EN 
Українська
Polski
Español
Deutsch
日本語
Italiano
Português (BR)
Français
العربية
Nederlands
中文
한국어
हिन्दी
Türkçe
Tiếng Việt
Русский
Bahasa Indonesia
فارسی
বাংলা
ไทย
Filipino
Kiswahili
اردو
Bahasa Melayu
தமிழ்
עברית
Ελληνικά
ਪੰਜਾਬੀ
አማርኛ
မြန်မာဘာသာ
ខ្មែរ
ქართული
नेपाली
Монгол хэл
Հայերեն
සිංහල
ລາວ
Čeština
Română
Magyar
Български
Svenska
Dansk
Suomi
Norsk
Slovenčina
Hrvatski
Српски
Slovenščina
Lietuvių
Latviešu
Eesti
Íslenska
Shqip
Македонски
Bosanski
Malti❄️ Crystal Growth — Dendritic Solidification
Phase-field simulation of crystal solidification. Watch dendritic arms grow with 6-fold symmetry, control undercooling, anisotropy strength, and thermal diffusivity to explore nucleation and crystal growth physics.
About Crystal Growth & Dendritic Solidification
When a liquid metal or water cools below its melting point, solidification begins through nucleation — the spontaneous formation of tiny crystalline embryos. Classical nucleation theory predicts that a nucleus must exceed a critical radius r* = 2γTeq/(ΔHf·ΔT) before it can grow stably, where γ is the solid-liquid surface energy, Teq is the equilibrium melting point, ΔHf is the latent heat of fusion, and ΔT is the undercooling. This simulator places a seed crystal at the centre and solves the phase-field equation ∂φ/∂t = M[∇²φ + φ(1−φ)(φ−1/2+λu)] on a finite-difference grid, where φ is the phase-field order parameter (0 = liquid, 1 = solid), M is interface mobility, λ is the thermal coupling constant, and u = T − Teq is the dimensionless temperature.
The coupled thermal diffusion equation ∂u/∂t = D·∇²u + (1/2)∂φ/∂t captures latent heat release as the solid front advances. Crystal anisotropy is encoded by modulating the interface energy with a(θ) = 1 − ε·cos(n·θ), where θ is the local interface normal angle, ε is the anisotropy strength, and n is the fold number (n=6 for hexagonal ice, n=4 for cubic metals). This produces the characteristic dendritic arms — elongated protrusions along energetically preferred crystallographic directions — that give snowflakes their six-fold symmetry and cast metals their branched microstructure.
Frequently Asked Questions
What is the phase-field model used in this simulation?
The simulation solves ∂φ/∂t = M[∇²φ + φ(1−φ)(φ−1/2+λu)], where φ is the order parameter (0=liquid, 1=solid), M is interface mobility, λ couples the thermal field u=T−Teq to the phase, and the double-well terms φ(1−φ)(φ−1/2) stabilise the two bulk states. The coupled equation ∂u/∂t = D·∇²u + (1/2)∂φ/∂t transports latent heat released during solidification.
Why do crystals grow in a dendritic (tree-like) pattern?
Dendritic growth arises from a morphological instability: a protrusion on the solid-liquid interface grows faster because it releases latent heat more efficiently into the undercooled melt. The competition between surface tension (which stabilises flat interfaces) and diffusion (which destabilises them) selects a characteristic dendrite tip radius and growth velocity known as the Ivantsov solution.
What is undercooling and why does it affect crystal shape?
Undercooling (ΔT) is how far the melt temperature is below the equilibrium melting point Teq. Low undercooling produces compact, slowly growing crystals; high undercooling drives fast dendritic growth with fine tip radii. This is why rapidly quenched metals have fine dendritic microstructures while slowly cooled melts form coarse equiaxed grains.
What does the anisotropy strength ε control?
Anisotropy strength ε controls how much the interface energy and mobility vary with crystallographic orientation. Higher ε sharpens the preferred growth directions (six arms of a snowflake) and suppresses off-axis growth. At ε=0 the interface is isotropic and the crystal grows as a circle; increasing ε towards 0.07 produces sharp, well-defined dendritic arms with narrow side-branching.
How is 6-fold crystal symmetry implemented?
The anisotropy function a(θ) = 1 − ε·cos(n·θ) is applied to the interface gradient angle θ = atan2(∂φ/∂y, ∂φ/∂x). Setting n=6 creates six energetically preferred directions 60° apart, matching hexagonal crystal symmetry (ice Ih). Choosing n=4 gives cubic (4-fold) symmetry matching metals like iron or aluminium; n=3 gives trigonal, n=8 quasi-crystalline patterns.
What is nucleation and how does it start crystal growth?
Nucleation is the formation of a tiny crystalline embryo in the melt. Classical nucleation theory (CNT) shows a nucleus must exceed a critical radius r* = 2γTeq/(ΔHf·ΔT) to be stable; smaller embryos dissolve back into the melt. In this simulation a small solid seed is placed at the centre to bypass the nucleation barrier and directly observe anisotropic dendritic growth.
What determines thermal diffusivity in this model?
Thermal diffusivity D controls how fast latent heat diffuses away from the growing interface via ∂u/∂t = D·∇²u + (1/2)∂φ/∂t. Higher D removes heat faster, reducing the thermal halo around the dendrite tip and allowing higher tip velocity. Lower D creates a larger thermal boundary layer that slows growth and can trigger tip-splitting or seaweed morphologies.
Why are snowflakes six-fold symmetric but all different?
The six-fold symmetry of snowflakes reflects hexagonal ice Ih crystal structure where water molecules form hydrogen-bonded rings with 120° bond angles. All six arms grow simultaneously in the same local environment so they develop identically. However, each arm traces a unique growth history as the snowflake tumbles through air with changing temperature and humidity — making every snowflake unique in detail while symmetric within itself.
What are real-world applications of dendritic solidification research?
Dendritic microstructure directly affects mechanical properties of cast metals, turbine blade alloys, and 3D-printed parts. Controlling dendrite arm spacing (DAS) through cooling rate optimises strength and ductility. Phase-field models like this one are used industrially to design casting processes, predict microsegregation in nickel superalloys, and optimise additive manufacturing parameters for aerospace components.
How does grain competition occur in solidifying metals?
When multiple nuclei form simultaneously, each grows dendrites aligned with its own crystallographic axes. Where two dendrite fields meet, the faster-growing grain (with preferred axes roughly aligned with the temperature gradient) blocks and consumes the slower-growing neighbour. This grain competition determines the final microstructure — columnar grains in directionally solidified alloys, equiaxed grains in rapidly cast materials.
A phase-field model of dendritic crystal solidification with adjustable undercooling and anisotropy.
3D · Three.js / WebGL renderer · 60 FPS target · runs fully client-side, no install