A mathematical model for the solute drag effect on recrystallization
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INTRODUCTION
THE quantitative theory of the solute drag effect was first developed by Lu¨cke and Detert.[1] This theory was further refined by Cahn[2] and Lu¨cke and Stu¨we.[3] The basic idea of the theory is that foreign atoms moving with the grain boundary or lagging behind the boundary exert a drag force on the boundary. It was suggested that the force could be calculated from the product between the force exerted by a foreign atom and the concentration of foreign atoms. In order to calculate this force, an interaction energy or a binding energy between a foreign atom and the boundary was introduced, and the distribution of the foreign atoms across the boundary was calculated from a diffusion equation and an interaction energy being a simple function of the distance from the center of the boundary. Cahn[2] obtained analytical expressions for the solute drag at very rapid and very slow migration rates and constructed an approximate equation for the solute drag, which could be applied for an entire range of velocities. On the other hand, Lu¨cke and Stu¨we[3] obtained an approximate equation for the solute drag. Their approximate equation can be used for very low and very high velocities and for intermediate velocities, as well if the interaction parameter is small. Hillert and Sundman[4] developed a model for the solute drag in binary systems from the thermodynamical point of view. In their model, the solute drag is evaluated from the Gibbs energy dissipation in the diffusion process across the grain boundary. Their model is not limited to dilute soluMASAYOSHI SUEHIRO is with the Yawata R & D Lab., Nippon Steel Corp., Fukuoka 804, Japan. ZI-KUI LIU, formerly with the Department of Materials Science and Engineering, Royal Institute of Technology, is with the Department of Materials Science and Engineering, University of ˚ GREN is with the Wisconsin-Madison, Madison, WI 53706. JOHN A Department of Materials Science and Engineering, Royal Institute of Technology, S-100 44 Stockholm, Sweden. Manuscript submitted January 6, 1997. METALLURGICAL AND MATERIALS TRANSACTIONS A
tions and is applicable for both moving grain boundaries and phase interfaces, while other models are inapplicable to moving phase interfaces and/or are limited to dilute solutions. They showed that the limiting case of their model to grain boundaries in dilute solutions is identical to the treatments by Cahn[2] and Lu¨cke and Stu¨we.[3] The approximation in the intermediate velocity range introduced by Cahn and Lu¨cke and Stu¨we was not needed. Thus, the treatment by Hillert and Sundman is more general than those by Cahn and Lu¨cke and Stu¨we. ˚ gren[5] further developed a simplified calculation A method of the growth rate of ferrite plates into austenite in multicomponent systems. This calculation method can reproduce the features of the treatment by Hillert and Sundman,[4] even though it is the simplified method. In the model, the solute drag was taken into account as the rotation of a tangent of the Gibbs energy curve at the representative co
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