# Interdisciplinary Applied Mathematics

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Thermodynamic analysis at the interface leads to the so-called Lippmann’s equation

7 = 7o — ^cV2,    (8.18)

where V is the voltage difference, 70 is the surface tension at zero voltage, and c is the capacitance per unit area. This equation is applicable to all three forms of electrowetting, but c relates to the EDL for CEW and EW, and to    the    dielectric    layer    for    EWOD.    Also,    for    CEW, 7 is    the    interfa

cial surface tension between the two media, whereas for EW and EWOD,

dielectric coating

gas

EDL

electrolyte

aqueous

liquid

‘■electrode

FIGURE 8.8. Electrocapillary principles for (a) continuous electrowetting; (b) electrowetting, and (c) electrowetting-on-dielectric (Lee et al., 2002). (Courtesy of C.-J. Kim.)

Y =    Ysi    is the    solid-liquid    surface tension.    In    the    following,    we    will    derive

the generalized Young-Lippmann equation that governs equilibrium based on energy minimization principles. An alternative derivation for constant surface tension and neglecting gravity is obtained using Young’s equation of equilibrium at the triple contact line (equation (8.6)) and substituting the Lippmann equation (8.18) to get

cV 2

cos в = cos во H—,    (8.19)

2Ygi

where 90 is the zero-voltage contact angle. We note the strong dependence of the contact angle on the voltage, which may lead to a switch between a hydrophobic and hydrophilic surface, and also its independence on DC or AC current. The above equation is a special case of the generalized

Teflon

Electrolyte EDL-

Substrate

Electrode (Au)