21-Mat-A6 Materials Selection and Design for Materials Processing · December 2017
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
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.
An interface between a precipitate and its surrounding matrix is classified by how completely the two crystal lattices remain in registry (atomic position matching) across the boundary, and how much registry CAN be maintained depends on the lattice misfit between the two phases, $\delta=(a_{\text{ppt}}-a_{\text{matrix}})/a_{\text{matrix}}$.
(i) The three interface types. A coherent interface has every lattice plane running continuously, atom-for-atom, across the boundary: the two lattices share a common set of atomic positions there, and any small lattice-parameter mismatch is accommodated entirely by uniform elastic strain in the surrounding lattice rather than by any structural discontinuity — achievable only for small misfit and/or a small particle. A semi-coherent interface still matches most lattice planes, but periodically inserts a misfit (edge) dislocation wherever the accumulated mismatch would otherwise become too large to keep straining elastically; between dislocations the interface is still locally coherent, but the array of interfacial dislocations partially relaxes the overall coherency strain. An incoherent interface has no continuity of lattice planes across the boundary at all — structurally similar to a general high-angle grain boundary — relieving all of the lattice mismatch at the cost of a disordered, higher-energy atomic arrangement.
(ii) Why the interfacial energies differ. Coherent interfaces have the lowest interfacial energy of the three (typically of order 10–200 mJ/m²) because atomic bonding across the boundary is essentially continuous — almost the entire energy cost is the ELASTIC coherency-strain energy stored in the surrounding lattice, not a true structural interfacial energy. Semi-coherent interfaces carry an intermediate interfacial energy (order 200–500 mJ/m²): between dislocations the low-energy coherent bonding is retained, but each misfit-dislocation core is a region of distorted bonding that adds a genuine structural contribution on top of the now-reduced residual elastic strain. Incoherent interfaces have the highest interfacial energy (order 500–1000 mJ/m², comparable to a random high-angle grain boundary), because essentially every bond crossing the boundary is mismatched relative to either lattice, with no elastic-strain "discount" at all.
(iii) Driving force for coherent→incoherent as the particle grows. For a fully coherent particle, the elastic coherency-strain energy stored around it scales with the particle's VOLUME ($\Delta G_{\text{elastic}}\approx A\mu\delta^2 V$ for shear modulus $\mu$, misfit $\delta$, volume $V$), since every unit volume of strained matrix around the particle adds its own elastic energy. The interfacial energy, by contrast, scales with SURFACE AREA. As radius $r$ increases, volume grows as $r^3$ while area grows only as $r^2$, so the elastic-strain penalty of staying fully coherent eventually outgrows the interfacial-energy cost of introducing misfit dislocations or losing registry altogether. Beyond a critical particle size, the system therefore lowers its TOTAL energy by paying a (per-unit-area) interfacial-energy increase to eliminate the faster-growing volumetric strain term — the same size threshold that governs, for example, the $\theta''\to\theta'$ transition in Al–Cu precipitation sequences.
Precipitate-free zones are narrow, precipitate-depleted bands adjacent to grain boundaries formed during aging of a precipitation-hardenable alloy, and arise from either of two mechanistically distinct causes.
Mechanism 1 — vacancy-denuded zone. Precipitate nucleation in the matrix depends on the excess quenched-in vacancy concentration retained from solution treatment, since those vacancies are what make solute clustering fast enough at the aging temperature. Grain boundaries are highly efficient vacancy sinks, so during quenching and early aging, vacancies within a diffusion distance of the boundary annihilate there faster than they can be replenished, leaving a boundary-adjacent zone whose LOCAL vacancy concentration falls below the level needed for matrix nucleation — even though the SOLUTE concentration there is unchanged from the bulk. The zone width scales with the vacancy-diffusion distance during the quench, $\sqrt{D_v t}$.
Mechanism 2 — solute-depletion zone. Grain boundaries are themselves excellent heterogeneous nucleation sites, so coarse precipitates nucleate and grow preferentially directly ON the boundary. As they grow, they draw solute by long-range diffusion from the adjacent matrix, depleting the matrix SOLUTE concentration in a boundary zone below the level needed to nucleate matrix precipitates — even though the vacancy concentration there may be entirely normal. Here the zone exists because there is not enough solute left locally, not because nucleation assistance is missing.
The two mechanisms are distinguished by whether the PFZ width tracks the vacancy-diffusion distance (mechanism 1, dominant at fast quench rates or low aging temperatures) or the solute-diffusion distance to the boundary precipitates (mechanism 2, dominant when boundary precipitation is coarse and vigorous). Because PFZs lack the dispersion-strengthening precipitates that harden the matrix, whichever mechanism operates leaves a preferential, lower-strength path for localized plastic flow and intergranular fracture — a practical concern in age-hardened Al alloys.
A migrating grain boundary is driven by its own curvature (capillary) pressure, $P_\gamma=2\gamma_b/D$, which shrinks small grains and grows large ones. A dispersion of fine, incoherent second-phase particles intersecting the boundary exerts an opposing retarding pressure — Zener pinning — because each particle the boundary must bypass costs it a small increment of boundary area that must be recreated: $P_z\approx 3f\gamma_b/2r$ for particle radius $r$ and volume fraction $f$. As long as the dispersion stays fine and thermally stable at the treatment temperature (resists Ostwald-ripening coarsening over the hold time), $P_z$ remains large relative to $P_\gamma$ for all but the very largest grains, and migration stalls once the two balance — the Zener limiting grain size $D_{\max}=4r/3f$ (derived in full in Question 3(c)). This is why FINE precipitates specifically are effective: a coarser dispersion of the same volume fraction gives a larger, less-restrictive $D_{\max}$, and if the particles themselves coarsen at the treatment temperature ($r$ increasing, $f$ roughly conserved), $D_{\max}$ grows with them and growth can eventually resume — the restriction is only as durable as the fine dispersion itself remains fine.
