pith. sign in

arxiv: 2605.31258 · v1 · pith:CBUKGNPJnew · submitted 2026-05-29 · ❄️ cond-mat.soft

Droplets sitting on thin elastic sheets: A study with the boundary element method

Salik Sultan , Josua Grawitter , Gon\c{c}alo C. Antunes , Holger Stark This is my paper

Pith reviewed 2026-06-28 20:23 UTC · model grok-4.3

classification ❄️ cond-mat.soft
keywords elasto-capillarityboundary element methodelastic sheetliquid lensdroplet wettingfocal lengthtension control
0
0 comments X

The pith

The equilibrium shape of a droplet lens on a clamped elastic sheet depends on sheet thickness, and isotropic stretching tunes its focal length via applied tension.

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

The paper formulates a boundary element method to compute the coupled elastic deformation and capillary forces for a droplet on a thin sheet. It tests the method on three boundary-condition protocols. For clamped sheets, various shape metrics establish that lens geometry changes with sheet thickness. Isotropic stretching introduces tension as a tunable parameter that alters radial stress profiles and focal length. Uniaxial stretching produces elongated droplets together with folds and dimples.

Core claim

The boundary element simulations show that for a clamped elastic sheet the lens shape of the droplet crucially depends on the sheet thickness; isotropic stretching supplies an extra control parameter that changes the droplet shape and the tension distribution in the sheet, allowing the focal length of the liquid lens to be adjusted by varying the applied tension; uniaxial stretching elongates the droplet while generating folds and dimples.

What carries the argument

The boundary element method extension that solves the coupled elasto-capillary problem for three distinct boundary-condition protocols on the elastic sheet.

Load-bearing premise

The extended boundary element method together with the three boundary-condition protocols correctly reproduces the equilibrium shapes of the coupled system.

What would settle it

Direct comparison of simulated lens morphology against experiments on clamped sheets of systematically varied thickness would falsify the thickness dependence if the measured shapes show no systematic change with thickness.

read the original abstract

Elasto-capillarity of a droplet wetting an elastic sheet provides an interesting system, both for fundamental and applied research. The droplet sinks into the sheet and assumes the shape of a lens. To determine the equilibrium shape in simulations, we formulate a boundary element method (BEM) extending our earlier approaches, and apply the BEM to three specific protocols for the boundary conditions of the sheet. For a clamped elastic sheet, we use various morphological metrics to demonstrate that the lens shape crucially depends on the sheet thickness. Stretching the sheet isotropically, allows for an additional control parameter to influence the droplet shape and the tension in the sheet, which we quantify by radial profiles of the azimuthal and radial elastic stresses. We further demonstrate how the focal length of a liquid lens can be tuned by varying the applied tension. Finally, stretching the sheet along one direction, elongates the droplet, and the sheet shows folds and dimples.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The manuscript formulates a boundary element method (BEM) extending prior approaches to compute equilibrium shapes of droplets on thin elastic sheets under three boundary-condition protocols. For clamped sheets it reports that lens morphology depends on sheet thickness via morphological metrics and radial stress profiles; isotropic in-plane stretching is shown to tune droplet shape and focal length through applied tension; uniaxial stretching produces elongated droplets with folds and dimples.

Significance. If the numerical solutions faithfully represent the coupled elasto-capillary equilibrium, the results establish thickness and isotropic tension as practical control parameters for liquid-lens geometry on elastic supports, with direct relevance to tunable optics and soft-matter patterning. The BEM framework itself constitutes a reusable computational tool for this class of problems.

major comments (1)
  1. [Abstract / Methods] Abstract and § (Methods/BEM formulation): the central claims on thickness dependence and tension-tuned focal length rest on the fidelity of the BEM extension for fluid-elastic coupling, yet no explicit validation is presented against analytical limits such as the rigid-substrate spherical-cap solution or the small-deformation plate-theory limit for thin sheets. Without such checks, discretization or contact-line artifacts cannot be ruled out as the source of the reported morphological trends.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their detailed review and for highlighting the importance of validating the BEM extension. We address the single major comment below and will incorporate the requested checks in a revised manuscript.

read point-by-point responses

  1. Referee: [Abstract / Methods] Abstract and § (Methods/BEM formulation): the central claims on thickness dependence and tension-tuned focal length rest on the fidelity of the BEM extension for fluid-elastic coupling, yet no explicit validation is presented against analytical limits such as the rigid-substrate spherical-cap solution or the small-deformation plate-theory limit for thin sheets. Without such checks, discretization or contact-line artifacts cannot be ruled out as the source of the reported morphological trends.

    Authors: We agree that explicit validation against analytical limits strengthens the manuscript. Our BEM formulation extends prior work on fluid-structure problems, but the current text does not include direct comparisons for the coupled droplet-sheet case. In the revised version we will add a dedicated validation subsection (likely in Methods) that (i) recovers the spherical-cap solution on a rigid substrate in the limit of large sheet thickness and (ii) reproduces the small-deformation plate-theory predictions for thin sheets under isotropic tension. These checks will be performed with the same discretization and contact-line treatment used for the reported results, thereby confirming that the observed thickness and tension trends are not numerical artifacts. revision: yes

Circularity Check

0 steps flagged

Minor self-citation for BEM extension; central morphological results are independent numerical outputs with no reduction to fitted inputs or self-citation chains

full rationale

The paper formulates and applies a boundary element method (BEM) to solve the coupled elasto-capillary problem for three boundary-condition protocols. Central claims (thickness dependence of lens shape for clamped sheets; tension-tuned focal length under isotropic stretch) are obtained directly as outputs of these numerical solutions. The abstract notes an 'extension of our earlier approaches,' constituting a self-citation, but this pertains only to the solver formulation and is not invoked to establish uniqueness theorems, forbid alternatives, or force any reported quantity by construction. No parameters are fitted to data subsets and then presented as predictions, no ansatzes are smuggled via citation, and no renaming of known results occurs. The derivation chain is a standard numerical discretization of the physical model, rendering the results independent of the inputs. This is the expected non-circular outcome for a simulation study.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the work rests on standard continuum elasto-capillary equations and the authors' prior BEM formulation.

pith-pipeline@v0.9.1-grok · 5703 in / 926 out tokens · 21128 ms · 2026-06-28T20:23:02.117897+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

58 extracted references · 39 canonical work pages

  1. [1]

    In: Fluid-Structure Interac- tions in Low-Reynolds-Number Flows, pp

    Duprat, C., Stone, H.A.: Chapter 6: Elas- tocapillarity. In: Fluid-Structure Interac- tions in Low-Reynolds-Number Flows, pp. 193–246. The Royal Society of Chemistry, Cambridge (2016). https://doi.org/10.1039/ 9781782628491-00193

    2016

  2. [2]

    Bico, J., Reyssat, ´E., Roman, B.: Elasto- capillarity: When surface tension deforms elastic solids. Annu. Rev. Fluid Mech. 50(1), 629–659 (2018) https://doi.org/10. 1146/annurev-fluid-122316-050130 14

    2018

  3. [3]

    Andreotti, B., Snoeijer, J.H.: Statics and dynamics of soft wetting. Annu. Rev. Fluid Mech.52(1), 285–308 (2020) https://doi.org/ 10.1146/annurev-fluid-010719-060147

    work page doi:10.1146/annurev-fluid-010719-060147 2020

  4. [4]

    Shuttleworth, R.: The surface tension of solids. Proc. Phys. Soc. Sect. A63(5), 444–457 (1950) https://doi.org/10.1088/ 0370-1298/63/5/302

    1950

  5. [5]

    M¨ uller, P., Sa´ ul, A.: Elastic effects on sur- face physics. Surf. Sci. Rep.54(5), 157– 258 (2004) https://doi.org/10.1016/j.surfrep. 2004.05.001

    work page doi:10.1016/j.surfrep 2004

  6. [6]

    EPL113, 66001 (2016) https://doi.org/ 10.1209/0295-5075/113/66001

    Andreotti, B., Snoeijer, J.H.: Soft wetting and the Shuttleworth effect, at the cross- roads between thermodynamics and mechan- ics. EPL113, 66001 (2016) https://doi.org/ 10.1209/0295-5075/113/66001

    work page doi:10.1209/0295-5075/113/66001 2016

  7. [7]

    Chen, L., Bonaccurso, E., Gambaryan- Roisman, T., Starov, V., Koursari, N., Zhao, Y.: Static and dynamic wetting of soft sub- strates. Curr. Opin. Colloid Interface Sci.36, 46–57 (2018) https://doi.org/10.1016/j.cocis. 2017.12.001

    work page doi:10.1016/j.cocis 2018

  8. [8]

    Park, S.J., Weon, B.M., Lee, J.S., Lee, J., Kim, J., Je, J.H.: Visualization of asymmet- ric wetting ridges on soft solids with x-ray microscopy. Nat. Comm.5(1), 4369 (2014) https://doi.org/10.1038/ncomms5369

    work page doi:10.1038/ncomms5369 2014

  9. [9]

    Xu, Q., Jensen, K.E., Boltyanskiy, R., Sarfati, R., Style, R.W., Dufresne, E.R.: Direct mea- surement of strain-dependent solid surface stress. Nat. Comm.8(1), 555 (2017) https: //doi.org/10.1038/s41467-017-00636-y

    work page doi:10.1038/s41467-017-00636-y 2017

  10. [10]

    Gorcum, M., Andreotti, B., Snoeijer, J.H., Karpitschka, S.: Dynamic solid surface ten- sion causes droplet pinning and depinning. Phys. Rev. Lett.121, 208003 (2018) https: //doi.org/10.1103/PhysRevLett.121.208003

    work page doi:10.1103/physrevlett.121.208003 2018

  11. [11]

    Style, R.W., Che, Y., Park, S.J., Weon, B.M., Je, J.H., Hyland, C., German, G.K., Power, M.P., Wilen, L.A., Wettlaufer, J.S., Dufresne, E.R.: Patterning droplets with durotaxis. Proc. Natl. Acad. Sci. U.S.A. 110(31), 12541–12544 (2013) https://doi. org/10.1073/pnas.1307122110

    work page doi:10.1073/pnas.1307122110 2013

  12. [12]

    Langmuir35(23), 7571–7577 (2019) https://doi.org/10.1021/ acs.langmuir.8b02037

    Alert, R., Casademunt, J.: Role of substrate stiffness in tissue spreading: Wetting transi- tion and tissue durotaxis. Langmuir35(23), 7571–7577 (2019) https://doi.org/10.1021/ acs.langmuir.8b02037

    2019

  13. [13]

    ACS Appl

    Lee, I.-N., Dobre, O., Richards, D., Ballestrem, C., Curran, J.M., Hunt, J.A., Richardson, S.M., Swift, J., Wong, L.S.: Photoresponsive hydrogels with photoswitchable mechanical properties allow time-resolved analysis of cellular responses to matrix stiffening. ACS Appl. Mater. Interfaces10(9), 7765–7776 (2018) https://doi.org/10.1021/acsami.7b18302

    work page doi:10.1021/acsami.7b18302 2018

  14. [14]

    Aland, S., Mokbel, D.: A unified numeri- cal model for wetting of soft substrates. Int. J. Numer. Methods Eng.122(4), 903–918 (2021) https://doi.org/10.1002/nme.6567

    work page doi:10.1002/nme.6567 2021

  15. [15]

    Gomez, H., Velay-Lizancos, M.: Thin-film model of droplet durotaxis. Eur. Phys. J. Special Top.229(2), 265–273 (2020) https: //doi.org/10.1140/epjst/e2019-900127-x

    work page doi:10.1140/epjst/e2019-900127-x 2020

  16. [16]

    Science354(6317), 1240–1241 (2016) https://doi.org/10.1126/science

    Huang, J., Juszkiewicz, M., Jeu, W.H., Cerda, E., Emrick, T., Menon, N., Rus- sell, T.P.: Capillary wrinkling of floating thin polymer films. Science317(5838), 650– 653 (2007) https://doi.org/10.1126/science. 1144616

    work page doi:10.1126/science 2007

  17. [17]

    Soft Matter9, 8289–8296 (2013) https://doi.org/10.1039/ C3SM50736J

    Toga, K.B., Huang, J., Cunningham, K., Russell, T.P., Menon, N.: A drop on a float- ing sheet: boundary conditions, topography and formation of wrinkles. Soft Matter9, 8289–8296 (2013) https://doi.org/10.1039/ C3SM50736J

    2013

  18. [18]

    Schroll, R.D., Adda-Bedia, M., Cerda, E., Huang, J., Menon, N., Russell, T.P., Toga, K.B., Vella, D., Davidovitch, B.: Capillary deformations of bendable films. Phys. Rev. Lett.111, 014301 (2013) https://doi.org/10. 1103/PhysRevLett.111.014301

    2013

  19. [19]

    Schulman, R.D., Dalnoki-Veress, K.: Liquid droplets on a highly deformable membrane. Phys. Rev. Lett.115, 206101 (2015) https: 15 //doi.org/10.1103/PhysRevLett.115.206101

    work page doi:10.1103/physrevlett.115.206101 2015

  20. [20]

    compass needles

    Schulman, R.D., Ledesma-Alonso, R., Salez, T., Rapha¨ el, E., Dalnoki-Veress, K.: Liquid droplets act as “compass needles” for the stresses in a deformable membrane. Phys. Rev. Lett.118, 198002 (2017) https://doi. org/10.1103/PhysRevLett.118.198002

    work page doi:10.1103/physrevlett.118.198002 2017

  21. [21]

    Smith-Mannschott, K., Xu, Q., Heyden, S., Bain, N., Snoeijer, J.H., Dufresne, E.R., Style, R.W.: Droplets sit and slide anisotropi- cally on soft, stretched substrates. Phys. Rev. Lett.126, 158004 (2021) https://doi.org/10. 1103/PhysRevLett.126.158004

    2021

  22. [22]

    Schulman, R.D., Dalnoki-Veress, K.: Droplets capped with an elastic film can be round, elliptical, or nearly square. Phys. Rev. Lett. 121, 248004 (2018) https://doi.org/10.1103/ PhysRevLett.121.248004

    2018

  23. [23]

    Science359(6377), 775–778 (2018) https:// doi.org/10.1126/science.aao1290

    Kumar, D., Paulsen, J.D., Russell, T.P., Menon, N.: Wrapping with a splash: High- speed encapsulation with ultrathin sheets. Science359(6377), 775–778 (2018) https:// doi.org/10.1126/science.aao1290

    work page doi:10.1126/science.aao1290 2018

  24. [24]

    Soft Mat- ter14, 4913–4934 (2018) https://doi.org/10

    Davidovitch, B., Vella, D.: Partial wetting of thin solid sheets under tension. Soft Mat- ter14, 4913–4934 (2018) https://doi.org/10. 1039/C8SM00323H

    2018

  25. [25]

    Shanahan, M.E.R.: Contact angle equilib- rium on thin elastic solids. J. Adhesion 18(4), 247–267 (1985) https://doi.org/10. 1080/00218468508080461

    1985

  26. [26]

    Shanahan, M.E.R.: Equilibrium of liq- uid drops on thin plates; plate rigidity and stability considerations. J. Adhesion 20(4), 261–274 (1987) https://doi.org/10. 1080/00218468708074946

    1987

  27. [27]

    Kozyreff, G., Davidovitch, B., Prasath, S.G., Palumbo, G., Brau, F.: Effect of external ten- sion on the wetting of an elastic sheet. Phys. Rev. E107, 035101 (2023) https://doi.org/ 10.1103/PhysRevE.107.035101

    work page doi:10.1103/physreve.107.035101 2023

  28. [29]

    Li, Z., Ren, W.: The motion of a thin drop on an elastic sheet. J. Fluid Mech. 1022, 4 (2025) https://doi.org/10.1017/jfm. 2025.10779

    work page doi:10.1017/jfm 2025

  29. [30]

    Brinker, M., Dittrich, G., Richert, C., Lakner, P., Krekeler, T., Keller, T.F., Huber, N., Huber, P.: Giant electrochemical actuation in a nanoporous silicon-polypyrrole hybrid material. Sci. Adv.6(40), 1483 (2020) https: //doi.org/10.1126/sciadv.aba1483

    work page doi:10.1126/sciadv.aba1483 2020

  30. [31]

    Brinker, M., Huber, P.: Wafer-scale electroactive nanoporous silicon: Large and fully reversible electrochemo- mechanical actuation in aqueous electrolytes. Adv. Mater., 2105923 (2021) https://doi.org/10.1002/adma.202105923

    work page doi:10.1002/adma.202105923 2021

  31. [32]

    Berge, B., Peseux, J.: Variable focal lens controlled by an external voltage: An appli- cation of electrowetting. Eur. Phys. J. E 3(2), 159–163 (2000) https://doi.org/epje/ v3/p159(e9028)

    2000

  32. [33]

    Biomicrofluidics4(3), 031501 (2010) https://doi.org/10.1063/1.3460392

    Nguyen, N.-T.: Micro-optofluidic lenses: A review. Biomicrofluidics4(3), 031501 (2010) https://doi.org/10.1063/1.3460392

    work page doi:10.1063/1.3460392 2010

  33. [34]

    Frontiers in Robotics and AI8, 166 (2021) https://doi.org/10.3389/frobt

    Chen, L., Ghilardi, M., Busfield, J.J.C., Carpi, F.: Electrically tunable lenses: A review. Frontiers in Robotics and AI8, 166 (2021) https://doi.org/10.3389/frobt. 2021.678046

    work page doi:10.3389/frobt 2021

  34. [35]

    In: 2020 IEEE Aerospace Conference, pp

    Fogle, F., Cierny, O., Vale Pereira, P., Kam- merer, W., Cahoy, K.: Miniature optical steerable antenna for intersatellite communi- cations liquid lens characterization. In: 2020 IEEE Aerospace Conference, pp. 1–13 (2020). https://doi.org/10.1109/AERO47225.2020. 9172448

    work page doi:10.1109/aero47225.2020 2020

  35. [36]

    Nature 591(7848), 142–146 (2021) https://doi.org/ 10.1038/s41586-020-2992-3

    Agudo-Canalejo, J., Schultz, S.W., Chino, H., Migliano, S.M., Saito, C., Koyama- Honda, I., Stenmark, H., Brech, A., May, A.I., Mizushima, N., Knorr, R.L.: Wet- ting regulates autophagy of phase-separated 16 compartments and the cytosol. Nature 591(7848), 142–146 (2021) https://doi.org/ 10.1038/s41586-020-2992-3

    work page doi:10.1038/s41586-020-2992-3 2021

  36. [37]

    Journal of Cell Biology220(10), 202103175 (2021) https://doi.org/10.1083/jcb.202103175

    Kusumaatmaja, H., May, A.I., Knorr, R.L.: Intracellular wetting mediates con- tacts between liquid compartments and membrane-bound organelles. Journal of Cell Biology220(10), 202103175 (2021) https://doi.org/10.1083/jcb.202103175

    work page doi:10.1083/jcb.202103175 2021

  37. [38]

    Nature634(8036), 1204–1210 (2024) https: //doi.org/10.1038/s41586-024-07990-0

    Wang, Y., Li, S., Mokbel, M., May, A.I., Liang, Z., Zeng, Y., Wang, W., Zhang, H., Yu, F., Sporbeck, K., Jiang, L., Aland, S., Agudo-Canalejo, J., Knorr, R.L., Fang, X.: Biomolecular condensates mediate bend- ing and scission of endosome membranes. Nature634(8036), 1204–1210 (2024) https: //doi.org/10.1038/s41586-024-07990-0

    work page doi:10.1038/s41586-024-07990-0 2024

  38. [39]

    Mokbel, M., Mokbel, D., Liese, S., Weber, C., Aland, S.: A simulation method for the wetting dynamics of liquid droplets on deformable membranes. SIAM J. Sc. Com- put46(6), 806–829 (2024) https://doi.org/ 10.1137/24M1641142

    work page doi:10.1137/24m1641142 2024

  39. [40]

    Soft Matter17, 2454 (2021) https://doi.org/ 10.1039/d0sm02082f

    Grawitter, J., Stark, H.: Steering droplets on substrates using moving steps in wettability. Soft Matter17, 2454 (2021) https://doi.org/ 10.1039/d0sm02082f

    work page doi:10.1039/d0sm02082f 2021

  40. [41]

    Soft Mat- ter17, 9469 (2021) https://doi.org/10.1039/ d1sm01113h

    Grawitter, J., Stark, H.: Droplets on sub- strates with oscillating wettability. Soft Mat- ter17, 9469 (2021) https://doi.org/10.1039/ d1sm01113h

    2021

  41. [42]

    Soft Matter20, 3161 (2024) https://doi.org/10.1039/d4sm00213j

    Grawitter, J., Stark, H.: Steering droplets on substrates with plane-wave wettability pat- terns and deformations. Soft Matter20, 3161 (2024) https://doi.org/10.1039/d4sm00213j

    work page doi:10.1039/d4sm00213j 2024

  42. [43]

    Cambridge University Press, Cam- bridge (1992)

    Pozrikidis, C.: Boundary Integral and Sin- gularity Methods for Linearized Viscous Flow. Cambridge University Press, Cam- bridge (1992)

    1992

  43. [44]

    Dover Publications, Mineola/NY (2005)

    Kim, S., Karrila, S.J.: Microhydrodynamics. Dover Publications, Mineola/NY (2005)

    2005

  44. [45]

    Soft Matter13, 3544–3555 (2017) https://doi

    Schaaf, C., Stark, H.: Inertial migration and axial control of deformable capsules. Soft Matter13, 3544–3555 (2017) https://doi. org/10.1039/C7SM00339K

    work page doi:10.1039/c7sm00339k 2017

  45. [46]

    Soft Matter17, 4804–4817 (2021) https://doi.org/10.1039/ D1SM00276G

    Patel, K., Stark, H.: A pair of particles in inertial microfluidics: effect of shape, softness, and position. Soft Matter17, 4804–4817 (2021) https://doi.org/10.1039/ D1SM00276G

    2021

  46. [47]

    Skalak, R., Tozeren, A., Zarda, R.P., Chien, S.: Strain energy function of red blood cell membranes. Biophys. J.13, 245 (1973) https: //doi.org/10.1016/S0006-3495(73)85983-1

    work page doi:10.1016/s0006-3495(73)85983-1 1973

  47. [48]

    Springer, ??? (2012)

    Kr¨ uger, T.: Computer Simulation Study of Collective Phenomena in Dense Suspen- sions of Red Blood Cells Under Shear. Springer, ??? (2012). https://doi.org/10. 1007/978-3-8348-2376-2

    2012

  48. [49]

    theory and possible experiments

    Helfrich, W.: Elastic properties of lipid bilay- ers. theory and possible experiments. Z. Naturforsch. C28, 693–703 (1973) https:// doi.org/10.1515/znc-1973-11-1209

    work page doi:10.1515/znc-1973-11-1209 1973

  49. [50]

    Kantor, Y., Nelson, D.R.: Crumpling transi- tion in polymerized membranes. Phys. Rev. Lett.58(26), 2774–2777 (1987) https://doi. org/10.1103/PhysRevLett.58.2774

    work page doi:10.1103/physrevlett.58.2774 1987

  50. [51]

    Gompper, G., Kroll, D.M.: Random surface discretizations and the renormalization of the bending rigidity. J. Phys. I6(10), 1305–1320 (1996) https://doi.org/10.1051/jp1:1996246

    work page doi:10.1051/jp1:1996246 1996

  51. [52]

    Moffatt, H.K.: Viscous and resistive eddies near a sharp corner. J. Fluid Mech. 18, 1–18 (1964) https://doi.org/10.1017/ S0022112064000015

    1964

  52. [53]

    Alinovi, E., Bottaro, A.: A boundary ele- ment method for stokes flows with interfaces. J. Comp. Phys.356, 261–281 (2018) https: //doi.org/10.1016/j.jcp.2017.12.004

    work page doi:10.1016/j.jcp.2017.12.004 2018

  53. [54]

    Langmuir13, 7293 (1997) https: //doi.org/10.1021/la970825v 17

    Ruijter, M.J., De Coninck, J., Blake, T.D., Clarke, A., Rankin, A.: Contact angle relax- ation during the spreading of partially wet- ting drops. Langmuir13, 7293 (1997) https: //doi.org/10.1021/la970825v 17

    work page doi:10.1021/la970825v 1997

  54. [55]

    Extreme Mech

    Lee, Y.-J., Lim, S.-M., Yi, S.-M., Lee, J.- H., Kang, S.-g., Choi, G.-M., Han, H.N., Sun, J.-Y., Choi, I.-S., Joo, Y.-C.: Aux- etic elastomers: Mechanically programmable meta-elastomers with an unusual poisson’s ratio overcome the gauge limit of a capacitive type strain sensor. Extreme Mech. Lett.31, 100516 (2019) https://doi.org/10.1016/j.eml. 2019.100516

    work page doi:10.1016/j.eml 2019

  55. [56]

    Course of Theoreti- cal Physics, Vol

    Landau, L.D., Lifshitz, E.M.: Theory of Elasticity, 3rd edn. Course of Theoreti- cal Physics, Vol. 7. Elsevier Butterworth- Heinemann, Oxford (1986). https://doi.org/ 10.1016/B978-0-08-057069-3.50003-6

    work page doi:10.1016/b978-0-08-057069-3.50003-6 1986

  56. [57]

    Fortes, M.A.: Microscopic and macroscopic contact angles. J. Chem. Soc., Faraday Trans. I78, 101–107 (1982) https://doi.org/10. 1039/F19827800101

    1982

  57. [58]

    Measurement Science and Technology8(6), 601–605 (1997) https: //doi.org/10.1088/0957-0233/8/6/003

    Rheims, J., K¨ oser, J., Wriedt, T.: Refractive- index measurements in the near-ir using an abbe refractometer. Measurement Science and Technology8(6), 601–605 (1997) https: //doi.org/10.1088/0957-0233/8/6/003

    work page doi:10.1088/0957-0233/8/6/003 1997

  58. [59]

    Cheerios effect

    Vella, D., Mahadevan, L.: The “Cheerios effect”. Am. J. Phys.73, 817–825 (2005) https://doi.org/10.1119/1.1898523 18

    work page doi:10.1119/1.1898523 2005