Morphodynamics of surface-attached active drops

Domain: physics
Published: 2026-02-12
Editorial outcome: major_revision

Abstract
Many biological and synthetic systems are suspensions of oriented actively-moving components. Unlike in passive suspensions, the interplay between orientational order, active flows, and interactions with boundaries gives rise to fascinating new phenomena in such active suspensions. Here, we examine the paradigmatic example of a surface-attached drop of an active fluid (an “active drop”), which has so far only been studied in the idealized limit of thin drops. We find that such surface-attached active drops can exhibit a wide array of stable steady-state shapes and internal flows that are far richer than those documented previously, depending on boundary conditions and the strength of active stresses. Our analysis uncovers quantitative principles to predict and even rationally control the conditions under which these different states arise — yielding design principles for next-generation active materials.

Introduction
Active matter refers to systems whose components consume energy from their environment and convert it into motion and/or biomass, thereby driving them far from equilibrium1–3. Many biological and synthetic active systems are compartmentalized and therefore resemble liquid drops, with an interface that separates a suspension of active units from the surrounding environment. These units can self-organize and acquire orientational order, exert stresses, and induce spontaneous flows that can deform the confining interface. In turn, these changes in shape can influence the organization of the active units. This interplay between order, flow, and shape plays a key role in the function of diverse forms of active matter, such as organoids4,5, bacterial and amoeba colonies6–12, the mitotic spindle13–15, microtubule vesicles16, droplets in active suspensions17,18, phase-separated fibril drops19, intracellular and developmental flows20–23, and colloidal and ferrofluid droplets24–26.

In both natural and synthetic contexts, active drops do not typically exist in isolation, but are attached to surfaces, such as in microbial colonies6–8,10–12, single-cell and tissue migration27–31, and active wetting phenomena24–26,29,32,33. Understanding the morphodynamics of these surface-attached active systems is of fundamental interest in active matter physics and has key implications for both biology and materials science. How does the coupling between orientational order, flow, and interactions with confining interfaces influence the shape of surface-attached active drops? A comprehensive answer to this question is still missing, despite the critical role of this interplay in the functioning of active systems. In particular, while current theoretical models provide useful intuition, they rely on the thin-film approximation12,34–41, which assumes that all variables describing the drop vary less in the direction parallel to the substrate than in the direction normal to it42,43. This strong approximation is not applicable to many real-world systems, for which the drop shape, ordering of active units, and self-generated flows can vary in all directions. As a result, current understanding is incomplete. Indeed, by relaxing the thin-film approximation, here, we show that the morphodynamics of surface-attached drops are far richer than was previously known.

We revisit the paradigmatic continuum model of an active drop—containing a suspension of active units that exhibit nematic order and spontaneously self-generate flow—attached to a rigid, impermeable, flat surface. Unlike previous work, we do not make the thin-film approximation, but instead perform time-dependent simulations of the complete conservation equations. We find that the drop can exhibit a rich array of steady-state shapes and internal flows, which are a function of the alignment of active units with the boundaries and the active Capillary number comparing the active stresses exerted by the units and the capillary pressure. We uncover all possible states of the drop and the necessary conditions for symmetry breaking, which only emerges spontaneously when (i) the units become polarized at the solid substrate and the liquid–air interface of the drop and (ii) when they adopt a particular alignment direction at those boundaries. These results are in stark contrast to those generated using the thin-film approximation, which predicts a unique equilibrium shape for a given absolute value of the Capillary number, independent of alignment conditions and the contractile or extensile nature of the active stresses. Finally, we show that these equilibrium drop shapes can be controlled reversibly via surface anchoring and through the bulk active stresses exerted by the units.

Altogether, our findings provide a deeper understanding of the morphodynamics of active drops, with implications for various biological processes such as cell migration, tissue morphogenesis, biofilm development, and intracellular flows. These insights not only shed light on the complex interplay between order, flow, and shape in living systems but also offer a framework for the design and control of synthetic active materials and living systems.

Results
Model of an active nematic drop attached to a solid substrate
We investigate the morphodynamics of surface-attached viscous nematic droplets: drops of fluid containing a suspension of elongated active units that acquire orientational order with head-tailsymmetry. These units tend to maintain such order and exert force dipoles that spontaneously generate flow, thereby deforming the interface between the active fluid inside the drop and the surrounding fluid, which we consider to be passive and quiescent (Fig. 1a). For simplicity, we consider a two-dimensional (2D) continuum model to describe a nematic droplet of radius R of an incompressible, Newtonian fluid of dynamic viscosity μ and surface tension γ associated with the liquid–air interface. We assume that the droplet is pinned to a rigid, impermeable substrate extending from 0 ≤ x ≤ 2R, with an initial semicircular shape, and contains a uniform suspension of active units. The nematic orientational order of the units is described by the director vector field p(r,t), where r is position and t is time. In this description, the active units can polarize at the boundaries of the drop (see Fig. 1; and Supplementary Information for a full nematic description) The flow generated by the active stress exerted by the units is described by the fluid isotropic pressure Π(r,t) and the velocity field u(r,t), assuming Stokes flow, i.e., low-Reynolds-number flow. The dimensionless volume and momentum conservation equations read:
∇·u=0, and 0=∇·σ, (1)
where σ is the stress tensor of the fluid. For simplicity, we consider that active units do not proliferate. The stress tensor takes into account the fluid isotropic pressure Π, which enforces flow incompressibility (Eq. (1)), the viscous stress, and the active stress generated by the active components in the drop. Assuming that the flow is Newtonian and neglecting any complex rheology of the fluid resulting from the presence of the active units, the stress tensor is given by:
σ= −ΠI+μ[∇u+(∇u)T]+αpp, (2)
where α is the signature of the active stress, which quantifies the strength of the force dipoles exerted by the active units. The sign of α characterizes the nature of the flow produced by the active units: for α > 0, the flow is extensile, while for α < 0, it is contractile, favoring stretching or contraction along the axis of the active unit, respectively.

In a biological context, swimming microorganisms can produce either contractile or extensile flow depending on their motility mode1. Similarly, microtubule suspensions can exhibit either extensile or contractile behavior16,44–46.

To determine the orientation field p describing the nematic order of active units in the drop, we consider the following dynamical equation, Γ(∂tp+u·∇p+Ω·p)= −δF/δp, where Γ is a rotational viscosity, Ω is the vorticity tensor, and the right-hand side corresponds to the first variation of the free-energy functional F with respect to p1,47. For simplicity, we assume that this order is established much more quickly than the flows generated within the drop, which yields δF/δp=0. This assumption implies that the orientation field relaxes instantaneously toward its free-energy minimum and is uncoupled from the flow dynamics, usually referred to as the strong elastic limit (see Supplementary Information). We consider the following free energy for the orientation field48–50:
F= ∫A dA K/2 (∇p:∇p), (3)
where K is the Frank elastic constant in the one-constant approximation48,49 and A is the area of the 2D drop. The first variation of F with respect to p yields the following equation for the instantaneous orientation field,
∇2p=0: (4)
At the interface of the drop we impose a kinematic condition specifying that the velocity of the interface is equal to the velocity of the fluid, thus precluding mass transfer across the interface, the stress balance, and the orientation of the units:
Fluid: (∂r·u)·nb=0, σ·nb= −γCnb at r=ri (5a)
Interfacial polarization of active units: p=Ri·bt at r=ri (5b)
where ri is the position of the interface, nb and bt are the unit normal and tangential vectors to the interface, respectively, and C=∇·nb is twice the mean curvature of the liquid–air interface. It is important to emphasize that while the active units exhibit nematic order in the bulk, satisfying the p → −p symmetry, they become polarized at the boundaries of the drop, as described by Eq. (5b) for the liquid–air interface (see Supplementary Information for a full nematic description of the active drop). Here, Ri is the rotation matrix prescribing the angle θi that the orientation of the units form with the tangent vector to the interface. For simplicity, we follow refs. 36,38 and consider only quarter turns in the orientation angle, i.e., θi=wiπ/2, where wi ∈ Z is the winding number (Fig. 1a).

At the solid substrate, we impose no-permeation and no-slip conditions for the velocity field, i.e., u=0 at y=0, and we prescribe the polar orientation of the active units with respect to the tangent vector to the substrate, equivalently to the condition at the interface, i.e., p=Rs·be, where Rs is the corresponding rotation matrix prescribing the angle θs at the substrate. For simplicity, we also assume that the orientation angle with the substrate only changes in quarter turns, and thus it is specified by an additional winding number ws. As initial conditions, we consider the shape of the drop to be semicircular and the fluid in the drop is at rest, i.e., u(r,t=0)=0 and Π(r,t=0)=γ/R.

Dimensionless parameters governing the morphodynamics of a surface-attached active drop
Before solving Eqs. (1–5), we non-dimensionalize these equations to reduce the number of parameters. To this end, we choose the initial drop radius R as the characteristic length scale, viscocapillary time μR/γ as the characteristic time scale, γ/μ as the characteristic velocity scale, and capillary pressure γ/R as the characteristic pressure scale. All results and variables hereafter are non-dimensionalized by these quantities. For simplicity, we retain the same notation for the dimensionless variables. Non-dimensionalization reveals only one governing dimensionless parameter, the active Capillary number:
Caα ≡ α/(γ/R) = αR/γ, (6)
which compares the active stress exerted by the active units with the capillary pressure36,51. Thus, our minimal model is described by three dimensionless parameters: the active Capillary number, Caα, and the two winding numbers wi and ws, which specify the orientation angles of the active units with respect to the liquid–air interface and solid substrate, respectively.

Previous works describe surface-attached nematic drops using the thin-film approximation. This approximation simplifies Eqs. (1–5) into a single equation describing the height of the drop h(r, t) by assuming that all variables change slowly along the direction of the substrate compared to the variation in the direction normal to the substrate: ∂th+∇·(h3∇∇2h/3+Caα,lub h2r). This approach reduces the number of dimensionless parameters to one, which is a modified active Capillary number that absorbs a winding number w, determining the orientational turns of the active units from the solid substrate to the liquid–air interface, Caα,lub ≡ α/(2πwγ/R). Here, by contrast, we do not a priori impose the slenderness of the drop; instead, we conduct full numerical simulations of the conservation Eqs. (1–5).

Surface-attached active drops reach stable steady states
How is the organization of active nematic units influenced by, and in turn influence the interfacial morphodynamics of a surface-attached nematic drop? To address this question, we perform numerical simulations of Eqs. (1–5). We first explore the case shown in Fig. 1b as an illustrative example, which corresponds to Caα=7.5, ws=0, and wi=2 (Supplementary Movie 1). In this case, the active units generate extensile flows and their orientation rotates by π radians from the substrate to the liquid–air interface. This spatial organization of the active units, combined with the high active stresses they exert, generates strong flows capable of inducing large deformations of the drop. However, after a transient period, the drop eventually reaches a stable, steady-state shape and internal flow, where active, viscous, and capillary forces are balanced, as shown in the snapshot in Fig. 1b. This equilibrium shape and flow contrasts with the thin-film prediction36,38. Under this approximation, the active droplet always breaks symmetry, regardless of the value of the slender active Capillary number Caα,lub or the orientation of the units relative to the substrate and liquid–air interface.

Motivated by this finding, we next explore whether a surface-attached nematic drop always reaches a stable steady state for all values of the governing parameters, and investigate the necessary conditions for symmetry breaking, which can ultimately lead to drop migration.

Surface-attached active drops adopt a rich array of equilibrium shapes and flows
What are the possible shapes that a surface-attached active drop can adopt? To address this question, we perform time-dependent simulations for all possible combinations of winding numbers wi and ws, extensile and contractile stresses, and values of the active Capillary number |Caα|. The possible anchoring configurations are ws=0 and wi={0,1,2} for planar substrate anchoring, and ws=1 and wi={1,2,3} for homeotropic substrate anchoring, as shown in the schematic in Fig. 1. We find that the drop adopts equilibrium shapes for all values of Caα and wi (Supplementary Movies 2-13). This result contrasts with previous works on active nematic drops that are not attached to a solid boundary, which have shown that such free drops undergo fingering instabilities for different anchoring conditions and values of Caα34,52 (Supplementary Movies 16 and 15). Moreover, the drop exhibits a wide variety of stable steady states, illustrated in the (Caα, wi) state diagram in Fig. 2. Among these, some states spontaneously break symmetry.

What conditions are necessary for this symmetry breaking?

Parallel substrate anchoring
We begin by analyzing cases where the units align parallel to the solid substrate, corresponding to ws=0 (Fig. 2a). Additionally, we consider the case of wi=0, where the units are also aligned parallel to the liquid–air interface; our results are shown in the bottom row of Fig. 2a and Supplementary Movies 2–5). These anchoring conditions generate a bipolar structure in the ordering of units, with two point defects at the triple contact points between the solid substrate, the fluid, and the surrounding environment. Consequently, the order of active units, the flow, and the shape of the drop are symmetric with respect to its central midplane. Interestingly, for −4.5 ≲ Caα ≲ 2.5, this nematic ordering produces two steady-state counter-rotating vortices, whose handedness is independent of the extensile or contractile nature of the active stresses. However, the drop shape is markedly different between the contractile (Caα < 0) and extensile (Caα > 0) cases. In the former, the drop contracts in the direction parallel to the substrate, resulting in a mushroom-like shape. In the latter, the extensile stresses tend to flatten the drop along the x direction. Intriguingly, for Caα ≳ 4.5, the extensile stresses become strong enough to elongate and shrink the drop at its center, generating a flat film that connects two lobes with multiple counter-rotating vortices. This shape transition coincides with the emergence of a spiral defect inside the two lobes. These results contrast with the thin-film prediction, in which the condition wi=ws=0 does not generate flow, and the drops remain undeformed. This is a consequence of neglecting the variation of p along the direction of the substrate.

We next focus on the case with ws=0 and wi=1, in which the units make a counterclockwise quarter turn from the substrate to the liquid–air interface (Fig. 2a, middle row and Supplementary Movies 6–9). Our results reveal that a mismatch in the orientation of boundary-polarized units is a necessary condition for symmetry breaking. The direction in which the drop breaks symmetry depends on the handedness of the units’ rotation and the extensile and contractile nature of the active stresses. If the units make a clockwise rotation (ws=0 and wi=3), the shape and flow of the drop are identical to those in the case of counterclockwise rotation in Fig. 2a under a x → −x mirror symmetry. Furthermore, for the same units’ rotation, the extensile and contractile cases lead to opposite flow vortex rotations. As a consequence, symmetry is broken in opposite directions. Intriguingly, the hedgehog defect in the units’ ordering at one of the poles of the drop, observed in the extensile case, does not emerge in the contractile case, where instead a stable spiral defect appears in the bulk of the drop. Again, the asymmetric steady-state shapes and flows obtained here are markedly different from the thin-film prediction. Under this approximation, the extensile and contractile cases are identical under a x → −x reflection, and changing the handedness of the units’ rotation is equivalent to changing the sign of Caα,lub, which does not hold in the complete model.

Next, we explore the case of ws=0 and wi=2 (Fig. 2, top row and Supplementary Movies 10–13), where the units rotate π radians from the substrate, where they are horizontally aligned, to the interface, where they are also tangentially aligned. As expected, this configuration does not break the symmetry of the drop, unlike the thin-film equation. In particular, these anchoring conditions generate a vortex defect in the ordering of active units. In contractile cases, this defect generates a flow with two pairs of counter-rotating vortices, and the drop achieves a mushroom-like shape, similar to the case with wi=ws=0. In extensile cases, the drop becomes more elongated, and high activity turns the vortex defect into a line defect, resulting in a complex flow structure with multiple vortices, similar to the case with wi=ws=0.

Homeotropic substrate anchoring
Thus far, we have explored cases in which the active units are aligned tangentially with the solid substrate (ws=0). To investigate whether other substrate anchoring configurations can alter the drop’s morphodynamics, we conduct numerical simulations of drops in which the units are oriented orthogonally to the substrate (ws=1). We find that the drop shapes obtained with (ws, wi) = {(0, 0), (0, 1), (0, 2)} are equivalent to those obtained by a π/2 rotation of the units, i.e., (ws, wi) = {(1, 1), (1, 2), (1, 3)}, under the transformation Caα → −Caα. This result makes sense intuitively, as the flow generated by an extensile unit oriented horizontally is equivalent to the one generated by a contractile unit oriented vertically, and vice versa (Fig. 1b). This finding is unlikely to hold if the strong elastic limit is relaxed, i.e., if the flow is coupled back with the orientation of the active units. Exploring the influence of this additional complexity will be a useful direction for future work.

Describing the nematic field using a nematic tensor
Describing a nematic through a director field p is not always equivalent to describing it through a nematic tensor Q. In our case, the key difference between the two descriptions arises from the boundary conditions: in the p-based formulation, the imposed boundary conditions do not respect the nematic symmetry (p → −p), causing the active units to become polarized at the substrate and liquid–air interfaces. To understand the implications of these differences in our surface-attached drop, we describe the nematic through the following dynamical equation for Q, Γ[∂tQ+u·∇Q−(Q·Ω−Ω·Q)]= −δF/δQ, where Γ is a rotational viscosity, and F a free-energy functional that reads1,53–56:
F= ∫A dA [a/2 Q:Q+ b/4 (Q:Q)2+ K/2 (∇Q)2], (7)
where Q=S(pp−1/2I), with S a scalar representing the strength of the order parameter, K the Frank elastic constant, and a and b parameters from a Landau expansion in the order parameter that determine the isotropic-nematic transition (a > 0 in the nematic phase and b > 0 for stability). These parameters define the maximum value the order parameter can take, Smax = sqrt(a/b). Here, we also assume purely relaxational dynamics, i.e., the nematic relaxation rate is much faster than the flow strain rates (see Supplementary Information), which yields,
0=[a−btr(Q2)]Q+K∇2Q: (8)
We numerically solve Eq. (8) together with Eqs. (1), where the active stress tensor is now given by σ = −αQ (see Supplementary Information)1,53–57. A key difference between the two frameworks is that the configurations (ws, wi) = (0, 0) and (0, 2) are identical for a true nematic described by Q (Fig. S1a and Supplementary Information), whereas they are distinct when the nematic is described by the orientational vector field p (Fig. 2a). Furthermore, symmetry breaking of the drop shape emerges only in the p description, where the units become polarized at the drop boundaries, while it remains symmetric in the full nematic Q-tensor description. Nonetheless, the physical principles and transformations under which the shape and hydrodynamics of parallel and orthogonal anchoring are equivalent remain the same for both descriptions (Figs. 2b and S1b and Supplementary Information). Moreover, in the Supplementary Information, we explore whether nematic advection plays a relevant role in the shape morphodynamics for both the p and Q descriptions—that is, when the nematic relaxation timescale toward the free-energy minimum is comparable to the flow timescale. While the transient dynamics with nematic advection differ from those in the strong-elastic limit, the final steady-state shapes and flows that the drop adopts are nearly identical in both cases (Figs. S2 and S3).

Flow and energy dissipation in active drops follow simple scaling laws
Despite the complexity of shapes and internal flows exhibited by a surface-attached active nematic drop, steady-state flow inside the drop follows simple scaling laws (Fig. 3). In particular, the maximum velocity, max(|u|) ≡ |u|max, follows identical linear scaling laws for both contractile and extensile cases when |Caα| ≲ 1, i.e., |u|max ∝ |Caα|, for any value of wi (Fig. 3a). These linear scaling laws make intuitive sense for |Caα| ≲ 1, as the only forcing in the system is the active stress, which is proportional to Caα, the fluid dynamics are governed by the linear Stokes equations, and nonlinear effects due to drop deformation are small. Interestingly, the maximum velocity is usually higher when the drop breaks symmetry (wi=1), except in certain cases at large values of |Caα|, where the drop undergoes large deformations (e.g., wi=0 and Caα ≳ 5).

What is the energy expenditure of the active drop? We compute the total entropy production rate as follows:
ρTs_tot = ρT dAs_tot = ∫A dA σ:∇u, (9)
where ρ is the fluid density, T is the temperature, and s_tot = ∂ts+u·∇s. The total entropy production can be decomposed into two contributions, i.e., the dissipation associated with viscous stresses, and the work done by the active units:
s_v = ∫A dA σv:∇u, s_a = ∫A dA σa:∇u, (10)
where σv = μ[∇u+(∇u)T] and σa = −αpp are the viscous and active components of the total stress tensor. We compute Eqs. (9 and 10) for the case where units are tangentially aligned with the substrate, ws=0. Viscous dissipation is always irreversible, increasing the system’s entropy, s_v > 0. In our system, we find that the dimensionless viscous dissipation scales as s_v ∝ Ca2α for all values of wi when |Caα| ≲ 1, and is maximum when symmetry is broken, i.e., wi=1. Conversely, the active contribution scales as s_a ∝ −Ca2α for all values of wi when |Caα| ≲ 1. Interestingly, only when wi=0 does viscous dissipation remains smaller than the work done by active forces to sustain the self-generated flow and ordering, implying a negative value of s_tot (Fig. 3b, bottom). This behavior reverses only in the extensile case, Caα > 0, when the drop undergoes large deformations—and the flow inside the drop consequently changes its handedness—i.e., when Caα ≳ 1, at which points s_tot becomes positive (Fig. 3b, top right).

Reversible shape and flow control through time-dependent anchoring of active units
Our results thus far highlight the pivotal importance of interfacial anchoring on drop morphodynamics—suggesting a potential route toward reversibly controlling drop shape and flows. To explore this possibility, we vary the anchoring of the active units at the liquid–air interface over time (Fig. 4a, bottom). The upper panel of Fig. 4a shows the initial condition of the drop when the units are tangentially aligned with both the substrate and the liquid–air interface (ws=wi=0), and Caα=2.5. The anchoring with the substrate remains fixed, while the winding number at the liquid–air interface varies over time, as specified in the bottom panel of Fig. 4b, which also shows the maximum velocity inside the drop, |u|max, to determine when the drop reaches steady state. At time t=10, the drop reaches a stable steady-state shape and flow (Fig. 4b). At t=11, we drastically change the interfacial winding number wi from wi=0 to wi=1, forcing the units to orient perpendicular to the liquid–air interface. After a transient period (11 ≲ t ≲ 15; Fig. 4c), the drop acquires a new steady state (Fig. 4d). To test if this process is reversible, at t=40, we drastically change the winding number back to wi=0. Although the transient states corresponding to wi=1 → wi=0 (Fig. 1e) are not identical to its opposite, wi=0 → wi=1 (Fig. 1c), implying that changing the boundary condition is not equivalent to time reversal, the system relaxes back to the same steady state over time (Fig. 1b). This result shows that this minimal active matter system allows for reversible control of stable steady-state shapes.

Methods

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "This manuscript presents a numerical investigation of the dynamics of surface-attached drops of an active nematic fluid. The\nmodel couples the configuration of the nematic constituents (in the strong elastic limit) to the fluid flow they induce, which\ncauses deformations of the drop surface. The focus is on understanding the spontaneous shapes and flows that emerge as a\nfunction of the anchoring boundary conditions applied on the nematic on both the flat substrate and drop interface.\nDepending on these boundary conditions, a variety of drop shapes are observed, some symmetry and some not. The\nauthors also show how transitions between shapes can be triggered by changing boundary conditions over time.\nOverall, the paper is well written and the findings are interesting and a useful addition to the existing literature on such\nsystems. However, I have several comments and questions that the authors should address before I can recommend\npublication.\nMajor comments:\n1. The system of interest is nematic but relies on a mathematical model that describes the orientation field in terms of a\ndirector field p(x,t). It is understood that the two orientations p and -p describe the same nematic field, and indeed the active\nstress and the relaxation equation (4) are invariant under the transformation p -> -p. However, describing a nematic field\nusing a vector director is not always equivalent to using a nematic Q tensor. In the present work, this is clear from the\nboundary conditions: for a true nematic, the two winding number w=0 and 2 describe the same orientation and therefore\nshould not differ. Another place where this is apparent is at the contact line when ws=wi=0: in that case, a defect is created\nat the contact line, which is an artefact of using a vector to describe the nematic. One more difference lies in the types of\ndefects that are allowed mathematically: using a vector director does not permit fractional charge defects (+ or – 1/2), unlike\nin a true nematic where these defects are in fact prevalent. This should be at the very least discussed, and the resulting\nlimitations should be acknowledged. In particular, this suggests that some of the findings reported in the paper may not be\nobserved in systems with true nematic symmetry (such as microtubule/kinesin motor suspensions for instance).\n2. The authors neglect elastic stresses in the momentum balance, and only account for viscous and active stresses. This a\nconvenient approximation, especially if the focus is on understanding the role of active stresses on system dynamics.\nHowever, this approximation seems inconsistent with the neglect of advective terms in equation (4). The current version of\nequation (4) assumes strong elasticity (so that advection and rotation by the flow is negligible; however, if elasticity is strong,\nso are elastic stresses so neglecting them in the momentum balance is questionable. A more rigorous discussion of these\napproximations is desirable.\n3. The two quantities in equation (8) are the rate of viscous dissipation and active power input in the fluid bulk. While it’s fine\nto interpret them as contributions to the rate of change of entropy, another way of looking at them would be to perform a\nmechanical energy budget starting from the momentum equation (by dotting it with the fluid velocity, integrating over the\nvolume of fluid, and performing integration by parts). This would lead to an energy budget where the sum of viscous\ndissipation and active power is balanced by surface terms (likely involving surface tractions, that can be related to capillary\nstresses and to the shape of the drop). This may provide a physical interpretation for the signs reported in figure 3(b) for the\nnet rate of entropy production.\n4. On page 5, the authors mention the “disappearance of the point defects at the triple contact point” when extensile stresses\nbecome strong. I don’t see how that’s possible. As mentioned in my point 1 above, the choice of ws=wi=0 for the surface\nwinding numbers must create a singularity in the director field at the contact point. How can a defect not be present there?\nThis should be explained. A zoomed-in picture showing the field lines in this region could be helpful to understand what’s\ngoing on (field lines are difficult to see in the current figures).\nMinor comments:\n5. The linear and quadratic scalings with capillary number observed in figures 3(a) and (b) are unsurprising, since the only\nforcing in the system is the active stress and the governing equations for the flow are linear.\n6. The idea of controlling drop morphology by varying the anchoring boundary conditions are interesting. Can the authors\nspeculate how this could be achieved in experiments?\n7. Can the authors speculate on how their findings might change in three-dimensional drops? What is the contact line was\nallowed to move?", "reviewer_2": "The present manuscript numerically investigates the steady-states of surface attached active drops. Activity arises from a\nstress-field (contractile or extensile) related to a nematic order parameter. The authors study the drop shapes and internal\nvelocity profiles depending on a so-called ”active capillary number” (an adimensional number which characterizes the\nstrength of the active stress compared to the capillary pressure) and the anchoring conditions of the nematic order parameter\nat the substrate and the free interface.\nThe originality of this manuscript lies in the systematic study of two-dimensional droplets of active liquid (aka active liquid\nridges) beyond the thin film limit. Here the variation of the directory field in the z-direction and the full velocity profiles (in the\nStokes limit) are explicitly resolved. The field of the nematic order parameter is treated in the strong elastic limit and is solved\nadiabatically as a boundary value problem coupled to the drop shape.\nOverall the manuscript is clearly laid out and nicely illustrated. My main criticism of this manuscript is the formulation of the\nboundary conditions for the director field, which are imposing a polarization of the liquid at the boundaries. In particular this\nchoice imposes a (i) symmetry-breaking, where the authors claim a spontaneous symmetry-breaking takes place. (ii) it is not\nclear to me, how the polarity is treated at the pinned triple contact points of the droplets, which constitute a singular point.\nThe anchoring conditions are crucial to this manuscript. I strongly recommend the authors provide a clearer explanation.\nBelow is a detailed criticism.\n1. I have a major concern with the formulation of the boundary conditions for the order\nparameter p. While at the substrate (y=0) the boundary condition is p = R_s.e_x, the anchoring condition at the free interface\nis expressed as p = R_i.t, with R_i and\nR_s denoting rotation matrices and t denoting the tangent of the interface. How are\nthe triple contact points treated? Taking as example the initial condition with a\nsemicircular shape, the imposed boundary conditions for p are ill posed and cannot\nbe fulfilled. For example, ω = 0 implies an anchoring at the free surface of p = e_z\nand p = −e_z at the left and right triple point, respectively (assuming e_z denotes a\nunit vector orthogonal to the substrate and the tangent vector at the free surface, t, has a direction as indicated in Fig. 1a),\nwhile at\nthe same time a substrate anchoring with ω = 0 implies that p = e_x all along the\nsubstrate. The authors should state clearly how the triple contact points are treated. On page 5 the authors mention the\nappearance of point defects in the director field at the triple contact points for the anchoring conditions ω_s = ω_i = 0. How\nare these defects consistent with the imposed boundary conditions?\n2. Another concern, still related to the anchoring conditions for the order parameter p, is the choice to specify the polar order\nat the interfaces, instead of only the nematic order. Since the authors refer in their manuscript only to active nematic\nparticles, how can this choice be justified within the same framework? I may be wrong, but the symmetry-breaking observed\nin Fig. 2a for ω_s = 0 and ω_i = 1 is imposed by the rigid (and to some point pathological) anchoring of p to the two different\ninterfaces and is not a spontaneous process as claimed by the authors on page 5 (left column, first paragraph).\n3. As far as I understood the triple contacts points are pinned to the substrate. A priori I do not have problem with this\nlimitation. However, the obtained steady-state drop shapes (Fig. 2a) are intriguing. Especially for the depicted combination\nof ω_s = 0, ω_i = 1 at high contractile stress the droplet contracts strongly close to the left contact point and is only attached\nvia a long flat foot to the right contact point. What does the velocity field and the director field look like in these flat\nextensions? A supplementary figure which shows the drop profile, velocity and p fields in the vicinity of the triple contact\npoints for the parameter combinations depicted in Fig. 2a is necessary. This could also clarify the impact of the anchoring\nconditions on the velocity and director field (see concerns raised above).\n4. Fig. 2a (bottom row, extensile stress, Ca_α = 4.5): The two lobes of the droplet extend into the substrate. Please comment\nhow this is possible? I assume the problem can be resolved by stating that the substrate only extends from x = 0 to x = 2?\n5. In Fig. 3: I am not sure I fully understand the physical relevance of the depicted scaling relations, can the authors make an\nattempt at explaining the scaling? Also, the curves do not look very smooth, it would be more appropriate to show a symbol\nat the parameters, where the simulations have been performed. The entropy production rate for extensile stresses and\nwinding number ω_i = 0 changes abruptly at Ca_α ≈ 1. At the same moment the linear scaling for u_max breaks down.\nPlease comment. Is this change related to a qualitative change in the drop profile, velocity field, etc? Finally, the steady\nstates shown in Fig. 2a correspond to an active capillary number, where the scaling laws are not valid anymore. Please\ninclude a (supplementary) figure with a capillary number where the scaling is valid (e.g. Ca_α = 0.1).\n6. I find the discussion of previous works (thin film models, e.g. on page 5, right col., first paragraph) a bit hand waving and\ntoo simplistic. Also the three-dimensional phase-field model by Tjhung et al. (Nat. Comms. 2015, doi: 10.1038/ncomms6420)\nshould discussed more thoroughly, since there droplet motion is found for contractile droplets in the absence of self-\npropulsion. Furthermore, the thin film model by Stegemerten et al. (Soft Matter 2022, doi: 10.1039/d2sm00648k) identifies\nbistable behavior for droplets with active stresses. I agree that most thin film models (like the one from Stegemerten et al.) do\nnot take into account the z-dependence of the director field. However, there, symmetry is broken spontaneously and not by\nthe imposed surface anchoring.\nMinor remarks/typos:\n• The use of the term ”active suspension” in the abstract (line 4) is misleading since it implies the presence of concentration\nfluctuations of the suspended particles, which is not considered in the manuscript. ”Active liquid” is maybe more\nappropriate?\n• Fig. 2 caption: ”... and the grey curves depict the orientation of the active units.” Some curves appear grey, but most are red.\n• Fig. 3: The color code is misleading. The font colors for ”Extensile” and ”Contractile” correspond to the colors used for\ncurves with winding number 1 and 2."}, "author_response": "Response to Reviewer comments—Morphodynamics of surface-attached active drops\nReviewer 1 comments:\nThis manuscript presents a numerical investigation of the dynamics of surface-attached drops of an active nematic fluid. The\nmodel couples the configuration of the nematic constituents (in the strong elastic limit) to the fluid flow they induce, which\ncauses deformations of the drop surface. The focus is on understanding the spontaneous shapes and flows that emerge as a\nfunction of the anchoring boundary conditions applied on the nematic on both the flat substrate and drop interface. Depending\non these boundary conditions, a variety of drop shapes are observed, some symmetry and some not. The authors also show how\ntransitions between shapes can be triggered by changing boundary conditions over time.\nOverall, the paper is well written and the findings are interesting and a useful addition to the existing literature on such systems.\nHowever, I have several comments and questions that the authors should address before I can recommend publication.\nResponse: We thank the Reviewer for the time spent reading our paper, and for providing such thoughtful and constructive\nfeedback. It is gratifying that they found our paper to be well written and the findings are interesting and a useful addition to the\nexisting literature on active matter. We also appreciate the Reviewer’s suggestion for more rigorous consideration of the problem\nformulation and boundary conditions, which has helped us better understand this active system and significantly improve the\nmanuscript. We have followed all of their helpful suggestions in revising the manuscript, as detailed in the responses below. In\nsummary, we have:\n• Performed new simulations for a true nematic, in which the nematic field is described by a nematic tensor Q rather\nthan a director vector p. These simulations show that, although no spontaneous symmetry breaking emerges, the main\nresults obtained using p remain conceptually equivalent. We have added a new subsection discussing these new results, a\nSupplementary Information section describing this framework in detail, and a new Supplementary Information figure.\n• Performed new simulations showing that nematic advection effects have a minimal impact on the steady-state drop shapes\nand internal flows, supporting the assumption that considering the strong elastic limit while neglecting elastic stresses is\nreasonable. We have updated both the Main Text and Supplementary Information to highlight these new findings and have\nadded two new supplementary figures where we consider the advection of the nematic director field p and the nematic\ntensor Q.\n• Characterized the flow and nematic orientation near the triple contact points where the drop is pinned, as well as in thin-\nfilm regions for certain parameter values, showing that both are accurately resolved. We have added a new supplementary\nfigure showing zoomed-in views of these regions.\n• Conducted new simulations in which the drop is not pinned to the solid substrate, by introducing a finite slip length\nbetween the active liquid and the substrate to model friction. We show that, under conditions where the active drop breaks\nsymmetry, it can migrate in a treadmilling fashion. We have included a new supplementary figure showing this new result.\nWe believe these new results have improved our work significantly, and we are grateful to the Reviewer for their guidance in\nobtaining these new results.\nMajor comments:\n1. The system of interest is nematic but relies on a mathematical model that describes the orientation field in terms of a director\nfield p(x,t). It is understood that the two orientations p and -p describe the same nematic field, and indeed the active stress\nand the relaxation equation (4) are invariant under the transformation p -> -p. However, describing a nematic field using\na vector director is not always equivalent to using a nematic Q tensor. In the present work, this is clear from the boundary\nconditions: for a true nematic, the two winding number w = 0 and 2 describe the same orientation and therefore should not\ndiffer. Another place where this is apparent is at the contact line when w = w = 0: in that case, a defect is created at the\ns i\ncontact line, which is an artifact of using a vector to describe the nematic. One more difference lies in the types of defects\nthat are allowed mathematically: using a vector director does not permit fractional charge defects (+ or – 1/2), unlike in a true\nnematic where these defects are in fact prevalent. This should be at the very least discussed, and the resulting limitations should\nbe acknowledged. In particular, this suggests that some of the findings reported in the paper may not be observed in systems\nwith true nematic symmetry (such as microtubule/kinesin motor suspensions for instance).\nResponse: We are grateful to the Reviewer for raising this important issue. As the Reviewer points out, describing a nematic\nthrough p is not always equivalent to describing it through a nematic tensor Q. It is indeed the case of our configuration. To\naddress this issue, we have conducted new numerical simulations in which the nematic is described by Q. We consider the\nfollowing equation for Q, under the same assumption as in Eq. (4) of the manuscript—nematic order is established much more\nquickly than the flows generated within the drop:\n0 = (a−btr(Q2)) Q+K ∇2Q, (1)\nQ\nwhere Q = S (pp− 1I), with S a scalar representing the strength of the order parameter, K the Frank elastic constant, and\na and b parameters from a Landau expansion in the order parameter that determine the isotropic–nematic transition (a > 0\nin the nematic phase and b > 0 for stability). These parameters define the maximum value the order parameter can take,\nS = 2a/b. Using Q, the active stress tensor becomes σ =−αQ. To make Eq. (1) dimensionless, we choose Q = a/b\nmax a c\nas characteristic quantity for the nematic tensor, which yields:\n0 = (1−tr(Q˜2)) Q˜ +K˜∇2Q˜, (2)\nwhere Q˜ = S˜(pp− 1I) is the dimensionless nematic tensor, K˜ = (K /a)/R2 is a dimensionless parameter that compares\nthe elastic coherence length ℓ = K/a and the drop radius R, and S˜=S/ a/b is the dimensionless strength of the nematic\nQ\norder parameter. Equation (3) of the manuscript describing the fluid stress tensor reads in dimensionless form:\nσ˜ =−Π˜I +∇u˜+(∇u˜)T−Ca Q˜, (3)\nα\nwhere, in this case, the active Capillary number reads: Ca ≡α a/b/(γ/R).\nα\nTo compare with the results of the Main Text solving for p, we consider both planar and orthogonal anchoring of the nematic\nwith the solid substrate, i.e., Q˜ = S˜(eˆ eˆ − 1I) and Q˜ = S˜(eˆ eˆ − 1I), respectively. At the liquid–air interface, we\nx x 2 y y 2\nimpose either tangential anchoring, Q˜ =S˜(tˆtˆ− 1I), or orthogonal anchoring, Q˜ =S˜(nˆnˆ − 1I), where tˆ and nˆ are the unit\n2 2\ntangential and normal vectors to the liquid–air interface, respectively. For simplicity, in all simulations we set S˜ = 1 at the\nboundaries, and K˜ =1. The figure attached below depicts the results of numerical simulations. Panel a is equivalent to Fig. 2a\nQ\nof the manuscript, corresponding to planar anchoring with the substrate (w = 0) and both tangential (w = 0) and orthogonal\ns i\n(w = 1) anchoring with the liquid-air interface. Panel b is equivalent to Fig. 2b of the manuscript, where we show that the\ni\nsteady states of a surface-attached active drop for planar (w =0) and orthogonal (w =1) unit anchoring with the substrate are\ns s\nequivalent under the transformations w →w+π/2 and Ca →−Ca . Here,|Ca |=2.5.\nα α α\nContractile Extensile\niw\nrebmun\ngnidniW\na\nwi = 1\nws = 0\nwi = 0\nws = 0\nCoordinate x Coordinate x Coordinate x Coordinate x\nActive capillary number Caα\nb π/2 &Caα→–Caα\n≡ ≡ Extensile Contractile\n≡\nContractile Extensile\n≡\ny etanidrooC\ny etanidrooC\n0 0.0250.05\nVelocity |u|\n3\n2 0 0.025 0.05 0 0.015 0 0.035\n1\n1\n0\n0 1 2 0 1 2 0 1 2 0 1 2\n3\n2 0 0.025 0.057 0 0.025 0 0.025 0.05 0 0.05 0.11\n0\n1\n0\n0 1 2 0 1 2 0 1 2 0 1 2\n-4.5 -2.5 2.5 4.5\nCoordinate x Coordinate x Coordinate x Coordinate x\ny etanidrooC\ny etanidrooC\n0 0.0250.05\nVelocity |u| 3\n2 0 0.025 0.05 0 0.015 0 0.035 1\n1\n0\n0 1 2 0 1 2 0 1 2 0 1 2\n3\n2 0 0.025 0.057 0 0.025 0 0.025 0.05 0 0.05 0.11 0\n1\n0\n0 1 2 0 1 2 0 1 2 0 1 2\n-4.5 -2.5 2.5 4.5\nCoordinate x Coordinate x Coordinate x Coordinate x\ny etanidrooC\ny etanidrooC\n0 0.0250.05 Velocity |u|\n3\n2 0 0.025 0.05 0 0.015 0 0.035 1\n1\n0\n0 1 2 0 1 2 0 1 2 0 1 2\n3\n2 0 0.025 0.057 0 0.025 0 0.025 0.05 0 0.05 0.11 0\n0\n0 1 2 0 1 2 0 1 2 0 1 2\n-4.5 -2.5 2.5 4.5\ny etanidrooC\ny etanidrooC\n0 0.0250.05 Velocity |u| 3\n2 0 0.025 0.05 0 0.015 0 0.035 1\n1\n0\n0 1 2 0 1 2 0 1 2 0 1 2\n3\n2 0 0.025 0.057 0 0.025 0 0.025 0.05 0 0.05 0.11 0\n1\n0\n0 1 2 0 1 2 0 1 2 0 1 2\n-4.5 -2.5 2.5 4.5\nCoordinate x Coordinate x Coordinate x Coordinate x\ny etanidrooC\ny etanidrooC\n0 0.0250.05\nVelocity |u|\n3\n2 0 0.025 0.05 0 0.015 0 0.035 1\n1\n0\n0 1 2 0 1 2 0 1 2 0 1 2\n3\n2 0 0.025 0.057 0 0.025 0 0.025 0.05 0 0.05 0.11\nC0oordinate x Coordinate x Coordinate x Coordinate x\n1\n0\n0 1 2 0 1 2 0 1 2 0 1 2\n-4.5 -2.5 2.5 4.5\ny etanidrooC\ny etanidrooC\nVelocity |u|\n0 0.025 0.05 0 0.025 1\n0 1 2 0 1 2\n0 0.025 0 0.025 0.05 0\n0 1 2 0 1 2\n-4.5 -2.5 2.5 4.5\nCoordinate x Coordinate x Coordinate x Coordinate x\ny etanidrooC\ny etanidrooC\nVelocity |u|\n0 0.025 0.05 0 0.025 1\n0 1 2 0 1 2\nCoordinate x Coordinate x Coordinate x Coordinate x\n0 0.025 0 0.025 0.05\n0\n0 1 2 0 1 2\n-4.5 -2.5 2.5 4.5\ny etanidrooC\ny etanidrooC\nVelocity |u|\n0 0.025 0.05 0 0.025 1\n0 1 2 0 1 2\n0 0.015 0 0.025 0.05 0 Coordinate x Coordinate x Coordinate x Coordinate x\n0 1 2 0 1 2\n-4.5 -2.5 2.5 4.5\ny etanidrooC\ny etanidrooC\nVelocity |u|\n0 0.025 0.05 0 0.025 1\n0 1 2 0 1 2\n0 0.015 0 0.025 0.05 0\n0 1 2 0 1 2\n-4.5 -2.5 2.5 4.5\nAs the Reviewer anticipated, within this framework there are only two anchoring conditions for fixed susbtrate anchoring\n(w = 0 in this case), i.e., w = 0 and w = 1. Thus, symmetry breaking is also a consequence of describing the nematic via\ns i i\np and not considering a true nematic. However, the fact that planar (w = 0) and orthogonal (w = 1) substrate anchoring are\ns s\nequivalent under the transformations w →w+π/2 and Ca →−Ca is also true when the nematic is described by Q. We have\nα α\nrevised the Main Text of the manuscript to include a careful discussion of both frameworks and the corresponding results (see\nnew subsection in the Main Text, lines 338–370). Additionally, we have added a new section to the Supplementary Information\ndescribing the derivation and non-dimensionalization of the equations above. The figure above has also been included in the\nSupplementary Information.\n2. The authors neglect elastic stresses in the momentum balance, and only account for viscous and active stresses. This a\nconvenient approximation, especially if the focus is on understanding the role of active stresses on system dynamics. However,\nthis approximation seems inconsistent with the neglect of advective terms in equation (4). The current version of equation (4)\nassumes strong elasticity (so that advection and rotation by the flow is negligible; however, if elasticity is strong, so are elastic\nstresses so neglecting them in the momentum balance is questionable. A more rigorous discussion of these approximations is\ndesirable.\nResponse: The Reviewer raises a very important issue. In the original manuscript, we considered the strong elastic limit for the\ndirector field p but neglected nematic elastic stresses in the momentum conservation equation. To address the role of nematic\nhydrodynamic advection and elastic relaxation, we conducted simulations in which the strong elastic limit is relaxed—that is,\nwe included nematic advective terms, and the only stresses acting on the fluid are active, with no elastic contributions. If we\ndescribe the nematic through the tensor Q˜, Eq. (2) now reads:\n(Γ˜ ∂ Q˜ +u˜·∇Q˜ −(Q˜ ·Ω˜ −Ω˜ ·Q˜) = (1−tr(Q˜2)) Q˜ +K˜ ∇2Q˜, (4)\nQ t˜ Q\nwhere 2Ω˜ = ∇u˜ −(∇u˜)T is the dimensionless vorticity tensor, and Γ˜ ≡ (Γ /µ)γ/(aR) is a dimensionless parameter that\nQ Q\ncompares the nematic rotational viscosity Γ with the liquid viscosity µ. We conducted new numerical simulations solving\nQ\nEq. (4), considering Γ˜ = 1. While the transient dynamics with nematic advection differ from those in the strong-elastic limit,\nQ\nthe final steady-state shapes and flows that the drop adopts are nearly identical in both cases, as shown by the figure attached\nbelow.\nContractile Extensile\niw\nrebmun\ngnidniW\nStrong-elastic limit With nematic advection\nContractile Extensile\nwi = 1\nws = 0\nwi = 0\nws = 0\nCoordinate x Coordinate x Coordinate x Coordinate x\nActive capillary number Caα\ny\netanidrooC\ny etanidrooC\n0 0.025 0.05 0 0.015 0 0.025 0.05 0 0.015\n2\n1\n1\n0\n0 1 2 0 1 2 0 1 2 0 1 2\n3\n2 0 0.025 0 0.025 0.05 0 0.025 0 0.025 0.05\n0\n1\n0\n0 1 2 0 1 2 0 1 2 0 1 2\n-2.5 2.5 -2.5 2.5\nWe conduct the same numerical calculations but now describing the nematic through the director field p, as in the original\nmanuscript, where now Eq. (4) of the manuscript reads:\n(Γ˜ ∂ p+u˜·∇p+Ω˜ ·p =∇2p, (5)\nt˜\nwhere Γ˜ ≡ (ΓR2/K)/t = (ΓR2/K)/(µR/γ), is a dimensionless parameter that compares the characteristic elastic relax-\nc\national time scale of the nematic with the viscous characteristic time scale of the liquid t = µR/γ. Similarly to the case of\nc\nEq. (4) of this rebuttal, here the transient dynamics considering p advection are also different compared to the strong-elastic\nlimit, but the final steady-state shapes are identical in both cases, as shown in the figure attached below.\nContractile Extensile\niw\nrebmun\ngnidniW\nStrong-elastic limit With nematic advection\nContractile Extensile\nwi = 1\nws = 0\nwi = 0\nws = 0\nCoordinate x Coordinate x Coordinate x Coordinate x\nActive capillary number Caα\ny etanidrooC\ny etanidrooC\ny etanidrooC\n0 0.05 0.09 0 0.1 0.17 0 0.05 0.09 0 0.1 0.17\nVelocity |u|\n1 wi = 2\n2\n0\n0 1 2 0 1 2 0 1 2 0 1 2\nws = 0\n0 0.15 0.34 0 0.1 0.19 0 0.15 0.32 0 0.1 0.19\n1\n1\n0\n0 1 2 0 1 2 0 1 2 0 1 2\n0 0.03 0.07 0 0.02 0.05 0 0.03 0.07 0 0.02 0.05\n1\n0\n0\n0 1 2 0 1 2 0 1 2 0 1 2\n-2.5 2.5 -2.5 2.5\nWe have included the two figures above as supplementary figures, added supplementary sections describing these results in\ndepth, and included a new sentence summarizing the main findings in the revised Main Text (lines 370-378).\n3. The two quantities in equation (8) are the rate of viscous dissipation and active power input in the fluid bulk. While it’s\nfine to interpret them as contributions to the rate of change of entropy, another way of looking at them would be to perform\na mechanical energy budget starting from the momentum equation (by dotting it with the fluid velocity, integrating over the\nvolume of fluid, and performing integration by parts). This would lead to an energy budget where the sum of viscous\ndissipation and active power is balanced by surface terms (likely involving surface tractions, that can be related to capillary\nstresses and to the shape of the drop). This may provide a physical interpretation for the signs reported in figure 3b for the\nnet rate of entropy production.\nResponse: We thank the Reviewer for this thoughtful suggestion. We indeed keep track of the rate of change of energy associated\nwith surface tension as a function of time, which reads:\n∂ E =− dAγu·n∇·n. (6)\nt γ\nA\nHowever, Fig. 3 of the manuscript only reports contributions to the energy budget at steady state, where the velocity at the\ninterface of the drop is zero, implying ∂ E = 0, and therefore only viscous dissipation rate and active power contribute to the\nt γ\nrate of change of the energy budget. Nonetheless, to better understand some of the changes in entropy production rate and |u|\nmax\nbeyond the linear regime, we have included new steady-state snapshots in Fig. 3. For example, the abrupt change at Ca ≈ 1 for\nα\nω = 0 is a consequence of the significant change in droplet shape as capillary pressure becomes comparable to active stresses.\ni\nIn this specific case, the handedness of the two vortices reverses due to the shape change, which causes the entropy production\nrate to change sign (see new sentence in lines 417-418).\n4. On page 5, the authors mention the “disappearance of the point defects at the triple contact point” when extensile stresses\nbecome strong. I don’t see how that’s possible. As mentioned in my point 1 above, the choice of w = w = 0 for the surface\ns i\nwinding numbers must create a singularity in the director field at the contact point. How can a defect not be present there? This\nshould be explained. A zoomed-in picture showing the field lines in this region could be helpful to understand what’s going on\n(field lines are difficult to see in the current figures).\nResponse: We thank the Reviewer for identifying this issue and apologize for the oversight. The Reviewer is correct—the\ndefects at the triple contact point persist throughout the simulation, as shown in the zoomed-in view below. We have revised\nthat sentence in the Main Text of the manuscript (lines 276-277) and included the figure below as a supplementary figure, which\nshows zoomed-in views of the triple contact points and the thin-film regions for the cases (w, Ca ) = (0,−4.5),(0,4.5), and\ni α\n(1,−4.5).\n0 0.45\ny etanidrooC\n0 0.1 0.22\n0 0.1 0.25\n1\n0 10−3\nVelocity |u| 0.5 0.1\n0\n0\n−0.1\n0 1 2 0 1 2 2 2.2 2.4 2.6\nCoordinate x Coordinate x Coordinate x\n0 0.4 0.8\n0.1\n1 0 0.02\n0.5 0.1\n0 0 0\n−0.1 0 0.1 0 1 2 1 1.5 2\nCoordinate x Coordinate x Coordinate x\ny etanidrooC\na b\n0.02\n0\n0 0.1 0.2 0.3\nCoordinate x\nc\nMinor comments:\n5. The linear and quadratic scalings with capillary number observed in figures 3a and b are unsurprising, since the only\nforcing in the system is the active stress and the governing equations for the flow are linear.\nResponse: We agree with the Reviewer. To emphasize it, we have included the following sentence in corresponding section of\nthe revised manuscript (lines 387-391): These linear scaling laws make intuitive sense for |Ca |≲1, as the only forcing in the\nα\nsystem is the active stress, which is proportional to Ca , the fluid dynamics are governed by the linear Stokes equations, and\nα\nnonlinear effects due to drop deformation are small.\n6. The idea of controlling drop morphology by varying the anchoring boundary conditions are interesting. Can the authors\nspeculate how this could be achieved in experiments?\nResponse: We thank the Reviewer for raising this important point. Potential ways to control the anchoring conditions over time\nin experiments include:\n• Using suspensions of microtubules and light-activatable motor proteins. These motor proteins can crosslink and organize\nthe microtubules into different structures upon illumination, thereby enabling temporal control (see Refs. [1–3]). While\nthis approach may not directly control anchoring, it can modulate the microtubule structure and flow at the surface, making\nit a promising candidate for testing our ideas in surface-attached droplets.\n• Using stimuli-responsive surface treatments (e.g., pH-responsive polymer brushes or photoresponsive molecules grafted\nto substrates) which could allow for reversible switching between different anchoring conditions via chemical or light\nstimuli. Temperature-responsive polymers like PNIPAM could also provide thermal control over surface interactions.\n• For systems with electrically/magnetically responsive active particles (e.g., charged colloids, magnetic bacteria), external\nfields could dynamically reorient particles near interfaces.\n• Receptor-ligand interactions to control cell e.g., using different surface proteins or local chemical gradients to modulate\nbacterial adhesion and orientation at surfaces.\nWe have added a few sentences to the Discussion section (lines 478-498) highlighting how some of these approaches could be\nused to implement the proposed time-varying anchoring conditions.\n7. Can the authors speculate on how their findings might change in three-dimensional drops? What if the contact line was\nallowed to move?\nResponse: We thank the Reviewer for raising these interesting ideas. Considering three-dimensional drops introduces additional\npossibilities for the orientation of active units along the now two-dimensional substrate and liquid–air interface, as well as the\ninclusion of new terms in the free energy functional [4], both of which are likely to result in richer drop morphodynamics. One\nintriguing possibility is the emergence of non-axisymmetric shapes in 3D, or, for instance, Rayleigh-Plateu instabilities when\nthe droplet is extruded from of the substrate, as in cases where (w , w)=(0,1) and Ca =−4.5 in both Q and p descriptions\ns i α\n(Figs. 2 and S1). We have added a sentence in the Discussion section to highlight that this is an interesting direction for future\nresearch (lines 472-475).\nTo examine how our results change when the contact line is allowed to move, we explore this possibility by considering the\nfollowing Navier slip boundary condition at the solid substrate:\nu=ℓ eˆ ·[∇u+(∇u)T]·(I −eˆ eˆ )=ℓ ∂ ueˆ , (7)\ns y u y s y x\nwhere ℓ denotes the local slip length. We have performed new numerical simulations incorporating this boundary condition\ns\nfor one representative case in which the nematic is described by the director field p. This condition is (w ,w) = (0,1) and\ns i\nCa = 4.5. As expected, the drop can migrate only when symmetry is broken, and it does so in a treadmilling fashion.\nα\nVelocity |u|\nWe have included the figure attached below as a supplementary figure in the revised manuscript, and added a new section in the\nSupplementary Information describing how we model the substrate slip condition.\n0 0.45 0.9\nTime t\ny\netanidrooC\n1\n0.5\n0\n0 2 4\nCoordinate x\n[1] T.D.Ross,H.J.Lee,Z.Qu,R.A.Banks,R.Phillips, andM.Thomson,“Controllingorganizationandforcesinactivematterthrough\nopticallydefinedboundaries,”Nature572,224–229(2019).\n[2] R. Zhang, S.A. Redford, P.V. Ruijgrok, N. Kumar, A. Mozaffari, S. Zemsky, A.R. Dinner, V. Vitelli, Z. Bryant, M.L. Gardel, et al.,\n“Spatiotemporalcontrolofliquidcrystalstructureanddynamicsthroughactivitypatterning,”Nat.Mater.20,875–882(2021).\n[3] F. Yang, S. Liu, H.J. Lee, R. Phillips, and M. Thomson, “Dynamic flow control through active matter programming language,” Nat.\nMater.,1–11(2025).\n[4] M.C.Marchetti,J.-F.Joanny,S.Ramaswamy,T.B.Liverpool,J.Prost,M.Rao, andR.A.Simha,“Hydrodynamicsofsoftactivematter,”\nRev.Mod.Phys.85,1143(2013).\n[5] F.Stegemerten,K.John, andU.Thiele,“Symmetry-breaking,motionandbistabilityofactivedropsthroughpolarization-surfacecou-\npling,”SoftMatter18,5823–5832(2022).\n[6] M.BenAmarandL.J.Cummings,“Fingeringinstabilitiesindriventhinnematicfilms,”Phys.Fluids13,1160–1166(2001).\n[7] J.F.JoannyandS.Ramaswamy,“Adropofactivematter,”J.FluidMech.705,46–57(2012).\n[8] A.Loisy,J.Eggers, andT.B.Liverpool,“Tractionlessself-propulsionofactivedrops,”Phys.Rev.Lett.123,248006(2019).\n[9] S.Trinschek,F.Stegemerten,K.John, andU.Thiele,“Thin-filmmodelingofrestingandmovingactivedroplets,”Phys.Rev.E101,\n062802(2020).\n[10] A.Loisy,J.Eggers, andT.B.Liverpool,“Howmanywaysacellcanmove:themodesofself-propulsionofanactivedrop,”SoftMatter\n16,3106–3124(2020).\n[11] S.Shankar,V.Raju, andL.Mahadevan,“Optimaltransportandcontrolofactivedrops,”Proc.Nat.Acad.Sci.U.S.A.119,e2121985119\n(2022).\n[12] A.Ioratim-Uba,A.Loisy,S.Henkes, andT.B.Liverpool,“Thenonlinearmotionofcellssubjecttoexternalforces,”SoftMatter18,\n9008–9016(2022).\n[13] H.Z. Ford, G.L. Celora, E.R. Westbrook, M.P. Dalwadi, B.J. Walker, H. Baumann, C.J. Weijer, P. Pearce, and J.R. Chubb, “Pattern\nformationalongsignalinggradientsdrivenbyactivedropletbehaviourofcellgroups,”bioRxiv,2024–04(2024).\nReviewer 2 comments:\nThe present manuscript numerically investigates the steady-states of surface attached active drops. Activity arises from a stress-\nfield (contractile or extensile) related to a nematic order parameter. The authors study the drop shapes and internal velocity\nprofiles depending on a so-called “active capillary number” (an adimensional number which characterizes the strength of the\nactive stress compared to the capillary pressure) and the anchoring conditions of the nematic order parameter at the substrate and\nthe free interface. The originality of this manuscript lies in the systematic study of two-dimensional droplets of active liquid (aka\nactive liquid ridges) beyond the thin film limit. Here the variation of the directory field in the z-direction and the full velocity\nprofiles (in the Stokes limit) are explicitly resolved. The field of the nematic order parameter is treated in the strong elastic limit\nand is solved adiabatically as a boundary value problem coupled to the drop shape.\nOverall the manuscript is clearly laid out and nicely illustrated. My main criticism of this manuscript is the formulation of the\nboundary conditions for the director field, which are imposing a polarization of the liquid at the boundaries. In particular this\nchoice imposes a (i) symmetry-breaking, where the authors claim a spontaneous symmetry-breaking takes place. (ii) it is not\nclear to me, how the polarity is treated at the pinned triple contact points of the droplets, which constitute a singular point. The\nanchoring conditions are crucial to this manuscript. I strongly recommend the authors provide a clearer explanation.\nResponse: We thank the Reviewer for the time spent reading our paper, and for providing such thoughtful and constructive\nfeedback. It is gratifying that they found our paper to be clearly laid out and nicely illustrated. We also appreciate the Re-\nviewer’s suggestion for more rigorous consideration of the problem boundary conditions, which has helped us better understand\nthis active system and significantly improve the manuscript. We have followed all of their helpful suggestions in revising the\nmanuscript, as detailed in the responses below. In summary, we have:\n• Performed new simulations for a true nematic, in which the nematic field is described by a nematic tensor Q rather\nthan a director vector p. These simulations show that, although no spontaneous symmetry breaking emerges, the main\nresults obtained using p remain conceptually equivalent. We have added a new subsection discussing these new results, a\nSupplementary Information section describing this framework in detail, and a new Supplementary Information figure.\n• Performed new simulations showing that nematic advection effects have a minimal impact on the steady-state drop shapes\nand internal flows, supporting the assumption that considering the strong elastic limit while neglecting elastic stresses is\nreasonable. We have updated both the Main Text and Supplementary Information to highlight these new findings and have\nadded two new supplementary figures where we consider the advection of the nematic director field p and the nematic\ntensor Q.\n• Characterized the flow and nematic orientation near the triple contact points where the drop is pinned, as well as in thin-\nfilm regions for certain parameter values, showing that both are accurately resolved. We have added a new supplementary\nfigure showing zoomed-in views of these regions.\n• Conducted new simulations in which the drop is not pinned to the solid substrate, by introducing a finite slip length\nbetween the active liquid and the substrate to model friction. We show that, under conditions where the active drop breaks\nsymmetry, it can migrate in a treadmilling fashion. We have included a new supplementary figure showing this new result.\nWe believe these new results have improved our work significantly, and we are grateful to the Reviewer for their guidance in\nobtaining these new results.\n1. I have a major concern with the formulation of the boundary conditions for the order parameter p. While at the substrate\n(y = 0) the boundary condition is p = R ·eˆ , the anchoring condition at the free interface is expressed as p = R ·t, with R\ns x i i\nand R denoting rotation matrices and t denoting the tangent of the interface. How are the triple contact points treated? Taking\ns\nas example the initial condition with a semicircular shape, the imposed boundary conditions for p are ill posed and cannot be\nfulfilled. For example, ω = 0 implies an anchoring at the free surface of p = eˆ and p = −eˆ at the left and right triple\ni z z\npoint, respectively (assuming eˆ denotes a unit vector orthogonal to the substrate and the tangent vector at the free surface, t,\nz\nhas a direction as indicated in Fig. 1a), while at the same time a substrate anchoring with ω = 0 implies that p = eˆ all along\ns x\nthe substrate. The authors should state clearly how the triple contact points are treated. On page 5 the authors mention the\nappearance of point defects in the director field at the triple contact points for the anchoring conditions ω = ω = 0. How are\ns i\nthese defects consistent with the imposed boundary conditions?\n2. Another concern, still related to the anchoring conditions for the order parameter p, is the choice to specify the polar order\nat the interfaces, instead of only the nematic order. Since the authors refer in their manuscript only to active nematic\nparticles,\nhow can this choice be justified within the same framework? I may be wrong, but the symmetry-breaking observed in Fig. 2a\nfor ω = 0 and ω = 1 is imposed by the rigid (and to some point pathological) anchoring of p to the two different interfaces\ns i\nand is not a spontaneous process as claimed by the authors on page 5 (left column, first paragraph).\nResponse: We thank the Reviewer for raising these important points, which we appreciate having the opportunity to clarify. We\nfully agree with the Reviewer that at the triple contact points, the substrate and interface boundary conditions cannot be simul-\ntaneously satisfied in a continuous manner. Our model handles this incompatibility by allowing topological defects to naturally\nemerge at these locations. Indeed, the resolution of this apparent inconsistency is that when w = w = 0, the boundary condi-\ns i\ntions for p to accommodate such surface orientations introduce a defect at the contact line at all times—i.e., this combination\nof winding numbers results in a singularity in the nematic director field p where the director field becomes undefined. But we\nemphasize that this defect at the pinned contact line is perfectly consistent with the imposed boundary conditions for p. These\nboundary conditions are imposed to the equations for both components of the director field, ∇2p = 0, and ∇2p = 0, where\nx y\np=(p ,p ).\nx y\nAn important aspect to emphasize here is that p represents a nematic director field, that is, we describe the active nematic using\na vector director. However, the system is not polar: the two orientations p and -p represent the same physical state, as both the\nactive stress and the relaxation equation [Eqs. (2) and (4) in the manuscript] are invariant under the transformation p → −p.\nThat said, describing a nematic field using the vector p is not always equivalent to using the nematic tensor Q.\nTo better clarify the differences between both frameworks, we have conducted new numerical simulations in which the nematic is\ndescribed by Q. We consider the following equation for Q, under the same assumption as in Eq. (4) of the manuscript—nematic\norder is established much more quickly than the flows generated within the drop:\n0= (a−btr(Q2)) Q+K ∇2Q, (8)\nQ\nwhere Q = S (pp− 1I), with S a scalar representing the strength of the order parameter, K the Frank elastic constant, and\na and b parameters from a Landau expansion in the order parameter that determine the isotropic–nematic transition (a > 0\nin the nematic phase and b > 0 for stability). These parameters define the maximum value the order parameter can take,\nS = 2a/b. Using Q, the active stress tensor becomes σ =−αQ. To make Eq. (8) dimensionless, we choose Q = a/b\nmax a c\nas characteristic quantity for the nematic tensor, which yields:\n0= (1−tr(Q˜2)) Q˜ +K˜∇2Q˜, (9)\nwhere Q˜ = S˜(pp− 1I) is the dimensionless nematic tensor, K˜ = (K /a)/R2 is a dimensionless parameter that compares\nthe elastic coherence length ℓ = K /a and the drop radius R, and S˜ = S/ a/b is the dimensionless strength of the\nQ Q\nnematic order parameter. Equation (3) of the manuscript describing the fluid stress tensor reads in dimensionless form:\nσ˜ =−Π˜I +∇u˜+(∇u˜)T−Ca Q˜, (10)\nα\nwhere, in this case, the active Capillary number reads: Ca ≡α a/b/(γ/R).\nα\nTo compare with the results of the Main Text solving for p, we consider both planar and orthogonal anchoring of the nematic\nwith the solid substrate, i.e., Q˜ = S˜(eˆ eˆ − 1I) and Q˜ = S˜(eˆ eˆ − 1I), respectively. At the liquid–air interface, we\nx x 2 y y 2\nimpose either tangential anchoring, Q˜ =S˜(tˆtˆ− 1I), or orthogonal anchoring, Q˜ =S˜(nˆnˆ − 1I), where tˆ and nˆ are the unit\n2 2\ntangential and normal vectors to the liquid–air interface, respectively. For simplicity, in all simulations we set S˜ = 1 at the\nboundaries, and K˜ =1. The figure attached below depicts the results of numerical simulations. Panel a is equivalent to Fig. 2a\nof the manuscript, corresponding to planar anchoring with the substrate (w = 0) and both tangential (w = 0) and orthogonal\ns i\n(w = 1) anchoring with the liquid-air interface. Panel b is equivalent to Fig. 2b of the manuscript, where we show that the\ni\nsteady states of a surface-attached active drop for planar (w =0) and orthogonal (w =1) unit anchoring with the substrate are\ns s\nequivalent under the transformations w →w+π/2 and Ca →−Ca . Here,|Ca |=2.5.\nα α α\nContractile Extensile\niw\nrebmun\ngnidniW\na\nwi = 1\nws = 0\nwi = 0\nws = 0\nCoordinate x Coordinate x Coordinate x Coordinate x\nActive capillary number Caα\nb π/2 &Caα→–Caα\n≡ ≡ Extensile Contractile\n≡\nContractile Extensile\n≡\ny etanidrooC\ny etanidrooC\n0 0.0250.05\nVelocity |u|\n3\n2 0 0.025 0.05 0 0.015 0 0.035\n1\n1\n0\n0 1 2 0 1 2 0 1 2 0 1 2\n3\n2 0 0.025 0.057 0 0.025 0 0.025 0.05 0 0.05 0.11\n0\n1\n0\n0 1 2 0 1 2 0 1 2 0 1 2\n-4.5 -2.5 2.5 4.5\nCoordinate x Coordinate x Coordinate x Coordinate x\ny etanidrooC\ny etanidrooC\n0 0.0250.05\nVelocity |u| 3\n2 0 0.025 0.05 0 0.015 0 0.035 1\n1\n0\n0 1 2 0 1 2 0 1 2 0 1 2\n3\n2 0 0.025 0.057 0 0.025 0 0.025 0.05 0 0.05 0.11 0\n1\n0\n0 1 2 0 1 2 0 1 2 0 1 2\n-4.5 -2.5 2.5 4.5\nCoordinate x Coordinate x Coordinate x Coordinate x\ny etanidrooC\ny etanidrooC\n0 0.0250.05 Velocity |u|\n3\n2 0 0.025 0.05 0 0.015 0 0.035 1\n1\n0\n0 1 2 0 1 2 0 1 2 0 1 2\n3\n2 0 0.025 0.057 0 0.025 0 0.025 0.05 0 0.05 0.11 0\n0\n0 1 2 0 1 2 0 1 2 0 1 2\n-4.5 -2.5 2.5 4.5\ny etanidrooC\ny etanidrooC\n0 0.0250.05 Velocity |u| 3\n2 0 0.025 0.05 0 0.015 0 0.035 1\n1\n0\n0 1 2 0 1 2 0 1 2 0 1 2\n3\n2 0 0.025 0.057 0 0.025 0 0.025 0.05 0 0.05 0.11 0\n1\n0\n0 1 2 0 1 2 0 1 2 0 1 2\n-4.5 -2.5 2.5 4.5\nCoordinate x Coordinate x Coordinate x Coordinate x\ny etanidrooC\ny etanidrooC\n0 0.0250.05\nVelocity |u|\n3\n2 0 0.025 0.05 0 0.015 0 0.035 1\n1\n0\n0 1 2 0 1 2 0 1 2 0 1 2\n3\n2 0 0.025 0.057 0 0.025 0 0.025 0.05 0 0.05 0.11\nC0oordinate x Coordinate x Coordinate x Coordinate x\n1\n0\n0 1 2 0 1 2 0 1 2 0 1 2\n-4.5 -2.5 2.5 4.5\ny etanidrooC\ny etanidrooC\nVelocity |u|\n0 0.025 0.05 0 0.025 1\n0 1 2 0 1 2\n0 0.025 0 0.025 0.05 0\n0 1 2 0 1 2\n-4.5 -2.5 2.5 4.5\nCoordinate x Coordinate x Coordinate x Coordinate x\ny etanidrooC\ny etanidrooC\nVelocity |u|\n0 0.025 0.05 0 0.025 1\n0 1 2 0 1 2\nCoordinate x Coordinate x Coordinate x Coordinate x\n0 0.025 0 0.025 0.05\n0\n0 1 2 0 1 2\n-4.5 -2.5 2.5 4.5\ny etanidrooC\ny etanidrooC\nVelocity |u|\n0 0.025 0.05 0 0.025 1\n0 1 2 0 1 2\n0 0.015 0 0.025 0.05 0 Coordinate x Coordinate x Coordinate x Coordinate x\n0 1 2 0 1 2\n-4.5 -2.5 2.5 4.5\ny etanidrooC\ny etanidrooC\nVelocity |u|\n0 0.025 0.05 0 0.025 1\n0 1 2 0 1 2\n0 0.015 0 0.025 0.05 0\n0 1 2 0 1 2\n-4.5 -2.5 2.5 4.5\nThe figure above shows that, within this framework, there are only two anchoring conditions for a fixed substrate anchoring\n(w =0 in this case), i.e., w = 0 and w = 1. Hence, while symmetry breaking is spontaneous and not the result of pathological\ns i i\nboundary conditions for p, it arises as a consequence of describing the nematic using a director vector field p. However, the\nfact that planar (w = 0) and orthogonal (w = 1) substrate anchoring are equivalent under the transformations w → w+π/2\ns s\nand Ca → −Ca is also true when the nematic is described by Q. Moreover, the figure above shows that for w = 0 and\nα α s\nw = 0, the same defect in the nematic orientation appears at the triple contact point as when the orientation is described by\ni\np, due to the imposed orientations at the substrate and liquid–air interface. We have revised the Main Text of the manuscript\nto include a careful discussion of both frameworks and the corresponding results (see new subsection in the Main Text, lines\n338–378). Additionally, we have added a new section to the Supplementary Information describing the derivation and non-\ndimensionalization of the equations above. The figure above has also been included in the Supplementary Information. We\nappreciate the Reviewer’s thoughtful questions, and have now explicitly clarified all these points in the revised manuscript,\nguided by their feedback.\n3. As far as I understood the triple contacts points are pinned to the substrate. A priori I do not have problem with this\nlimitation. However, the obtained steady-state drop shapes (Fig. 2a) are intriguing. Especially for the depicted combination of ω = 0,\ns\nω = 1 at high contractile stress the droplet contracts strongly close to the left contact point and is only attached via a long flat\ni\nfoot to the right contact point. What do the velocity field and the director field look like in these flat extensions? A supple-\nmentary figure which shows the drop profile, velocity and p fields in the vicinity of the triple contact points for the parameter\ncombinations depicted in Fig. 2a is necessary. This could also clarify the impact of the anchoring conditions on the velocity and\ndirector field (see concerns raised above).\nResponse: We thank the Reviewer and apologize for not having carefully characterized and visualized the velocity and director\nfields near the contact point. The reviewer is also correct regarding the boundary conditions—the contact line is pinned to the\nsubstrate, as specified by the boundary condition described on line 160 of the manuscript: u = 0 at y = 0. Following their\nadvice, we have included the figure below as supplementary figure, which shows zoomed-in views of the triple contact points\nand the thin-film regions for the cases (w, Ca ) = (0,−4.5),(0,4.5), and (1,−4.5). Moreover, we have explicitly mentioned\ni α\nthat the drop is pinned to the substrate in line 101 of the revised manuscript.\n0 0.45\ny etanidrooC\n0 0.1 0.22\n0 0.1 0.25\n1\n0 10−3\nVelocity |u| 0.5 0.1\n0\n0\n−0.1\n0 1 2 0 1 2 2 2.2 2.4 2.6\nCoordinate x Coordinate x Coordinate x\n0 0.4 0.8\n0.1\n1 0 0.02\n0.5 0.1\n0 0 0\n−0.1 0 0.1 0 1 2 1 1.5 2\nCoordinate x Coordinate x Coordinate x\ny etanidrooC\na b\n0.02\n0\n0 0.1 0.2 0.3\nCoordinate x\nc\n4. Fig. 2a (bottom row, extensile stress, Ca = 4.5): The two lobes of the droplet extend into the substrate. Please comment\nα\nhow this is possible? I assume the problem can be resolved by stating that the substrate only extends from x = 0 to x = 2?\nResponse: We thank the Reviewer for bringing up this important point and apologize for not being clear in the original\nmanuscript regarding the extension of the substrate. The Reviewer is correct, the substrate only extends from x = 0 to x = 2,\nand therefore the liquid–air interface does not interact with the solid substrate at any point during the simulation. To clarify this\nissue, we have explicitly mentioned that the substrate only extends from 0≤x≤2R in lines 101-102 of the revised manuscript.\n5. In Fig. 3: I am not sure I fully understand the physical relevance of the depicted scaling relations, can the authors make an\nattempt at explaining the scaling? Also, the curves do not look very smooth, it would be more appropriate to show a symbol\nat the parameters, where the simulations have been performed. The entropy production rate for extensile stresses and\nwinding number ω = 0 changes abruptly at Ca ≈ 1. At the same moment the linear scaling for |u| breaks down. Please comment.\ni α max\nIs this change related to a qualitative change in the drop profile, velocity field, etc? Finally, the steady states shown in Fig. 2a\ncorrespond to an active capillary number, where the scaling laws are not valid anymore. Please include a (supplementary)\nfigure with a capillary number where the scaling is valid (e.g., Ca = 0.1).\nα\nResponse: We thank the Reviewer for bringing up these points. First, the linear scaling of the velocity arises from the fact that\nthe only forcing in the system is the active stress, and the governing Stokes equations for the flow are linear. This scaling holds\nonly when the drop does not deform significantly—that is, at sufficiently small values of the active Capillary number. At higher\nvalues of Ca and thus larger deformations, nonlinearities introduced by a non-semispherical shape cause the breakdown of the\nα\nlinear scaling. We have revised the main text of the manuscript to clarify these points (see lines 387-391).\nRegarding the abrupt change in entropy production rate and |u| at Ca ≈ 1 for ω = 0—this is a consequence of the significant\nmax α i\nchange in droplet shape as capillary pressure becomes comparable to active stresses. In this specific case, the handedness\nof the two vortices reverses due to the shape change, which causes the entropy production rate to change sign. To clarify this\npoint, we have included a new steady-state snapshot in Fig. 3 for Ca = 0.5 and added a corresponding sentence in the revised\nα\nmanuscript (lines 417-418).\nWe have also followed the Reviewer’s advice and added symbols to the curves to indicate the parameter values at which the\nsimulations were conducted. Additionally, we have conducted more simulations to make curves smoother.\nContractile\nWinding ∼|Ca| α number w i 0 1 2\nActive Capillary number |Ca|\nα\n|u|\nyticolev\nmumixaM\nxam Extensile 1\n∼|Ca| 10−3 α\n10−6\n∼|Ca|2\nα\ns etar\n.dorp\nyportnE\n)+(\ntot\n1\n10−3\n∼|Ca|2\nα\n10−6\n10−9\n∼|Ca|2 α s etar\n.dorp\nyportnE\n)–(\ntot\nContractile Extensile\n10−9\n∼|Ca|2 10−6 α\n10−3\n1\n102 101 100 10−1 10−2 10−3 10−3 10−2 10−1 100 101 102\nw rebmun gnidniW\nw = 2\nw = 1\nw = 0\nCoordinate x Coordinate x Coordinate x Coordinate x\nActive capillary number Caα\ny etanidrooC\ny etanidrooC\ny etanidrooC\nContractile Extensile 0 0.07 0.14 0 0.05 0.09 0 0.1 0.17 0 0.1 0.23 Velocity |u| 1 2 0.5\n0\n0 1 2 0 1 2 0 1 2 0 1 2\n0 0.4 0.8 0 0.15 0.34 0 0.1 0.19 0 0.15 0.35 1 1 0.5\n0 0 1 2 0 1 2 0 1 2 0 1 2\n0 0.1 0.22 0 0.03 0.07 0 0.02 0.05 0 0.1 0.25 1 0 0.5\n0 0 1 2 0 1 2 0 1 2 0 1 2\n-4.5 -2.5 2.5 4.5\nw\nrebmun gnidniW\nw = 2\nw = 1\nw = 0\nCoordinate x Coordinate x Coordinate x Coordinate x\nActive capillary number Caα\ny etanidrooC\ny etanidrooC\ny etanidrooC\nContractile Extensile\n0 0.07 0.14 0 0.05 0.09 0 0.1 0.17 0 0.1 0.23 Velocity |u| 1 2 0.5\n0 0 1 2 0 1 2 0 1 2 0 1 2\n0 0.4 0.8 0 0.15 0.34 0 0.1 0.19 0 0.15 0.35 1 1 0.5 0\n0 1 2 0 1 2 0 1 2 0 1 2\n0 0.1 0.22 0 0.03 0.07 0 0.02 0.05 0 0.1 0.25 1 0 0.5\n0\n0 1 2 0 1 2 0 1 2 0 1 2\n-4.5 -2.5 2.5 4.5\nw\nrebmun\ngnidniW\nw = 2\nw = 1\nw = 0\nCoordinate x Coordinate x Coordinate x Coordinate x\nActive capillary number Caα\ny etanidrooC\ny etanidrooC\ny etanidrooC\nCo0ntr0a.0c7tile0.14 Velocity |u| 0 0.05 0.09 Exte 0 nsile 0.1 0.17 0 0.1 0.23 1 2 0.5 0\n0 1 2 0 1 2 0 1 2 0 1 2\n0 0.4 0.8 0 0.15 0.34 0 0.1 0.19 0 0.15 0.35\n1 1 0.5 0 0 1 2 0 1 2 0 1 2 0 1 2\n0 0.1 0.22 0 0.03 0.07 0 0.02 0.05 0 0.1 0.25\n1 0 0.5 0\n0 1 2 0 1 2 0 1 2 0 1 2\n-4.5 -2.5 2.5 4.5\nw rebmun gnidniW\nw = 2\nw = 1\nw = 0\nCoordinate x Coordinate x Coordinate x Coordinate x\nActive capillary number Caα\ny etanidrooC\ny etanidrooC\ny etanidrooC\n0 0.07 0.14 0 0.05 0.09 0 0.1 0.17 0 0.1 0.23 Velocity |u| 1 2 0.5\n0\n0 1 2 0 1 2 0 1 2 0 1 2\n0 0.4 0.8 0 0.15 0.34 0 0.1 0.19 0 0.15 0.35 a 1 1 0.5\n0 0 1 2 0 1 2 0 1 2 0 1 2\n0 0.1 0.22 0 0.03 0.07 0 0.02 0.05 0 0.1 0.25 1 0 0.5\n0 b 0 1 2 0 1 2 0 1 2 0 1 2 Coordinate x Coordinate x Coordinate x Coordinate x\n-4.5 -2.5 2.5 4.5\ny etanidrooC\ny etanidrooC\ny etanidrooC\nVelocity |u| 2\n1\n0 10−3 3×10−3 0\n0 1 2\n-4.5 -2.5 0.5 4.5\n6. Ifindthediscussionofpreviousworks(thinfilmmodels, e.g., onpage5, rightcol., firstparagraph)abithandwavingand\ntoosimplistic. Alsothethree-dimensionalphase-fieldmodelbyTjhungetal. (Nat. Comms. 2015,doi: 10.1038/ncomms6420)\nshoulddiscussedmorethoroughly,sincetheredropletmotionisfoundforcontractiledropletsintheabsenceofself-propulsion.\nFurthermore, the thin film model by Stegemerten et al. (Soft Matter 2022, doi: 10.1039/d2sm00648k) identifies bistable be-\nhavior for droplets with active stresses. I agree that most thin film models (like the one from Stegemerten et al.) do not take\nintoaccountthez-dependenceofthedirectorfield. However,there,symmetryisbrokenspontaneouslyandnotbytheimposed\nsurfaceanchoring.\nResponse: We thank the Reviewer for raising this important point and apologize for not providing a sufficiently thorough\ndiscussion of previous studies, especially the work by Tjhung et al. (Nat. Commun. 2015). We have revised the Discussion\nsection to include a more balanced consideration of related models (lines 472–475).\nRegarding the work of Stegemerten et al. (Stegemerten et al. Soft Matter, 2022 [5]) and similar thin-film models of surface-\nattached active nematic drops (e.g., [5–13]), it is the opposite—the nematic director is allowed to vary only in the vertical y\ndirection and is assumed to remain uniform along the substrate (x direction). This assumption is central to the thin-film (or\nlubrication) approximation: owing to the slenderness of the drop, gradients of p and u·eˆ in the vertical direction dominate,\nx\nand thus only these terms are retained, with errors of order O[(∂ h)2], where h is the height of the drop. Consequently, sym-\nx\nmetry breaking also arises due to surface anchoring in the thin-film approximation (see e.g., [8]), which makes sense, since the\nthin-film equation is derived from the full equations considered in our work. However, the strong assumptions of the thin-film\napproximation lead to notable differences with respect to the complete equations when ∂ h ∼ 1. For example, due to the\nx\nassumption of p being x-independent, when the nematic units are tangentially aligned with both the substrate and the free\ninterface (w =w =0), no flow is generated inside the drop and it remains undeformed, unlike in the full model. Additionally,\ns i\nin the thin-film framework, the droplet shape is identical for both w = 1 and w = 2, despite differences in internal flow and\ni i\nnematic organization [8]. This outcome is an artifact of neglecting variations along x and the vertical component of the flow\nu·eˆ , and depth-averaging to obtain an equation for h. We extensively discussed the limitations of the thin-film approximation\ny\nand compare thin-film predictions (e.g., from [8]) with results from the full equations in several parts of the original manuscript\n(see e.g., section Surface-attached active drops reach stable steady states, Surface-attached active drops adopt a rich array of\nequilibrium shapes and flows, or lines 193-205 of the revised manuscript).\nRegarding the work by Tjhung et al. (Nat. Comms. 2015), one of the key differences from our study is that they consider\nan active polar suspension, rather than an active nematic suspension. Although we cited this work in the original manuscript,\nwe did not discuss it in detail due to this distinction. Moreover, they considered additional complexities that we disregarded,\nsuch as effective friction with the substrate introduced via a finite slip length, which lead to drop self-propulsion. In our case,\nintroducing a slip condition at the solid substrate similarly results in drop self-propulsion (as previously found in the thin-film\nlimit, e.g., [8]). We explore this effect in the full equations by considering the following Navier slip boundary condition at the\nsolid substrate:\nu=ℓ eˆ ·[∇u+(∇u)T]·(I −eˆ eˆ )=ℓ ∂ ueˆ , (11)\ns y u y s y x\nwhere ℓ denotes the local slip length. We have performed new numerical simulations incorporating this boundary condition\ns\nfor one representative case in which the nematic is described by the director field p. This condition is (w ,w) = (0,1) and\ns i\nCa = 4.5. As expected, the drop can migrate only when symmetry is broken, and it does so in a treadmilling fashion,\nα\nreaching a constant traveling-wave velocity. We have included the figure attached below as a supplementary figure in the revised\nmanuscript, and added a new section in the Supplementary Information describing how we model the substrate slip condition.\nVelocity |u|\n0 0.45 0.9\nTime t\ny\netanidrooC\n1\n0.5\n0\n0 2 4\nCoordinate x\nIn addition, Tjhung et al. (Nat. Comms. 2015) also consider elastic stresses, fluid inertia both inside and outside the drop,\nand advection of the polar field, all of which we disregarded in our work. In the original manuscript, we considered the strong\nelastic limit for the director field p but neglected nematic elastic stresses in the momentum conservation equation. To address\nthe role of nematic hydrodynamic advection and elastic relaxation, we conducted simulations in which the strong elastic limit\nis relaxed—that is, we included nematic advective terms, and the only stresses acting on the fluid are active, with no elastic\ncontributions. If we describe the nematic through the tensor Q˜, Eq. (9) now reads:\n(Γ˜ ∂ Q˜ +u˜·∇Q˜ −(Q˜ ·Ω˜ −Ω˜ ·Q˜) = (1−tr(Q˜2)) Q˜ +K˜ ∇2Q˜, (12)\nQ t˜ Q\nwhere 2Ω˜ = ∇u˜ −(∇u˜)T is the dimensionless vorticity tensor, and Γ˜ ≡ (Γ /µ)γ/(aR) is a dimensionless parameter that\nQ Q\ncompares the nematic rotational viscosity Γ with the liquid viscosity µ. We conducted new numerical simulations solving\nQ\nEq. (12), considering Γ˜ = 1. While the transient dynamics with nematic advection differ from those in the strong-elastic limit,\nQ\nthe final steady-state shapes and flows that the drop adopts are nearly identical in both cases, as shown by the figure attached\nbelow.\nContractile Extensile\niw\nrebmun\ngnidniW\nStrong-elastic limit With nematic advection\nContractile Extensile\nwi = 1\nws = 0\nwi = 0\nws = 0\nCoordinate x Coordinate x Coordinate x Coordinate x\nActive capillary number Caα\ny\netanidrooC\ny etanidrooC\n0 0.025 0.05 0 0.015 0 0.025 0.05 0 0.015\n2\n1\n1\n0\n0 1 2 0 1 2 0 1 2 0 1 2\n3\n2 0 0.025 0 0.025 0.05 0 0.025 0 0.025 0.05\n0\n1\n0\n0 1 2 0 1 2 0 1 2 0 1 2\n-2.5 2.5 -2.5 2.5\nWe conduct the same calculations but now describing the nematic through the director field p, as in the original manuscript,\nwhere now Eq. (4) of the manuscript reads:\n(Γ˜ ∂ p+u˜·∇p+Ω˜ ·p =∇2p, (13)\nt˜\nwhere Γ˜ ≡ (ΓR2/K)/t = (ΓR2/K)/(µR/γ), is a dimensionless parameter that compares the characteristic elastic relax-\nc\national time scale of the nematic with the viscous characteristic time scale of the liquid t = µR/γ. Similarly to the case of\nc\nEq. (12) of this rebuttal, here the transient dynamics considering p advection are also different compared to the strong-elastic\nlimit, but the final steady-state shapes are identical in both cases, as shown in the figure attached below.\nContractile Extensile\niw\nrebmun\ngnidniW\nStrong-elastic limit With nematic advection\nContractile Extensile\nwi = 1\nws = 0\nwi = 0\nws = 0\nCoordinate x Coordinate x Coordinate x Coordinate x\nActive capillary number Caα\ny etanidrooC\ny etanidrooC\ny etanidrooC\n0 0.05 0.09 0 0.1 0.17 0 0.05 0.09 0 0.1 0.17\nVelocity |u|\n1 wi = 2\n2\n0\n0 1 2 0 1 2 0 1 2 0 1 2\nws = 0\n0 0.15 0.34 0 0.1 0.19 0 0.15 0.32 0 0.1 0.19\n1\n1\n0\n0 1 2 0 1 2 0 1 2 0 1 2\n0 0.03 0.07 0 0.02 0.05 0 0.03 0.07 0 0.02 0.05\n1\n0\n0\n0 1 2 0 1 2 0 1 2 0 1 2\n-2.5 2.5 -2.5 2.5\nWe have included the two figures above as supplementary figures, added supplementary sections describing these results in\ndepth, and included a new sentence summarizing the main findings in the revised Main Text (lines 370-378).\nMinor remarks/typos:\n7. The use of the term “active suspension” in the abstract (line 4) is misleading since it implies the presence of concentration\nfluctuations of the suspended particles, which is not considered in the manuscript. “Active liquid” is maybe more appropriate?\nResponse: We agree with the Reviewer and have replaced the term “active suspension” with “active fluid” in the abstract.\n8. Fig. 2 caption: “... and the grey curves depict the orientation of the active units.” Some curves appear grey, but most are red.\nResponse: We thank the Reviewer for pointing out the typo, which we have corrected in the revised manuscript.\n9. Fig. 3: The color code is misleading. The font colors for “Extensile” and “Contractile” correspond to the colors used for\ncurves with winding number 1 and 2.\nResponse: We thank the Reviewer for pointing out the confusion caused by the color scheme. We have revised Fig. 3 using a\nblack-and-white color palette for clarity.\n[1] T.D.Ross,H.J.Lee,Z.Qu,R.A.Banks,R.Phillips, andM.Thomson,“Controllingorganizationandforcesinactivematterthrough\nopticallydefinedboundaries,”Nature572,224–229(2019).\n[2] R. Zhang, S.A. Redford, P.V. Ruijgrok, N. Kumar, A. Mozaffari, S. Zemsky, A.R. Dinner, V. Vitelli, Z. Bryant, M.L. Gardel, et al.,\n“Spatiotemporalcontrolofliquidcrystalstructureanddynamicsthroughactivitypatterning,”Nat.Mater.20,875–882(2021).\n[3] F. Yang, S. Liu, H.J. Lee, R. Phillips, and M. Thomson, “Dynamic flow control through active matter programming language,” Nat.\nMater.,1–11(2025).\n[4] M.C.Marchetti,J.-F.Joanny,S.Ramaswamy,T.B.Liverpool,J.Prost,M.Rao, andR.A.Simha,“Hydrodynamicsofsoftactivematter,”\nRev.Mod.Phys.85,1143(2013).\n[5] F.Stegemerten,K.John, andU.Thiele,“Symmetry-breaking,motionandbistabilityofactivedropsthroughpolarization-surfacecou-\npling,”SoftMatter18,5823–5832(2022).\n[6] M.BenAmarandL.J.Cummings,“Fingeringinstabilitiesindriventhinnematicfilms,”Phys.Fluids13,1160–1166(2001).\n[7] J.F.JoannyandS.Ramaswamy,“Adropofactivematter,”J.FluidMech.705,46–57(2012).\n[8] A.Loisy,J.Eggers, andT.B.Liverpool,“Tractionlessself-propulsionofactivedrops,”Phys.Rev.Lett.123,248006(2019).\n[9] S.Trinschek,F.Stegemerten,K.John, andU.Thiele,“Thin-filmmodelingofrestingandmovingactivedroplets,”Phys.Rev.E101,\n062802(2020).\n[10] A.Loisy,J.Eggers, andT.B.Liverpool,“Howmanywaysacellcanmove:themodesofself-propulsionofanactivedrop,”SoftMatter\n16,3106–3124(2020).\n[11] S.Shankar,V.Raju, andL.Mahadevan,“Optimaltransportandcontrolofactivedrops,”Proc.Nat.Acad.Sci.U.S.A.119,e2121985119\n(2022).\n[12] A.Ioratim-Uba,A.Loisy,S.Henkes, andT.B.Liverpool,“Thenonlinearmotionofcellssubjecttoexternalforces,”SoftMatter18,\n9008–9016(2022).\n[13] H.Z. Ford, G.L. Celora, E.R. Westbrook, M.P. Dalwadi, B.J. Walker, H. Baumann, C.J. Weijer, P. Pearce, and J.R. Chubb, “Pattern\nformationalongsignalinggradientsdrivenbyactivedropletbehaviourofcellgroups,”bioRxiv,2024–04(2024).\nReviewer 2 comments:\nThe authors have undertaken extensive numerical calculations to investigate i.e., the effect of the (formulation of the) boundary\nconditions, elastic stresses and advection of the nematic field on the steady state profiles (Figs. S1 - S3). I feel that all points\nfrom my previous report, except for point 2, have been sufficiently addressed in the revised manuscript.\nConcerning point 2 from my previous report, I agree with the authors that the bulk equations respect nematic symmetry. How-\never, the boundary conditions do not, since they explicitly expressed by Dirichlet conditions on the polarity field p. Furthermore,\nsince the droplet is not self-polarizing, polarization in the bulk only arises from the boundary conditions (free boundary and\ncontact with the substrate).\nFrom the mathematical formulation (Eq. 5b) and line 163 it is clear that the boundary condition act directly on the polarization\nand do not respect nematic symmetry. However, this important detail is not sufficiently highlighted in the accompanying text\nnor is it clearly stated in the new paragraph (lines 338–378, Describing the nematic field using a nematic tensor). The double\narrow for p used in Fig. 1 are misleading. I might be wrong, but my impression is that the loss of symmetry-breaking for the\ncase (ω = 0, ω = 1, Fig. S1) can be attributed to the formulation of the boundary conditions respecting nematic symmetry\ns i\nand not to the introduction of the nematic tensor Q for the bulk equations. The authors should discuss more openly the origin of\nsymmetry-breaking for the case (ω = 0, ω = 1) in the paragraph dedicated to Fig. 2a (line 282 onwards). Furthermore, Fig. 1\ns i\nshould be modified to reflect the true nature of the boundary conditions.\nResponse: We thank the Reviewer for the time spent reading the new version of our paper, and we are pleased that they found\nall their points were addressed and that the manuscript improved considerably after implementing their suggestions. We also\nthank them again for providing new, thoughtful, and constructive feedback that has further improved our work. In particular, we\nagree with the Reviewer that, in the p-description of the nematic, the active units become polar at the boundaries by virtue of\nthe imposed boundary conditions. We also agree that this specific point was not sufficiently emphasized in the previous version\nof the manuscript as a necessary condition for symmetry breaking. To address this issue, we have followed all of their helpful\nsuggestions in revising the manuscript, as detailed in the responses below. In summary, we have:\n• Revised both the main text and the Supplementary Information to clarify that the results shown in the main text correspond\nto the p-description, in which the active units are polarized at the drop boundaries due to the imposed boundary condition\n(Eq. 5b). These changes are highlighted in blue in lines 70–73, 107–110, Eq. 5, 158–163, 295, 349–356, 379–381,\n729–735, and in the Fig. 1 caption.\n• Modified Figs. 1, 2, and 4 to highlight the direction of polarization of the active units at the drop boundaries.\nFurthermore I have a technical question concerning Fig. S1a (case ω = 1). The red lines marking the orientation of the active\ni\nunits indicate the presence of a defect on the free boundary in the center of the droplet at position x = 1. Is that impression\ncorrect? I would have expected that a formulation of the boundary conditions in terms of a Dirichlet condition on Q with S˜=1\n(lines 730–745) does not lead to a defect on the boundary. Please comment or rectify.\nResponse: We thank the Reviewer for this insightful question. The observation is correct—a defect indeed appears at the center\nof the liquid–air interface. This arises because the imposed boundary conditions enforce the nematic director to be parallel to the\nsubstrate and orthogonal to the liquid–air interface. Satisfying both anchoring conditions simultaneously requires the presence\nof a defect at the interface center to accommodate the resulting mismatch in orientation.\nMinor points:\nPlease make sure that all supplementary figures are referenced from the main text, e.g., it seem that Fig. S4 is not referenced\nfrom the main text.\nResponse: We thank the Reviewer for pointing out this issue. All Supplementary Figures are now referenced either in the\nSupplementary Information or in the main text (e.g., see lines 847-849).\nPlease indicate the winding number at the substrate ω in the caption of Fig. S4.\ns\nResponse: We have indicated the substrate winding number (w = 0) in the updated caption of Fig. S4.\ns\nWe believe that with the improvements we have made, the manuscript is now suitable for publication in Nature Communications.\nThank you for your consideration.\nReviewer 4 comments:\nReviewer 1 has raised a couple of valid and important points. I believe the authors have adequately addressed most of these\ncomments by extending their model to incorporate advection terms and a Q-tensor description, as well as by performing addi-\ntional simulations and analysis. However, I am still confused with the new results which they presented as part of the response\nto Comment 1 and I think the authors should provide more clarifications to my concerns below.\nNematic liquid crystals can both be described by a director field: p(r,t) (using the same notation as in the manuscript) or by a\nQ-tensor: Q(r,t). The dynamics for p can be described by the Leslie-Ericksen equation (line 131 in the manuscript), while the\ndynamics for Q can be described by the Beris-Edwards equation (line 343 in the manuscript). Far from any topological defects,\nsimulations using either p(r,t) or Q(r,t) will give identical results, and one can interpret p as the largest eigenvector of the\nQ-tensor.\nIn the revised manuscript, the authors performed additional simulations using a Q-tensor description to compare with the\noriginal results (done using director field p). The results for w = 0 and w = 1 are qualitatively different for both Q-tensor\ns i\ndescription and p-field description. In particular the result with p-field has a symmetry-breaking steady state which can cause\nthe droplet to migrate. On the other hand, the steady state solution for Q-tensor remains symmetric about the central axis. I don’t\nthink the authors have explained sufficiently why they have very significant difference. Is it due to the fact that in p-description,\nthe p-field can form +1 defects at the contact lines and these defects become unstable in Q-tensor description?\nResponse: We thank the Reviewer for the time and effort spent carefully reading our revised manuscript and for acknowledging\nour extensions to include advection terms, a Q-tensor formulation, and additional simulations and analyses. We appreciate their\nthoughtful comments and the opportunity to clarify the origin of the differences between the p- and Q-based descriptions.\nIn the p-description, the boundary conditions explicitly impose the orientation of the active units at both the solid and liquid–air\ninterfaces, which leads to the units becoming polar at the drop boundaries. A mismatch between these boundary orientations\nconstitutes a necessary condition for symmetry breaking. In contrast, in the Q-tensor description, the boundary conditions are\nformulated in terms of the nematic director field and do not prescribe a net polarity at the interfaces. As a result, the Q-tensor\nsolution remains symmetric about the central axis. The apparent difference between the two formulations therefore arises from\nthe boundary conditions rather than from the instability or suppression of +1 defects in the Q-tensor case.\nWe agree with the Reviewer that this point was not discussed or emphasized in the previous version of our manuscript. To clarify\nthis important aspect, we have revised both the main text and the Supplementary Information as follows:\n• Revised both the main text and the Supplementary Information to clarify that the results shown in the main text correspond\nto the p-description, in which the active units are polarized at the drop boundaries due to the imposed boundary condition\n(Eq. 5b). These changes are highlighted in blue in lines 70–73, 107–110, Eq. 5, 158–163, 295, 349–356, 379–381,\n729–735, and in the Fig. 1 caption.\n• Modified Figs. 1, 2, and 4 to highlight the direction of polarization of the active units at the drop boundaries.\nWe believe that with the improvements we have made, the manuscript is now suitable for publication in Nature Communications.\nThank you for your consideration.\n5"}, {"round": 2, "reviewer_comments": {"reviewer_2": "The authors have satisfactorily addressed all the points raised in my previous review. The manuscript is now suitable for\npublication in its present form.", "reviewer_4": "In the revised manuscript, the authors have sufficiently addressed my comments and also the original concerns raised by\nreviewer #1"}, "author_response": ""}]

Decision letter