Question 5 of 8: Grain Growth and Zener Pinning

Question 5: Grain Growth and Zener Pinning (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.

5.1 — (a) Why ≥6-sided grain configurations promote growth

In a 2-D polycrystalline network, boundary triple junctions meet at (very close to) 120° for equilibrium surface tension balance. A grain with exactly six sides can have all of its boundaries perfectly straight while still meeting its neighbours at 120° everywhere — the topological "neutral" case. A grain with FEWER than six sides has polygon angles smaller than 120°, so to meet its neighbours at 120° its boundaries must bow OUTWARD (convex, bulging into the neighbours), with their centres of curvature inside the grain. A grain with MORE than six sides has larger polygon angles, so its boundaries bow INWARD (concave as seen from outside, bulging into the grain itself), with their centres of curvature in the neighbouring grains. Since a boundary always migrates toward its own centre of curvature (to reduce its area), the boundaries of a grain with more than six sides move outward into its smaller neighbours — it grows — while those of a <6-sided grain move inward, and it shrinks and eventually disappears. A six-sided grain with straight boundaries is only in metastable balance; configurations of six or more intersections next to fewer-sided grains are what sustain growth. At high temperature, boundary mobility is thermally activated and high enough for this curvature-driven migration to proceed at an appreciable rate, so the effect is only observed as significant grain growth when both a temperature high enough for boundary mobility AND sufficient TIME are available.

5.2 — (b) Two grain-boundary motion mechanisms: recrystallization vs. grain growth

1. Recrystallization (strain-driven) boundary migration. The driving force is the STORED STRAIN ENERGY difference across the boundary: a newly nucleated, strain-free grain grows by consuming the surrounding heavily strained, dislocation-dense cold-worked matrix. The boundary always migrates FROM the low-dislocation-density (strain-free) side TOWARD the high-dislocation-density (strained) side, i.e. it moves so as to consume strain energy, and its velocity is proportional to the stored strain-energy difference $\Delta G_{\text{strain}}$ (which can be orders of magnitude larger than typical grain-boundary curvature driving forces) — direction is set entirely by WHERE the strain is, independent of boundary curvature.
2. Grain growth (curvature-driven) boundary migration. Once recrystallization is complete (the strained matrix is fully consumed and strain energy is no longer available as a driving force), the only remaining driving force is the reduction of grain-boundary AREA/energy itself, via the curvature argument of part (a): a boundary migrates toward its own centre of curvature, from the convex side into the concave side, regardless of which side has more or less stored strain (there isn't any left). Because grain growth boundaries move toward their centre of curvature while recrystallization boundaries move toward the region of HIGHER stored energy, the two mechanisms can, for the same physical boundary segment, drive migration in opposite senses depending on which stage of the anneal is in progress — exactly the apparent reversal the question describes.

5.3 — (c) Deriving $D_{\max}=4r/3f$ from the pinning-force balance

Given. Grain-boundary driving pressure $P_{\text{boundary}}=2\gamma/D$; a dispersion of spherical precipitates of radius $r$ at volume fraction $f$.

Find. $D_{\max}$ as a function of $f$ and $r$.

Approach. Derive the Zener drag pressure from the energy a boundary saves by intersecting a particle, then set drag pressure equal to the driving pressure (the definition of the pinned equilibrium).

1. Drag force per particle. A spherical particle of radius $r$ intersected by a boundary of energy $\gamma$ (per unit area) removes a boundary area up to $\pi r^2$ (the particle's maximum great-circle cross-section) from the total boundary energy budget. The maximum RESTRAINING FORCE the particle can exert follows from the boundary tension $\gamma$ acting around the contact circle of circumference $2\pi r\cos\phi$, resolved along the direction of motion ($\sin\phi$): $F=2\pi r\gamma\cos\phi\sin\phi=\pi r\gamma\sin2\phi$, maximised at $\phi=45^{\circ}$ giving $F_{\max}=\pi r\gamma$.
2. Number of particles intersecting unit boundary area. For a random dispersion of volume fraction $f$ and particle radius $r$, the number of particle CENTRES per unit volume is $N_v=f/(\tfrac43\pi r^3)$, and the number of particles intersecting a unit area of boundary (a slab of thickness $2r$ centred on the boundary) is $N_s=N_v\cdot2r=\dfrac{f}{\tfrac43\pi r^3}\cdot2r=\dfrac{3f}{2\pi r^2}$.
3. Total pinning (drag) pressure. Multiplying the force per particle by the number of particles per unit boundary area: $$P_{\text{drag}}=N_s\cdot F_{\max}=\dfrac{3f}{2\pi r^2}\cdot\pi r\gamma=\dfrac{3f\gamma}{2r}$$
4. Balance against the driving pressure at $D=D_{\max}$. Grain growth halts when the curvature-driven pressure can no longer exceed the drag pressure, i.e. at equality: $$\dfrac{2\gamma}{D_{\max}}=\dfrac{3f\gamma}{2r}\ \Longrightarrow\ \boxed{D_{\max}=\dfrac{4r}{3f}}$$

5.4 — (d) Conditions for minimum grain growth with Nb(C,N) in HSLA steel

Since $D_{\max}=4r/3f$, minimum grain growth (smallest possible $D_{\max}$) at a given exposure temperature requires:

1. Small precipitate radius $r$. A fine, closely spaced Nb(C,N) dispersion (achieved by controlled Nb microalloying additions and a thermomechanical schedule that promotes strain-induced precipitation during hot rolling rather than coarse precipitation during slow reheating) minimises $D_{\max}$ directly, and the drag pressure scales as $1/r$, so this is the more powerful lever of the two.
2. High volume fraction $f$. Sufficient dissolved Nb, C and N (within the solubility-product limit of Question 2) to precipitate a high number density of Nb(C,N) particles.
3. Thermal stability of the precipitate at the exposure temperature (the binding condition on both 1 and 2). $r$ and $f$ must stay small/high, respectively, AT TEMPERATURE — if the reheating or welding temperature exceeds the Nb(C,N) solvus, the particles dissolve ($f\rightarrow0$) and pinning is lost entirely (rapid, uncontrolled grain growth follows); even below the solvus, particles coarsen by Ostwald ripening over time (r increases, f roughly constant), so $D_{\max}$ rises with holding time. Minimum grain growth is therefore obtained by keeping the reheat/soak temperature safely below the Nb(C,N) solvus temperature and limiting time at temperature, so that the dispersion stays both fine and undissolved.