- Email: [email protected]
Precipitation kinetics of ordered γ′ phase and microstructure evolution in a NiAl alloy
Materials Chemistry and Physics 182 (2016) 125e132
Contents lists available at ScienceDirect
Materials Chemistry and Physics journal homepage: www.elsevier.com/locate/matchemphys
Precipitation kinetics of ordered g0 phase and microstructure evolution in a NieAl alloy Xingchao Wu, Yongsheng Li*, Mengqiong Huang, Wei Liu, Zhiyuan Hou School of Materials Science and Engineering, Nanjing University of Science and Technology, Nanjing 210094, China
h i g h l i g h t s Kinetics of g0 phase from nucleation to coarsening in NieAl alloy is studied. Non-stoichiometric ordered phase and initial cluster are quantitatively described. Time exponent of particle radius decreases from nucleation and growth to coarsening. Average aspect ratio of g0 phase shows three stages with respect to average radius.
a r t i c l e i n f o
a b s t r a c t
Article history: Received 17 June 2015 Received in revised form 22 June 2016 Accepted 9 July 2016 Available online 18 July 2016
The precipitation kinetics of ordered g0 (L12-Ni3Al) phase from nucleation and growth to coarsening in a NieAl alloy is studied by phase field simulation. The initial nucleation of the g0 phase is quantitatively described by the variations of composition and order parameter profiles of the initial Al-enriched cluster, and the coarsening kinetics is clarified with the time exponent of the average particle radius of g0 phase. The precipitation of the g0 phase from the supersaturated g solid solution goes through the nonstoichiometric ordered phase, stoichiometric ordered phase, growth and coarsening. The time exponent of the average radius of the g0 phase with coherent equilibrium is smaller than that indicated by the LifshitzeSlyozoveWagner theory at the coarsening stage, and it decreases from 1.35 of the nucleation and growth stage to 0.22 of the coarsening stage. The peak of the particle size distribution decreases and the width of the particle size distribution increases as the aging progresses. As the average radius of the g0 phase increases, the average aspect ratio of the g0 phase undergoes a slowly increase to a fast increase from nucleation to growth stage, then goes into a relatively stable state at the coarsening stage. © 2016 Elsevier B.V. All rights reserved.
Keywords: Alloys Annealing Computer modelling and simulation Phase transitions
1. Introduction Nickel-based superalloys are the preferred structural materials in commercial and military jet engines and land-based gas turbines because of their excellent high-temperature properties [1e6]. In nickel-based superalloys, the coherent g0 (L12-Ni3X) phase embedded in the disordered g (fcc-Ni) matrix, the coherent elastic strain energy, and the interactions between the g0 phase and dislocations lead to a strengthening of the alloys. Therefore, the kinetics evolution of phases’ size and the spatial distribution of the ordered g0 phase during thermal aging and high-temperature service will affect the properties of nickel-based superalloys. To study the phase transformation of the g0 phase in nickel-
* Corresponding author. E-mail address: [email protected] (Y. Li). http://dx.doi.org/10.1016/j.matchemphys.2016.07.013 0254-0584/© 2016 Elsevier B.V. All rights reserved.
based alloys, thermodynamic calculation of the phase diagram [7e16] and dynamic description of the growth and coarsening by simulation and experiments [17e19] have been conducted in past decades. Wendt [17] studied the decomposition of Ni-14 at.% Al during isothermal aging at 823 K using atomic probe field-ion microscopy (AP-FIM); the results showed that the particles grow in proportion to t1/3 at later aging stages. Transmission electron microscopy (TEM) study of the coarsening behavior of the g0 phase in elastically constrained NieAleTi alloys indicated that the average particle size increases in proportion to t1/3 at first, after which the coarsening remarkably decelerates [18]. Vaithyanathan and Chen [19] investigated the coarsening of ordered intermetallic phase with coherency stress using a diffuse-interface phase field model; they demonstrated that the particle size distribution (PSD) of the intermetallic phase becomes broader as the volume fraction increases.
126
X. Wu et al. / Materials Chemistry and Physics 182 (2016) 125e132
In addition to the growth and coarsening, the nucleation of the
g0 phase, as a critical and extreme short-time phenomenon, has attracted much attention as well. Zhang [20,21] combined a diffusion-interface model with the minimax technique to predict the morphology of critical nuclei during solid-to-solid phase transformation in both two and three dimension; the results indicated that strong elastic interactions may lead to critical nuclei with a wide variety of shapes, including plates, needles, and cuboids with non-convex interfaces. The maximum composition within a critical nucleus can be either higher or lower than that of the equilibrium phase, while the morphology of an equilibrium phase may exhibit lower symmetry than the critical nuclei resulting from elastic interactions. Miyazaki [22,23] studied the critical nuclei sizes of the g0 phase in a NieAl alloy with Al content ranging from 11.7 to 12.6 (at.%) using the macroscopic composition gradient method; the critical nucleus size shows a steep increase with decreasing composition in the vicinity of the phase boundary, whereas little composition dependence is revealed for the case when the composition is far from the phase boundary. However, the previous studies focused on either growth and coarsening kinetics or the morphology of the g0 phase at the nucleation stage. It is theoretically meaningful to study the precipitation microstructure and kinetics evolution of the ordered g0 phase from nucleation to coarsening. As a process of atomic ordering and composition clustering, the initial nucleation and growth of the g0 phase may present some novel properties. Besides, the coarsening time exponent are mostly by fitting the LifshitzSlyozov-Wagner (LSW) theory, the actual time exponent is expected for the coarsening kinetics. Therefore, the present work studies the precipitation kinetics of the ordered g0 phase from nucleation to coarsening with the verified thermodynamics parameters; the initial nucleation is detected by the variation of composition and order parameter profiles of the g0 phase, the kinetics laws of growth and coarsening, particle size distribution (PSD) and average aspect ratio of the g0 phase are also studied.
2. Model and method The free energy Ftotal of a NieAl alloy can be expressed as the sum of the chemical free energy, the interfacial energy, and the elastic strain energy, which is expressed by the extended CahneHilliard free energy function [24e27]:
Z " Ftotal ¼ V
# 3 X kc kf 2 2 ðVfi Þ þ fel dV; f þ ðVcÞ þ 2 2
vfi ðr; tÞ dF ¼ Mf total þ xi ðr; tÞ dfi ðr; tÞ vt
ði ¼ 1; 2; 3Þ;
(3)
where Mc and Mf are the chemical mobility and the interface mobility, respectively, and xc ðr; tÞ and xi ðr; tÞ are the random Langevin noise items [29], simulating fluctuations of composition and LRO parameters at the initial stage, respectively. The amplitude of noise item is chosen to be small enough to trigger the nucleation, and it will be removed when the system can develop automatically. According to the sub-lattice model and the CALPHAD parameters [30,31], the Gibbs energy of the NieAl alloy can be expressed as follows [24e26]:
h 2 Ni G ¼ cGAl 0 þ ð1 cÞG0 þ cð1 cÞ L0 þ L1 ð2c 1Þ þ L2 ð2c 1Þ 3 3 i X X þ L3 ð2c 1Þ3 þ 4U1 c2 f2i þ 12U4 ð1 2cÞc2 f2i i¼1
i¼1
48U4 c3 f1 f2 f3 þ ðRT=4Þ 9 8 ½cð1 þ f1 þ f2 þ f3 Þln½cð1 þ f1 þ f2 þ f3 Þþ > > > > > > > ½1 cð1 þ f1 þ f2 þ f3 Þln½1 cð1 þ f1 þ f2 þ f3 Þþ > > > > > > > > > f þ f Þln½cð1 f f þ f Þþ ½cð1 f > > 1 2 3 1 2 3 > > = < ½1 cð1 f1 f2 þ f3 Þln½1 cð1 f1 f2 þ f3 Þþ ; > > > > ½cð1 f1 þ f2 f3 Þln½cð1 f1 þ f2 f3 Þþ > > > > > > þ f f Þln½1 cð1 f þ f f Þþ ½1 cð1 f 1 2 3 1 2 3 > > > > > > > > ½cð1 þ f1 f2 f3 Þln½cð1 þ f1 f2 f3 Þþ > > ; : ½1 cð1 þ f1 f2 f3 Þln½1 cð1 þ f1 f2 f3 Þ (4) Ni where GAl 0 and G0 are the Gibbs energy of pure Al and Ni, respectively, L0 , L1 , L2 , and L3 are the interaction parameters of excess energy, U1 and U4 are the parameters of bond energy, R is the gas constant, and T is the absolute temperature. The chemical free energy density is given byf ¼ G=Vm, where Vm is the mole volume. According to equation (4), the chemical free energy of the g phase can be obtained by setting all three order parameters fi (i ¼ 1, 2, 3) to zero, while the chemical free energy of g0 phase is calculated by minimizing the free energy. The chemical free energy curves of the g phase and g0 phase as a function of Al content XAl at 1073 K are plotted in Fig. 1. The equilibrium compositions of the g g g0 and g0 phases at the stress-free state are XAl ¼0.141 and XAl ¼ 0.222 (atom fraction), respectively. The typical free energy DG0 for the g to g0 phase transformation is 173 Jmol1 at 1073 K, where the two phases have the same free energy, as denoted by the arrow in Fig. 1.
(1)
i¼1
where V is the volume of the system, fi is the long-range order (LRO) parameter field, c is the composition field of Al, f is the volume free energy density, which defines the basic thermodynamic properties of the system, and fel is the elastic energy density. kc and kf are the gradient energy coefficients of the composition and order parameters, respectively, which can be determined from the total interface energy. The temporal evolution of the composition field can be described by the non-linear CahneHilliard diffusion equation, while the spatial-temporal evolution of the LRO parameters can be obtained by solving the time-dependent GinzburgeLandau equations [28]:
vcðr; tÞ dF ¼ V$ Mc V total þ xc ðr; tÞ; vt dcðr; tÞ
(2) Fig. 1. Chemical free energy of the g and g0 phases in a NieAl alloy at 1073 K.
X. Wu et al. / Materials Chemistry and Physics 182 (2016) 125e132
The elastic energy density can be calculated by [32,33]:
1 el fel ¼ Cijkl εel ij εkl ; 2
U1 ¼ 13415:515 þ 2:0819247 T; (5)
0 where Cijkl is the elastic constant tensor, εel ij ¼ εij εij is the elastic strain induced by the lattice mismatch and composition inhomogeneity, where εij is the total strain and its detailed solution can be found in Refs. [32,33], ε0ij is the eigenstrain and given by ε0ij ¼ ε0 dij dc , ε0 ¼ ð1=aÞðda=dcÞ is the composition expansion coefficient of the lattice parameter, dij is the Kronecker delta function, dc ¼ c c0 , with c0 being the nominal composition of the alloy. The parameter ε0 can be calculated from the g/g0 phase lattice misfit. The numerical calculation is performed by substituting equation (1) into equations (2) and (3), then transforming into the dimensionless form:
!# " * * 2 dfel* vc r* ; t * * * df * k ¼ V $ c cð1 cÞV c þ V c dc dc vt * * * * þ xc r ; t ; * 2 vfi r* ; t * df * ¼ k*f V* fi þ xi r* ; t * * dfi vt
ði ¼ 1; 2; 3Þ
127
(6)
(7)
where t * ¼ tMf DG0 , r* ¼ r=l, l represents the length scale of the system, V* ¼ v=vðr=lÞ, k*c ¼ kc Vm =ðDG0 l2 Þ, k*f ¼ kf Vm =ðDG0 l2 Þ, f * ¼ fVm =DG0 , fel* ¼ fel Vm =DG0 ,c ¼ Mc =Mf l2 is a dimensionless * * diffusivity, xc ðr ; t * Þ and xi ðr ; t * Þ are the dimensionless forms of xðr; tÞ and xi ðr; tÞ, respectively. The semi-implicit Fourier-spectral method is used for solving the kinetics equations (6) and (7) [34,35]. The computations were carried out in a 512 512 grid cell, l ¼ 1.5 nm, c ¼ 4 104 , and Vm ¼ 1.48 105 m3mol1 The dimensionless grid length and the time step adopted are Dx* ¼ Dy* ¼ 0:3 and Dt * ¼ 5 103 , respectively. The noise values for the composition and order parameters are [0.06, 0.06] and [0.05, 0.05], respectively. The gradient coefficient of the composition and order parameters are kc ¼ 2:5 109 Jm1 and kf ¼ 6:0 1012 Jm1 , corresponding to the interface energy of 0.05 Jmol1 [26]. The lattice misfit of the g and g0 phases is 0.0047 [36], and the elastic constants of the g and g0 g g g phases at 1073 K [36] are C11 ¼ 205:6 GPa, C12 ¼ 148:2 GPa, C44 ¼ g’ g’ 93:7 GPa and C11 ¼ 205:8 GPa, C12 ¼ 149:5 GPa, and g’ C44 ¼ 99:3 GPa, respectively. The thermodynamic parameters in equation (4) are as follows [30,31]:
U4 ¼ 7088:736 3:6868954 T:
3. Results and discussions 3.1. Phase boundary of g and g0 phases Using the sub-lattice model and the thermodynamic CALPHAD parameters, we calculated the phase boundary of the disordered g (fcc-Ni) phase and the ordered g0 (L12eNi3Al) phase in NieAl alloy, as shown in Fig. 2, which is a reproduction of the results in Ref. [31]. The previous experimental data [7e10], and calculated data [11e16,25] for temperatures ranging from 800 to 1300 K are plotted for comparison, where the data from Refs. [7e11,25] are for the coherent equilibrium, and the data from Refs. [12e16] are for the incoherent equilibrium. However, the incoherent phase boundary of the g and g0 phases calculated by the sub-lattice model and CALPHAD parameters agrees well with both the experimental and the theoretical calculation results. Therefore, the thermodynamic free energy and parameters can support a qualified phase field simulation for the precipitation in NieAl alloys. It is worth noting that the previous calculations for the phase diagram data were given by different methods: the cluster variation method (CVM) used by Sanchez [11] and Wang [25], the CALPHAD method used by Du [12] and Dupin [13], and Mishin’s Monte Carlo (MC) simulation with a slightly underestimated interface free energy [16]. In addition, Li and Ardell [37] studied the coherent solubility limitation of the g0 phase in NieAl alloy with different initial contents of Al; their results showed that the solute concentration of the g phase in the thermodynamic equilibrium increases with increasing initial composition. They also studied the incoherent g/g0 solvus in NieAl alloys [38,39]; however, the differences between the incoherent and coherent solubility limits as a function of temperature are very small and of essentially no consequence. Also, the incoherent equilibrium between the g and g0 phases is rarely observed for normal heat-treatment, which leads to the fact that nearly all of the previous investigations of the g/g0 equilibrium in NieAl alloys have been measured with coherent solubility limits [38].
GAl 0 ¼ 11278:378 þ 188:684153 T 31:748192 T lnðTÞ 1:23 1028 T 9 ; GNi 0 ¼ 5179:159 þ 117:854 T 22:096 T lnðTÞ 4:84 103 T 2 ; L0 ¼ 162407:75 þ 16:212965 T; L1 ¼ 73417:798 34:914 T; L2 ¼ 33471:014 9:837 T; L3 ¼ 30758:01 þ 10:253 T;
Fig. 2. Phase boundary of the g and g0 phases calculated by the sub-lattice model for temperatures ranging from 800 to 1300 K; the previous experimental data (green) and calculated data (purple) are plotted for comparison. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
128
X. Wu et al. / Materials Chemistry and Physics 182 (2016) 125e132
3.2. Initial cluster and microstructure evolution of the g0 phase As can be seen from Fig. 2, the phase boundary of the g and g0 phases determines the equilibrium composition of the g and g g0 0 phases at 1073 K, which are approximately XAl ¼0.141 and g
XAl ¼ 0.222, respectively. We choose an intermediate composition XAl ¼ 0.172, namely Ni-17.2 at.% Al alloy, to study the precipitation kinetics. Because some low composition alloys have been studied in NieAl alloys [17,22,23], the intermediate composition also has a more accurate thermodynamics parameters than that is near the phase boundaries. The temporal microstructures of the g0 phase in a Ni-17.2 at.% Al alloy aged at 1073 K are presented in Fig. 3, where the white regions represent the g0 phase and the black regions represent the g phase. It can be seen that the g0 phase is nearly spherical at the nucleation and initial growth stages (see Fig. 3(a) and (b)), after which it transforms to a cuboidal shape at the growth and early coarsening stages (see Fig. 3(c) and (d)). At the quasi-stationary state coarsening stage (see Fig. 3(e) and (f)), some g0 phases change to rectangular shapes along the elastic soft directions [10] and [01] by a connection coarsening mechanism [28]. The shape transformation of the g0 phase during the precipitation depends on the volume free energy and surface energy when the particle is small at the early stage of precipitation; as the particles coarsen, the elastic strain energy dominates the shape transformation [25]. The coarsening of the g0 phase usually proceeds by means of Ostwald ripening and coalescence of neighboring particles [28], which are designated as B and B* (Ostwald ripening) and C and C* (coalescence) in Fig. 3(e) and (f). The initial precipitation process of the g0 phase can be seen from Fig. 3(a) and (b), where some Al-enriched cluster evolve to large g0 phase particles as the aging progresses and some metastable Alenriched clusters are dissolved, as seen inside the rectangles designated by A and A*. As is known, the transformation of a metastable nucleus to a stable one is a thermally active process, so the nucleation of a new phase needs to overcome an energy barrier
[40,41]. The evolution of the Al-enriched cluster (or g0 nuclei) inside the rectangle A in Fig. 3(a) are presented in Fig. 4. The cluster designated D grows to a nearly spherical g0 phase particle at t* ¼ 25, while the cluster designated by letter E shrinks and is dissolved gradually. Fig. 5 displays the evolution of Al content XAl and order parameter profiles of the Al-enriched cluster shown in Fig. 4. The Al content of the cluster designated D rises gradually and reaches the maximum of 0.208 at approximately t* ¼ 21, as shown in Fig. 5(a), where the maximum composition shows little deviation from the thermodynamic phase diagram (~0.222) at 1073 K, as shown in Fig. 2. The composition deviation in the initial nucleus is declared in previous studied [20,21]. However, the order parameter of particle D nearly reaches the maximum of 0.98 at approximately t* ¼ 3. The variations of the composition and order parameter indicate that the atomic ordering is faster than the composition clustering at the initial precipitation of the g0 phase, which also be showed in the previous studied of Ni75Al7.5V17.5 (at.%) alloy [42]. If the initial emerged clusters cannot evolve to a stable nucleus, such as the particle designated E, its composition and order parameter fall to the initial state again as the aging progresses, as shown in Fig. 5(c) and (d). Therefore, the magnitude of the initial thermal noise is suitable for the automatic development of the system by the nucleation and growth of some Al-enriched clusters and dissolving again of the metastable clusters. The initial nucleation of g0 can be detected by the composition and order parameters of phase field simulations. Comparing Fig. 5(a) with Fig. 5(b), it can be seen that the order parameter of particle D nearly stays at the stable value from t* ¼ 3 to t* ¼ 25, while the Al content gradually increases from t* ¼ 3 to t* ¼ 21. At this stage, the g0 phase is a non-stoichiometric ordered phase. Note that the Al content at t* ¼ 21 is nearly equal to the Al content at t* ¼ 25; the g0 phase becomes a stoichiometric ordered phase from then on. It can be concluded that the evolution of the g0 phase at the nucleation stage is the precipitation from supersaturated solid solution to non-stoichiometric ordered phase and
Fig. 3. Microstructure evolution of the simulated g0 phase in Ni-17.2 at.% Al alloy aged at 1073 K, (a) t* ¼ 10, (b) t* ¼ 30, (c) t* ¼ 80, (d) t* ¼ 160, (e) t* ¼ 300, and (f) t* ¼ 1000.
X. Wu et al. / Materials Chemistry and Physics 182 (2016) 125e132
129
Fig. 4. Initial Al-enriched cluster growing up to a g0 phase or dissolving inside the rectangle A in Fig. 3, (a) t* ¼ 3, (b) t* ¼ 12, (c) t* ¼ 21, and (d) t* ¼ 25.
0.22
1.2
t*=3 t*=12 t*=21 t*=25
(a)
1.0
Order Parameter
0.21
XAl
0.20 0.19 0.18
t*=3 t*=12 t*=21 t*=25
(b) 0.8 0.6 0.4 0.2
0.17
0.0
0.16 325
330
335
340
345
350
325
330
335
Grid
340
345
350
Grid 1.0
(c)
t*=3 t*=12 t*=21 t*=25
0.8
XAl
Order Parameter
0.19
0.18
t*=3 t*=12 t*=21 t*=25
(d)
0.6 0.4 0.2
0.17
0.0 315
320
325
330
Grid
315
320
325
330
Grid
Fig. 5. Evolution of Al content XAl (a) and (c), and order parameter of Al-enriched clusters (b) and (d), (a) and (b): an Al-enriched cluster grows into a stable g0 particle, (c) and (d): the dissolution of an Al-enriched cluster.
stoichiometric ordered phase. Additionally, the gradual composition increase is different from the classical nucleation theory, where the emerged nucleus reaches an equilibrium composition [43]; therefore, the g0 phase precipitation is by means of a nonclassical nucleation and growth mechanism. The evolution of the Al content XAl and order parameter of the g0 phase from nucleation to coarsening in Ni-17.2 at.% Al alloy aged at 1073 K are shown in Fig. 6. It can be seen from Fig. 6(a) that there is a composition difference between the center and the edge of the g0 phase from t* ¼ 80, namely inhomogeneous composition. The reason for the inhomogeneity is the misfit strain, and the inhomogeneity becomes stronger as the misfit strain increases, which has been discussed in Ref. [44]. However, the order
parameter of the g0 phase stays at the stable value after t* ¼ 20. As the growth and coarsening progresses, the Al content and order parameter become wider than before. The non-classical nucleation followed by growth and coarsening in present simulation is different to that proposed by Staron et al., their study on the Ni13 at.% Al aging at 450 C showed a coarsening reaction from an initial small cluster of very beginning of decomposition, which is different from a classical nucleation reaction also [45]. 3.3. Size and shape evolutions of the g0 phase Fig. 7 illustrates the temporal average particle radius rave of the
g0 phase in Ni-17.2 at.% Al alloy aged at 1073 K for t* ¼ 1000, in
130
X. Wu et al. / Materials Chemistry and Physics 182 (2016) 125e132
0.22 0.21
1.2
(a)
t*=20 t*=160
t*=80 t*=1000
0.19
XAl
1.0
Order parameter
0.20
0.18 0.17 0.16
t*=20 t*=160
(b)
t*=80 t*=1000
0.8 0.6 0.4 0.2
0.15
0.0 310
320
330
340
350
360
310
320
Grid
330
340
350
360
Grid
Fig. 6. Evolution of Al content XAl (a) and order parameter (b) of the g0 phase from nucleation to coarsening in Ni-17.2 at.% Al alloy aged at 1073 K.
may cause a statistical error when counting the number and calculating the size of the g0 phase. To minimize the error, the simulation cell is chosen to be large enough to ensure a sufficient number of g0 phases for calculation. Fig. 8 shows the temporal evolution of the normalized particle size distribution (PSD) of r/rave for the g0 phase in a Ni-17.2 at.% Al alloy aged at 1073 K, r is the radius of g0 particle at a aging time and rave is the average of g0 particle radius. For the PSDs presented here,
3.0 (a)
(b)
(c)
(d)
2.5
PSD
which logarithmic coordinates are adopted. The average particle radius has a rapid increase at the initial stage, t* < 60, then it slows down in the following two stages, as denoted by the dashed and solid lines in Fig. 7, respectively. The three stages of rave correspond to nucleation and initial growth, growth and coarsening, and quasistationary state coarsening, respectively. The variations of rave in the three stages have also been presented in both experimental results [17] and simulation results [46] for a Ni-14 at.% Al alloy aged at 823 K. We fitted the data of rave for the three obvious stages with a linear function, and the time exponents of rave∝(t*)n are n ¼ 1.35, 0.42, and 0.22, respectively. It should be noted that the time exponent 0.22 for the coarsening is less than the 0.33 predicted by the LSW theory [47,48], while it is close to the 0.20 reported in Wen’s result for Ni-14 at.% Al alloy aged at 823 K for 1400 min [46]. Mushongera’s recent results in multi-component Ni-based alloys also showed a time exponent ranging from 2.2 to 2.35 with elastic effects [49]. The time exponent of the g0 phase in elastically constrained NieAleTi alloys shows a remarkable decrease from 1/3 after entering the coarsening stage [18]. Therefore, the elastic strain induced by the lattice mismatch of the g and g0 phases results in a different coarsening kinetics to the classical LSW theory, in which a dilute solution without elastic interactions is assumed. It should be noted that the 2D approximation
2.0 1.5 1.0 0.5
2.5
PSD
2.0 1.5 1.0 0.5 0.0 Fig. 7. Evolution of average particle radius of the g0 phase in Ni-17.2 at.% Al alloy aged at 1073 K; the nucleation and initial growth, growth and coarsening, and quasistationary state coarsening stages are fitted with dash-dot, dashed, and solid lines, respectively.
0.5
1.0
1.5
0.5
1.0
1.5
r/rave Fig. 8. Particles size distribution (PSD) evolution of the g0 phase in Ni-17.2 at.% Al alloy aged at 1073 K, (a) t* ¼ 20, (b) t* ¼ 80, (c) t* ¼ 160, and (c) t* ¼ 1000.
(a)
1.6
Average Aspect Ratio
Average Aspect Ratio
X. Wu et al. / Materials Chemistry and Physics 182 (2016) 125e132
1.4 1.2 1.0 0
2
4
6
8
10
1.6
131
(b)
1.4
1.2
1.0 0
200
400
600
800
1000
t*
rave (nm)
Fig. 9. Average aspect ratio as a function of the average particle radius of the g0 phase (a), and the aging time (b) in Ni-17.2 at.% Al alloy aged at 1073 K.
the interval size is 0.1 r/rave, the distribution was normalized by dividing the total particle number nt with the particle number ni in an individual interval, and with the scaled size of the interval width. It can be seen that the peak of the PSD decreases and the width of the PSD increases as the aging progresses, and the positions of the peak values of the PSD shifts from r/rave ¼ 0.9 to 1.0 as the aging progresses. The decrease in the peak and increase in the PSD width occur because of stable-state coarsening, the peaks of the PSD lies at the 1.0 at later stage indicates the kinetics scaling behavior of coarsening. To describe the spatial characteristics of the g0 phase morphology, the shape of the g0 phase has been analyzed with various shape parameters [50e53]. In this work, we use the average aspect ratio, P, to characterize the shape evolution of the g0 phase, which is defined as the ratio of the long axis to the short axis of the g0 phase. The variations of P with the average particle radius rave and the aging time t* are plotted in Fig. 9 (a) and (b), respectively. These results are similar to the variation of the average radius, and the average aspect ratio also shows three stages with increasing aging time t* and average particle radius rave. The average aspect ratio stays at approximately 1.03 when the average particle radius rave is less than 3.5 nm; this stage corresponds to the nucleation and initial growth. The average aspect ratio increases rapidly at the growth and early coarsening stage, then it slows down when entering the quasi-stationary state coarsening. The average aspect ratio remains at approximately 1.59 after t* ¼ 600. 4. Conclusions The kinetics of nucleation, growth, and coarsening of the g0 phase in a NieAl alloy were studied using the phase field model with the thermodynamic free energy and parameters. The nucleation process was detected by the initial variations of composition and order parameters of the newly emerged g0 phase; the atomic ordering in the nucleus of the g0 phase is faster than the composition clustering for the Ni-17.2 at.% Al alloy aged at 1073 K. The precipitation of the g0 phase from the supersaturated solid solution is in the following order: non-stoichiometric ordered phase/ stoichiometric ordered phase/growth and coarsening. The average particle radius of the g0 phase and the aging time show an exponential relationship, rave∝(t*)n, at the nucleation and initial growth, growth and coarsening, and later coarsening stages, with exponents n ¼ 1.35, 0.42, and 0.22, respectively. The variation of the average aspect ratio also shows obvious three stages; it remains at approximately 1.03 when the average particle radius rave is less than 3.5 nm, then it increases rapidly at the growth and coarsening stage of the g0 phase, and maintains a stable value again when the g0 phase goes into the later coarsening stage. The peak of
the PSD decreases and the width of the PSD increases as the aging progresses, and the positions of the peak values of the PSDs are located at r/rave ¼ 0.9 to 1.0. The phase field simulations with the thermodynamic parameters of the NieAl alloy give an effective kinetic description for the precipitation of the ordered g0 phase from nucleation to coarsening. Acknowledgments The authors acknowledge the financial support from the National Natural Science Foundation of China (No. 51571122), the Fundamental Research Funds for the Central Universities (No. 30920130121012), and the Graduate Innovation Project of Jiangsu Province (No. SJLX_0157). Nomenclature
Symbol and meaning PSD particle size distribution total free energy of the NieAl alloy Ftotal c composition field of Al f volume free energy density kc gradient energy coefficient of composition LRO long-range order t time Mf interface mobility xi ðr; tÞ random Langevin noise item of order parameter GNi Gibbs free energy of pure Ni 0 Li interaction parameter of excess energy T absolute temperature Vm mole volume g0 XAl equilibrium composition of g0 phase Cijkl elastic constant tensor εij total strain dij Kronecker delta function r* dimensionless position l length scale of the system k*c dimensionless gradient energy coefficient of composition f* dimensionless volume free energy density x*c ðr ; t * Þ dimensionless random Langevin noise item of composition Dx* dimensionless grid length of x axis Dt * dimensionless time step rave average particle radius r/ rave normalized particle size LSW LifshitzeSlyozoveWagner V the volume of the system
132
X. Wu et al. / Materials Chemistry and Physics 182 (2016) 125e132
long-range order parameter field elastic energy density gradient energy coefficient of order parameter positon chemical mobility random Langevin noise item of composition Gibbs energy of pure Al parameter of bond energy gas constant Gibbs energy of the NieAl alloy equilibrium composition of g phase typical free energy when g and g0 phases have same free energy εel elastic strain ij ε0ij eigenstrain c0 nominal composition of Al t* dimensionless time c dimensionless diffusivity k*f dimensionless gradient energy coefficient of order parameter fel* dimensionless elastic energy density x*i ðr ; t * Þ dimensionless random Langevin noise item of order parameter Dy* dimensionless grid length of y axis XAl Al content n time exponent of rave∝(t*)n P average aspect ratio
fi fel kf r Mc xc ðr; tÞ GAl 0 Ui R G g XAl DG0
References [1] M. Durand-Charre, The Microstructure of Superalloys, Gordon and Breach Science Publishers, Amsterdam, 1997. [2] R.C. Reed, The Superalloys: Fundamentals and Applications, Cambridge University Press, Cambridge, 2006. [3] H.X. Li, M. Qiao, C.G. Zhou, Mat. Chem. Phys. 143 (2014) 915e920. [4] S. Keshavarz, S. Ghosh, Acta Mater 61 (2013) 6549e6561. [5] F. Sun, S.C. Mao, J.X. Zhang, Mat. Chem. Phys. 147 (2014) 483e487. [6] T. Tanimoto, Md. Moniruzzaman, Y. Murata, N. Miura, Y. Kondo, Y. Tsukada, T. Koyama, Comp. Mat. Sci. 93 (2014) 56e61. [7] P.K. Rastogi, A.J. Ardell, Acta Mater 17 (1969) 595e602. [8] M.M.P. Jansshen, Metall. Trans. 4 (1973) 1623e1633. [9] H. Okamoto, J. Phase Equili. Diff. 14 (1993) 257e259. [10] M. Watanabe, Z. Horita, D.J. Smith, M.R. Mccartney, T. Sano, M. Nemoto, Acta Mater 40 (1994) 3381e3387. [11] J.M. Sanchez, J.R. Barefoot, R.N. Jarrett, J.K. Tien, Acta Mater 32 (1984)
[12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43]
[44] [45] [46] [47] [48] [49] [50] [51] [52] [53]
1519e1525. Y. Du, N. Clavaguera, J. Alloys Comp. 237 (1996) 20e32. N. Dupin, I. Ansara, B. Sundman, Calphad 25 (2001) 279e298. Y. Ma, A.J. Ardell, Z. Metallk 94 (2003) 972e975. H. Okamoto, J. Phase Equilib. Diff. 25 (2004) 257e259. Y. Mishin, Model. Simul. Mat. Sci. Eng. 22 (2014) 045001. H. Wendt, P. Hassen, Acta Mater 31 (1993) 1649e1659. T. Maebashi, M. Doi, Mat. Sci. Eng. A 373 (2004) 72e79. V. Vaithyanathan, L.Q. Chen, Acta Mater 50 (2002) 4061e4073. L. Zhang, L.Q. Chen, Q. Du, Acta Mater 56 (2008) 3568e3576. L. Zhang, L.Q. Chen, Q. Du, Commun. Comput. Phys. 7 (2010) 674e682. T. Miyazaki, S. Kobayashi, T. Koyama, Metall. Mat. Trans. A 30A (1999) 2783e2789. T. Miyazaki, T. Kozakai, G.C. Schoen, Mat. Sci. Forum 502 (2005) 139e144. J.Z. Zhu, Z.K. Liu, V. Vaithyanathan, L.Q. Chen, Scr. Mater 46 (2002) 401e406. J.C. Wang, M. Osawa, T. Yokokawa, H. Harada, M. Enomoto, Comp. Mat. Sci. 39 (2007) 871e879. J.W. Cahn, J.E. Hilliard, J. Chem. Phys. 28 (1958) 258e267. Y.S. Li, Y.X. Pang, X.C. Wu, W. Liu, Model. Simul. Mat. Sci. Eng. 22 (2014) 035009. Y. Wang, D. Banerjee, C.C. Su, A.G. Khachaturyan, Acta Mater 46 (1998) 2983e3001. L.D. Landau, E.M. Lifshitz, Course of Theoretical Physics, Statistical Physics, third ed., Pergamon, Oxford, 1980 part 1. A.T. Dinsdale, Calphad 15 (1991) 317e425. I. Ansara, N. Dupin, H.L. Lukas, B. Sundman, J. Alloys Comp. 247 (1997) 20e30. M. Doi, Prog. Mat. Sci. 40 (1996) 79e180. P. Fratzl, O. Penrose, J.L. Lebowitz, J. Stat. Phys. 95 (1999) 1429e1503. L.Q. Chen, J. Shen, Comp. Phys. Commun. 108 (1998) 147e158. J.Z. Zhu, L.Q. Chen, Phys.. Rev. E 60 (1999) 3564e3572. A.J. Ardell, Phi. Mag. 94 (2014) 2101e2130. F. Li, A.J. Ardell, Scr. Mater 137 (1997) 1123e1128. A.J. Ardell, Model. Simul. Mat. Sci. Eng. 8 (2000) 277e286. F. Li, A.J. Ardell, J. Phase Equilib. Diff. 19 (1998) 334e339. T.W. Heo, L.Q. Chen, JOM 66 (2014) 1520e1528. T.W. Heo, L. Zhang, Q. Du, L.Q. Chen, Scr. Mater 63 (2010) 8e11. Y.S. Li, Z. Cheng, Y.L. Lu, Y.X. Wang, Prog. Nat. Sci. 14 (2004) 1099e1103. R. Wagner, R. Kampmann, P.W. Voorhees, Homogeneous second-phase precipitation, in: G. Kostorz (Ed.), Phase Transformations in Materials. Weinheim, Wiley-VCH, New York, Chichester, 2001, p. 309. Y. Wang, A.G. Khachaturyan, Acta Metall. Mater 43 (1995) 1837e1857. P. Staron, M. Haese-Seiller, R. Kampmann, R. Wagner, Phys. B 234e236 (1997) 989e991. Y.H. Wen, B. Wang, J.P. Simmons, Y. Wang, Acta Mater 54 (2006) 2087e2099. I.M. Lifshitz, V.V. Slyozov, J. Phys. Chem. Solids 19 (1961) 35e50. C. Wagner, Z. Elektrochem. 65 (1961) 581e591. L.T. Mushongera, M. Fleck, J. Kundin, F. Querfurth, H. Emmerich, Adv. Eng. Mater 17 (2015) 1149e1157. M. F€ ahrmann, P. Fratzl, O. Paris, E. F€ ahrmann, W.C. Johnson, Acta Mater 43 (1995) 1007e1022. S.V. Prikhodko, A.J. Ardell, Acta Mater 51 (2003) 5021e5036. Y. Ma, A.J. Ardell, Acta Mater 55 (2007) 4419e4427. J.P. MacSleyne, J.P. Simmons, M.D. Graef, Acta Mater 56 (2008) 427e437.
Contents lists available at ScienceDirect
Materials Chemistry and Physics journal homepage: www.elsevier.com/locate/matchemphys
Precipitation kinetics of ordered g0 phase and microstructure evolution in a NieAl alloy Xingchao Wu, Yongsheng Li*, Mengqiong Huang, Wei Liu, Zhiyuan Hou School of Materials Science and Engineering, Nanjing University of Science and Technology, Nanjing 210094, China
h i g h l i g h t s Kinetics of g0 phase from nucleation to coarsening in NieAl alloy is studied. Non-stoichiometric ordered phase and initial cluster are quantitatively described. Time exponent of particle radius decreases from nucleation and growth to coarsening. Average aspect ratio of g0 phase shows three stages with respect to average radius.
a r t i c l e i n f o
a b s t r a c t
Article history: Received 17 June 2015 Received in revised form 22 June 2016 Accepted 9 July 2016 Available online 18 July 2016
The precipitation kinetics of ordered g0 (L12-Ni3Al) phase from nucleation and growth to coarsening in a NieAl alloy is studied by phase field simulation. The initial nucleation of the g0 phase is quantitatively described by the variations of composition and order parameter profiles of the initial Al-enriched cluster, and the coarsening kinetics is clarified with the time exponent of the average particle radius of g0 phase. The precipitation of the g0 phase from the supersaturated g solid solution goes through the nonstoichiometric ordered phase, stoichiometric ordered phase, growth and coarsening. The time exponent of the average radius of the g0 phase with coherent equilibrium is smaller than that indicated by the LifshitzeSlyozoveWagner theory at the coarsening stage, and it decreases from 1.35 of the nucleation and growth stage to 0.22 of the coarsening stage. The peak of the particle size distribution decreases and the width of the particle size distribution increases as the aging progresses. As the average radius of the g0 phase increases, the average aspect ratio of the g0 phase undergoes a slowly increase to a fast increase from nucleation to growth stage, then goes into a relatively stable state at the coarsening stage. © 2016 Elsevier B.V. All rights reserved.
Keywords: Alloys Annealing Computer modelling and simulation Phase transitions
1. Introduction Nickel-based superalloys are the preferred structural materials in commercial and military jet engines and land-based gas turbines because of their excellent high-temperature properties [1e6]. In nickel-based superalloys, the coherent g0 (L12-Ni3X) phase embedded in the disordered g (fcc-Ni) matrix, the coherent elastic strain energy, and the interactions between the g0 phase and dislocations lead to a strengthening of the alloys. Therefore, the kinetics evolution of phases’ size and the spatial distribution of the ordered g0 phase during thermal aging and high-temperature service will affect the properties of nickel-based superalloys. To study the phase transformation of the g0 phase in nickel-
* Corresponding author. E-mail address: [email protected] (Y. Li). http://dx.doi.org/10.1016/j.matchemphys.2016.07.013 0254-0584/© 2016 Elsevier B.V. All rights reserved.
based alloys, thermodynamic calculation of the phase diagram [7e16] and dynamic description of the growth and coarsening by simulation and experiments [17e19] have been conducted in past decades. Wendt [17] studied the decomposition of Ni-14 at.% Al during isothermal aging at 823 K using atomic probe field-ion microscopy (AP-FIM); the results showed that the particles grow in proportion to t1/3 at later aging stages. Transmission electron microscopy (TEM) study of the coarsening behavior of the g0 phase in elastically constrained NieAleTi alloys indicated that the average particle size increases in proportion to t1/3 at first, after which the coarsening remarkably decelerates [18]. Vaithyanathan and Chen [19] investigated the coarsening of ordered intermetallic phase with coherency stress using a diffuse-interface phase field model; they demonstrated that the particle size distribution (PSD) of the intermetallic phase becomes broader as the volume fraction increases.
126
X. Wu et al. / Materials Chemistry and Physics 182 (2016) 125e132
In addition to the growth and coarsening, the nucleation of the
g0 phase, as a critical and extreme short-time phenomenon, has attracted much attention as well. Zhang [20,21] combined a diffusion-interface model with the minimax technique to predict the morphology of critical nuclei during solid-to-solid phase transformation in both two and three dimension; the results indicated that strong elastic interactions may lead to critical nuclei with a wide variety of shapes, including plates, needles, and cuboids with non-convex interfaces. The maximum composition within a critical nucleus can be either higher or lower than that of the equilibrium phase, while the morphology of an equilibrium phase may exhibit lower symmetry than the critical nuclei resulting from elastic interactions. Miyazaki [22,23] studied the critical nuclei sizes of the g0 phase in a NieAl alloy with Al content ranging from 11.7 to 12.6 (at.%) using the macroscopic composition gradient method; the critical nucleus size shows a steep increase with decreasing composition in the vicinity of the phase boundary, whereas little composition dependence is revealed for the case when the composition is far from the phase boundary. However, the previous studies focused on either growth and coarsening kinetics or the morphology of the g0 phase at the nucleation stage. It is theoretically meaningful to study the precipitation microstructure and kinetics evolution of the ordered g0 phase from nucleation to coarsening. As a process of atomic ordering and composition clustering, the initial nucleation and growth of the g0 phase may present some novel properties. Besides, the coarsening time exponent are mostly by fitting the LifshitzSlyozov-Wagner (LSW) theory, the actual time exponent is expected for the coarsening kinetics. Therefore, the present work studies the precipitation kinetics of the ordered g0 phase from nucleation to coarsening with the verified thermodynamics parameters; the initial nucleation is detected by the variation of composition and order parameter profiles of the g0 phase, the kinetics laws of growth and coarsening, particle size distribution (PSD) and average aspect ratio of the g0 phase are also studied.
2. Model and method The free energy Ftotal of a NieAl alloy can be expressed as the sum of the chemical free energy, the interfacial energy, and the elastic strain energy, which is expressed by the extended CahneHilliard free energy function [24e27]:
Z " Ftotal ¼ V
# 3 X kc kf 2 2 ðVfi Þ þ fel dV; f þ ðVcÞ þ 2 2
vfi ðr; tÞ dF ¼ Mf total þ xi ðr; tÞ dfi ðr; tÞ vt
ði ¼ 1; 2; 3Þ;
(3)
where Mc and Mf are the chemical mobility and the interface mobility, respectively, and xc ðr; tÞ and xi ðr; tÞ are the random Langevin noise items [29], simulating fluctuations of composition and LRO parameters at the initial stage, respectively. The amplitude of noise item is chosen to be small enough to trigger the nucleation, and it will be removed when the system can develop automatically. According to the sub-lattice model and the CALPHAD parameters [30,31], the Gibbs energy of the NieAl alloy can be expressed as follows [24e26]:
h 2 Ni G ¼ cGAl 0 þ ð1 cÞG0 þ cð1 cÞ L0 þ L1 ð2c 1Þ þ L2 ð2c 1Þ 3 3 i X X þ L3 ð2c 1Þ3 þ 4U1 c2 f2i þ 12U4 ð1 2cÞc2 f2i i¼1
i¼1
48U4 c3 f1 f2 f3 þ ðRT=4Þ 9 8 ½cð1 þ f1 þ f2 þ f3 Þln½cð1 þ f1 þ f2 þ f3 Þþ > > > > > > > ½1 cð1 þ f1 þ f2 þ f3 Þln½1 cð1 þ f1 þ f2 þ f3 Þþ > > > > > > > > > f þ f Þln½cð1 f f þ f Þþ ½cð1 f > > 1 2 3 1 2 3 > > = < ½1 cð1 f1 f2 þ f3 Þln½1 cð1 f1 f2 þ f3 Þþ ; > > > > ½cð1 f1 þ f2 f3 Þln½cð1 f1 þ f2 f3 Þþ > > > > > > þ f f Þln½1 cð1 f þ f f Þþ ½1 cð1 f 1 2 3 1 2 3 > > > > > > > > ½cð1 þ f1 f2 f3 Þln½cð1 þ f1 f2 f3 Þþ > > ; : ½1 cð1 þ f1 f2 f3 Þln½1 cð1 þ f1 f2 f3 Þ (4) Ni where GAl 0 and G0 are the Gibbs energy of pure Al and Ni, respectively, L0 , L1 , L2 , and L3 are the interaction parameters of excess energy, U1 and U4 are the parameters of bond energy, R is the gas constant, and T is the absolute temperature. The chemical free energy density is given byf ¼ G=Vm, where Vm is the mole volume. According to equation (4), the chemical free energy of the g phase can be obtained by setting all three order parameters fi (i ¼ 1, 2, 3) to zero, while the chemical free energy of g0 phase is calculated by minimizing the free energy. The chemical free energy curves of the g phase and g0 phase as a function of Al content XAl at 1073 K are plotted in Fig. 1. The equilibrium compositions of the g g g0 and g0 phases at the stress-free state are XAl ¼0.141 and XAl ¼ 0.222 (atom fraction), respectively. The typical free energy DG0 for the g to g0 phase transformation is 173 Jmol1 at 1073 K, where the two phases have the same free energy, as denoted by the arrow in Fig. 1.
(1)
i¼1
where V is the volume of the system, fi is the long-range order (LRO) parameter field, c is the composition field of Al, f is the volume free energy density, which defines the basic thermodynamic properties of the system, and fel is the elastic energy density. kc and kf are the gradient energy coefficients of the composition and order parameters, respectively, which can be determined from the total interface energy. The temporal evolution of the composition field can be described by the non-linear CahneHilliard diffusion equation, while the spatial-temporal evolution of the LRO parameters can be obtained by solving the time-dependent GinzburgeLandau equations [28]:
vcðr; tÞ dF ¼ V$ Mc V total þ xc ðr; tÞ; vt dcðr; tÞ
(2) Fig. 1. Chemical free energy of the g and g0 phases in a NieAl alloy at 1073 K.
X. Wu et al. / Materials Chemistry and Physics 182 (2016) 125e132
The elastic energy density can be calculated by [32,33]:
1 el fel ¼ Cijkl εel ij εkl ; 2
U1 ¼ 13415:515 þ 2:0819247 T; (5)
0 where Cijkl is the elastic constant tensor, εel ij ¼ εij εij is the elastic strain induced by the lattice mismatch and composition inhomogeneity, where εij is the total strain and its detailed solution can be found in Refs. [32,33], ε0ij is the eigenstrain and given by ε0ij ¼ ε0 dij dc , ε0 ¼ ð1=aÞðda=dcÞ is the composition expansion coefficient of the lattice parameter, dij is the Kronecker delta function, dc ¼ c c0 , with c0 being the nominal composition of the alloy. The parameter ε0 can be calculated from the g/g0 phase lattice misfit. The numerical calculation is performed by substituting equation (1) into equations (2) and (3), then transforming into the dimensionless form:
!# " * * 2 dfel* vc r* ; t * * * df * k ¼ V $ c cð1 cÞV c þ V c dc dc vt * * * * þ xc r ; t ; * 2 vfi r* ; t * df * ¼ k*f V* fi þ xi r* ; t * * dfi vt
ði ¼ 1; 2; 3Þ
127
(6)
(7)
where t * ¼ tMf DG0 , r* ¼ r=l, l represents the length scale of the system, V* ¼ v=vðr=lÞ, k*c ¼ kc Vm =ðDG0 l2 Þ, k*f ¼ kf Vm =ðDG0 l2 Þ, f * ¼ fVm =DG0 , fel* ¼ fel Vm =DG0 ,c ¼ Mc =Mf l2 is a dimensionless * * diffusivity, xc ðr ; t * Þ and xi ðr ; t * Þ are the dimensionless forms of xðr; tÞ and xi ðr; tÞ, respectively. The semi-implicit Fourier-spectral method is used for solving the kinetics equations (6) and (7) [34,35]. The computations were carried out in a 512 512 grid cell, l ¼ 1.5 nm, c ¼ 4 104 , and Vm ¼ 1.48 105 m3mol1 The dimensionless grid length and the time step adopted are Dx* ¼ Dy* ¼ 0:3 and Dt * ¼ 5 103 , respectively. The noise values for the composition and order parameters are [0.06, 0.06] and [0.05, 0.05], respectively. The gradient coefficient of the composition and order parameters are kc ¼ 2:5 109 Jm1 and kf ¼ 6:0 1012 Jm1 , corresponding to the interface energy of 0.05 Jmol1 [26]. The lattice misfit of the g and g0 phases is 0.0047 [36], and the elastic constants of the g and g0 g g g phases at 1073 K [36] are C11 ¼ 205:6 GPa, C12 ¼ 148:2 GPa, C44 ¼ g’ g’ 93:7 GPa and C11 ¼ 205:8 GPa, C12 ¼ 149:5 GPa, and g’ C44 ¼ 99:3 GPa, respectively. The thermodynamic parameters in equation (4) are as follows [30,31]:
U4 ¼ 7088:736 3:6868954 T:
3. Results and discussions 3.1. Phase boundary of g and g0 phases Using the sub-lattice model and the thermodynamic CALPHAD parameters, we calculated the phase boundary of the disordered g (fcc-Ni) phase and the ordered g0 (L12eNi3Al) phase in NieAl alloy, as shown in Fig. 2, which is a reproduction of the results in Ref. [31]. The previous experimental data [7e10], and calculated data [11e16,25] for temperatures ranging from 800 to 1300 K are plotted for comparison, where the data from Refs. [7e11,25] are for the coherent equilibrium, and the data from Refs. [12e16] are for the incoherent equilibrium. However, the incoherent phase boundary of the g and g0 phases calculated by the sub-lattice model and CALPHAD parameters agrees well with both the experimental and the theoretical calculation results. Therefore, the thermodynamic free energy and parameters can support a qualified phase field simulation for the precipitation in NieAl alloys. It is worth noting that the previous calculations for the phase diagram data were given by different methods: the cluster variation method (CVM) used by Sanchez [11] and Wang [25], the CALPHAD method used by Du [12] and Dupin [13], and Mishin’s Monte Carlo (MC) simulation with a slightly underestimated interface free energy [16]. In addition, Li and Ardell [37] studied the coherent solubility limitation of the g0 phase in NieAl alloy with different initial contents of Al; their results showed that the solute concentration of the g phase in the thermodynamic equilibrium increases with increasing initial composition. They also studied the incoherent g/g0 solvus in NieAl alloys [38,39]; however, the differences between the incoherent and coherent solubility limits as a function of temperature are very small and of essentially no consequence. Also, the incoherent equilibrium between the g and g0 phases is rarely observed for normal heat-treatment, which leads to the fact that nearly all of the previous investigations of the g/g0 equilibrium in NieAl alloys have been measured with coherent solubility limits [38].
GAl 0 ¼ 11278:378 þ 188:684153 T 31:748192 T lnðTÞ 1:23 1028 T 9 ; GNi 0 ¼ 5179:159 þ 117:854 T 22:096 T lnðTÞ 4:84 103 T 2 ; L0 ¼ 162407:75 þ 16:212965 T; L1 ¼ 73417:798 34:914 T; L2 ¼ 33471:014 9:837 T; L3 ¼ 30758:01 þ 10:253 T;
Fig. 2. Phase boundary of the g and g0 phases calculated by the sub-lattice model for temperatures ranging from 800 to 1300 K; the previous experimental data (green) and calculated data (purple) are plotted for comparison. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
128
X. Wu et al. / Materials Chemistry and Physics 182 (2016) 125e132
3.2. Initial cluster and microstructure evolution of the g0 phase As can be seen from Fig. 2, the phase boundary of the g and g0 phases determines the equilibrium composition of the g and g g0 0 phases at 1073 K, which are approximately XAl ¼0.141 and g
XAl ¼ 0.222, respectively. We choose an intermediate composition XAl ¼ 0.172, namely Ni-17.2 at.% Al alloy, to study the precipitation kinetics. Because some low composition alloys have been studied in NieAl alloys [17,22,23], the intermediate composition also has a more accurate thermodynamics parameters than that is near the phase boundaries. The temporal microstructures of the g0 phase in a Ni-17.2 at.% Al alloy aged at 1073 K are presented in Fig. 3, where the white regions represent the g0 phase and the black regions represent the g phase. It can be seen that the g0 phase is nearly spherical at the nucleation and initial growth stages (see Fig. 3(a) and (b)), after which it transforms to a cuboidal shape at the growth and early coarsening stages (see Fig. 3(c) and (d)). At the quasi-stationary state coarsening stage (see Fig. 3(e) and (f)), some g0 phases change to rectangular shapes along the elastic soft directions [10] and [01] by a connection coarsening mechanism [28]. The shape transformation of the g0 phase during the precipitation depends on the volume free energy and surface energy when the particle is small at the early stage of precipitation; as the particles coarsen, the elastic strain energy dominates the shape transformation [25]. The coarsening of the g0 phase usually proceeds by means of Ostwald ripening and coalescence of neighboring particles [28], which are designated as B and B* (Ostwald ripening) and C and C* (coalescence) in Fig. 3(e) and (f). The initial precipitation process of the g0 phase can be seen from Fig. 3(a) and (b), where some Al-enriched cluster evolve to large g0 phase particles as the aging progresses and some metastable Alenriched clusters are dissolved, as seen inside the rectangles designated by A and A*. As is known, the transformation of a metastable nucleus to a stable one is a thermally active process, so the nucleation of a new phase needs to overcome an energy barrier
[40,41]. The evolution of the Al-enriched cluster (or g0 nuclei) inside the rectangle A in Fig. 3(a) are presented in Fig. 4. The cluster designated D grows to a nearly spherical g0 phase particle at t* ¼ 25, while the cluster designated by letter E shrinks and is dissolved gradually. Fig. 5 displays the evolution of Al content XAl and order parameter profiles of the Al-enriched cluster shown in Fig. 4. The Al content of the cluster designated D rises gradually and reaches the maximum of 0.208 at approximately t* ¼ 21, as shown in Fig. 5(a), where the maximum composition shows little deviation from the thermodynamic phase diagram (~0.222) at 1073 K, as shown in Fig. 2. The composition deviation in the initial nucleus is declared in previous studied [20,21]. However, the order parameter of particle D nearly reaches the maximum of 0.98 at approximately t* ¼ 3. The variations of the composition and order parameter indicate that the atomic ordering is faster than the composition clustering at the initial precipitation of the g0 phase, which also be showed in the previous studied of Ni75Al7.5V17.5 (at.%) alloy [42]. If the initial emerged clusters cannot evolve to a stable nucleus, such as the particle designated E, its composition and order parameter fall to the initial state again as the aging progresses, as shown in Fig. 5(c) and (d). Therefore, the magnitude of the initial thermal noise is suitable for the automatic development of the system by the nucleation and growth of some Al-enriched clusters and dissolving again of the metastable clusters. The initial nucleation of g0 can be detected by the composition and order parameters of phase field simulations. Comparing Fig. 5(a) with Fig. 5(b), it can be seen that the order parameter of particle D nearly stays at the stable value from t* ¼ 3 to t* ¼ 25, while the Al content gradually increases from t* ¼ 3 to t* ¼ 21. At this stage, the g0 phase is a non-stoichiometric ordered phase. Note that the Al content at t* ¼ 21 is nearly equal to the Al content at t* ¼ 25; the g0 phase becomes a stoichiometric ordered phase from then on. It can be concluded that the evolution of the g0 phase at the nucleation stage is the precipitation from supersaturated solid solution to non-stoichiometric ordered phase and
Fig. 3. Microstructure evolution of the simulated g0 phase in Ni-17.2 at.% Al alloy aged at 1073 K, (a) t* ¼ 10, (b) t* ¼ 30, (c) t* ¼ 80, (d) t* ¼ 160, (e) t* ¼ 300, and (f) t* ¼ 1000.
X. Wu et al. / Materials Chemistry and Physics 182 (2016) 125e132
129
Fig. 4. Initial Al-enriched cluster growing up to a g0 phase or dissolving inside the rectangle A in Fig. 3, (a) t* ¼ 3, (b) t* ¼ 12, (c) t* ¼ 21, and (d) t* ¼ 25.
0.22
1.2
t*=3 t*=12 t*=21 t*=25
(a)
1.0
Order Parameter
0.21
XAl
0.20 0.19 0.18
t*=3 t*=12 t*=21 t*=25
(b) 0.8 0.6 0.4 0.2
0.17
0.0
0.16 325
330
335
340
345
350
325
330
335
Grid
340
345
350
Grid 1.0
(c)
t*=3 t*=12 t*=21 t*=25
0.8
XAl
Order Parameter
0.19
0.18
t*=3 t*=12 t*=21 t*=25
(d)
0.6 0.4 0.2
0.17
0.0 315
320
325
330
Grid
315
320
325
330
Grid
Fig. 5. Evolution of Al content XAl (a) and (c), and order parameter of Al-enriched clusters (b) and (d), (a) and (b): an Al-enriched cluster grows into a stable g0 particle, (c) and (d): the dissolution of an Al-enriched cluster.
stoichiometric ordered phase. Additionally, the gradual composition increase is different from the classical nucleation theory, where the emerged nucleus reaches an equilibrium composition [43]; therefore, the g0 phase precipitation is by means of a nonclassical nucleation and growth mechanism. The evolution of the Al content XAl and order parameter of the g0 phase from nucleation to coarsening in Ni-17.2 at.% Al alloy aged at 1073 K are shown in Fig. 6. It can be seen from Fig. 6(a) that there is a composition difference between the center and the edge of the g0 phase from t* ¼ 80, namely inhomogeneous composition. The reason for the inhomogeneity is the misfit strain, and the inhomogeneity becomes stronger as the misfit strain increases, which has been discussed in Ref. [44]. However, the order
parameter of the g0 phase stays at the stable value after t* ¼ 20. As the growth and coarsening progresses, the Al content and order parameter become wider than before. The non-classical nucleation followed by growth and coarsening in present simulation is different to that proposed by Staron et al., their study on the Ni13 at.% Al aging at 450 C showed a coarsening reaction from an initial small cluster of very beginning of decomposition, which is different from a classical nucleation reaction also [45]. 3.3. Size and shape evolutions of the g0 phase Fig. 7 illustrates the temporal average particle radius rave of the
g0 phase in Ni-17.2 at.% Al alloy aged at 1073 K for t* ¼ 1000, in
130
X. Wu et al. / Materials Chemistry and Physics 182 (2016) 125e132
0.22 0.21
1.2
(a)
t*=20 t*=160
t*=80 t*=1000
0.19
XAl
1.0
Order parameter
0.20
0.18 0.17 0.16
t*=20 t*=160
(b)
t*=80 t*=1000
0.8 0.6 0.4 0.2
0.15
0.0 310
320
330
340
350
360
310
320
Grid
330
340
350
360
Grid
Fig. 6. Evolution of Al content XAl (a) and order parameter (b) of the g0 phase from nucleation to coarsening in Ni-17.2 at.% Al alloy aged at 1073 K.
may cause a statistical error when counting the number and calculating the size of the g0 phase. To minimize the error, the simulation cell is chosen to be large enough to ensure a sufficient number of g0 phases for calculation. Fig. 8 shows the temporal evolution of the normalized particle size distribution (PSD) of r/rave for the g0 phase in a Ni-17.2 at.% Al alloy aged at 1073 K, r is the radius of g0 particle at a aging time and rave is the average of g0 particle radius. For the PSDs presented here,
3.0 (a)
(b)
(c)
(d)
2.5
PSD
which logarithmic coordinates are adopted. The average particle radius has a rapid increase at the initial stage, t* < 60, then it slows down in the following two stages, as denoted by the dashed and solid lines in Fig. 7, respectively. The three stages of rave correspond to nucleation and initial growth, growth and coarsening, and quasistationary state coarsening, respectively. The variations of rave in the three stages have also been presented in both experimental results [17] and simulation results [46] for a Ni-14 at.% Al alloy aged at 823 K. We fitted the data of rave for the three obvious stages with a linear function, and the time exponents of rave∝(t*)n are n ¼ 1.35, 0.42, and 0.22, respectively. It should be noted that the time exponent 0.22 for the coarsening is less than the 0.33 predicted by the LSW theory [47,48], while it is close to the 0.20 reported in Wen’s result for Ni-14 at.% Al alloy aged at 823 K for 1400 min [46]. Mushongera’s recent results in multi-component Ni-based alloys also showed a time exponent ranging from 2.2 to 2.35 with elastic effects [49]. The time exponent of the g0 phase in elastically constrained NieAleTi alloys shows a remarkable decrease from 1/3 after entering the coarsening stage [18]. Therefore, the elastic strain induced by the lattice mismatch of the g and g0 phases results in a different coarsening kinetics to the classical LSW theory, in which a dilute solution without elastic interactions is assumed. It should be noted that the 2D approximation
2.0 1.5 1.0 0.5
2.5
PSD
2.0 1.5 1.0 0.5 0.0 Fig. 7. Evolution of average particle radius of the g0 phase in Ni-17.2 at.% Al alloy aged at 1073 K; the nucleation and initial growth, growth and coarsening, and quasistationary state coarsening stages are fitted with dash-dot, dashed, and solid lines, respectively.
0.5
1.0
1.5
0.5
1.0
1.5
r/rave Fig. 8. Particles size distribution (PSD) evolution of the g0 phase in Ni-17.2 at.% Al alloy aged at 1073 K, (a) t* ¼ 20, (b) t* ¼ 80, (c) t* ¼ 160, and (c) t* ¼ 1000.
(a)
1.6
Average Aspect Ratio
Average Aspect Ratio
X. Wu et al. / Materials Chemistry and Physics 182 (2016) 125e132
1.4 1.2 1.0 0
2
4
6
8
10
1.6
131
(b)
1.4
1.2
1.0 0
200
400
600
800
1000
t*
rave (nm)
Fig. 9. Average aspect ratio as a function of the average particle radius of the g0 phase (a), and the aging time (b) in Ni-17.2 at.% Al alloy aged at 1073 K.
the interval size is 0.1 r/rave, the distribution was normalized by dividing the total particle number nt with the particle number ni in an individual interval, and with the scaled size of the interval width. It can be seen that the peak of the PSD decreases and the width of the PSD increases as the aging progresses, and the positions of the peak values of the PSD shifts from r/rave ¼ 0.9 to 1.0 as the aging progresses. The decrease in the peak and increase in the PSD width occur because of stable-state coarsening, the peaks of the PSD lies at the 1.0 at later stage indicates the kinetics scaling behavior of coarsening. To describe the spatial characteristics of the g0 phase morphology, the shape of the g0 phase has been analyzed with various shape parameters [50e53]. In this work, we use the average aspect ratio, P, to characterize the shape evolution of the g0 phase, which is defined as the ratio of the long axis to the short axis of the g0 phase. The variations of P with the average particle radius rave and the aging time t* are plotted in Fig. 9 (a) and (b), respectively. These results are similar to the variation of the average radius, and the average aspect ratio also shows three stages with increasing aging time t* and average particle radius rave. The average aspect ratio stays at approximately 1.03 when the average particle radius rave is less than 3.5 nm; this stage corresponds to the nucleation and initial growth. The average aspect ratio increases rapidly at the growth and early coarsening stage, then it slows down when entering the quasi-stationary state coarsening. The average aspect ratio remains at approximately 1.59 after t* ¼ 600. 4. Conclusions The kinetics of nucleation, growth, and coarsening of the g0 phase in a NieAl alloy were studied using the phase field model with the thermodynamic free energy and parameters. The nucleation process was detected by the initial variations of composition and order parameters of the newly emerged g0 phase; the atomic ordering in the nucleus of the g0 phase is faster than the composition clustering for the Ni-17.2 at.% Al alloy aged at 1073 K. The precipitation of the g0 phase from the supersaturated solid solution is in the following order: non-stoichiometric ordered phase/ stoichiometric ordered phase/growth and coarsening. The average particle radius of the g0 phase and the aging time show an exponential relationship, rave∝(t*)n, at the nucleation and initial growth, growth and coarsening, and later coarsening stages, with exponents n ¼ 1.35, 0.42, and 0.22, respectively. The variation of the average aspect ratio also shows obvious three stages; it remains at approximately 1.03 when the average particle radius rave is less than 3.5 nm, then it increases rapidly at the growth and coarsening stage of the g0 phase, and maintains a stable value again when the g0 phase goes into the later coarsening stage. The peak of
the PSD decreases and the width of the PSD increases as the aging progresses, and the positions of the peak values of the PSDs are located at r/rave ¼ 0.9 to 1.0. The phase field simulations with the thermodynamic parameters of the NieAl alloy give an effective kinetic description for the precipitation of the ordered g0 phase from nucleation to coarsening. Acknowledgments The authors acknowledge the financial support from the National Natural Science Foundation of China (No. 51571122), the Fundamental Research Funds for the Central Universities (No. 30920130121012), and the Graduate Innovation Project of Jiangsu Province (No. SJLX_0157). Nomenclature
Symbol and meaning PSD particle size distribution total free energy of the NieAl alloy Ftotal c composition field of Al f volume free energy density kc gradient energy coefficient of composition LRO long-range order t time Mf interface mobility xi ðr; tÞ random Langevin noise item of order parameter GNi Gibbs free energy of pure Ni 0 Li interaction parameter of excess energy T absolute temperature Vm mole volume g0 XAl equilibrium composition of g0 phase Cijkl elastic constant tensor εij total strain dij Kronecker delta function r* dimensionless position l length scale of the system k*c dimensionless gradient energy coefficient of composition f* dimensionless volume free energy density x*c ðr ; t * Þ dimensionless random Langevin noise item of composition Dx* dimensionless grid length of x axis Dt * dimensionless time step rave average particle radius r/ rave normalized particle size LSW LifshitzeSlyozoveWagner V the volume of the system
132
X. Wu et al. / Materials Chemistry and Physics 182 (2016) 125e132
long-range order parameter field elastic energy density gradient energy coefficient of order parameter positon chemical mobility random Langevin noise item of composition Gibbs energy of pure Al parameter of bond energy gas constant Gibbs energy of the NieAl alloy equilibrium composition of g phase typical free energy when g and g0 phases have same free energy εel elastic strain ij ε0ij eigenstrain c0 nominal composition of Al t* dimensionless time c dimensionless diffusivity k*f dimensionless gradient energy coefficient of order parameter fel* dimensionless elastic energy density x*i ðr ; t * Þ dimensionless random Langevin noise item of order parameter Dy* dimensionless grid length of y axis XAl Al content n time exponent of rave∝(t*)n P average aspect ratio
fi fel kf r Mc xc ðr; tÞ GAl 0 Ui R G g XAl DG0
References [1] M. Durand-Charre, The Microstructure of Superalloys, Gordon and Breach Science Publishers, Amsterdam, 1997. [2] R.C. Reed, The Superalloys: Fundamentals and Applications, Cambridge University Press, Cambridge, 2006. [3] H.X. Li, M. Qiao, C.G. Zhou, Mat. Chem. Phys. 143 (2014) 915e920. [4] S. Keshavarz, S. Ghosh, Acta Mater 61 (2013) 6549e6561. [5] F. Sun, S.C. Mao, J.X. Zhang, Mat. Chem. Phys. 147 (2014) 483e487. [6] T. Tanimoto, Md. Moniruzzaman, Y. Murata, N. Miura, Y. Kondo, Y. Tsukada, T. Koyama, Comp. Mat. Sci. 93 (2014) 56e61. [7] P.K. Rastogi, A.J. Ardell, Acta Mater 17 (1969) 595e602. [8] M.M.P. Jansshen, Metall. Trans. 4 (1973) 1623e1633. [9] H. Okamoto, J. Phase Equili. Diff. 14 (1993) 257e259. [10] M. Watanabe, Z. Horita, D.J. Smith, M.R. Mccartney, T. Sano, M. Nemoto, Acta Mater 40 (1994) 3381e3387. [11] J.M. Sanchez, J.R. Barefoot, R.N. Jarrett, J.K. Tien, Acta Mater 32 (1984)
[12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43]
[44] [45] [46] [47] [48] [49] [50] [51] [52] [53]
1519e1525. Y. Du, N. Clavaguera, J. Alloys Comp. 237 (1996) 20e32. N. Dupin, I. Ansara, B. Sundman, Calphad 25 (2001) 279e298. Y. Ma, A.J. Ardell, Z. Metallk 94 (2003) 972e975. H. Okamoto, J. Phase Equilib. Diff. 25 (2004) 257e259. Y. Mishin, Model. Simul. Mat. Sci. Eng. 22 (2014) 045001. H. Wendt, P. Hassen, Acta Mater 31 (1993) 1649e1659. T. Maebashi, M. Doi, Mat. Sci. Eng. A 373 (2004) 72e79. V. Vaithyanathan, L.Q. Chen, Acta Mater 50 (2002) 4061e4073. L. Zhang, L.Q. Chen, Q. Du, Acta Mater 56 (2008) 3568e3576. L. Zhang, L.Q. Chen, Q. Du, Commun. Comput. Phys. 7 (2010) 674e682. T. Miyazaki, S. Kobayashi, T. Koyama, Metall. Mat. Trans. A 30A (1999) 2783e2789. T. Miyazaki, T. Kozakai, G.C. Schoen, Mat. Sci. Forum 502 (2005) 139e144. J.Z. Zhu, Z.K. Liu, V. Vaithyanathan, L.Q. Chen, Scr. Mater 46 (2002) 401e406. J.C. Wang, M. Osawa, T. Yokokawa, H. Harada, M. Enomoto, Comp. Mat. Sci. 39 (2007) 871e879. J.W. Cahn, J.E. Hilliard, J. Chem. Phys. 28 (1958) 258e267. Y.S. Li, Y.X. Pang, X.C. Wu, W. Liu, Model. Simul. Mat. Sci. Eng. 22 (2014) 035009. Y. Wang, D. Banerjee, C.C. Su, A.G. Khachaturyan, Acta Mater 46 (1998) 2983e3001. L.D. Landau, E.M. Lifshitz, Course of Theoretical Physics, Statistical Physics, third ed., Pergamon, Oxford, 1980 part 1. A.T. Dinsdale, Calphad 15 (1991) 317e425. I. Ansara, N. Dupin, H.L. Lukas, B. Sundman, J. Alloys Comp. 247 (1997) 20e30. M. Doi, Prog. Mat. Sci. 40 (1996) 79e180. P. Fratzl, O. Penrose, J.L. Lebowitz, J. Stat. Phys. 95 (1999) 1429e1503. L.Q. Chen, J. Shen, Comp. Phys. Commun. 108 (1998) 147e158. J.Z. Zhu, L.Q. Chen, Phys.. Rev. E 60 (1999) 3564e3572. A.J. Ardell, Phi. Mag. 94 (2014) 2101e2130. F. Li, A.J. Ardell, Scr. Mater 137 (1997) 1123e1128. A.J. Ardell, Model. Simul. Mat. Sci. Eng. 8 (2000) 277e286. F. Li, A.J. Ardell, J. Phase Equilib. Diff. 19 (1998) 334e339. T.W. Heo, L.Q. Chen, JOM 66 (2014) 1520e1528. T.W. Heo, L. Zhang, Q. Du, L.Q. Chen, Scr. Mater 63 (2010) 8e11. Y.S. Li, Z. Cheng, Y.L. Lu, Y.X. Wang, Prog. Nat. Sci. 14 (2004) 1099e1103. R. Wagner, R. Kampmann, P.W. Voorhees, Homogeneous second-phase precipitation, in: G. Kostorz (Ed.), Phase Transformations in Materials. Weinheim, Wiley-VCH, New York, Chichester, 2001, p. 309. Y. Wang, A.G. Khachaturyan, Acta Metall. Mater 43 (1995) 1837e1857. P. Staron, M. Haese-Seiller, R. Kampmann, R. Wagner, Phys. B 234e236 (1997) 989e991. Y.H. Wen, B. Wang, J.P. Simmons, Y. Wang, Acta Mater 54 (2006) 2087e2099. I.M. Lifshitz, V.V. Slyozov, J. Phys. Chem. Solids 19 (1961) 35e50. C. Wagner, Z. Elektrochem. 65 (1961) 581e591. L.T. Mushongera, M. Fleck, J. Kundin, F. Querfurth, H. Emmerich, Adv. Eng. Mater 17 (2015) 1149e1157. M. F€ ahrmann, P. Fratzl, O. Paris, E. F€ ahrmann, W.C. Johnson, Acta Mater 43 (1995) 1007e1022. S.V. Prikhodko, A.J. Ardell, Acta Mater 51 (2003) 5021e5036. Y. Ma, A.J. Ardell, Acta Mater 55 (2007) 4419e4427. J.P. MacSleyne, J.P. Simmons, M.D. Graef, Acta Mater 56 (2008) 427e437.