This simulation shows how austenite in a eutectoid (0.77 wt% C) steel transforms to pearlite, bainite, or martensite, depending on the thermal path.
Isothermal (TTT): the steel is quenched instantly from above the temperature A1 to the hold temperature, held for the hold time, then quenched to room temperature. The three black curves give the time needed at each hold temperature for the austenite to be 1% transformed (solid curve on the left, start), 50% transformed (dashed curve), and 99% transformed (solid curve on the right, finish).
The shaded regions show the microstructure: gray is austenite, blue is pearlite, orange is bainite, and purple is martensite. The lighter band between the start and finish curves is partly transformed austenite. In continuous cooling, the shading at a point is the microstructure of steel that reaches that point while cooling at a constant rate.
Continuous cooling (CCT): the steel cools at a constant rate. Time is measured from the moment the steel cools through A1 = 727 °C. The blue curves are the CCT start, 50%, and finish curves; check the box to compare them with the gray TTT curves. The dotted green curve is the critical cooling rate, the slowest rate that avoids pearlite and bainite.
The alloying shift slider multiplies all transformation times, the way alloying elements such as Cr, Mo, and Ni slow the diffusional transformations and increase hardenability.
The panel on the right shows the room-temperature microstructure and an estimated hardness. Press play to watch the microstructure develop along the path. The path is drawn only when you press play; changing a slider clears it. Press reset to clear the path without changing the settings.
Martensite is what austenite becomes when it is cooled too fast for carbon to diffuse. Austenite is FCC iron with all 0.77 wt% carbon dissolved in it. Below Ms, the FCC lattice becomes unstable, and it shears cooperatively into a body-centered structure without any atoms diffusing. The carbon has no time to leave, so it stays trapped in the new lattice. That trapped carbon distorts the lattice into body-centered tetragonal (BCT) and makes martensite hard. The driving force is the free-energy difference between this supersaturated FCC phase and BCC iron. Only austenite has that driving force and the FCC structure that can shear.
Bainite has already done what martensite only partly does. It is ferrite, which is BCC iron holding almost no carbon (about 0.02 wt%), plus cementite (Fe3C) carrying the rest of the carbon. The carbon has already been sorted into its equilibrium homes, and the iron is already BCC, so there is no FCC lattice to shear. Ferrite plus cementite also has a lower free energy than martensite. Martensite is a metastable, high-energy, supersaturated phase, and converting bainite into it would raise the free energy, so it can’t happen spontaneously at any cooling rate. The same reasoning applies to pearlite, which is also ferrite plus cementite, just in a lamellar arrangement.
The possible transformations all go downhill:
None go from ferrite + cementite back to martensite. The only route is to reheat above A1, which dissolves the cementite and makes austenite again, and then quench.
At a constant temperature, the fraction of austenite transformed by diffusional processes follows the Avrami equation:
\[ f = 1 - \exp\!\left[-(k\,t)^{n}\right], \qquad n = 2.5 \]The rate constant is the sum of a pearlite and a bainite contribution:
\[ k = \frac{1}{S}\left(k_P + k_B\right) \] \[ k_P = C_P\,(A_1 - T)^{3.799}\exp\!\left(-\frac{Q_P}{R\,T}\right) \quad (T < A_1) \] \[ k_B = C_B\,(B_s - T)^{2.5}\exp\!\left(-\frac{Q_B}{R\,T}\right) \quad (T < B_s) \]The undercooling terms \((A_1 - T)\) and \((B_s - T)\) represent the thermodynamic driving force for nucleation, and the Arrhenius terms represent diffusion. Their product produces the C shape. The fraction of transformed material that is pearlite is \(k_P/(k_P+k_B)\); pearlite and bainite form at equal rates at 540 °C, the nose of the curve, so pearlite dominates above the nose and bainite below it.
The times for a transformed fraction \(X\) (the TTT curves) are
\[ t_X = \frac{\left[-\ln(1-X)\right]^{1/n}}{k} \]\(A_1 = 727\) °C, \(B_s = 638\) °C, \(Q_P = 231.5\) kJ/mol, \(Q_B = 107.3\) kJ/mol, \(\ln C_P = 11.808\), \(\ln C_B = 1.955\) (with \(T\) in K inside the exponentials and in °C inside the undercooling terms). These values were fit to the standard textbook TTT diagram for eutectoid steel (nose near 540 °C at about 0.7 s), with the requirements that \(k_P = k_B\) at the nose, pearlite forms at least 10 times faster than bainite at 620 °C, and bainite forms at least 10 times faster than pearlite at 460 °C. \(S\) is the alloying shift factor.
During continuous cooling, the transformation is integrated in small temperature steps. At each step, the fictitious time \(t^*\) that would produce the current fraction at the current temperature is calculated, the time step is added, and the Avrami equation gives the new fraction:
\[ t^{*} = \frac{\left[-\ln(1-f)\right]^{1/n}}{k(T)}, \qquad f_{\text{new}} = 1 - \exp\!\left[-\left(k(T)\,(t^{*} + \Delta t)\right)^{n}\right] \]This is the Scheil–Avrami additivity approach. Because the steel spends only a short time at each temperature, the CCT curves lie below and to the right of the TTT curves.
Below \(M_s = 220\) °C, the austenite remaining when \(M_s\) is reached transforms athermally (no diffusion, no time dependence) according to the Koistinen–Marburger equation:
\[ f_M = f_{\gamma,M_s}\left\{1 - \exp\left[-0.011\,(M_s - T)\right]\right\} \]About 12% of that austenite is retained at room temperature. Diffusional transformation is not allowed below \(M_s\) in this model.
In TTT mode, each point \((t, T)\) is shaded by the isothermal transformed fraction \(f\): austenite for \(f < 0.01\), the product phase for \(f > 0.99\), and a lighter mixed shade in between. The product is pearlite where \(k_P > k_B\) and bainite otherwise. In CCT mode, the point is shaded by the fraction calculated along the constant-rate cooling curve that passes through it, with rate \((A_1 - T)/t\). Below \(M_s\), austenite that has not transformed becomes martensite.
Pearlite that forms above 650 °C is labeled coarse, and below 650 °C fine. Bainite that forms above 350 °C is labeled upper, and below 350 °C lower. The hardness is a rule-of-mixtures estimate: pearlite 15–42 HRC and bainite 42–60 HRC (both increasing as the formation temperature decreases), martensite 65 HRC, retained austenite 20 HRC. Values below about 20 HRC are outside the normal Rockwell C range and are only indicative.
| \(A_1\) | eutectoid temperature (727 °C) |
| \(B_s\) | bainite start temperature in the rate expression (638 °C) |
| \(C_P,\ C_B\) | pre-exponential factors for pearlite (s\(^{-1}\) °C\(^{-3.799}\)) and bainite (s\(^{-1}\) °C\(^{-2.5}\)) |
| \(f\) | fraction of austenite transformed by diffusion (pearlite + bainite) |
| \(f_{\text{new}}\) | transformed fraction at the end of a time step |
| \(f_M\) | fraction of the steel that is martensite |
| \(f_{\gamma,M_s}\) | fraction of austenite remaining when the steel reaches \(M_s\) |
| \(k\) | overall rate constant (s\(^{-1}\)) |
| \(k_P,\ k_B\) | rate constants for pearlite and bainite formation (s\(^{-1}\)) |
| \(M_s\) | martensite start temperature (220 °C) |
| \(n\) | Avrami exponent (2.5) |
| \(Q_P,\ Q_B\) | activation energies for pearlite and bainite formation (kJ/mol) |
| \(R\) | ideal gas constant (8.314 J/(mol K)) |
| \(S\) | alloying shift factor; multiplies all transformation times |
| \(T\) | temperature (°C; K in the Arrhenius terms) |
| \(t\) | time (s) |
| \(t_X\) | time to reach transformed fraction \(X\) at constant temperature (s) |
| \(t^{*}\) | fictitious time that gives the current transformed fraction at the current temperature (s) |
| \(\Delta t\) | time step (s) |
| \(X\) | transformed fraction that defines a curve (0.01 start, 0.50, 0.99 finish) |
| HRC | Rockwell C hardness |
This simulation was created in the Department of Chemical and Biological Engineering at University of Colorado Boulder for LearnChemE.com by John L. Falconer using Claude AI. It was prepared with financial support from the National Science Foundation (DUE 2336987 and 2336988). Address any questions or comments to LearnChemE@gmail.com.