Case 1 — Semi-Infinite Surface Diffusion

Diffusion from a constant-concentration surface into a semi-infinite solid (infinite along the Positive X-axis).
Scroll down for the analytical solution.

Case 2 — Planar Source in Infinite Medium

An thin film of atoms at the origin spreads symmetrically through an infinite medium along the Positive and Negative X-axis.
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Case 3 — Planar Source in Semi-Infinite Medium

A thin film of atoms diffuses into a semi-infinite body along the Positive X-axis.
Scroll down for the analytical solution.

3D Atom View

t = 0.00 s

2000 atoms

Controls

Jumping Frequency (Γ) 20 Hz

Jump Length (λ) 0.50 µm

Number of Atoms 2000

Sim Speed 1.0×

D = Γλ²/60.8333 µm²/s

√(2Dt)0.00 µm

Concentration Profile — Analytical vs Simulated

Analytical erfc

Simulated (histogram)

Parameters

C(x,t)— concentration profile

C₀— surface concentration

D— diffusion coefficient

√(2Dt)— diffusion length estimate

Solution$C(x,t)=C_0\;\text{erfc}\!\left(\dfrac{x}{2\sqrt{Dt}}\right)$

Diffusivity$D=\dfrac{\Gamma\,\lambda^2}{6}$

$C(0,t)=C_0$ (fixed surface) $C(x,0)=0$ (initial) $C(\infty,t)=0$

Parameters

C(x,t)— concentration profile

M— mass per unit area

D— diffusion coefficient

√(2Dt)— diffusion length estimate

Solution$C(x,t)=\dfrac{M}{2\sqrt{\pi D t}}\;\exp\!\left(-\dfrac{x^2}{4Dt}\right)$

Diffusivity$D=\dfrac{\Gamma\,\lambda^2}{6}$

$C(x,0)=0$ $C(\pm\infty,t)=0$ $\int_{-\infty}^{+\infty}C\,dx=M$

Parameters

C(x,t)— concentration profile

M— mass per unit area

D— diffusion coefficient

√(2Dt)— diffusion length estimate

Solution$C(x,t)=\dfrac{M}{\sqrt{\pi D t}}\;\exp\!\left(-\dfrac{x^2}{4Dt}\right)$

Diffusivity$D=\dfrac{\Gamma\,\lambda^2}{6}$

$C(x,0)=0$ $C(\infty,t)=0$ $\int_{0}^{\infty}C\,dx=M$