Difference between revisions of "Capillary rise between elastic sheets"

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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.  
 
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
+
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) l<sub>J</sub> =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.  
rise length (or Jurin length) l<sub>J</sub> =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.
+
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.  
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
+
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 [2].
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.
+
  
 
[[Image:Fig-01.jpg|thumb|center|600px| '''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.
 
[[Image:Fig-01.jpg|thumb|center|600px| '''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.

Revision as of 02:15, 8 March 2009

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). 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 [2].

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.

References

(not done yet)

Bico, J., Roman, B., Moulin, L. & Boudaoud, A. 2004 Nature 432, 690.

Cohen, A. E. & Mahadevan, L. 2003 Proc. Natl Acad. Sci. 100, 12141–12146.

Grotberg, J. B. & Jensen, O. E. 2004 Annu. Rev. Fluid Mech. 36, 121–147.

Hosoi, A. E. & Mahadevan, L. 2004 Phys. Rev. Lett. 93, 137802.

Jurin, J. 1718 Phil. Trans. 30, 739–747.

Landau, L. D. & Levich, B. 1942 Acta Physicochim. URSS 17, 42–54.

Landau, L. D. & Lifshitz, E. M. 1986 Theory of Elasticity, 3rd Edn. Pergamon.

Senturia, S. D. 2001 Microsystem Design. Kluwer.