Sources and References

• Porter, Easterling, and Sherif, Phase Transformations in Metals and Alloys, 3rd ed., CRC Press.
• Reed-Hill and Abbaschian, Physical Metallurgy Principles, 4th ed., Cengage.
• Kou, Welding Metallurgy, 2nd ed., Wiley.
• Campbell, Complete Casting Handbook, 2nd ed., Butterworth-Heinemann.
• Callister and Rethwisch, Materials Science and Engineering: An Introduction, 10th ed., Wiley.
• Dantzig and Rappaz, Solidification, 2nd ed., EPFL Press.

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Chapter 1: Atomic Transport in Solids

Physical metallurgy connects the processing history of a metal to its microstructure and, through microstructure, to its mechanical and physical properties. Every processing step that modifies structure—casting, heat treatment, welding, forming—proceeds through atomic-scale events: atoms move, phases nucleate and grow, defects multiply or anneal. The starting point is therefore solid-state diffusion.

1.1 Fick’s Laws and Mechanisms

The flux of species \( A \) in a concentration gradient is

\[ J_A = -D \frac{\partial C_A}{\partial x}, \]

with diffusivity \( D \) measured in m²/s. For transient diffusion, mass conservation gives

\[ \frac{\partial C}{\partial t} = \frac{\partial}{\partial x}\!\left(D \frac{\partial C}{\partial x}\right). \]

Diffusion in crystalline metals proceeds chiefly through vacancy and interstitial mechanisms. Substitutional atoms exchange places with vacancies; small solutes such as carbon, nitrogen, and hydrogen jump between interstitial sites. The diffusivity follows Arrhenius behaviour,

\[ D = D_0 \exp\!\left(-\frac{Q}{RT}\right), \]

where \( Q \) is the activation energy for migration plus, for the substitutional case, the vacancy formation energy.

1.2 Grain Boundaries and Short-Circuit Paths

Boundaries, dislocation cores, and free surfaces all present paths with reduced activation energy. At low homologous temperatures, short-circuit diffusion dominates kinetics; at high homologous temperatures bulk diffusion takes over. The crossover drives process choices: carburizing and nitriding rely on bulk diffusion at elevated temperature, whereas grain-boundary segregation and sensitization are low-temperature phenomena.

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Chapter 2: Solidification

Casting and welding are fundamentally solidification processes. The microstructures produced—grain size, dendrite morphology, segregation, porosity—are set during the liquid–solid transition.

2.1 Nucleation of Solid from Liquid

Homogeneous nucleation requires overcoming an energy barrier

\[ \Delta G^{*} = \frac{16 \pi \gamma_{sl}^{3}}{3\,(\Delta G_v)^2}, \]

where \( \Delta G_v \) is the volumetric free-energy driving force and \( \gamma_{sl} \) the solid–liquid interfacial energy. Real castings nucleate heterogeneously on mould walls, inclusions, and grain refiners; the same expression applies with \( \gamma_{sl}^3 \) replaced by an effective product that depends on the wetting angle.

2.2 Constitutional Supercooling and Morphology

In an alloy, solute is partitioned at the liquid–solid interface with partition coefficient \( k = C_s/C_l \). For \( k < 1 \), solute is rejected into the liquid, depressing its local liquidus. If the temperature gradient ahead of the interface is shallower than the liquidus-temperature gradient, constitutionally supercooled liquid exists ahead of the front and the planar interface becomes unstable. The Tiller criterion is

\[ \frac{G}{R} < \frac{m_l C_0 (1-k)}{D_l k}, \]

with \( G \) the thermal gradient, \( R \) the growth velocity, \( m_l \) the liquidus slope, \( C_0 \) the bulk composition, and \( D_l \) the liquid diffusivity. Below this threshold the interface breaks down successively into cellular, cellular-dendritic, and fully dendritic morphologies.

2.3 Microsegregation

Non-equilibrium partitioning during dendritic growth leaves a composition profile described by the Scheil equation,

\[ C_s = k C_0 (1 - f_s)^{k-1}, \]

where \( f_s \) is fraction solidified. The last-to-freeze interdendritic liquid is highly enriched in solute and is the preferred site of eutectic phases, inclusions, and shrinkage porosity. Homogenization heat treatment reduces microsegregation by diffusion at a rate that scales as \( D t / \lambda^2 \), with \( \lambda \) the secondary dendrite arm spacing.

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Chapter 3: Phase Transformations in the Solid State

3.1 Diffusional Transformations

Eutectoid decomposition, precipitation from supersaturated solution, and recrystallization share the nucleation-and-growth framework. The Johnson–Mehl–Avrami–Kolmogorov expression

\[ X = 1 - \exp(-k t^n) \]

describes the transformed volume fraction for constant nucleation and growth conditions. The exponent \( n \) reflects dimensionality and time-dependence of nucleation.

In steels, the eutectoid decomposition of austenite into pearlite at temperatures just below \( A_1 \) produces lamellae of ferrite and cementite whose interlamellar spacing decreases with increasing undercooling. Bainite forms at lower temperatures through a combination of diffusional and displacive events and presents distinctive sheaf morphologies.

3.2 Martensitic Transformations

When austenite is cooled rapidly enough that diffusion is suppressed, it transforms displacively to martensite—a body-centred-tetragonal phase supersaturated in carbon. The transformation is athermal, starting at \( M_s \) and proceeding as temperature drops toward \( M_f \). Hardness in steel martensite scales with carbon content; toughness recovers on tempering, during which carbide precipitates and dislocation density decreases.

3.3 Precipitation Hardening

Aluminium, nickel, and high-strength steel alloys rely on coherent or semi-coherent precipitates to pin dislocations. The sequence typically runs solution treatment → quench → natural or artificial aging, producing Guinier–Preston zones, metastable precipitates, and finally the equilibrium phase. Strength rises to a peak and falls during overaging as precipitates coarsen and lose coherency. The Orowan expression

\[ \tau = \frac{Gb}{L - 2r} \]

gives the stress to bow a dislocation between incoherent particles of spacing \( L \) and radius \( r \).

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Chapter 4: Manufacturing Processes and Microstructure

4.1 Casting

Shape castings experience thermal gradients that produce a chill zone of fine equiaxed grains at the mould wall, a columnar zone of grains aligned with the heat-flow direction, and, if conditions permit, an interior equiaxed zone. Grain refinement, mould coatings, and chills manipulate this structure. Porosity arises from shrinkage and from dissolved-gas rejection; Niyama and similar criteria quantify the likelihood of shrinkage porosity based on the local ratio \( G/\sqrt{T} \).

4.2 Solid-State Heat Treatments

Annealing, normalizing, quenching, and tempering are the workhorses. The iron–carbon diagram together with continuous-cooling-transformation (CCT) or time–temperature-transformation (TTT) diagrams predicts the phase mixture from a given cooling path. Hardenability is measured by Jominy end-quench and correlated with composition through Grossman and DI parameters; it governs the thickness that can be through-hardened for a given quench severity.

4.3 Laser Processing

Laser heat sources deliver concentrated energy with very high gradients and cooling rates (up to \( 10^6 \) K/s). Applications include surface hardening, welding, and additive manufacturing. The Rosenthal solution for a moving point heat source in a thick plate,

\[ T - T_0 = \frac{Q}{2\pi k r}\exp\!\left(-\frac{v(r+x)}{2\alpha}\right), \]

gives a first approximation to the temperature field and is the starting point for more sophisticated finite-element analyses.

4.4 Joining Operations

Fusion welding produces a heat-affected zone (HAZ) whose microstructure varies steeply from the fusion boundary outward. In low-alloy steels the coarse-grained HAZ next to the fusion line may transform to martensite on cooling, producing high hardness and susceptibility to hydrogen-assisted cracking. Preheat, interpass control, and low-hydrogen consumables mitigate this risk. Solid-state joining such as friction-stir welding avoids melting entirely and produces refined, thermomechanically processed microstructures with modest HAZ effects.

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Chapter 5: Case Studies and Design Application

5.1 Hardenability-Limited Design

Consider a shaft to be quenched and tempered to a uniform 35 HRC through a diameter of 75 mm. The combination of section size and quench severity sets a minimum ideal critical diameter \( D_I \), which in turn constrains the alloy composition through Grossman’s multiplying factors. Specifying a leaner steel than hardenability allows produces a soft core and premature fatigue failure.

5.2 Weld-Joint Failure Analysis

A fractured pressure-vessel weld is sectioned, polished, and etched. Optical microscopy reveals a coarse columnar fusion zone, a narrow coarse-grained HAZ, and a cracked region initiated at a subsurface slag inclusion. Hardness traverses show a peak of 380 HV in the HAZ—consistent with martensite. Corrective action combines preheat, low-hydrogen electrodes, and revised welding procedure specification supported by procedure qualification records.

5.3 Additive Manufacturing of a Turbine Blade

Laser powder-bed fusion of a nickel superalloy produces columnar grains aligned with the build direction, a texture that is strong but anisotropic. Post-build hot isostatic pressing closes residual porosity, solution treatment homogenizes segregation, and aging develops \( \gamma' \) precipitates. The resulting microstructure is compared with conventionally cast-and-heat-treated counterparts, and creep performance at service temperature is mapped against grain-boundary character.