W.W. Mullins

Theory of Thermal Grooving

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.

Morphological Stability of a Particle Growing by Diffusion or Heat Flow

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 Laplaces 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.

Flattening of a Nearly Plane Solid Surface due to Capillarity

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. Greens 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.

Two‐Dimensional Motion of Idealized Grain Boundaries

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.

Morphological Changes of a Surface of Revolution due to Capillarity‐Induced Surface Diffusion

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 ...

The effect of thermal grooving on grain boundary motion

Abstract A theory is presented which describes the dynamics of thermal groove formation at a moving grain boundary. The treatment is based on the Gibbs-Thompson formula relating curvature to chemical potential, and assumes surface diffusion to be the mechanism of groove development. It is proved that a boundary will become stuck at the surface if the magnitude of the angle it makes with the surface normal is less than a critical value θc. This result is combined with certain consequences of the soap film model of grain boundaries to deduce an explanation of the specimen thickness effect of grain growth. Additional experimental evidence is presented and interpreted showing that thermal grooves do retard boundaries that terminate on a surface causing their migration to be spasmodic. Finally, a discussion is given of the hypothesis that a certain type of exaggerated grain growth is caused by inequalities in gas—metal inter facial free energies.

On the theory of anisotropy of crystalline surface tension

Abstract Application of fundamental geometrical and statistical mechanical principles to the terrace ledge kink (TLK) model of crystalline surfaces leads to a two-parameter theory of the anisotropy of surface tension (specific free energy). The contribution to ledge free energy due to the entropy associated with kinks and the effect of neighboring ledges in restricting the kink density as the ledge spacing decreases are calculated by treating each ledge as a cooperative chain. The results show clearly the importance of the kink entropy contribution to ledge free energy. Examples of the predicted anisotropy of surface tension are given for copper. The predicted variations in surface tension are such that rounded areas can appear on the equilibrium form. A critical underpressure is predicted to be necessary to force ledges together on an evaporating crystal.

The statistical self‐similarity hypothesis in grain growth and particle coarsening

The time dependence of particle size (e.g., mean volume) in normal grain growth, bubble growth, and late‐stage coarsening is deduced from a statistical self‐similarity (SSS) hypothesis, according to which consecutive configurations of the system in the self‐similar mode are geometrically similar in a statistical sense, and from the scaling characteristic of v, the rate of change of the volume of a given particle. It is shown that if v scales as vα under uniform magnification, where v is the mean particle volume and α a constant depending on the controlling kinetics and geometry of the system, then v1−α is a linear function of the time. Values of α are obtained for a number of cases. The treatment is free of many of the approximations and geometrical simplifications used in most theories. Evidence for the SSS hypothesis is discussed. Special new results are obtained for two‐dimensional bubble and idealized grain growth that should be testable by computer simulation or by observation of a well‐behaved soap‐bubble array.

The kinetics of grain boundary grooving in copper

Abstract In an earlier paper the diffusion equation associated with grain boundary grooving was solved and it was concluded that surface diffusion was the dominant transport mechanism for copper. For this mechanism the theory predicts that the shape of the groove profile has a time independent form and that the groove width and depth increase linearly with t 1 4 . These two predictions have been tested by studying the grooving of tilt boundaries in copper annealed in dry hydrogen at 930°C and 1035°C. Both predictions agree with the experimental results. The validity of the theory allows one to calculate the surface diffusion coefficient, Ds. The advantages of this method of determining Ds are discussed.

Magnetically induced grain-boundary motion in bismuth

Abstract The various free energies which a magnetic field generates in the grains of a Bi polycrystal give rise to calculable forces on the grain boundaries. These forces produce a strong preferred orientation in a Bi sample undergoing grain-growth. The study of single boundaries driven by the known magnetic forces has permitted the estimation of an upper limit to the boundary free energy. In a number of samples the boundaries exhibit a type of sticking that does not seem to be due to impurities. A new hypothesis is considered which explains this variation of mobility in terms of changes in the concentration of vacancies at the grain boundaries.