Mesoscale Self-Assembly: Capillary Bonds and Negative Menisci
Zach Wissner-Gross (February 23, 2009)
Mesoscale Self-Assembly: Capillary Bonds and Negative Menisci 
Ned Bowden, Scott R. J. Oliver, and George M. Whitesides
The Journal of Physical Chemistry B, 2000, 104 (12), 2714-2724
Soft matter Keywords
Capillary forces, self-assembly, menisci, capillary length
In their paper, Whitesides and coworkers float a layer of millimeter-sized PDMS  hexagons between perfluorodecalin (PFD) and water. They further pre-treat different edges of the hexagons, making them either hydrophilic (by oxidizing the edges with a plasma cleaner) or hydrophobic (by protecting the edges from oxidation with an additional cured layer of PDMS). By carefully agitating the solutions, the authors are able to induce self-assembly over the course of minutes to hours, and observe how structure varies with different patterns of hydrophobicity/hydrophilicity.
This article was written as a sister article to another publication . In that paper, the authors used PDMS with a density of 1.05 g/cm<math>^3</math>, barely greater than than of water. Here, the authors load their PDMS hexagons with aluminum oxide to a density of 1.86 g/cm<math>^3</math>, just less than that of PFD. The authors spend much of the paper discussing the theoretical and experimental differences observed between these two setups (i.e., in which case hydrophilic or hydrophobic interactions dominate). But, as the authors conclude in their abstract: "The arrays that formed from the heavy (1.86 g/cm<math>^3</math>) hexagons with a particular pattern of hydrophilic faces were analogous to the arrays that formed from the light (1.05 g/cm<math>^3</math>) hexagons with that pattern of hydrophobic faces."
Soft matter discussion
The paper is really composed of two parts: why the hexagons self-assemble, and what structures they can assemble into. All the physics takes place is the former, so I will go through their discussion on why self-assembly occurs.
The physics behind self-assembly is shown in Figures 1 and 2. Two forces are responsible for the observed self-assembly. First, energy is released when interfacial area between the PFD and water decreases -- we can call this surface energy. Second, energy is released when liquid returns to the level of the interface -- this is simply gravitational energy. Since the authors are working at the millimeter (i.e., meso-) scale, they are in fact working in a regime where the forces produced be these potential energy gradients are comparable.
Now let's look closer at Figure 1. The "light" hexagons are largely submerged in the water due to their lower density. Therefore, the PFD will creep up farther into water layer to cover hydrophobic faces than the water creeps down into the PFD layer to cover hydrophilic faces. Since the PFD is migrating farther across the interface, more surface energy and gravitation energy are being stored in this case, so the forces between hydrophobic faces should be stronger. When two hydrophobic faces then come into contact, the force is attractive, since by moving closer together the hexagons reduce the total contact area between water and PFD, and there is less penetration across the interface for both liquids.
As for the "heavy" hexagons, the logic works equally well, but with the hydrophilic and hydrophobic faces reversed (see Figure 2). Thus, a "light" [1,2] hexagon (i.e., a hexagon with adjacent hydrophobic edges 1 and 2 and hydrophilic edges 3, 4, and 5) should assemble similarly to a [1,2,3,4] heavy hexagon. The authors report