# Interdisciplinary Applied Mathematics

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dE

~dR

0.

dE

dR +

d9

Neglecting any evaporation, the above equation states that at equilibrium, if the contact angle increases, then the radius must also change. Specifically, assuming a constant droplet volume, this implies that the radius should decrease. This volume v also depends on R, 9, and thus dv =    dR+ |^d9.

From geometric considerations we have that, v = nR3 (2/3 — (3/4) cos 9 + (1/12) cos 39), and since volume is constant

dv = 0 ^ dR = R

2cos2(0/2) cot(0/2) 2 + cot 9

d9 = Rq(9) d9,

(8.7)

where the last equation defines the function q(9). By substituting equation (8.7) into the equilibrium equation (dE = 0), we obtain the generalized Young’s equation:

2 +cos9 (dE    5Я

27ri?2 sin 9 ) dRRq( ^ + d9

0.

(8.8)

This equation can accommodate modeling of any energy function with multiple contributions due to interfacial effects, temperature or voltage effects, gravity, etc.    In    the    case in    which E    contains    contributions    only    due    to

interfaces between gas-liquid-solid, we recover the standard Young equation    (8.6). To    illustrate    this    we    compute    this    interfacial    energy,    assuming

constant surface tensions, from

Ei = YglAgl + (7ls 7sg ) Als,

where Agl = 2nR2(1 — cos 9) is the surface area of the gas-liquid interface and Als = nR2 sin29 is the surface area of the liquid-solid interface. The interfacial potential energy is then

Ei = R2[2nygl(1 — cos 9) + n sin2 9(yis — Ysg)].    (8.9)

By substituting equation (8.9) into the generalized Young’s equation (8.8), we obtain the classical Young’s equation (8.6).

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