A. Nucleation

According to the classical nucleation theory, nucleation is defined as the formation of particles from subcritical size to supercritical size.^{Reference Mittemeijer22} During nucleation, the production of new phase releases chemical Gibbs energy whereas the creation of the interfaces costs Gibbs energy. Precipitation could be governed by various nucleation modes, including continuous nucleation, site saturation nucleation, and the intermediate nucleation called mixed nucleation or Avrami nucleation.^{Reference Liu, Sommer, Bos and Mittemeijer12,Reference Avrami23–Reference Avrami25} The nucleation of M₂₃C₆ is typically heterogeneous, which occurs along martensite laths or prior austenite grain boundaries. In this case, the interface energy is lower and the critical size is larger than that in homogeneous nucleation. As the nucleation rate decreases drastically with the decline of solute saturation caused by solute redistribution,^{Reference Fan, Liu, Liu, Zhang, Yang and Zhou20} the nucleation in diffusion-controlled transformation, such as the precipitation of M₂₃C₆, is usually assumed to be dominated by site saturation nucleation, which indicates that there already are a number of preexisting nuclei (supercritical particles of the new phase) at t = 0 and the further nucleation rate is zero.^{Reference Liu, Nitsche, Sommer and Mittemeijer7}

B. Growth

Generally, two kinds of growth are considered: volume-diffusion-controlled growth which occurs in phase transformations where long-range compositional changes take place; and interface-diffusion-controlled growth which occurs as the growth is determined by atomic processes at and near the interfaces. The growth of M₂₃C₆ is a typical volume-diffusion-controlled one, which is characterized by solute diffusion in the growth front (precipitation/matrix interface). The growth of M₂₃C₆ in T/P92 steel can be decelerated arising out of the segregation of substitutional alloying elements such as chromium (Cr), tungsten (W), and molybdenum (Mo)etc.,^{Reference Chen and van der Zwaag26} at and near the interfaces. The retardation effect had been found in γ-α transformation in ternary Fe–C–M alloys by applying the local equilibrium (LE) model, in which both carbon (C) and M can have partitioning, where M denotes the sum of carbide-forming metal elements.^{Reference Chen, Appolaire and van der Zwaag27,Reference Chen and van der Zwaag28} However, it is complex to deal with the LE model because there are so many kinds of substitutional alloying elements in this steel. Therefore, this work is based on the assumption that substitutional alloying elements are immobile, i.e., the precipitation of M₂₃C₆ is controlled only by the diffusion of C, and the model would possibly overestimate the transformation kinetics, as the segregation of alloying element into interface will cause dissipation of Gibbs energy.

Accordingly, two types of precipitation are classified by the diffusion direction: positive precipitation, where the solution atoms diffuse into the growth front; and negative precipitation, where the solute atoms diffuse away from the moving interface.^{Reference Zener21} The carbide precipitation in α-Fe is typically positive, the schematic diagram of the diffusion profile is illustrated in Fig. 1, where C _β and C _α are the equilibrium concentration in the after-transformation precipitate and matrix, respectively, C _m is the saturated composition in the nontransformed single phase alloy, and r ₀ is the position of the growth front.

FIG. 1.

Schematic image of diffusion profile in positive precipitation.

The growth of two adjacent particles is divided into three types by the number of dimensions, as illustrated in Fig. 2. In one-dimensional case, r ₀ is the half width of the plate, in two-dimensional case, r ₀ is the radius of the cylinder, and in the case of three-dimensional growth, r ₀ is the radius of the spherical particle with 2l being the distance of the two particles.

FIG. 2.

Schematic image of the three kinds of growth mode: (a) 1D case; (b) 2D case; (c) 3D case.

The diffusion of solute atoms can be described by the classical diffusion equation:

(1)$${{\partial C} \over {\partial t}} = D\nabla ^2 C + \left( {\left( \,{p - 1} \right)\left( {{D \over r}} \right)} \right)\left( {\nabla C} \right)\,\quad,$$

where p equals 1, 2, or 3 for 1D, 2D, and 3D growth, respectively and D is the temperature-dependent diffusion coefficient of solute atoms, which is constant in isothermal transformation. C is the concentration of solute atoms, which varies as a function of the position of growth front r and the transformation time t. Meanwhile, r also varies as a function of time. The numerical solution of Eq. (1) can be obtained under boundary conditions: C(r ₀, t) = C _α, C(r, 0) = C _m and $\left( {{{\partial C} \mathord{\left/ {\vphantom {{\partial C} {\partial r}}} \right. \kern-\nulldelimiterspace} {\partial r}}} \right)_{r = l} \,\,= 0$, with l representing the position of impingement and being equal to half distance of two adjacent particles. Using this numerical solution and the value of growth coefficient λ (obtained by fitting, see more details in Sec. IV), the real concentration of M₂₃C₆ precipitation in steels can be obtained, as shown in Fig. 3. With time prolonging, the position of growth front advances and the slope of concentration curve in front of the interface declines as a function of distance away from the interface. If time duration is sufficiently long, the concentration of solute atoms at r = l is less than C _m, suggesting the occurrence of the overlapping of diffusion fields, i.e., the occurrence of soft impingement.

FIG. 3.

The diffusion profile of M₂₃C₆ precipitation in T92 steel, which is obtained by combination of numerical solution of diffusion equation and growth coefficient.

C. Soft impingement model with linear concentration profile

For simplifying the soft impingement model, a linear approximation of concentration profile needs to be conducted. Figure 4 illustrates the linear approximation of the concentration profile in front of the interface and the transition of diffusion fields from nonoverlapping (i.e., nonimpingement) to overlapping (i.e., impingement). r _c is the critical particle size in soft impingement. Before impingement, the position of growth front does not reach r _c, and the growth of precipitates follows the square root rule.^{Reference Wert and Zener29} When the position of growth front exceeds r _c, soft impingement takes place, and the overlapping of diffusion fields will retard the growth velocity due to the interference between particles. After the occurrence of soft impingement, C _m (t) will change from C _m to C_α with prolongation of time. When the position of growth front reaches r _f, C _m (t) equals C_α, and the transformation will cease as a result of the disappearance of supersaturation.

FIG. 4.

Schematic image of linear approximation of the diffusion profile, where r _c is the critical point of impingement, r _f is the end point of impingement t _c and t _f are the time corresponding to r _c and r _f respectively.

As shown in Fig. 4, the initiation of soft impingement at r = r _c can be obtained according to the conservation law of mass requiring^{Reference Fan, Liu, Liu, Zhang, Yang and Zhou20}:

(2)$$r_{\rm{c}} ^{\,\,\,p} \left( {C_{\rm{\beta }} - C_m } \right) = \left( {l^{\,p} - r_{\rm{c}} ^{\,\,\,p} } \right)\left( {{{C_m - C_{\rm{\alpha }} } \over 2}} \right)\quad,$$

where p equals 1, 2, or 3 for 1D, 2D, and 3D growth, respectively. A rearrangement of Eq. (2) gives

(3)$$r_{\rm{c}} = \left( {{{C_m - C_{\rm{\alpha }} } \over {2C_{\rm{\beta }} - C_{\rm{\alpha }} - C_m }}} \right)^{{1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p}} l\quad.$$

Equation (3) can be used to determine the initial point of soft impingement.

Similarly, the end of transformation at r = r _f can also be obtained according to the conservation law of mass requiring:

(4)$$r_{\rm{f}} ^{\,\,\, p} \left( {C_{\rm{\beta }} - C_m } \right) = \left( {l^{\,p} - r_{\rm{f}} ^{\,\,\,p} } \right)\left( {C_m - C_{\rm{\alpha }} } \right)\,\quad,$$

where p equals 1, 2, or 3 for 1D, 2D, and 3D growth, respectively. And a rearrangement of Eq. (4) gives

(5)$$r_{\rm{f}} = \left( {{{C_m - C_{\rm{\alpha }} } \over {C_{\rm{\beta }} - C_{\rm{\alpha }} }}} \right)^{{1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p}} l\,\quad.$$

Taking the supersaturation degree as δ = (C _m − C _α)/(C _β − C _α) as a reference, Eq. (5) can also be rewritten as:

(6)$$r_{\rm{f}} = {\rm{\delta }}^{{1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p}} l\,\quad.$$

Thus, Eq. (6) can be used to determine the initial point of soft impingement.

D. Growth before soft impingement

As there is no overlapping of diffusion fields, the growth before soft impingement follows the square root rules, and can be described by Zener’s equation^{Reference Zener21}:

(7)$$r = {\rm{\lambda }}_p \left( {Dt} \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} \,\quad,$$

where λ_p is the time-independent growth coefficient with p as 1, 2, or 3 for 1D, 2D, and 3D growth. Zener had hypothesized that the growth coefficient λ_p depended only upon the supersaturation degree δ. However, λ_p is not a simple function of δ, and it varies from 0 to ∞ as δ varies from 0 to 1. If applied to a complex system, the value of λ_p would be difficult to obtain.

According to Eq. (7), the initial time of soft impingement t _c can be calculated by:

(8)$$t_{\rm{c}} = {{\left( {{{r_{\rm{c}} } \mathord{\left/ {\vphantom {{r_{\rm{c}} } {\lambda _p }}} \right. \kern-\nulldelimiterspace} {\lambda _p }}} \right)^2 } \over D}\,\quad.$$

The transformed fraction f before impingement can be written as:

(9)$$f = \left( {{r \over l}} \right)^p = \left( {{{{\rm{\lambda }}_p (Dt)^{{1 \over 2}} } \over l}} \right)^p \quad\,.$$

By substituting and rearranging Eq. (8) into Eq. (10), another form of f can be obtained with r _f as a reference:

(10)$$f = \left( {{{{\rm{\lambda }}_p (Dt)^{{1 \over 2}} } \over {r_{\rm{f}} }}} \right)^p {1 \over {\rm{\delta }}}\,\quad,$$

where f varies from 0 to 1.

According to Eqs. (3) and (9), the transformed fraction at the initial point of impingement can be given as:

(11)$$f_{\rm{c}} = \left( {{{r_{\rm{c}} } \over l}} \right)^p = {{C_m - C_{\rm{\alpha }} } \over {2C_{\rm{\beta }} - C_{\rm{\alpha }} - C_m }} = {{\rm{\delta }} \over {2 - {\rm{\delta }}}}\,\quad,$$

suggesting that the transformed fraction at the initial point of impingement depends only on the supersaturation degree δ.

E. Growth after soft impingement

When the position of growth front exceeds r _c, i.e., r ₀ ≥ r _c, soft impingement will take place. The conservation law of mass is still applicable to the determination of r ₀:

(12)$$r_0 ^{\,\,\,p} \left( {C_{\rm{\beta }} - C_m } \right) = \left( {l^{\,p} - r_0 ^{\,\,\,p} } \right)\left( {{{2C_m - C_{\rm{\alpha }} - C_m \left( t \right)} \over 2}} \right)\quad\,.$$

A rearrangement of Eq. (12) gives:

(13)$$r_0 = \left( {{{2C_m - C_{\rm{\alpha }} - C_m \left( t \right)} \over {2C_{\rm{\beta }} - C_{\rm{\alpha }} - C_m \left( t \right)}}} \right)^{{1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p}} l\,\quad,$$

where C _m (t) is a complex function of time. However, Fan et al.^{Reference Fan, Liu, Liu, Zhang, Yang and Zhou20} have developed the expression for C _m (t). Thus if we take r _f as a reference, the transformed fraction f after soft impingement can be given as:

(14)$$f = 1 - \left( {1 - \left( {{{r_{\rm{c}} } \over {r_{\rm{f}} }}} \right)^p } \right)\exp \left( { - k\left( {t - t_{\rm{c}} } \right)} \right)\,\quad,$$

where $k = \left( {{\textstyle{{pD} \over 2}}} \right)\left( {{\textstyle{{{\rm{\lambda }}_p } \over {r_{\rm{f}} }}}} \right)^p \left( {Dt_1 } \right)^{{{\left( {p - 2} \right)} \over 2}} \left( {1 - \left( {{\textstyle{{r_{\rm{c}} } \over {r_{\rm{f}} }}}} \right)^p } \right)^{ - 1}\quad.$

F. Transformed fraction of the entire transformation

With the combination of Eqs. (10) and (14), the model of transformed fraction as a function of time during the whole transformation process can be written as a piecewise function:

(15)$$\!\!\!\left\{ \matrix{ f = {1 \over {\rm{\delta }}}\left( {{{{\rm{\lambda }}_p (Dt)^{{1 \over 2}} } \over l}} \right)^p ,\,{\rm{ }}t \,\le\, t_{\rm{c}} \hfill \cr f = 1 - \left( {1 - \left( {{{r_{\rm{c}} } \over {r_{\rm{f}} }}} \right)^p } \right)\exp \left( { - k\left( {t - t_{\rm{c}} } \right)} \right),\,{\rm{ }}t > t_{\rm{c}} \hfill \cr} \right.\,\quad,$$

where $k = \left( {{\textstyle{{pD} \over 2}}} \right)\left( {{\textstyle{{{\rm{\lambda }}_p } \over {r_{\rm{f}} }}}} \right)^p \left( {Dt_1 } \right)^{{{\left( {p - 2} \right)} \over 2}} \left( {1 - \left( {{\textstyle{{r_{\rm{c}} } \over {r_{\rm{f}} }}}} \right)^p } \right)^{ - 1}$, and f varies from 0 to1.

A. Effect of l on the precipitation of M₂₃C₆

l is an important parameter in the precipitation model. A higher l value means a lower nuclei number existing in site saturation at the beginning of the isothermal transformation, according to the definition of site saturation nucleation.^{Reference Liu, Nitsche, Sommer and Mittemeijer7} The value of l measured from TEM images is approximately equal to the width of martensite lath. As the M₂₃C₆ carbides are nucleated along martensite laths and former austenite boundaries, the value of l varies with different initial microstructure conditions, such as different lath width or prior austenite grain size. The smaller the lath width or austenite grain size is, the smaller the value of l is. Upon Eq. (15), and fitted λ₂ in the case of 2D, the transformed fraction as a function of time can be predicted with different values of l, as shown in Fig. 6, and the calculated t _c and r _f are listed in Table VI. A higher value of l leads to a longer transformation time, i.e., the postponing of the initial time of soft impingement and the entire transformation time. Moreover, the position of growth front at the end of transformation r _f increases significantly as l increases, suggesting that the final particle size of precipitates at the end of transformation is larger due to a larger grain size or lath width, and meanwhile, the distribution density is lower according to the basis of site saturation nucleation. Therefore, fine grain size is conducive to the dispersion of M₂₃C₆ precipitates.

FIG. 6.

Predicted curve of f-t with different value of l in case of 2D.

TABLE VI.

The change of calculated parameters as a function of different l.

B. Prediction of M₂₃C₆ precipitation at different temperatures

Zener had suggested that the growth coefficient λ_p is temperature-independent,^{Reference Zener21} and accordingly, the precipitation of M₂₃C₆ can be predicted using Eq. (17) with the fitted value of λ_p. The prediction of transformed fraction as a function of time at different temperatures is shown in Fig. 7(a). The transformation time is prolonged with decreasing of holding temperature because of the lowering of the diffusion coefficient D of carbon. Then, the time-temperature-transformation (TTT) diagram of M₂₃C₆ precipitation can be obtained from the f-t curve under different temperatures, as shown in Fig. 7(b): the onset and the end time are corresponding to the precipitation fraction 2% and 98% respectively. However, comparing with the data calculated by Robson and Bhadeshia,^{Reference Robson and Bhadeshia33} the slope of the predicted TTT diagram is steeper, which means that the onset time of precipitation is delayed at high temperatures and advanced at low temperatures. This phenomenon is due to the temperature-independence of growth. The classic growth coefficient of partitioning phase transformations in binary alloys is derived by solving the diffusion equations^{Reference Chen and van der Zwaag14}:

(16)$${\rm{\lambda }} = \sqrt {{{2\left( {n - 1} \right)\left( {1 - {\rm{\delta }}} \right)^2 } \over {n{\rm{\delta }}}}} ,\,n > 1\,\quad,$$

FIG. 7.

Prediction of M₂₃C₆ precipitation in T92 steel: (a) The f-t curve with different temperature in case of 2D, where the growth coefficient is time-independent; (b) TTT diagram of M₂₃C₆ precipitation based on the f-t curve and the comparison with the data in Ref. Reference Robson and Bhadeshia33, where an obvious discrepancy can be found.

when n = 2, it would become linear approximation, when n = 3, the diffusion field would be quadratic. This equation suggests that the growth coefficient is dependent only on the degree of supersaturation.

Therefore, a temperature-dependent growth coefficient ${\rm{\lambda }}_p^\prime$ is taken into account for reducing the discrepancy. The growth coefficient at different temperatures can be described by Arrhenius equation:

(17)$${\rm{\lambda }}_p^\prime = {\rm{\lambda }}_0 \exp \left( {{{ - Q_{\rm{\lambda }} } \over {RT}}} \right)2\,\quad,$$

where λ₀ is the preexponential factor, and Q _λ is the activation energy of growth. It is obvious that ${\rm{\lambda }}_p^\prime$ increases with the elevation of temperature by comparing Eqs. (16) and (17). The final equation with temperature-dependent growth coefficient can be obtained by substituting Eq. (17) into Eq. (15):

(18)$$\!\!\!\left\{ \matrix{ f = {1 \over {\rm{\delta }}}\left( {{{{\rm{\lambda }}_0 \exp \left( {{{ - Q_{\rm{\lambda }} } \over {RT}}} \right)\left( {Dt} \right)^{{1 \over 2}} } \over l}} \right)^p ,\,t \,\le\, t_{\rm{c}} \hfill \cr f = 1 - \left( {1 - \left( {{{r_{\rm{c}} } \over {r_{\rm{f}} }}} \right)^p } \right)\exp \left( { - k\left( {t - t_{\rm{c}} } \right)} \right),\,t > t_{\rm{c}} \hfill \cr} \right.\quad\,.$$

By fitting Eq. (18) with experimental data in 2D case, the value of λ₀ and Q _λ can be obtained, and the calculated initial time of soft impingement t _c and the SD are listed in Table VII. As the growth of M₂₃C₆ is controlled by carbon diffusion, the activation energy of M₂₃C₆ growth is nearly equal to the diffusion activation energy of carbon in martensite. Similarly, the prediction of f-t curve can be given by Eq. (18) and then the TTT diagram is obtained based on the f-t curve at different temperatures, as shown in Fig. 8. It is shown that, the onset time of precipitation advanced at elevated temperature and delayed at low temperature, by comparing with Fig. 7. With the change of growth coefficient ${\rm{\lambda }}_p\!\!\!\prime\,$, the predicted TTT diagram is consistent with the data calculated by Ref. Reference Robson and Bhadeshia33. The results suggest that, the assumption for the temperature-dependent growth coefficient is reasonable.

FIG. 8.

Prediction of M₂₃C₆ precipitation in T92 steel with temperature-dependent ${\rm{\lambda }}_p^\prime$: (a) The f-t curve with different temperature in case of 2D; (b) TTT diagram of M₂₃C₆ precipitation based on the f-t curve and the comparison with the data in Ref. Reference Robson and Bhadeshia33, where an agreement is achieved.

TABLE VII.

The calculated parameters with a temperature-dependent growth coefficient by fitting Eq. (17) with experimental data at 650 °C in 2D case. Notes that, Q _G is approximately equal to diffusion coefficient of carbon in martensite, and ${\rm{\lambda }}_p^\prime$ is approximately equal to λ_p at 650 °C.