Page:Scientific Papers of Josiah Willard Gibbs.djvu/296

In the preceding paragraph it is shown that the surface of contact of phases $A$ and $B$ is stable under certain circumstances, with respect to the formation of a thin sheet of the phase $C$. To complete the demonstration of the stability of the surface with respect to the formation of the phase $C$, it is necessary to show that this phase cannot be formed at the surface in lentiform masses. This is the more necessary, since it is in this manner, if at all, that the phase is likely to be formed, for an incipient sheet of phase $C$ would evidently be unstable when $\sigma _{AB}<\sigma _{AC}+\sigma _{BC}$, and would immediately break up into lentiform masses.

It will be convenient to consider first a lentiform mass of phase $C$ in equilibrium between masses of phases $A$ and $B$ which meet in a plane surface. Let figure 10 represent a section of such a system through the centers of the spherical surfaces, the mass of phase $A$ lying on the left of $DEH'FG$, and that of phase $B$ on the right of $DEH''FG$. Let the line joining the centers cut the spherical surfaces in $H'$ and $H''$, and the plane of the surface of contact of $A$ and $B$ in $I$. Let the radii of $EH'F$ and $EH''F$ be denoted by $r',r''$, and the segments $IH',IH''$, by $x',x''$. Also let $IE$, the radius of the circle in which the spherical surfaces intersect, be denoted by $R$. By a suitable application of the general condition of equilibrium we may easily obtain the equation

$\sigma _{AC}{\frac {r'-x'}{r'}}+\sigma _{BC}{\frac {r''-x''}{r''}}=\sigma _{AB},$

(561)

which signifies that the components parallel to $EF$ of the tension $\sigma _{AC}$ and $\sigma _{BC}$ are together equal to $\sigma _{AB}$. If we denote by $W$ the amount of work which must be expended in order to form such a lentiform mass as we are considering between masses of indefinite extent having the phases $A$ and $B$, we may write

$W=M-N,$

(562)

where $M$ denotes the work expended in replacing the surface between $A$ and $B$ by the surfaces between $A$ and $C$ and $B$ and $C$, and $N$ denotes the work gained in replacing the masses of phases $A$ and $B$ by the mass of phase $C$. Then

$M=\sigma _{AC}s_{AC}+\sigma _{BC}s_{BC}-\sigma _{AB}s_{AB},$

(563)

where $s_{AC},s_{BC},s_{AB}$ denote the areas of the three surfaces concerned; and

$N=V'(p_{C}-p_{A})+V''(p_{C}-p_{B}),$

(564)