14.1 Crystal Structures, Phase Diagrams & Iron-Iron Carbide Equilibrium

13.1 Crystal Structures, Phase Diagrams & Iron-Iron Carbide Equilibrium

Understanding the atomic architecture of crystalline solids and the thermodynamic principles governing phase transformations is fundamental to materials science and mechanical engineering. In industrial applications—ranging from heavy mining machinery in Coal India Limited (CIL) operations to high-pressure thermal power equipment—the mechanical properties (tensile strength, ductility, hardness, and fracture toughness) of metallic components are directly dictated by their underlying crystal structures and equilibrium microconstituents.

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1. Crystallography & Metallic Unit Cells

Metals solidify into long-range, periodic three-dimensional arrays of atoms termed crystal lattices. The smallest repeating geometric entity that completely describes the symmetry and atomic positions of the entire crystal is the unit cell.

Fundamental Crystallographic Parameters

1. Coordination Number (CN): The number of nearest neighboring atoms in direct physical contact with a central atom within the crystal lattice.
2. Atomic Packing Factor (APF): The volumetric fraction of solid sphere atoms occupying the total unit cell volume:

$\text{APF} = \frac{N_{\text{atoms}} \times V_{\text{atom}}}{V_{\text{unit cell}}} = \frac{N_{\text{atoms}} \times \left(\frac{4}{3}\pi r^3\right)}{V_{\text{unit cell}}}$

where $N_{\text{atoms}}$ is the effective number of lattice atoms belonging exclusively to one unit cell, and $r$ is the atomic radius.

  Simple Cubic (SC)         Body-Centered Cubic (BCC)      Face-Centered Cubic (FCC)
     +-------+                     +-------+                     +-------+
    /|      /|                    /|  (•) /|                    /| (•)  /|
   +-------+ |                   +-------+ |                   +-------+ |
   | |     | |                   | | (•) | |                   | |(•)(•| |
   | +-----|-+                   | +-----|-+                   | +-----|-+
   |/      |/                    |/      |/                    |/ (•)  |/
   +-------+                     +-------+                     +-------+
  Atoms: 8×(1/8) = 1            Atoms: 8×(1/8)+1 = 2          Atoms: 8×(1/8)+6×(1/2) = 4
  Coord No: 6                   Coord No: 8                   Coord No: 12
  APF: 0.524                    APF: 0.680                    APF: 0.740

Comparative Analysis of Metallic Unit Cells

Interstitial Voids & Carbon Solubility in Iron

A critical metallurgical paradox frequently tested in technical recruitment exams is why FCC austenite ($\gamma$-Fe) dissolves up to 2.11 wt% Carbon at 1147 °C, whereas BCC ferrite ($\alpha$-Fe) dissolves a maximum of only 0.022 wt% Carbon at 727 °C, even though BCC is more loosely packed ($\text{APF} = 0.680$) than FCC ($\text{APF} = 0.740$):

• In FCC (Austenite): The interstitial voids are large, symmetric octahedral sites located at the unit cell center $\left(\frac{1}{2}, \frac{1}{2}, \frac{1}{2}\right)$ and edge midpoints $\left(\frac{1}{2}, 0, 0\right)$. The maximum sphere radius that fits without lattice distortion is:

$r_{\text{oct, FCC}} = (\sqrt{2} - 1)r_{\text{Fe}} \approx 0.414 r_{\text{Fe}} \approx 0.053\text{ nm}$

• In BCC (Ferrite): The octahedral voids are situated at face centers $\left(\frac{1}{2}, \frac{1}{2}, 0\right)$ and edge midpoints $\left(\frac{1}{2}, 0, 0\right)$. These sites are highly asymmetric (squashed along one axis), offering an opening radius of only:

$r_{\text{oct, BCC}} = \left(\frac{2}{\sqrt{3}} - 1\right)r_{\text{Fe}} \approx 0.154 r_{\text{Fe}} \approx 0.019\text{ nm}$

Because a carbon interstitial atom has an atomic radius $r_{\text{C}} \approx 0.071\text{ nm}$, forcing carbon into BCC ferrite produces severe tetragonal strain, limiting its maximum equilibrium solid solubility to a minute $0.022\text{ wt%}$.

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2. Thermodynamics of Phase Equilibria & Gibbs Phase Rule

A phase is a physically homogeneous, structurally distinct, and mechanically separable portion of a material system. In 1876, J. Willard Gibbs formulated the fundamental thermodynamic criterion for multi-component, heterogeneous equilibrium:

$P + F = C + 2$

where:

• $P$ = Number of equilibrium phases present simultaneously.
• $F$ = Degrees of freedom (variance), representing the number of independent intensive variables (temperature, pressure, composition) that can be varied without altering the number of equilibrium phases.
• $C$ = Number of chemically independent components in the system.

Condensed Phase Rule for Metallurgical Systems

Because solid-state and liquid-state metallurgical transformations occur at constant atmospheric pressure ($1\text{ atm}$), the pressure variable is fixed, eliminating one degree of freedom. The condensed (isobaric) Gibbs Phase Rule applies:

$P + F = C + 1$

For a binary alloy system ($C = 2$, e.g., Iron-Carbon):

$P + F = 2 + 1 = 3 \implies F = 3 - P$

• In a single-phase field ($P = 1$): $F = 2$ (Both temperature and alloy composition can vary independently).
• In a two-phase field ($P = 2$): $F = 1$ (Specifying temperature automatically locks the equilibrium compositions of both coexisting phases along tie-lines).
• At an invariant point ($P = 3$): $F = 0$ (Three phases coexist in equilibrium at a single fixed temperature and invariant composition).

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3. The Iron-Iron Carbide ($\text{Fe-Fe}_3\text{C}$) Equilibrium Phase Diagram

Pure elemental iron exhibits allotropy (polymorphism), undergoing reversible crystallographic phase changes upon heating:

$\alpha\text{-Fe (BCC)} \xrightarrow{912^{\circ}\text{C}} \gamma\text{-Fe (FCC)} \xrightarrow{1394^{\circ}\text{C}} \delta\text{-Fe (BCC)} \xrightarrow{1538^{\circ}\text{C}} \text{Liquid Fe}$

(Note: $\alpha$-ferrite is ferromagnetic up to its Curie temperature $A_2 = 768^{\circ}\text{C}$, becoming paramagnetic above $768^{\circ}\text{C}$ without structural change).

Temperature (°C)
  1538°C +----------------------------------------+
  1495°C |  L + δ  |    Liquid (L)                |
         |---------+                              |
         | δ+γ |  γ (Austenite) + Liquid          |
  1147°C +-----+----------------------------------+ <--- Eutectic (1147°C, 4.30% C)
         |     |                                  |      Liquid -> Austenite + Fe3C
         |  γ  |    Austenite (γ) + Cementite     |      (Ledeburite)
         |     |                                  |
   912°C +-----+                                  |
         |α+γ /|                                  |
   727°C +---+-+----------------------------------+ <--- Eutectoid (727°C, 0.77% C)
   (A1)  | α |                                    |      Austenite -> Ferrite + Fe3C
         |---|     Ferrite (α) + Cementite (Fe3C) |      (Pearlite)
         |α+Fe3C                                  |
     0°C +---+------------------------------------+
        0.022% C  0.77% C        2.11% C        4.30% C          6.67% C
        (Pure Fe) (Eutectoid)   (Max Steel)    (Eutectic)     (Cementite)

The Three Invariant Isothermal Reactions

Critical Transformation Lines & Designations

• $A_1$ Line ($727^{\circ}\text{C}$): Lower critical temperature line. Below $A_1$, all austenite decomposes into ferrite and cementite under equilibrium cooling.
• $A_3$ Line ($912^{\circ}\text{C} \to 727^{\circ}\text{C}$): Upper critical temperature line for hypoeutectoid steels ($<0.77%\text{ C}$). Represents the onset of proeutectoid ferrite separation from austenite.
• $A_{\text{cm}}$ Line ($727^{\circ}\text{C} \to 1147^{\circ}\text{C}$): Upper critical line for hypereutectoid steels ($0.77% \text{ to } 2.11%\text{ C}$). Represents the solubility limit of carbon in austenite, above which proeutectoid cementite precipitates along austenite grain boundaries.
• $A_2$ Line ($768^{\circ}\text{C}$): Magnetic transition line (Curie temperature of $\alpha$-ferrite).
• $A_4$ Line ($1394^{\circ}\text{C} \to 1495^{\circ}\text{C}$): Allotropic transformation line between $\gamma$-austenite and $\delta$-ferrite.

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4. Quantitative Phase Calculations: The Lever Rule

Within any two-phase equilibrium region bounded by phase boundaries at a given temperature, a horizontal tie-line is constructed. The overall alloy composition $C_0$ divides the tie-line into two segments. The mass fraction of each phase is inversely proportional to the length of the adjacent tie-line segment.

$\text{Mass Fraction of Phase } \alpha: \quad W_{\alpha} = \frac{C_{\beta} - C_0}{C_{\beta} - C_{\alpha}}$

$\text{Mass Fraction of Phase } \beta: \quad W_{\beta} = \frac{C_0 - C_{\alpha}}{C_{\beta} - C_{\alpha}}$

Worked Step-by-Step Numerical Example

Problem: Consider a plain carbon steel containing $0.45\text{ wt% C}$ (AISI 1045 medium carbon steel) slowly cooled under equilibrium conditions to just below the eutectoid temperature ($727^{\circ}\text{C} - \Delta T$). Calculate:

1. The total mass fraction of ferrite ($\alpha$) and cementite ($\text{Fe}_3\text{C}$).
2. The mass fraction of proeutectoid ferrite ($\alpha_{\text{pro}}$) and eutectoid pearlite ($P$).

Step 1: Total equilibrium phase fractions just below $727^{\circ}\text{C}$

• At $727^{\circ}\text{C}$, the tie-line extends from $C_{\alpha} = 0.022%\text{ C}$ to $C_{\text{Fe}_3\text{C}} = 6.67%\text{ C}$ with overall composition $C_0 = 0.45%\text{ C}$:

$W_{\alpha, \text{total}} = \frac{6.67 - 0.45}{6.67 - 0.022} = \frac{6.22}{6.648} = 0.9356 \implies 93.56\%$

$W_{\text{Fe}_3\text{C}, \text{total}} = \frac{0.45 - 0.022}{6.67 - 0.022} = \frac{0.428}{6.648} = 0.0644 \implies 6.44\%$

Step 2: Proeutectoid ferrite vs. Pearlite fraction just above $727^{\circ}\text{C}$

• Just above $727^{\circ}\text{C}$, the tie-line spans between proeutectoid ferrite ($C_{\alpha} = 0.022%\text{ C}$) and remaining austenite ($C_{\gamma} = 0.77%\text{ C}$):

$W_{\alpha, \text{pro}} = \frac{0.77 - 0.45}{0.77 - 0.022} = \frac{0.32}{0.748} = 0.4278 \implies 42.78\%$

$W_{\gamma} = \frac{0.45 - 0.022}{0.77 - 0.022} = \frac{0.428}{0.748} = 0.5722 \implies 57.22\%$

Upon crossing $727^{\circ}\text{C}$, the $57.22%$ remaining austenite undergoes the invariant eutectoid transformation into 100% Pearlite. Therefore:

• Proeutectoid Ferrite: $42.78%$
• Eutectoid Pearlite: $57.22%$

(Note: Inside the pearlite constituent, eutectoid ferrite contributes $0.5722 \times \frac{6.67 - 0.77}{6.67 - 0.022} = 50.78%$, which when added to $42.78%$ proeutectoid ferrite matches the total $93.56%$ ferrite exactly).

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5. Microconstituents & Phase Morphology

1. Ferrite ($\alpha$-iron): BCC interstitial solid solution of carbon in iron. Soft, ductile, highly magnetic, tensile strength $\approx 300\text{ MPa}$, hardness $\approx 80\text{ HB}$.
2. Austenite ($\gamma$-iron): FCC solid solution. Ductile, malleable, non-magnetic, high work-hardening capacity; unstable below $727^{\circ}\text{C}$ under slow cooling.
3. Cementite ($\text{Fe}_3\text{C}$): Intermetallic iron carbide containing $6.67\text{ wt% C}$. Orthorhombic crystal structure, extremely hard ($>800\text{ HV}$ or $65\text{ HRC}$) and brittle, negligible tensile ductility.
4. Pearlite: Lamellar aggregate of alternating thin plates of $\alpha$-ferrite ($88\text{ wt%}$) and $\text{Fe}_3\text{C}$ ($12\text{ wt%}$). Formed by cooperative sideways diffusion during the eutectoid transformation. Yield strength $\approx 400\text{--}600\text{ MPa}$.
5. Bainite: Non-lamellar microconstituent formed by isothermal transformation below the pearlite range ($250\text{--}550^{\circ}\text{C}$). Consists of a fine dispersion of cementite needles/platelets embedded in a ferrite matrix (Upper Bainite: feathery at $350\text{--}550^{\circ}\text{C}$; Lower Bainite: acicular, high-toughness plate structure at $250\text{--}350^{\circ}\text{C}$).
6. Martensite: Diffusionless, athermal shear transformation product formed when austenite is quenched rapidly below the Martensite Start ($M_s$) temperature. Possesses a Body-Centered Tetragonal (BCT) crystal structure supersaturated with trapped carbon atoms. Features extreme hardness ($60\text{--}68\text{ HRC}$), high yield strength, and severe brittleness in the as-quenched state.