Influence of coupling with calculation of phase diagrams on microsegregation forming simulation of Al-4.5%Cu alloy
LIU Yong-gang(刘永刚)1, CHEN Guang(陈 光)1, SUN Guo-xiong(孙国雄)2
1. Joint Laboratory of Nanostructured Materials and Technology,
Nanjing University of Science and Technology, Nanjing 210094, China;
2.Department of Materials Science and Engineering, Southeast University, Nanjing 210096, China
Received 20 April 2006; accepted 30 June 2006
Abstract: The effect of coupling with calculation of phase diagrams on microsegregation forming simulation was investigated. The traditional simplified phase diagram and calculated phase diagram were introduced into the numerical models respectively and simulation on microsegregation forming of the Al-4.5%Cu alloy ingot was also presented. The simulation results were both compared with the experiment results. The results show that the calculated sencondary arm spacing with these two kinds of phase diagram are almost the same because relationship between the coarsening model and the information of phase diagram is not close. The calculated eutectic phase volume fractions of different locations in the ingot coupled with different phase diagrams are discrepant. The calculated volume fractions are consistent with the experiment results when calculated phase diagram couples, but are far from the experiment results and obviously inacceptable when traditional simplified phase diagram couples. So, coupling with accurate calculated phase diagrams is very significant for microsegregation forming simulation since much information of the phase diagram is used in the models and it can improve the precision of simulation results.
Key words: microsegregation; simulation; calculation of phase diagrams
1 Introduction
Since simulation on solidification process turned to micro-scale calculation in the late 1980s especially in the last decade of the 20th century, it has been one of the focuses in the material field. And as one orientation of the micro-scale simulation, forecasting of microsegregation forming has important effect on direct materials processing. The aim of microsegregation forming is to forecast microstructure and properties of materials and directly carry out latter processing through calculating dendritic segregation parameters which include sencondary arm spacing, solute distributing in secondary arm, volume fraction and amount of the eutectic phase[1]. In recent years, some advanced micro models have been reported[2-8], and such kinetic effects which can influence microsegregation as solid state back diffusion, secondary dendrite arm coarsening, undercooling were considered in these models. The effects of coupling with calculation of phase diagrams and providing accurate information of the phase diagram on microsegregation forming simulation were mainly discussed. Through comparing simulation results with experiment results, the significance of coupling with calculation of phase diagrams on microsegregation forming simulation was demonstrated.
2 Models and nemerical method
2.1 Main mathematical and physical models
The kinetic coarsening equation[9] was applied in the following models:
(1)
where λ(t) is the half of secondary arm spacing; B is a geometric factor and it takes a particular value for one class of alloys (for aluminum alloys, its value is 13.125[10, 11] in this paper). The coarsening parameter M is defined as
(2)
where ΔH is the latent heat of fusion; γ is the solid/liquid interface energy; DL is diffusion coefficient of liquid phase; T is temperature; m is slope of liquidus curve in phase diagram; k is partition coefficient, and 
is concentration of alloy in liquid phase at solid/liquid interface.
Redistribution of the solute in the solid obeys Fick’s second law:
(3)
where Cs is concentration of alloy in solid phase; Ds is the diffusion coefficient of solid phase.
The mass balance yield is
(4)
where X(t) is the length of the volume element; Xs is the length of the solid phase; C0 is the nominal concentration of alloy.
The KGT model[12] was used to calculate primary tip undercooling. It was founded based on researching of solute equilibrium at the dendrite tip[12]. It describes the relationship among the dendrite tip radius rT, the growth velocity of dendrite tip vg and the dendrite tip undercooling ΔTt, and it consists of the following equations:
(5)
(6)
(7)
(8)
where Ω is the solute supersaturation of dendrite tip; Pe is the Péclet number of solute; and Iv is the Ivantsov function which is defined as
(9)
Ei is a exponential integral function, 
; Γ is the Gibbs-Thomson coefficient, 
.
2.2 Numerical method of micro models
After cooling curves of all the nodes were calculated from macro calculation, these curves were input into micro models of microsegregation simulation. Some important kinetic effects of solidification process were considered in micro models and so microsegregation parameters of every node can be calculated iteratively. The method for micro models is based on approaches that Sasikumar[13] and Kraft[14] have explored. The numerical method of fixed time step was applied and the deforming space grid technique that increased node numbers by steps was employed to track the solid/liquid front. A flow chart showing the scheme of microsegregation calculations is given in Fig.1.
2.3 Coupling with calculation of phase diagrams
The foundation of the models for solidification must base on describing the phase equilibrium at solid/liquid interface rightly. And any incorrect supposition may lead to false simulation results. At present, the best and the most accurate method to treat the phase equilibrium at the interface is coupling the microsegregation calcula- tions with phase diagram calculations according to the CALPHAD method. Using this method the tie lines can be calculated on the fly, i.e. at each time step the kinetic equations are solved. The software of phase diagram calculations Pandat 2.0 was coupled with micro models. For binary alloys, in the calculated phase diagrams the liquidus and solidus are not perfect lines any more but curves consising of a series of discrete points. All the phase diagrams’ information used in micro models was calculated by linear interpolation method from calculated phase diagrams’ data and then was input into the program. For the Al-Cu alloy discussed in this study, the calculated phase diagram given by Pandat 2.0 is illustrated in Fig.2. It proves that this thermodynamic description for the Al-Cu system yielded results is in better agreement with experimental data.
In order to study the effect of calculated phase diagram on simulation results, the traditional simplified phase diagram for Al-Cu system was also applied into microsegregation calculations. In the simplified phase diagram, liquidus and solidus were supposed to be lines and the partition coefficient is a constant. The data of simplified phase diagram for Al-Cu system can be seen in Table 1[9,11].
Table 1 Data of Al-Cu system for simplified phase diagram
3 Results and discussion
As an example, simulation on microsegregation
Fig.1 Scheme of numerical method of micro model (I is the number of nodes)
Fig.2 Calculated phase diagram of Al-Cu system by phase diagram calculation software Pandat2.0
forming of an Al-4.5%Cu alloy ingot in water-cooled Cu mould was carried out. In this system, the size of the ingot is 50 mm×100 mm, the water-cooled temperature is 300 K, and the pour temperature is 980 K. The upper surface of the ingot directly contact with the air and the undersurface is adiabatic. Considering the symmetry of the cylinder ingot, a planar vertical section was chosen for the calculation area. The physical data for Al-Cu alloys used in the model calculations are given in Table 2[15].
Some locations that are in different distance from the center of a planer 60 mm lower than the upper surface in the ingot were chosen for study. The microsegregation parameters after solidification of these locations, including sencondary arm spacing and volume fraction of eutectic phase, were given in Table 3. All the experiment data in Table 3 are statistical results. The images and testing of the data were achieved through the commercial automatic image analysis system for metallograph CIMAS. The reliability of the software was validated.
Table 2 Physical data for Al-Cu alloys used in model calculations
Table 3 Influence of calculated phase diagram on microsegregation simulation results after solidification
It can be seen from Table 3 that the secondary arm spacings with simplified phase diagram is in better agreement with testing results. However, these results are close to the calculated secondary arm spacings with calculated phase diagram. Taking the testing errors into consideration, it can be thought that the calculated secondary arm spacings with two different phase diagrams are in the same precision range, i.e. coupling with calculation of phase diagrams has no remarkable effects on simulation results of the secondary arm spacing. Calculated results of volume fraction of eutectic phase are very discrepant while two kinds of phase diagram are applied. When the traditional simplified phase diagram is used, the calculated data of volume fraction of eutectic phase is not in good agreement with the experimental results, but they are in good agreement with each other when the calculated phase diagram is coupled. The reason is that the relationship between the coarsening model and the data of phase diagram is not so close, seen from Eqn.(2). However, much information of the phase diagram is used to solve the diffusion equation and the solute balance equation, so the disadvantages of applying the simplified phase diagram to solve these equations are obvious. Furthermore, it is just for the classical Al-Cu binary alloy system, if phase diagrams of ternary or multicomponent systems extrapolated from these simplified phase diagrams are used in microsegregation forming simulations, more unimaginable calculated results will be obtained.
In conclusion, the results of volume fraction of eutectic phase calculated with the simplified phase diagram are not consistent with the experiment results and the simulation is not satisfied. Therefore, precise forecasting of microsegregation is unimaginable without exactly treating the phase equilibrium at the solid/liquid interface and calculation of phase diagrams.
4 Conclusions
1) Relationship between the coarsening model and the information of phase diagram is not close. Coupling with calculation of phase diagrams has no remarkable effects on simulation results of the secondary arm spacing.
2) The calculated data of volume fraction of eutectic phase is not in good agreement with the experimental results when the traditional simplified phase diagram is used and the simulation is not satisfied. Coupling with calculated phase diagrams can improve the precision of simulation results greatly.
3) Coupling with calculated phase diagrams is very significant for microsegregation forming simulation. Precise prediction of microsegregation is unimaginable without exactly treating the phase equilibrium at the solid/liquid interface and calculation of phase diagrams.
Acknowledgements
The authors are grateful for financial support from the National Natural Science Foundation of China (59974011). Heartfelt thanks are also given to the CompuTherm LLC group of Wisconsin-Madison University for providing the software program to calculate the phase equilibria.
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(Edited by CHEN Can-hua)
Foundation item: Project (59974011) supported by the National Natural Science Foundation of China
Corresponding author: LIU Yong-gang, PhD, Instructor; Tel: +86-25-84315159; E-mail: lygnjust@mail.njust.edu.cn
