21-Mat-A5 Phase Transformations and Thermal Treatment · December 2018

Question 3 of 8: Precipitation Hardening, Spinodal Decomposition and Ordering

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Notes on this paper

Paper format. National Exams, December 2018 — 12-Mtl-A5, Phase Transformations of Metals, Glasses and Ceramics. Three hours, closed book, approved Casio or Sharp calculator only. Eight questions of 20 marks each; the rubric states that five questions constitute a complete paper and only the first five appearing in the answer book are marked. All eight are answered here, because this set is a study resource rather than an exam script. Several sub-parts explicitly call for an essay-format answer, and the rubric rewards clarity and organisation, so those answers are written as structured prose with supporting sketches rather than as note form.

Note on the exam code

This December 2018 sitting's printed header reads 12-Mtl-A5, Phase Transformations of Metals, Glasses and Ceramics, covering the Fe-C phase diagram and microstructural design, precipitate solubility, precipitation hardening and spinodal decomposition, interfaces and precipitate-free zones, grain growth and Zener pinning, nucleation mechanisms, classical nucleation theory and constitutional supercooling, and glass and glass-ceramic processing.

Reference texts. The answers below are keyed to the works normally recommended for this syllabus code:

D. A. Porter, K. E. Easterling and M. Y. Sherif, Phase Transformations in Metals and Alloys, 3rd ed. — nucleation and growth theory (Ch. 4), precipitation hardening and spinodal decomposition (Ch. 5), solidification (Ch. 4), recrystallization and grain growth (Ch. 3).
W. D. Callister and D. G. Rethwisch, Materials Science and Engineering: An Introduction, 10th ed. — the Fe-Fe₃C phase diagram and microstructural development (Ch. 9–10), glasses and glass-ceramics (Ch. 12–13).
R. E. Reed-Hill and R. Abbaschian, Physical Metallurgy Principles, 4th ed. — binary phase diagrams and the lever rule, ordering and superlattices, nucleation kinetics.
F. C. Campbell (ed.), Phase Diagrams: Understanding the Basics, ASM International — precipitation sequences, coherent/semi-coherent/incoherent interfaces.
ASM Handbook, Vol. 4 (Heat Treating) and Vol. 9 (Metallography and Microstructures) — microconstituent identification and heat-treatment practice.
J. E. Shelby, Introduction to Glass Science and Technology, 2nd ed. — glass transition, glass-ceramic nucleation and crystallization schedules.

Question 3: Precipitation Hardening, Spinodal Decomposition and Ordering (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

3.1 — (a) A range of metastable precipitates from one alloy: age-hardening Al-Cu

Take the classic precipitation-hardening system Al-4 wt%Cu. The equilibrium phase diagram has a terminal α (Al-rich FCC) solid solution whose solvus falls steeply with decreasing temperature, from roughly 5.6 wt%Cu at the eutectic temperature (548°C) to under 0.5 wt%Cu at room temperature, with the equilibrium precipitate being the intermetallic $\theta$ (CuAl₂).

Precipitation-hardening heat treatment on the α-solvus of a schematic Al-Cu-type diagram: (1) solution treat above the solvus to dissolve all Cu into a single-phase α; (2) quench to retain a supersaturated solid solution; (3) age below the solvus, during which the SAME single alloy composition can be driven through a whole sequence of increasingly stable, increasingly incoherent precipitate structures.

A single alloy composition, aged at a single temperature below the solvus, does not jump directly to the equilibrium $\theta$ precipitate. Instead the free-energy landscape favours a sequence of intermediate, metastable structures that trade interfacial energy against strain energy and driving force:

GP (Guinier–Preston) zones. Fully coherent, solute-rich, Cu-atom-plane clusters just 1–2 atomic layers thick on $\{100\}_\alpha$, with no distinct crystal structure of their own — essentially a local composition fluctuation still on the matrix lattice. Lowest interfacial energy, highest coherency strain.
$\theta''$ (intermediate). A distinct tetragonal structure, still fully coherent with the matrix on $\{100\}_\alpha$, larger than GP zones and giving the peak coherency-strain hardening (commercial peak-aged tempers target this stage).
$\theta'$ (intermediate). Semi-coherent tetragonal $\text{CuAl}_2$-related structure; the coherency strain is partially relieved by interfacial misfit dislocations, so the particle is larger and less effective per particle at blocking dislocations, but there are typically more of them.
$\theta$ (equilibrium) Incoherent, body-centred tetragonal $\text{CuAl}_2$, nucleated usually at grain boundaries or on existing $\theta'$; coarse and widely spaced (overaged condition), giving the lowest hardness of the sequence.

The same alloy composition therefore produces a whole family of metastable microstructures purely as a function of ageing time and temperature, because each successive structure is thermodynamically more stable (lower $\Delta G$) but kinetically slower to nucleate (higher interfacial energy penalty to overcome) than the one before it — the classic Ostwald step rule for precipitation sequences.

3.2 — (b) Precipitation versus spinodal decomposition

Both processes decompose a single supersaturated solid solution into two compositionally distinct phases, but they differ fundamentally in mechanism, driving force, and the microstructure produced:

Thermodynamic route. Classical precipitation occurs inside the metastable region of the phase diagram, where $\partial^2 G/\partial c^2>0$ everywhere — a homogeneous solution is a local free-energy minimum with respect to small composition fluctuations, so a fluctuation RAISES free energy locally and must be overcome by a nucleation barrier ($\Delta G^*$, exactly the barrier derived in Question 7(a)). Spinodal decomposition occurs inside the truly unstable region, where $\partial^2 G/\partial c^2<0$ — any infinitesimal composition fluctuation LOWERS the free energy immediately, so there is no nucleation barrier at all.
Mechanism. Precipitation proceeds by classical nucleation and growth: a discrete, sharply-bounded new phase forms at discrete sites (homogeneously or, more often, heterogeneously) and then grows by long-range diffusion down a normal (positive) concentration gradient. Spinodal decomposition proceeds by continuous, gradual amplification of small, diffuse composition waves throughout the whole volume simultaneously, driven by uphill (negative-diffusivity) diffusion — atoms diffuse UP an existing concentration gradient because doing so lowers the system's free energy, the opposite sense to Fick's-law-normal diffusion.
Resulting microstructure. Precipitation gives discrete particles with sharp interfaces, a distinct new crystal structure, and (usually) a nucleation incubation period before anything is detectable. Spinodal decomposition gives a continuous, interconnected, wavelength-selected modulated structure (no sharp particle/matrix boundary, at least initially) with NO incubation time — decomposition begins immediately upon entering the spinodal region, and coarsens progressively rather than nucleating and then growing discrete particles.
Compositional continuity. In classical precipitation the new phase has a composition that can, in principle, differ hugely and discontinuously from the matrix from the moment it appears. In spinodal decomposition the two regions differ only slightly in composition at first and diverge continuously and gradually toward the two equilibrium compositions as decomposition proceeds.

Topic. Spinodal vs. nucleation-and-growth decomposition of a supersaturated solid solution.

Key relations. Spinodal boundary defined by $\partial^2 G/\partial c^2=0$; inside it, the interdiffusion coefficient becomes effectively negative, $\tilde D=D(\partial^2G/\partial c^2)/RT<0$, which is exactly what drives uphill diffusion.

Why this works. The sign of $\partial^2G/\partial c^2$ is the whole story: it is positive in the metastable region between the binodal (solvus) and spinodal curves (nucleation barrier exists), and negative inside the spinodal (no barrier, spontaneous decomposition).

Common pitfall. Treating "spinodal decomposition" as just a faster or finer version of ordinary precipitation, rather than as a mechanistically distinct process with no nucleation step at all — the absence of an incubation time and the modulated (not discrete-particle) microstructure are the diagnostic signatures examiners look for.

Source. Porter, Easterling & Sherif, Phase Transformations in Metals and Alloys, 3rd ed., §5.3 (spinodal decomposition) & §5.5 (particle coarsening).

3.3 — (c) Ordered domains: why they form, and an example

An "ordered domain" forms when, below a critical ordering temperature $T_c$, unlike-atom nearest-neighbour bonds ($A$-$B$) are energetically more favourable than the average of like-atom bonds ($A$-$A$ and $B$-$B$), i.e. the ordering energy $\omega=E_{AB}-\tfrac12(E_{AA}+E_{BB})<0$. Below $T_c$ the enthalpy gain from maximizing unlike-neighbour bonds outweighs the configurational-entropy loss of giving up a random arrangement, so atoms preferentially occupy one of two (or more) distinct sublattices — the alloy adopts a superlattice with long-range order.

Disordered vs. ordered (B2, CsCl-type) arrangement of a 50:50 A-B alloy such as β-brass (CuZn): in the ordered state, A atoms occupy the cube-corner sublattice and B atoms the body-centre sublattice.

Because ordering nucleates independently at many points in the crystal simultaneously (there is no preferred single origin), each nucleation event can choose EITHER of the two sublattices as "the A sublattice" with equal probability — these are the two symmetrically equivalent, degenerate ordered variants. Where two such regions, ordered with opposite sublattice choices, grow and meet, they cannot merge into a single continuous superlattice: the boundary between them is an antiphase domain boundary, and each internally-consistent ordered region is an ordered domain. Long-range order is therefore only "long-range" within a single domain; a bulk sample below $T_c$ is a polydomain structure, with domain size set by how much time/temperature was available for domain coarsening (analogous to grain growth, driven by antiphase-boundary energy).

Topic. Order-disorder transformations and antiphase domains.

Key relations. Bragg-Williams long-range order parameter $S=(r-x_A)/(1-x_A)$, where $r$ is the fraction of "correct"-sublattice occupation; $S=1$ perfect order, $S=0$ fully random.

Why this works. A negative ordering energy makes the ordered state a true equilibrium ground state below $T_c$ (not just a metastable one), and ordering proceeds by short-range diffusive exchange of atoms. For second-order orderings such as β-brass (B2) there is no classical $\Delta G^*$ barrier — order develops continuously, much as spinodal decomposition does — whereas first-order orderings such as Cu₃Au (L1₂) nucleate and grow ordered regions; either way, independent ordering events produce antiphase domains.

Common pitfall. Describing an ordered structure without mentioning antiphase domains misses the reason ordered alloys are polycrystalline-within-a-single-grain at the ordering scale, and why domain boundaries (not just grain boundaries) can be strengthening or embrittling features.

Source. Reed-Hill & Abbaschian, Physical Metallurgy Principles, 4th ed., §11 (order-disorder transformations); Porter, Easterling & Sherif §3 (ordering reactions).

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