Diffusion

Watch concentration profiles evolve through a slab and into a semi-infinite solid. Drag the time slider to see Fick's 2nd Law spread atoms by an error-function curve, then apply it to a real case: carburizing steel.

Concentration Profile · Steady State

At steady state, the concentration profile through a slab is linear. The slope dC/dx is constant, so the flux J is the same at every depth — atoms flow downhill at a uniform rate from the high-concentration face to the low-concentration face.

Fick's First Law

J = - D \cdot \frac{d C}{d x}

flux [kg·m⁻²·s⁻¹] = − diffusivity × concentration gradient

D (m²/s) 1.0×10⁻¹¹

L (mm) 2.0

C₁ (kg/m³) 4.0

C₂ (kg/m³) 0.5

dC/dx: −1.75×10³
J: +1.75×10⁻⁸
units: kg·m⁻²·s⁻¹

A bigger gradient or higher diffusivity → bigger flux. Sign convention: J is positive when atoms move in the +x direction.

C(x,t) · Non-steady Profile

A semi-infinite solid starts at uniform C₀. The surface is held at Cₛ. Drag t to watch the profile evolve. At t = 0 the curve is a step; as time grows, atoms diffuse inward and the front advances proportional to $\sqrt{D t}$ .

Fick's Second Law

\frac{\partial C}{\partial t} = D \cdot \frac{\partial^{2} C}{\partial x^{2}}

semi-infinite solid, constant surface concentration

\frac{C - C_{0}}{C_{s} - C_{0}} = 1 - \erf (\frac{x}{2 \sqrt{D t}})

Cₛ (wt%) 1.20

C₀ (wt%) 0.20

D (m²/s) 3.2×10⁻¹²

t (s) 3600

$\sqrt{D t}$: —
Diffusion length: —
C at x = 0.5 mm: —

The error function erf(z) rises from 0 to 1. Where $\frac{x}{2 \sqrt{D t}}$ ≈ 1, the local concentration is about halfway between C₀ and Cₛ. That distance — $2 \sqrt{D t}$ — is the practical "diffusion depth" of the process.

Carburization · Carbon Profile in Steel

Heating 925 °C target C at case = 0.80 wt%

Low-carbon steel is held in a carbon-rich atmosphere at high temperature. Carbon diffuses in from the surface, producing a hard, wear-resistant case over a tough core. Case depth is the distance at which the carbon content reaches the target value.

Process Controls

T (°C) 925

t (h) 4.0

Cₛ (wt%) 1.20

C₀ (wt%) 0.20

C target 0.80

D(T): —
$\sqrt{D t}$: —
Case depth: —
z = $\frac{x}{2 \sqrt{D t}}$: —

Arrhenius

D = D_{0} \cdot \exp (- \frac{Q}{R T})

C in γ-Fe: D₀ = 2.3×10⁻⁵ m²/s, Q = 148 kJ/mol

Push temperature up and case depth grows fast — D is exponential in 1/T. Doubling time only grows depth by $\sqrt{2}$ ; adding 50 °C can do far more.