Development Center, Warren, MI , USA 3 State Key Laboratory of Automotive Safety and Energy, Tsinghua University, Beijing , China

Transcription

1 EPD Congress 2013 TMS (The Minerals, Metals & Materials Society), 2013 STUDY ON EFFECTS OF INTERFACIAL ANISOTROPY AND ELASTIC INTERACTION ON MORPHOLOGY EVOLUTION AND GROWTH KINETICS OF A SINGLE PRECIPITATE IN Mg-Al ALLOY BY PHASE FIELD MODELLING Guomin Han 1, Zhiqiang Han 1, Alan A. Luo 2, Anil K. Sachdev 2, Baicheng Liu 1, 3 1 Key Laboratory for Advanced Materials Processing Technology (Ministry of Education), Department of Mechanical Engineering, Tsinghua University, Beijing , China 2 Chemical and Materials Systems Laboratory, General Motors Global Research and Development Center, Warren, MI , USA 3 State Key Laboratory of Automotive Safety and Energy, Tsinghua University, Beijing , China Keywords: precipitation simulation, elastic inhomogeneity, interface anisotropy, morphology evolution, growth kinetics, phase-field model, Mg-Al alloy Abstract The growth of precipitate phase -Mg 17 Al 12 in Mg-Al alloy is closely related to interface anisotropy and elastic interaction. The aim of this work is to investigate the effects of the interface anisotropy and elastic interaction on the morphology evolution and growth kinetics of the precipitate. In the work, the precipitation of was studied using phase field modeling. In the model, the chemical free energy of the precipitate and the matrix were obtained via thermodynamic calculation. It is demonstrated that the interface anisotropy results in lath-shaped precipitate and the elastic interaction affects the shape of the end of the precipitate. The interface anisotropy causes uneven growth in different directions and the elastic interaction influences its preferential growth orientation. By combining the interface anisotropy and elastic interaction in the phase field modeling, the results showed that the precipitate phase has a lath-shape with lozenge ends, which is in agreement with experimental observations. 1. Introduction Magnesium alloys possess important application potential in automobile and aerospace industries due to their light weight and high specific strength [1]. Mg-Al series alloy is a typical commercial magnesium alloy. Based on the binary phase diagram, -Mg 17 Al 12 precipitate forms during aging process of Mg-Al alloy that can be strengthened with heat treatment. The phase is a complex intermetallic compound with -Mn body-centered cubic structure [2-3]. Its morphology evolution and growth kinetics have effects on the mechanical property of components [2-4]. So an understanding on the precipitate evolution and its growth characteristics is vitally important. In general, the morphology and growth of the precipitate are considerably affected by the 97

2 interface characteristics and misfit strains due to the interaction between the precipitate and the matrix [5]. The interface between the precipitate and the matrix is composed of coherent, non-coherent and semi-coherent interfaces. The coherent interface has a low interfacial energy, while the non-coherent interface has a higher interfacial energy. A strong anisotropy exists in the interface growth if the interfacial energy has a large difference in various directions of the precipitate growth. The misfit strain is related to the elastic interaction that can be calculated by elastic strain energy. Some knowledge about the morphology and crystallography of the precipitates in Mg-Al alloys has been acquired by SEM and TEM observations [2-4]. However, the mechanisms of the precipitate evolution are still not very clear. Phase field method based on energy relaxation has certain advantages in studying the morphology evolution and growth kinetics of precipitate, especially in revealing the effects of the governing variables quantitatively. Heo et al. [6] studied the effect of elastic interaction on the phase transformation in poly crystals system by using phase field modeling. Gao et al. [7] simulated the evolution of 1 precipitate of a magnesium rare earth alloy using phase field model and investigated the effect of the elastic interaction on the precipitation process. Hu et al. [8] simulated the morphology and growth kinetics of Te-precipitate in CdTe crystals with the consideration of the effects of the elastic interaction and interface anisotropy. While for Mg-Al alloy, related research works were reported less. The work carried out by Li et al. [9] was an early attempt, and they simulated the precipitation process of phase in Mg-Al alloy using phase field method. A comprehensive understanding on the effects of the interface anisotropy and the elastic interaction on the evolution of the precipitates is still required. In the present paper, a phase field model was presented to simulate the morphology evolution and growth kinetics of a single precipitate of Mg-Al alloy. In the model, the interfacial energy and elastic interaction were taken into account, and the chemical free energy for the precipitate phase and the matrix were directly obtained via thermodynamic calculation. The simulation results of the precipitate morphology were compared with experimental observations. The effects of the interfacial energy anisotropy and elastic interaction on the morphology evolution and growth kinetics were discussed. 2. Phase Field Model Some modifications were made based on KKS phase field model [10]. The chemical free energy and potential were directly obtained via thermodynamic calculation, and the elastic strain energy was included in the total energy. The modified KKS phase field model was used to simulate the precipitation process of -Mg 17 Al 12 in Mg-Al alloy during aging process. These precipitates usually grow from the supersaturated matrix with the specific orientation relationship [2]. According to the symmetry, there are 12 precipitate variants. For simplicity, we only consider 3 individual precipitate variants. Thus we need to use only one order parameter, which is defined as 0 in the matrix and 1 in the precipitate, as non-conserved quantity in the phase field model. 98

3 In the model, the total energy F containing the contribution of the chemical free energy, interface gradient energy and the elastic strain energy is expressed as follows, where,, 2 ( ) 2 ela F c,, T f c,, T d ve (1) v 2 f c T is the density of the chemical free energy. () is the gradient energy ela coefficient related to interfacial energy. is the order parameter. energy. The density of the chemical free energy,, E is the elastic strain f c T is defined as follows [10], f h( ) f ( c, T) (1 h( )) f ( c, T) wg( ) (2) where c and c are the concentration of solute Al (molar fraction) in the matrix and precipitate, respectively. f ( c, T) and f ( c, T) are the free energy densities of the matrix and precipitate, respectively, which can be obtained directly by thermodynamic calculation, as shown in Figure 1. h () is a monotonic function valued between 0 and 1, g () is a double-well potential, and w is the height of the double-well potential. h () and g () are calculated as follows, 2 3 h ( ) 3 2 (3) 2 2 g ( ) (1 ) (4) Figure 1. The molar free energy of the matrix and precipitate in Mg-Al alloy with different composition 99

4 For the calculation of the elastic strain energy E ela, we must calculate the stress-free transformation strain or eigenstrain 0 ij ( p). According to the crystallographic features of the HCP-BCC transformation in Mg-Al ally, the eigenstrain can be calculated by the approach described in reference [11]. Due to the low volume fraction of the precipitate and the lack of the modulus data for the precipitate, we assumed that the precipitate phase and matrix have the same elastic modulus. The elastic modulus of the matrix can be defined as, C C C C C C C C ijkl C C 0 44 C C where C 11 =58GPa, C 12 =25GPa, C 13 =20.8GPa, C 33 =61.2GPa, C 44 =16.6GPa [9]. Thus the elastic strain energy E ela can be obtained based on Khachaturyan's elastic strain theory [12], 3 ela 1 d g * 2 2 E ( ) 3 2 B n (5) (2 ) g g where g is a vector in the Fourier space, n g g. 2 * is complex conjugate of g 2. g Bn ( ) is expressed as, 2 is Fourier transform of g 2, Bn ( ) C n( p) ( n) ( pn ) (6) ijkl ij kl i ij jk kl l where 0 ( p) ij is the stress tensor and ( p) C ( p). 0 0 ij ijkl kl 0 ij is defined as, ( p ) (7) ij ij Thus the variational derivative of the elastic strain energy E ela can be calculated, ela E 2 2 ( ) Bn g x (8) 100

5 where Bn 2 ( ) g is the inverse transformation of 2 Bn ( ) g x in the Fourier space. 3. Results and Discussion In the simulation, the precipitate variants in the direction of 0, 60 and 120 degrees are referred to as variant-1, variant-2 and variant-3, respectively. We only investigated the evolution of a single variant each time. The simulation domain was divided into cells. For each cell, the initial concentration of the matrix is and the initial concentration of the precipitate phase is It was assumed that the solute diffusion coefficient D and the temperature T are constant in all simulations with the value of e-18m 2 /s and 441K, respectively. The evolution of the precipitates in Mg-Al alloy showed the characteristics of strong interface anisotropy [3]. In this paper, the interface between the precipitate and matrix was assumed to be composed of semi-coherent and non-coherent interfaces. The tips of the precipitate are non-coherent interface, while the two sides of the precipitate are semi-coherent interface. The interfacial energy of the semi-coherent was assumed to be 60mJ/m 2, and the interfacial energy of the non-coherent was assumed to be 300mJ/m 2. Thus the interface anisotropy can be defined as, ( ) f( ) semi-coherent non-coherent semi-coherent where non-coherent is the interfacial energy of the non-coherent interface, semi-coherent is the interfacial energy of the semi-coherent interface. f () is the interface anisotropy coefficient. For variant-1, the value of interface anisotropy coefficient in polar coordinate system can be shown in Figure 2. For the other two variants, we can construct the interface anisotropy coefficient similarly. (9) Figure 2. The value of interface anisotropy coefficient in polar coordinate system 101

6 In accordance with the KKS phase field model [10], the interface mobility and the gradient energy coefficient can be calculated by (), so the interface mobility and the gradient energy coefficient are also anisotropic. Figure 3 shows the morphology evolution of the precipitate with consideration of the interface anisotropy. In the Figure, the blue denotes the matrix, and the red denotes the precipitate. It can be seen that the precipitates have a lath-shaped morphology, and their ends are about semi-circle. Due to the effects of the interface anisotropy on the precipitate with different orientation, the single variant-1 evolves into lath along the direction of 0 degree, while for the single variant-2 along the direction of 60 degrees and for the single variant-3 along the direction of 120 degrees. (a) (b) (c) (d) (e) (f) (g) (h) (i) Figure 3. The lath morphology of the precipitate with only consideration of the interface anisotropy, (a-c) variant-1, (d-f) variant-2, (g-i) variant-3 The effect of elastic interaction was taken into account after considering the interface anisotropy. Figure 4 shows the morphology evolution of the precipitate. The figure indicates that the precipitates have lath-shaped morphology and the main difference as compared with the morphology shown in Figure 3 is that the precipitates have lozenge ends. The simulation results show the orientation characteristics of different single precipitate variant. As we know, the precipitate would grow into a round shape without consideration of the interface anisotropy and elastic interaction. The morphology of the precipitate is lath-shaped when only the effect of the interface anisotropy is taken into account. With the consideration of the effect of elastic interaction, the morphology of the ends changes into lozenge shape. So we 102

7 can conclude that the effect of the interface anisotropy on the precipitate morphology is to form the lath shape, and the effect of the elastic interaction is to form the lozenge ends. In fact, the morphology characteristics of lath-shape with lozenge ends are in agreement with the results of the authors TEM observation (see Figure 5). (a) (b) (c) (d) (e) (f) (g) (h) (i) Figure 4. The morphology evolution of the precipitate with consideration of the interface anisotropy and elastic strain energy, (a-c) variant-1, (d-f) variant-2, (g-i) variant-3 (a) Lozenge (b) Lozenge Figure 5. The morphology of the precipitate (a) simulation, (b) TEM bright-field image For the growth kinetics of the precipitate, we can take the variant-2 as an example to do 103

8 analysis, and similar analysis can be made for the other two variants. Figure 6 shows the growth kinetics of the variant-2 in different directions with and without consideration of the elastic energy. It can be seen that the growth of the precipitates in all directions are about linear. The growth in longitudinal direction is much faster than that in the transverse direction. The non-coherent interface at the tips with higher interfacial energy is much more unstable than the semi-coherent interface at the sides of the precipitates, which is regarded the main reason of the strong anisotropy of the precipitate growth. The elastic interaction slightly speeds up the growth of the width direction (see W1 and W2 in the figure). The preferential growth direction was changed from L4 to L3 as a result of the effect of the elastic interaction. Figure 6. The growth kinetics of the precipitate with and without consideration of the elastic energy 4. Conclusion 1. A phase field model was developed to simulate the morphology evolution and growth kinetics of single precipitate variants in Mg-Al alloy. Both the interface anisotropy and elastic interaction were considered in the model. 2. The simulation results demonstrated that the morphology and growth kinetics of the single precipitates are affected by the interface anisotropy and elastic interaction. The morphology characteristics of the precipitate with lath-shape and lozenge-ends are in agreement with the TEM observations. 3. The lath-shape morphology is attributed to the effect of the interface anisotropy, and the elastic interaction is the main contribution to the formation of the lozenge ends. From the viewpoint of growth kinetics, the interface anisotropy results in the anisotropic growth characteristics of the precipitate, and the elastic interaction affects the preferential growth orientation of the precipitates. 104

9 Acknowledgement This work is funded by the National Natural Science Foundation of China (Grant No ), the General Motors Global Research and Development Center, Tsinghua University Initiative Scientific Research Program, and the MoST (Ministry of Science and Technology) of China under the contracts of No.2010DFA72760 and 2011DFA References [1] B. L. Mordike and T. Ebert, Magnesium: properties-applications-potential, Materials Science and Engineering A, 302 (2001), [2] S. Celotto, TEM study of continuous precipitation in Mg-9wt%Al-1wt%Zn alloy, Acta Materialia, 48(2000), [3] C. R. Hutchinson, J. F. Nie and S. Gorsse, Modeling the precipitation processes and strengthening mechanisms in a Mg-Al-(Zn) AZ91 Alloy, Metallurgical and Materials Transactions A, 36(2005), [4] J. B. Clark, Age hardening in a Mg-9wt% Al alloy, Acta Metallurgica, 16(1968), [5] D. A. Porter, K. E. Easterling and M. Y. Sherif, Phase Transformations in Metals and alloys (UK, UK: CRC Press, 2009), 118. [6] G. Sheng, S. Bhattacharyya, H. Zhang, K. Chang, S. L. Shang, S. N. Mathaudhu, Z. K. Liu and L. Q. Chen, Effective elastic properties of polycrystals based on phase-field description, Materials Science and Engineering A, 554(2012), [7] Y. Gao, H. Liu, R. Shi, N. Zhou, Z. Xu, Y. M. Zhu, J. F. Nie and Y. Wang, Simulation study of precipitation in an Mg Y Nd alloy, Acta Materialia, 60 (2012), [8] S. Y. Hu, Charles H and Henager Jr, Phase-field simulations of Te-precipitate morphology and evolution kinetics in Te-rich CdTe crystals, Journal of Crystal Growth, 311(2009), [9] M. Li, R. J. Zhang and J. Allison, Modeling casting and heat treatment effects on microstructure in super vacuum die casting (SVDC) AZ91 magnesium alloy, Magnesium Technology 2010, [10] S. G. Kim, W. T. Kim and T. Suzuki, Phase-field model for binary alloys, Physical Review E, 60(1999), [11] N. Katarzyna and M. Braszczynska, Precipitates of -Mg 17 Al 12 phase in AZ91 alloy, Magnesium Alloys-Design, Processing and Properties, (2011), [12] A. G. Khachaturyan, Theory of Structural Transformation in Solid, (America, New York: Wiley-Interscience 1983),