Spherification: Reaction-Diffusion Gelation Profile

Bath & Droplet

Sodium alginate 1.0%

Calcium chloride bath 1.0%

Droplet radius 8.0 mm

Time acceleration 40×

PDE Gelation State

Droplet suspended above bath

Immersion time

0.0 s

Shell (φ≥0.5 front)

0.00 mm

Liquid core radius

8.00 mm

Volumetric gelled %

Pure-diffusion estimate

0.00 mm

PDE / analytic ratio

–

How this 2D model differs from the 3D scene

The 3D version assumes a closed-form parabolic front x(t) = √(2·D·t). This 2D simulator instead numerically solves the real reaction-diffusion PDE — a version of the shrinking-core model used in chemical engineering for gas/fluid–solid reactions (ore leaching, oxidation) — for the calcium concentration C(r,t) and gel fraction φ(r,t) fields inside a spherically-symmetric droplet:

∂C/∂t = D·(∂²C/∂r² + (2/r)·∂C/∂r) − k·C·(1−φ)
∂φ/∂t = k·C·(1−φ)
boundary: C(R,t) = C_bath   (surface clamp)
symmetry: ∂C/∂r|_{r=0} = 0

This is integrated on a radial finite-difference grid every frame (explicit FTCS, sub-stepped for stability), so the concentric-ring view on the left and the radial profile plot on the right show an actual computed field, not an assumed shape. Because free Ca²⁺ is consumed as it binds ("egg-box" cross-linking), the front the PDE produces runs measurably behind the pure-diffusion estimate shown for comparison — a reaction-limited correction the closed-form model can't capture.

Sodium alginate % — raises both the binding-site density (faster local reaction, k) and mesh density (slower diffusion, D).
Calcium bath % — raises the boundary concentration C_bath and the diffusion coefficient D (steeper gradient).
Droplet radius — sets the grid extent R; gelling a bigger sphere solid takes roughly R² longer.
PDE / analytic ratio — stays near 1 only while reaction is fast compared to diffusion is negligible; watch it drop below 1 as consumption starts limiting the front.

Liquid core (φ<0.5) Gelled shell (φ≥0.5) C(r) — Ca²⁺ concentration φ(r) — gel fraction

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