eduKateSG · How Science Works
Choose your reading route
A small bump on a solidification front can grow when transport and local undercooling favour its advance more than neighbouring regions. Diffusion can amplify some disturbances, while capillarity suppresses sufficiently fine ones.
Full chapter index · Check your understanding · Materials Science · Science master hub
A mechanism-focused enrichment guide for curious students, parents and teachers. Worked numbers are explicitly hypothetical; research links identify supporting scientific studies. This is not an examination-syllabus claim or a laboratory operating procedure.
Inside this guide
Chapters 1–3 · Understand the mechanism
1. Watch a flat front become patterned
Chapters 4–5 · Work through the model
Chapters 6–8 · Evaluate the evidence
6. Calculate growth without claiming a final pattern
When a liquid alloy solidifies, the boundary between solid and liquid need not remain flat. A nearly planar front can develop shallow cells and later branching structures. Mullins–Sekerka instability concerns the growth or decay of small disturbances to such an interface.
Focus first on the beginning: does a small bump shrink back or grow? The later shape of a mature dendrite is a more complicated question involving nonlinear growth and crystallographic anisotropy.
This guide treats directional alloy solidification. Our snow-crystal guide explains ice shapes grown from vapour. Both involve transport and interface growth, but an alloy’s solute boundary layer gives this article a different physical and practical focus.
In a common dilute-alloy case, the growing solid incorporates less solute than the adjacent liquid. With partition coefficient k = Cs/Cl below one, solidification rejects solute into the liquid ahead of the front.
Diffusion spreads that solute away, but continuing growth replenishes it. A concentration boundary layer can form. Because composition affects the local equilibrium freezing temperature, the liquid’s melting conditions now vary with position.
Research on the onset of dendritic patterns in alloy solidification follows the development of this instability. The useful picture is not just a cold front advancing: temperature and composition fields develop together.
CHAPTER 3 OF 8 · Understand the mechanism
3. Understand constitutional undercooling
Liquid can be above the interface temperature yet still be below its own local equilibrium freezing temperature. That apparently surprising condition is constitutional undercooling: the relevant equilibrium temperature changes because composition changes.
A small protrusion can reach liquid with a different solute concentration and greater local driving force for solidification. If it advances faster, the protrusion becomes larger. This feedback can destabilise the initially flat front.
But a quick undercooling test is not the whole stability theory. Mullins and Sekerka’s planar-interface study analyses whether individual disturbance components grow or decay. Interface curvature and transport alter the full result.
A useful diffusion–growth scale is ℓD ≈ D/V, where D is liquid solute diffusivity and V is front speed. The dimensions are metres: (m²/s)/(m/s). Faster advance leaves less distance for diffusion to smooth the solute field.
With invented D = 1.0 × 10−9 m²/s and V = 1.0 × 10−5 m/s, ℓD is about 100 μm. Raising V to 2.0 × 10−5 m/s gives 50 μm; lowering it to 5.0 × 10−6 m/s gives 200 μm.
This scale is not a predicted dendrite spacing. It describes a transport length under a simple steady-growth picture. A spacing prediction additionally needs thermal conditions, capillarity, material properties and the relevant stability or pattern-selection model.
| Invented front speed | D/V transport scale |
|---|---|
| 5 μm/s | 200 μm |
| 10 μm/s | 100 μm |
| 20 μm/s | 50 μm |
Curved interfaces have a different equilibrium condition from flat ones. Surface energy makes sharply curved features costly. Capillarity therefore tends to suppress fine-scale disturbances, competing with destabilising transport feedback.
A linear stability analysis can assign a growth rate s(q) to a disturbance with wavenumber q. Positive s means its small amplitude grows; negative s means it decays. A most rapidly growing wavelength, if present, belongs to the specified conditions.
Interferometric experiments on directional ice growth in solutions tested instability predictions and found differences between solution systems. That is a reminder that a successful theory–experiment match in one material does not validate every simplification in another.
CHAPTER 6 OF 8 · Evaluate the evidence
6. Calculate growth without claiming a final pattern
An illustrative linear disturbance obeys A(t) = A₀ exp(st). If an invented mode begins at A₀ = 1 μm with s = 0.020 s−1, after 50 seconds its amplitude is about 2.72 μm.
For a second invented mode with s = −0.020 s−1, the same initial amplitude becomes about 0.368 μm. The first grows and the second decays even though both begin with the same size.
Once the growing disturbance becomes large, the linear approximation stops being sufficient. It cannot promise that amplitude grows exponentially forever or determine the finished dendrite from one number. The calculation identifies early amplification, not a complete casting simulation.
CHAPTER 7 OF 8 · Evaluate the evidence
7. Read a solidification experiment carefully
Report pulling speed, thermal gradient, composition and the evolving front shape. A polished final cross-section records the product; time-resolved imaging can reveal when the planar interface first lost stability.
Convection can redistribute solute and alter the boundary layer. Research on a stirred alloy melt explicitly studies this complication. An experiment with mixing should not automatically be interpreted using a no-flow diffusion model.
High growth speeds can introduce additional physics. Work extending stability theory to rapid solidification revisits conventional criteria. Avoid turning “faster growth destabilises” into a universal rule across all speeds and materials.
Is constitutional undercooling simply a colder liquid? No. It compares actual temperature with a composition-dependent local equilibrium freezing temperature.
Is D/V the finger spacing? No. It is a diffusion–growth scale. Pattern spacing requires further physics and a suitable model.
Why does this matter? Solidification patterns influence microstructure and solute distribution, which help shape material properties. Continue with Materials Science to connect a changing interface with the material produced.