---
title: "Grain Growth by Curvature: Why Small Grains Vanish Interactive Lab | Sandbox Physics"
description: "Why do grains with fewer than six sides shrink while grains with more grow?"
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language: "en"
page_type: "lab"
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  zh-CN: "https://sandboxphysics.com/zh/labs/curvature-grain-growth/"
last_modified: "2026-10-31"
---

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.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

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

**Background** A 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 matters** Grain 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 misconception** Big grains grow and small grains shrink because of their size.

**Better mental model** In 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

1. 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.

2. 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.

3. 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.

Physical conclusion

Curvature drives grain boundaries, triple junctions fix their angles, and together they make a grain's fate in two dimensions depend only on how many neighbours it has.

## Related experiments and articles

Crystal defects and microstructure · Reading path

### From the Burgers vector to alloy dendrites

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **Grain boundaries move by curvature, so a round grain shrinks at a steady rate**](https://sandboxphysics.com/stories/crystal-defects-microstructure/grain-boundaries-move-by-curvature/)
2. [Article **The von Neumann–Mullins law says grains with fewer than six sides shrink**](https://sandboxphysics.com/stories/crystal-defects-microstructure/von-neumann-mullins-and-coarsening/)

Fields & matter · Starter path

## From a field to an electron microscope

Map an electric field and steer a charged particle. Then find the electron's wavelength and use electron beams to see a crystal and a surface.

[Start here → **Make an Electric Field Visible** Move the probe around a dipole and check that the field always points downhill on the potential landscape. Observe · 8 min](https://sandboxphysics.com/labs/electric-field/)

1. [1 Make an Electric Field Visible](https://sandboxphysics.com/labs/electric-field/)
2. [2 A Charge Bending through Fields](https://sandboxphysics.com/labs/charged-particle/)
3. [3 Electron Wavelength & Voltage](https://sandboxphysics.com/labs/electron-wavelength/)
4. [4 Electron Diffraction & Reciprocal Space](https://sandboxphysics.com/labs/electron-diffraction/)
5. [5 TEM Bright Field, Dark Field & SAED](https://sandboxphysics.com/labs/tem-mode-selector/)
6. [6 SEM Secondary-Electron Topography](https://sandboxphysics.com/labs/sem-topography/)

[View the full path →](https://sandboxphysics.com/topics/fields-and-matter/#starter-path)