# Interdisciplinary Applied Mathematics

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FIGURE 8.7. Top    view    profiles of    liquid    droplets    as    a    function    of    the    liquid

volume for 0 = 60° in (a) and 0 = 30° in (b). (Courtesy of S. Troian.)

breakup. The critical value of the volume per unit length of the stripe is (Darhuber et al., 2000)

V    w2

— <5

l ~ 4sin20

в — — sin 2(9

where w is the channel width and the contact angle 0 is measured in radians. Exceeding the critical value of Vjl will effectively increase the value of the contact angle, and thus it may delay breakup, but it may create bulges, as in the work of (Gau et al., 1999).

Such effects    were    studied    by    (Darhuber    et    al.,    2000),    who    performed

numerical simulations using the program SURFACE EVOLVER, which is based on energy minimization techniques; this program was developed by (Brakke, 1992). The liquid surface was triangulated, and the total energy of the system was expressed as a function of the vertices of the triangular elements. Standard minimization techniques were employed (e.g., conjugate gradients), and different boundary conditions were incorporated in the minimization process. Curved interfaces produce a pressure excess inside the liquid (see equation (8.2)). Figure 8.7 shows profiles from the simulations of Darhuber et    al.    for    a single microdroplet residing    on    a    1 fim    x    8 fim    hy

drophilic region for contact angles 0 = 60° and 0 = 30°, while the rest of the surrounding area is completely hydrophobic, i.e., 0 = 180°. For the same volume of liquid, the smaller liquid with contact angle spreads faster, as expected. As the volume of the microdroplet increases the liquid fills up the stripe but remains confined within the stripe, although it forms a bulge for the larger contact angle. This implies incomplete wetting of the hydrophilic strip, which can be interpreted as surface energy imbalance between the liquid-vapor and the liquid-solid systems, with the latter acquiring higher energy levels. The morphological features of liquid microdroplets residing over heterogeneous substrates depend critically on the surface tension value. In (Darhuber et al., 2000), parametric studies for different surface tensions but constant liquid volume were also performed. By reducing the surface tension, complete spreading of the microdroplet over the hydrophilic patch is obtained. This is expected, since reduction in the surface tension leads to a reduction of the corresponding contact angle; see equation (8.6).

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