# Interdisciplinary Applied Mathematics

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Microparticles, from 20 nm to about 3 pm can also be used to fabricate microdevices, such as pumps and valves, which in turn can be used for microfluidic control. Several studies have focused on fabricating self-assembled structures using paramagnetic particles carried by liquids in microchannels (Hayes et    al.,    2001; Doyle et    al.,    2002).    The    ability    to    form    supraparticle

structures and precisely control their arrangement and motion externally by magnetic fields could lead to many novel applications such as microoptical filters and gratings, but also to new materials and new micro- and

FIGURE 1.6. Colloidal micropumps using 3-micron silica microspheres. (a) Lobe movement of a gear pump. (b) Peristaltic pump. The channel is 6 microns, and the motion is induced by optical traps. (Courtesy of D. Marr.)

nanofabrication protocols (Furst et al., 1998; Hayes et al., 2001; Whitesides and Grzybowski, 2002).

Colloidal micropumps and colloidal microvalves are already in existence and have been used for active microfluidic control. For example, in (Terray et al., 2002), latex microspheres were manipulated by optical traps to pump fluids. These devices are    about    the    size    of    a    human red    blood    cell;    see

Figure 1.6. These colloidal micropumps are based on positive-displacement pumping techniques and operate by imparting forward motion to small volumes of fluid. The two micropumps shown in Figure 1.6 induce motions of 2 to 4 p,m/s with corresponding flow rate of 0.25 nl/hour; see (Terray et al., 2002) for details.

##### 1.2 The Continuum Hypothesis

Important details of the operation of micromachines involve complex dynamical processes and unfamiliar physics. The dynamics of fluids and their interaction with surfaces in microsystems are very different from those in large systems. In microsystems the flow is granular for liquids and rarefied for gases, and the walls “move.” In addition, other phenomena such as thermal creep, electrokinetics, viscous heating, anomalous diffusion, and even quantum and chemical effects may be important (Chan et al., 2001). In particular, the material of the wall is very important in the dynamics; for example, a simple graphite submicron bearing exhibits complex vibrational modes and interacts differently with the fluid than does a diamondoid submicron bearing. Similarly, for gas microflows the material surface, i.e., its type and roughness, determines the fluid-wall interactions, which lead to definition of thermal and momentum accommodation coefficients (see Section 2.2.2).

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