Modeling Menisci and Capillary Forces from the Millimeter to the Micrometer Size Range
Original Entry by Michelle Borkin, AP225 Fall 2009
Bartosz A. Grzybowski, Ned Bowden, Francisco Arias, Hong Yang, and George M. Whitesides. J. Phys. Chem. B, 2001, 105 (2), pp 404–412.
The research presented in this paper investigates capillary interactions from the millimeter to sub-millimeter scales (0.1 to 10 mm) to gain a better understanding of self-assembly driven by capillary forces at the macroscopic scale. They examine the shape of PFD (perfluorodecalin) and water interfaces (i.e. immiscible fluid-fluid interface) and its interactions with hydrophobic and hydrophilic materials. By studying the shapes of these interfaces, energy profiles can be derived to characterize the different types of interactions (e.g. attraction, repulsion). The investigators use both computer simulations as well as experimental evidence to draw conclusions. For the computer modeling, when finding an analytical solution using the Laplace equation applied to two infinite surfaces was impossible, the menisci shapes were determined numerically using Surface Evolver which applies a Finite Element Method (FEM) to model the contours of the menisci. For the experimental measurements of the menisci shapes, the water layer was solidified with gelatin, then a cast was made using a UV-curable polymer (NOA), and then the cast was imaged with both optical and SEM microscopes. Menisci of varying sizes from millimeter to submillimeter were created and computer modeled in order to determine how the decay length scaled with the menisci dimensions.
Self-assembly, a very important aspect of many biological processes at the molecular scale, is investigated in this paper at the "mesoscale". Self-assembly in this regime is produced primarily through capillary forces which arise from the interactions between the menisci. As shown in Figure 1, two positive (i.e. concave) menisci will attract, two negative (i.e. convex) menisci will attract, and one positive with one negative meniscus will repel. The positive meniscus is produced with a hydrophobic surface (i.e. the water is not attracted to the non-polar surface thus resulting in a high contact angle), and a negative meniscus is produced with a hydrophilic surface (i.e. the water is attracted to the polarized surface). The two primary contributions to the free energy in this set-up are the surface energy (proportional to the surface area of the interface) and the gravitational energy (unfavorable at large separations). Since the strength and direction of the forces are determined by the meniscus shape, the primary objective of the computer modeling and experiments were to determine the shapes and resulting energy profiles.
Figures 9 and 10 show some of the experimental and computational results. Figure 9 displays some of the sample microscope images along side their computer models counter-parts for comparison. Figure 10 shows the resulting FEM derived energy profiles including the specific contributions from capillary and gravitational energies. Based on their observations, the length/size scale of the menisci scale linearly the length/size scale of the hydrophobic/hydrophilic surface objects. The capillary forces at the interface scale linearly with the length/size scale as well. However it is important, as the authors say, to note that these results only apply at the millimeter to submillimeter regime since at smaller scales gravity is insignificant compared to the capillary forces.