Thermodynamic-kinetic simulation of constrained dendrite growth in steels
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I. INTRODUCTION
DURING the past 20 years, great advances have been made in developing the theory of constrained growth of columnar dendrites.[1–8] The theory has primarily been applied to binary metallic alloys, but an approximation to multicomponent alloys is also available.[7,8] Due to the increased interest in rapid solidification processes, the theory has been developed to take into account the effect of a high growth rate upon the microstructure.[5,6,8] A typical application is the modeling of dendritic growth in welding processes.[8,9,10] The theory, however, is not yet capable of reliably predicting the mode of solidification (i.e., which solid phase is formed first from the liquid). This would be of great value for stainless steels, in which the mode of solidification is known to affect the solidification-cracking susceptibility. One way to approach the problem is to determine the phase stabilities during growth with a thermodynamic solution model. In this study, the problem is solved by coupling an approximate dendritic growth model of multicomponent steels[7,8] to the necessary set of chemical potential–equality equations, based on the classical substitutional solution model.[11] Depending on the growth kinetics, this set of equations determines the thermodynamic equilibrium at the ferrite/liquid and austenite/ liquid interfaces and, consequently, the primary phase of solidification. In the present treatment also, the deviation in the local thermodynamic equilibrium caused by interface friction,[12] capillarity,[12] and solute trapping,[12–15] which has an effect at high growth rates, is taken into account. Several calculations are presented to test the model and to validate it with experimental data. II. DESCRIPTION OF MODEL During dendritic growth, solute pileups will be formed ahead of the dendrite tips (Figure 1), which will cause the
JYRKI MIETTINEN, Research Scientist, is with the Laboratory of Metallurgy, Helsinki University of Technology, FIN-02015 Espoo, Finland. Manuscript submitted September 7, 1999. METALLURGICAL AND MATERIALS TRANSACTIONS B
liquid phase to become undercooled. Under steady-state conditions, the solute supersaturation, which indicates the relative height of the pileup, can be described as[6] Vi 5
nom xLs i * 2 xi Ls xi * 2 xSi*
[1]
where Vi is defined for a paraboloid dendrite tip as `
Vi 5 Pi exp (Pi)
e exp z(2z) z
[2]
VRt 2DLi
[3]
Pi
with Pi 5
is the nominal composition; xLs In these equations, xnom i i * and xSi* are the liquid and solid pileup compositions at the dendrite tip, respectively; Pi is the Peclet number; V is the growth rate of the tip (in cm/s), DLi is the diffusion coefficient of solute i in liquid (in cm2/s), and Rt is the dendrite tip radius (in centimeters). In order to calculate the dendrite tip undercooling in multicomponent steels, a specific model of constrained dendrite growth (CGD) was developed. In this model, Eq. [1], for all solutes (where i 5 2, . . , n), is solved with the modified[12] chemical potential–equality equation Ls S S S S m
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