Page:Scientific Papers of Josiah Willard Gibbs.djvu/364

General Relations.—For any constant state of strain of the surface of the solid we may write

$d\epsilon _{S(1)}=td\eta _{S(1)}+\mu _{2}d\Gamma _{2(1)}+\mu _{3}d\Gamma _{3(1)}+{\text{etc.,}}$

(674)

since this relation is implied in the definition of the quantities involved. From this and (659) we obtain

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

(675)

which is subject, in strictness, to the same limitation that the state of strain of the surface of the solid remains the same. But this limitation may in most cases be neglected. (If the quantity $\sigma$ represented the true tension of the surface, as in the case of a surface between fluids, the limitation would be wholly unnecessary.)

Another method and notation.—We have so far supposed that we have to do with a non-homogeneous film of matter between two homogeneous (or very nearly homogeneous) masses, and that the nature and state of this film is in all respects determined by the nature and state of these masses together with the quantities of the foreign substances which may be present in the film. (See page 314.) Problems relating to processes of solidification and dissolution seem hardly capable of a satisfactory solution, except on this supposition, which appears in general allowable with respect to the surfaces produced by these processes. But in considering the equilibrium of fluids at the surface of an unchangeable solid, such a limitation is neither necessary nor convenient. The following method of treating the subject will be found more simple and at the same time more general.

Let us suppose the superficial density of energy to be determined by the excess of energy in the vicinity of the surface over that which would belong to the solid, if (with the same temperature and state of strain) it were bounded by a vacuum in place of the fluid, and to the fluid, if it extended with a uniform volume-density of energy just up to the surface of the solid, or, if in any case this does not sufficiently define a surface, to a surface determined in some definite way by the exterior particles of the solid. Let us use the symbol $(\epsilon _{S})$ to denote the superficial energy thus defined. Let us suppose a superficial density of entropy to be determined in a manner entirely analogous, and be denoted by $(\eta _{S})$. In like manner also, for all the components of the fluid, and for all foreign fluid substances which may be present at the surface, let the superficial densities be determined, and denoted by $(\Gamma _{2}),(\Gamma _{3})$, etc. These superficial densities of the fluid components relate solely to the matter which is fluid or movable. All matter which is immovably attached to the solid mass is to be regarded as a part of the same. Moreover, let y be defined by the equation

$\varsigma =(\epsilon _{S})-t(\eta _{S})-\mu _{2}(\Gamma _{2})-\mu _{3}(\Gamma _{3})-{\text{etc.}}$

(676)