Crystal Growth: Diffusion-Limited Aggregation in 3D

A random walk that never lets go

Diffusion-limited aggregation (DLA) models how crystals, soot, mineral deposits and metal electroplates grow when the limiting factor is not the sticking itself but getting a new particle to the surface at all. The recipe is almost embarrassingly simple: drop a seed particle in the centre, release a new particle far away, let it wander in a random walk — a Brownian path with no preferred direction — and the instant it touches the growing cluster, freeze it in place forever. Repeat thousands of times.

live demo · a random-walking particle about to stick to the cluster● LIVE

This was formalised by Thomas Witten and Leonard Sander in 1981, and it reproduces something real crystals and lightning-like discharges both do: branching, fractal growth rather than smooth, dense growth. A particle wandering toward a bumpy cluster is statistically far more likely to hit an outward-pointing tip than to wander all the way into a deep concave gap — the tip "screens" the gap from incoming particles. That single asymmetry, repeated over thousands of arrivals, is what turns a random process into a self-similar branching fractal with a measured dimension of about 2.5 in 3D (roughly 1.71 in 2D) — well below the dimension 3 of solid, space-filling growth.

Real crystals add a rulebook: the lattice

Pure DLA sticks particles wherever they land, which gives realistic-looking soot and dendrites but not the crisp facets of an actual mineral. Real crystals also obey a lattice — a repeating arrangement of atoms fixed by the chemistry of the material. This simulation snaps each arriving particle to the nearest site of a chosen lattice before freezing it:

cubic     — 6 nearest neighbours, 90° angles      (rock salt, pyrite)
hexagonal — 6-fold symmetry in a plane             (quartz, ice, snowflakes)
FCC       — 12 nearest neighbours, close-packed     (many metals, diamond-like nets)

Constraining growth to a lattice is why snowflakes are famously six-sided: water molecules bond into a hexagonal lattice, so every arm of a growing ice crystal branches at multiples of 60°, and because all six arms experience nearly identical temperature and humidity as they grow, they tend to develop in near-perfect step with one another.

Nucleation: why growth needs a head start

Before any of this can happen, a crystal needs a nucleus to grow from. Classical nucleation theory treats the very first cluster of atoms as competing energies: forming a solid surface costs energy (surface tension), while the bulk transition from liquid/gas to solid releases energy. Below a critical radius r*, the surface cost dominates and the tiny cluster is more likely to dissolve back away than to grow; above r*, the bulk energy release wins and growth becomes self-sustaining. This is exactly why breath fogs a cold window only where there is already a speck of dust or a scratch to nucleate on, and why supercooled water can stay liquid well below 0°C until something — a vibration, a dust grain, an ice chip — gives it a nucleus to start from.

Layer by layer, tip by tip

The colour-by-layer-depth view used on this page's simulation is a direct visualisation of growth history: particles that stuck early sit at the fractal's core, particles that stuck recently sit on its outermost, most exposed tips. Because those tips are the parts most exposed to incoming random walkers, they are also the parts that keep growing fastest — a rich-get-richer feedback loop that is the real mathematical reason DLA clusters end up branching instead of filling in solid.

Where diffusion-limited growth shows up outside the lab

The same random-walk-then-stick mechanism governs mineral dendrites growing inside rock cracks, the branching patterns of electrodeposited metal in a battery gone wrong, soot aggregates in a candle flame, and even some models of bacterial colony growth on a nutrient-starved plate. Wherever growth is capped not by the material's willingness to bond but by how fast new material can randomly diffuse in from a distance, you get the same fractal branching signature, whether the "particles" are atoms, ions, or entire cells.

Frequently asked questions

Why do DLA clusters look like branching trees instead of solid blobs?

Because an outward-pointing tip intercepts far more random-walking particles than a deep concave gap does — the tip effectively shields the gap from incoming growth. That feedback compounds over thousands of arrivals, producing a fractal with dimension around 2.5 in 3D rather than a solid, space-filling shape.

Why does the lattice choice change the crystal's shape so much?

The lattice fixes which directions new atoms can bond in. A hexagonal lattice only allows 60°-spaced bonds, which is why ice and quartz crystals show six-fold symmetry, while a cubic lattice restricts bonding to 90° angles, producing the blocky look of salt or pyrite crystals — the same random-walk arrival process produces visibly different crystals depending purely on this underlying geometric rule.

What has to happen before a crystal can start growing at all?

A nucleus above the critical radius has to form first. Classical nucleation theory shows that below this size, the energy cost of creating new surface outweighs the energy released by solidifying, so tiny clusters are more likely to dissolve than survive — which is why supercooled or supersaturated solutions often need a seed, a dust particle or a disturbance to trigger crystallisation at all.

Try it live

Everything above runs in your browser — open Crystal Growth and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.

▶ Open Crystal Growth simulation

Crystal Growth: Random Walks That Build Fractal Crystals