A theory is presented which describes the development of surface grooves at the grain boundaries of a heated polycrystal. The mechanisms of evaporation-condensation and surface diffusion are discussed with the use of the Gibbs-Thompson formula and the assumption that the properties of an interface do not depend on its orientation. For the idealized case in which only one of the mechanisms is operative, the groove profile is shown to have a time-independent shape whose linear dimensions are proportional to t½ for evaporation-condensation, and to t½ for surface diffusion. The proportionality constants are evaluated, and criteria are developed which permit one to estimate which process predominates in practice. Order of magnitude agreement is obtained with estimates of actual grooving speeds and profiles.

The stability of the shape of a moving planar liquid-solid interface during the unidirectional freezing of a dilute binary alloy is theoretically investigated by calculating the time dependence of the amplitude of a sinusoidal perturbation of infinitesimal amplitude introduced into the planar shape. The calculation is accomplished by using gradients of the steady-state thermal and diffusion fields satisfying the perturbed boundary conditions (capillarity included) to determine the velocity of each element of interface, a procedure justified in some detail. Instability occurs if any Fourier component of an arbitrary perturbation grows; stability occurs if all components decay. A stability criterion expressed in terms of growth parameters and system characteristics is thereby deduced and is compared with the currently used stability criterion of constitutional supercooling; some very marked differences are discussed.

The stability of the shape of a spherical particle undergoing diffusion-controlled growth into an initially uniformly supersaturated matrix is studied by supposing an expansion, into spherical harmonics, of an infinitesimal deviation of the particle from sphericity and then calculating the time dependence of the coefficients of the expansion. It is assumed that the pertinent concentration field obeys Laplace's equation, an assumption whose conditions of validity are discussed in detail and are often satisfied in practice. A dispersion law is found for the rate of change of the amplitude of the various harmonics. It is shown that the sphere is stable below and unstable above a certain radius Rc, which is just seven times the critical radius of nucleation theory; analogous conclusions are obtained for the solidification problem. The results for the sphere are used to discuss the stability of nonspherical growth forms.

To represent ideal grain boundary motion in two dimensions, a rule of motion of plane curves is considered whereby any given point of a curve moves toward its center of curvature with a speed that is proportional to the curvature. A general theorem is deduced concerning the change of area enclosed by such a curve. Three families of curves are found that obey the curvature rule of motion while undergoing the shape preserving transformations of uniform magnification, translation, and rotation respectively. Pieces of these curves represent the steady shapes of idealized grain boundaries under certain symmetrical conditions.

The partial differential equation describing morphological changes of a surface of revolution due to capillarity-induced surface diffusion has been derived under the assumption of isotropy of surface tension and surface self-diffusion coefficient. A stable, convergent finite-difference method has been developed for the general case of an arbitrary surface of revolution and solutions have been obtained for the specific problems of the blunting of field-emission tips and the sintering of spheres. Spheroidization of cylindrical rods, as well as field-emission tips with taper below a certain critical value, is predicted; for tapers above the critical value, steady-state shapes are predicted and equations describing the blunting and recession of the tips are presented. If the sintering results for spheres are represented by a plot of log x/a vs log t, it is found that the inverse slope varies from approximately 5.5 to approximately 6.5 for the range 0.05≤x/a≤0.3, in contrast with the constant value of 7 found by Kuczynski from an order-of-magnitude analysis. At higher values of x/a, n increases steadily and without bound.

The relaxation of a nearly plane surface to flatness is discussed under the assumption that all surface properties are independent of orientation. A general solution is obtained for the combined action of the transport processes of viscous flow, evaporation-condensation (in a closed system), volume diffusion, and surface diffusion. Green's function solutions are developed for each of the transport processes separately, and criteria are obtained to decide which process dominates. The initial forms of these solutions represent point concentrations (particles), or line concentrations (wires) of material set upon an infinite plane. The progressive topographical developments described by the formulas are idealized representations of the latter stages of the sintering of small wires and particles to a plane.

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