SOLIDIFY — watch metal freeze

Live: solidify.frankcai.dev (needs WebGPU — Chrome/Edge, Safari 26+, recent Firefox)

A real-time phase-field solidification instrument that runs entirely in your browser on WebGPU. Undercool a melt, tap to nucleate crystals, and watch dendrites grow, branch, collide, and become grains — then read the result like a metallographer: etched micrograph view, grain-size histogram, live ASTM grain number.

Switch the crystal symmetry from cubic (×4) to hexagonal (×6) and the same equations grow a snowflake — shown here in cross-polarised (orientation) view:

The landing page opens with a scroll-driven descent — from the GPU running the solver, down through the die, an electron column, the specimen, and out to a synchrotron — every scene a live 3D wireframe:

What it simulates

The Kobayashi (1993) anisotropic phase-field model for a pure undercooled melt, extended to many grains:

• φ (order parameter) — anisotropic Allen–Cahn dynamics with a j-fold surface-energy anisotropy ε(θ) = ε̄(1 + δ cos j(θ − θ₀)) and stochastic interface noise for side-branching.
• T (temperature) — heat diffusion with latent-heat release K·∂φ/∂t. The glowing halo around every growing tip is the latent heat; growth stalls when recalescence warms the interface back to the melting point.
• Grains — each nucleus carries its own crystallographic orientation θ₀ in a grain-ID field that propagates just ahead of the front; grain boundaries appear where fronts collide, with no extra model terms.

Numerics: explicit Euler on a 512²–2048² grid, compact 9-point Laplacians (checkerboard-free), divergence-form anisotropy, all in WGSL compute shaders — roughly a billion cell-updates per second on a mid-range discrete GPU, with GPU-fence backpressure so slow devices throttle gracefully instead of freezing.

TRUE 3D mode

Flip one switch and the instrument solves the full volumetric phase-field — up to 192³ ≈ 7.1 million voxels — and raymarches it live:

• Real 3D crystallography — per-grain quaternion orientations; cubic ⟨100⟩ anisotropy grows six-armed dendrites, the hexagonal K₆ + c-axis form grows plates or needles (habit slider). Shift-tap seeds a Σ3 twin pair (60° about a shared ⟨111⟩).
• CAD-style camera — orbit/dolly/pan plus a Fusion-style ViewCube with face/edge/corner snapping, tap-to-nucleate at depth.
• Nine lenses on the volume — MELT, ORIENT, SLICE, FIELD (x-ray), SEM, RINGS, THERM, NEON, CURV.
• Serial sectioning — a free section plane (depth/tilt/turn) with a CT sweep mode; the cut face renders as Nital/Klemm's/Beraha's etches, an EBSD IPF map, or a Niyama porosity-risk map.
• Shrinkage porosity — a generation-stamped feed flood from the riser marks starved liquid; pockets that solidify unfed become pores that x-ray dark in FIELD, with live porosity % and the Niyama criterion recorded at every freezing voxel. As of v6.2 the record is honest to its own definition (the non-latent environment rate, −1 where no front ever passed, gated against a discrete CPU recount) and converts to real K·s^½·mm⁻¹ — judged against Niyama's cited 0.775 steel radiographic threshold for steel, and refusing by name to judge any other alloy class. The lab card adds the Clyne–Davies hot-tearing timing ratio off the pour's own f_s(t), labelled as a timing ratio: RDG needs mechanics this solver does not carry.
• Stereology + IPF panels — grain size measured on the section plane vs the true 3D census (the classic stereological underestimate, live), and an inverse-pole-figure texture scatter.
• Take it home — export the crystal as a watertight STL (surface-nets mesh, printable), record a 6-second 360° turntable webm, or share the whole setup as a link.
• Guided tour part III — "Into the volume": five chapters from the first six-armed dendrite to porosity NDT and the STL export.
• The full instrument (v3.0) — everything from 2D now runs in the volume: the dilute-alloy solute field + composer (lazily-allocated solute textures, +57 MB only while on), Bridgman directional growth up the z-axis, a steerable top-surface laser weld, stochastic Σ3 growth twins spawned GPU-side at the moving front, cusped {100} faceted growth, the icosahedral quasicrystal answer to the forbidden 5-fold, all nine presets, the cooling probe / Scheil overlay / SDAS ruler / pole figure instruments, retro voxel + 8-bit looks, and 3D share links that carry the whole setup.
• The grain selector (3D-only showpiece) — a helical pigtail channel under the Bridgman pull: dozens of chill-floor grains race in, exactly one exits into the blade cavity — the real mechanism behind single-crystal turbine blades, verified headlessly (64 grains → 1).
• Shaped moulds (v6.2) — the lab's mould is a rasterized geometry library (shell, plate, step block, wedge) behind one public voxel-mask entry point, with the feed flood, chill floor and nucleation staging all made mask-aware so a sealed chamber can no longer be fed or seeded through a wall. The step block is the classic foundry teaching casting: four section thicknesses fed from one common pour, each freezing at its own rate from wall-conduction geometry alone, each independently measured by a per-section census (thickness · local d̄ · σ_y, thinnest first) on the report card.

Budget: 57 B/voxel over seven textures — ~403 MB VRAM at 192³ (+57 MB solute while alloy is on), with an OOM ladder down through 160³/128³/96³, all four selectable in the ENGINE row; 2D mode is untouched at 60 fps.

The instrument

• Ten lenses — MELT (incandescent blackbody), ORIENT (cross-polarized grains), ETCH (micrograph + scale bar), FIELD (T + isotherms), RINGS (solidification-time isochrones), THERM (FLIR ironbow), SEM (secondary-electron look), NEON (glowing contours), XRAY (synchrotron-radiograph absorption, shows solute), CURV (Gibbs–Thomson curvature).
• Scenarios — free growth, Bridgman directional solidification (pulled gradient frame), and a steerable/auto-raster laser weld that remelts and resolidifies the microstructure.
• Alloy mode — Warren–Boettinger-type dilute solute (qualitative): constitutional undercooling, solute halos, frozen-in microsegregation, composition/partition/liquidus sliders.
• Materials — ten qualitative identities (model metal, Al–Cu, Fe–C steel, Ni superalloy, Co alloy, copper, Mg AZ91, Zn spangle, ice, succinonitrile). Crystal structure picks the dendrite symmetry (FCC/BCC → 4-fold, HCP → 6-fold — and yes, cobalt freezes FCC), and each material sets anisotropy, latent heat, alloy bundle, and how brightly its melt actually glows: steel pours white-hot, zinc at 420 °C is just liquid silver.
• Alloy composer — build your own composition: pick a base (Al/Fe/Ni/Mg/Cu/Zn), add elements in wt% with live at% conversion, using approximate textbook dilute-limit binary coefficients (liquidus slope m, partition k per element). The composer reports the real chemistry — liquidus shift ΔT_L = Σmᵢcᵢ and growth restriction factor Q = Σmᵢcᵢ(kᵢ−1) — then collapses the mix onto the model's pseudo-binary solute field (k_eff = the mᵢcᵢ-weighted mean partition), with every clamp labelled. Famous-alloy quick-fills (A356, AA2024, 4340, IN718, AZ91, bronze…) and shareable #alloy=… deep links.
• Twinning — stochastic growth twins nucleate at the front in twin registry (θ₀ + π/j) and must out-grow their parent to survive, like real feathery grains in aluminum DC casting; Shift+click stamps a twinned seed pair — in hexagonal mode that grows the rare 12-branched snowflake. Twin boundaries etch faint, as in real metallography.
• Nucleation you cannot cheat — there is no "nuclei per second" slider, because that is not a thing you set. You charge the melt with an inoculant: n_max potential sites, each with its own activation undercooling drawn from a Gaussian, each firing once when the melt first gets that cold (Thévoz–Rappaz site distribution + Greer free-growth activation). Sites only fire on a new maximum undercooling, so recalescence stops nucleation by itself, and cooling the same charge faster reaches a deeper undercooling before that happens — more sites fire, finer casting. The test suite asserts exactly that coupling.
• Lab mode — the instrument's other half: instead of dragging sliders at a running melt, you specify the experiment first (charge + inoculant, hold time above the liquidus, atmosphere, pour superheat, mould temperature and shape (3D: shell/plate/step/wedge), a furnace/air/quench/soak cooling programme, and optionally a pre-pour strength spec), pour it, and read a report card. As of v6.1 the card is read the way a foundry reads a cast cup: T_L / T_N / recalescence / T_S extracted from the cooling curve by real thermal analysis (with the method's own error against the measured f_s printed, not tuned away), dissolved hydrogen by Sievert's law with the Ransley–Neufeld solubilities (atmosphere-ordered, zero under vacuum), refiner fade over the hold (settling is why you pour promptly), and the as-cast census with its Hall–Petch σ_y — judged pass/fail against the spec as dialled at the pour, at the precision the card prints, with the same one verdict logic the furnace card uses. Change a dial mid-pour and the card says so. Pour the step block and the card gains a per-section table — thickness · local d̄ · σ_y, thinnest first — the same census machinery run four times over four regions of one casting.
• Heat treatment — the second clock — a real schedule (°C, hours) on a separate clock ~11 orders longer than solidification: Arrhenius integrals over the whole trajectory set a budget, measured Potts kinetics spend it; annealing twins in the volume (Cu and Co twin, Al and Ni refuse with their Murr-1975 numbers), homogenization at frozen φ, oxide/decarb card lines, and a Hall–Petch σ_y row judged against a spec you commit before the run. What it cannot honestly run it refuses, each with its own sentence — incipient melting, and the domain limit with the analytic answer still printed.
• Process controls — undercooling, cooling rate, inoculant charge (+ potency and spread), chill wall, reheat-to-remelt, symmetry, anisotropy, noise, latent heat, brush size, and a pro panel (ε̄, γ, α, τ, k) for power users. Reset arms a staged melt; run/pause plus a ×1/×2/×4 fast-forward multiplier on the transport.
• Looks & navigation — scroll-zoom + right-drag pan in any lens, pixel mode (chunky nearest-neighbour cells) and an 8-bit dithered palette toggle.
• Live metallography — fraction solid, interface undercooling, grain count, grain-size histogram, ASTM G number, computed by GPU reduction while the sim runs.
• Analysis instruments — a cooling-curve probe (foundry thermal analysis: watch the recalescence arrest at the liquidus; ctrl-tap moves the probe), a Scheil overlay (analytic T(fs) path vs the measured interface temperature), and an SDAS ruler that measures secondary-arm spacing by dragging a linear-intercept line, metallographer-style.
• Recorder — one-button ⏺ capture of the canvas to a .webm clip.
• Touch — pinch to zoom, two-finger pan, tap to nucleate.
• The science page — /science/ documents the equations, the numerics, what's quantitative vs qualitative, and the references.
• Guided tour — three parts, 31 chapters: the physics from the Mullins–Sekerka instability through twinning, casting CET, directional growth, welding, alloys and heat treatment; a control-by-control walk through every panel of the instrument; and "out of the plane" for the volume.
• Engineer it & challenge — a separable CMA-ES optimizer runs casting after casting, measures each ASTM grain number, and learns the schedule — every attempt pinned to a lab-notebook strip. Challenge mode deals you the same target first and scores you against it.

Running

Needs a browser with WebGPU (Chrome/Edge, Safari 26+, recent Firefox).

npm install
npm run dev      # local dev server
npm run build    # static build in dist/

No frameworks, no external assets: Vite + TypeScript + raw WebGPU, ~2.5k lines.

Physics sanity checks

• Single-crystal morphology reproduces the canonical Kobayashi '93 figures (parabolic tips, side-branches only with noise, arm count follows j).
• Liquid is metastable: no growth without a nucleus; homogeneous noise cannot freeze the melt.
• Slider ranges are clamped to the numerically stable envelope of the explicit scheme.
• ASTM G from mean grain area (E112), measured against the model resolution set in the SCALE panel.

Real units, and what they cost

The solver is dimensionless. Three factors convert it to SI, and only one of them is a choice:

factor	value	where it comes from
kelvin per unit	≈249 K for Al, ≈44 K for water	forced — it is (L/c_p)/K, the heat equation's own latent coupling
µm per cell	you set it	the model's physical resolution; the domain is n × µm/cell
seconds per unit	derived	forced by whichever diffusivity is transporting

So the app reads in °C, K, K/s, seconds and µm, and the SCALE panel shows each factor with its provenance — nothing is presented as a result when it was a choice, or as a choice when it was pinned. It also names the dimensionless groups the model fails: the Stefan number is matched by construction, but the Lewis number is ~1 where a real alloy is ~10⁴, and the capillary ratio is undefined because the Kobayashi interface has no calibrated surface energy. That last one is why tip radius and arm spacing here are shapes rather than predictions — under the default solver.

The calibrated solver

There is a second solver, and you turn it on in the SCALE panel. Karma–Rappel thin-interface asymptotics tie the phase-field parameters to two numbers the material actually has, the capillary length d0 = Γ/ΔT0 and the diffusivity D:

d0 = a1 · W0 / λ          a1 = 5√2/8 = 0.8839
τ0 = a2 · λ · W0² / D     a2 = 0.6267

Pick λ and everything else is forced — the interface width, the relaxation time, the physical size of a cell, the length of a timestep. Seven dials grey out, because they are no longer choices. For Al–4.5Cu that is d0 = 3.2 nm, W0 = 109 nm, a 0.087 µm cell and an 89 µm field of view at 1024², so the SDAS ruler reports arm spacings a micrograph would give.

λ is the one control left, and it is a convergence knob rather than a physics one: it sets how many capillary lengths wide the diffuse interface is, and the asymptotics are exact only as that goes to zero. So the app prints W0/d0 next to it, and the test suite checks that the answer does not depend on it.

In an alloy the model also carries an anti-trapping current. Without it, an interface of finite width traps solute it should have rejected, which looks exactly like a larger partition coefficient and grows a completely convincing dendrite anyway. Measured here: with the current, k_eff sits on the real k and does not care about the interface width; without it, k_eff runs 24–40 % high and the excess grows with the width.

What it is checked against — real numbers from the literature, not self-consistency:

measurement	reference	result
steady tip velocity V·d0/D	Karma–Rappel 1998, 2D solvability, Δ = 0.55, ε₄ = 0.05: 0.0170	0.01679 (1.2 %)
parabolic tip radius ρ/d0	Tong, Beckermann, Karma & Li 2001: 27.6	28.8 (4.4 %)
critical nucleus radius	Gibbs–Thomson R* = d0/Δ	0.5 %
independence of interface width	same answer at W0/d0 = 1.8 → 3.6	6.4 % spread
effective partition coefficient	the real k, width-independent	0.135–0.150 vs k = 0.15

One consequence arrives free. Under the calibrated alloy path one dimensionless degree is the alloy's freezing range instead of L/c_p, so the app's own shipped nucleation potency — the same dial, untouched — reads 11 K where the Kobayashi scaling made it 37 K. That is the band real castings occupy, reached by fixing the temperature scale rather than by tuning nucleation.

Limits, stated: it is 2D only (the volume still runs Kobayashi, and the switch says so instead of appearing and doing nothing), the Lewis mismatch is unchanged, and the expansion has its own validity bound τ0·V/W0 ≲ 0.2 that a deeply undercooled pure melt reaches by λ ≈ 4.

Two consequences worth knowing. The undercooling slider's own maximum is deeper than any real aluminium melt reaches (249 K against a Turnbull limit near 187 K) — it turns red there. And the same dial on water tops out at 44 K, which is essentially exactly water's homogeneous nucleation limit.

Heat treatment — the second clock

Everything above runs on the solidification clock: under the calibrated solver a timestep is of order 10⁻⁷ s, so a long run is a few milliseconds of metal time. A four-hour soak is 1.4·10⁴ s — eleven orders of magnitude away — and the phase-field solver cannot be integrated through one, not slowly, not on a bigger GPU, not ever. So heat treatment is a separate model on a separate clock, and src/heattreat.ts owns the map between them exactly as units.ts owns the dimensionless↔SI map: real schedule (seconds, °C) → Arrhenius integral over the whole trajectory → a budget → a GPU pass that consumes it. φ is frozen for the duration — that is what solid state means — so the two clocks never have to be reconciled.

There is no process switch. You set an environment — a temperature schedule — and the model reports what happened: grain growth, homogenization, twinning and oxidation all fall out of the same schedule through their own integrals. "Stress relief" is not a mode; it is the schedule where every integral comes back negligible, and the card says so because the arithmetic said so.

Two growth laws, one for each side of the map:

D^n − D₀^n = ∫ k(T) dt       the material's law — sourced n and k(T), fixes the ENDPOINT
D^m − D₀^m = K_MC · S        the model's law — measured m and K_MC, spends S Monte Carlo sweeps

They meet at the endpoint and nowhere else: the material law says where the grain finishes, the Potts pass spends whatever sweeps its own kinetics need to get there, and the trajectory between is the model's. m and K_MC are measured properties of this implementation, not assumptions — m = 2.44, K_MC = 4.79 in the plane (three casts: 2.38 / 2.44 / 2.61, overlapping bands) and m = 2.25, K_MC = 1.28 in the volume (K stable to 1.8 % across casts at the pinned exponent). Ideal curvature-driven growth is parabolic; a finite-state lattice Potts model is not, and assuming m = 2 was measured to cost 9 499 sweeps against the correct 1 980 — a 4.8× budget error in a number nothing else in the app would have contradicted.

What it is checked against:

measurement	reference	result
anneal endpoint, end-to-end (12 h / 520 °C, Al, 1 595 grains)	the sourced growth law: d̄ → 50.4 µm	measured 14.0 → 48.6 µm — ratio 0.963
homogenization decay	discrete DCT-II eigenvalue (1 − 2D(1 − cos k))^I	rel-err 9·10⁻⁷ (2D and 3D), conservation to 10⁻⁷
Σ3 twin registry	exact 60° about ⟨111⟩, checked against real volume adjacency	25/26 exact; twins survive further annealing
oxide card lines	parabolic x = √(∫k_p dt), sourced constants	Al passive film 2.3 nm after 14.5 h at 520 °C — steel scales in mm
Hall–Petch demo (1 h at 0.85 T_m)	σ_y = σ0 + k_HP/√d̄, sourced constants	Al 12 → 20.1 µm (40 → 36 MPa); steel 12 → 295.9 µm (243 → 105 MPa)

The report card's last row is σ_y by Hall–Petch on the measured census — grain-size strengthening alone, no precipitates, no work hardening, and the card says so. Dial a spec (σ_y ≥ N MPa) and the met/missed verdict is judged against the spec as it stood when the run started — a spec you can only set after the furnace is a spec you can move. The arrow is one-way: this furnace can only coarsen, and coarser is softer, so an anneal can only soften; a spec above the as-cast strength is called unreachable up front, because meeting a higher spec takes a finer pour, not a schedule.

Limits, stated: φ stays frozen — the phase field is never re-solutioned, so a treatment cannot dissolve or regrow the solid itself. T, c and age remain the as-cast record; the treatment is isothermal by construction and the card says so, rather than showing a cold casting labelled 540 °C. The oxide scale is a card number, never painted into the fields. The twin plate is the one inserted thing — per-cell twin spawns were built and measured dead (3/512 alive at 300 sweeps, none at 450), because a {111} stacking event is sub-grid for a per-cell Potts flip; the plate is stamped in exact Σ3 registry and everything after birth is the pass's physics. There is no precipitate aging and no T6 — MaterialSI has no precipitate kinetics, and inventing them is the one thing this instrument does not do. Grain growth is unpinned — no particles, no solute drag — where a real specimen stalls. And grain statistics on a 188 µm volume stop meaning anything past ~64 µm, so a schedule that would go there is refused with the law's answer still printed.