Equilibrium of an elastically confined liquid drop

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Kwon Drop Diagram.
Diagrams of the liquid drop between the substrate and the elastic surface, with photographs from experiment. In case (a), the glass sheet is relatively stiff due to its large height (1.85 mm) above the parafilm substrate. In case (b), the sheet acts more elastically because its left end is only 0.28 mm above the substrate, and as a result its right end is in contact with the substrate.

The authors examine the behavior of a liquid drop confined between a rigid substrate from below and an elastic plate from above. The drop will spread and thin between the plates, while also pulling the free end of the flexible sheet down. The authors work out theoretical predictions for the shape of the elastic sheet subject to physical and geometric boundary conditions. A few interesting cases are noted, which correspond to different boundary conditions. Experimental data is provided which corroborates the theoretical predictions, and the authors discuss the application of the analysis to subjects as varied as Chinese calligraphy and the stiction of microcantilevers.

General Information

Keywords: surface force, meniscus, surface tension, contact angle, stiction

Authors: Hyuk-Min Kwon, Ho-Young Kim, Jerome Puell, and L. Mahadevan.

Date: May 7, 2008.

School of Mechanical and Aerospace Engineering, Seoul National University, Seoul 151-744, Republic of Korea

School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, USA

Journal of Applied Physics 103, 093519 (2008). [1]


The authors first develop the theory corresponding to a drop of liquid in equilibrium between a rigid flat plate and another sheet that is allowed to bend. The length scales are assumed sufficiently small that gravitational effects are negligible. Only the equilibrium case is considered, so that it is assumed the system is at rest and free energy is minimal. It is assumed that both the substrate and the elastic glass sheet are hydrophilic, and one end of the flexible plate is fixed at a height H above the substrate. The interfacial forces due to the liquid will bend the elastic plate toward the substrate, until either the free edge of the plate is slightly closer to the substrate than H, or the free edge is actually in contact with the substrate. The analysis of these cases is very similar, except for some different geometric boundary conditions.

By considering volume conservation, contact angle geometry, curvature of the liquid and the corresponding pressure, shear force, and surface tension, the authors determine that the boundary condition problem simplifies to a key equation:

<math> h' = \frac{L^4}{l_a^2 H R_0}S(x - x_m) </math>

where <math>S</math> is the Heaviside function, as well as a number of geometric boundary conditions. Here <math>h</math> is the deformation of the flexible plate, <math>L</math> is the length of the plates, and <math>R_0</math> is the radius of curvature of the liquid at its edge where the plates are furthest apart. <math>l_a</math> is the adhesion or bending length and <math>x_m</math> is the point along the horizontal axis where the meniscus of the liquid is found.

The authors used MATLAB to solve the above equation, subject to ten boundary conditions corresponding to the case that the free end of the flexible plate is not in contact with the substrate. They showed that the theory is consistent with the notion that more spreading of the liquid occurs when there is high interfacial tension and the top plate is soft and close to the substrate. Spreading is also greater when the contact angles are smaller and when the volume of liquid is larger, or when a dimensionless "stiffness" parameter <math>\eta</math> is smaller. Experimental data is provided that supports the equation derived theoretically.

Kwon Sheet Shapes.
Theoretically predicted plate shapes and meniscus locations for the cases that (I) the top plate is not long enough to come into contact with the substrate, (II) the top plate is long enough to come into contact with the substrate but not long enough to lie tangent to it, and (III) the top plate is long enough that its free end lies tangent to the substrate.

The authors then explore the interesting case of decreasing the stiffness parameter <math>\eta</math> or, equivalently, increasing the length of the plates <math>L</math> until the top plate comes into contact with the substrate. One boundary condition changes in that the distance between one end of the plate and the substrate is now zero, but if we continue to extend this parameter, the angle between the top plate and the substrate will also decrease until it is zero. At this special condition, the flexible plate is tangent to the substrate at the point where the two are in contact and also, presumably, at all points farther to the right along the substrate. This case too has slightly different boundary conditions. The theoretically predicted shapes of the flexible plate corresponding to these different cases can be seen in the figure at right. The authors provided experimental data that supports their theory concerning these other geometric cases as well.

The authors comment on the simplicity of the geometry considered for this analysis, and suggest that much more work could be done on similar problems for more complex geometries, such as the wetting of a canvas by paint on the hairs of a paintbrush, for instance.

Connection to soft matter

The theory and experimental work presented in this article tackles a simple and fundamental problem in soft matter concerning capillary forces and surface tension. The behavior of a drop of liquid between two plates is in some ways an even simpler problem than that of the classic vertical capillary. The drop wets the surfaces by spreading until it has minimized the surface energy. The contact angles are relevant in determining how far the drop will spread.

This problem is also interesting in that it takes a new look at the classic problem of a meniscus. In most cases, a meniscus location is predicted assuming solid surfaces in the problem are rigid and static. Here, one of the plates is allowed to bend, and the authors have demonstrated that the location of a meniscus can still be predicted quite accurately. Experimental results support their theoretical predictions.

Additionally, the fact that one of the surfaces is elastic in the problem means that ideas from soft matter are needed to analyze the equilibrium behavior of the flexible glass sheet in this scenario. A dimensionless "stiffness" parameter can be derived in terms of geometric and physical parameters that corresponds to how flexible a glass sheet is. This parameter could be applied to various other problems in soft matter involving flexible plates.