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

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260
EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES.

In the preceding paragraph it is shown that the surface of contact of phases and is stable under certain circumstances, with respect to the formation of a thin sheet of the phase . To complete the demonstration of the stability of the surface with respect to the formation of the phase , 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 would evidently be unstable when , and would immediately break up into lentiform masses.

It will be convenient to consider first a lentiform mass of phase in equilibrium between masses of phases and 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 lying on the left of , and that of phase on the right of . Let the line joining the centers cut the spherical surfaces in and , and the plane of the surface of contact of and in . Let the radii of and be denoted by , and the segments , by . Also let , the radius of the circle in which the spherical surfaces intersect, be denoted by . By a suitable application of the general condition of equilibrium we may easily obtain the equation

(561)

which signifies that the components parallel to of the tension and are together equal to . If we denote by 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 and , we may write

(562)

where denotes the work expended in replacing the surface between and by the surfaces between and and and , and denotes the work gained in replacing the masses of phases and by the mass of phase . Then

(563)

where denote the areas of the three surfaces concerned; and

(564)