Page:Scientific Papers of Josiah Willard Gibbs.djvu/351

This page has been proofread, but needs to be validated.

EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES.

315

other component substances may then be denoted by our usual symbols (see page 235),

\epsilon _{S(1)},

⁠

\eta _{S(1)}

⁠

\Gamma _{2(1)}

⁠

\Gamma _{3(1)},

⁠etc.

Let the quantity or be defined by the equation

\sigma =\epsilon _{S(1)}-t\eta _{S(1)}-\mu _{2}\Gamma _{2(1)}-\mu _{3}\Gamma _{3(1)}-{\text{etc.,}}

(659)

in which $t$ denotes the temperature, and $\mu _{2},\mu _{3}$ , etc. the potentials for the substances specified at the surface of discontinuity. As in the case of two fluid masses (see page 257), we may regard $\sigma$ as expressing the work spent in forming a unit of the surface of discontinuity—under certain conditions, which we need not here specify—but it cannot properly be regarded as expressing the tension of the surface. The latter quantity depends upon the work spent in stretching the surface, while the quantity or depends upon the work spent in forming the surface. With respect to perfectly fluid masses, these processes are not distinguishable, unless the surface of discontinuity has components which are not found in the contiguous masses, and even in this case (since the surface must be supposed to be formed out of matter supplied at the same potentials which belong to the matter in the surface) the work spent in increasing the surface infinitesimally by stretching is identical with that which must be spent in forming an equal infinitesimal amount of new surface. But when one of the masses is solid, and its states of strain are to be distinguished, there is no such equivalence between the stretching of the surface and the forming of new surface.^[1]

↑ This will appear more distinctly if we consider a particular case. Let us consider a thin plane sheet of a crystal in a vacuum (which may be regarded as a limiting case of a very attenuated fluid), and let us suppose that the two surfaces of the sheet are alike. By applying the proper forces to the edges of the sheet, we can make all stress vanish in its interior. The tensions of the two surfaces are in equilibrium with these forces, and are measured by them. But the tensions of the surfaces, thus determined, may evidently have different values in different directions, and are entirely different from the quantity which we denote by $\sigma$ , which represents the work required to form a unit of the surface by any reversible process, and is not connected with any idea of direction.
In certain cases, however, it appears probable that the values of a and of the superficial tension will not greatly differ. This is especially true of the numerous bodies which, although generally (and for many purposes properly) regarded as solids, are really very viscous fluids. Even when a body exhibits no fluid properties at its actual temperature, if its surface has been formed at a higher temperature, at which the body was fluid, and the change from the fluid to the solid state has been by insensible gradations, we may suppose that the value of $\sigma$ coincided with the superficial tension until the body was decidedly solid, and that they will only differ so far as they may be differently affected by subsequent variations of temperature and of the stresses applied to the solid. Moreover, when an amorphous solid is in a state of equilibrium with a solvent, although it may have no fluid properties in its interior, it seems not improbable that the particles at its surface, which have a greater degree of mobility, may so arrange themselves that the value of <r will coincide with the superficial tension, as in the case of fluids.

[1] This will appear more distinctly if we consider a particular case. Let us consider a thin plane sheet of a crystal in a vacuum (which may be regarded as a limiting case of a very attenuated fluid), and let us suppose that the two surfaces of the sheet are alike. By applying the proper forces to the edges of the sheet, we can make all stress vanish in its interior. The tensions of the two surfaces are in equilibrium with these forces, and are measured by them. But the tensions of the surfaces, thus determined, may evidently have different values in different directions, and are entirely different from the quantity which we denote by $\sigma$ , which represents the work required to form a unit of the surface by any reversible process, and is not connected with any idea of direction.
In certain cases, however, it appears probable that the values of a and of the superficial tension will not greatly differ. This is especially true of the numerous bodies which, although generally (and for many purposes properly) regarded as solids, are really very viscous fluids. Even when a body exhibits no fluid properties at its actual temperature, if its surface has been formed at a higher temperature, at which the body was fluid, and the change from the fluid to the solid state has been by insensible gradations, we may suppose that the value of $\sigma$ coincided with the superficial tension until the body was decidedly solid, and that they will only differ so far as they may be differently affected by subsequent variations of temperature and of the stresses applied to the solid. Moreover, when an amorphous solid is in a state of equilibrium with a solvent, although it may have no fluid properties in its interior, it seems not improbable that the particles at its surface, which have a greater degree of mobility, may so arrange themselves that the value of <r will coincide with the superficial tension, as in the case of fluids.

[1]