Calculation of the product phase grain edge length and quadruple points per unit volume during solid state transformations

Abstract

General expressions are derived for the calculation of the total product phase grain edge length per unit volume L ^βββ_V and the total number of product phase quadruple points per unit volume QV (i.e., ββββ quadruple points) at any given time during solid state transformations occurring by nucleation and growth process. It is shown that, {

$$L_V^{\beta \beta \beta } = 0.155\int_0^{V_{V_{ex} } } {(S_{V_{ex} } )^2 Exp(--V_{V_{ex} } ){\text{ }}dV_{V_{ex} } } $$

} and, {

$$Q_V = \tfrac{\pi }{{96}}\int_0^{V_{V_{ex} } } {(S_{V_{ex} } )^3 Exp(--V_{V_{ex} } ){\text{ }}dV_{V_{ex} } } $$

} where VVex is the extended volume fraction of the product phase, and SVex is the total extended product phase-matrix interfacial area per unit volume. It is assumed that the spatial distribution of the product phase nuclei is random. The analysis is applicable to any arbitrary time dependent nucleation rate, and any arbitrary time and/or size dependent growth rate, provided that the product phase particles have an equiaxed shape in the extended structure. The analysis is applicable to isothermal transformations as well as nonisothermal and continuous cooling transformations. It is shown that L ^βββ_V and QV are basically determined by the path of microstructural evolution described by the variation of product phase-matrix interfacial area per unit volume with the volume fraction of the product phase. The ASTM grain size number of the transformed microstructure and average grain shape can be calculated from these results.