Physical Metallurgy: Characterisation, Defects, Diffusion, Phase Transformations, Heat Treatment and Corrosion

Point defects are vacancies, self-interstitials and solute atoms, substitutional or interstitial. Vacancies are equilibrium defects, because the configurational entropy they add outweighs their formation energy at any temperature above 0 K: their fraction is n_v/N = exp(−Q_v/kT), with Q_v about 1 eV per atom for many metals, rising to about 10⁻⁴ near the melting point. Line defects are dislocations, whose geometry and mechanics belong to the next chapter: an edge dislocation has its Burgers vector perpendicular to its line, a screw dislocation parallel. Surface (planar) defects include the free surface, grain boundaries (low-angle boundaries are arrays of dislocations; high-angle boundaries, above roughly 10-15°, have a disordered core and a higher energy), twin boundaries (a mirror plane, with very low energy when coherent) and stacking faults (an error in the stacking sequence, ABCABABC in FCC, whose energy per unit area — the stacking-fault energy — controls how widely dislocations dissociate).

An interface between two phases is classified by how well their lattices match across it. The lattice misfit is δ = (a_β − a_α)/a_α. A coherent interface is one in which the planes run continuously across, with the small misfit taken up by elastic coherency strain: low interfacial energy (below about 200 mJ/m²) but a strain energy that grows with particle size — GP zones in Al-Cu are the example. When the particle grows, the strain is relieved by an array of misfit dislocations spaced D = b/δ apart, and the interface becomes semi-coherent, with intermediate energy (about 200-500 mJ/m²). When the misfit is large (above about 0.25) or the structures unrelated, no matching is possible and the interface is incoherent, with a high energy (about 500-1000 mJ/m²) similar to a high-angle grain boundary but no coherency strain. The sequence coherent → semi-coherent → incoherent is followed by a growing precipitate, and it controls both where precipitates nucleate and how they strengthen.

Small atoms — C, N, H, B in iron — diffuse interstitially, jumping between interstitial sites of a lattice that has plenty of them; the activation energy is only the migration barrier and diffusion is fast. Substitutional atoms, and the host atoms themselves, can move only into a neighbouring vacancy, so the activation energy is the sum of the vacancy formation and migration energies and diffusion is orders of magnitude slower. In both cases D = D₀ exp(−Q/RT). Fick’s second law with a constant surface concentration C_s imposed on a semi-infinite solid of initial uniform concentration C₀ — the carburising case — has the solution (C_x − C₀)/(C_s − C₀) = 1 − erf(x/(2√(Dt))), equivalently C_x = C_s − (C_s − C₀) erf(x/(2√(Dt))). Useful values: erf(0.5) = 0.5205 and erf(z) = 0.5 at z ≈ 0.477. Because the solution depends on x and t only through x/√(Dt), a given concentration is reached at a depth that grows as √(Dt): doubling the case depth takes four times as long, and the characteristic diffusion distance is x ≈ √(Dt).

Diffusion runs faster along short-circuit paths where the structure is more open: Q_surface < Q_grain boundary < Q_volume, so surface and grain-boundary diffusion dominate at low temperatures and in fine-grained material, and volume (lattice) diffusion at high temperatures, where the far larger cross-section of the lattice wins. The Kirkendall effect proved the vacancy mechanism: in a diffusion couple of copper and α-brass with inert molybdenum markers at the interface, the markers moved towards the brass, because zinc diffuses out of the brass faster than copper diffuses in, and the net flux of atoms is balanced by an opposite flux of vacancies, which can condense as Kirkendall voids on the faster-diffusing side. Darken’s equations describe it with intrinsic diffusivities, D̃ = x_B D_A + x_A D_B for the interdiffusion coefficient. Uphill diffusion — against the concentration gradient — occurs because the true driving force is the gradient of chemical potential, not of concentration: in Darken’s classic couple, carbon moved from a lower-carbon steel containing silicon (which raises the activity of carbon) into a higher-carbon steel without it; and spinodal decomposition inside a miscibility gap is uphill diffusion throughout.

Below the melting point the solid has the lower free energy, and for small undercooling ΔT the driving force per unit volume is ΔG_v ≈ ΔH_f ΔT/T_m (ΔH_f the latent heat per unit volume). A spherical embryo of radius r changes the free energy by ΔG = −(4/3)πr³ΔG_v + 4πr²γ: the surface term wins at small r, so the curve rises to a maximum at the critical radius r* = 2γ/ΔG_v, with a barrier ΔG* = 16πγ³/(3ΔG_v²). Both fall as the undercooling rises (r* ∝ 1/ΔT, ΔG* ∝ 1/ΔT²), which is why homogeneous nucleation in pure metals needs undercoolings of the order of 0.2 T_m and is almost never seen. Heterogeneous nucleation on a mould wall, an inoculant particle or an inclusion forms a spherical cap with contact angle θ, and the barrier is reduced by the shape factor S(θ) = (2 + cos θ)(1 − cos θ)²/4: S = 0 at θ = 0 (complete wetting, no barrier), 0.5 at 90° and 1 at 180°. The critical radius is the same; only the volume of the critical nucleus, and so the barrier, shrinks. Growth after nucleation, and the overall fraction transformed, follow the Johnson-Mehl-Avrami-Kolmogorov equation f = 1 − exp(−ktⁿ), which gives the S-shaped transformation curves from which TTT diagrams are built.

In an isomorphous alloy the first solid to form is richer in the high-melting component; the equilibrium partition coefficient k = C_S/C_L (less than 1 when the solute lowers the melting point) describes the rejection of solute into the liquid. Under equilibrium the solid re-homogenises by diffusion as it grows; in real castings diffusion in the solid is too slow and the result is coring — dendrite cores poor in solute and interdendritic regions rich in it. The limiting case of no solid diffusion and a perfectly mixed liquid is the Scheil equation, C_S = kC₀(1 − f_S)^(k−1), which predicts that the last liquid becomes infinitely enriched — in practice until a eutectic forms — and so explains why even a nominally single-phase alloy can contain some non-equilibrium eutectic. Coring (microsegregation) is removed by homogenisation annealing, whose time scales with the square of the dendrite arm spacing over D. A eutectic reaction L → α + β gives a lamellar or rod two-phase structure; a peritectic L + α → β forms β as a layer around α that then chokes the reaction, so peritectic alloys are rarely at equilibrium as cast.

The solute rejected ahead of a growing solid raises the local liquidus, so the liquid just ahead of the interface can be below its own liquidus even though the actual temperature rises into the liquid: constitutional supercooling. A planar front is stable only if the temperature gradient G at the interface and the growth rate R satisfy G/R > ΔT₀/D_L, with ΔT₀ = mC₀(1 − k)/k the freezing range. As G/R falls the front breaks down from planar to cellular to columnar dendritic, and far enough ahead of it equiaxed grains nucleate. The product G·R is the cooling rate, which sets the scale of the structure: dendrite arm spacings fall as the cooling rate rises (secondary arm spacing roughly ∝ (G·R)^(−1/3)). A casting shows this as three zones — a thin chill zone of fine equiaxed grains at the mould wall, a columnar zone of grains growing along the heat-flow direction, and a central equiaxed zone. Macrosegregation is composition variation on the scale of the casting: normal segregation (solute pushed to the last-freezing centre), inverse segregation (solute-rich liquid sucked back to the surface by shrinkage), gravity segregation, and the channel defects (A- and V-segregates) of large ingots.

Precipitation from a supersaturated solid solution, α′ → α + β, usually proceeds through metastable intermediates because they nucleate more easily: their coherent interfaces with the matrix have low energy. In Al-4% Cu the sequence is GP zones (copper-rich discs one or two atomic planes thick, fully coherent) → θ″ (coherent) → θ′ (semi-coherent) → θ (CuAl₂, incoherent). The eutectoid reaction in steel, γ → α + Fe₃C, forms pearlite by cooperative growth of ferrite and cementite lamellae behind a common front, with carbon diffusing sideways; the interlamellar spacing is inversely proportional to the undercooling below 727 °C, so faster cooling or lower transformation temperatures give finer, harder pearlite. Below the pearlite nose, bainite — ferrite plates with cementite between (upper) or within (lower) them — forms instead.

Martensite forms when austenite is cooled so fast that carbon cannot diffuse out: the FCC lattice shears into a body-centred tetragonal one (the Bain distortion plus a lattice-invariant shear by slip or twinning) with all the carbon trapped. The transformation is diffusionless and athermal — the fraction formed depends on how far below the martensite-start temperature M_s the steel is taken, not on time — and it ends at M_f. The c/a ratio of the tetragonal cell, and the hardness, rise with carbon content; M_s falls with carbon and alloying, so high-carbon steels retain austenite at room temperature. The plates or laths form on a specific habit plane with a shape change that produces surface relief. After a precipitate has formed, the total interfacial energy is lowered by coarsening (Ostwald ripening): small particles dissolve and large ones grow, because by the Gibbs-Thomson effect the solubility next to a particle of radius r is raised, C(r) = C∞ exp(2γV_m/(rRT)) ≈ C∞(1 + 2γV_m/(rRT)). Lifshitz-Slyozov-Wagner theory gives r³ − r₀³ = kt for diffusion-controlled coarsening, with k ∝ Dγ C∞ — so a creep-resistant alloy chooses precipitates of low solubility, low diffusivity or low interfacial energy.

The glass transition is not a phase change in the thermodynamic sense but a kinetic one. On cooling a liquid that does not crystallise — a silicate melt, or an amorphous polymer — the viscosity rises steeply until, at the glass transition temperature T_g, molecular rearrangement can no longer keep up with the cooling rate and the structure freezes as a rigid, disordered solid, with a change of slope in volume (and a step in heat capacity) rather than the discontinuity of melting. T_g depends on the cooling rate. For oxide glasses the viscosity at T_g is about 10¹² Pa·s; working (about 10³ Pa·s), softening and annealing points are defined on the same viscosity scale. For polymers, T_g separates the glassy (brittle, stiff) from the rubbery or leathery state; semicrystalline polymers show both a T_g for their amorphous fraction and a T_m for their crystals.

Steels. Annealing (full annealing: austenitise about 30-50 °C above A₃ and furnace cool) gives coarse pearlite and maximum softness; normalising (air cooling from the same range) gives finer pearlite and a refined grain; spheroidising (long holding just below A₁) turns cementite into spheres for machinability and formability; hardening quenches from the austenite field to martensite; and tempering reheats martensite (stages: carbide precipitation, decomposition of retained austenite, cementite formation and coarsening, then recovery) to trade hardness for toughness. The isothermal transformation (TTT) diagram records start and finish times at each temperature after an instantaneous quench; the continuous cooling transformation (CCT) diagram records what happens along real cooling curves, and its curves lie lower and to the right. The critical cooling rate is the slowest rate that misses the nose and gives full martensite. Hardenability — the depth to which a steel hardens — is raised by most alloying elements (Mn, Cr, Mo, Ni, B) and by a coarser austenite grain, which move the curves to the right; it is measured by the Jominy end-quench test and expressed as an ideal critical diameter. Austempering holds just above M_s to form bainite without quench distortion; martempering (marquenching) holds just above M_s to equalise temperature, then cools slowly through the martensite range.

Surface hardening gives a hard, wear-resistant case on a tough core. Thermochemical methods change the surface composition: carburising (diffusing carbon into a low-carbon steel at about 900-950 °C, then quenching and tempering), nitriding (nitrogen at about 500-550 °C into steels with Al, Cr, Mo, V, giving hard nitrides with no quench and little distortion), and carbonitriding. Thermal methods change only the structure: flame and induction hardening heat a thin surface layer of a medium-carbon steel into austenite and quench it, the induction case depth falling as frequency rises (skin effect). Cold-worked metal stores energy as dislocations; on heating it passes through recovery (dislocation rearrangement into low-energy subgrain walls; stress relief with little change in strength), recrystallisation (nucleation and growth of new strain-free grains, driven by the stored energy; the recrystallisation temperature is roughly 0.3-0.5 T_m and falls with more prior cold work and higher purity; there is a critical minimum strain below which it does not occur) and grain growth (driven by grain-boundary energy, d² − d₀² = kt ideally, and pinned by second-phase particles).

Cast irons (above about 2.1% C) are classified by the form of their carbon. White iron has all its carbon as cementite and is hard and brittle; grey iron has graphite flakes, which give good damping and machinability but act as cracks; ductile (spheroidal-graphite, SG) iron has graphite nodules produced by treating the melt with magnesium or cerium, and is tough; malleable iron is white iron annealed for many hours so that cementite decomposes into irregular temper-carbon nodules. Heat treatments include stress relief, annealing to a ferritic matrix, normalising to a pearlitic one, hardening, and austempering ductile iron to ADI, whose ausferrite matrix gives a strength-toughness combination rivalling steels. Precipitation-hardening aluminium alloys (2xxx, 6xxx, 7xxx) are solution treated in the single-phase field, quenched to retain a supersaturated solution, and aged — naturally at room temperature (T4) or artificially at about 150-190 °C (T6). Hardness rises as GP zones and coherent precipitates form, peaks, then falls on overageing as the precipitates coarsen and lose coherency; ageing at a higher temperature reaches a lower peak sooner.

The general methods of prevention follow from the electrochemistry. Material selection and design remove the conditions (dissimilar metals, crevices, stagnant water). Coatings separate the metal from the electrolyte: barrier coatings (paint, polymers, enamel), noble metallic coatings (nickel, chromium — protective only while unbroken), and sacrificial coatings (zinc in galvanising, which protects steel even at a scratch because zinc is the anode). Cathodic protection makes the whole structure a cathode, either with sacrificial anodes of zinc, magnesium or aluminium, or with an impressed current from a rectifier through inert anodes; it is used on pipelines, ships’ hulls and buried tanks. Anodic protection holds a passivating metal (stainless steel in sulphuric acid) inside its passive range with a potentiostat. Inhibitors added to the environment adsorb on anodic or cathodic sites or scavenge oxygen.