Skip to main content Accessibility help

Deformation of an elastic substrate due to a resting sessile droplet



On a sufficiently soft substrate, a resting fluid droplet will cause significant deformation of the substrate. This deformation is driven by a combination of capillary forces at the contact line and the fluid pressure at the solid surface. These forces are balanced at the surface by the solid traction stress induced by the substrate deformation. Young's Law, which predicts the equilibrium contact angle of the droplet, also indicates an a priori radial force balance for rigid substrates, but not necessarily for soft substrates that deform under loading. It remains an open question whether the contact line transmits a non-zero force tangent to the substrate surface in addition to the conventional normal (vertical) force. We present an analytic Fourier transform solution technique that includes general interfacial energy conditions, which govern the contact angle of a 2D droplet. This includes evaluating the effect of gravity on the droplet shape in order to determine the correct fluid pressure at the substrate surface for larger droplets. Importantly, we find that in order to avoid a strain singularity at the contact line under a non-zero tangential contact line force, it is necessary to include a previously neglected horizontal traction boundary condition. To quantify the effects of the contact line and identify key quantities that will be experimentally accessible for testing the model, we evaluate solutions for the substrate surface displacement field as a function of Poisson's ratio and zero/non-zero tangential contact line forces.



Hide All
[1] Andreotti, B., Bäumchen, O., Boulogne, F., Daniels, K. E., Dufresne, E. R., Perrin, H., Salez, T., Snoeijer, J. H. & Style, R. W. (2016) Soft capillarity: When and how does surface tension deform soft solids? Soft Matter 12, 29932996.
[2] Andreotti, B. & Snoeijer, J. H. (2016) Soft wetting and the Shuttleworth effect, at the crossroads between thermodynamics and mechanics. Europhys. Lett. 113, 66001.
[3] Bostwick, J. B., Shearer, M. & Daniels, K. E. (2014) Elastocapillary deformations on partially-wetting substrates: Rival contact-line models. Soft Matter 10, 73617369.
[4] Chaudhury, M. K. & Whitesides, G. M. (1992) How to make water run uphill. Science 256, 15391541.
[5] Dhir, V., Gao, D. & Morley, N. B. (2004) Understanding magnetic field gradient effect from a liquid metal droplet movement. J. Fluids Eng. 126, 120124.
[6] Gomba, J. M. & Perazzo, A. P. (2012) Closed-form expression for the profile of partially wetting two-dimensional droplets under gravity. Phys. Rev. E 86, 23752.
[7] Harland, B., Walcott, S. & Sun, S. X. (2011) Adhesion dynamics and durotaxis in migrating cells. Phys. Biol. 8, 6665.
[8] Hui, C. Y. & Jagota, A. (2014) Deformation near a liquid contact line on an elastic substrate. Proc. R. Soc. A 470, 20140085.
[9] Jerison, E. R., Xu, Y., Wilen, L. A. & Dufresne, E. R. (2011) Deformation of an elastic substrate by a three-phase contact line. Phys. Rev. Lett. 106, 186103.
[10] Limat, L. (2012) Straight contact lines on a soft, incompressible solid. Eur. Phys. J. E 35, 113.
[11] Marichev, V. A. (2011) The Shuttleworth equation: Its modifications and current state. Prot. Metals Phys. Chem. Surf. 47, 2530.
[12] Onuki, A. & Kanatani, K. (2005) Droplet motion with phase change in a temperature gradient. Phys. Rev. E 72, 27844.
[13] Park, S. J., Weon, B. M., Lee, J. S., Lee, J. & Kim, J. (2014) Visualization of asymmetric wetting ridges on soft solids with X-ray microscopy. Nature Commun. 5, 4369.
[14] Pericet-Cámara, R., Best, A., Butt, H. J. & Bonaccurso, E. (2008) Effect of capillary pressure and surface tension on the deformation of elastic surfaces by sessile liquid microdrops: An experimental investigation. Langmuir 24, 1056510568.
[15] Soutas-Little, R. W. (1999) Elasticity. Dover Publications, Mineola, NY.
[16] Style, R. W. & Dufresne, E. R. (2012) Static wetting on deformable substrates, from liquids to soft solids. Soft Matter 8, 71777184.
[17] 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. (2013) Patterning droplets with durotaxis. Proc. Nat. Acad. Sci. 110, 1254112544.
[18] Style, R. W., Jagota, A., Hui, C. Y. & Dufresne, E. R. (2017) Elastocapillarity: Surface tension and the mechanics of soft solids. Annu. Rev. Condens. Matter Phys. 8, 99118.
[19] Teh, S. Y., Lin, R., Hung, L. H. & Lee, A. P. (2008) Droplet microfluidics. Lab on a Chip 8, 198200.
[20] Weijs, J. H., Andreotti, B. & Snoeijer, J. H. (2013) Elasto-capillarity at the nanoscale: On the coupling between elasticity and surface energy in soft solids. Soft Matter 9, 84948503.
[21] Weijs, J. H., Snoeijer, J. H. & Andreotti, B. (2014) Capillarity of soft amorphous solids: A microscopic model for surface stress. Phys. Rev. E 89, 042408.


Type Description Title
Supplementary materials

Bardall supplementary material
Bardall supplementary material 1

 PDF (165 KB)
165 KB


Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed