Computer Simulation of Phase Decomposition in Fe Cu Mn Ni Quaternary Alloy Based on the Phase-Field Method

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1 Materials Transactions, Vol. 46, No. 6 (5) pp. 87 to 92 Special Issue on Computer Modeling of Materials and Processes #5 The Japan Institute of Metals Computer Simulation of Phase Decomposition in Fe Quaternary Alloy Based on the Phase-Field Method Toshiyuki Koyama* and Hidehiro Onodera Computational Materials Science Center, National Institute for Materials Science, Tsukuba 35-47, Japan Recently, the phase-field method is becoming a powerful tool to simulate and predict complex microstructure evolutions in interdisciplinary fields of materials science. In this study, the phase-field simulations are demonstrated on the phase decomposition in the (bcc) phase during isothermal aging in Fe quaternary system, which is a base alloy system of the light-water reactor pressure vessel. Since the CALPHAD method based on a thermodynamic database of equilibrium phase diagrams is used for the evaluation of a chemical free energy in this simulation, the calculated microstructure changes are directly linked to the phase diagram of the Fe system. At the early stage of phase decomposition, the -rich zone with bcc structure begins to nucleate, and the component X (¼, ) is partitioned to the -rich phase. When the composition in the precipitate reaches almost the equilibrium value, the component X inside the precipitates moves to the interface region between the precipitate and the matrix. Finally, there appears the shell structure that the precipitates are surrounded by the thin layer with high concentration of component X. This microstructure change is reasonably explained by considering the local equilibrium at the compositionally diffused interface region of -rich nano-particles surface. (Received December, 4; Accepted February 7, 5; Published June 5, 5) Keywords: phase-field method, phase transformation, phase decomposition, spinodal decomposition, reactor pressure vessel, diffusion equation. Introduction During last decade, the phase-field method is becoming a powerful tool to simulate and predict complex microstructure evolutions in interdisciplinary fields of materials science. 5) The objective of this study is to model the spinodal decomposition of the (bcc) phase in Fe quaternary alloy, which is a base alloy system of the lightwater reactor pressure vessel, using the phase-field modeling. 6,7) The reason for the neutron irradiation induced embrittlement 8 ) of the light-water reactor pressure vessel has been considered that the presence of the nano-scale copper precipitates and/or the point defect clusters formed during the neutron irradiation. They act as obstacles of the dislocation movement, resulting in the hardening of the materials. Therefore, the -rich cluster formation and the role of solute elements such as and, which will effect on the formation and stability of the -rich cluster, have been investigated in this alloy system. According to the experimental results of the three-dimensional atom probe analysis by Miller et al., ) the atoms segregate at the interface region between -rich cluster and matrix. Recently, Seko et al. 2) figured out the atoms prefer to segregate at the -rich cluster surface on the basis of the atomistic theoretical calculation. In this study, the behavior of the composition fields of the components and during spinodal decomposition in Fe quaternary alloy is investigated using the phase-field simulation, and the reason for the segregation of and atoms at the surface of the -rich phase is discussed using the simulation results. 2. Calculation Method Since we focus on the microstructure changes during the *Corresponding auther, KOYAMA.Toshiyuki@nims.go.jp diffusion-controlled phase decomposition in Fe quaternary alloy system in this simulation, the invariant variables describing the microstructure developments are the composition fields only, and therefore, the phase-field method used in this work is equivalent to the computer simulation based on the non-linear diffusion equation. 3 5) The details of the non-linear diffusion equation and the evaluation of the total free energy of microstructure are described in the following sections. 2. Non-linear diffusion equation The non-linear diffusion equations governing the phase decomposition in the Fe quaternary alloy system are written 2 ðr; tþ ¼ M 22 r 2 2 ðr; tþþm 23 r 2 3 ðr; þ M 24 r 2 4 ðr; 3 ðr; tþ ¼ M 32 r 2 2 ðr; tþþm 33 r 2 3 ðr; þ M 34 r 2 4 ðr; tþ; 4 ðr; tþ ¼ M 42 r 2 2 ðr; tþþm 43 r 2 3 ðr; þ M 44 r 2 4 ðr; tþ; where the local composition field of component i, c i ðr; tþ,isa function of the spatial position r and time t. The integer values i ¼, 2, 3, and 4 are corresponding to Fe,,, and, respectively. M ij is a dimensionless mobility of atom diffusion defined by M 22 ¼ c 2 ð c 2 Þ; M 33 ¼ c 3 ð c 3 Þ; M 44 ¼ c 4 ð c 4 Þ; M 23 ¼ M 32 ¼ c 2 c 3 ; M 24 ¼ M 42 ¼ c 2 c 4 ; M 34 ¼ M 43 ¼ c 3 c 4 : From a physical point of view, the quantity M ij should be functions of local composition and temperature, but we ð2þ

2 88 T. Koyama and H. Onodera assume M ij are functions of average compositions c i of component i in eq. (2), for simplicity. Furthermore, since we simulate the isothermal phase decomposition only in the following, i.e. the aging temperature is constant, the dependence of M ij on temperature is not considered explicitly. The diffusion potential of component i, i ðr; tþ, is calculated by 2 ðr; tþ G sys c 2 3 ðr; tþ G sys c 3 4 ðr; tþ G sys c 4 2 c ð2r 2 c 2 þr 2 c 3 þr 2 c 4 Þ; 3 c ðr 2 c 2 þ 2r 2 c 3 þr 2 c 4 Þ; c c ðr 2 c 2 þr 2 c 3 þ 2r 2 c 4 4 Z G sys ¼ ðgc þ E surfþdr: r where G sys is a total free energy of microstructure, which is defined by a sum of the chemical free energy Gc and the composition gradient energy E surf. The elastic strain energy is ignored in this study for simplicity. Because we focus on the early stage of phase decomposition in this simulation, the elastic strain energy will crucially not influence on the behavior of microstructure changes. Gc is a Gibbs free energy density of the (bcc) phase in Fe quaternary system, and c is a composition gradient energy coefficient that is assumed to be a constant in this work. Since the CALPHAD method 6) based on a thermodynamic database of equilibrium phase diagrams is used for the evaluation of a chemical free energy Gc, the calculated microstructure changes are directly linked to the phase diagram of the Fe system. The detail expression of the chemical free energy function and the composition gradient energy equation are explained in the next section. ð3þ ð4þ 2.2 Total free energy of microstructure The chemical free energy (Gibbs free energy) of the (bcc) phase in Fe--- quaternary system with magnetic contribution is described by the sub-regular solution approximation, 6) G c ðc i; TÞ ¼ X G i c i þ E G þ mg G þ RT X c i ln c i ; ð5þ i i where G i is the Gibbs energy of pure element i in the case of the crystal structure is bcc, which is expressed as a function of temperature. 7) E G is the excess free energy corresponding to the heat of mixing, and mg G is the magnetic contribution to the Gibbs energy. R and T are the gas constant and the absolute temperature, respectively. The functions of E G and mg G are defined as E G X X L i;j c ic j þ X X X L i;j;k c ic j c k ; ð6þ i j>i i j>i k>j mg G RT lnð þ Þf ðþ; T=T C ; ð7þ where the interaction parameters L i;j and L i;j;k, the rie temperature TC, and the atomic magnetic moment are available from the thermodynamic database of equilibrium phase diagrams. These parameters for the Fe system have been determined by Miettinen, 8) and ones for the Fe system are available from the SSOL database in ThermoCalc, 9) then we used the following data: L ;2 ðl ;2 L ;3 L ;4 ¼ 433: 6:22T; ¼ 39257:976 4:49834T: simulation of Fe{{Þ ¼ 2759: þ :237T; ¼ 956:63 :28726T þð789:3 :9292TÞðc c 4 Þ; L 2;3 ¼ 9: 6:T 9865:ðc 2 c 3 Þ; L 2;4 used only for the ¼ 8366: þ 2:82T; L 3;4 ¼ 5638:3 þ 3:64T þ 6276:ðc 3 c 4 Þ; L ;2;3 ¼ 3:; L ;2;4 ¼ L ;3;4 ¼ L 2;3;4 ¼ ; ðj/molþ TC ¼ 43c 58c 3 þ 575c 4 þ 23c c 3 ; ðkþ ¼ 2:22c :27c 3 þ :85c 4 : The function f ðþ in eq. (7) is given by Hillert and Jarl ) as a function of : f ðþ 79 D p þ p 3 6 þ 9 35 þ 5 ; ð Þ; 6 f ðþ 5 D þ 5 35 þ 25 ; ð Þ; ð9þ 5 D þ p ; ðp ¼ :4 for bcc phase, and p ¼ :28 for othersþ: Figures (a) and (b) show the metastable phase diagrams calculated at 873 K under considering the (bcc) phase only in the Fe and Fe ternary alloy systems, respectively. It is noted that the ( þ 2 ) two-phase region of Fe binary phase diagram is extended with increasing or content. The average composition of the alloy used in the simulation is denoted by the solid circle. The thin straight lines inside the two-phase region are calculated tie lines. The density of composition gradient energy, E surf, of the Fe quaternary system is expressed based on the spinodal theory developed by Cahn and Hilliard 3,4) as Fe at% 6 8 a) b) Fe at% Fig. Metastable phase diagrams of Fe and Fe ternary systems where the (bcc) phase is considered only. 6 8 ð8þ

3 Computer Simulation of Phase Decomposition in Fe Quaternary Alloy Based on the Phase-Field Method 89 E surf ¼ X i 2 cðrc i Þ 2 ¼ 2 X X c ð ij þ Þðrc i Þðrc j Þ i2 j2 ¼ c ðrc 2 Þ 2 þ c ðrc 3 Þ 2 þ c ðrc 4 Þ 2 þ c ðrc 2 Þðrc 3 Þþ c ðrc 2 Þðrc 4 Þþ c ðrc 3 Þðrc 4 Þ: ðþ The numerical value of the composition gradient energy coefficient c is taken as c ¼ : 4 (Jm 2 /mol). This value is roughly estimated using the relation, 3) c ¼ ð=2þd 2, where is an interaction parameter between atoms and we used ¼ L ;2ðT ¼ KÞ ¼433: (J/mol) (see eq. (8)). d is an effective interaction distance ) and the value of which is assumed as d ¼ :7 (nm) that is about a half length of the interface region between the -precipitate and the matrix. 3. Simulation Results 3. Two-dimensional simulation of isothermal phase decomposition Figure 2 shows the two-dimensional simulation of the isothermal phase decomposition of the -bcc phase in Fe 5 at% ternary alloy at 873 K. The upper layer and the lower one in Fig. 2 indicate the and composition fields, respectively. The local composition is represented by gray scale, where the pure and the at% are represented by white (see the gray scale at the right hand side of Fig. 2). The numerical values in Fig. 2 indicate the dimensionless normalized aging time (The unit s means the dimensionless time). Figure 2(a) is the initial stage of the supersaturated solid solution where the small composition fluctuation is imposed followed by the Gaussian noise using the random number. At the early stage of aging (Fig. (b)), the -rich zone is nucleated from the supersaturated solid solution, and the component is mainly partitioned to the -rich phase. When the composition in the precipitate reaches the equilibrium value (Fig. (c)), the component inside the precipitates moves to the interface regions between the -precipitate and the matrix. Finally, there appears the shell structure that the precipitates are surrounded by the thin layer with high concentration of component (Fig. (d)). Figure 3 demonstrates the simulation of the isothermal phase decomposition of the -bcc phase in Fe 5 at% ternary alloy at 873 K. The drawing style of figure is the same as Fig. 2, and the upper layer and the lower one in Fig. 3 represent the and composition fields, respectively. At the initial stage of aging, the -rich zone is nucleated, and is mainly partitioned to the -rich phase. When the composition inside the precipitate increases, the component starts moving toward the interface region. Finally, the shell structure also appears in this alloy. Comparing Fig. 2 and Fig. 3, we note the global microstructure changes are almost the same, but the tendency of the segregation of atoms at the surface of the -rich phase is larger than that of atoms. Figure 4 shows the simulation for the isothermal phase decomposition of the -bcc phase in Fe 5 at% 5 at% quaternary alloy at 873 K. The upper, middle, and the lower layers indicate the, and composition fields, respectively. The simulated microstructure changes look like almost the combination between Fig. 2 and Fig. 3. At the early stage, the -rich zone is nucleated from the super-saturated solid solution, and the components and are partitioned to the -rich phase. With progress of aging, the components and inside the precipitates move to the interface region between the precipitate and matrix, then the shell structure appears. The tendency of the segregation of atoms at the surface of -rich phase is larger than that of atoms, and the high region is located inside the -shell. Fe--5at% aged a)6s' b)s' c)8s' d)3s' nm at% at% Fig. 2 Two-dimensional simulation of phase decomposition in Fe 5 at% alloy at 873 K.

4 9 T. Koyama and H. Onodera Fe--5at% aged a)6s' b)s' c)8s' d)3s' nm at% at% Fig. 3 Two-dimensional simulation of phase decomposition in Fe 5 at% alloy at 873 K. Fe--5at%-5at% aged a)6s' b)s' c)8s' d)3s' at% at% nm at% at% Fig. 4 Two-dimensional simulation of phase decomposition in Fe 5 at% 5 at% alloy at 873 K. 3.2 Temporal changes of the composition profiles in Fe system Figure 5 shows the one-dimensional simulation of the composition profile development in Fe 5 at% 5 at% ally at 873 K. The vertical axis is a composition and the abscissa is a distance, and numerical value in each figure is the dimensionless aging time. The thick, dotted, and thin curves correspond to the,, and composition profiles, respectively. The and compositions are magnified by twice so as to see the segregation behavior easily. At the early stage of aging, the -rich zone is nucleated and the components and are mainly

5 Composition Computer Simulation of Phase Decomposition in Fe Quaternary Alloy Based on the Phase-Field Method 9 partitioned to the -rich phase (see the profile at s ). When the composition in the precipitate reaches almost the equilibrium value, the components and inside the precipitates move to the interface between the precipitate and matrix (see the profiles at s and 2 ks ). In particular, component starts moving faster than. When we see the composition peak denoted by (A), which disappears by Ostwald ripening, the and compositions at the center of the precipitate gradually rise up during the -particle resolving. 4. Discussion In this section, the reason for the segregation of and atoms at the surface of the -rich phase is discussed. Using the simulation results of the composition profiles of Fig. 5, we can obtain the local composition value at each spatial point in the microstructure. Plotting the local composition given from Fig. 5 at t ¼ s and t ¼ 2 ks on the phase diagrams, we are able to draw the dotted and solid curves inside the two-phase region of the phase diagrams (Fig. 6). The solid circle indicates the alloy composition. At the early stage of phase decomposition of t ¼ s (see the dotted curves), the phase separation proceeds almost along the tie line direction. However, at the late stage of t ¼ 2 ks, the trajectory of the composition profile on the phase diagram is deviated largely from the tie line (see the thick solid curves), s' s' s' 2ks' Fe--5at%-5at% aged Distance c 2c 2c (A) Distance 8ks'.6ks' ks' 2ks' nm Fig. 5 One-dimensional simulation of the temporal development of composition profiles in Fe 5 at% 5 at% alloy at 873 K. Fe at% 6 8 a) b) s' s' 2ks' 2ks' Fe at% Fig. 6 The dotted curve and thick solid one inside the two-phase region are the trajectory of the composition profile at t ¼ s and t ¼ 2 ks in Fig. (5), respectively. 6 8 Fe-- G /RT c Chemical free energy, - Fe (B) (b) (A) (a) Fig. 7 Gibbs energy surface of the Fe ternary alloy system at 873 K. The thick solid curve denoted by (A) is the trajectory of the local composition on the Gibbs free energy surface. which corresponds to the segregation of both and atoms at the interface region, i.e. the composition of and takes high value when composition is about 5 that is almost the center of the interface region. The reason why this segregation phenomenon is taken place is reasonably explained by considering the local equilibrium at the compositionally diffused interface region of -rich nano-particles surface. Figure 7 shows the Gibbs energy surface plane of Fe ternary system at 873 K. When we look at the global shape of the free energy surface, there is a convex part at the Fe side and the convex produces a large driving force for the phase separation between -rich phase and Fe-rich one. This driving force for phase separation decreases rapidly with increasing content. The thick dotted curve denoted by (a) on the bottom plane is the thick solid curve in the two-phase region of the Fe phase diagram of Fig. 6(a). The thin dotted line denoted by (b) on the bottom plane is a tie line, and the solid circle is a position of the alloy composition. The thick solid curve (A) plots the trajectory along the curve (a) on the free energy surface, and the thin solid curve (B) on the free energy surface corresponds to the trajectory along the tie line (b). If there is no segregation of and atoms at the interface region between the -precipitate and the matrix, the trajectory of composition profile on the free energy surface should be along the tie line indicated as (B). But if the trajectory expands toward the -rich side represented as (A), it is clearly understood that the case of (A) is energetically more stable than that of (B). Therefore, the reason why the solute elements and segregate at the interface region is that the composition profile is self-regulated so as to reduce the chemical free energy at the interface region of precipitate. Therefore, this is an origin for the shell microstructure appearing. On the other hand, the atomistic qualitative explanation of this behavior is as follows: Since the interchange energy between Fe and atoms is fairly large, the phase separation between -rich phase and Ferich one is take place so as to reduce the number of the Fe bonding pare. However, the Fe pare exists inherently at the interface between matrix and precipitate. If or

6 92 T. Koyama and H. Onodera atoms segregate at the interface region, the number of Fe pare will decrease, and the interchange energy of Fe or Fe pare takes negative value. Therefore, the segregation of and atoms at interface region lower the internal energy in the chemical free energy. The above-mentioned segregation behavior seems to be a common phenomenon, when the third elements that has an ordering tendency with bulk component is added to binary alloy system that shows a large phase separation. For instance in the Al Mg Ag system, it is known that Mg and Ag atoms move to the surface region of -zone at the later stage of phase decomposition. 2) The similar mechanism may be applicable to this case. 5. Conclusions The isothermal phase decomposition of the (bcc) phase in the Fe quaternary system was simulated on the basis of the phase-field method, and the segregation behavior of the and atoms is analyzed using the simulation result. The results obtained are as follows: () At the initial stage of phase decomposition, and atoms are partitioned to the -rich phase. When the composition in the precipitate reaches the equilibrium value, the components and inside the precipitates moves to the interface region between the -precipitate and matrix. Finally, there appears the shell structure that the precipitates are surrounded by the thin layer with high concentration of components and. (2) The segregation tendency of atoms at the surface of -particles is larger than that of. (3) The shell structure of the -particle appears because the composition profile is self-regulated so as to reduce the chemical free energy at the local interface region between precipitate and matrix. Acknowledgements This work was partly supported by a NEDO International Joint Research Grant on Structuring Knowledge, Science and Technology for Nano Material Processing and Nanometal Technology Project, and by the Special Coordination Funds for Promoting Science and Technology on Nanohetero Metallic Materials from the Ministry of Education, lture, Sport, Science and Technology. This work was supported also by NAREGI Nanoscience Project; Ministry of Education, lture, Sports, Science and Technology, and the CREST; Japan Science and Technology Agency. REFERENCES ) T. Koyama: Materia Japan (Bull. of the Japan Inst. Metals) 42 (3) ) T. Koyama: Ferrum (Bulletin of The Iron and Steel Institute of Japan) 9 (4) 2, 3, 376, ) L.-Q. Chen: Annu. Rev. Mater. Res. 32 (2) 3. 4) W. J. Boettinger, J. A. Warren, C. Beckermann and A. Karma: Annu. Rev. Mater. Res. 32 (2) 63. 5) M. Ode, S. G. Kim and T. Suzuki: ISIJ Int. 4 () 76. 6) T. Koyama and H. Onodera: Mater. Trans. 44 (3) ) T. Koyama and H. Onodera: Mater. Trans. 44 (3) ) KINZOKU(Materials Science & Technology), AGNE GIJUTSU CENTER, 73 (3) ) M. K. Miller, P. Pareige and M. G. Burke: Mater. Charact. 44 () 235. ) Y. Nagai, T. Toyama, Z. Tang, M. Hasegawa, S. Yanagita, T. Ohkubo and K. Hono: Mater. Sci. Forum (4). ) M. K. Miller, B. D. Wirth and G. R. Odette: Mater. Sci. Eng. A 353 (3) 33. 2) A. Seko, N. Odagaki, S. R. shitani, I. Tanaka and H. Adachi: Mater. Trans. 45 (4) ) J. W. Cahn: The Selected Works of J. W. Cahn, ed. by W. C. Carter and W. C. Johnaon, (TMS, 998) 29. 4) J. E. Hilliard: Phase Transformation, ed. by H. I. Aaronson, (ASM, Metals Park, Ohio, 97) ) T. Koyama and H. Onodera: Met. Mater. Int. (4) 32. 6) N. Saunders and A. P. Miodownik: CALPHAD, (Pergamon, 998). 7) A. T. Dinsdale: CALPHAD 5 (99) 37. 8) J. Miettinen: Computer Coupling of Phase Diagrams and Thermochemistry 27 (3) 4. 9) SSOL database in ThermoCalc (ver.m), Thermo-Calc Software AB. ) M. Hillert and M. Jarl: CALPHAD 2 (978) ) K. Hono: Prog. Mater. Sci. 47 (2) 62.