Fick’s second law of non-steady-state diffusion calculator

In non-steady-state mass transfer, concentration changes as a function of both spatial position x and elapsed time t. Governed by Fick’s second law (∂c/∂t = D ∂²c/∂x²), the diffusion coefficient D follows an Arrhenius relationship with absolute temperature. This calculator evaluates exact analytical solutions for semi-infinite geometries, resolving the evolution of transient concentration profiles and the decaying surface flux.

Input variables

Boundary condition model

Temperature (T) 950 °C (1223 K)

Elapsed diffusion time (t) 2.0 h

Surface concentration (c_s) 1.20 wt%

Initial uniform concentration (c₀) 0.20 wt%

Arrhenius pre-exponential factor (D₀) 2.30 × 10⁻⁵ m²/s

Activation energy (Q) 140.0 kJ/mol

Diffusion coefficient (D) 2.42 × 10⁻¹¹ m²/s

Characteristic diffusion length (2√Dt) 0.835 mm

Instantaneous surface flux (J_surface) 4.09 × 10⁻⁵ kg/(m²·s)

Concentration profile across penetration depth

Surface flux decay over time

Mathematical formulation and physical principles

One-dimensional non-steady-state diffusion is described by the partial differential equation:

∂c/∂t = ∂/∂x (D ∂c/∂x) = D (∂²c/∂x²)

The temperature dependence of the diffusion coefficient D is governed by the Arrhenius equation:

D = D₀ exp(-Q / RT)

where R = 8.314 J/(mol·K) is the universal gas constant and T is the absolute temperature in kelvin.

Constant surface concentration: For initial condition c(x, 0) = c₀ and boundary condition c(0, t) = c_s, the solution is:
(c(x, t) - c₀) / (c_s - c₀) = 1 - erf(x / (2√Dt)) = erfc(x / (2√Dt))
The instantaneous surface flux follows from Fick’s first law evaluated at x = 0:
J(0, t) = -D (∂c/∂x)|_x=0 = D (c_s - c₀) / √(πDt)
Thin-film planar source: For an instantaneous planar deposit of mass M per unit area at x = 0 into a semi-infinite medium:
c(x, t) = (M / √(πDt)) exp(-x² / (4Dt))