FR39 · Crystal defects

Grain Growth by Curvature: Why Small Grains Vanish

Why do grains with fewer than six sides shrink while grains with more grow? Each boundary moves toward its centre of curvature, and where three boundaries meet the angles settle near 120 degrees. A grain with fewer than six sides then has boundaries bowed outward, so it shrinks; one with more than six has them bowed inward, so it grows. In two dimensions the rate depends on the side count alone, not on the grain size.

Anneal a phase-field polycrystal and test the von Neumann–Mullins law: each grain changes area at a rate set by its number of sides alone.

Grains in the box $N=70$

Mean number of sides $\langle n\rangle=5.94$

Measured von Neumann–Mullins slope $\frac{\mathrm{slope}}{Lk}=1.016\pm0.017$

Mean grain area $\langle A\rangle=234\ \text{cells}$

Anneal a polycrystal and watch it coarsen

Colour grains by

Grains at the start

70

N_0

Radius of a dropped round grain

24

R/\Delta x

Boundary mobility

1

L

Gradient energy coefficient

1

k

Drag the ring onto any grain to follow it, or drag the plus sign into the box to drop a new round grain.

Playback speed

Time steps per frame

4

n_{\mathrm{step}}

Model, method and sources

$\frac{\partial\eta_i}{\partial t}=-L\Bigl(-\eta_i+\eta_i^3+2\eta_i\sum_{j\neq i}\eta_j^2-k\nabla^2\eta_i\Bigr),\qquad \frac{dA}{dt}=\frac{\pi}{3}Lk\,(n-6)$

Every grain owns its own order parameter, and all of them relax down the free-energy gradient of the multi-order-parameter model of Kazaryan, Wang, Dregia and Patton (1999) with isotropic boundary energy and mobility. A boundary then moves at a speed proportional to its curvature, so a grain with fewer than six sides shrinks and one with more grows at a rate fixed by its side count alone. The box is periodic, and sides are counted between grains that share at least two cell faces.

Kazaryan, Wang, Dregia & Patton (1999), arXiv:cond-mat/9912044v1 · Mason, Lazar, MacPherson & Srolovitz (2015), arXiv:1507.03379v1

Derive the von Neumann–Mullins law

Model assumptions

Each grain has its own order parameter, and the field relaxes by the multi-order-parameter model of Kazaryan, Wang, Dregia and Patton (1999) with isotropic boundary energy and mobility. Boundaries move at a speed proportional to their curvature; where three meet, the angles settle near 120 degrees. The scatter plot collects the area rate of every grain that keeps its side count over a short window. The dashed line is the von Neumann–Mullins law for isotropic two-dimensional grain growth, a reference rather than a fit. The box is periodic and 128 cells wide; the microstructure is simulated, not a micrograph.

Area rate against number of sides

Area of the picked grain

More numbers and how they are computed

Sides of the picked grain: $0$
Area rate of the picked grain: $0$
Von Neumann–Mullins slope: $0$
Grain rates collected: $0$
Model time: $0$
Free energy of the field: $0$
Run and seed: $0$

A grain's area is the sum over cells of its share of the order parameters, so it changes smoothly. Its rate is a straight-line fit over a window in which its side count stays the same, and the slope readout is a weighted fit of all such rates against the number of sides minus six. The exported file lists the seed, the settings and every collected rate.

Physics tutorial

Grain boundaries straighten, and small grains pay for it

BackgroundA metal or ceramic is usually a polycrystal: many small crystals, the grains, each with its own lattice orientation. The boundaries between them cost energy, so when the material is hot enough for atoms to hop across, the boundaries move to reduce their total length.

Why it mattersGrain size sets strength, toughness and creep resistance, so annealing schedules are designed around how fast grains coarsen. In two dimensions that growth obeys one of the few exact laws in materials science.

Start with the essentials

Focus question: Why do grains with fewer than six sides shrink while grains with more grow?
One-sentence intuition: Each boundary moves toward its centre of curvature at a speed proportional to the curvature. Where three boundaries meet, the angles relax to 120 degrees. Adding up the turning of a grain's perimeter then gives a rate that depends on the number of sides alone: $dA/dt=(\pi/3)\,M\gamma\,(n-6)$ .

Core mathematical model

Curvature-driven boundary motion

v_n=M\gamma\,\kappa,\qquad \frac{dA}{dt}=-2\pi M\gamma\ \ \text{(isolated round grain)}

A round grain of radius $R$ has $\kappa=1/R$ , so its area falls at a constant rate whatever its size. In the phase-field model the product $M\gamma$ becomes $Lk$ .

Von Neumann–Mullins law

\frac{dA_n}{dt}=-M\gamma\Bigl(2\pi-n\,\frac{\pi}{3}\Bigr)=\frac{\pi}{3}\,M\gamma\,(n-6)

The tangent of a closed perimeter turns through 2π in total. Each triple junction supplies a jump of π/3, and the curved sides supply the rest, which is what drives the area change.

Common difficulties

Size is not what decides

Typical misconceptionBig grains grow and small grains shrink because of their size.

Better mental modelIn two dimensions the rate depends on the number of sides, not the area. Small grains tend to have few sides, which is why they usually shrink, but a small seven-sided grain still grows.

Run the experiment

01
Watch a polycrystal coarsen
Start with A polycrystal coarsens and follow the scatter plot of area rate against number of sides.
What to observe: The points line up on the dashed von Neumann–Mullins line, and the grain count drops while the mean area grows.
02
Shrink a lone round grain
Choose A round grain shrinks at a steady rate and watch the picked-grain chart.
What to observe: Its area falls along a straight line until it vanishes, as the constant rate for a round grain predicts.
03
Follow a defect in a honeycomb
Choose One defect in a honeycomb and drag the ring onto the small grain in the middle.
What to observe: Six-sided grains barely change, the small grain with fewer sides shrinks and vanishes, and its neighbours lose a side each.

Grain Growth by Curvature: Why Small Grains Vanish

Grain boundaries straighten, and small grains pay for it

Core mathematical model

Curvature-driven boundary motion

Von Neumann–Mullins law

Common difficulties

Size is not what decides

Run the experiment

Watch a polycrystal coarsen

Shrink a lone round grain

Follow a defect in a honeycomb

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