Analysis of grain growth in a two-phase gamma titanium aluminide alloy
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Pg 5 C2 (gs /D)
INTRODUCTION
THE importance of second-phase particles/precipitates in controlling the kinetics of recrystallization and grain growth has long been recognized and exploited in industrial processing of materials. Some of the well-known examples of grain-size control using a dispersion of fine particles are as follows: (1) carbides/nitrides of Ti, V, and Nb in microalloyed steels; (2) Al3Ti and Al3Zr particles in aluminum alloys; and (3) ThO2 particles and dopants such as K2O in tungsten filament wires. Similarly, particulate additions are used to delay the matrix grain growth during sintering and thereby achieve high densities in powder metallurgy products. Grain growth in single-phase alloys has been studied extensively through theoretical and experimental approaches.[1–4] Under isothermal heat-treatment conditions, normal grain growth is well described by the following empirical equation:[4] D p 2 D 0p 5 C1 t exp (2Q/RT)
[1]
where D and D0 are the current and initial values of the mean grain size, respectively; Q is the apparent activation energy for grain growth; R is the gas constant; T is the temperature; t is the annealing time; p is the grain growth exponent; and C1 is a kinetic constant. For most singlephase materials, the value of p ranges from 2 to 10 due to the drag force exerted by solute atoms on grain boundaries.[3] Only in the case of ultra-high-purity metals annealed at temperatures very close to the melting point does p approach 2. This corresponds to the limiting case where the grain boundary migration rate is linearly proportional to the driving pressure Pg given by V. SEETHARAMAN, Senior Scientist, is with the Materials and Processes Division, UES Inc., Dayton, OH 45432-1894. S.L. SEMIATIN, Senior Scientist, is with Materials Processing/Materials Science, Wright Laboratory Materials Directorate, WL/MLLM, Wright-Patterson Air Force Base, OH 45433. Manuscript submitted July 30, 1996. METALLURGICAL AND MATERIALS TRANSACTIONS A
[2]
where gs denotes the grain boundary interfacial energy, and C2 is a geometrical constant which is characteristic of the grain shape and can vary by at least a factor of 3, depending on the assumptions regarding the relationship between the grain size and the radius of curvature.[1,3] The retardation of grain growth by second-phase particles/precipitates was first theoretically investigated by Zener.[5] The calculation of the drag pressure (also known as Zener drag) exerted by the particles on a moving grain boundary is a complicated statistical problem and has been solved only approximately.[5–11] The drag pressure, Pz, is mainly determined by the volume fraction, f, and the mean diameter, d, of the particles. Other factors which influence Pz include the particle morphology, degree of coherency between the particles and the matrix, and the nature of spatial distribution (random vs nonrandom) of the particles.[12] Despite these complications, a generalized expression for the drag pressure can be written as Pz 5 C3 (gs f/d)
[3]
where C3 is a constant a
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