# Interdisciplinary Applied Mathematics

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ade(0) « 0.0592 + 0.00120 + 0.0022 tan(1.71 — в),

where в is in radians. The following Young equation is obtained for this case by energy minimization:

Ysg — Yis . V2 2 + cos в

COS вb Q w —-:—V-

Ygi    2YglR 2n sin в

аёе(в)д(в)

~df

0. (8.21)

The quantity q(9) has been defined in equation (8.7). We note that there is 1/R dependence in this equation, unlike all the other cases we have presented so far. In general, the contact angle variation with the voltage is very weak for dielectric droplets.

A    more    realistic    model    for    the microdroplets    is    that    of    an imperfect

conductor residing on an imperfect insulator. This case was studied numerically in (Shapiro et al., 2003a), and it leads to the contact angle saturation phenomenon observed in experiments. Many physical mechanisms can cause such saturation, including charge-trapping, liquid resistivity, and electrolysis. The model developed in (Shapiro et al., 2003a), identifies liquid resistivity as the leading cause of contact angle saturation. For sufficiently small в the interfacial energy beats the electrical energy, and thus the total energy goes to infinity; i.e., a minimum is never achieved around the complete wetting state. This model agrees very well with the experimental data reported in (Moon et al., 2002), where a single and a double layer of Teflon and silicon dioxide were used for the solid dielectric layer. In Figure 8.11 a comparison of the numerical results of Shapiro et al. is presented against several sets of experimental data for a certain value of the solid-to-liquid resistance. If the value of this ratio is infinity, then we recover the perfect conducting droplet described by the standard Young-Lippmann formula; however, no contact angle saturation is predicted for that case. The nondimensional voltage used in the plot is defined as U = esV2/(hygl), where h is the thickness of the dielectric layer. Also, A0 measures the resistivity ratio of solid to liquid.

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