1 Introduction

With the increasing demand of energy conservation and environmental protection, raising the steam pressure and temperature has been recognized as the most effective approach, which can effectively contribute to improving the fuel thermal efficiency and reducing pollutant emissions [1, 2]. In order to meet the urgent requirement for heat resistance materials brought by the elevated steam parameters, 9%–12% Cr martensitic steels have been vigorously developed and widely applied in ultra-supercritical coal-fired power plants as structural materials. Their excellent comprehensive performance such as creep rupture strength, toughness, and oxidation resistance can better deal with the complicated service environments including high temperature, creep stress and steam corrosion. Among them, G115 steel [3, 4] is one of the representative 9Cr–3W–3Co heat resistance steels and has been designed by the China Iron and Steel Research Institute in order to replace the traditional P91 and P92 steels at the service temperature of 630–650 °C.

As the crucial microstructural characteristics in heat resistance steels, the nucleation, growth, and coarsening behaviors of precipitates are directly related to the microstructure degeneration, eg., dislocation annihilation and subgrain coarsening, and thereby affect the final creep resistance. Common precipitates in martensitic heat resistance steels mainly consist of MX carbonitrides, M23C6 carbides, Laves phase and Z phase [5]. Among them, MX carbonitrides exhibit extremely high stability and almost no coarsening occurs during creep [6,7,8]. The growth and coarsening of M23C6 carbides do harm to the low-temperature toughness [9] and decrease their pinning effects on boundary migration as well as dislocation gliding, though the addition of boron has been verified to significantly improve the stability of M23C6 carbides [10,11,12]. Fine Laves phases are believed to benefit the creep strength of heat resistance steels; however, their coarsening and clustering will greatly reduce the creep strength and impact toughness [13, 14]. Z phases form accompanied by the dissolution of finely dispersed MX phases and hence have a detrimental effect on the creep strength [15, 16]. Consequently, mastering the precipitation behaviors of nucleation, growth, and coarsening is necessary to understand and improve the service performance of heat resistance steels and ensure the long-term service security of thermal power plants.

In terms of the latest generation of martensitic heat resistance steel G115, considerable research efforts have been devoted to the design of heat treatment process [17, 18], the connection between microstructure evolution and mechanical properties [3, 19, 20], the creep deformation behavior [21,22,23] as well as the fracture failure mechanism [24, 25]. However, the numerical simulation of nucleation–growth–coarsening precipitation kinetics has not been achieved. In this paper, the classical nucleation theory (CNT) and the modified Langer–Schwartz (MLS) model are adopted to systematically simulate the precipitate evolution of nucleation rate, number density, average radius, and volume fraction during long-term aging. By verifying with the experimental statistics data, the mechanisms behind precipitation behaviors are discussed in detail. The purpose is to provide theoretical basis for the precipitate design and the creep performance improvement of heat resistance steels.

2 Experimental material and procedures

2.1 Experimental material

A modified G115 heat resistance steel with the chemical composition of Fe–8.88Cr–2.63W–3.02Co–0.088C–0.0088N–0.20V–0.05Nb–0.52Mn–0.23Si–0.015B (wt.%) was chosen as the experimental material. The decreased W amount can increase the incubation time of Laves phase and postpone its coarsening rate. Raw materials were melted in a vacuum induction furnace and casted into an ingot of 50 kg. The ingot was then normalized at 1100 °C for 1 h and tempered at 780 °C for 3 h. The creep tests were conducted using the INSTRON 5582 tensile testing machine on the as-received G115 specimens (ϕ5 mm, 25 mm long) under 180 and 140 MPa at 650 °C, and the creep rupture lifetimes were measured as 659 and 8791 h, respectively. The specimens taken from the grip portion of creep rupture specimens are equivalent to the aging state because the loading and strain accumulation can be ignored under the action of the threads [26]. Thus, the heat treatment and aging process can be summarized in Fig. 1.

Fig. 1
figure 1

Heat treatment and aging process

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2.2 Microstructural characterization

The matrix microstructure and precipitates in G115 steel were characterized by optical microscopy (OM, Leica MEF4M), scanning electron microscopy (SEM, Zeiss Gemini500), transmission electron microscopy (TEM, JEOL JEM 2010F), and scanning transmission electron microscopy (STEM, FEI Talos F200X). The specimens for OM and SEM were mechanically polished and etched in a mixed solution of 1 g picric acid, 5 mL hydrochloric acid and 100 mL ethanol. The thin foils for TEM and STEM investigation were prepared by mechanically thinning to 50 µm in thickness and twin jet polishing in the electrolyte of 15% perchloric acid and 85% ethanol at − 25 °C and 20 V. The chemical compositions of precipitates were analyzed by energy-dispersive spectroscopy (EDS). The method of combining secondary electron (SE) images with backscattered electron (BSE) images was adopted in order to distinguish M23C6 and Laves phases, on account that Laves phases containing more large-sized W atoms are brighter under the BSE mode. For non-spherical particles, the equivalent diameters were calculated by averaging two perpendicular axes. The statistics on the precipitate size distributions were performed based on more than 200 particles from 20 random zones with the Image-Pro Plus software.

2.3 Numerical simulation

The CNT and the MLS model were adopted to simulate the nucleation rate, number density, average radius, and volume fraction of Laves phase and M23C6 phase in two alloy systems of Fe–W and Fe–Cr–C. The numerical simulation of precipitation kinetics was carried on by programming on the MATLAB software, and for the detailed calculation processes, see Sect. 3.3. The equilibrium chemical compositions of precipitates were calculated using the Thermo-calc database TCFE7.

3 Results and discussion

3.1 Characterization of matrix microstructure and precipitates

Figure 2 shows the microstructure characteristics of G115 heat resistance steel after tempering treatment (0 h aging) at different magnifications of OM, SEM and TEM. Like other 9%–12% Cr heat resistance steels, typical tempered martensitic microstructure can be observed, as shown in Fig. 2a, b. The prior austenite grains (PAGs) are divided into numerous substructures by the boundaries of PAGs, martensitic packets and blocks. Figure 2c illustrates that there is a quite large number density of dislocation networks distributing in martensitic laths. The average PAG size and martensitic lath width are measured as 60.6 ± 12.0 μm and 339.3 ± 18.5 nm, respectively. Moreover, plenty of precipitates can be observed at boundaries and in the interior of grain in both SEM and STEM images. As shown in Fig. 2d, approximately spherical and rod-shaped precipitates with the particle size below 100 nm show a tendency to locate at boundaries, which can serve as effective obstacles for pinning dislocations and boundaries [27, 28] and guarantee the microstructural stability during long-term service under high temperature. The preferential distribution of precipitates along grain boundaries is usually affected by the combination of element segregation ability, particle interfacial energy, and the lattice mismatch with the matrix.

Fig. 2
figure 2

Microstructural characteristics under OM (a), SEM (b) and TEM (c, d) in G115 steel after tempering treatment

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In order to investigate the precipitate characteristics in G115 heat resistance steel further, the precipitate morphology after 659 h aging at 650 °C was observed under TEM, as shown in Fig. 3. Three types of precipitates are detected, namely rod-like M23C6 carbides, bulky or irregular-shaped Laves phase, and nearly spherical MX carbonitrides. Most M23C6 carbides and Laves phase are observed near boundaries and dislocations; in contrast, MX carbonitrides are dispersed in the martensitic matrix and have finer particle size. The number density of MX phase is quite less compared with the other two precipitates, which results from the relatively low addition of C and N elements in G115 steel. Therefore, the precipitation behaviors of M23C6 phase and Laves phase will be mainly discussed in this paper.

Fig. 3
figure 3

Precipitate morphology of M23C6 phase (a), Laves phase (b) and MX phase (c) under TEM

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The chemical compositions of main precipitates in G115 steel are identified by EDS analysis, as illustrated in Table 1. M23C6 carbides contain relatively high Cr, Fe, W, C as well as a small amount of V. It is worth pointing out that B element was also detected, but it was not listed in the final statistical table due to its large fluctuation resulted from the background peak. The phenomenon that B element prefers to aggregate in M23C6 phase has also been verified by other scholars [5, 11, 12]; therefore, the molecular formula of M23C6 phase in G115 steel can be summarized as (Fe, Cr, W, V)23(C, B)6. Laves phase is primarily rich in Fe, Cr, W and corresponds to the common molecular formula of (Fe, Cr)2W. As for MX phase, only (V, Nb)C carbides were found in G115 steel and the signals of Fe and Cr from the matrix were also detected owing to their tiny particle size below 50 nm. It has been reported that MX carbides and MX nitrides usually exhibit the nearly spherical and needle-like morphology respectively [5], which is in good consistent with our TEM observations in Fig. 3. In order to grasp the alloying element distribution in M23C6 phase and Laves phase more intuitively, as shown in Fig. 4, the STEM mapping indicates the significantly different enrichment degree of Cr, W and C in two precipitates. It is noteworthy that Laves phases have been reported to prefer to nucleate adjacent to M23C6 phase at micrograin boundaries, which may arise from the segregation of W and Si along boundaries and the enrichment of Si and P around growing M23C6 carbides [29, 30].

Table 1 Chemical composition of main precipitates in G115 steel measured by EDS analysis
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Fig. 4
figure 4

Distribution of alloying elements in M23C6 phase and Laves phase by STEM

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3.2 Statistical analysis of precipitation kinetics

To systematically investigate the nucleation-growth-coarsening kinetics of M23C6 phase and Laves phase during long-term aging exposure, SEM images under SE and BSE modes for the G115 steel specimens aging for 0, 659 and 8791 h at 650 °C are compared, as illustrated in Fig. 5. Two precipitates can be distinguished easily based on the discrepancy in brightness contrast. Only M23C6 phases can be observed after 0 h aging (namely the tempered state), which is determined by the lower precipitation temperature and longer incubation time of Laves phase in comparison to M23C6 phase. Laves phases have occurred at boundaries and near M23C6 phases after aging for 659 h, and its number density is relatively lower than that of M23C6 phase. As the aging time increases to 8791 h, the particle sizes of both M23C6 phase and Laves phase have increased to some extent. Meanwhile, the number density of Laves phase is significantly reduced, which implies the occurrence of precipitate coarsening.

Fig. 5
figure 5

SEM images of M23C6 phase and Laves phase after different aging time at 650 °C. a–c SE mode; d–f BSE mode

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In order to quantitatively analyze the precipitation evolution in G115 steel, the precipitate size distributions (PSDs) of M23C6 phase and Laves phase after different aging times at 650 °C have been measured. Figure 6 shows the PSD curves of M23C6 phase and Laves phase, and it can be noted that both precipitates are in great concordance with the lognormal distribution. Multiple regression analysis [31] has been carried on in order to evaluate the fitting degree of PSD curves with three parameters, namely the average equivalent diameter \(\overline{d}\), the standard deviation σ and the determination coefficient R2, as shown in Table 2. It is worth mentioning that a truncation correction of mean equivalent diameter in terms of a perfect cut has been adopted because the observed precipitates by SEM are truncated by the specimen surface [32]:

$$\overline{d}_{{{\text{corr}}}} = \frac{4}{\uppi }\overline{d}_{{{\text{obs}}}}$$
(1)

where \(\overline{d}_{{{\text{corr}}}}\) and \(\overline{d}_{{{\text{obs}}}}\) represent the corrected average equivalent diameter and the observed average equivalent diameter, respectively.

Fig. 6
figure 6

PSDs of M23C6 phase and Laves phase after different aging time (t) at 650 °C

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Table 2 Evolution of observed average equivalent diameter, corrected average equivalent diameter, standard deviation and determination coefficient with aging time at 650 °C
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According to the fitting results, the corrected average equivalent diameters of M23C6 phase after 0, 659 and 8791 h aged at 650 °C are 78.6 ± 1.5, 102.9 ± 5.8 and 179.7 ± 7.7 nm, respectively. In comparison, those of Laves phase are 0, 172.1 ± 12.1 and 360.5 ± 52.5 nm, respectively. Meanwhile, the standard deviations of M23C6 phase are 0.22 ± 0.02, 0.33 ± 0.07 and 0.32 ± 0.05 nm, respectively, and those of Laves phase are 0, 0.27 ± 0.09 and 0.46 ± 0.17 nm, respectively. Both two types of precipitates exhibit the increasing tendency of standard deviations with aging time, which indicates a greater dispersion degree of particle diameters. Furthermore, the coefficients of determination for M23C6 phase are 0.95, 0.82 and 0.88, respectively, and those of Laves phase after 659 and 8791 h aging are 0.84 and 0.83, respectively. The decline in R2 reflects the decrease in the fitting reliability with the lognormal distribution. The lower R2 of Laves phase could be explained by the appearance of the bimodal phenomenon and the enlarged statistical errors caused by the dramatic decline in the total number density of Laves phase after long-term aging.

After stable nuclei form in the supersaturated matrix, precipitates start to grow up, the driving force of which is the solute diffusion owing to the concentration gradient. The coarsening of precipitates in heat resistance steels is commonly achieved by consuming smaller ones driven by the release of excess interfacial energy and the precipitate size distributions change accordingly, which obeys the Ostwald ripening theory [33]:

$$r_{{2}}^{3} - r_{{1}}^{3} = Kt$$
(2)

where \(r_{{1}}\) and \(r_{{2}}\) stand for the average particle diameters before and after coarsening, respectively; and K is the coarsening rate constant.

Assuming that the mobility matrix is diagonal and the concentration difference of other high mobility components is neglected, K can be theoretically described as:

$$K \sim \frac{{8V_{{\text{m}}}^{\upbeta } \sigma_{\upalpha \upbeta } M_{p} }}{{9(\overline{C}_{p}^{\upbeta } - \overline{C}_{p}^{\upalpha } )^{2} }}$$
(3)

where \(V_{{\text{m}}}^{\upbeta }\), \(\sigma_{\upalpha \upbeta }\), \(M_{{p}}\), and \((\overline{C}_{{p}}^{\upbeta } - \overline{C}_{{p}}^{\upalpha } )\) represent the molar volume of precipitate, the interfacial energy, the mobility of low mobility species \(p\), and the composition gradient at interface, respectively.

The average coarsening rate constants of M23C6 phase and Laves phase with the increasing aging time are calculated further, as shown in Fig. 7. The K values of M23C6 phase during 0–659 and 659–8791 h are calculated as \(3.2 \times 10^{ - 29}\) and \({2}{\text{.0}} \times {10}^{{ - 29}}\) m3/s, respectively. In contrast, the K values of Laves phase are \({2}{\text{.7}} \times {10}^{{ - 28}}\) and \({1}{\text{.8}} \times {10}^{{ - 28}}\) m3/s, respectively. The results indicate that the coarsening rate constants of both M23C6 phase and Laves phase decrease as the aging time increases, which is determined by the reduction of composition gradient at interface due to the stepwise consumption of precipitate-forming atoms. It can be noticed that K values of Laves phase are around an order of magnitude higher than those of M23C6 phase, although Laves phase occurs until the aging stage. The coarsening mechanism of M23C6 phase in 9Cr–W steels has been studied by Abe [34] by comparing two Ostwald ripening modes with experimental results, and the volume diffusion was determined as the dominating factor. In terms of Laves phase, the diffusion of W atoms is generally regarded as the rate controlling process and decides its longer incubation time [35, 36]. Hald and Korcakova’s research [37] implied that the ripening mode controlled by volume diffusion is not sufficient to explain the coarsening behavior of Laves phases in P92 steel, which demonstrates that the higher mobility of W atoms along grain boundaries could accelerate their ripening. Furthermore, the higher interfacial energy together with the lower composition gradient at interface may also contribute to the higher K values of Laves phase according to Eq. (3).

Fig. 7
figure 7

Average coarsening rate constants of M23C6 phase and Laves phase with increasing aging time

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3.3 Numerical simulation of nucleation–growth–coarsening behavior of precipitates

In order to achieve a better understanding on the nucleation–growth–coarsening behavior of precipitates in G115 martensitic heat resistance steel, the precipitation kinetics of Laves phase and M23C6 phase are simulated in two simple alloy systems of Fe–W and Fe–Cr–C, respectively. The MLS model [38, 39] is adopted in this work due to its advantage of less computational load for long-term creep exposure.

The Gibbs free energy of forming unit critical cap-shaped cluster on grain boundary (\(\Delta G\)) can be expressed based on the classical nucleation theory as follows [40]:

$$\Delta G = \frac{4}{3}\uppi r^{3} \Delta G_{{\text{V}}} K(\theta_{{\text{W}}} ) + 2\uppi r^{2} \sigma_{\upalpha \upbeta } L(\theta_{{\text{W}}} ) - \uppi r^{2} \Delta \sigma S(\theta_{{\text{W}}} )$$
(4)

where \(\Delta G_{{\text{V}}}\) is the free energy change of clusters per unit volume, namely the nucleation driving force; \(r\) is the cluster radius; \(\Delta \sigma\) is the difference of the grain boundary energy and the interfacial energy between clusters and grain boundary; and \(K(\theta_{{\text{W}}} )\), \(L(\theta_{{\text{W}}} )\) and \(S(\theta_{{\text{W}}} )\) are functions related to the wetting angle \(\theta_{{\text{W}}}\).

Assuming that the chemical compositions of two precipitates are constant, and they can be determined as \({\text{(Fe}}_{{{0}{\text{.2}}}} {\text{Cr}}_{{{0}{\text{.8}}}} {)}_{{{23}}} {\text{C}}_{{6}}\) and \({\text{Fe}}_{{2}} {\text{W}}\) by Thermo-Calc for 650 °C. The nucleation driving force of precipitate AxBy can be calculated by

$$\Delta G_{{\text{V}}} { = }\frac{{1}}{{V_{{\text{m}}} }}\left[ {\frac{x}{x + y}(\mu_{{\text{A}}}^{\phi } - \mu_{{\text{A}}}^{0} ) + \frac{y}{x + y}(\mu_{{\text{B}}}^{\phi } - \mu_{{\text{B}}}^{0} )} \right]$$
(5)

where \(V_{{\text{m}}}\) is the molar volume; \(\mu_{{\text{A}}}^{\phi }\) and \(\mu_{{\text{B}}}^{\phi }\) are the chemical potentials of A and B in precipitates; \(\mu_{{\text{A}}}^{0}\) and \(\mu_{{\text{B}}}^{0}\) are the chemical potentials of A and B in the matrix; and x and y are the atom numbers of A and B in the chemical formula of precipitate, respectively. According to the regular solution model, Eq. (5) can be further expressed as

$$\Delta G_{{\text{V}}} = \frac{{R_{{\text{g}}} T}}{{V_{{\text{m}}} }}\left[ {\frac{x}{x + y}\ln \left( {\frac{{\overline{c}_{{\text{A}}} }}{{c_{{\text{A}}}^{{\text{e}}} }}} \right){ + }\frac{y}{x + y}\ln \left( {\frac{{\overline{c}_{{\text{B}}} }}{{c_{{\text{B}}}^{{\text{e}}} }}} \right)} \right]$$
(6)

where \(R_{{\text{g}}}\) is the gas constant; \(T\) is the aging temperature; \(\overline{c}_{{\text{A}}}\) and \(\overline{c}_{{\text{B}}}\) are the average matrix compositions; and \(c_{{\text{A}}}^{{\text{e}}}\) and \(c_{{\text{B}}}^{{\text{e}}}\) are the equilibrium matrix compositions.

The time-dependent nucleation rate \(J^{*}\) can be calculated from the equation [41, 42],

$$J^{*} = N_{{\text{V}}} \beta^{*} Z{\text{exp}}\left( { - \frac{{\Delta G^{*} }}{kT}} \right){\text{exp}}\left( { - \frac{\tau }{t}} \right)$$
(7)

where \(N_{{\text{V}}}\), \(\beta^{*}\), \(Z\), \(\Delta G^{*}\), k and \(\tau\) stand for the number of available nucleation sites per unit volume, the attachment rate of atoms to adjacent clusters, the Zeldovich coefficient, the critical nucleation barrier, the Boltzmann constant and the incubation time, respectively.

As the precipitation progresses, the number of available nucleation sites in the martensite tends to decrease, which can be expressed as

$$N_{{\text{V}}} { = }\frac{{1}}{{a^{2} }}(S_{{\text{G}}} - \uppi \overline{R}^{2} N)$$
(8)
$$S_{{\text{G}}} { = }\frac{{1}}{L} + \frac{{1}}{W} + \frac{{1}}{TH}$$
(9)

where \(a\), \(S_{{\text{G}}}\), \(\overline{R}\) and \(N\) correspond to the lattice constant, the lath boundary area per unit volume, the average radius of precipitates, and the number of precipitates, respectively; and \(L\), \(W\) and \(TH\) are the average length, width, and thickness of martensitic laths, respectively.

The MLS model is established on the non-linearized Gibbs-Thomson equation [42], and the precipitate growth rate is

$$\frac{{{\text{d}}\overline{R}_{{{\text{LS}}}} }}{{{\text{d}}t}} = \frac{1}{{c_{{p}} - c_{\upalpha }^{\text{e}} }}\frac{D}{{\overline{R}_{{{\text{LS}}}} }} \cdot \left[ {\overline{c} - c_{\upalpha }^{{{\text{e}}}} \exp \left( {\frac{{2\sigma_{\upalpha \upbeta } V_{{\text{m}}}^{\upbeta } }}{{R_{{\text{g}}} T}}\frac{1}{{\overline{R}_{{{\text{LS}}}} }}} \right)} \right]$$
(10)

where \(c_{{p}}\), \(c_{\upalpha }^{{\text{e}}}\), and \(\overline{c}\) represent the precipitate composition, the equilibrium solute concentration in the matrix and the average solute concentration in the matrix, respectively; \(D\) is the volume diffusion coefficient in the matrix; and \(\overline{R}_{{{\text{LS}}}}\) is the mean size of particles.

The nucleation, growth, and coarsening of precipitates are treated as concomitant processes in the MLS model, and the kinetic parameters can be gained by solving three basic equations [42]:

$$(\xi_{{p}} - \xi )\rho^{3} \cdot n = \xi_{0} - \xi$$
(11)
$$\frac{{{\text{d}}\rho }}{{{\text{d}}\tau ^{\prime}}} + \frac{\rho }{3}\left( {\frac{1}{{\xi_{0} - \xi }} + \frac{{bk_{\sigma } }}{{\rho \ln \xi - k_{\sigma } }} \cdot \frac{1}{\xi \ln \xi }} \right)\frac{{{\text{d}}\xi }}{{{\text{d}}\tau ^{\prime}}} = - j^{*} \cdot \frac{{\rho^{4} }}{3} \cdot \frac{{\xi_{p} - \xi }}{{\xi_{0} - \xi }}$$
(12)
$$\begin{aligned} \frac{{{\text{d}}\rho }}{{{\text{d}}\tau ^{\prime}}} + \frac{{bk_{\sigma } }}{{\xi (\ln \xi )^{2} }} \cdot \frac{{{\text{d}}\xi }}{{{\text{d}}\tau ^{\prime}}} & = \frac{1}{{\xi_{{p}} - 1}} \cdot \frac{1}{\rho }\left[ {\xi - \exp \left( {\frac{{k_{\sigma } }}{\rho }} \right)} \right] +\\ & \rho^{3} \frac{{\xi_{p} - \xi }}{{\xi_{0} - \xi }} \cdot j^{*} \cdot \left( {\frac{{k_{\sigma } }}{\ln \xi } + \delta \rho^{*} - \rho } \right) \\ \end{aligned}$$
(13)

where \(\rho\), \(\rho^{*}\), \(\delta \rho^{*}\), \(\xi\), \(\xi_{{p}}\) and \(\xi_{0}\) are the dimensionless formation of the particle radius, the critical nucleation radius, the nucleation radius difference, the average solute concentration in the matrix, the precipitate composition, and the initial solute concentration in the matrix, respectively; \(n\), \(\tau ^{\prime}\), \(k_{\sigma }\) and \(j^{*}\) are the dimensionless formation of the particle number, the aging time, the interfacial energy and the particle nucleation rate, respectively; \(b\) is the parameter of the apparent density function reflecting the PSDs of particles.

Figure 8 displays the simulation results of nucleation rate and number density of Laves phase and M23C6 phase. As shown in Fig. 8a and c, the nucleation rates first rapidly increase to a certain value and remain stable and then gradually decrease in the later period of aging process. This is because the precipitation continues to consume alloying elements in the matrix, which results in the consequent decline in the chemical driving force of precipitation. Meanwhile, the occupation of available nucleation sites also causes the decrease in nucleation rate. Three different interfacial energies are adopted for both Laves phase and M23C6 phase. The higher interfacial energy of Laves phase has been reported in many experimental and theoretical computational studies as a result of larger lattice fit with the martensite matrix [37, 43]. It can be noticed that both the nucleation rate and particle number density exhibit the tendency of increasing with the decline in interfacial energy. This is determined by the reduced critical nucleation work, which will enhance the nucleation rate exponentially in accordance with Eq. (7). As shown in Fig. 8b and d, the particle number density per unit volume tends to increase during nucleation and growth, though a downward shift emerges accompanied by the Ostwald ripening effect. The transition point occurs later for Laves phase, which is determined by the slow diffusion of W atoms. Furthermore, the features that Laves phase has the longer incubation time and lower maximum number density are well consistent with the SEM observations in Fig. 5.

Fig. 8
figure 8

Simulated nucleation rate (a), number density (b) of Laves phase, nucleation rate (c), and number density (d) of M23C6 phase with increasing aging time

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As illustrated in Fig. 9, the average particle radius and volume fraction of Laves phase and M23C6 phase with the increasing aging time can also be gained. In order to validate the results of numerical simulations, the experimental statistics data as marked with green pentagrams are compared with the prediction curves calculated with the MLS model. Both the average radii of Laves phase and M23C6 phase continue to increase during the aging process, and the elevated interfacial energy corresponds to the larger particle size. Combined with Eq. (3), the increase in interfacial energy is capable of raising the coarsening rate constant and thereby significantly accelerate the coarsening behavior of precipitates. As shown in Fig. 9a, the predicted average radii of Laves phase are well consistent with the experimental values when the interfacial energy takes 0.10 J/m2, which demonstrates that the precipitation kinetics of Laves phase in G115 steel can be explained with the MLS model in Fe-W binary alloy system. It should be mentioned that the interfacial energy is not always a fixed value, which is depended on multiple factors such as precipitate tomography, precipitate size, and the orientation relationship between precipitates and the matrix [44]. Therefore, the interfacial energy here refers to an average value associated with precipitation kinetics using the mean field approach.

Fig. 9
figure 9

Simulated average radius (a), volume fraction (b) of Laves phase, average radius (c), and volume fraction (d) of M23C6 phase with increasing aging time

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As can be seen from Fig. 9c, however, the simulated average particle radius of M23C6 phase in G115 steel is grossly overestimated for the interfacial energies varying from 0.05 to 0.15 J/m2. One of the essential reasons is that the inhibition effect of boron on M23C6 phase coarsening [5, 11] is not considered in the Fe–Cr–C system. In fact, the mechanisms of boron are still controversial until now. Except for segregation around grain boundaries, B atoms also tend to distribute in M23C6 phase by forming more stable M23(C,B)6 phase, which has been verified to have a lower formation energy [10]. Besides, the interfacial energy of Fe(110)/Fe23(C,B)6(111) shows a decreasing tendency with the boron concentration in carbides [10]. Boron is also considered to occupy the vacancies which migrate from the matrix, and suppresses the accommodation of local volume change around M23C6 carbides near PAGBs and hence stabilizes M23C6 phase [45]. Considerable research results [5, 10,11,12] have shown that the coarsening rate of M23C6 phase can be effectively controlled by adding a small amount of boron, though it remains challenging to grasp the inner mechanism. The modification of diffusion coefficients of solute atoms is necessary if predicting the precipitation kinetics of M23C6 phase with the MLS model. Figure 9b and d shows the simulation results of volume fraction for Laves phase and M23C6 phase, respectively. The predicted volume fraction of Laves phase maintains increasing as aging goes on and eventually tends to be steady after around 104 h.

4 Conclusions

  1. 1.

    Typical tempered martensitic microstructure can be observed in G115 steel after heat treatment, and the average PAG size and martensitic lath width are measured as 60.6 ± 12.0 μm and 339.3 ± 18.5 nm, respectively. Three types of precipitates including rod-like M23C6 carbides, bulky or irregular-shaped Laves phase, and nearly spherical MX carbides are detected after 659 h aging at 650 °C. The molecular formulas of M23C6 phase, Laves phase and MX phase are summarized as (Fe, Cr, W, V)23(C, B)6, (Fe, Cr)2W and (V, Nb)C by EDS analysis.

  2. 2.

    Laves phases can only be observed in G115 steel until 659 h aging at 650 °C, and their longer incubation time is associated with the slow diffusion of W atoms. Both the PSDs of M23C6 phases and Laves phases obey the lognormal distribution. The multiple regression analysis demonstrates that Laves phases have larger average equivalent diameters, larger standard deviations, and lower coefficients of determination. Besides, the average coarsening rate constant of Laves phases is calculated as approximately one order of magnitude higher than M23C6 carbides, which could be accelerated by the rapid grain boundary diffusion.

  3. 3.

    The nucleation–growth–coarsening kinetic parameters including nucleation rate, number density, average radius, and volume fraction of Laves phase and M23C6 phases are simulated and compared in Fe–W and Fe–Cr–C system, respectively. The kinetic evolution of Laves phase can be well predicted with the interfacial energy of 0.10 J/m2. Nonetheless, the neglect of the inhibitory effect from boron on coarsening causes the overestimated average radius and volume fraction of M23C6 carbides.