# Interdisciplinary Applied Mathematics

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spreading front follows the periodicity imposed by the pattern unless the width of the hydrophilic stripe is below about 50 p,m.

In general, for heterogeneous substrates the wetting of a liquid on a solid is quite complex. If a liquid droplet is residing partially on a hydrophilic stripe and partially on a hydrophobic patch, there may be a variation of the contact angle along the contact line. Typically, the droplet may migrate to regions with small contact angle. This was demonstrated in the experiments of    (Darhuber    et    al.,    2000)    with    glycerol    on    patterned    silicon

surfaces. The chemical modulation leading to alternating hydrophilic and hydrophobic stripes was achieved using a self-assembled monolayer of octadecyltrichlorosilane (OTS) 3 nm in thickness. The contact angle of glycerol on OTS was measured in (Darhuber et al., 2000), to be about 95°, while that    on    SiO2    was    less    than    5°.    The    hydrophilic    SiO2    stripes in    the

experiments were 7 to 15 p,m in width, while the width of the microstructures formed varied between 10 and 47 p,m. If the heterogeneous stripes are much larger than    the    droplet    radius,    then the    contact    angle    will    take    the

appropriate value defined by Young’s equation for the relevant stripe.

In (Gau et al., 1999), the equilibrium shape of a liquid droplet in contact with a completely hydrophilic surface was examined as a function of the liquid volume per unit length. It was found that if it exceeds a critical value, the droplet forms a pronounced bulge along the contact line. In general, the shape and distribution of liquid droplets on a microchannel wall depend on the contact angle, the surface tension, and as the volume of liquid deposited. If the amount of liquid deposited on a relatively long microchannel is subcritical, then the contact lines will stay within the hydrophilic stripe, and it will behave as if the substrate were homogeneous; i.e., it is susceptible to the natural instability we described earlier manifested as capillary

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