Journal of Materials Science & Technology  2019 , 35 (11): 2582-2590 https://doi.org/10.1016/j.jmst.2019.05.064

Orginal Article

Mechanism for the multi-stage precipitation of Fe-Ni based alloy

Yan Jiang, Qiang Zuo, Feng Liu*

State Key Laboratory of Solidification Processing, Northwestern Polytechnical University, Xi’an 710072, China

Corresponding authors:   *Corresponding author.E-mail address: liufeng@nwpu.edu.cn (F. Liu).

Received: 2019-04-30

Revised:  2019-05-21

Accepted:  2019-05-30

Online:  2019-11-05

Copyright:  2019 Editorial board of Journal of Materials Science & Technology Copyright reserved, Editorial board of Journal of Materials Science & Technology

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Abstract

This work lays great emphasis on the distribution evolution of sigma (σ) phase in Fe-Ni based N08028 alloy during aging process, the result of which could provide new insights into phase change and precipitation mechanism. It is found that the σ phase, in any case, tends to separate out with a granular shape at grain boundaries (GBs) primarily, and with the increment of time, it is obliged to precipitate in grain interiors (GIs) with a lamellar structure. The mechanisms for the evolving volume fraction and morphology of σ phase are discussed, and a model appropriate for the multi-stage behavior transforming from intergranular to intragranular precipitation is derived, as well as revealing the thermodynamics and transformation kinetics corresponding to this process. The results reveal that the occurrence of multi-stage precipitation is correlated with the redistribution of solute atoms and with the difference in coupling effect of thermodynamic driving force and kinetic activation energy between the intergranular and intragranular precipitation.

Keywords: Precipitation ; Intergranular precipitation ; Intragranular precipitation ; Thermodynamics ; Transformation kinetics

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Yan Jiang, Qiang Zuo, Feng Liu. Mechanism for the multi-stage precipitation of Fe-Ni based alloy[J]. Journal of Materials Science & Technology, 2019, 35(11): 2582-2590 https://doi.org/10.1016/j.jmst.2019.05.064

1. Introduction

As a typical solid-solution-strengthened Fe-Ni based alloy, N08028 alloy has attracted a great attention with regard to its application in chemical and petroleum industry as oil pipes for its favorable combination of mechanical and corrosion resistance properties [1,2]. Although the N08028 alloy has many outstanding attributes, it often suffers a lot from embrittlement due to the precipitation of σ phase, which affects mechanical and corrosion properties [3], in particular impact toughness, even in low amount [4]. Regarding that the precipitation behavior of σ phase is of great importance, many efforts have consequently, been spent in investigating the σ phase. For instance, Lee et al. [5] investigated the formation mechanism of σ phase and concluded that the precipitation sequence is to be grain boundary Cr2N, cellular Cr2N and σ phase and that it is the nitrogen-depleted zone near Cr2N that induces the nucleation of σ phase. Niewolak et al. [6] found that both the presence of molybdenum and the increased Cr/Fe-ratio at the ferrite/austenite interface can promote σ phase formation. Villanueva et al. [7] demonstrated that σ phase occur both at austenite (γ) GBs and inside delta ferrite (δ) islands in austenitic stainless steel and the precipitation level of the σ phase: Duplex stainless steel > Superferritic stainless steel > Austenitic stainless steel. Further, the morphology [8], crystallographic feature [9], volume fraction [10], dissolution [11] and precipitation behavior [12], with respect to the σ phase have also been explored.

However, since most studies [[6], [7], [8], [9], [10],[12], [13], [14]] were focused on steels containing δ phase or more than σ precipitate and laid their emphasis on σ phase formation or orientation relationships between the interfacialσprecipitate and adjacent matrix phases, information about the distribution evolution of σ phase in austenitic stainless steels is scanty, albeit the distribution position of second phases can directly affect the physical and mechanical properties of materials [15,16]. Take, for example, the phenomenon that the intragranular particles of La2O3 serve as obstacles to moving dislocations in GIs while the intergranular La2O3 particles will do great damage to the ductility through the way of inducing intense stress/strain concentration during plastic deformation [16]. According to the report by Lewis [17], the grain boundary facilitates the formation of σ phase at an early stage of aging and at the expense of the previously precipitated large TiC particles, while σ phase in the γ matrix grows in the form of thin plates which appear to be nucleated around TiC particles. Doubtlessly, his study systematically investigated the σ phase in atomic scale, less attention was, however, paid to the multi-stage behavior transforming from intergranular to intragranular precipitation and thus the behind mechanism kept unclear. Recently, similar results arose in austenitic stainless steel S32654 [18], where σ phase was found following a sequence of intergranular, cellular and intragranular with aging temperature and inducing the precipitation of Cr2N ahead of cell boundary. This study does shed some light on the transition mechanism of σ phase, which is thought to be closely related with free energy reduction and faster precipitation kinetics from high Cr and Mo contents. However, this theory has not been verified yet due to the lack of an appropriate way to quantify the evolving driving force and activation energy corresponding to the transition process of σ phase. Besides, the kinetics of σ phase precipitation studied by Barcik [19] was only attempt to express the precipitation rate of σ phase and could not be exerted to express the multi-stage precipitation. Furthermore, the mechanisms for the morphology and volume fraction evolution of σ phase distributed at GBs or in GIs still remain unclear yet.

To this end, it is still a necessity of paying attention to the mechanism for the multi-stage precipitation of Fe-Ni-based alloy N08028. It is herein aimed at studying the evolving distribution of σ phase during aging at different temperatures. On this basis, the mechanism of intergranular and intragranular precipitation is discussed, and a multi-stage precipitation model appropriate for the transition behavior is derived, as well as revealing the thermodynamics and transformation kinetics corresponding to this process. The present research will help to understand the multi-stage precipitation behavior of σ phase, control the precipitation process of Fe-Ni based alloys, and improve the precipitation theories with regard to the so-called thermo-kinetic model on a whole.

2. Experimental

A commercial Fe-Ni based alloy (N08028), with a nominal chemical composition of Fe-32.18Ni-27.91Cr-3.93Mo-0.81 Mn (wt%), was used. The as-received material was cut into dimensions of 8 mm × 8 mm × 6 mm and followed by a solution treatment for 2 h at 1200 °C with water quenching to guarantee a supersaturated solid solution. Afterwards, the specimens were aged at 900, 950 and 1000 °C for a time duration ranging from 0.5 h to 16 h. The heat treatments were conducted in a muffle furnace and terminated by water quenching.

The microstructure of specimens was examined by an Axiovert 40 MAT metallographic optical microscope (OM). Phase analysis was performed with a Bruker AXS D8 X-ray diffractometer (XRD) using a Co Kα radiation with a scanning angle from 42° to 100° and a scanning speed of 2°/min. The phase morphology was observed by a Tescan VEGA3 scanning electron microscopy (SEM) and FEI Talos F200X TEM. Sample preparation for the OM and SEM involves mechanical grinding followed by etched in a solution of saturated acid with a volume fraction of H2O2:HCl:H2O = 1:2:2. The volume fraction of intermetallic phase was calculated by Image-Pro Plus software based on SEM images. Thin foil specimens for TEM observation were prepared using an ion milling machine (Model 600, Gatan) equipped with a liquid nitrogen cooling system. The evolving precipitate size was measured by quantitative image analysis, while the evolving fraction of precipitate was obtained by quantitative XRD analysis.

3. Results and discussion

3.1. The precipitation behavior

Fig. 1 gives the XRD patterns of the alloys in different conditions. In contrast to the solution-treated specimen where only γ phase is detected, in the specimens aged at 900 °C for 16 h, new peaks appear at positions within a range of 2θ = 45° and 2θ = 60° (as the blue frame indicated), which correspond to σ phase. Since the volume fraction of second phase is proportional to the integrated intensity of diffraction peaks [20], XRD is used to analyze σ phase evolution qualitatively in Fig. 1(b)-(d). The phenomenon that the increased diffraction peaks of σ phase appear at a decreased aging temperature demonstrates that a lower aging temperature produces a larger quantity of σ phase. This is contrary to the result in ref. [15], where a larger volume fraction of precipitates appear in specimens with a higher aging temperature, and can be explained by combined effects of the diffusion behavior of atoms, the solid solubility of σ phase in γ matrix, and the supercooling degree of phase transition. The diffusion behavior of atoms can be calculated by the Arrhenius equation below:

D=D0exp(-QD/(RT)) (1)

where D0 is the pre-exponential factor for diffusion and QD is the activation energy of solute diffusion, they all hinge on the chemical composition and the structure of alloy, R is a gas constant and T the temperature. In this work, since all the specimens come from the same alloy, the effects of D0, QD and R can be excluded. The diffusion behavior of atoms during aging process is only affected by the temperature: a lower aging temperature results in a smaller diffusion rate, and in return a smaller volume fraction of precipitates. Obviously, this is contrary to the experimental results, so the changes in σ phase solid solubility and supercooling degree over the aging temperature dominate the precipitation process. Generally, a lower heat-treatment temperature results in a smaller solid solubility of the precipitate and a higher degree of supercooling. In other words, a decreased aging temperature for a same alloy will stimulate a higher degree of supersaturation and a smaller critical nucleation radius, hence a higher nucleation rate for the σ precipitate [21]. Due to a smaller diffusion speed, the σ phase formed at a lower aging temperature exhibits a smaller size, which is in accord with the OM results in Fig. 2.

Fig. 1.   XRD patterns obtained from specimen solution treated at 1200 °C for 2 h and aged at 900 °C for 16 h (a) and σ peaks obtained from specimens aged for various hours at (b) 1000 °C, (c) 950 °C and (d) 900 °C.

Fig. 2.   OM photographs from specimens aged at temperatures of 1000 °C (a), 900 °C and 950 °C (b) for different times.

Fig. 2 shows a series OM images of the specimens aged at 1000 °C, 950 °C and 900 °C for various aging time. Owing to the identification by XRD, it can be known that the matrix is γ phase while the second phase with various morphologies is σ phase. As shown in the Fig. 2(a), the σ precipitate tends to separate out in isolation with a granular shape at GBs primarily (0.5 h, 1 h), and then grows along the GBs (2.5 h or 2 h) as the time goes on. Finally, it starts to arise in GIs with a needle-like morphology (3-16 h) when all the GBs are covered with σ phase. Similar behaviors occur in specimens with different aging temperatures (Fig. 2(b) and (c)). This indicates that the transition from intergranular to intragranular precipitation is not an accident but an inevitable result closely related with the material itself. Obviously, this is distinct from the previous studies where σ phase formed through the way of eutectic reaction and appeared only in the interdendritic regions [22]. A closer observation reveals that the σ phase formed at a lower temperature exhibit a smaller size, as shown in Fig. 2(a)-(c), in accord with the previous discussion. Due to the small depth of field of OM images, SEM is performed on specimens in different conditions. According to Fig. 3, it is further verified that the solution-treated specimens are composed of only γ matrix while both γ matrix and σ phase are emerged in aged specimens. One thing to be noted is that the detailed SEM observation reveals that the σ phase in GIs is, more precisely, lamellar structure rather than needle-like, as shown in Fig. 3(b).

Fig. 3.   SEM images of alloy after solution treatment (a) and the morphology of needle-like σ phases in GIs (b).

TEM observations (Fig. 4) are performed on specimens aged at 900 °C for 16 h, with an extra attention paid to the γ/σ interfaces. It is apparent that the σ phase at GBs exhibits discrete granular particles (Fig. 4(a)), while in GIs it presents a lamellar structure. HRTEM images, showing the micro-structure of γ/σ interfaces at GBs (Fig. 4(c)) and in GIs (Fig. 4(d)), indicate that the σ phase and the γ phase, in any cases, adopt an incomplete coherent relationship. This is due to the different crystal structures and lattice constants between the σ and γ phases. Compound use of the XRD result and SAED pattern finds that the σ phase in the studied alloy possesses a tetragonal lattice (a = b =8.96 Å, c =4.64 Å) while the γ phase a face-centered cubic structure (a = b = c =3.66 Å). The selected area diffraction (SAD) patterns reveal that the σ and γ phases in GIs follow a traditional Nenno relationship [12,23], namely (111)γ//(001)σ, [1(_)10]γ//[11(_)0]σ (Fig. 4(d)), while those at GBs follow a relationship of (200)γ//(410)σ, [001]γ//[1(_)40]σ (Fig. 4(c)).

Fig. 4.   Bright field TEM images of the precipitates at GBs (a) and in GIs (b) and HRTEM images of γ/σ interface locating at GBs (c) and GIs (d), with the electron diffraction pattern and enlarged view from the selected area.

It is well known that the morphology of a new phase formed during a solid-state phase transformation is determined by the competition of interface energy and strain energy, which acts as resistance to nucleation. For the σ phase at GBs, the interface between the σ and γ phase is noncoherent (Fig. 4(c)). Similar results was found in Ref. [17], where the grain boundary σ grows from nuclei, which do not have a coherent interface with the matrix, and exhibits no definite orientation with either grain. As a result, the σ phase at GBs contains a low density of stacking faults in comparison with lamellar precipitating within the grains [17], and hence, the precipitation process at GBs is dominated by the interface energy [24], derived from a high chemical energy change and a low matching degree to form σ nuclei. In this way, the crystal nucleus of σ phase here is similar to a sphere. Due to the criterion of minimum energy path and the effect of boundary diffusion, the σ nuclei at GBs elongate and thicken quickly and result in isolated particles [15], in consistent with the experiment result in Fig. 2. As for the ones in GIs, the interface is semi-coherent (Fig. 4(d)), produced by the orientation relationship of (111)γ//(001)σ, [1(_)10]γ//[11(_)0]σ. Because of the high mismatch degree calculated as about 10.3%, the elastic strain yielded to sustain the coherent interface finally leads the precipitation process and the σ nucleus tends to be a thin sheet. Since the nucleus grows along the close alignment direction of γ matrix, the σ phase in GIs finally have a lamellar structure, as shown in Fig. 3(b).

With regard to the phenomenon of the intergranular precipitation occurring prior to the intragranular precipitation, this can be explained by the huge defects in GBs, which can facilitate the precipitation of σ phase in three ways: firstly, the GBs can act as high-energy nucleation sites [25] and stimulate a lower nucleation energy for the intergranular precipitation as compared to that for the intragranular precipitation [26]. Furthermore, the GBs can act as a collection plate of solute atoms, which induce the segregation of σ phase forming elements like Ni, Fe, Cr, Mo in the vicinity of GBs [27], and a higher degree of supersaturation at GBs finally results in an increased nucleation rate of σ phase. Last but not least, the diffusion rates of σ phase forming elements within the austenite grains are greatly lower than those along GBs for the abundant vacancies in GBs [28]. All the three reasons make the σ phase take longer time to nucleate and grow up in GIs.

3.2. Modeling derivation and the determination of parameters

In this study, the intergranular and intragranular precipitations are nucleation and growth type transformations, which are controlled by solute diffusion and influenced by soft impingement [29,30]. Therefore, a model based on the Johoson-Mehr-Avrami (JMA) theory [31] is established to approximately describe the transition behavior.

During intergranular precipitation, the nucleation of a precipitate starts by a volume diffusion of solute atoms in GIs entering GBs, and then turns into a grain boundary diffusion of transporting those solute atoms to the nucleus to hold the precipitation process [32] (Fig. 5(a)). Since the solute atoms diffuse more slowly in GIs than in GBs, the process is dominated by volume diffusion, namely, homogenous nucleation [21] dominates this process. So the nucleation rate Ig(t) can be given as [32,33]:

$I_g(t)=I_gexp(-\frac{t}{τ_1})$. (2)

where Ig is the homogenous nucleation rate, being obtained by Ig = ωN1exp (-QN1/(RT)), with QN1 estimated as NA × (QmGg*). N1 is the number of potential nucleation sites, ω the oscillation frequency of atoms estimated by ω = kT/h, h the Planck constant, NA the Avogadro constant and Qm the activation energy for atomic migration. ΔGg* is the corresponding critical Gibbs energy of nucleus formation, given as 16πγγσ3/(3ΔGv2) and τ1 is introduced for a rapidly decay factor in nucleation rate due to the saturation of GBs being occupied by the nuclei. The extent of solute atoms in GIs diffusing to GBs can be quantified by thickening the laminar GBs, so an assumption of 1-D planar precipitate is adaptive to describe the growth of precipitates. Then the volume of a precipitate nucleated at time τ can be expressed as [34]:

Ve1=4Aλ1$\sqrt{D(t-τ)}$ (3)

where A is the grain boundary area estimated by A=3V/(2Rγ), V the average volume of grains, Rγ the average radius of grains and λ1 the parameter related with concentrations C0, Cσ and Cγ through the saturation degree ks = 2(Cγ-C0)/(Cσ-Cγ). Though the nucleation mechanism in this cause is neither site saturation (Ig(t) = 0) nor continuous nucleation (Ig(t) = constant), it can reduce to continuous nucleation (t/τ1→0) and to site saturation (t/τ1→∞). This means the current nucleation mechanism is analogous to Avrami nucleation. On this basis, a JMA type function is adopted to describe the precipitation kinetics, and the extended transformed volume can be given as [35]:

Ve=$∫_0^tV⋅Ig(τ)⋅V_{e1}dτ=∫_0^t\frac{6I_gVλ_1D_0^{1/2}}{R_γ}exp(-\frac{QD/2}{RT})exp(-\frac{τ}{τ_1})(t-τ)^{1/2}dτ$ (4)

and the kinetics can be described as:

f1=1-exp$(-\frac{V_e}{V})$. (5)

Fig. 5.   Schematic diagrams showing the diffusion process of intergranular precipitation (a) and the ellipse used to obtain the assumed morphology of intragranular precipitate (b).

Introducing a growth exponent n1 = 1/2 + 1/(1 + r2/r1) with r2/r1 being proportional to the ratio of volume parameter for continuous nucleation to that for site saturation, the total effective activation energy for the intergranular precipitation including nucleation and growth can be expressed by:

$Q_{e1}=\frac{\frac{1}{2}Q_D+(n_1-\frac{1}{2})Q_{N1}}{n_1}$ (6)

Obviously, the growth exponent n1 and activation energy Qe1 are functions of time and change with the processing of transformation. For τ1→∞, continuous nucleation prevails with n1 = 1.5 and Qe1 = (2QN1 + QD)/3. For τ1→0, pure site saturation prevails with n1 = 0.5 and Qe1 = QD. Since n1 and Qe1 are constant throughout the transformation, the change of Qe1 is derived from the decay of nucleation rate.

With the proceeding of intergranular precipitation, the GBs are occupied by σ precipitates and the intragranular precipitation dominates the process gradually. Since the precipitation in GIs is homogeneous, the corresponding nucleation rate I can be obtained by I = ωN2exp(-QN2/(RT)) [32], where N2 is the number of potential nucleation sites. QN2 is the activation energy for intragranular precipitation and can be obtained by NA× (QmG*), with ΔG* given as 16πγγσ3/(3(ΔGvGs)2). Because the σ phase in GIs has a lamellar structure (Fig. 3(b)), the precipitate is herein assumed to be an oblate ellipsoid obtained by rotating an ellipse around its minor axis, with OA = OB = a, OC = c, a >> c (Fig. 5(b)). On this basis, the volume of an ellipsoidal precipitate nucleated at time τ can be expressed as [36,37]:

$V_{e2}=\frac{4}{3}πa^2b=\frac{32}{3}π⋅k_rK_e^3(D(t-τ))^{3/2}$ (7)

where kr (=c/a) is the aspect ratio of the oblate ellipsoid, Ke is a parameter dependent on the saturation degree ks and aspect ratio kr. Then the extended transformed volume can be given as:

$V_{e'}=∫_0^tV⋅I(τ)⋅V_{e2}dτ=∫_0^t\frac{32π⋅Vk_rK_e^3I⋅D_0^{3/2}}{3}exp(-\frac{3Q_D/2}{RT})exp(-\frac{τ}{τ_2})(t-τ)^{3/2}dτ$ (8)

where τ2 is brought in as a decay factor due to the nucleation rate decreases drastically as the degree of solute saturation descends with the solute redistribution [38]. The kinetics of the intragranular precipitation can be described as:

$f_2=1-exp(-\frac{V_{e'}}{V})$ (9)

By introducing a growth exponent n2 = 3/2 + 1/(1+r2/r1), the overall effective activation energy for the intragranular precipitation can be expressed by:

$Q_{e2}=\frac{\frac{3}{2}Q_D+(n_2-\frac{3}{2}Q_{N2})}{n_2}$ (10)

Analogously, the growth exponent n2 and activation energy Qe2 change with the transformation. When τ2→∞, only the continuous nucleation prevails with values of n2 ( = 2.5) and Qe2 (= (2QN2+3QD)/5) being constants. With the processing of transformation, τ2 decreases progressively. When τ2→0, pure site saturation prevails and the values of n2 and Qe2 are 1.5 and QD respectively. Since n2 and Qe2 are constants throughout the transformation, the change of Qe2 is derived from the decay of nucleation rate.

Provided that the total volume fraction of σ phase after precipitation is f0, the contributions from the intergranular and the intragranular precipitation are denoted as f10 and f20, respectively. The evolving volume fraction of σ phase during the intergranular precipitation can be described by modifying the expression of the transformed fraction Eq. (5) as:

$f_1=f_{10}⋅(1-exp(-\frac{V_e}{Vf_{10}}$ (11)

and that during the intragranular precipitation can be described as:

Note that, t0 is introduced into Eq. (12) as a lag time for the postponement of the occurrence of intragranular stage to the intergranular precipitation. In consequence, the evolving volume fraction of σ phase during the multi-stage precipitation can be expressed as:

f=f1+f2 (13)

In this model, the change of chemical volume free energy ΔGv can be obtained by ΔGv= ΔGm/Vm, where ΔGm is the molar Gibbs free energy difference between the precipitate and the matrix and Vm is the molar volume of the precipitate. To estimate the ΔGm, the classical thermodynamic calculations are applied, where the values of ΔGm at varied temperatures are obtained by Eq. (14) as [39,40]:

together with a Thermo-Calc software, as listed in Table 1. The port side superscripts ‘0’ and ‘1’ in equations represent the initial and final state, while the ‘σ’ and ‘γ’ denote the σ precipitate and γ matrix, respectively. The strain energy ΔGs can be found in Ref. [22] and the Vm is estimated by Thermo-Calc software as 4.11 × 10-6 m3/mol. Several key parameters are crucial important for the model implementation, including D0, QD, Qm, γγγ, γγσ, Rγ, kr,f0, f10, f20, t0, N1, τ1, N2 and τ2. D0, QD and Qm are found in Ref. [41], γγγ and γγσ in Ref. [10], and Rγ and kr is measured by statistical analysis of the experimental images, as listed in Table 2, Table 3, respectively. f0 is deduced from the thermodynamic equilibrium by Thermo-Calc software, while f10 and f20 are measured by quantitative image analysis using Image-Pro Plus software. By fitting the evolving precipitate size with time to the following equation:

2a=4Ke$\sqrt{D(t-t_0)}$ (15)

t0 can be obtained, with a good fit as shown in Fig. 6. Likewise, the values of N1, τ1, N2, and τ2 can be obtained by fitting the current model simultaneously to the evolving fraction of σ phase upon aging at different temperatures (Fig. 7).

Table 1   Equilibrium phases and corresponding compositions for alloy N08028 at different temperatures calculated with Thermo-calc and the calculated values of ΔGm and ΔGv.

Temperature (ºC)PhaseFe (wt%)Ni
(wt%)
Cr (wt%)Mn (wt%)Mo (wt%)ΔGm
(J/mol)
ΔGv (J/m3)
900γ36.2435.3125.30.932.221313.443.19 × 108
σ29.1214.442.720.1113.65
950γ36.0234.5925.950.912.531190.082.89 × 108
σ29.0414.8242.020.1213.99
1000γ35.833.9226.530.882.86984.592.39 × 108
σ28.9815.2141.350.1314.33

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Table 2   Values of parameters used for the current model prediction for precipitation in N08028 alloy.

ParameterValueParameterValue
k (J/K)1.38 × 10-23D0 (m2/s)1 × 10-4
NA (mol-1)6.02 × 1023QD (kJ/mol)277
R (J/(mol K))8.314γγγ (J/m2)0.4
h (J s)6.626 × 10-34γγσ (J/m2)0.7
Qm (J)4.59 × 10-19Rγ (μm)39.25

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Table 3   Measured and fitted values of parameters used for the current model prediction for precipitation at various temperatures (T: temperature).

T (ºC)f0f10f20krt0 (s)N1τ1N2τ2
10001.11 × 10-16.11 × 10-25.02 × 10-21.22 × 10-28.77 × 1035.73 × 10232.06 × 10-141.57 × 10261.21 × 10-10
9501.55 × 10-16.94 × 10-28.57 × 10-21.81 × 10-27.33 × 103
9002 × 10-17.41 × 10-21.26 × 10-12.79 × 10-25.81 × 103

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Fig. 6.   Comparison of the predicted (line) and experimental (symbols) size (length) of precipitates in GIs at various aging temperatures.

Fig. 7.   Comparison of the experimental (symbols) and predicted (line) σ phase fractions during the entire precipitation, f, intergranular precipitation, f1, and intragranular precipitation, f2, at aging temperatures of (a) 1000 °C, (b) 950 °C and (c) 900 °C.

3.3. Modeling application and thermo-kinetic analysis

During intragranular precipitation, as the aging temperature goes down, the solute diffusion speed drops off and the size of precipitates becomes smaller, leading to a larger reduction in the semi-major axis of the oblate spheroid than that in the semi-minor axis, i.e. the lower aging temperature produces a larger kr and a smaller 2a, satisfying with Table 3 and Fig. 6. As for t0, a lower aging temperature results in a smaller critical nucleation radius and a smaller nucleation energy, and in return a smaller t0, see Table 3. This is also tally with Fig. 7, where the transition from the intergranular to the intragranular precipitation occurs earlier upon a lower aging temperature.

The evolving volume fraction of σ phases during the whole precipitation (f) and each stage (f1, f2) are heavily dependent on the aging temperature, as predicted in Fig. 7. A deeper observation reveals that the mutation for f curves, increasing rapidly to about 0.06% and followed by a slow growth to a peak value ranging from 0.1% to 0.2%, is derived from the transition from intergranular to intragranular precipitation: f1 for the intergranular precipitation where heterogeneous nucleation and 1-D planar diffusion-controlled growth are adopted as the nucleation and growth mechanism, and f2 for the intragranular precipitation where homogeneous nucleation and 3-D ellipsoidal diffusion-controlled growth are adopted. Compared to f1, f2 increases more significantly with time due to the spatial limitations of GBs upon intergranular precipitation. As a result, the intragranular precipitation plays a more important role in the increment of σ phase fraction on the whole.

Since the model is established on the classical nucleation and growth theory combined with JMA type function, the good agreement between the model prediction and the experimental results indicates the occurrence of multi-stage precipitation is ascribed to the difference combinations of thermodynamic driving force and kinetic activation energy (or energy barrier) between the intergranular and the intragranular precipitations.

During the intergranular precipitation, the effective driving force is ΔGv and the effective activation energy Qe1 is composed of two parts, i.e. QN1 for the nucleation and QD for the growth. However, during the intragranular precipitation, the effective driving force is (ΔGvGs) and the effective activation energy Qe2 is composed of two parts, i.e. QN2 for the nucleation and QD for the growth. To explain the multi-stage precipitation, the thermodynamic driving force and kinetic activation energy are calculated as shown in Fig. 8(a), where, it is noticeable that the thermodynamic driving force in both stages decreases with increasing the aging temperature, in contrast with a contradictory trend for the kinetic energy barrier. The lower aging temperature corresponds to a higher thermodynamic driving force and a lower kinetic energy barrier for precipitation. This is compatible with Fig. 2, Fig. 6 where a large number of precipitates with smaller size are obtained by aging at lower temperature. Obviously, in both stages, a higher aging temperature stimulates a higher effective activation energy, as shown in Fig. 8. For any given temperature, however, the effective activation energy is always decreasing with the transformation (Fig. 8(b)), due to the decay of nucleation rate.

Fig. 8.   Evolutions of driving force and activation energy versus aging temperatures (a) and evolutions of total effective activation energy with transition (b).

Regardless of the aging temperatures, the thermodynamic driving force for the intergranular precipitation is always about 500 J/mol higher than that for the intragranular precipitation. This is because the strain energy, caused by the strain field between the nucleus and its surrounding matrix, is consciously ignored during the intergranular precipitation due to the relax of stress field by crystal defects in the GBs [24], as shown in Fig. 4(c), where the preferential orientation relationship between σ phase and γ phase only emerge upon intragranular precipitation rather than intergranular precipitation. Besides, as calculated by the model, for all the aging temperatures, the kinetic activation energy for nucleation upon intergranular precipitation is always 600-1000 J/mol lower than that for intragranular precipitation due to different nucleation mechanisms (Fig. 8(a)). Overall, both the higher thermodynamic driving force and the lower kinetic activation energy, have played a role in the intergranular precipitation occurring prior to the intragranular precipitation.

4. Conclusions

This study systematically investigated the evolving distribution of σ phase in N08028 alloy, which provides new insights into the phase change and precipitation mechanisms. Some important conclusions can be summarized as follows:

(1) The σ phase always tends to separate out primarily at GBs, and then arise at GIs. This multi-stage behavior is correlated with the redistribution of solute atoms and with the difference in coupling effect of thermodynamic driving force and kinetic activation energy between the intergranular and the intragranular precipitation.

(2) The volume fraction of σ phase increases with a decrease in aging temperature and this can be explained by the effects of atom diffusion, solid solubility and supercooling degree of phase transition.

(3) The distribution position can greatly influence the morphology of σ phase through affecting the competition of interface energy and strain energy, as well as the criterion of minimum energy path.

(4) A thermo-kinetic model based on classical nucleation and growth theory and JMA type function is established, which can well describe the multi-stage behavior and help unveil the relevant mechanisms.

Acknowledgements

This work was supported by the Natural Key Research and Development Program of China (Nos. 2017YFB0305100 and 2017YFB0703001), the National Natural Science Foundation of China (Nos. 51431008, 51790481 and 51801157), the Fundamental Research Funds for the Central Universities (Nos. 3102017jc01002 and 3102018gxc022) and the Research Fund of the State Key Laboratory of Solidification Processing of Northwestern Polytechnical University (No. 117-TZ-2015).


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