Dendritic Crystal Growth

A flat front that can't stay flat

Drop a cold object into humid air and frost grows on it not as a smooth glaze but as feathery ferns. Pour molten aluminium into a mould and slice it open under a microscope: the metal froze as a forest of microscopic trees. Charge a lithium battery too fast and metallic lithium plates out as needles instead of a flat film. These look like different phenomena. They are the same instability wearing different clothes.

The picture to hold in your head: a crystal is growing into a liquid, and the only thing that lets it grow is getting rid of something at the interface. When water freezes it releases latent heat (about 6.0 kJ/mol, 334 J/g for ice) that has to diffuse away before more ice can form. When an alloy freezes the solid rejects solute that has to diffuse away. Either way, the growth speed is limited by how fast a diffusion field can clear the interface.

Now suppose the interface is almost flat but has a tiny bump. The bump pokes a little further into the liquid — into colder, fresher, less-contaminated material — so it can dump its heat and solute more easily, so it grows faster than the flat regions beside it. The bump runs ahead. That tip then develops its own bumps, which run ahead again. Within microseconds you have a branching tree. The word is dendrite, from the Greek δένδρον, "tree."

The Mullins–Sekerka instability

In 1964 William Mullins and Robert Sekerka made this intuition quantitative. They asked: if you take a flat solidification front and add a sinusoidal ripple of wavelength λ, does the ripple grow or decay? Two effects fight:

• The diffusion field destabilizes. The bumps of the ripple stick into a steeper part of the diffusion gradient, so they grow faster. This destabilizing push gets stronger for short wavelengths (sharp ripples poke into a gradient that varies more steeply).
• Surface tension stabilizes. A curved interface has a higher energy. Through the Gibbs–Thomson effect, high curvature depresses the local melting point, so the tips of sharp ripples melt back. This restoring force also gets stronger for short wavelengths (sharper curvature).

Because both effects strengthen as wavelength shrinks but with different powers, there is a crossover. Ripples longer than a critical wavelength λ* grow; shorter ones are smoothed away. The fastest-growing wavelength sits a little above λ*, and it sets the spacing of the eventual branches:

λ* ≈ 2π · √( d₀ · D / V )

  d₀ = capillary length  = γ·Tm·cp / L²    (~ 0.1–1 nm for metals)
  D  = diffusion coefficient in the liquid   (~ 10⁻⁹ m²/s)
  V  = interface growth velocity
  γ  = solid–liquid surface energy           (J/m²)
  L  = latent heat per unit volume           (J/m³)
  cp = heat capacity per unit volume         (J/m³·K)
  Tm = melting temperature                    (K)

The geometric mean of a nanometre-scale capillary length and a micrometre-to-millimetre diffusion length lands the branch spacing in the micrometre range — exactly what we see. Notice the velocity dependence: grow faster (larger V) and λ* shrinks, so fast freezing gives finer, more closely-spaced branches.

          liquid (cold, supersaturated)
   ┌───────────────────────────────────────┐
   │      ▲        ▲        ▲        ▲       │  ← bumps run AHEAD
   │     ╱ ╲      ╱ ╲      ╱ ╲      ╱ ╲      │     into fresh liquid
   │ ───┘   └────┘   └────┘   └────┘   └─── │  ← original flat front
   │ ░░░░░░░░░░ SOLID (crystal) ░░░░░░░░░░░ │
   └───────────────────────────────────────┘
        wavelength λ* sets the branch spacing

Two ways to make a melt unstable: thermal and constitutional

There are two distinct reasons the liquid just ahead of the interface can be "hungrier" than the interface itself, and both drive dendrites.

Thermal undercooling. In a pure substance frozen from an undercooled melt (the bulk liquid is below its freezing point), the latent heat released at the interface flows backward into the cold liquid. A protruding tip sits in colder liquid and sheds heat faster, so it grows faster. Pure undercooled water and pure metals like nickel freeze this way; the dendrite arms are essentially heat-flow fingers.

Constitutional supercooling. In an alloy, the solid rejects solute. With a partition coefficient k = C_solid/C_liquid < 1, a solute-rich boundary layer of thickness ≈ D/V builds up ahead of the interface. Solute lowers the local freezing point (follow the liquidus line, slope m). So even if the actual temperature rises smoothly into the melt, the local freezing point rises even faster, and a zone of liquid ends up below its own freezing point. Tiller's criterion says the front goes unstable when

G / V  <  m · C₀ · (1 − k) / (k · D)

  G  = temperature gradient in the liquid (K/m)
  V  = growth velocity (m/s)
  C₀ = nominal solute concentration
  m  = liquidus slope (K per wt%)

The ratio G/V is the master control knob of casting. Keep G/V high (steep gradient, slow pull) and the front stays planar or cellular. Let G/V fall (shallow gradient, fast freezing) and you get cells, then dendrites, then equiaxed dendritic grains. This single inequality explains why the chilled rim of a casting is fine and planar while the slow-cooled centre is coarsely dendritic.

What sets the tip: the solvability puzzle

Knowing the front is unstable is not the same as predicting the shape. For decades a paradox stood: diffusion theory (the Ivantsov solution) allowed a smooth paraboloid tip of any radius R growing at velocity V, as long as the product was fixed by undercooling — it gave a relationship for R·V but not R and V separately. Yet real dendrites pick one definite tip radius and one definite speed. What breaks the tie?

The answer, worked out in the 1980s (microscopic solvability theory, by Langer, Kessler, Levine, Pomeau and others), is crystalline anisotropy. Surface energy is not perfectly isotropic; it is a few percent lower along certain crystal axes. That tiny anisotropy is singular — it selects a unique operating state out of the continuum. The selection rule is written

σ* = 2 d₀ D / (R² V)  ≈ constant   (depends on anisotropy ε)

  ⇒  R² · V ≈ const

So faster growth means a sharper tip (R falls as V rises), and the anisotropy that you might think is a minor correction is in fact the thing that makes a dendrite a dendrite rather than a featureless blob. Turn the anisotropy off in a simulation and the branches lose their crisp directions and degenerate into a "seaweed" structure — which is exactly what you see in materials with nearly isotropic surface energy.

By the numbers

System	Driving force	Tip radius R	Tip speed V	Arm spacing
Snowflake (ice Ih in air)	Supersaturation ~10–30 %	~1–5 µm	~1–10 µm/s	10–100 µm
Undercooled pure Ni	ΔT ≈ 50–250 K	~1 µm	up to ~50 m/s	—
Al–Cu casting alloy	Constitutional	~5–20 µm	~10⁻⁴–10⁻² m/s	10–500 µm (SDAS)
Succinonitrile (lab analog)	ΔT ≈ 0.5–2 K	~10–50 µm	~µm/s	~100 µm
Lithium electrodeposit	Overpotential / Li⁺ depletion	~0.1–1 µm	~µm/s–nm/s	µm scale

A few anchors worth memorizing. The capillary length d₀ for most metals is sub-nanometre, around 0.1–1 nm. Liquid diffusivities D cluster near 10⁻⁹ m²/s (10⁻⁵ cm²/s). The latent heat of fusion of ice is 6.01 kJ/mol; of aluminium, 10.7 kJ/mol; of nickel, 17.5 kJ/mol. Secondary dendrite arm spacing (SDAS) in castings scales with local solidification time t_f roughly as SDAS ≈ A·t_f^1/3 — so a part that freezes in 1 s versus 1000 s has arms about 10× finer, and finer arms mean higher strength and toughness.

Where dendrites show up

• Snowflakes and frost. Ice grows from water vapour by attachment, and the hexagonal lattice plus the diffusion instability gives six branched arms. The Nakaya diagram maps the morphology onto temperature and supersaturation: thin plates near −2 °C, hollow columns and needles near −5 °C, and the iconic six-pointed stellar dendrites in the humid band near −15 °C. Window frost is the same physics on a 2D surface.
• Cast and welded metals. Almost every solidified alloy — engine blocks, weld beads, ingots — froze dendritically, and the dendrite arm spacing is a forensic record of how fast it cooled. Metallurgists read SDAS to reconstruct cooling rates and to predict strength. Single-crystal turbine blades are grown by deliberately defeating dendritic branching with directional solidification and a grain selector.
• Lithium-metal battery dendrites. On fast charge, Li⁺ is reduced to metal faster than it can spread, so it plates as needles. They pierce the separator and short the cell — a leading safety and cycle-life problem for next-generation high-energy batteries. Suppressing them (solid electrolytes, pulsed charging, host architectures, artificial SEI layers) is one of the most active areas in energy storage.
• Manganese dendrites in agate and limestone. The black "fern" patterns on dendritic agate and on limestone bedding planes are manganese- and iron-oxide deposits that crystallized in fluid-filled cracks. They are mineral dendrites, often mistaken for fossil plants.
• The electrochemistry classroom demos. A silver or lead "tree" (Diana's tree, the lead tree) grown by displacement plating in a metal-salt solution is a dendrite you can grow on a bench in an afternoon.

Common misconceptions and pitfalls

• "Dendrites are a kind of crystal defect." No — a single dendrite is usually one continuous single crystal; all its arms share the same lattice orientation. The branching is a shape instability of the growth front, not a stacking or point defect.
• "Snowflakes are symmetric because they're crystals." The six-fold symmetry comes from the lattice, but the matching of the six arms is a coincidence of shared environment — each arm falls through nearly the same temperature and humidity history. Break that symmetry of environment (turbulence, collisions) and you get the lopsided flakes that make up most real snow.
• "Faster cooling means bigger crystals." Backwards. Faster cooling (higher V, lower G/V) gives finer structure — smaller λ*, tighter arm spacing, more nucleation sites. Slow cooling gives coarse dendrites and big grains.
• "Surface tension just rounds things off." Surface tension is not a passive smoother here; through anisotropy it actively selects the operating point. Without anisotropic surface energy you do not get clean dendrites at all — you get disordered seaweed.
• "Dendrite = fractal." Diffusion-limited aggregation (DLA) is a fractal with no preferred directions; a true crystalline dendrite has a definite orientation and a selected tip radius set by anisotropy. They look superficially alike but are different objects.
• "If you just charge a battery slowly, lithium can never dendrite." Slow charging helps a lot but does not guarantee a flat deposit. Local current crowding at SEI cracks and surface roughness can still seed needles even at modest rates; that is why the engineering fight is about uniform interfaces, not just rate limits.