Mathematical Modelling of Isothermal Decomposition of Austenite in Steel

: The main goal of this paper is mathematical modelling and computer simulation of isothermal decomposition of austenite in steel. Mathematical modelling and computer simulation of isothermal decomposition of austenite nowadays is becoming an indispensable tool for the prediction of isothermal heat treatment results of steel. Besides that, the prediction of isothermal decomposition of austenite can be applied for understanding, optimization and control of microstructure composition and mechanical properties of steel. Isothermal decomposition of austenite is physically one of the most complex engineering processes. In this paper, methods for setting the kinetic expressions for prediction of isothermal decomposition of austenite into ferrite, pearlite or bainite were proposed. After that, based on the chemical composition of hypoeutectoid steels, the quantiﬁcation of the parameters involved in kinetic expressions was performed. The established kinetic equations were applied in the prediction of microstructure composition of hypoeutectoid steels.


Introduction
The research of the mathematical simulation of microstructure distribution in steel is one of the highest-priority research areas in the simulation of phenomena of the heat treatment of steel. By using the additivity rule and kinetic equations of isothermal decomposition of austenite, it is possible to calculate kinetics of austenite decomposition at continuous cooling of steel. The prediction of isothermal decomposition of austenite can be applied for understanding, optimization and control of microstructure composition and mechanical properties of steel [1][2][3][4].
The most common method of computer prediction of isothermal decomposition of austenite results is based on the chemical composition of steel by using time-temperaturetransformation (TTT) diagrams [5].
Studies of the kinetics of isothermal decomposition of austenite have been intensified in the course of some pioneering studies on the isothermal decomposition of austenite [6][7][8].
The prediction of microstructure composition is usually based on semi-empirical methods derived from kinetic equations of microstructure transformation [9]. To describe the transformation kinetics by mathematical methods, a semi-empirical approach is employed using the Johnson-Mehl-Avrami-Kolmogorov (JMAK) equation together with additivity rule [10,11].
The phase transformations can be categorized into two categories: reconstructive phase transformations and displacive phase transformations. Decompositions of austenite into ferrite and pearlite in steels are typical examples of reconstructive phase transformations, while martensite, bainite, and Widmanstatten ferrite phase transformations can be recognized as displacive phase transformations [12]. The formation of ferrite occurs by nucleation at the austenite grain boundaries. After that the ferrite grows inside the austenite grains. The rate of volume fraction of the ferrite is a function of the nucleation rate and the velocity of the ferrite/austenite interface. The nucleation rate is primarily a function of the undercooling below the A e3 temperature and the grain size of austenite [13,14].
The nucleation mechanism of pearlite involves the formation of two phases, ferrite and cementite. The nucleation of cementite is a rate-limiting step in hypoeutectoid steels. The proeutectoid ferrite nucleates first and continues to grow with the same crystallographic orientation during the pearlite formation. For hypereutectoid steels, the role of the nucleation of ferrite is a limited process in comparison with the roles of the cementite nucleation. In eutectoid steel, the pearlite nucleation is assumed to occur at the austenite grain corners, edges, and boundaries.
Two different theories are proposed for the growth of pearlite. The Zener-Hillert theory assumes that the volume diffusion of carbon in the austenite is the rate-controlling mechanism [15,16]. In addition, Hillert theory assumes that grain boundary diffusion of the carbon atoms is the rate-controlling mechanism. The nucleation rate of pearlite follows the general nucleation theory [16].
The Johnson-Mehl-Avrami-Kolmogorov (JMAK) theory predicts the overall transformation rate on the basis of nucleation and growth rates. It is the most widely used model to describe the austenite-pearlite transformation kinetics [15].
Bainite was discovered nearly eight decades ago [17]. The research work carried out in the field of bainite is immense [17][18][19][20]. A qualitative theory to explain bainite formation still remains a subject of controversy [18,21]. One theory suggests a diffusion-controlled transformation where bainitic growth occurs by a diffusional ledge mechanism, while the other suggests that the bainite reaction is a displacive transformation [18]. Both theories have assumed models to predict the transformation kinetics [22,23]. The growth of bainite and Widmanstatten ferrite requires the partitioning of interstitial carbon. Because of this reason, their growth is controlled by diffusion of interstitial atoms of carbon [12].
Computer simulation of isothermal decomposition of austenite in steel is still a complex problem. The dependence of the physical quantities involved in the kinetic expressions of austenite decomposition has not yet been sufficiently defined in the literature. Efforts are being made to predict the dependence of physical quantities in the kinetic expressions of austenite decomposition on chemical composition. This work proposes inversion methods for quantification of kinetic parameters and setting the kinetic expressions in the prediction of isothermal decomposition of austenite into ferrite, pearlite, or bainite.
The proposed method of setting kinetic relations can be used in the calculation of characteristic kinetic parameters for other groups of steel. The established model can be used for computer simulation of austenite decomposition in other steels with similar chemical compositions.

Materials and Methods
where X is transformed part of the microstructure, t is time, and k and n are kinetic parameters. By extracting the time component, Equation (1) can be written as: In computer-based mathematical analysis, it is convenient to define the kinetics of austenite decomposition in an incremental form. By differentiating Avrami s equation, it follows that: After introducing Equation (2) in Equation (3) and a short rearrangement, it follows that: Equation (4) can be written in an incremental form, and the volume fraction ∆X (N) of austenite transformed in the time interval ∆t (N) can be calculated as [3]: where X (N−1) is the volume fraction of austenite transformed in previous N − 1 time intervals. Kinetic parameters k and n can be evaluated inversely by using data of time of isothermal transformation. The total volume fraction of austenite transformed during isothermal decomposition can be calculated as:

Ferrite Transformation
The ferrite transformation takes place by the mechanism of nucleation and growth, with the following assumed kinetic parameters [13]: where S is the surface of austenite grain suitable for nucleation, while I F is the nucleation rate and G F is the growth rate defined as: In Equations (9) and (10), T is temperature, ∆T is the undercooling below the critical temperature A e3 , D 0 is the material constant, R is the universal gas constant, Q dif is the diffusion activation energy, while c 0 , c α , and c γ are the concentrations of steel, ferrite and austenite at the boundary with ferrite, respectively. The effective diffusion length is defined as [7,13]: where k 1 , k 2 and n 1 are the kinetic parameters dependent on chemical composition of steel. After introducing Equations (9)- (11) in Equation (8), and after some modification, Equation (8) can be rewritten as [14]: To determine the values of the constants k 1 , k 2 and n 1 , it is first necessary to determine the value of the coefficient k F for three temperatures, and then to solve a system of three equations with three unknowns (k 1 , k 2 , n 1 ).
It was assumed that the ferrite transformation does not take place to the end, but to the maximum volume V max = V rF × V, when the normalized volume fraction of ferrite can be defined as [18]: In Equation (13), X F is the volume fraction of ferrite and V rF is the relative volume of ferrite. The linear temperature dependence of the volume V rF can be evaluated using an Fe-Fe 3 C diagram with the following assumptions: at temperatures A e3 and B s , the volume V rF is equal to 0, while at temperature A e1 , it takes the maximum value V rF = c 0 /0.8 ( Figure 1). where: where: (19) Equation (8) can be rewritten as [14]: To determine the values of the constants k1, k2 and n1, it is first necessary to determine the value of the coefficient kF for three temperatures, and then to solve a system of three equations with three unknowns (k1, k2, n1).
It was assumed that the ferrite transformation does not take place to the end, but to the maximum volume Vmax = VrF × V, when the normalized volume fraction of ferrite can be defined as [18]: In Equation (13), XF is the volume fraction of ferrite and VrF is the relative volume of ferrite. The linear temperature dependence of the volume VrF can be evaluated using an Fe-Fe3C diagram with the following assumptions: at temperatures Ae3 and Bs, the volume VrF is equal to 0, while at temperature Ae1, it takes the maximum value VrF = c0 / 0.8 ( Figure  1). where: where: The real volume of ferrite can be written as: The real volume of ferrite can be written as: The extended volume of ferrite is defined as: where t i is the incubation time. After introducing Equation (22) in Equation (21), it follows that: After integrating Equation (25), it follows that: For small values of the normalized volume fraction of ferrite, Equation (26) can be written as: At any temperature, knowing an incubation time of ferrite transformation, found out from the IT diagram, the value of the kinetic parameter k F can be written as: From Equation (26), follows the normalized volume fraction of ferrite which is: Equation (29) can be written in an incremental form, when the normalized volume fraction of ferrite formed by the mechanism of nucleation and growth in the time interval ∆t (N) can be calculated as: At any time of transformation, the real volume fraction of ferrite can be calculated as (13)).

Bainite Transformation
The bainite transformation begins at a temperature B s . Like ferrite transformation, it does not take place to the end, but to the maximum volume V max = V rB × V, when the normalized volume fraction of bainite can be defined as [18]: where V rB is the relative volume of bainite, while X B is the volume fraction of bainite defined as: where u is the volume of the structural unit of bainite. The nucleation rate is defined as [14]: where k 6 and k 7 are the kinetic parameters dependent on chemical composition of steel. After introducing Equation (33) to Equation (32), it follows that: If the following is accepted [24]: Equation (36) can be rewritten as: where k 8 is the kinetic parameters. The time of bainite transformation can be expressed by: For small values of the normalized volume fraction of bainite it can be taken that ln(1 − ξ B )V rB ≈ ln(1 − X B ); therefore, Equation (38) can be rewritten as: As a rule, at low temperatures austenite is completely transformed into bainite, when it can be assumed that V rB ≈ 1 and ξ B ≈ X B. With this assumption, the kinetic parameter n B can be defined as: where t i is the incubation time and t 0.99 is the finish time of the isothermal bainite transformation found out from the IT diagram. The denominator of Equation (39) is a function of temperature; therefore, for the incubation time and constant temperature, Equation (39) can be rewritten as: At any temperature, knowing an incubation time of bainite transformation, found out from the IT diagram, the value of the kinetic parameter k B can be expressed by: For 99% of austenite transformed into bainite, Equation (38) can be written as: where the linear temperature dependence of the volume V rB is assumed: Coefficients a 5 and a 6 can be determined by corresponding values of the volume V rB on two different temperatures in IT diagram. Based on Equations (39) and (41), the kinetic parameter k B can be written as: With the previously determined kinetic parameter n B , the defined temperature dependence of the volume V rB and with the known values of the constants k 6 , k 7 and k 8 , the kinetics of the bainite transformation is completely defined. To determine the values of the constants k 6 , k 7 and k 8 , it is first necessary to determine the value of the coefficient k B for three temperatures, and then to solve a system of three equations with three unknowns.
Based on Equations (46) and (47), the volume fraction of bainite and the normalized volume fraction of bainite can be determined by: Equation (47) can be written in an incremental form, when the normalized volume fraction of bainite formed in the time interval ∆t (N) can be calculated as: At any time of transformation, the real volume fraction of bainite can be calculated as X B = ξ B V rB (Equation (31)).

Pearlite Transformation
In the remaining undercooled austenite that has not transformed into ferrite or bainite, at temperatures lower than A e1 , the pearlite transformation takes place. The kinetics of pearlite transformation is independent of the kinetics of the previous ferrite or bainite transformation. At temperatures A e1 > T ≥ B s the remaining volume available for pearlitic transformation is V rP1 ·V, while at temperatures T < B s the remaining volume is For pearlite transformation by the mechanism of nucleation and growth, the following kinetic parameters are assumed: where S is surface of austenite grain suitable for nucleation, while I P is the nucleation rate and G P is the growth rate defined as [14]: In Equations (51) and (52), ∆T is the undercooling below the critical temperature A e3 , while c γα and c γFe3C are the concentrations of austenite at the boundary with ferrite and cementite, respectively. k 3 is the kinetic parameters dependent on chemical composition. After introducing Equations (51) and (52) into Equation (50) and after some modifications, it can be rewritten: To determine the values of the constants k 3 , k 4 and k 5 , it is first necessary to determine the value of the coefficient k P for three temperatures, and then to solve a system of three equations with three unknowns.
The real volume of pearlite can be written as: where the normalized volume fraction can be expressed by: The extended volume of pearlite is defined as: After introducing Equation (56) into Equation (55), it follows that: After integrating Equation (59), it follows that: For small values of the normalized volume fraction of perlite, Equation (60) can be written as: At any temperature, knowing the incubation time of pearlite transformation, found out from the IT diagram, the value of the kinetic parameter k P can be written as: From Equation (60) follows the normalized volume fraction of pearlite, which is: Metals 2021, 11, 1292 9 of 14 As for ferrite and bainite transformation, Equation (63) can be written in an incremental form. The normalized volume fraction of pearlite formed by the mechanism of nucleation and growth in the time interval ∆t (N) can be calculated by: The presented method for estimation of k P and ξ P is also valid at temperatures lower than B s . In that case, in the above equations, the relative volume V rP1 should be replaced by the relative volume V rP2 .

Materials
With the aim of qualitatively and quantitatively defining the influence of chemical composition on the isothermal decomposition of austenite, the values of kinetic parameters were investigated on a number of hypoeutectoid, low-alloy steels [25]. Their composition is shown in Table 1.

Results
Section 2.1 presents methods for estimating kinetic parameters, which completely define the kinetics of austenite isothermal decomposition into ferrite, pearlite and bainite. The calculated values of the kinetic parameters depend on the chemical composition, i.e., they are valid only for one steel.
The dependence of kinetic parameters of ferrite, pearlite and bainite transformation on the content of carbon, chromium, molybdenum and nickel was estimated by regression analysis (Equations (68)-(78)). Because of the similar content of manganese and silicon in studied steels, these elements were not included in the regression analysis. Based on the proposed equations, the kinetic parameters involved in mathematical model of ferrite, pearlite and bainite transformation can be calculated for any other chemical composition of hypoeutectoid, low-alloy steels ( Table 2).

Discussion
The values of kinetic parameters given in Table 2 were verified by comparing the modeled curves of the isothermal transformation (IT) diagram of steel 42CrMo4, 36Cr6, Ck45, and 28NiCrMo74 with those obtained experimentally. In Figures 2-5, the dashed lines show the experimental IT diagram, while the mathematically determined times of start (incubation time) and times of finish of the isothermal austenite decomposition, t 0.01 and t 0.99 , are shown by solid lines. Additionally, Figure 2 shows curves corresponding to bainite volume fraction of 25%, 50%, 75% and 90%.             Kinetic parameters of ferrite, bainite and pearlite transformation, k F , k B and k P , were calculated by Equations (12), (45) and (53), respectively. Other physical quantities used in the developed mathematical model are shown in Table 3. Table 3. Physical quantities used in modelling of austenite isothermal decomposition.

Quantity Value
Units Description Since the developed model is written in incremental form, it is suitable for predicting austenite decomposition during the continuous cooling of steel using Scheil s additivity rule. Additionally, it is very easy to extend this approach in the prediction of the kinetics of austenite decomposition for other types of steel.

Conclusions
In this paper, the equations for the estimation of microstructure constituents' volume fractions after the isothermal decomposition of austenite have been proposed. Isothermal decomposition of austenite implies quenching of steel from the austenite range to the temperature of isothermal transformation where all austenite decomposes at a constant temperature.
The inversion methods for the calculation of characteristic variables in the mathematical model of kinetics of austenite decomposition were developed.
The mathematical model was verified by the comparison of experimentally and mathematically determined IT diagrams of steel. It can be concluded that characteristic parameters included in the mathematical model of ferrite, pearlite and bainite transformation can be successfully evaluated by the proposed method.