Capillary rise between elastic sheets

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By Sung Hoon Kang

Title: Capillary rise between elastic sheets

Reference: Ho-Young Kim AND L. Mahadevan, J. Fluid Mech. 548, 141-150 (2006).

Soft matter keywords

capillary rise, surface tension, hydrophilic, three-phase junction, contact angle, Jurin’s law

Abstract from the original paper

When a paintbrush is dipped into a pot of paint and pulled out, surface tension forces cause the individual hairs in the brush to coalesce even as the brush becomes impregnated with paint. We study a simple model of this elastocapillary interaction in the context of the surface-tension-driven vertical rise of a liquid between two long flexible hydrophilic sheets that are held a small distance apart at one end. We provide an analytic theory for the static shapes of the sheets as well as the liquid rise height which is different from that of the classical law of Jurin, and show that our experiments are quantitatively consistent with the theory.

Soft matter example

(not done yet) When a paintbrush is taken out of a liquid, the hairs of the brush coalesce because the surface tension at the air-water-hair interface wants to minimize the energy of the system. The total energy of the system is the sum of the elastic energy of the deformed hairs, gravity and the capillary energy of the liquid-vapor interface. For a same liquid-vapor interface, the pointedness of the tip of the brush is determined by the relative stiffness of the bristles. The rise of the liquid through the brush hair is related to the length of a paint stroke by influencing the holding power of a brush. In this paper, the authors studied how the stiffness of elastic sheets influences the capillary rise.

The rise and fall of the liquid is determined by the balance of the hydrostatic pressure with surface tension, which is known as a Jurin's law [1]. For a rigid wettable capillary of radius r in a reservoir of liquid (density ρ and interfacial tension σ), the capillary rise length (or Jurin length) lJ =2σ/ρgr (g: the acceleration due to gravity).

For a capillary of radius r ∼100 μm wetted by water, this leads to lJ ∼15 cm. If the capillary tube is sufficiently flexible, Jurin’s law is modified because of the ability of the interfacial forces to deform the tube. A variant of this problem has been well-studied as a model for how flexible capillaries or airways may collapse as a bubble moves through them (Grotberg & Jensen 2004). In this regime, tension in the tube dominates the elastic response to hydrodynamic forces as fluid mensicus moves through it. However, if the tube is replaced by flexible sheets or hairs with free ends as in a paint brush, the dominant elastic contribution is due to sheet or hair bending. Here we focus on this qualitatively different bending-dominated regime and consider a simple model brush: two flexible sheets of length L clamped at a distance w apart, and free at the other end. When this system is dipped in a liquid, capillary forces associated with the high curvature of the mensicus between the sheets lead to a negative pressure that causes the liquid to rise between the sheets. Simultaneously, the sheets come together under the influence of this same negative pressure. For short and stiff sheets, the liquid rise is assisted by a slight decrease in the gap between the sheets, suggesting but a perturbative correction to the law of Jurin. However, when the sheets are long and flexible, they are nearly stuck together at the free ends with almost no liquid in between, showing a qualitatively different behaviour from the Jurin regime. After this work was underway, we became aware of similar work exploring this latter regime using a combination of experiments and scaling arguments (Bico et al. 2004). Our approach complements this by investigating all the regimes quantitatively from both a theoretical and experimental viewpoint.

Fig. 1 Schematic and shape of the sheets when (a) the sheets are relatively stiff so that the ends are separate: coordinate system and experimental image of glass cover slips, 24mm long and initially 1mm apart, after they were slowly withdrawn out of water; (b) the sheets are relatively soft so that ends are in contact: coordinate system and experimental image of glass sheets, 42mm long and initially 0.6mm apart, slowly withdrawn out of water.


(not done yet)

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