Questions: Grain Boundaries and Interfacial Defects

Question 1

An engineer reduces the average grain size of a steel alloy from 100 μm to 25 μm. According to the Hall-Petch relationship, by what factor does the grain-boundary strengthening contribution (k/√d) change?

• A. It increases by a factor of 2 — grain size decreased by 4, so √d decreased by 2
• B. It increases by a factor of 4 — strength scales directly with 1/d
• C. It decreases by a factor of 2 — smaller grains have fewer dislocations to contribute to strengthening
• D. It doubles — the number of grain boundaries doubles when diameter is halved

Answer: (A)

The Hall-Petch relationship is σ_y = σ₀ + k/√d. The grain size decreased from 100 to 25 μm — a factor of 4. Since strength scales as 1/√d, a 4× decrease in d gives a 2× increase in the 1/√d term (√4 = 2). The strengthening contribution doubles. Note that option B is the common error of assuming linear rather than square-root dependence.

Question 2

A stainless steel component has been held at 600°C for several hours, causing chromium carbide precipitation at grain boundaries. The material is then exposed to a corrosive environment. What failure mode is most likely?

• A. Uniform corrosion across the entire surface because carbides increase overall reactivity
• B. Intergranular corrosion preferentially attacking the chromium-depleted zones adjacent to grain boundaries, leaving grain interiors intact
• C. Pitting corrosion at the center of grains where carbide precipitation is highest
• D. Stress corrosion cracking driven by the increased dislocation density near carbides

Answer: (B)

Sensitization — chromium carbide precipitation at grain boundaries — depletes the adjacent matrix of chromium below the ~12% threshold needed for passivation. The grain interiors retain their full chromium content and resist corrosion, while the boundary regions are electrochemically active. The result is intergranular corrosion: rapid attack along the grain boundary network that can cause catastrophic failure (knife-line attack) while the bulk material appears intact.

Question 3

The Hall-Petch effect arises because finer grains contain fewer dislocations, making it harder for plastic deformation to initiate.

• T. True
• F. False

Answer: False

The Hall-Petch mechanism is about dislocation MOTION, not dislocation density. In a polycrystalline material, dislocations glide along slip planes until they pile up at grain boundaries, where the crystallographic mismatch prevents easy cross-boundary slip transfer. The stress concentration at the pile-up tip must build high enough to nucleate slip in the adjacent grain. Finer grains limit pile-up length, requiring higher applied stress for slip propagation. The effect is a barrier effect, not a source-density effect.

Question 4

Low-angle grain boundaries (misorientation < ~15°) can be modeled as an ordered array of edge dislocations, with dislocation spacing decreasing as the misorientation angle increases.

• T. True
• F. False

Answer: True

This is the Frank formula relationship: an array of parallel edge dislocations produces a tilt of the crystal lattice across the boundary, and the dislocation spacing D is related to misorientation angle θ by D = b/θ (for small angles, where b is the Burgers vector). As misorientation increases, dislocations pack more closely, increasing boundary energy. Above ~15°, the dislocations overlap and this model breaks down — high-angle boundaries are more disordered and have higher, angle-independent energy.

Question 5

How does grain boundary energy drive grain growth during high-temperature annealing, and why do engineers sometimes want to inhibit this process?

Model answer: Grain boundaries are high-energy surfaces — atoms at the interface are in distorted, elevated-energy positions compared to atoms in the grain interior. The total grain boundary energy of a polycrystalline material is proportional to total boundary area. At elevated temperature, atoms have enough thermal energy to diffuse, allowing boundaries to migrate toward their center of curvature. This reduces boundary curvature and total area, causing large grains to grow at the expense of small ones — lowering the system's total energy. Engineers want to inhibit grain growth when fine-grained microstructures are desired for high strength (Hall-Petch effect). Second-phase particles (Zener pinning) physically obstruct boundary migration, stabilizing grain size during processing or high-temperature service.

This explains why many engineering processes (age hardening, controlled cooling, alloying with grain-boundary-pinning elements like niobium or vanadium in steels) specifically target grain size stability — the microstructure-property relationship is only as useful as the microstructure's stability under service conditions.