Iron and Steel Division - Grain Boundary Grooving by Volume Diffusion

The development, by the mechanism of volume diffusion, of a grain boundary groove on an interface separating a solid phase and a saturated fluid phase is calculated under the following assumptions: 1) isotropy of the interfacial free energy 's, 2) applicability of the Gibbs-Thompson formula relating curvature and chemical potential, 3) a nearly planar groove surface. The groove profile is found to have a .fixed shape and linear dimensions that are proportional to (time)1/3. The constant of proportionality is evaluated and involves ?s in such a way that the latter my be evaluated from grooving kinetics. A grain boundary that intersects an interphase interface will tend to produce a groove along the line of intersection (Fig. 1). The ultimate motivation for the formation of the groove is the reduction in interfacial free energ that occurs as the grain boundary contracts. Smith has discussed the way in which the free energy of the interfaces determines their dihedral angles at the groove root and has given a detailed analysis of the effect of the dihedral angles upon the nature the microstructures developed in polyphase alloys. Herring2 has extended the considerations to the case in which the free energy of an interface depends on its orientation and has thus added torque terms to the condition for interface equilibrium. Although the grain boundary and the interphase interface may quickly achieve the proper equilibrium angles where they intersect, as determined by the balance of equivalent tensions and torques,"2 the groove will generally continue to grow (though with decreasing speed). Thermodynamically, this continued growth reflects the fact that the lowest free energy cannot be achieved until the entire grain boundary has been eliminated; mechanistically, the growth occurs in response to inhomogeneities of chemical potential that are caused by the constraint of fixed dihedral angles at the groove root. The latter point is illustrated in Fig. 1, which shows that atoms* near the groove are on the convex side of a cylindrically curved surface and therefore have a higher chemical potential than atoms on the flat interface beyond. Thus atoms tend to leave the curved groove shoulders and force the groove to deepen. When both phases are solid additional chemical potential gradient may arise from whatever strain fields are necessary to insure continuity of material across the interface. In the discussion that follows, this complication due to strain will not occur since we always assume one phase to be a fluid and the other a solid. The transport processes that may operate to permit growth of the groove are surface (interface)* diffusion, volume diffusion in the phase on either side of the interface, and evaporation-condensation when one phase is gaseous. (Also a melting-freezing process is responsible for groove enlargement on a solid-melt interface as discussed in Sec. VI). The theory of grain boundary grooving has been discussed in detail for the processes of surface diffusion and of one type of evaporation-condensation.3'4 The theoretical predictions for the mechanism of surface diffusion have been confirmed in an experimental study of grooving kinetics in copper.5 In this paper, the theory of groove development by volume diffusion will be presented. Volume diffusion can occur either by atomic migration in the solid S or by diffusion of S atoms in the fluid phase, or solvent, adjacent to the groove surface. As discussed in Sec. V, volume diffusion in the solid should be less important than surface diffusion in forming a groove. Apart from verifying this statement then, the main interest of the present analysis centers on diffusion in an external solvent. The solvent may be either a liquid (e.g., water, or-