The nucleation kinetics of grain boundary allotriomorphs of proeutectoid ferrite at austenite grain faces have been measured in three high purity Fe-C alloys as a function of isothermal reaction time and temperature. Several correction techniques, including discrimination between different nucleation sites and the effect of carbon diffusion fields on further nucleation of ferrite, were incorporated into a stereological procedure utilizing the SchwartzJSaltykov size distribution analysis. This analysis enabled the number of ferrite particles per unit unreacted grain boundary area to be obtained as a function of isothermal reaction time, and thus the time-dependent nucleation kinetics to be obtained as a function of temperature and carbon concentration. These rates were then compared with those predicted by classical heterogeneous nucleation theory using various models for the critical nucleus. It was concluded that viable critical nuclei must have predominately low energy interphase boundaries. Only a very small fraction of the austenite grain face area appears to be capable of supporting nucleation.

The kinetics of ferrite nucleation at austenite grain boundaries in Fe-C alloys

Geometrical constructions, such as the tangent construction on the molar free energy for determining whether a particular composition of a solution is stable, are related to similar tangent constructions on the orientation-dependent interfacial energy for determining stable interface orientations and on the orientation dependence of the crystal growth rate which tests whether a particular orientation appears on a growing crystal. Subtle differences in the geometric constructions for the three fields arise from the choice of a metric (unit of measure). Using results from studies of extensive and convex functions, we demonstrate that there is a common mathematical structure for these three disparate topics and use this to find new uses for well-known graphical methods for all three topics. Thus, the use of chemical potentials for solution thermodynamics is very similar to known vector formulations for surface thermodynamics and the method of characteristics which tracks the interfaces of growing crystals; the Gibbs-Duhem equation is analogous to the Cahn-Hoffman equation. The Wulff construction for equilibrium crystal shapes can be modified to construct a “phase shape” from solution free energies that is a potentially useful method of numerical calculations of phase diagrams from known thermodynamical data.

Crystal Shapes and Phase Equilibria: A Common Mathematical Basis

The interfacial energies between α and γ iron were calculated for Kurdjumov-Sachs (K-S), Nishiyama-Wassermann (N-W), and one orientation relationship (OR), which may represent irrational OR, with varying interface orientation. The relaxation of atomic structure in the vicinity of the interface was performed by the Monte Carlo method using an embedded atom method (EAM) potential. Whereas the polar plot of calculated interfacial energies exhibited small and large cusps for the K-S and N-W ORs, an almost equiaxed energy surface with shallow cusps was obtained for the irrational ORs. The equilibrium shape of an α particle, N-W oriented with the γ matrix, was a thick rectangular plate with broad facets containing monatomic ledges. In contrast, the equilibrium shape of a particle, K-S oriented with the γ matrix, was elongated nearly in the closed-packed directions with {112}y type and another broad facet that achieves relatively good matching. The volume of these shapes in the Wulff space tends to be smaller for K-S than that of the N-W OR. The equilibrium shape of the grain boundary α particle at the γ grain boundary was calculated under the assumption that the α particle has K-S OR with one of the two γ grains using the modified Wulff construction proposed earlier. The variation of the shape of the α particle with γ grain boundaries at an early stage of precipitation may primarily be ascribed to the change in the variant of the smallest nucleation activation energy and less prominently to the change in the γ grain boundary energy.

Calculation of the interfacial energies between α and γ iron and equilibrium particle shape

Grain boundary precipitation of M23C6 in an austenitic steel

The accelerated nucleation of precipitates at GP zones is explained using heterogeneous nucleation theory. Nucleation at zone : matrix boundaries is encouraged by several factors: 1) the chemical interfacial energy of zone : matrix boundaries can significantly decrease the interfacial energy barrier to nucleation; 2) destruction of quenched-in excess vacancies at incoherent portions of the nucleus surface may make the change in the volume free energy significantly more negative; 3) the crystal structures of the zone and matrix are identical and parallel which permits the nucleus to be faceted in both phases. Some additional assistance to nucleation at GP zones is provided by: 4) the accelerated diffusivity resulting from the presence of excess vacancies and 5) the large area of zone : matrix boundary per unit volume of matrix. These factors can more than compensate for the decreased solute supersaturation due to the formation of GP zones and provide an explanation for the enhanced nucleation of precipitates in the presence of GP zones.

Jong K. Lee

Papers

Application of the modified Gibbs-Wulff construction to some problems in the equilibrium shape of crystals at grain boundaries