10.2 Critical Temperatures, Invariant Reactions & the Lever Rule

Critical Transformation Temperatures & Boundaries

Phase transitions during equilibrium heating and cooling are identified by critical temperature designators (from the French arrêt, meaning arrest):

• $A_1$ (Lower Critical Temperature, $727^\circ\text{C}$): The eutectoid isotherm ($P-S-K$ line). Represents the lower boundary of the austenite phase field. Below $A_1$, austenite cannot exist under equilibrium conditions; it transforms completely into ferrite and cementite.
• $A_3$ (Upper Critical Temperature for Hypoeutectoid Steels, $912^\circ\text{C} \to 727^\circ\text{C}$): The $G-S$ line. Represents the boundary separating the single-phase austenite ($\gamma$) field from the two-phase ferrite + austenite ($\alpha + \gamma$) field. On cooling, proeutectoid ferrite begins to nucleate at $A_3$.
• $A_{cm}$ (Upper Critical Temperature for Hypereutectoid Steels, $727^\circ\text{C} \to 1148^\circ\text{C}$): The $S-E$ line. Represents the solubility limit of carbon in austenite. On cooling hypereutectoid austenite across $A_{cm}$, proeutectoid cementite precipitates along prior austenite grain boundaries.
• $A_4$ (Delta-to-Gamma Transition, $1394^\circ\text{C}$): The $N-J$ line. Pure iron transforms from $\gamma$-austenite to $\delta$-ferrite.
• $A_2$ (Curie Temperature, $770^\circ\text{C}$): The magnetic transition temperature ($M-O$ line). Below $770^\circ\text{C}$, BCC $\alpha$-ferrite is ferromagnetic; above $770^\circ\text{C}$, it is paramagnetic. This transition involves electron spin randomization without changes in crystallographic structure or lattice parameter.

Thermal Hysteresis in Welding: $Ae$ vs. $Ac$ vs. $Ar$

Under real welding thermal cycles, heating and cooling rates deviate significantly from infinite-time equilibrium, introducing kinetic superheating and undercooling:

• $Ae_1, Ae_3$: Thermodynamic equilibrium temperatures (derived from zero-rate dilatometry or thermodynamic modeling).
• $Ac_1, Ac_3$ (chauffage): Transformation temperatures observed during heating. Diffusion requires thermal activation; hence, rapid welding heating rates ($100\text{ to }>1000^\circ\text{C/s}$) displace the onset of austenitization upward: $Ac_1 > Ae_1$ and $Ac_3 > Ae_3$.
• $Ar_1, Ar_3$ (refroidissement): Transformation temperatures observed during cooling. Transformation requires undercooling (thermodynamic driving force $\Delta G_v$); hence, continuous weld cooling displaces the onset of decomposition downward: $Ar_3 < Ae_3$ and $Ar_1 < Ae_1$.

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Invariant Equilibrium Reactions in Steels

At specific invariant points on the binary $\text{Fe}-\text{Fe}_3\text{C}$ diagram, three phases exist in thermodynamic equilibrium at fixed temperature, pressure, and chemical composition (Gibbs phase rule: $F = C - P + 1 = 2 - 3 + 1 = 0$ degrees of freedom):

                                INVARIANT REACTIONS IN THE Fe-Fe3C SYSTEM

     1. PERITECTIC (1495°C, 0.17 wt% C)      2. EUTECTIC (1148°C, 4.30 wt% C)       3. EUTECTOID (727°C, 0.77 wt% C)

             L (0.53% C)                            Liquid (4.30% C)                      Austenite (0.77% C)
                  +                                        |                                      |
         δ-Ferrite (0.09% C)                               | (Cooling)                            | (Cooling)
                  | (Cooling)                              v                                      v
                  v                               γ (2.14%) + Fe3C (6.70%)               α (0.022%) + Fe3C (6.70%)
          Austenite (0.17% C)                           [Ledeburite]                            [Pearlite]

1. The Peritectic Reaction ($1495^\circ\text{C}$, $0.17\text{ wt}\%\text{ C}$)

$$L\ (0.53\text{ wt}\%\text{ C}) + \delta\text{-ferrite}\ (0.09\text{ wt}\%\text{ C}) \xrightleftharpoons[\text{heating}]{\text{cooling}} \gamma\text{-austenite}\ (0.17\text{ wt}\%\text{ C})$$

• Welding Significance & Hot Tearing Susceptibility: During weld metal solidification of carbon and low-alloy steels, alloys containing between $0.09\text{ and }0.17\text{ wt}\%$ carbon undergo the peritectic transition. The transformation from BCC $\delta$-ferrite (density $\rho \approx 7.35\text{ g/cm}^3$) to FCC $\gamma$-austenite (density $\rho \approx 7.65\text{ g/cm}^3$) involves an abrupt solid-state volume contraction of approximately $0.5\%$ linear shrinkage ($1.5\%$ volumetric). When this solid-state contraction occurs in the presence of terminal interdendritic liquid films, severe tensile strains are concentrated across thin liquid films, triggering solidification cracking (hot tearing). Steels with carbon contents centered around $0.12\text{ to }0.15\text{ wt}\%$ exhibit the highest sensitivity to solidification cracking in arc welds.

2. The Eutectic Reaction ($1148^\circ\text{C}$, $4.30\text{ wt}\%\text{ C}$)

$$L\ (4.30\text{ wt}\%\text{ C}) \xrightleftharpoons[\text{heating}]{\text{cooling}} \gamma\text{-austenite}\ (2.14\text{ wt}\%\text{ C}) + \text{Fe}_3\text{C}\ (6.70\text{ wt}\%\text{ C})$$

• Welding Significance: Governs cast iron solidification. The resulting microstructural constituent is ledeburite, a fine mechanical mixture of austenite and cementite. Upon subsequent cooling to room temperature, the austenite in ledeburite transforms into pearlite.

3. The Eutectoid Reaction ($727^\circ\text{C}$, $0.77\text{ wt}\%\text{ C}$)

$$\gamma\text{-austenite}\ (0.77\text{ wt}\%\text{ C}) \xrightleftharpoons[\text{heating}]{\text{cooling}} \alpha\text{-ferrite}\ (0.022\text{ wt}\%\text{ C}) + \text{Fe}_3\text{C}\ (6.70\text{ wt}\%\text{ C})$$

• Welding Significance: The cooperative, diffusion-controlled decomposition of austenite into alternating, interpenetrating lamellae of $\alpha$-ferrite and cementite produces pearlite. The interlamellar spacing ($\lambda_p$) is inversely proportional to the degree of undercooling below $A_1$ ($\lambda_p \propto 1/\Delta T$). Finer lamellar spacing increases yield strength and Charpy V-notch toughness per the Hall-Petch relationship.

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Proeutectoid Phase Evolution & The Lever Rule

Steels are classified by nominal carbon concentration relative to the eutectoid composition ($0.77\text{ wt}\%\text{ C}$):

                               EQUILIBRIUM PHASE FRACTION LEVER RULE

                 0.022% C                     C_0                   6.70% C
                    |--------------------------*-----------------------|
                 α-Ferrite                   Bulk                  Cementite
                    |<-------- L_α ----------->|<------- L_Fe3C ------>|

                    Weight Fraction α = L_Fe3C / (L_α + L_Fe3C) = (6.70 - C_0) / (6.70 - 0.022)
                    Weight Fraction Fe3C = L_α / (L_α + L_Fe3C) = (C_0 - 0.022) / (6.70 - 0.022)

Hypoeutectoid Steels ($C_0 < 0.77\text{ wt}\%\text{ C}$)

Upon slow cooling from the single-phase $\gamma$-austenite region:

1. At temperature $T = A_3$, proeutectoid $\alpha$-ferrite nucleates heterogeneously at prior austenite grain boundaries.
2. Between $A_3$ and $A_1$, proeutectoid ferrite grows into grain boundary allotriomorphs. If the cooling rate is moderately elevated, it projects crystallographically aligned Widmanstätten ferrite sideplates into the austenite grain interiors.
3. Ferrite rejects carbon into the untransformed austenite because its solubility is limited to $<0.022\%$. The carbon concentration of the remaining austenite enriches along the $G-S$ ($A_3$) boundary until it reaches $0.77\text{ wt}\%$ at $727^\circ\text{C}$.
4. At $A_1$ ($727^\circ\text{C}$), all remaining enriched austenite decomposes into pearlite.

Hypereutectoid Steels ($C_0 > 0.77\text{ wt}\%\text{ C}$)

Upon cooling across $A_{cm}$:

1. Proeutectoid cementite nucleates at austenite grain boundaries, forming a continuous, brittle intergranular network.
2. Cementite extraction depletes the adjacent austenite of carbon along the $S-E$ line until it reaches $0.77\text{ wt}\%$ at $727^\circ\text{C}$.
3. At $A_1$, remaining austenite transforms into pearlite. In welded joints, continuous grain-boundary cementite networks act as low-energy cleavage pathways, causing extreme intergranular embrittlement.

Quantitative Lever Rule Formulation

To calculate equilibrium mass fractions at a temperature just below $727^\circ\text{C}$:

1. Total Equilibrium Phase Fractions:

   $$W_\alpha = \frac{C_{\text{Fe}_3\text{C}} - C_0}{C_{\text{Fe}_3\text{C}} - C_\alpha} = \frac{6.70 - C_0}{6.70 - 0.022}$$ $$W_{\text{Fe}_3\text{C}} = \frac{C_0 - C_\alpha}{C_{\text{Fe}_3\text{C}} - C_\alpha} = \frac{C_0 - 0.022}{6.70 - 0.022}$$

2. Microstructural Constituent Fractions (for Hypoeutectoid Steels):

   $$W_{\alpha,\text{pro}} = \frac{C_{\text{eutectoid}} - C_0}{C_{\text{eutectoid}} - C_\alpha} = \frac{0.77 - C_0}{0.77 - 0.022}$$ $$W_{\text{pearlite}} = \frac{C_0 - C_\alpha}{C_{\text{eutectoid}} - C_\alpha} = \frac{C_0 - 0.022}{0.77 - 0.022}$$

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