Application of the Mixed-Mode Model for Numerical Simulation of Pearlitic Transformation

Phase transformation of austenite into pearlite begins when temperature drops below A₁ (Fig. 1a). The driving force of this process is the difference between the free energy of austenite and the free energy of the mixture of ferrite and cementite. The carbon concentration at the ferrite/austenite boundary c_γα and at the austenite/cementite boundary c_γcem for steel containing x_γ is determined using extrapolation of the A₃ and A_cm curve, respectively. In the eutectoid steels first phase which nucleates is cementite and ferrite is the second.

Typical structure of pearlitic steel is shown in Fig. 2(a). One grain of pearlite usually consists of few pearlite colonies. Within each colony, plates of ferrite and cementite have the same crystallographic orientation. Distance between two plates of ferrite, referred to as interlamellar spacing, is about seven times greater than width of cementite plates. Pearlite typically forms heterogeneously on the austenite grain boundaries and grain corners (Fig. 2b). As the system always strives to minimize energy, and the surface tensions associated with nucleation depend on the crystallographic orientation, we can expect dependence between the orientation of the embryo and the orientation of the grain of austenite in which it grows. Meanwhile, microscopic observation shows that interface of the pearlite colony created with austenite is an incoherent high-energy interface. However, we can distinguish two different crystallographic relationships between ferrite and cementite within pearlite colony. Independent of carbon concentration and degree of undercooling, nucleation on clear austenite boundary exhibits the Pitsch/Petch relationship. Nucleation on the proeutectoid ferrite or cementite results in the formation of Bagaryatski relationship (Ref 8). Microscopic observation shows that single plates can cross austenite grain without a perceptible change in the direction. The rate of perlite nucleation depends on the rate of cooling, the structure of austenite and its chemical homogeneity. The finer grain of austenite and the greater variation in chemical composition, the faster is nucleation. Consequently, smaller colonies are created. The increase in carbon content in austenite causes the increase in the driving force of nucleation of cementite and the decrease in the driving force of nucleation of ferrite. When the concentration of carbon decreases, the effect is reversed (Ref 9).

Further growth of the pearlite phase occurs due to the movement of the interface boundary connected with the transport of carbon and alloying elements across the interface and their long-range diffusion in the austenite. This process simultaneously leads to the reconstruction of FCC austenite crystal lattice into a BCC ferrite lattice or orthorhombic lattice of cementite. The factors control the rate at which the interface moves are varied. According to Hillert’s (Ref 10) theory, when the interface boundary has a high mobility and can move as fast as substitutional solutes can be redistributed, the growth is controlled by volume diffusion. This assumption was adopted in the numerical models presented in the papers (Ref 4, 11). However, if the boundary is characterized by low mobility (even at high driving forces) and long-range diffusion processes are fast, the change is controlled by the interface diffusion.

The results of experimental tests and extended theoretical analysis indicate a significant influence of interface diffusion on the kinetics of the phase transformation (Ref 12). The values of activation energy determined for the interface diffusion are more realistic, less than that for volume diffusion in austenite and greater than for volume diffusion in ferrite. This effect is particularly well seen at high temperatures. Consequently, a mixed-mode model, which takes into account both volume diffusion as well as interface diffusion, was proposed by Sietsma and van der Zwaag (Ref 13) and was further developed by Pandit and Bhadeshia (Ref 14). In the presented work, this approach was used for the creation of the numerical model of pearlitic transformation.

The classic physical models of pearlitic transformation were improved as the research methods of the microstructure progressed. Detected anomalies: such as the creation of needles, branching of plates, were attributed to elastic stress (Ref 5, 15, 16), resulting from the mismatch of the lattice near the front of the transformation. The formation of spheroidal particles (Ref 17, 18), including autocatalytic mechanisms, for metastable conditions was considered. A detailed description of the process physics not directly reflected in possibility of predicting structure parameters, including: the interlamelar spacing and the size of the pearlite colony for varying temperature conditions. In addition, the computation time prevents the use of models for designing real technological processes.

Searching for a balance between model’s predictive capabilities and computing time was the main motivation for the work. Developed mixed-mode model was based on the simplified physical model of pearlite forming, assuming that such approach accelerates prediction of the final structure and mechanical properties of steel. In contrast to existing solutions, model allows to determine the interlamellar spacing for the conditions of cyclic cooling.

The models were analyzed for the possibility of describing the kinetics of pearlitic transformation and predicting the morphological parameters of the structure, which determine the strength properties. These parameters are the interlamellar spacing and the size of the pearlite colonies.

The kinetics of the pearlitic phase transformation is correctly described by models based on the solution of JMAK equation (Ref 19-21), see, for example (Ref 22). The need to apply the Scheil additivity rule (Ref 23) to account for temperature change is the main disadvantage of this model. The alternative approach is the Leblond model (Ref 24) and its upgrades (Ref 25), in which kinetics of phase transformation is described by a differential equation with respect to time. Both models provide information about volume fraction of phases in near real time, but they require experimental identification of parameters.

On the basis of theories of growth controlled by volume diffusion and interface diffusion, two relationships to determine the size of the interlamellar spacing as a function of undercooling were created. Zener (Ref 26) assumed that the system stabilized at a spacing for which the growth rate is a maximum, and this hypothesis leads to:where T_e—the eutectoid temperature, σ_αcem—interfacial energy per unit area of the ferrite-cementite boundary, ρ—density, Q—the heat of the transformation.

On the other hand, Puls and Kirkaldy (Ref 27) formulated the following relationship for interface diffusion-controlled growth and the maximum rate of entropy criterion:

Experimental tests (Ref 28) confirm the better predictive capabilities of the Zener model. However, it should be noted that the above relationships can be used for isothermal conditions only.

Among the models which predict kinetics of pearlitic transformation and morphology of the final structure, model based on the Phase Field Method (Ref 5) should be distinguished. This model describes growth of pearlite colony and takes into account divergence of the plates. The interlamellar spacing is a function of temperature and time of the isothermal transformation. Meanwhile, the ability to determine the size of this parameter for variable temperature conditions is crucial due to use of the model in practice.

Taking into account continuous cooling conditions or cyclic temperature changes requires the use of more advanced models. The empirical relationship presented in (Ref 22) allows determining the constant interlamellar spacing within each colony for weighted average temperature of the transformation:where T_p—weighted average of temperature of pearlitic transformation in °C, [Mn], [Cr], [Si], [C]—manganese, chromium, silicon and carbon content in wt.%.

Meanwhile, due to its strong influence on the strength properties, it is necessary to determine distribution of the interlamellar spacing in the whole volume. Such a possibility is provided by models based on the solution of the diffusion equation. In addition, that model allows determining the size of the pearlite colony and distribution of carbon in the austenite.

Model describes sidewise and frontal growth of few pearlite colonies (Fig. 3) in a single grain of austenite, in which size was determined experimentally. The concentration at the interface between austenite/cementite and austenite/ferrite corresponded to the thermodynamic equilibrium conditions for a given temperature. Sites of nucleation on the edges of the austenite grain were selected randomly. Nucleation of first and next plates of cementite resulted from the mass balance, which took into account the carbon content in cementite at 6.67 wt.% (Fig. 3a). Cementite grows until the content of carbon in the distances dx and dy (size of the step) drops below c_b (0.05%C); then, the first plates of ferrite are symmetrically created (Fig. 3b). Value of parameter c_b was determined using inverse analysis for the experimental data and sensitivity analysis. Growth of ferrite plates causes pushing carbon into austenite and increase in its concentration, which leads to creation of new plates of cementite in a distance greater than 7 times the thickness of previously created plates of cementite. The whole cycle was repeated until the disappearance of the austenitic phase. The plates stop growing after the impingement with other plates.

Growth of ferrite and cementite plates is controlled by both the volume diffusion and the interface mobility. This approach is commonly referred to as a mixed-mode model, and its use in numerical modeling of pearlitic transformation can be found in (Ref 14, 29).

Volume diffusion of carbon in the austenite is described by the second Fick’s law:where D—diffusion coefficient, c—carbon concentration, t—time.

In Eq 4, much slower redistribution of substitutional alloying elements was omitted. The initial and boundary conditions at the interfacial boundary were determined for local thermodynamic equilibrium at γ/α and γ/cementite interfaces using ThermoCalc software. In consequence, the initial and boundary conditions were given by the following equations:where c_γ—carbon concentration in austenite, c_γα—equilibrium carbon concentration at γ/α boundary, c_γcem—equilibrium carbon concentration at γ/cementite boundary, n—unit vector normal to the surface, Ω—solution domain, Γ₁—γ/cementite boundary, Γ₂—γ/α boundary, Γ₃—edge of the solution domain.

The volume diffusion coefficient of carbon in austenite, which is the function of temperature and average carbon concentration, was calculated using Agren (Ref 30) relationship in the form:where y_c = x_c(1 − x_c), x_c—mole fraction in the steel, T—temperature in K.

The Fick’s Eq 4 was solved in 2D using FE method. Application of the variational principle gives:where c—vector of nodal values of concentration, H—diffusion matrix, C—geometrical matrix, p—boundary conditions vector. Matrices H, C and vector p are calculated as:where n_i—shape functions in the FE method.

Assumption of the quasi-stationary state during time step Δt and application of the Galerkin time integration scheme yields the final set of equations:where

Solution of Eq 9 allows to calculate distribution of the carbon concentration c in the austenite grain at the end of the time step Δt when this distribution at the beginning of the time step c_i is known.

At each time step, the carbon concentration profile in the austenite was calculated by the solution of the second Fick’s law (4) and on the basis of this solution the interface concentration \(c_{\gamma }^{\text{int}}\) was determined. The interface velocity was proportional to the available chemical driving force with the interface mobility (Ref 31):where M—interface mobility, ΔG—driving force for the phase transformation calculated using ThermoCalc, Q—the activation energy, R—the gas constant, d—the jump distance across the interface, v_D—Debye frequency, k—Boltzmann constant, M*—fitting parameter equal 0.8.

The driving force ΔG was proportional to the deviation from the equilibrium concentration:where \(c_{\gamma }^{\text{int}}\)—carbon content at γ/α and γ/cementite interface.

Process of diffusion evolves until equilibrium is reached in terms of carbon composition and phase fraction. The mixed-mode character is quantified by the parameter S, according to which if S = 0 phase transformation is diffusion controlled, whereas S = 1 means interface-controlled solution (Ref 32).

All parameter values used in the developed model are listed in Table 1. Equilibrium carbon concentrations for investigated steels were determined using ThermoCalc software, while temperatures of bainite start B_s and martensite start M_s were calculated on the basis of dilatometric tests.

The main goal of this part of experimental tests was to provide results, which will enable to evaluate correctness of the model in terms of prediction of the volume fraction of pearlite. Experiment concerned cold-worked tool steel C80U, in which pearlitic structure guarantees high strength and high utility properties, including hardness, abrasion resistance and durability. Tests were performed for various cooling rates and two temperatures of austenitization: 800 and 820 °C. The progress of the transformation was stopped by quenching in water—phase of austenite was represented by martensite. The metallographic analysis was performed to determine transformation kinetics. Microstructure of steel is shown in Fig. 4. The stages of pearlitic colonies formation during cooling at a rate of 20 °C/min are presented in Fig. 5.

In the first set of experimental tests, the cementite mesh method was used to determine average size of prior austenite grains which amounted to 34.8 μm for temperature of austenitization equal 800 °C and 39.2 μm for 820 °C. In the second set, the tests samples were heated to the austenitization temperature of 800 °C and then were cooled at various rates in the range 2÷10 °C/s to the room temperature. The samples were prepared using a standard metallographic technique. Selected microstructures showing perlite colonies on the background of martensite are presented in Fig. 6 and 7. It is seen that the volume fraction of pearlite decreases with the increase in cooling rate (Table 3). Volume fractions of phases were determined using digital image analysis.

Further investigations allowed determining the influence of austenitization temperature on the phase fraction in the final structure. For this purpose, the samples were heated to a temperature of 820 °C and cooled at 2 °C/s. Analysis of the microstructures showed a smaller volume fraction of perlite than for cooling at the same rate from 800 °C. Selected microstructures of pearlite colonies are presented in Fig. 4. The average interlamellar spacing determined on the basis of the tests was about 0.9 μm.

The verification of correctness of developed numerical model in determining the interlamellar spacing of the perlite was carried out based on the results of the experimental tests which were performed at the Institute for Ferrous Metallurgy in Gliwice, Poland. The research concerned the process of accelerated cooling of high-strength rails in the air and polymer mixture. The main goal of the heat treatment of rails was to obtain a fine structure of pearlite in the whole cross section of the head, which guarantees the best mechanical and exploitation properties. Rapid cooling after rolling of rails promotes fragmentation of the structure, but it can lead to an occurrence of unfavorable components of degenerated pearlite and bainite. The precise selection of parameters of the heat treatment is therefore a key task.

Experiment, which details were presented in (Ref 33), was performed for the steel 900A with the chemical composition given in Table 2. The research involved the physical simulations of the process of continuous cooling of the rail head at the rate of 0.25, 0.5 and 5 °C/s. The measured interlamellar spacing was, respectively, 0.143, 0.129 and 0.094 μm. Selected microstructures are shown in Fig. 8.

Numerical simulations of the kinetics of the pearlitic transformation were performed for steel C80U with a composition given in Table 2. Subsequent stages of pearlitic transformation are shown in Fig. 9. One can observe absorption of carbon from the austenite by growing plate of cementite (Fig. 9a). Decrease in carbon content to a level of c_b < 0.2%C leads to nucleation and growth of ferrite. During sidewise and frontal growth of ferrite, carbon is pushed into the austenite (Fig. 9b).

Growth of pearlite colonies during cooling at various rates from the temperature of austenitization equal 800 and 820 °C is shown, respectively, in Fig. 10 and 11. The volume fraction of pearlite corresponded to 20, 50 and 80% of the progress of phase transformation. The nucleation sites were chosen randomly. The nucleation rate increases with increasing overcooling below A_c1. At lowest cooling rate of 1 °C/s, the thicker plates of cementite and greater interlamellar spacing equal to 0.87 μm were observed. The smallest interlamellar spacing equal 0.52 μm was obtained for cooling rate of 10 °C/s. According to the assumed boundary conditions, absorption of carbon from the austenite by growing plate of cementite was noticeable. Meanwhile, during sidewise and frontal growth of ferrite carbon was pushed to austenite. The steepest gradient of the carbon concentration was observed for the thinnest plate.

Kinetics of pearlitic transformation, which was represented by the changes in pearlite volume fraction with time, is presented in Fig. 12. Dashed lines created by extrapolation of experimental results, marked by filled symbols, have the S-type shape. Results of simulations presented by open symbols remain within the margin of error, what confirms the correctness of the model.

On the basis of results of numerical simulation performed for cooling at 5 °C/s and presented in Fig. 12, graphical representation of microstructure with 50% pearlite content was created (Fig. 13), which enables discussions on the kinetics and mechanism of phase transformation.

The main task of the developed model was a correct prediction of the size of the interlamellar spacing. Simulations of perlite growth during continuous cooling and cyclic temperature changes were performed.

In the first case, rails made of A900 steel were cooled with the rates of 0.25, 0.5 and 5 °C/s. Stages of pearlite creation are shown in Fig. 14. Nuclei of cementite were formed randomly at the edges of austenite, and their number increased with the degree of undercooling. The ferrite nucleated symmetrically when the carbon concentration fell below 0.1. The growth of perlite led to the saturation of carbon to the level of 0.02÷0.22%. The smallest interlamellar spacing equal to 0.11 μm was obtained for the highest rates of cooling 5 °C/s, whereas the pearlite with the largest thickness of plates 0.154 μm was created for the cooling equal 0.25 °C/s. The results of simulations compared with the results of experimental tests confirm the correctness of the solution (Fig. 15).

Further numerical simulation concerned accelerated cooling of the rail head by a cyclic immersion in the polymer solution. The developed model included the finite element (FE) solution for heat transport in the macroscale and the solution of mass transport in the microscale. Single-point models were solved at each Gauss point of the FE mesh, while mixed-mode models were used in three points located in the main axis of the cross section of the rail head. The model assumes the heat transfer coefficient equal to 1700 W/m²K for temperatures exceeding 700 °C and reaching 2400 W/m²K at 300 °C. At lower temperatures, coefficient decreases and reaches 1200 W/m²K at the ambient temperature. Other parameters such as conductivity, density and specific heat were introduced as functions of temperature.

Process of cyclic cooling of the rail head was simulated according to the schedules shown in Table 4. The input to the model was the temperature distribution at the exit from the last rolling stand calculated in (Ref 34).

Results of simulation of cyclic cooling according to schedules I and II for 3 points located at a distance of 2.5, 12.5, 25 mm from the surface of the rail head are presented in Fig. 16. In both cases, at the point 1, which was situated close to the surface of the rail head, cyclic increase and decrease in temperature were observed, while at points 2 and 3 temperature decreases continuously. Cyclic cooling in the polymer solution and heating due to the latent heat leads to the formation of temperature fields characterized by different symmetries than at the exit from the last rolling stand.

After the cycle I, which was characterized by a shorter total time of cooling in the polymer mixture, temperature reaches 540 °C at the surface area and 600-675 °C in the center of the rail head. At the cross section of the rail head, we can observe four symmetrical temperature fields: 520.3-543.6 °C, 543.6-566.9 °C, 566.9-590.2 °C and 590.2.5-613.5 °C. Cycle II allows to get lower temperatures in the whole volume of the rail head, i.e., 490 °C at the surface area and 535-624 °C in the center of the rail head. Temperature fields range from 477.8 to 504.6 °C close to the surface to 531.5-692.6 °C in the center of the head. Obtained time–temperature profiles were used as entry data in the developed model of pearlitic transformation.

Results of simulations are presented in Fig. 17. The smallest interlamellar spacing equal to 0.21 μm was obtained for time–temperature profile according to cycle II at point 1 located close to the head surface. This distance at point 2 reached average value of 0.28 μm. The largest interlamellar spacing equal to 0.42 μm was obtained for point 3, in which pearlite transformation occurred at the highest temperatures.