The Conditions of Equilibrium where Several Surfaces Intersect

Consider the case where three different phases make contact:

Figure 35-6: Intersection, or triple line, where three phases make contact in space.

Considering

$$d\ensuremath{{U}^{\mbox{surf}}}= \gamma^{\alph... ...{\beta \zeta} dA^{\beta \zeta} + \gamma^{\zeta \alpha} dA^{\zeta \alpha}$$

must be a minimum, one may derive two relations for the angles of contact:

$$\input{equations/YE-gen}$$ (35-27)

which is the general equation the angles at a triple line, called Young's equation, where

Figure 35-7: The definitions of the terms in Young's Equation, Equation 35-27.

which is equivalent to the ``force balance'' where each of the $\gamma$ are considered the forces applied to the vertex.

For the special case in which one interface is constrained to be flat, as in

Figure 35-8: o

One can derive (by force balance most simply)

$$\input{equations/YE-flat}$$ (35-28)

(note--the above is two separate equations that appear to be run together: cos phi is the ratio of a difference over the liquid-vapor surface tension)
which is Young's equation for flat surfaces.

$$\phi \equiv 0 < \phi < 180^\circ$$ (35-29)