# Interdisciplinary Applied Mathematics

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versibility. It was reported in (Verheijen and Prins, 1999), that charge trapping may degrade the surface and affect electrowetting adversely. However, it was recommended that silicone oil be used to treat the surface in order to minimize the undesired contact angle hysteresis caused by the trapped charge. Another limitation of electrowetting is the saturation in the change of contact angle caused by applying an electrical potential. We discuss this in the next section, but we first present a more rigorous derivation of the Young-Lippmann equation.

###### 8.6.1 Generalized Young-Lippmann Equation

The Young-Lippmann equation can be derived rigorously based on the energy minimization equation that leads to the generalized Young equation (8.8). To this end,    following    the    analysis    of    (Shapiro    et    al.,    2003a),    we

consider a conducting liquid droplet residing on a dielectric solid and obtain the total energy of the system. It consists of the interfacial energy given in equation (8.9) as well as the dielectric energy stored in the solid and the externally applied charging source. The latter is twice the energy stored in the dielectric solid but with opposite sign. Therefore, we need to compute the potential energy only in the very thin solid dielectric layer. An electric field almost perpendicular to the surface area Als is proportional to the voltage V and inversely proportional to the layer thickness h. The energy stored in the solid dielectric is then

1    e V2

Ede = 7tes{V/h)2hAls = 7rR2 ~777~ sin2 0,

2    2h

where es is the dielectric constant of the solid. The total energy in this case is then

E(R, в) = R2(n sin2d(Yis — Ysg — esV2/(2h)) + 2nYgi(1 — cos в)) ,

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