Interdisciplinary Applied Mathematics

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Using the above concepts, self-assembled monolayer chemistry was used in (Zhao et al., 2001), to pattern surface-free energies in microchannel networks. In particular, it was found that when the pressure was maintained at sufficiently low levels, the liquid followed the hydrophilic pathways but above a threshold, the liquid crossed the boundary between the hydrophilic and hydrophobic regions. Two liquid streams separated by a gas membrane were transported side by side, allowing reactive components to be diffused from one stream to the other. This type of configuration can be used as a network    of    microchannels    with    “virtual”    walls.    The    condition    for    rup

ture of the virtual wall is that the angle 9b at the hydrophilic-hydrophobic boundary be equal to the (advancing) contact angle of the liquid on the nonpolar surface 9n. The maximum pressure that these virtual walls can sustain for a straight stream is


APmax = -j- sin(6»„ — 90°), h

where h is the film thickness. For curved pathways there is a limit on the curvature of the flow determined by the virtual wall rupture condition and the extra pressure difference due to curvature. An ultraviolet photopatter-ing method was developed in (Zhao et al., 2001), to pattern surface-free energies inside microchannels in situ within minutes, and applications for gas-liquid reactions in microchips as well as for pressure-sensitive switches were demonstrated. Virtual walls can lead to multiple and diverse functionality on microchips that may be difficult to achieve with other methods.

8.6 Electrocapillary

Similar to thermocapillary, where temperature is the controlling mechanism, in electrocapillary, or electrowetting electric potential can be used to change the surface tension and thus cause flow motion. Compared to thermocapillary, electrocapillary is much more energy efficient (power consumption is about 10 mW), and various applications have demonstrated induced speeds over 100 mm/s, in contrast to less than about 1 mm/s in thermocapillary. Some of the applications have demonstrated addressable liquid handling, i.e., droplet routing but also droplet cutting and merging (Cho et al., 2003). Other applications include optical switches (Beni et al., 1982), rotating micromotors (Lee and Kim, 2000), and liquid lenses (Kwon and Lee, 2001). From the fundamental standpoint, electrowetting is the low-frequency limit of the electromechanical response of an aqueous liquid to an electric field. In contrast, dielectrophoresis can be thought of as the high-frequency limit. (We note that, in general, the phenomenon of dielectrophoresis requires only a nonuniform field, and it exists also under DC conditions.) An analysis of the two mechanisms and a related experiment were reported in (Jones et al., 2003).

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