💎 Crystal Diffusion

Watch solute atoms diffuse through a crystal lattice via a vacancy mechanism. The diffusion coefficient follows the Arrhenius law D = D₀·exp(−Eₐ/kT). Observe how temperature, activation energy, and time shape the concentration profile.

Temperature T 800 K

Activation energy Eₐ 1.5 eV

Pre-exp. D₀ 1.0 ×10⁻&sup5; m²/s

Time step scale 1.0×

D: - m²/s

D/D₀: -

Sim time: 0 s

√(Dt): - nm

Peak C: 1.000

How it works

Vacancy mechanism: Diffusion in crystalline solids primarily occurs when an atom jumps into an adjacent vacant site. The probability of a successful jump depends on the thermal energy available relative to the migration barrier E_a.

Arrhenius diffusivity: D(T) = D₀ · exp(−E_a / k_BT), where k_B = 8.617×10⁻⁵ eV/K is the Boltzmann constant. Doubling T dramatically increases D.

Fick's 2nd law: ∂C/∂t = D · ∂²C/∂x². The concentration profile C(x,t) evolves from an initial spike (delta function) into a Gaussian of width σ = 2√(Dt).

Analytical solution: For a thin initial layer: C(x,t) = C₀ / [2√(πDt)] · exp(−x²/4Dt). The half-width scales as √(Dt), called the diffusion length.

Crystal lattice view: The top panel shows a 2-D slice of the crystal; atoms colored by local solute concentration. The bottom panel plots C(x,t) with the analytical Gaussian overlay.