Thermodynamics-Based Semi-Empirical Description of Liquidus Surface and Partition Coefficients in Ternary Al-Mg-Si alloy

Transcription

1 Materials Science Forum Vols pp. -8 online at 00 Trans Tech Publications, Switzerland Thermodynamics-ased Semi-Empirical Description of iquidus Surface and Partition oefficients in Ternary l--si alloy ndrás Roósz 1, János Farkas, 1 György Kaptay 1 University of Miskolc, Faculty of Metallurgical and Materials Engineering, Physical Metallurgy Department, 515 Miskolc-Egyetemváros University of Miskolc, Faculty of Metallurgical and Materials Engineering, Physical hemistry Department, 515 Miskolc-Egyetemváros Keywords: equilibrium phase diagram, binary, ternary, liquidus surface, partition coefficient, calculation. bstract. simple but thermodinamically well-founded method is described for the approximating calculation of liquidus surfaces and partition coefficient of multicomponent equilibrium phase diagrams. The applicability of method is shown by describing of the l-si, l- binary and l-si- ternary alloy systems. Introduction t the simulation of the solidification processes of solid solutions it is necessary to know the data of liquidus and solidus temperatures numerically as well as the so called partition coefficients as function of concentrations. In case of binary alloys it is enough to know the liquidus and solidus curves because concentrations of liquid and solid phases being in equilibrium with each other at the given temperature can be determined from the equilibrium phase diagrams. Thus, the partition coefficient k is known as the quotient of these two concentrations. In case of multicomponent ternary, quaternary, alloys, the partition coefficients cannot be determined from the equilibrium phase diagrams if the processes have more than one degree of freedom in case of N components, there are N-1 independent partition coefficients, though their knowledge is of basic importance for the simulation of solidification processes. Numerical data of liquidus and solidus surface and the partition coefficients can be deduced either from calculation results of a software operating on thermodynamical basis or from the equilibrium phase diagrams determined by experiments. It is necessary to note that the aforementioned data can be calculated from the thermodynamic functions only in case if the so - called fit parameters of the - system have been determined by comparing the calculated and measured equilibrium phase diagrams, i.e., phase diagrams determined previously by measurements can only be calculated by means of thermodynamic functions. However, it is a very significant advantage of thermodynamic calculations that their give the partitions coefficients as well in case of ternary or multicomponent alloy systems even for the processes having more than one variable e.g., the solidification of solid solution from the melt in case of ternary alloys. Thermodynamic calculations have a very significant disadvantage, as they are quite complicated, the softwares and data fields are very expensive and the PU time of softwares exceeds significantly by an order of magnitude the PU time of softwares simulating the solidification processes. iquidus and solidus temperatures can be determined in a very complicated way from the measured multicomponent equilibrium phase diagrams, and the partition coefficients cannot be directly determined from them. In [1] a method is shown for estimating the partition coefficient from measured data. In our present paper, a method serving for the approximation of liquidus and solidus surfaces and partition coefficients are described in case of l--si alloy system.. The details of this method can be found in []. ll rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of the publisher: Trans Tech Publications td, Switzerland, ID: /10/06,1::50

2 4 Materials Science, Testing and Informatics I Determining the constants of functions describing the liquidus curve and partition coefficient in case of the binary l-si and l- alloy systems iquidus curve. The liquidus and solidus curves of equilibrium phase diagrams of l-si and l- alloy systems have been digitalized such a way that the liquid and corresponding solid phase concentrations have been determined at 10 o steps from the melting point of aluminium o to the eutectic temperature l-si: 577 o, l-: 450 o. The partition coefficient has been obtained by the quotient of concentration of solid and liquid phases. In case of binary alloys, Equation 1 has the following simple form: T = T /1 + F, 1 where : is the mass concentration of Si and, 0 is the concentration of the pure solid phase. y rearranging the above equation, the following formula is obtained: T / T 1 = F. y using a third-order polynomial: F = The constants of third-order polynomial have been determined by regression 0 =0. Table 1 contains the values of constants as well as the value of R. Table I. onstants of equation of liquidus surface 0 T Si Si E-.096E E-7 inary l-si Si E- -.5E E E E E-7 inary l- R = Si R = T Ternary l--si R =0.957 In case of l-si alloy the equations of liquidus curves of both the primary l O =0 and the primary Si O =1 phases have been determined, while in case of the l- alloy, only the equation of liquidus curve of primary l phase O =0 has been determined. Partition coefficient. The following simple equation gives the value of partition coefficient as a function of the alloying concentration of liquid phase: ln k = ln k. gain, by using a third-order polynomial for approximation, the following equation can be obtained: ln k = Even in this case the constants of polynomical have been determined only for the l phase by regression. Tables lnk Si and lnk contain the values of constants and the value of R.

3 Materials Science Forum Vols Table II. onstants of equation of Si partition coefficients 0 lnk Si Si Si E-.0507E E-5 inary l-si Si E E E E E E-4 Si R = lnk Si Ternary l--si R =0.996 Table III. onstants of equation of partition coefficients 0 lnk Si Si 1 Si Si E E E E E E- -.81E E-4 1.6E-06 inary l- R =0.995 lnk Ternary l--si R =0.987 Fig 1. l-si and Fig. l- show the measured liquidus and solidus curves as well as the calculated liquidus and solidus curves. The solidus curve was calculated by the product of calculated liquidus concentration and calculated partition coefficient. The calculated and measured values are equal within error of the measurement. 700 l-si equilibrium phase diagram liquidus and solidus curves 650 T Si mass% Fig 1.

4 6 Materials Science, Testing and Informatics I l- equilibrium phase diagram liquidus and solidus curves T mass% Fig. Determination of constants of functions describing the liquidus surface and partition coefficients in case of the ternary l-si- alloy system iquidus surface. The liquidus surface given by isotherms of the equilibrium phase diagram of ternary l--si alloy system had been digitalized on the following way. 0 secants were indicated on the surface in uniform partition which secants where crossing the corner point indicating the pure l. The concentration data Si, belonging to the point of intersection of isotherms have been determined. In the case of ternary sytem Equation 1. can be described as follows: T = T /1 + F + F + F,. 5 fter rewriting the above equation, the following one is obtained: F, = T / T 1 F F, 6 where: F, = 1, , , The constants of equations have been determined by regression. The constants are contained in Table 1. The value of 0 is zero. The measured and calculated liquidus temperatures are shown in Fig.. In most cases the deviation is less than 1 K. The calculated liquidus surface can be seen in Fig. 4. Partition coefficients. The partition coefficients have been calculated according to the method described in 1 with the deviation that the following functions ln k which concern the binary alloys have been determined only from the binary alloys. The value of lnk containing the effect of third element has been obtained from the lnk data of a ternary alloy according to the following equation: ln k, = ln k ln k. 8

5 Materials Science Forum Vols gain, by approximating the functions by a third-order polynomial: ln k, = 1, , , Measured and estimated liquidus temperature Estimated liquidus temperature Measured liquidus temperature Fig. T Si % % Fig 4. The liquidus surface Table contains the constants of functions as well as the value of R for Si, while Table shows the constants of functions and the value of R for. The measured and calculated values of partition coefficients are compared in Fig. 5.

6 8 Materials Science, Testing and Informatics I Estimated partition coefficient Measured and estimated partition coefficient of in the ternary system y = x R = Measured partition coefficient Estimated partition coefficient Measured and estimated partition coefficient of Si in ternary l--si system y = x R = Measured partition coefficient Fig 5. Summary, conclusions In our paper the summary of a new method is shown for the mathematical description of equilibrium phase diagrams of multicomponent alloys. The temperature- and partition coefficient data serving for the basis of calculation can be obtained from the measurement or thermodynamic calculations. The applicability of method is shown in case of l-si and l- binary as well as l- Si- ternary alloys. The functions obtained by the described method are suitable for simplifying the solidification simulations in a great extent. References [1]. Roósz, J. Sz ke and M. Rettenmayr : Z. Metallk , []. Roósz, G. Kaptay : The concept of the ESTimated Phase Diagram ESTPHD system Z. Met. to be published