Interdisciplinary Applied Mathematics

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which upon minimization leads to the Young-Lippmann equation (Lipp-mann, 1875)


cos в — 7sg ~ 7lses    =0.    (8.20)


Ygi    2Ygih


We note that this equation does not depend on the radius of the droplet R, as in the standard Young equation that expresses triple-line force balance (equation (8.6)).


The above analysis is not valid for apolar liquids, e.g., a silicone oil atop a conducting solid surface. This case is analyzed in (Shapiro et al., 2003a), assuming that the droplet is an insulator with dielectric constant e;. The electric field varies in this case as V/R, so the stored electric energy is


Contact Angle versus U for Experiment and Theory


140



130


THEORY (matched): Ao = 100


—Theory (High liq resistance): Ao=50


. — . Theory (Low liq resistance): Ao = 350 — Young-Lipp (no resistance): Ao = <*>

70


60


1    2    3


Electrowetting Number U = es V2 / h


0


4


FIGURE 8.11. Contact angle saturation: The experimental data are taken from four different devices (see (Moon et al., 2002)). The theoretical curves are taken from (Shapiro et al., 2003a). They show energy-minimization-based results for three different values of the electrical resistivity in the liquid. The middle value is appropriate for water and closely matches the observed experimental results. (Courtesy of B. Shapiro.)


1/26, R3(V/R)2ade(e) <x R. Here, ade(0) is the shape factor, which cannot be computed analytically as for the previous cases; however, the following empirical fit was developed in (Shapiro et al., 2003a):

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