Condensed Matter
How a Snowflake Grows: Dendritic Solidification
How a Snowflake Grows is one of the oldest puzzles in physics and one of the most elegant. A snowflake is not a frozen raindrop — it is a single crystal built molecule by molecule from water vapour, then sculpted by a diffusion instability into a six-armed tree. The same runaway that branches a snowflake, called dendritic solidification, shapes cast metals, solder joints and turbine blades. Here is the mechanism, the numbers, and why the shape depends so sharply on temperature.- Also calledDendritic crystal growth (vapour deposition)
- Six-fold symmetry fromHexagonal ice Ih (a ≈ 0.452 nm, c ≈ 0.737 nm)
- Classic stellar dendrites at≈ −15 °C, high supersaturation
- Governing instabilityMullins–Sekerka (1963–64)
- First grown in the lab byUkichiro Nakaya, 1936
- Typical size / fall speed1–5 mm, ~0.3–1 m/s
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Grown from vapour, not frozen from water
A snow crystal is deposited, not frozen: water-vapour molecules attach directly to ice, one at a time, building a single ordered lattice. Inside a cloud at, say, −15 °C, supercooled droplets and ice crystals coexist. Crucially, the equilibrium vapour pressure over ice is lower than over liquid water — e_ice < e_water, a ratio of about 1.16 at −15 °C. So air that is merely saturated for droplets is strongly supersaturated for ice: vapour streams onto the crystal while nearby droplets evaporate. This is the Wegener–Bergeron–Findeisen process (1911–1938), the engine of most mid-latitude snow.
Each molecule locks into ice Ih, a hexagonal lattice (a ≈ 0.452 nm, c ≈ 0.737 nm) set by the tetrahedral hydrogen bonds of water. Its six equivalent prism faces are why the crystal is six-sided. Kepler asked why six? in his 1611 essay; the true answer — hydrogen-bond geometry — had to wait three centuries for atomic theory.
Why a flat face won't stay flat: the Mullins–Sekerka instability
Here is the heart of it. A slowly growing, perfectly flat ice face is unstable. To grow, the crystal must pull vapour out of the surrounding air, depleting it near the surface — the crystal sits inside a thin "halo" of vapour-poor air. In steady state that concentration field obeys Laplace's equation, ∇²c = 0, which is mathematically identical to the electrostatic potential around a charged conductor.
And just as the electric field crowds at a sharp point — the lightning-rod effect — the concentration gradient, and therefore the deposition flux, is steepest at any protruding corner or edge. W. F. Berg noted this "point effect of diffusion" in 1938. So a chance bump pokes into richer vapour and steeper gradients, grows faster, pokes out further, grows faster still: a runaway. William Mullins and Robert Sekerka made this rigorous in 1963–64, showing that perturbations above a threshold wavelength always amplify. That runaway is the seed of every branch on a snowflake.
What tames the runaway: capillarity and tip selection
If pointy grows faster, why isn't a snowflake infinitely spiky? Surface tension. A sharply curved surface has a raised equilibrium vapour pressure — the Gibbs–Thomson effect — so the very finest bumps are penalised and effectively smoothed away. Two effects now fight: diffusion destabilises, favouring ever-finer spikes; capillarity stabilises, killing the finest ones.
The winner is an intermediate scale — roughly the geometric mean of the capillary length d₀ (~1 nm for ice) and the diffusion length l_D = D ÷ v: λ ≈ 2π√(d₀ · l_D). This sets the branch spacing and the tip radius. Ivantsov (1947) found the family of paraboloid tip solutions, but it was the anisotropy of the ice surface that uniquely selects one tip (microscopic solvability theory, 1980s). Measured dendrite tips are ~1–10 μm across, with side-branches spaced tens of μm apart.
The Nakaya diagram: temperature writes the shape
Ukichiro Nakaya grew the first artificial snow crystals on a rabbit hair in 1936 and mapped their shape against temperature and supersaturation — the morphology (Nakaya) diagram. He called snow crystals "letters sent from the sky," because the shape encodes the conditions of its growth.
The startling result: the primary habit flips twice as the air gets colder. Near 0 to −3 °C ice grows as thin plates; around −5 °C as slender needles and hollow columns; near −15 °C — where the ice/water vapour-pressure gap peaks and growth is fastest — as the classic six-armed stellar dendrites; and below about −22 °C, back to columns and prisms. High supersaturation branches; low supersaturation facets. Why the plate–column–plate alternation happens is still not fully understood; the leading idea ties it to a temperature-dependent quasi-liquid "premelted" layer that changes whether the basal or prism face attaches molecules faster.
Why the six arms match — and why they usually don't
The hexagonal lattice fixes six equivalent growth directions, 60° apart, so branches emerge as a six-fold star. The arms of one crystal look alike not because they communicate, but because the tumbling crystal is only a few millimetres across: all six tips share the same temperature and humidity, moment by moment, as the flake falls, so they grow in near-lockstep. A different fall path writes a different record — hence "no two alike."
But the popular image oversells the symmetry. Kenneth Libbrecht's measurements show most natural crystals are lopsided or irregular; the flawless six-fold specimens are the rare, photogenic minority selected for calendars. And strict uniqueness is really molecular bookkeeping — exact defect and isotope patterns — not something the eye could ever resolve.
A worked estimate: why branches are inevitable
Take a stellar dendrite that reaches R ≈ 2.5 mm during its ~15-minute fall through the cloud. Mean tip advance is v ≈ R ÷ t = 2.5×10⁻³ m ÷ 900 s ≈ 2.8 μm/s. With the diffusivity of water vapour in air D ≈ 2.1×10⁻⁵ m²/s, the diffusion length is l_D = D ÷ v ≈ 7 m — thousands of times larger than the crystal itself.
So the whole flake sits deep inside its own vapour-depletion halo, and the only places touching fresh vapour are the outermost tips. Growth must concentrate there: branching is unavoidable. Feeding d₀ ≈ 1 nm into λ ≈ 2π√(d₀ · l_D) gives a characteristic bump scale of order 10² μm — the right order for observed side-branch spacings. (These are order-of-magnitude estimates; real selection also involves latent heat: every deposited molecule releases ~2.83 MJ/kg of sublimation heat that warms the tip and throttles it.)
Dendrites everywhere: from turbine blades to Kepler
"Dendrite" is Greek for little tree, and the Mullins–Sekerka instability shapes far more than snow. Cast metals and alloys freeze as forests of dendrites whose spacing and orientation set a part's strength — which is exactly why single-crystal turbine blades are grown to suppress dendritic grain boundaries. The same physics governs solder joints, weld pools, semiconductor and pharmaceutical crystallisation, and mineral growth.
The snowflake is simply the most photographed example, and its study is a lineage: Kepler's 1611 On the Six-Cornered Snowflake, sketches by Descartes (1637) and Hooke (1665), Wilson Bentley's first snowflake photomicrograph in 1885 (~5,000 in all), Nakaya's lab crystals, Mullins and Sekerka's theory, and Libbrecht's modern quantitative physics. Kepler's question — why six? — turned out to be a doorway into how ordered matter assembles itself.
| Feature | Faceted growth (plates & columns) | Dendritic growth (branched stars) |
|---|---|---|
| What limits growth | Slow surface attachment kinetics (needs steps) | Diffusion of vapour toward the crystal |
| Supersaturation | Low | High |
| Interface shape | Flat facets, sharp 60°/120° edges | Runaway tips with side-branches |
| Governing idea | 2-D nucleation on smooth faces | Mullins–Sekerka / Berg point effect |
| Typical temperature | ≈ 0 to −4 °C, and below −22 °C | ≈ −13 to −17 °C (peak supersaturation) |
| Everyday example | Hexagonal plate, hollow column | Classic six-pointed stellar snowflake |
Frequently asked questions
Do snowflakes form from frozen raindrops?
No — a frozen droplet is sleet, or the core of graupel. A snow crystal grows by deposition: water-vapour molecules attach directly to ice, one by one, so it is built as a single ordered lattice rather than a frozen blob.
Why do snowflakes have six sides?
Because ice Ih crystallises in a hexagonal lattice fixed by the tetrahedral hydrogen bonds between water molecules. That gives six equivalent growth directions 60° apart, so branches sprout in a six-fold star. Kepler first posed the question in 1611.
Are no two snowflakes really alike?
At the molecular level, essentially yes — each crystal's exact defect and isotope pattern is unique. But simple small crystals can look identical, and even the six arms of one flake are usually not perfectly symmetric; the flawless specimens are a rare, photogenic minority.
Why do the most elaborate snowflakes form near −15 °C?
Because the gap between the vapour pressure over supercooled water and over ice peaks near −15 °C (~16% supersaturation for ice), giving the strongest driving force. High supersaturation triggers the branching instability and stellar dendrites; colder or drier air makes plates or columns instead.
What makes the six arms so similar to each other?
Shared history, not communication. The whole crystal is only a few millimetres across, so all six tips feel the same temperature and humidity as it tumbles and falls, and grow in near-lockstep. A different fall path gives a different — but still six-fold — pattern.
Does dendritic solidification happen in anything besides snow?
Yes. Cast metals, alloys, solder, semiconductors and minerals all solidify into tree-like dendrites via the same Mullins–Sekerka instability. Controlling dendrites is central to metallurgy — single-crystal turbine blades are grown specifically to suppress them.