Page:Scientific Papers of Josiah Willard Gibbs.djvu/298

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262

EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES.

of insensible magnitude. {The diminution of the radii with increasing values of $p_{C}-p_{A}$ is indicated by equation (565).} Hence, no mass of phase $C$ will be formed until one of these limits is reached. Although the demonstration relates to a plane surface between $A$ and $B$ , the result must be applicable whenever the radii of curvature have a sensible magnitude, since the effect of such curvature may be disregarded when the lentiform mass is sufficiently small.

The equilibrium of the lentiform mass of phase $C$ is easily proved to be unstable, so that the quantity $W$ affords a kind of measure of the stability of plane surfaces of contact of the phases $A$ and $B$ .^[1]

Essentially the same principles apply to the more general problem in which the phases $A$ and $B$ have moderately different pressures, so that their surfaces of contact must be curved, but the radii of curvature have a sensible magnitude.

In order that a thin film of the phase $C$ may be in equilibrium between masses of the phases $A$ and $B$ , the following equations must be satisfied:—

\sigma _{AC}(c_{1}+c_{2})=p_{A}-p_{C},

\sigma _{BC}(c_{1}+c_{2})=p_{C}-p_{B},

where $c_{1}$ and $c_{2}$ denote the principal curvatures of the film, the centers of positive curvature lying in the mass having the phase $A$ . Eliminating $c_{1}+c_{2}$ , we have

\sigma _{BC}(p_{A}-p_{C})=\sigma _{AC}(p_{C}-p_{B}),

or

p_{C}={\frac {\sigma _{BC}p_{A}+\sigma _{AC}p_{B}}{\sigma _{BC}+\sigma _{AC}}}\cdot

(571)

It is evident that if $p_{C}$ has a value greater than that determined by this equation, such a film will develop into a larger mass; if $p_{C}$ has a less value, such a film will tend to diminish. Hence, when the phases $A$ and $B$ have a stable surface of contact.

↑ If we represent phases by the position of points in such a manner that coexistent phases (in the sense in which the term is used on page 96) are represented by the same point, and allow ourselves, for brevity, to speak of the phases as having the positions of the points by which they are represented, we may say that three coexistent phases are situated where three series of pairs of coexistent phases meet or intersect. If the three phases are all fluid, or when the effects of solidity may be disregarded, two cases are to be distinguished. Either the three series of coexistent phases all intersect,—this is when each of the three surface tensions is less than the sum of the two others,—or one of the series terminates where the two others intersect,—this is where one surface tension is equal to the sum of the others. The series of coexistent phases will be represented by lines or surfaces, according as the phases have one or two independently variable components. Similar relations exist when the number of components is greater, except that they are not capable of geometrical representation without some limitation, as that of constant temperature or pressure or certain constant potentials.

[1] If we represent phases by the position of points in such a manner that coexistent phases (in the sense in which the term is used on page 96) are represented by the same point, and allow ourselves, for brevity, to speak of the phases as having the positions of the points by which they are represented, we may say that three coexistent phases are situated where three series of pairs of coexistent phases meet or intersect. If the three phases are all fluid, or when the effects of solidity may be disregarded, two cases are to be distinguished. Either the three series of coexistent phases all intersect,—this is when each of the three surface tensions is less than the sum of the two others,—or one of the series terminates where the two others intersect,—this is where one surface tension is equal to the sum of the others. The series of coexistent phases will be represented by lines or surfaces, according as the phases have one or two independently variable components. Similar relations exist when the number of components is greater, except that they are not capable of geometrical representation without some limitation, as that of constant temperature or pressure or certain constant potentials.

[1]