Behaviour of flexible superhydrophobic striped surfaces during (electro-)wetting of a sessile drop

Arvind Arun Dev; Frieder Mugele; Ranabir Dey

REVIEW 4 major objections 4 minor 13 references

Behaviour of flexible superhydrophobic striped surfaces during (electro-)wetting of a sessile drop

T0 review · 4 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Flexible lamellae on a superhydrophobic surface deform according to the net horizontal capillary force at the droplet contact line, and their shape requires the nonlinear elastica equation rather than the usual small-deflection formula.

desk verdict Careful experimental study, but the headline claim about small-deflection theory is not supported because the comparison omits the vertical force that the model itself says is comparable. read the letter →

arxiv 1908.05742 v1 pith:B6D3A56N submitted 2019-08-12 cond-mat.soft physics.flu-dyn

Arvind Arun Dev , Ranabir Dey , Frieder Mugele This is my paper · ORCID

classification cond-mat.softphysics.flu-dyn

keywords superhydrophobicsurfaceselectrowettingelasticacapillaryforcelamelladeformationconfocalmicroscopyCassiestatesmall-deflectionassumption

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

This paper tries to establish what controls the microscopic bending of soft elastic lamellae that make up a flexible superhydrophobic substrate when a sessile droplet wets it. Using confocal microscopy of droplets in the Cassie state under electrowetting, the authors find that the measured deflection profile of the lamella at the droplet contact line is determined by the net horizontal component of the capillary forces exerted by the pinned liquid-air interface on the lamella top. Electrowetting plays only an indirect role: it moves the contact line and changes the curvature of the liquid-vapour interface, thereby changing the direction and magnitude of those capillary forces. The quantitative conclusion is that the frequently used small-deflection formula fails to describe the profiles, while the full nonlinear elastica equation, solved numerically, reproduces them across different stiffnesses, aspect ratios, and applied voltages.

What carries the argument

The load-bearing object is the nonlinear elastica equation derived from a variational principle, $B\theta''+F_x\cos\theta-F_z\sin\theta=0$, where $\theta(s)$ is the local tangent angle, $B=EI/(1-\nu^2)$ is the flexural rigidity of the lamella as a planar structure, and $F_x,F_z$ are the net horizontal and vertical capillary loads. The capillary loads are obtained from the measured angles $\psi$ and $\varphi$ via the trigonometric relations in Eq. (1); the angles are read from confocal images, not treated as fitting parameters. The equation is solved numerically with boundary conditions $\theta(0)=0$ and $\theta'(L)=0$, and it is the nonlinearity in $\cos\theta$ and $\sin\theta$ that lets the theory match the strongly bent lamellae, where the small-deflection formula fails.

What would settle it

Measure the interface angles $\psi$ and $\varphi$ with an independent optical or interferometric technique on the same droplet-lamella configuration, then compute the elastica profile; if the predicted $\delta(z)$ deviates systematically from the confocal profile, the force-balance picture is wrong. Alternatively, image lamellae in the drop interior away from the contact line: the model predicts they remain undeformed because horizontal forces cancel, so any regular finite deflection there would refute the claim.

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Extended reading notes

Core claim

The central claim is that the local deformation profile $\delta(z)$ of an elastic lamella is governed by the net horizontal capillary force $F_x$ pulling on its top, and that the equilibrium shape is the solution of the nonlinear elastica equation $B\theta''+F_x\cos\theta-F_z\sin\theta=0$ with clamped base and free top boundary conditions. The horizontal force arises from surface tension acting along the pinned three-phase contact line; its direction depends on the measured angles $\psi(\eta)$ and $\varphi(\eta)$ between the liquid-air interface and the lamella edges. Comparing confocal images with theory, the authors show that this equation captures the dependence of the deflection on Young's modulus, aspect ratio, and electrowetting number, including the smaller deflection when the contact line pins to both top edges instead of one. They also show that the closed-form small-deflection expression $\delta(z)=F_xL^3/(6B)(3z^2/L^2-z^3/L^3)$ does not describe the measured shapes. A corollary is that lamellae away from the contact line do not deform because the horizontal forces on their two top edges cancel.

Load-bearing premise

The force directions are not measured independently: the angles $\psi$ and $\varphi$ are read from the same confocal images in which the lamella deflection is observed, so the theory-experiment agreement is partly a consistency check.

Editorial extensions

If this is right

Quantitative analysis of flexible superhydrophobic microstructures must solve the nonlinear elastica equation; the closed-form small-deflection formula is not a reliable substitute.
Lamella deformation is strongest where the contact line pins to only one top edge, so surface designs that force symmetric two-edge pinning should reduce local bending.
Electrowetting changes lamella shapes only by moving the contact line and altering interface curvature; it exerts no directly detectable Maxwell-stress load on the lamella.
Lamellae under the drop away from the contact line stay upright because equal-and-opposite horizontal capillary forces on their two top edges cancel; this is a testable prediction.
Because the small-deflection model fails here, earlier micro-pillar experiments that used it to extract forces from deflections may need re-analysis.

Reading between the lines

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

One could replace the confocal-image angle estimates with independent interface-shape measurements, turning the elastica match into a predictive force-balance test rather than a consistency check.
The results suggest that local bending compliance, not just global contact angle, controls droplet mobility on flexible superhydrophobic surfaces; hysteresis may be set by the stiffest or softest lamella along the contact line.
For sufficiently soft or tall lamellae, the nonlinear equation predicts that increased capillary force will drive large rotations and possible collapse; such a bending-driven Cassie-to-Wenzel transition is a natural extension to test.
The same force-balance and elastica approach could be applied to micropillar arrays used for cell traction force measurements, where deflections may exceed the small-deflection regime.

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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

You should know two things about this paper. First, it's a genuinely useful experimental study: the authors measure, by confocal microscopy, the full deflection profiles of flexible PDMS lamellae under electrowetting, varying stiffness, aspect ratio, and applied voltage. That dataset is new and nontrivial. Second, the paper's central conclusion—that the frequently used small-deflection model is quantitatively inadequate and a nonlinear elastica solution is required—is not actually established by the evidence they present.

The paper's own force balance (Eqs. 1a–1e) gives vertical forces Fz of order γ, comparable to the horizontal components Fx. The nonlinear equation they solve (Eq. 2) includes both. But the small-deflection formula they compare against (Eq. 3) is not the small-deflection limit of Eq. 2 when Fz is present. The correct linearized equation is Bθ'' + Fx − Fzθ = 0, a beam-column equation where Fz acts as an axial force. The paper never solves this proper linearized equation, so the failure of Eq. 3 in Fig. 5 could be due entirely to the omission of Fz, not to geometric nonlinearity. This is a load-bearing gap in the argument, not a minor omission.

Two other soft spots are worth naming, in proportion. The angles ψ and φ, which determine the force magnitudes and directions, are measured from the same confocal images in which the deflection is observed. That makes the theory-experiment agreement partly a consistency check rather than an independent validation. And there are no error bars, no repeat measurements, and no goodness-of-fit statistic. The claimed electrowetting effect is small (3.28 vs 2.87 μm) and would need error analysis to be convincing.

To be fair, the experimental work is solid and the model is parameter-free and standard. The qualitative agreement in Figs. 5 and 6 does support the idea that net horizontal capillary force controls the deformation shape. But the quantitative conclusion about small-deflection theory being inadequate remains unproven until the corrected linearized comparison is done. The paper deserves a serious referee, but it should be returned for revision, not accepted as is. I'd want to see the comparison against the full linear beam-column equation, error bars on the measured profiles, and ideally angles obtained from an independent measurement or at least a sensitivity analysis.

Referee Report

4 major / 4 minor

Summary. The manuscript reports confocal-microscopy measurements of the bending of flexible PDMS lamellae at the contact line of a sessile drop in Cassie state, under ac electrowetting. The authors propose that the deformation profile is set by the net horizontal capillary force acting on the lamella top, with the force components reconstructed from measured angles ψ and φ via Eqs. (1a)-(1e). They solve the nonlinear elastica equation, Eq. (2), with clamped-free boundary conditions, compare the numerical shapes with experimental profiles in Figs. 5 and 6, and contrast the results with the small-deflection cantilever formula, Eq. (3). Their central claim is that the small-deflection model fails quantitatively and that the full nonlinear solution is required to describe the measured profiles.

Significance. If established, the result is useful for the wetting mechanics of soft microstructured superhydrophobic surfaces: it provides direct measurements of lamella deformation, uses no fitted parameters, and identifies the indirect role of electrowetting through the change in interface orientation. The paper also gives a clear warning against routine use of the linear cantilever formula for micro-pillar and lamella deflections. However, the comparison used to support the headline conclusion is currently not the correct small-deflection limit of the model being advocated, because the vertical force component is omitted from Eq. (3). The experimental validation is also qualitative, without error bars or repeat measurements. These issues are fixable, and the underlying dataset may well support the conclusion after a corrected linearized comparison.

major comments (4)

[§3, Eqs. (2) and (3)] The small-deflection formula Eq. (3) is not the small-deflection limit of Eq. (2) when Fz is nonzero. Linearizing Eq. (2) about θ = 0 gives Bθ'' + Fx − Fz θ = 0, with θ = δ'; Eq. (3) follows only if Fz is neglected entirely. The authors' own decomposition (1a)-(1c) gives Fz comparable to Fx at A-A': Fzl = γ cos(π−ψ) and Fzt = γ, so for ψ > 90° the net vertical force can exceed Fxl. The failure of Eq. (3) shown in Fig. 5 could therefore be caused by the omission of the axial/vertical force term rather than by geometric nonlinearity. To support the central claim that the small-deflection assumption is quantitatively inadequate, the authors must compare the data with the correct linearized solution of Eq. (2) that retains the Fz θ term and show that it too fails.
[Figs. 5 and 6; §3] The quantitative claim that the nonlinear solution 'describes the experimental results very well' is not supported by any quantitative metric. The figures show single profiles without error bars, no repeat measurements are reported, and no residual, R², or confidence interval is given. The reported difference between η = 0 and η = 0.19 (3.28 μm vs 2.87 μm at z ≈ 30 μm) is smaller than typical confocal uncertainties in soft materials, so the statement in the text that deformation decreases with η needs an uncertainty estimate to be meaningful.
[Eqs. (1a)-(1e); Sec. 7 of ESI] The angles ψ and φ are measured from the same confocal images in which the lamella deflection is observed. Since the deformed shape and the interface angles are coupled observables of one equilibrium state, the agreement in Figs. 5 and 6 is partly a consistency check rather than an independent test of the force-balance model. The paper should quantify the sensitivity of the predicted profiles to reasonable uncertainties in ψ and φ, or provide an independent determination of at least one of these angles, to strengthen the evidence that the model is not absorbing the deformation response through its inputs.
[§2.2 and Table 1] The Young's moduli E = 2.1, 0.9, and 0.6 MPa are reported from extensometer stress-strain curves, but no uncertainty is given. Since the theoretical deflection scales as 1/B, the comparison in Fig. 5 is sensitive to the value of E; reporting the measurement uncertainty of E and a propagation of that uncertainty into the theoretical profiles would make the comparison between theory and experiment more convincing.

minor comments (4)

[Fig. 4 caption] The word 'coloumns' should be 'columns'.
[Eqs. (1a)-(1e) and Fig. 3 caption] The definitions of ψ and φ should be stated unambiguously in the main text: it would help to specify explicitly that both angles are measured in the xz plane and to state the sign convention for Fz (positive upward or downward).
[References] Reference 18 contains a garbled author name, 'v. doris'; this needs correction.
[§3, paragraph after Eq. (3)] The manuscript refers to 'Sec. 7 in ESI' for the estimated values of ψ and φ, but the main text does not list any representative values. A short table of ψ(η) and φ(η) for the cases shown in Figs. 5 and 6 would make the force balance more transparent.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the elastic deformation model uses measured interface angles as boundary-force inputs, not as fitted parameters, and the predicted deflection profile is not equivalent to those inputs.

full rationale

The paper's derivation chain is not circular. The lamella deflection is predicted by solving the nonlinear elastica equation (Eq. 2) with boundary conditions (2a)-(2b), using net capillary forces from Eqs. (1a)-(1e). The angles ψ and φ entering those forces are measured from confocal images of the liquid-air interface and contact-line geometry, and the paper explicitly states that they are estimated from images rather than used as fitting parameters. The output quantity is the full deflection profile δ(z), which is not algebraically determined by the two measured angles; obtaining it requires numerical solution of a boundary-value problem. Agreement between this solution and the measured profiles in Figs. 5 and 6 is therefore a nontrivial consistency check, not a reduction of the output to the inputs. The comparison against the small-deflection formula Eq. (3) is also not a circular step: Eq. (3) is a standard cantilever formula, not fitted to the data. A possible objection that Eq. (3) omits the vertical force Fz that Eq. (2) includes, so the failure of Eq. (3) may reflect neglect of an axial-force term rather than geometric nonlinearity, is a correctness or validity concern, not a circularity concern. There is no load-bearing self-citation chain, no imported uniqueness theorem, and no result that is merely a renaming of its own inputs. Hence the circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The model has no fitted free parameters: E is measured independently, and psi and phi are measured from images. The load-bearing assumptions are the 2D plane-strain beam description, the neglect of direct Maxwell stress on the lamella, the accuracy of the measured interface angles, and Cassie-state wetting.

assumptions (5)

domain assumption The lamella is modeled as a linear-elastic Euler-Bernoulli beam with plane-strain flexural rigidity B = EI/(1-nu^2) and Poisson's ratio nu=0.5.
Used to derive Eq. (2); ignores shear deformation, finite-width effects, and viscoelasticity of PDMS.
domain assumption The only forces on the deflected lamella are the capillary line forces at the pinned contact line; direct Maxwell stress on the lamella is negligible.
Writes Eqs. (1a)-(1e) and Eq. (2); the paper states no direct Maxwell-stress effect was detected and expects it to be small.
domain assumption The angles psi and phi measured from confocal images give the local liquid-air interface tangent that determines the capillary force directions.
These measured angles are used as inputs to Eq. (2); because they come from the same deformed images as the deflection, the input and output are not fully independent.
domain assumption The droplet remains in the Cassie state for all voltages used.
Needed for the force model; verified visually in Fig. 3.
standard math The Euler-Lagrange equation derived from the bending energy and work of tip forces gives the equilibrium shape.
Standard variational mechanics of elastic rods (ref. 35).

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Cite this review

Pith. "Pith review of Behaviour of flexible superhydrophobic striped surfaces during (electro-)wetting of a sessile drop." pith.science (2026). https://pith.science/paper/B6D3A56N

@misc{pith2026190805742,
  author       = {Pith},
  title        = {Pith review of: Behaviour of flexible superhydrophobic striped surfaces during (electro-)wetting of a sessile drop},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/B6D3A56N}},
  note         = {Machine review of arXiv:1908.05742}
}

read the original abstract

We study here the microscopic deformations of elastic lamellae constituting a superhydrophobic substrate under different wetting conditions of a sessile droplet using electrowetting. The deformation profiles of the lamellae are experimentally evaluated using confocal microscopy. These experimental results are then explained using a variational principle formalism within the framework of linear elasticity. We show that the local deformation profile of a lamella is mainly controlled by the net horizontal component of the capillary forces acting on its top due to the pinned droplet contact line. We also discuss the indirect role of electrowetting in dictating the deformation characteristics of the elastic lamellae. One important conclusion is that the small deflection assumption, which is frequently used in the literature, fails to provide a quantitative description of the experimental results; a full solution of the non-linear governing equation is necessary to describe the experimentally obtained deflection profiles.

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Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [3]

Based on the aforemen- tioned discussion, it can be now intuitively concluded that the lamellae underneath the droplet but away from the macroscopic contact line do not deform because the horizontal components of the capillary forces acting on the two top edges of each are almost equal in magnitude but opposite in direction thereby resulting in zero net h...

work page 2011

[2] [17]

b Current afﬁliation: Dynamics of Complex Fluids, Max Planck Institute for Dynamics and Self-organization, Am Fassberg 17, 37077 Goettingen, Germany

Owing to such a wide range of technological applications, ex- a Current afﬁliation: Department of magnetism of nanostructured objects (DMONS), Institut de Physique et Chimie des Matèriaux de Strasbourg (IPCMS), CNRS UMR 7504, Universitè de Strasbourg, 23 Rue du Loess, 67034 Strasbourg, France. b Current afﬁliation: Dynamics of Complex Fluids, Max Planck I...

work page

[3] [27]

How- ever, the behaviour of the soft micropillars was analyzed assum- ing small deﬂections of the elastic pillars, which may not be the + P V S O B M / B N F < Z F B S > < W P M > 1–9 | 1 arXiv:1908.05742v1 [cond-mat.soft] 12 Aug 2019 general case considering the wide range of applications involving different magnitudes of capillary forces and structure g...

work page Pith review arXiv 1908

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23 D. Xia, L. S. M. Johnson and G. P. López, Advanced materials, 2012, 24 10, 1287–302. 24 Y.-H. Yeh, K.-H. Cho and L.-J. Chen, Soft Matter , 2012, 8, 1079–1086. 25 Y.-C. Chuang, C.-K. Chu, S.-Y. Lin and L.-J. Chen, Soft Matter, 2014, 10, 3394–3403. 26 G. Whyman, E. Bormashenko and T. Stein, Chemical Physics Letters, 2008, 450, 355 –

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Schellenberger, N

18 F. Schellenberger, N. Encinas, v. doris and H.-J. Butt, 2016, 116,. 19 P. Papadopoulos, L. Mammen, X. Deng, D. Vollmer and H.-J. Butt, Proceedings of the National Academy of Sciences , 2013, 110, 3254–3258. 20 R. J. Daniello, N. E. Waterhouse and J. P. Rothstein, Physics of Fluids, 2009, 21, 085103. 21 H. J. Im and J. H. Lee, Physics of Fluids, 2017, 2...

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Papadopoulos, B.-E

27 P. Papadopoulos, B.-E. Pinchasik, M. Tress, D. Vollmer, M. Kappl and H.-J. Butt, Soft Matter, 2018, 14, 7429–7434. 28 F. Mugele and J. Heikenfeld, Electrowetting: Fundamental 8 | 1–9 + P V S O B M / B N F < Z F B S > < W P M > Principles and Practical Applications, Wiley ,

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Otten and S

3 A. Otten and S. Herminghaus, Langmuir, 2004, 20, 2405–

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[8] [1999]

38 D. W. Inglis, Biomicroﬂuidics, 2010, 4, 026504. 39 E. Manias, J. Chen, N. Fang and X. Zhang, Applied Physics Letters, 2001, 79, 1700–1702. 40 A. Rabodzey , P. Alcaide, F. W. Luscinskas and B. Ladoux,Bio- physical journal, 2008, 95 3, 1428–38. 41 J. M. Gere, Mechanics of Material , Brooks/Cole–Thomson Learning,

work page 2010