Structure & Transformations · Grain growth

Grain growth and Zener pinning: why grains grow and how particles stop them

The first simulation below is already running. Watch the small grains with few sides shrink and vanish, then drag the particle sliders in the second one to stop the growth.

Why do grains grow when a metal is heated?

Each colour is one grain, a crystal with its own orientation. A grain boundary costs energy, about 0.3 J/m² in aluminium and 0.9 J/m² in nickel. Hot metal lowers its energy by losing boundary: a curved boundary moves toward its centre of curvature, small grains shrink away and the others get bigger. This is a 200 × 200 site Potts Monte Carlo model (Anderson et al. 1984).

Time
0 MCS
Grains left
0
Mean size d now
0 px
Start size d₀
0 px
Rate k (slope of d² − d₀²)
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Growth exponent n
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Sides where area stops changing (von Neumann-Mullins says 6)
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Try it: switch "Colour grains by" to number of sides. Red grains (fewer than 6 sides) shrink; the bottom plot shows area loss for n below 6 and gain above 6. Then set M to 0.5 and press New start: the rate k drops to about half.

How do particles stop grain growth (Zener pinning)?

Black dots are second-phase particles, for example oxides or carbides. A boundary that touches a particle loses the boundary area the particle covers, so it must pay energy to leave. Many particles hold the boundaries still and growth stops at a limiting size. Smith and Zener (1948) estimated it as d_lim = 4r/(3f) for round particles of radius r and volume fraction f.

Growing
Time
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Mean size d
0 px
Zener 4r/(3f)
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Simulated d ÷ Zener
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Slope of your points, d_lim ∝ f^(-m)
run 2 or more fractions
Real example: same f with r = 50 nm particles
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Try it: press "Run f = 0.02, 0.04, 0.08" and wait a minute or two. Each run stops at a size that falls as f rises. The fitted slope m comes out near 1/2, not the 1 of 4r/(3f), because this is a flat 2D model.

Which wins: boundary curvature or particle pinning?

A curved boundary is pushed toward its centre by the pressure P = 2γ/R (a sphere-shaped boundary; γ/R for a cylinder). Particles on the boundary hold it back with up to P_Z = 3fγ/(2r). The boundary moves only while P is larger. The two are equal at R* = 4r/(3f), which is where the Zener limit comes from.

Boundary moves
Curvature pressure P
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Zener pressure P_Z
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Net push P − P_Z
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Balance radius R*
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Particle spacing on the boundary, λ = r√(2π/(3f))
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Try it: start at R = 5 µm (steel-like: γ = 0.5 J/m², f = 0.01, r = 50 nm): P = 200 kPa beats P_Z = 150 kPa. Drag R to 10 µm: P drops to 100 kPa and the boundary stops. Halve r to 25 nm: P_Z doubles.

What is abnormal grain growth?

Sometimes one grain grows far bigger than all the others. It needs an edge the others lack: faster boundaries, lower-energy boundaries, or a spot where the pinning particles are missing. Here the other grains are held back by particles (f = 0.04 to start). The red grain starts about 2 times the mean size, so you can see that size alone is not enough.

Normal
Time
0 MCS
Grains left
0
Red grain size D
0 px
Mean of the others d̄
0 px
D ÷ d̄ (abnormal when above about 3)
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Try it: the red grain has boundary energy half the others' and eats the pinned grains. Switch to "None": it shrinks back below the mean size and can vanish. "Uneven pinning" with patch radius 40: the largest grain in the particle-free middle passes 3 times the mean after about 2000 MCS.

What should I remember about grain growth?

Boundaries cost energyAbout 0.3 to 0.9 J/m² in pure metals. Heating lets the metal remove boundary area, so the mean grain size goes up and the number of grains goes down.
Ideal growth is parabolicd² − d₀² = kt. Pure 2D models give an exponent near 2 to 2.5. Real metals often give n between 2 and 4, because solute and particles slow the boundaries.
Fewer than 6 sides: shrinkIn 2D, a grain with n sides changes area at a rate set by (n − 6) (von Neumann-Mullins). Small grains have few sides, so they vanish.
Particles set a size limitFine particles at a high fraction give small grains: d_lim ≈ 4r/(3f) in 3D. One grain with an advantage can escape and grow abnormally.
More detail: equations, the model, and its limits

Growth law (Burke and Turnbull 1952). Boundary speed is v = M P, with mobility M and pressure P = 2γ/R. If R scales with the mean grain size d, then d(dd)/dt is a constant, so d² − d₀² = k t with k proportional to Mγ. Mobility M rises steeply with temperature, so k does too. In real metals, fits of dⁿ − d₀ⁿ = k t often give n from 2 to 4 (Humphreys and Hatherly 2004), because of solute drag, particles, texture and a spread of boundary types.

von Neumann-Mullins (Mullins 1956). In 2D with equal boundary energies and 120° triple junctions, dA/dt = (π/3) M γ (n − 6). A grain's area rate depends only on its number of sides n. The bottom plot of widget 1 measures dA/dt for each n every 10 MCS and fits a line; the readout gives where the line crosses zero.

Zener pinning (Smith 1948). One sphere of radius r can pull back on a boundary with a force of at most π r γ. A flat boundary cuts n_s = 3f/(2πr²) particles per unit area, so the pinning pressure is P_Z = 3fγ/(2r). Setting 2γ/R = P_Z gives R* = 4r/(3f). Here R is the radius of curvature of the boundary. Textbooks (Porter, Easterling and Sherif 2009) take R as about the grain diameter, which gives d_lim = 4r/(3f); if you take R as the grain radius instead, the limiting diameter doubles to 8r/(3f). That factor of 2 is smaller than the spread between later models. Later models and experiments change the number in front and sometimes the power of f; see the review by Manohar, Ferry and Chandra (1998). Treat 4r/(3f) as a first estimate.

Why 2D gives a different scaling. In a flat model, particles are discs that pierce the plane, and a boundary line meets them at a different rate than a boundary surface meets spheres. 2D Monte Carlo and vertex studies find d_lim ∝ r/√f (Srolovitz et al. 1984; Kad and Hazzledine 1997), while 3D ones find about r/f^(1/3) at the fractions they could simulate. So the "d ÷ Zener" readout is not expected to be 1.

The model. Each site holds a grain number. One Monte Carlo step (MCS) tries, on average, one change per site: pick a site, pick one of its 8 neighbours, and try giving the site that neighbour's grain number. The energy is J for each unlike neighbour pair (J = 1). The change is kept if the energy does not rise, or with probability exp(−ΔE/kT) if it does, and the result is scaled by the mobility M in widget 1. kT is 0.5 in widget 1, 0.1 by default in widget 2 (so pinned boundaries stay put), and 0.3 in widget 4. Particle sites never change, and a grain next to a particle always counts as unlike, so particle and boundary energies are equal (the incoherent case). For widget 4, the red grain starts as a disc of radius 6 sites (about 2 times the mean size), and its boundaries have energy γ_red/γ = 0.4 to 1, or a mobility 1 to 16 times higher (all other flips are slowed by that factor). In this model a larger grain with no advantage shrinks back toward the mean; abnormal growth needs an edge in energy, mobility or pinning (Rollett et al. 1989 used energy and mobility edges).

Limits. Time is in MCS, not seconds; sizes are in lattice sites. The square lattice gives slightly faceted boundaries at low kT. The box is 200 × 200 with wrap-around edges, so runs stop being meaningful when fewer than about 25 grains are left. At the low kT of widget 2 (0.1) boundaries move more slowly but do not freeze: with f = 0 the mean size still doubles between 250 and 1500 MCS. Real particles also coarsen by Ostwald ripening, so d_lim slowly rises in real alloys; here they are fixed. Grain size d here is √(4Ā/π), from the mean grain area Ā.

Numbers used. γ = 0.5 J/m² is a round value inside the measured range for high-angle boundaries in pure metals, about 0.32 J/m² (Al) to 0.87 J/m² (Ni) (Porter, Easterling and Sherif 2009; Rollett lecture notes), with about 1 J/m² for Fe and Ni near their melting points (Roth 1975).

What else do people ask about grain growth and pinning?

What is the grain growth equation?

d² − d₀² = k t for ideal growth, where d₀ is the starting size and k depends on boundary mobility and energy. Real data are often fitted with dⁿ − d₀ⁿ = k t, with n between 2 and 4.

Why do grains with fewer than six sides shrink?

With 120° angles at the triple junctions, a grain with fewer than 6 sides must have boundaries that bulge outward. Each boundary moves toward its centre of curvature, so the grain shrinks. More than 6 sides means the boundaries curve inward and the grain grows.

What is the Zener pinning equation?

d_lim = 4r/(3f), from balancing the curvature pressure 2γ/R with the pinning pressure 3fγ/(2r). For r = 50 nm and f = 0.01 it gives about 6.7 µm. Other models use a different number in front.

What causes abnormal grain growth?

A few grains grow much larger than the rest when normal growth is held back (by particles or texture) and those grains have an edge: faster or lower-energy boundaries, or a region where particles dissolved or coarsened. It is also called secondary recrystallization.

Why does grain size matter for strength?

Smaller grains mean more boundaries that block dislocations, so the yield strength goes up (the Hall-Petch relation). Pinning particles keep grains small during hot working. See strengthening mechanisms.

How do I measure grain size after annealing?

Use line intercepts or grain areas from an optical image or an EBSD map, and state the method, since they give different numbers. See measuring grain size.

Why does a Monte Carlo Potts model show real grain growth?

Its only rule is to lower the total boundary length, which is also what drives real grain growth. It reproduces the shrinking of few-sided grains and a nearly parabolic growth law. It does not give real times; those need a mobility in m⁴/(J·s).

Does grain growth ever stop without particles?

Not fully in a thick piece of pure metal; it only slows, because the push 2γ/R falls as grains get bigger. Solute atoms that drag on boundaries slow it further, and in thin sheets and films growth nearly stops once grains are a few times as wide as the sheet is thick, because grooves where boundaries meet the surface hold them.

Related pages: Ostwald ripening of particles · boundary energy and motion · Hall-Petch strengthening · measuring grain size · diffusion couples · boundary types in EBSD

Where do these numbers come from?

Show the 12 references
  1. Humphreys, F.J., Hatherly, M. Recrystallization and Related Annealing Phenomena, 2nd ed. Elsevier, Oxford, 2004.
  2. Burke, J.E., Turnbull, D. Recrystallization and grain growth. Progress in Metal Physics 3 (1952) 220-292.
  3. Smith, C.S. Grains, phases, and interfaces: an interpretation of microstructure. Transactions of the AIME 175 (1948) 15-51 (reporting Zener's estimate).
  4. Mullins, W.W. Two-dimensional motion of idealized grain boundaries. Journal of Applied Physics 27 (1956) 900-904. doi:10.1063/1.1722511
  5. Anderson, M.P., Srolovitz, D.J., Grest, G.S., Sahni, P.S. Computer simulation of grain growth, I. Kinetics. Acta Metallurgica 32 (1984) 783-791.
  6. Srolovitz, D.J., Anderson, M.P., Grest, G.S., Sahni, P.S. Computer simulation of grain growth, III. Influence of a particle dispersion. Acta Metallurgica 32 (1984) 1429-1438.
  7. Manohar, P.A., Ferry, M., Chandra, T. Five decades of the Zener equation. ISIJ International 38 (1998) 913-924. doi:10.2355/isijinternational.38.913
  8. Kad, B.K., Hazzledine, P.M. Monte Carlo simulations of grain growth and Zener pinning. Materials Science and Engineering A 238 (1997) 70-77.
  9. Rollett, A.D., Srolovitz, D.J., Anderson, M.P. Simulation and theory of abnormal grain growth: anisotropic grain boundary energies and mobilities. Acta Metallurgica 37 (1989) 1227-1240.
  10. Porter, D.A., Easterling, K.E., Sherif, M.Y. Phase Transformations in Metals and Alloys, 3rd ed. CRC Press, Boca Raton, 2009 (Ch. 3: grain boundary energies and grain growth).
  11. Roth, T.A. The surface and grain boundary energies of iron, cobalt and nickel. Materials Science and Engineering 18 (1975) 183-192.
  12. Rollett, A.D. Grain boundary properties: energy. Course 27-750 lecture notes, Carnegie Mellon University, 2014.
Cite this page: Tripathy, Manisha. “Grain Growth and Zener Pinning.” untethered atom, 2026, https://untetheredatom.com/phase-transformations/grain-growth-and-zener-pinning.
BibTeX 
@misc{tripathy2026graingrowth,
  author = {Tripathy, Manisha},
  title  = {Grain Growth and Zener Pinning},
  year   = {2026},
  howpublished = {\url{https://untetheredatom.com/phase-transformations/grain-growth-and-zener-pinning}},
  note   = {Interactive web tool}
}