The usefulness of integral mean curvature measurements in the study of the kinetics of coarsening
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I.
INTRODUCTION
Two
strategies can be used to study microstructural evolution during phase transformation in multiphase systems involving a particulate dispersoid. One can either characterize the microstructure in terms of the location, size, etc. of the individual precipitate particles, or describe the structure in terms of global properties s u c h as the volume fraction, surface area, etc. of the entire particulate ensemble. Theories attempting to characterize transformation processes are generally based upon local descriptors of the microstructure. For example, the kinetics of phase transformation during growth are based upon models of nucleation and growth of individual particles of the second phase. The global descriptors of the microstructure, on the other hand, are easier to measure; relationships are therefore sought between the two types of descriptors to enable the testing of the theoretical model. These relationships are not always straightforward and must be applied judiciously in order to avoid confusion. 1.2,3 Typically, the two most commonly measured properties are the volume fraction and the surface area of the second phase. These measurements, by themselves, may not be adequate to distinguish between different models, as emphasized previously.t'~'3 The integral mean curvature* defined *See Appendix.
and discussed in detail by DeHoff~ is a property that can be measured unambiguously and will therefore provide additional insight into the kinetics. The purpose of this paper is to explore the usefulness of this measurement in the study of coarsening phenomena. Coarsening of the dispersoid in two-phase structures is a process driven by excess surface energy. The decrease in surface energy is accomplished by qualitatively shifting the size distribution of the particles to higher size values, resulting in a lower surface-to-volume ratio. The volume R.T. DeHOFF, Professor, and C.V. ISWARAN, Graduate Research Assistant, are both with the Department of Materials Science and Engineering, University of Florida, Gainesville, FL 32611. Manuscript submitted August 24, 1981. METALLURGICAL TRANSACTIONS A
fraction of the particulate phase remains essentially constant throughout the process. Therefore, the path of microstructural change remains undefined if one measures only the decrease in interfacial area during the process. Consequently, much of the experimental work involves the rather tedious process of measuring particle size distributions as a function of coarsening times. On the theoretical side, there exists substantial literature beginning with the classical work of Greenwood, s Lifshitz and Sloyozov,6 and Wagner 7 (the GLSW theory). These early theories were developed explicitly for the limiting case of small volume fraction of the second phase. Several researchers 8-12have since introduced modified versions that account for variations in volume fractions due to the composition of the alloy studied. Each theory invokes a rate controlling mechanism and arrives at an expression for the growth rate of p
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