Elastocapillarity: When Surface Tension Deforms Elastic Solids

Abstract

Although negligible at large scales, capillary forces may become dominant for submillimetric objects. Surface tension is usually associated with the spherical shape of small droplets and bubbles, wetting phenomena, imbibition or the motion of insects at the surface of water. However, beyond liquid interfaces, capillary forces can also deform solid bodies in their bulk as observed in recent experiments with very soft gels. Capillary interactions, which are responsible for the cohesion of sand castles, can also bend slender structures and induce the bundling of arrays of fibres. Thin sheets can finally spontaneously wrap liquid droplets within the limit of the constraints dictated by di↵erential geometry. The aim of this review is to describe the di↵erent scaling parameters and characteristic lengths involved in "elastocapillarity". We focus successively on three main configurations: • 3D, deformations induced in bulk solids • 1D, bending and bundling of rod-like structures • 2D, bending and stretching of thin sheets Although each configuration would deserve a detailed review, we hope our broad description will provide a general view on elastocapillarity.

Self‐Folding Using Capillary Forces

“…The orientation of the force (per unit length) at the contact line ensues from the local balance of the interfacial tensions associated with the three interfaces (i.e., liquid/solid (LS), liquid/vapor (LV), and solid/vapor (SV) interfaces in the example above). Such balance is captured by the Neumann relation, notably for liquid/liquid interfaces (e.g., droplets floating on liquid beds) and for liquids resting on very soft substrates . For droplets sitting on substrates whose rigidity makes the force component normal to the substrate inconsequential (i.e., Young's modulus E → ∞), the interfacial force balance at the triple contact line can be simplified and is typically expressed via the horizontal projection of the Neumann force triangle, better known as the Young–Dupré equation γ SV = γ SL + γ cos θ.…”

“…In this size range (e.g., below ≈2.7 mm for water at room temperature), the capillary force can additionally become comparable and even larger than the shear and/or bending stiffness of soft materials. As a result, capillary forces can microscopically deform soft substrates in close proximity of contact lines; and capillary torques can macroscopically bend or fold unconstrained substrates larger than a corresponding (bending) elastocapillary length ensuing from the ratio of bending stiffness B and surface tension . For the common case of a curved sheet or membrane of thickness t and Poisson's ratio ν, B = Et 3 /12(1 − ν 2 ).…”

“…Such balance is captured by the Neumann relation, notably for liquid/liquid interfaces (e.g., droplets floating on liquid beds) and for liquids resting on very soft substrates. [59] For droplets sitting on substrates whose rigidity makes the force component normal to the substrate inconsequential (i.e., Young's modulus E → ∞), the Adv. Mater.…”

“…As a result, capillary forces can microscopically deform soft substrates in close proximity of contact lines; [75] and capillary torques can macroscopically bend or fold unconstrained substrates larger than a corresponding (bending) elastocapillary length L B/ EC γ ≈ ensuing from the ratio of bending stiffness B and surface tension. [59,76,77] For the common case of a curved sheet or membrane of thickness t and Poisson's ratio ν, B = Et 3 /12(1 − ν 2 ). Under the rubric of elastocapillarity, [59] the latter effects are relevant for the micromechanics and applications of self-folding reviewed below.…”

Packing transitions in the elastogranular confinement of a slender loop

“…Properly accounting for global shape changes, specifically the coupling between large deformations of the thin loop structure and the evolution of its granular containing space, requires that we bring material considerations into the analysis. Drawing on similarities with other physical systems utilizing a thin loop [7,32,[37][38][39] hints at the existence of an additional length scale originating with the slender structure.…”