow. All distances are measured in units of the nematic correlation length ξ N , time is measured in units of the intrinsic nematic time scale τ N , and activity ζ in the units of L/ξ 2 N where L is the single elastic constant of the material. Equally, activity can be also described with the active length ξ ζ = L/ζ. Note that the nematic correlation length measures the effective size (thickness) of the defects and is given as a relative strength of the nematic elasticity vs. variations in nematic order, which arebeside the activity and confinement -the key energetic mechanisms that affect the formation, structure and dynamics of topological defects. For more on the approach, please see Methods. Finally, note that this work focuses on the defect phenomena in three-dimensional active nematic, i.e. active nematic in the whole bulk of the spherical droplet, which is different to many current works which consider a thin layer of active nematic material, for example at the surface of a droplet [18,60] A. Active regimes in spherical droplet Depending on the activity of the material, two active regimes with distinct behaviour of topological defects are found for the three-dimensional bulk active nematic in the confinement of a spherical droplet with perpendicular alignment: for low activity, structure with an offset stationary point defect is observed (Fig. 1a) whereas for higher activities, a regime of three-dimensional active turbulence with spatially-and time-varying defect loops is observed (Fig. 1c). These two regimes are separated by a (hysteretic) structural transition (Fig. 1b).
In the limit of no activity (i.e. also no material flow), the considered spherical confinement would exhibit a radial nematic director structure with a single +1 radial hedgehog point defect at the center, as imposed by the confining surface conditions; indeed, such profile is observed for activities below the activity ζ onset = 1.6 × 10 −3 L/ξ 2 n (see also Fig. 2). Note that the hedgehog point defect appears in the form of a small ring of local winding number +1/2, with diameter of the order of nematic correlation length, and is topologically equivalent to a point [55]; the exact small-scale structure will in practice depend on the microscopic building blocks of the material. Therefore, in view of our work, loops with radius of the order of the nematic correlation length will be called point defects. Above the activity ζ onset a self-sustained structure of a twisted director profile and a vortex flow appears, characterised by the orientation and magnitude of the angular momentum vector (Fig. 1a). Similar as in (passive) cholesteric nematic droplets [42], twisted director field pushes the point defect away from the droplet center. Since our system has no preferred chirality, the shift of the point defect can be both along or against the angular momentum vector. Additionally, there is a velocity field component of roughly ∼ 2.5-times smaller magnitude then the circular vortex flow which goes along the center of the droplet towards the point defect and then near the surface back to the opposite end of the spherical droplet confinement. For increasing activity, the magnitude of the vortex flow increases and the defect shifts more and more toward the surface of the droplet, and as a critical activity is reached, the small defect ring opens into a larger loop (see Fig. 1b). At this value of the activity, the activity-induced flow can overcome the elastic barrier of the nematic field to transform the point defect into a topologically equivalent large defect loop. This process involves contributions of defect core line tension, nematic elasticity, advection, backflow, shear active flow as well as coupling to confining geometry and actual local and global defect topology. Above the critical activity, Activity is the main control parameter for the transition between the point defect regime and the active turbulence; however, this transition could be also driven for example by changing the active nematic elastic constant or the size of the confining cavity. Figure 2b presents a phase diagram of active regimes with respect to activity and droplet size, showing that larger droplets transition into active turbulent regime at lower activities, whereas smaller droplets require larger activites to undergo structural transition from regular to irregular dynamics. Indeed, such behaviour can be well understood by considering the dimensionless activity number Ac = ζR 2 /L which gives the ratio between the confinement size R and active length ξ ζ . In Fig. 2b, the transition from point defect to active turbulence is observed roughly at Ac ∼ 20.
The structural transition between the point defect structure and the active turbulence exhibits a clear hysteresis, which is a signature of discontinuous (first-order) transition between the two structures (Fig. 2a). The ordered phase with the offset point defect can be "overactivated" (i.e. with activity above the transition value), which we simulate by gradually increasing activity in small steps from the low activity state until the turbulence appeared, assuring for each step in the activity change that the system reaches the dynamic steady state. Alternatively, the turbulent state can be "under-activated" (i.e. with activity below the transition value), before the turbulent defect loop collapses into a point defect, which we simulate by gradually decreasing activity in small steps. The turbulent state can only appear when the flow is strong enough to overcome the elastic tension force of the defect line. Once the defect line is extended, it can remain extended even at lower activities. In the turbulent regime the system behaves chaotically, where changes in the shape of defects can lead to a very different future evolution of the system. The system has no static equilibrium, but reaches a dynamical steady state in which macroscopic variables, such as the defect length, angular momentum and average number of loops settle to fluctuating around an average value.

B. Topology of defect loops in active turbulence
Topology is a natural tool for studying topological defects that goes beyond exact detailed material characteristics, but rather focuses on the overall structure, and we use topological analysis for the characterisation of active turbulent regime under the confinement of active nematic into spherical droplets. We focus on topological properties of the actual defect loops -their number, how they are connected to each other, and their topological charge -as well as on the local variation of the nematic director surrounding the defect loops, i.e. considering defect loops not only as lines but rather as ribbon-like objects. Figure 3 shows the dynamics of active nematic turbulence in the spherical droplet, analysed from the perspective of the topology of three-dimensional active defect loops. In Fig. 3a from 2D active nematic systems that +1/2 defect profiles are generally strong generators of flow and strongly propel this type of (in 2D) point defects, whereas −1/2 defect profiles do not generate flow and are less motile [15]. Indeed, it is this variability of the director field that we now observe in the local cross-sections of the three-dimensional defect loops (see Supplementary Fig. 3), which then affects the dynamics of not only itself but all defect loops, through both nematic elasticity and hydrodynamics. The role of local cross-sections is known to affect the dynamics of reconnection events between defect line segments [22,23].
In our system, immediately before and after topological reconnection events the director orientations of both defect line segments generally lie in the same plane, and rewire in a way that makes the director between the defect lines uniform (without introducing topological solitons). For example, two defect line segments with locally twisted profile may reconnect into another pair of twisted profiles, a pair of +1/2 and −1/2 local profiles, or other intermediate profiles with a net neutral winding number (see Supplementary Fig. 2). Finally, the fact that the defect lines in spherical droplet confinement with perpendicular conditions are closed into loops imposes that this local director profile (+1/2, −1/2 or twisted) must come back to the same orientation when encircling the loop, which encodes the total topological charge carried by the loop [62].
At the level of the topology of the loop as a whole with surrounding director, we observe that one of the loops always carries a +1 topological charge -set by the +1 topological charge imposed by the perpendicular (homeotropic) surfaces -whereas the rest are topologically neutral (i.e. charge zero) and can annihilate without contacting another loop (see Fig. 3d), or alternatively, also merge back with another loop. From purely topological perspective, also pairs of oppositely charged loops could be formed (not observed in our simulations), especially in larger systems and with larger activities, while still preserving the net charge conservation set by confinement. However, the formation of oppositely charged pairs is -at least in the considered material regime -energetically even more unfavourable as the formation of zero-charged defect loops. Finally, a possible further approach to classify the active defect loops topologically could involve quaternionic approach [62] or construction of Pontryagin-Thom surfaces [6]. of interesting surface-to-bulk coupled defect dynamics. Under degenerate planar anchoring, the defect lines within the bulk are not any more necessarily closed into loops, but can also terminate at the surface, as shown in Fig. 4. Indeed, in addition to closed defect loops observed for droplets with perpendicular surface alignment, defect lines with surface-to-surface spanning ends and surface boojum defects are observed, resulting in the turbulent defect dynamics which in the bulk is generally similar to the dynamics in homeotropic droplets but it is additionally coupled to the topological events at the droplet surface. shown in Fig. 4d), or (ii) a merger of two ends of defect line segments into one segment and the following detachment of the defect segment from the surface. In Fig. 4c, a +1 surface boojum interacts (and merges) with a section of a bulk defect, which results in splitting of the boojum into two +1/2 defects, transforming the surface director profile with two +1/2 defects and the +1 boojum into a surface field with four +1/2 defects. In the bulk of the droplet, the defect line that merged with the boojum splits into two parts, each of them terminating at one of the newly created +1/2 defects at the surface, as shown in Fig. 4e.
Additionally, not shown here, we observe creation of surface defect pairs when an existing bulk defect loop segment is pushed from the bulk to the surface, where it is split into two ends (seen at the surface as a pair of ±1/2 defects), or also alternatively, a reconfiguration of the surface charges by merging of combinations of −1/2 defects and +1 boojums.
Topologically, the observed active turbulent defect dynamics in droplets with inplane alignment is determined by the director on the surface where the total winding number (2D topological charge) must be equal to the Euler characteristic of the confining surface χ.
Specifically, in our droplets, the net 2D surface topological surface charge must equal χ = 2, which in the limit of zero activity, is realized by a pair of +1 boojums at the opposite poles of the droplet. However, for non-zero activity, as shown if Fig. 4, the surface boojums are only rarely observed and the surface charge is rather distributed between four +1/2 defects and additional pairs of ±1/2 defects, which are actually endings of the bulk +1/2 or −1/2 defect lines touching the droplet surface. More generally, the topological dynamics of the defect lines (and loops) in the 3D bulk of active nematic induces a lower dimensional active defect dynamics at the confining surfaces.
The turbulent defect dynamics at the surface of the droplet shares similarities with the 2D active nematic turbulence (e.g. in films or shells); however, it is different from the point that it is actually coupled to the active flow and defect reconfiguration of the entire bulk.

III. DISCUSSION
Confinement is today seen as one the major routes for controlling active matter -including active nematics -with multiple geometries explored [25,29,58,59]. From the perspective of this work, we should emphasize that we purposely chose the regime where the droplet radius R is generally comparable but larger than the active length ξ ζ = L/ζ, i.e.
R/ξ ζ < ∼ 35, and the surface interactions (surface anchoring) are strong, such that the confinement has a profound role. The main effect of confinement with the strong homeotropic boundary condition is that it enforces existence of at least one bulk topological defect (loop) by imposing non net-zero topological charge in the bulk of the droplet, and similarly strong planar anchoring imposes non-zero surface topological charge, thus enforcing existence of surface defects. This confinement-imposed existence of defects in the limit of low activity has a notable effect on the onset of active turbulence as it circumvents the need for spontaneous creation of defect pairs. Instead of a spontaneous defect creation, existing defect or defects get transformed -e.g. existing defect splits into a defect pair. As a result of this lower energy barrier mechanism, in this work we do not observe events where new defects would emerge from an instability of the homogeneous nematic field and would involve spontaneous appearance of a single loop with zero topological charge, but novel defect loops are instead created by a breakup from an existing defect loop. In similar manner, one could envisage that some surface anchoring profiles or structures with sharp edges, that would cause strong local distortions of nematic could perform as nucleating regions for the emergence of the active nematic defects. Finally, the topology and geometry of the confining surfaces thus can play an important part in generating distinct active regimes.
The major body of current work in active nematics is for two-dimensional or quasi-twodimensional regimes and geometries [4,45]. From the perspective of topological defects, the key difference between 2D and 3D active nematics is that the defects change from defect points in plane to defect lines and loops in three-dimensional space, which introduces a notable increase in the complexity of tracking and monitoring apporaches for the defects.
Topologically, the key difference is that in 2D nematic fields only +1/2 and −1/2 defects are possible, whereas in 3D nematics the defects can be in the from of points, loops or walls, and even lines if these can terminate appropriatelly like on surfaces. Also, solitonic solutions are starting to be seen in objects like active skyrmions [35]. These three-dimensional nematic defect structures can be characterised with different topological invariants (in 2D nematics there is only 2D topological charge) that account for the topology of the defects as a whole, such as 3D topological charge, linking and self-linking number. Defects can be characterized also locally through their cross-section, for example with the winding number. At the level of orientational order fields, active nematics are topologically equivalent to the passive (i.e. not active) nematics, in which 3D defect structures as complex as knots and links were demonstrated [33,52]. It is an interesting question if similarly complex three-dimensional topological defect structures could emerge also in active nematics and moreover, if they could have some role in real living matter.
In conclusion, using numerical modelling, we demonstrate the topology of three-dimensional active turbulence in droplets with fixed spherical shape of active nematic, considering both droplets with perpendicular and inplane imposed surface orientation. Structural transi- We apply mesoscopic continuum description of (dense) active nematic, that is based on coupled dynamic equations for orientational order and the material flow field [9,18,20].
Roughly, this approach is based on the material flow generation via the active forces caused by the distortions in the orientational ordering of the active nematic, and the back-coupled response of the active nematic orientation to the generated flow. The nematic ordering is described by a traceless tensor order parameter Q ij whose largest eigenvalue is the degree of order S (also called scalar order parameter) and the corresponding eigenvector gives the main ordering axis, called the director n. The dynamics of the Q-tensor is given by the adapted Beris-Edwards equation [3] where ∂ t is a derivative over time t, ∂ k is a derivative over k-th spatial coordinate, u is fluid velocity, and Γ bulk the rotational viscosity coefficient. Molecular field H ij drives the system towards the equilibrium of Q ij and can be interpreted from the free energy F (but also alternatively, see Ref. [9]): where F is written in the Landau-de Gennes form as: Derivatives of Q ij in Eq. 3 describe the effective elastic behavior of the director field, where L is the elastic constant. Summation over the repeated indices is implied. A, B, and C are parameters that can be used to tune the nematic phase behavior.
The advection term S ij couples the velocity and nematic ordering: where D ij is the symmetric, and Ω ij the antisymmetric part of the velocity gradient tensor The alignment parameter χ depends on the molecular shape and defines the flow-aligning or flow-tumbling regime.
Surface alignment is modelled as follows. For droplets with perpendicular surface alignment (homeotropic anchoring), strong anchoring with fixed radial director at the droplet surface is assumed. Droplets with inplane degenerate planar alignment (degenerate planar anchoring) are modelled by the surface free energy density: , and ν is the surface normal. Surface Q-tensor field follows the dynamicṡ where and f vol is the free energy density expressed in Eq. 3.
The fluid velocity obeys the incompressibility condition and the Navier-Stokes equation where ρ is the fluid density and Π ij the stress tensor, which consists of a passive and an P is fluid pressure, η the isotropic viscosity contribution, and ζ is the activity parameter characterizing the strength of force dipoles of contractile (ζ < 0) and extensile (ζ > 0) objects.
Coupled equations for the fluid velocity u i and the nematic order Q ij are solved numerically by the hybrid lattice-Boltzmann algorithm [10,18]. The hybrid algorithm consists of an explicit finite difference method for the Q-tensor evolution (Eq. 1), and the D3Q19 lattice Boltzmann model for the Navier-Stokes equation and the compressibility condition (Eqs. 8,9). The nematic stress tensor is implemented in the lattice Boltzmann algorithm as a force contribution with the half-force correction [28]. The spherical droplet cavity is allocated on a rectangular grid and a no-slip boundary condition is implemented by the bounce-back rule. A radial surface normal is allocated on the surface nodes, and it is used for the calculation of the surface Q-tensor field either for homeotropic or for planar degenerate anchoring. We performed multiple test to verify that results are not affected by spurious velocities and numerical method induced symmetry or symmetry breaking. For example, in Fig. 1a, for different random initial conditions, the defect is displaced from the center in different (arbitrary) directions that are typically not along the (rectangular) simulation grid axes. The results of the simulations are expressed in the units of elastic constant L, nematic correlation length ξ N = L/(A + BS eq + 9 2 CS 2 eq ) where S eq is the equilibrium nematic degree of order, and nematic intrinsic time scale τ N = ξ 2 N /Γ bulk L. The phase parameters are set to A = −0.190 L/ξ 2 N , B = −2.34 L/ξ 2 N and C = 1.91 L/ξ 2 N , the nematic is in alignment regime with χ = 1, isotropic viscosity contribution equals η = 1.38 ξ 2 N /Lτ N , and the strength of the planar degenerate anchoring is W deg = 6.6 · 10 −4 L/ξ N with surface rotational viscosity parameter Γ surf = 0.67 Γ bulk /ξ N . Grid resolution is set to ∆x = 1.5 ξ N and time resolution to ∆t = 0.057 τ N . Velocity field of the active nematic can be analysed also by calculating the total angular momentum Γ = ρr × udV , where r is the distance from the droplet centre and integration is performed over the whole volume of the active nematic droplet.
The parameters of the considered 3D active nematic system can be summarised by introducing two selected dimensionless numbers (according to Eqs. 1 and 9): Ericksen number Er = uR/Γ bulk L comparing the viscous terms to the elastic terms and activity number Ac = ζR 2 /L characterizing the relative strength of activity vs. confinement, where u is maximum velocity in droplets of radius R and activity ζ. In our simulations we take activities of up to ζ < ∼ 0.06 L/ξ 2 N and droplet radius of R = 136 ξ N which corresponds to Er < ∼ 100 and Ac < ∼ 35.

Tracing and visualisation of defects
Modelling of active nematics gives a continuous three-dimensional Q-tensor field and a velocity field on a discrete grid at each time step. To perform the topological analysis, geometry of defect lines is extracted from the Q field. Defects are detected by first finding the point in the sample that has the lowest degree of order S. A small nearby sphere with 3 data points in radius is searched for the lowest S and the line is propagated until it closes the defect loop by meeting the initial point. Visited points are removed from the search space, and the algorithm is repeated until no regions with sufficiently low order are found. This procedure produces polygonal lines that approximate the defect loops, and an associated coordinate frame is constructed, consisting of the tangentt and two perpendicular vectorŝ t 1 andt 2 . The choice of these two vectors is in principle arbitrary, as long as they vary continuously around the loop. For each defect loop segment, the surrounding director field is analysed on a small circle spanned by the perpendicular vectorst 1 andt 2 . Two mutually perpendicular normalized vectorsn 1 andn 2 are found which are used to parametrise the circle on the unit sphere of all directions, visited by the director as traversing the circle.
These directionsn 1 andn 2 are flipped head-to-tail if needed so that they vary continuously along the loop. Keeping the signs and orientation of the local coordinate frame consistent with respect to the initial reference point overcomes the ambiguities inherent in the line field topology. Note that we trace two coordinate frames over the entire loop. The first frame consists oft,t 1 , andt 2 and the second frame consists ofn 1 ,n 2 , andn 1 ×n 2 . The orientations of these frames are represented with quaternions instead of rotational matrices, to retain the full topological information in a closed loop [62]. The pairn 1 andn 2 define the discs, drawn in Fig. 3b to represent the cross section, while the colour represents the quantityt · (n 1 ×n 2 )/2 that distinguishes the winding number, from pure −1/2 profile in red, to pure +1/2 in blue. Splay-bend parameter is used to visualise local director around the active defect loops [38].
Velocity field in Fig. 1 is shown by blue arrows above selected cut-off magnitudes of Probability density of defect line segments per unit volume, with respect to relative distance from the center of the droplet. The probability is normalized to unity when integrated over the whole volume of the droplet. All three graphs are in the active turbulent regime. At lower activities, the defects are more likely found in the middle of the droplet, whereas with increasing activity the probability becomes more uniform. Immediately adjacent to the surface, probability is somewhat larger due to the flow spreading and extending the defects against the surface.