Analogies between elastic and capillary interfaces^*

In this paper we exploit some analogies between flows near capillary interfaces and near elastic interfaces. We first consider the elastohydrodynamics of a ball bearing and the motion of a gas bubble inside a thin channel. It is shown that there is a strong analogy between these two lubrication problems, and the respective scaling laws are derived side by side. Subsequently, the paper focuses on the limit where the involved elastic interfaces become extremely soft. It is shown that soft gels and elastomers, like liquids, can be shaped by their surface tension. We highlight some recent advances on this class of elastocapillary phenomena.

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Figure 1

Dimple formation in fluid and elastic interfaces. Shown on the left is a dimple below an impacting liquid drop. The air between the drop and the rigid wall is squeezed out of the thin gap and the resulting lubrication pressure induces a dimple measured by interferometry. (Reprinted with permission from [1]. Copyright 2012 by the American Physical Society.) On the right are very similar dimple profiles observed when a rigid sphere approaches an elastically compliant substrate. (Reprinted with permission from [2]. Copyright 2015 by the American Physical Society.)

Figure 2

Three similar lubrication problems. (a) Long bubble moving inside a channel [8]. (Adapted from [8], reproduced with permission from Cambridge University Press.) (b) Flexible plate that leaves a thin liquid film [23]. (Adapted from [23], reproduced with permission from Cambridge University Press.) (c) Sliding Hertzian contact [12, 16]. (Reproduced from [16] with the permission of AIP Publishing.) The red dashed zone in all three panels indicates the inner region of height $H$ and length $L$, which is described by a lubrication flow.

Figure 3

Zoom of the lubrication zone, shown as the dashed regions in Fig. 2. This inner region has a horizontal scale $L$ and a vertical scale $H$. In the frame comoving with the deformable interface, denoted by $h\left(x\right)$, the wall is moving with velocity $U$.

Figure 4

Surface tension effects in elastic solids. (a) Thin cylinders of very soft agar gel can develop a Rayleigh-Plateau instability driven by the solid surface tension. The stiffness is decreased from top ($E=81$ Pa) to bottom ($E=36$ Pa). (Reprinted with permission from [17]. Copyright 2010 by the American Physical Society.) (b) Rounding of sharp edges on a soft gelatine after it is withdrawn from a PDMS mould. (Reproduced from [19] with permission from the Royal Society of Chemistry.)

Figure 5

Wetting a soft solid. (a) X-ray image of a three-phase contact line of a water drop on a soft silicon gel. The water drop induces substantial elastic deformation of the substrate. The scale bar indicates $5\mu \mathrm{m}$. (Reprinted from [25], Nature Publishing Group, licensed under a Creative Commons Attribution 4.0 International Licence.) (b) Calculated profile of a wetting ridge that moves with a prescribed velocity $V$. The contact line motion induces a rotation of the wetting ridge by an angle $\phi$. (c) Rotation angle $\phi$ as a function of contact line velocity $V$, for water drops spreading on a silicon gel, comparing experiments (symbols) and theory (red solid line). [Panels (b) and (c) reprinted from [45], Nature Publishing Group, licensed under a Creative Commons Attribution 4.0 International Licence.]

Figure 6

Contact between a rigid indenter and a soft elastic layer. (a) Gels of varying stiffness indented by a rigid sphere, imaged by confocal microscopy. (Reprinted with permission from [48], Nature Publishing Group.) (b) Theoretical profiles of varying stiffness, including the definition of the contact width $\ell$. All profiles were computed with the very same Young angle at the three-phase contact line. [Image courtesy of Karpitschka (see [52] for details).] Stiff layers follow the classical JKR law (18) for solid adhesion (${\gamma }_s/E\ll \ell$), while soft layers behave as wetting liquids (${\gamma }_s/E\gg \ell$). These results suggest that Young's angle applies also for gels of finite stiffness.