Capillary rise between elastic sheets
Original entry: Sung Hoon Kang, APPHY 226, Spring 2009
Title: Capillary rise between elastic sheets
Reference: Ho-Young Kim AND L. Mahadevan, J. Fluid Mech. 548, 141-150 (2006).
Soft matter keywords
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
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 . 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). If the capillary tube is sufficiently flexible, the interfacial force can deform the tube which results in modification of Jurin’s law. In this case, as the meniscus of the fluid move through a capillary tube, the surface tension governs the elastic response to hydrodynamic forces. If we replace the tube by flexible sheets or hairs with free ends as the case of a paint brush, the bending of the sheet or the hair gives the dominant elastic contribution.
In order to study different bending-dominated regime, the authors did experiments using flexible sheets of two different stiffness with a same length L, fixed apart with distance a as shown in Fig. 1. As discussed above, when this model system is put in a liquid, capillary force causes the liquid to rise between the sheets and the sheets are brought together. For stiff sheets (Fig. 1 (a)), the liquid rise is facilitated by a slight decrease in the gap between the sheets, suggesting a small perturbation to Jurin's law. However, for flexible sheets (Fig. 1 (b)), they are nearly attached together at the free ends with almost no liquid between, showing a qualitatively different behaviour from the Jurin's model.
Then, they did scaling analysis to understand the phenomena and showed good agreement between their theoretical results and the experimental results. The details of the scaling analysis can be found in the paper and similar study was also conducted by another group using a combination of experiments and scaling arguments .
This study was very interesting to me showing how the mechanical property can affect capillary effects and it gave us insights to understand how the mechanical property and a geometry of a material can influence on complex hierarchical assembly behaviors of polymeric nanostructures .
 J. Jurin, Phil. Trans. 30, 739–747 (1718).
 J. Bico, B. Roman, L. Moulin, A. Boudaoud, Nature 432, 690 (2004).
 B. Pokroy, S. H. Kang, L. Mahadevan, J. Aizenberg, Science 323, 237-240 (2009).