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Examination of the Zener Relationship between Grain Size and Particle Dispersion

Zener-Smith proposed the original relation R = 4r 3f v in 1948, where R and r are the mean radii of the matrix grains and dispersed particles, and f v is the volume fraction of dispersed particles. They derived the relation under the assumption that the configuration of the grain boundary was independent of particle distribution. In the present work a modified relation R=4r/3f 2/3 v has been obtained by assuming that (1) the grain boundary is roughened by particle pinning, and (2) the amplitude of the roughness is λ/2, where λ is the inter-particle distance. Experimental data on the radii of austenite grains and the dispersed cementite particles in high C high Mn steels are shown to agree well with the modified relation.