Morphodynamics of surface-attached active drops

Alejandro Martinez-Calvo; Sujit S. Datta

REVIEW 3 major objections 5 minor 48 references

Morphodynamics of surface-attached active drops

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read A full 2D model of an active nematic drop on a rigid substrate reveals stable steady-state shapes and flows whose symmetry is set by boundary anchoring alone.

desk verdict Useful beyond-thin-film step, but the director equation is wrong for a nematic, so the state diagram as presented doesn't yet describe the claimed system. read the letter →

arxiv 2411.14636 v1 pith:BNEV3LHV submitted 2024-11-22 cond-mat.soft cond-mat.mtrl-sciphysics.bio-phphysics.flu-dyn

Alejandro Martinez-Calvo , Sujit S. Datta This is my paper · ORCID

classification cond-mat.softcond-mat.mtrl-sciphysics.bio-phphysics.flu-dyn

keywords activematternematicdropsthin-filmapproximationCapillarynumberwindingboundaryanchoringsymmetrybreakingsurface-attacheddrop

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Most theoretical work on surface-attached active drops has relied on the thin-film approximation, which assumes the drop is shallow and slowly varying. This paper drops that assumption and simulates the full two-dimensional equations for a nematic active drop on a rigid substrate. It finds that such drops settle into a wide variety of stable steady-state shapes and internal flows, selected by the angle (winding number) the active units make with the substrate and the liquid-air interface and by the active Capillary number. The paper claims that whether the drop breaks left-right symmetry is determined solely by a mismatch between substrate and interface anchoring, and that drops can be switched reversibly between distinct shapes by changing the interfacial anchoring. These behaviors are absent from thin-film predictions, which give a unique shape set by $|\mathrm{Ca}_\alpha|$ alone.

What carries the argument

The central object is the full 2D continuum model of an active nematic drop: the incompressible Stokes equations with an active stress $\sigma_a = -\alpha \mathbf{p}\mathbf{p}$, coupled to a Laplacian orientation field $\nabla^2\mathbf{p}=0$ (the strong elastic limit) and to a deformable interface with capillary stress. The behavior is governed by three dimensionless parameters: the active Capillary number $\mathrm{Ca}_\alpha = \alpha/(\gamma/R)$ comparing active stress to capillary pressure, and two winding numbers $w_s$ and $w_i$ that specify, in quarter turns, the angle the director makes with the solid substrate and the liquid-air interface. These winding numbers act as boundary conditions that determine whether the director field has defects at the contact points or in the bulk, and thus control whether the resulting flow is symmetric or not. The key methodological step is relaxing the thin-film (lubrication) approximation and solving the full equations numerically with a finite-element method on a deforming mesh, which is what reveals the multiplicity of stable states.

What would settle it

Run a high-resolution simulation of the same model at $\mathrm{Ca}_\alpha=100$ with $w_s=w_i=0$: the paper's state diagram predicts a mirror-symmetric steady state, so observing spontaneous left-right symmetry breaking or persistent oscillations instead would contradict the claim that boundary mismatch is necessary for symmetry breaking.

Watch

Extended reading notes

Core claim

The paper's central claim is that a surface-attached two-dimensional drop of a nematic active fluid—described by Stokes flow with an active stress $-\alpha \mathbf{p}\mathbf{p}$ and an orientation field that relaxes instantly to its free-energy minimum, so $\nabla^2 \mathbf{p}=0$—reaches a stable steady-state shape and flow for every tested combination of the active Capillary number $\mathrm{Ca}_\alpha=\alpha/(\gamma/R)$ and boundary winding numbers $(w_s,w_i)$. The steady states are far richer than thin-film theory predicts: they include symmetric mushroom-like contractile shapes, flattened extensile lobes connected by a thin film, and asymmetric shapes with single vortices or spiral defects. Symmetry breaking occurs only when the orientation imposed at the substrate differs from that at the interface, and the handedness of the resulting asymmetry is set by the rotation sense of the units and the sign of activity. The paper also establishes an equivalence between planar and homeotropic substrate anchoring, $(w_s,w_i)\to(w_s+1,w_i+1)$ with $\mathrm{Ca}_\alpha\to-\mathrm{Ca}_\alpha$, and demonstrates that changing the interfacial winding number over time reversibly toggles the drop between its stable states.

Load-bearing premise

The paper's results assume that the orientation field of the active units relaxes instantly to its lowest-energy configuration and is never dragged or rotated by the flow, so it obeys Laplace's equation rather than evolving with the fluid.

Editorial extensions

If this is right

Thin-film-based predictions for surface-attached active drops are incomplete: they miss the stable symmetric states, the symmetry-breaking criterion, and the multiplicity of shapes that the full 2D model produces.
Boundary anchoring alone—not the magnitude or sign of the active stress—determines whether a surface-attached active drop breaks symmetry, so surface chemistry can be used as a control knob.
The equivalence $(w_s,w_i)\to(w_s+1,w_i+1)$ with $\mathrm{Ca}_\alpha\to-\mathrm{Ca}_\alpha$ means that a contractile drop with homeotropic substrate anchoring behaves like an extensile drop with planar anchoring, halving the number of state-diagram quadrants that need to be computed or measured.
Reversible switching of the interfacial anchoring toggles the drop between stable shapes even after large deformations, suggesting a route to reconfigurable active droplets that behave as soft actuators.
For $|\mathrm{Ca}_\alpha|\lesssim1$, internal flow speed and energy dissipation follow simple scaling laws ($|\mathbf{u}|_{\max}\sim|\mathrm{Ca}_\alpha|$, $\dot s_v\sim\mathrm{Ca}_\alpha^2$, $\dot s_a\sim-\mathrm{Ca}_\alpha^2$), so the cost of driving a given shape can be estimated from activity alone.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the strong elastic limit is relaxed to include flow-alignment coupling, the clean equivalence between anchoring configurations and the criterion that symmetry breaking requires a boundary mismatch could break down; the paper itself flags this as unlikely to hold, and it is the first assumption worth testing.
The predicted reversible switching suggests an experimental protocol: pattern or photocontrol the anchoring at the liquid-air interface of a surface-attached active nematic drop (for example, with light-sensitive surfactants) and observe whether the drop toggles between the corresponding steady states with the reported time scales.
The scaling laws found here could be used to infer the active stress magnitude in biological systems: measuring the maximum internal flow speed and drop radius in a microbial colony or cell aggregate would give $\alpha$ from $|\mathbf{u}|_{\max}\sim\mathrm{Ca}_\alpha$ in the low-activity regime.
Because the model is two-dimensional, the corresponding 3D surface-attached drop might show additional instabilities (e.g., azimuthal symmetry breaking) that the 2D state diagram cannot capture; extending these simulations to 3D would test whether the boundary-mismatch criterion survives.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

First: the paper has a load-bearing error in the director model. The authors write a one-constant Frank energy (Eq. 3) for a nematic director and claim the first variation gives ∇²p = 0 (Eq. 4). For a unit director, that is not the Euler–Lagrange equation; the constraint |p|=1 changes it to ∇²p + |∇p|² p = 0, equivalently ∇²θ=0 for p=(cosθ, sinθ). The weak form in the SI (S5a) confirms the code actually solves the unconstrained vector Laplace problem, so |p| varies inside the drop and the active stress −αpp is not the intended nematic stress. This isn't a cosmetic detail: the state diagram, symmetry-breaking rule, and the (ws,wi)→(ws+1,wi+1) mapping are all computed with this director field. The central claim that these are results for surface-attached active nematic drops is not yet supported.

What is genuinely useful: the paper takes a well-studied thin-film problem and does full 2D Stokes-flow simulations with a deformable interface, showing stable steady-state shapes, symmetry breaking set by boundary anchoring, and reversible switching via time-dependent anchoring. That is a real step beyond Loisy et al. and the presentation is clear. The qualitative contrast with thin-film predictions is likely to survive a corrected director equation, even if the details change.

Other soft spots: 'all possible states' is based on two substrate winding numbers, three interfacial winding numbers, and a limited |Caα| range. There's no convergence study, no error bars, no code or data. The strong-elastic-limit assumption is acknowledged by the authors and is a separate concern.

Who is this for: people working on active wetting and biofilm/colony models will want to know the thin-film results may be incomplete. It deserves a serious referee, but the director equation has to be fixed—or the model must be explicitly redefined as an unconstrained vector order parameter with the free energy extended accordingly—before the results can be taken at face value. I'd send it to review, with a strong request for a corrected model and a convergence check. Worth a discussion in our group as a lesson in variational constraints.

Referee Report

3 major / 5 minor

Summary. This paper studies a two-dimensional Stokes-flow model of a surface-attached drop of an active nematic fluid. The model combines incompressibility and Stokes equations with an active stress -alpha p p, kinematic and stress boundary conditions at the free surface, and a director field that is assumed to relax instantaneously to the minimum of a one-constant Frank energy. The authors simulate the full equations without a thin-film approximation, sweep over extensile/contractile activity and discrete quarter-turn anchoring conditions at the substrate and interface, and report steady-state shapes and flows, a state diagram, scaling laws for maximum velocity and dissipation, and a demonstration of reversible switching by changing the interfacial anchoring. The central claims are that stable steady states exist for all tested parameters, that symmetry breaking is controlled by the boundary-alignment mismatch, and that (ws, wi) states are equivalent under (ws, wi) -> (ws+1, wi+1) with Ca_alpha -> -Ca_alpha.

Significance. If the results are correct, they substantially extend the thin-film theory of active drops by showing multiple stable morphologies and quantitative scaling laws in a simple parameter-free model. The design-principle claim, especially the reversible anchoring switching, is potentially important for experiments on active droplets and colonies. The paper has genuine strengths: the model is minimal, the parameters are few, the simulations are time-dependent with interface tracking, and many results are presented as falsifiable predictions, including the state diagram, the scaling exponents, and the symmetry-breaking rule. However, the significance is conditional on correcting the director-field equation described below; as written, the computation may solve a different model from the one stated, and the absence of a convergence study and of deposited code or data weakens the quantitative claims.

major comments (3)

[Model section, Eq. (4) and Supplementary Eq. (S5a)] The director field is treated as an unconstrained vector. Eq. (4) is presented as the first variation of F in Eq. (3), but for a nematic director with |p| = 1 the first variation under the unit-vector constraint gives p x (curl p) = 0, equivalently Laplace's equation for the angle theta, not the vector Laplace equation del^2 p = 0. The weak form in Eq. (S5a) confirms that the code solves the unconstrained vector Laplace problem, with unit magnitude imposed only on the Dirichlet boundaries. Because the active stress in Eq. (2) is -alpha p p, an uncontrolled interior magnitude of p directly changes the flow forcing, so the shapes, flows, state diagram in Fig. 2, the "symmetry breaking is set by boundary alignment only" rule, and the period-2 equivalence under Ca_alpha -> -Ca_alpha are computed for a different order-parameter model than the nematic director model described. Please rerun the parameter sweep with the constrained director equation, or explicitly state that p is an unconstrained order-parameter vector and justify that choice, and report whether the state diagram, the equivalence mapping, and the scaling laws survive.
[Fig. 3 and the section on scaling laws] The claims that the drop adopts equilibrium shapes for all values of Ca_alpha and w, and that the state diagram in Fig. 2 uncovers all possible states, are stronger than the actual sampling. The simulations use only ws in {0,1}, wi in {0,1,2,3}, and a limited range of |Ca_alpha| (up to 4.5 in Fig. 2 and up to 10^2 in Fig. 3). No grid-convergence study, temporal-refinement check, or uncertainty estimates are reported for the phase boundaries or the scaling exponents. Since the scaling laws in Fig. 3 are a central quantitative result, please provide a resolution study, report the parameter ranges actually simulated, and state the numerical uncertainty behind each claimed "all possible states" statement.
[Homeotropic substrate anchoring paragraph] The paper correctly notes that the equivalence (ws, wi) -> (ws+1, wi+1) with Ca_alpha -> -Ca_alpha is unlikely to hold if the strong elastic limit is relaxed. This caveat is not limited to that mapping: the entire state diagram, the symmetry-breaking rule, and the scaling laws are computed with a director field that is slaved to the free-energy minimum and never coupled back to the flow. Please either add a test of flow-alignment coupling for a subset of parameters or explicitly restrict the "quantitative principles" and "all possible states" claims to the strong-elastic, flow-uncoupled model, including in the abstract and conclusions.

minor comments (5)

[Reversible shape and flow control section] The text refers to "Fig. 1b" through "Fig. 1e" when describing the time-dependent switching sequence; these references should be to Fig. 4b-e.
[Eq. (7) and Fig. 3b] The quantity rho T s_dot_tot is called the total entropy production rate, but it can be negative in the contractile case (Fig. 3b), so it is a balance of viscous dissipation and active work rather than a total entropy production rate; please define it more precisely or use a less misleading name.
[Abstract and Fig. 2 introduction] The phrase "all possible states" overstates the discrete anchoring set that was sampled; please rephrase as "all states in the sampled anchoring set" or otherwise qualify the claim.
[Supplementary Eq. (S5c)] The boundary term is written as an integral of sigma dot n dot phi_u and then stated to be expressed as an integral of the surface gradient; the equivalence should be shown explicitly in the supplement.
[Throughout] There are small typographical issues, such as "4th-order" instead of "fourth-order" and an incompletely parenthesized thin-film equation in the section comparing with previous work; please correct these.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the state diagrams, symmetry-breaking rules, and mapping symmetries are numerical consequences of the stated minimal model, with no fitted parameters and no load-bearing self-citations.

full rationale

The paper's derivation chain is self-contained: Eq. (2) defines the active stress in terms of the director p; Eq. (3) defines the Frank free energy; Eq. (4) gives the strong-elastic-limit orientation equation; Eq. (5) supplies boundary conditions; Eq. (6) defines the sole dimensionless active Capillary number. The state diagrams in Fig. 2, the 'symmetry breaking is determined by the boundary alignment' rule, and the mapping (ws, wi) -> (ws+1, wi+1) with Ca_alpha -> -Ca_alpha are all computed consequences of this closed set of equations, not restatements of the inputs. No parameter is fitted to the outputs it later 'predicts'; no experimental data subset is used to infer another subset. The quarter-turn restriction on winding numbers is introduced as an explicit 'for simplicity' modeling choice following Refs. [34,36] (Loisy, Eggers, and Liverpool), which are not self-citations, and it is not disguised as a derived result. The authors' admitted caveat that the period-2 equivalence 'is unlikely to hold if the strong elastic limit is relaxed' is an honest scope limitation, not a circular step. The skeptic concern that Eq. (4) is the unconstrained minimizer rather than the constrained unit-director minimizer is a potential correctness or consistency issue with the model as stated, but it does not make any prediction reduce to its inputs by construction: the simulated flows and shapes are still nontrivial outputs of the stated PDE system. Therefore the circularity score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No fitted constants and no new entities are introduced. All parameters are control parameters scanned in simulation; the central results are computed consequences of the stated equations. The main ledger burden is the set of modeling simplifications enumerated in the axioms.

assumptions (4)

domain assumption Stokes flow with Newtonian viscosity, active stress -αpp, one-constant Frank energy, strong elastic limit ∇²p=0
Model section, Eqs. (1)-(4). Neglects inertia, complex rheology, flow-driven reorientation, and advection of the director; these are stated as simplifications in the Discussion.
ad hoc to paper Boundary anchoring is limited to quarter turns, θi = wiπ/2 for integer winding numbers wi and ws
Model section after Eq. (5) and Fig. 1, following Refs. [34,36]. This discretization is needed for the "all possible states" enumeration but excludes arbitrary anchoring angles.
domain assumption The drop is two-dimensional, the exterior fluid is passive and quiescent, surface tension is constant, and active units neither proliferate nor vary in density
Model section and Discussion. These choices make the minimal model tractable but omit 3D curvature, Marangoni stresses, and density feedback.
domain assumption Steady states are sought from a semicircular, quiescent initial condition
Numerical simulations section: initial conditions are semicircular with u=0 and Π=γ/R. The reported state diagram may miss branches reachable only from other initial shapes.

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Cite this review

Pith. "Pith review of Morphodynamics of surface-attached active drops." pith.science (2026). https://pith.science/paper/BNEV3LHV

@misc{pith2026241114636,
  author       = {Pith},
  title        = {Pith review of: Morphodynamics of surface-attached active drops},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BNEV3LHV}},
  note         = {Machine review of arXiv:2411.14636}
}

read the original 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 suspension (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.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

48 extracted references · 46 canonical work pages

[1]

M. C. Marchetti, J.-F. Joanny, S. Ramaswamy, T. B. Liverpool, J. Prost, M. Rao, and R. A. Simha, Rev. Mod. Phys. 85, 1143 (2013)

2013
[2]

Zhang, A

R. Zhang, A. Mozaffari, and J. de Pablo, Nat. Rev. Mater. 6, 437 (2021)

work page 2021
[3]

Hallatschek, S

O. Hallatschek, S. S. Datta, K. Drescher, J. Dunkel, J. Elgeti, B. Waclaw, and N. S. Wingreen, Nat. Rev. Phys. , 1 (2023)

work page 2023
[4]

Fernández, B

P. Fernández, B. Buchmann, A. Goychuk, L. Engelbrecht, M. Raich, C. Scheel, E. Frey, and A. Bausch, Nat. Phys. 17, 1130 (2021)

work page 2021
[5]

Ishihara, A

K. Ishihara, A. Mukherjee, E. Gromberg, J. Brugués, E. Tanaka, and F. Jülicher, Nat. Phys.19, 177 (2023)

work page 2023
[6]

Beroz, J

F. Beroz, J. Yan, Y . Meir, B. Sabass, H. Stone, B. Bassler, and N. Wingreen, Nat. Phys. 14, 954 (2018)

work page 2018
[7]

Pearce, B

P. Pearce, B. Song, D. J. Skinner, R. Mok, R. Hartmann, P. K. Singh, H. Jeckel, J. S. Oishi, K. Drescher, and J. Dunkel, Phys. Rev. Lett. 123, 258101 (2019)

work page 2019
[8]

Hartmann, P

R. Hartmann, P. Singh, P. Pearce, R. Mok, B. Song, F. Díaz- Pascual, J. Dunkel, and K. Drescher, Nat. Phys.15, 251 (2019)

work page 2019

Show all 48 references

[9]

B. Qin, C. Fei, A. A. Bridges, A. A. Mashruwala, H. A. Stone, N. S. Wingreen, and B. L. Bassler, Science (2020), 10.1126/science.abb8501

2020 doi
[10]

Nijjer, C

J. Nijjer, C. Li, Q. Zhang, H. Lu, S. Zhang, and J. Yan, Nat. Commun. 12, 6632 (2021)

2021
[11]

Nijjer, C

J. Nijjer, C. Li, M. Kothari, T. Henzel, Q. Zhang, J.-S. Tai, S. Zhou, T. Cohen, S. Zhang, and J. Yan, Nat. Phys. , 1 (2023)

2023
[12]

H. Ford, G. Celora, E. Westbrook, M. Dalwadi, B. Walker, H. Baumann, C. Weijer, P. Pearce, and J. Chubb, bioRxiv , 2024 (2024)

2024
[13]

Brugués and D

J. Brugués and D. Needleman, Proc. Nat. Acad. Sci. U.S.A. 111, 18496 (2014)

2014
[14]

Oriola, D

D. Oriola, D. Needleman, and J. Brugués, Annu. Rev. Biophys. 47, 655 (2018)

2018
[15]

Oriola, F

D. Oriola, F. Jülicher, and J. Brugués, Proc. Nat. Acad. Sci. U.S.A. 117, 16154 (2020)

2020
[16]

Keber, E

F. Keber, E. Loiseau, T. Sanchez, S. DeCamp, L. Giomi, M. Bowick, M. Marchetti, Z. Dogic, and A. Bausch, Science 345, 1135 (2014)

2014
[17]

Adkins, I

R. Adkins, I. Kolvin, Z. You, S. Witthaus, M. Marchetti, and Z. Dogic, Science 377, 768 (2022)

2022
[18]

Tayar, F

A. Tayar, F. Caballero, T. Anderberg, O. Saleh, M. Marchetti, and Z. Dogic, Nat. Mater. 22, 1401 (2023)

2023
[19]

H. Fu, J. Huang, J. van der Tol, L. Su, Y . Wang, S. Dey, P. Zi- jlstra, G. Fytas, G. Vantomme, P. Dankers, et al., Nature 626, 1011 (2024)

2024
[20]

Wensink, J

H. Wensink, J. Dunkel, S. Heidenreich, K. Drescher, R. Gold- stein, H. Löwen, and J. Yeomans, Proc. Nat. Acad. Sci. U.S.A. 109, 14308 (2012)

2012
[21]

Khuc Trong, H

P. Khuc Trong, H. Doerflinger, J. Dunkel, D. St Johnston, and R. Goldstein, eLife 4, e06088 (2015)

2015
[22]

Saintillan, M

D. Saintillan, M. Shelley, and A. Zidovska, Proc. Nat. Acad. Sci. U.S.A. 115, 11442 (2018)

2018
[23]

Needleman and M

D. Needleman and M. Shelley, Phys. Today 72, 6 (2019)

2019
[24]

Khalil, S

K. Khalil, S. Mahmoudi, N. Abu-Dheir, and K. Varanasi, App. Phys. Lett. 105 (2014)

2014
[25]

Aggarwal, E

A. Aggarwal, E. Kirkinis, and M. De La Cruz, J. Fluid Mech. 955, A10 (2023)

2023
[26]

Aggarwal, E

A. Aggarwal, E. Kirkinis, and M. Olvera de la Cruz, Phys. Rev. Lett. 131, 198201 (2023)

2023
[27]

Tjhung, D

E. Tjhung, D. Marenduzzo, and M. Cates, Proc. Natl. Acad. Sci. U. S. A. 109, 12381 (2012)

2012
[28]

Tjhung, A

E. Tjhung, A. Tiribocchi, D. Marenduzzo, and M. E. Cates, Nature communications 6, 1 (2015)

2015
[29]

Pérez-González, R

C. Pérez-González, R. Alert, C. Blanch-Mercader, M. Gómez- González, T. Kolodziej, E. Bazellieres, J. Casademunt, and 7 Morphodynamics of surface-attached active drops X. Trepat, Nat. Phys. 15, 79 (2019)

2019
[30]

Casademunt, Langmuir 35, 7571 (2018)

Alert, R and J. Casademunt, Langmuir 35, 7571 (2018)

2018
[31]

Liese, X

S. Liese, X. Zhao, C. Weber, and F. Jülicher, arXiv preprint arXiv:2312.07239 (2023)

2023 arXiv
[32]

Ben Amar and L

M. Ben Amar and L. J. Cummings, Phys. Fluids 13, 1160 (2001)

2001
[33]

J. F. Joanny and S. Ramaswamy, J. Fluid Mech.705, 46 (2012)

2012
[34]

Loisy, J

A. Loisy, J. Eggers, and T. B. Liverpool, Phys. Rev. Lett. 123, 248006 (2019)

2019
[35]

Trinschek, F

S. Trinschek, F. Stegemerten, K. John, and U. Thiele, Phys. Rev. E 101, 062802 (2020)

2020
[36]

Loisy, J

A. Loisy, J. Eggers, and T. B. Liverpool, Soft Matter 16, 3106 (2020)

2020
[37]

Shankar, V

S. Shankar, V . Raju, and L. Mahadevan, Proceedings of the National Academy of Sciences 119, e2121985119 (2022)

2022
[38]

Ioratim-Uba, A

A. Ioratim-Uba, A. Loisy, S. Henkes, and T. B. Liverpool, Soft Matter 18, 9008 (2022)

2022
[39]

A. Oron, S. H. Davis, and S. G. Bankoff, Rev. Mod. Phys. 69, 931 (1997)

1997
[40]

R. V . Craster and O. K. Matar, Rev. Mod. Phys.81, 1131 (2009)

2009
[41]

Needleman and Z

D. Needleman and Z. Dogic, Nat. Rev. Mater. 2, 1 (2017)

2017
[42]

G. Lee, G. Leech, M. Rust, M. Das, R. McGorty, J. Ross, and R. Robertson-Anderson, Sci. Adv. 7, eabe4334 (2021)

2021
[43]

Berezney, B

J. Berezney, B. Goode, S. Fraden, and Z. Dogic, Proc. Nat. Acad. Sci. U.S.A. 119, e2115895119 (2022)

2022
[44]

M. J. Stephen and J. P. Straley, Rev. Mod. Phys.46, 617 (1974)

1974
[45]

P. G. De Gennes and J. Prost, The physics of liquid crystals , V ol. 83 (Oxford university press, 1993)

1993
[46]

Prost, F

J. Prost, F. Jülicher, and J.-F. Joanny, Nat. Phys.11, 111 (2015)

2015
[47]

Giomi and A

L. Giomi and A. DeSimone, Phys. Rev. Lett. 112, 147802 (2014)

2014
[48]

Alert, J

R. Alert, J. Phys. A: Mathematical and Theoretical 55, 234009 (2022). 8 Morphodynamics of surface-attached active drops SUPPLEMENTARY INFORMATION Non-dimensionalization. To perform two-dimensional (2D) numerical simulations of the system of equations presented in the main text...

2022

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

M. C. Marchetti, J.-F. Joanny, S. Ramaswamy, T. B. Liverpool, J. Prost, M. Rao, and R. A. Simha, Rev. Mod. Phys. 85, 1143 (2013)

2013

[2] [2]

Zhang, A

R. Zhang, A. Mozaffari, and J. de Pablo, Nat. Rev. Mater. 6, 437 (2021)

work page 2021

[3] [3]

Hallatschek, S

O. Hallatschek, S. S. Datta, K. Drescher, J. Dunkel, J. Elgeti, B. Waclaw, and N. S. Wingreen, Nat. Rev. Phys. , 1 (2023)

work page 2023

[4] [4]

Fernández, B

P. Fernández, B. Buchmann, A. Goychuk, L. Engelbrecht, M. Raich, C. Scheel, E. Frey, and A. Bausch, Nat. Phys. 17, 1130 (2021)

work page 2021

[5] [5]

Ishihara, A

K. Ishihara, A. Mukherjee, E. Gromberg, J. Brugués, E. Tanaka, and F. Jülicher, Nat. Phys.19, 177 (2023)

work page 2023

[6] [6]

Beroz, J

F. Beroz, J. Yan, Y . Meir, B. Sabass, H. Stone, B. Bassler, and N. Wingreen, Nat. Phys. 14, 954 (2018)

work page 2018

[7] [7]

Pearce, B

P. Pearce, B. Song, D. J. Skinner, R. Mok, R. Hartmann, P. K. Singh, H. Jeckel, J. S. Oishi, K. Drescher, and J. Dunkel, Phys. Rev. Lett. 123, 258101 (2019)

work page 2019

[8] [8]

Hartmann, P

R. Hartmann, P. Singh, P. Pearce, R. Mok, B. Song, F. Díaz- Pascual, J. Dunkel, and K. Drescher, Nat. Phys.15, 251 (2019)

work page 2019