Page:Scientific Papers of Josiah Willard Gibbs.djvu/352

With these preliminary notions, we now proceed to discuss the condition of equilibrium which relates to the dissolving of a solid at the surface where it meets a fluid, when the thermal and mechanical conditions of equilibrium are satisfied. It will be necessary for us to consider the case of isotropic and of crystallized bodies separately, since in the former the value of $\sigma$ is independent of the direction of the surface, except so far as it may be influenced by the state of strain of the solid, while in the latter the value of $\sigma$ varies greatly with the direction of the surface with respect to the axes of crystallization, and in such a manner as to have a large number of sharply defined minima.^[1] This may be inferred from the phenomena which crystalline bodies present, as will appear more distinctly in the following discussion. Accordingly, while a variation in the direction of an element of the surface may be neglected (with respect to its effect on the value of $\sigma$) in the case of isotropic solids, it is quite otherwise with crystals. Also, while the surfaces of equilibrium between fluids and soluble isotropic solids are without discontinuities of direction, being in general curved, a crystal in a state of equilibrium with a fluid in which it can dissolve is bounded in general by a broken surface consisting of sensibly plane portions.

For isotropic solids, the conditions of equilibrium may be deduced as follows. If we suppose that the solid is unchanged, except that an infinitesimal portion is dissolved at the surface where it meets the fluid, and that the fluid is considerable in quantity and remains homogeneous, the increment of energy in the vicinity of the surface will be represented by the expression

$\int [\epsilon _{V}'-\epsilon _{V}''+(c_{1}+c_{2})\epsilon _{S(1)}]\delta NDs$

where $Ds$ denotes an element of the surface, $\delta N$ the variation in its position (measured normally, and regarded as negative when the solid is dissolved), $c_{1}$ and $c_{2}$ its principal curvatures (positive when their centers lie on the same side as the solid), $\epsilon _{S(1)}$ the surface-density of energy, $\epsilon _{V}'$ and $\epsilon _{V}''$ the volume-densities of energy in the solid and fluid respectively, and the sign of integration relates to the elements $Ds$. In like manner, the increments of entropy and of the quantities of the several components in the vicinity of the surface will be

$\int [\eta _{V}'-\eta _{V}''+(c_{1}+c_{2})\eta _{S(1)}]\delta NDs,$ $\int [\gamma _{1}'-\gamma _{2}']\delta NDs,$ $\int [-\gamma _{2}''+(c_{1}+c_{2})\Gamma _{2(1)}]\delta NDs,$ etc.

The entropy and the matter of different kinds representd by these

1. The differential coefficients of $\sigma$ with respect to the direction-cosines of the surface appear to be discontinuous functions of the latter quantities.