Formula & Calculator
Fick's First Law of Diffusion (Materials)
Relates the steady-state diffusive flux of atoms in a solid to the concentration gradient driving the diffusion.
Interpretation
J = −D(dC/dx). Flux proportional to concentration gradient. Steady‑state diffusion. D is diffusivity. Used in materials processing, corrosion, and semiconductor fabrication.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| J | Diffusive flux | atoms/m2.s |
| D | Diffusion coefficient | m2/s |
| dC/dx | Concentration gradient | atoms/m3 per m |
What it means
Fick’s first law describes steady‑state diffusion: the flux J (amount crossing area per time) is proportional to the concentration gradient dC/dx, with D as the diffusion coefficient. The negative sign indicates flow from high to low concentration. This law is the foundation for modelling diffusion in solids, liquids, and gases. In materials science, it is applied to understand diffusion during heat treatment, oxidation, and doping of semiconductors. Engineers use it to calculate the rate at which atoms move, enabling predictions of case depth in carburising, decarburisation, and other diffusion‑controlled phenomena. The law assumes steady state; for transient diffusion, Fick’s second law is used. Understanding Fick’s first law is fundamental for materials processing, corrosion control, and thin‑film technology.
Worked example
Fick's First Law – Two Examples
Real‑World| Parameter | Value |
|---|---|
| D | 1×10⁻¹⁴ m²/s |
| dC/dx | 1×10²⁴ atoms/m⁴ |
| Parameter | Value |
|---|---|
| D | 5×10⁻¹³ m²/s |
| dC/dx | 5×10²³ atoms/m⁴ |
Common mistakes
- Same as ID 30 – see that entry (materials context).
- Note: In materials, D is often in m²/s, concentration gradient in atoms/m⁴ or kg/m⁴ – ensure consistent units.
Applications
Fick's first law (J = −D·dC/dx) governs steady‑state diffusion, relating the diffusive flux to the concentration gradient. It is essential for understanding mass transport in materials processing, from alloying to corrosion. Engineers use it to design diffusion barriers, to model the growth of oxide layers, and to optimise coating processes. In semiconductor manufacturing, it determines the doping profiles. In food packaging, it models the diffusion of moisture and gases. By applying Fick's law, materials engineers can predict concentration profiles, design diffusion couples, and control the rate of diffusion to achieve desired material properties.
- Design of diffusion barriers and protective coatings
- Oxidation and corrosion modelling
- Semiconductor doping and diffusion process design
- Hydrogen embrittlement and hydrogen diffusion analysis
- Food packaging and pharmaceutical controlled‑release systems
Frequently Asked Questions
Fick's first law states that the diffusive flux (J) is proportional to the concentration gradient: J = –D · (dC/dx), where D is the diffusion coefficient, C is the concentration, and x is distance. It describes steady‑state diffusion.
The negative sign indicates that flux occurs from high concentration to low concentration (down the gradient). It ensures mass moves in the direction that reduces the concentration gradient.
Flux J has units of (atoms/m²·s) or (kg/m²·s). Concentration C is (atoms/m³) or (kg/m³). D is (m²/s). The product gives J.
For a plane wall of thickness L with concentrations C₁ and C₂ on the two surfaces, the flux is J = –D · (C₂ – C₁) / L. If D is constant, the concentration profile is linear.
First law is for steady‑state (no time dependence). Second law describes transient diffusion: ∂C/∂t = D · ∂²C/∂x².
Using it for transient conditions. Also, assuming D is constant when it depends on concentration (which is common for some systems).
In a steady‑state diffusion experiment (e.g., using a diffusion couple), measuring the flux and the concentration gradient allows the calculation of D from J = –D·dC/dx.
- Steady‑state.
- One‑dimensional diffusion.
- No convection or other transport mechanisms.
- Constant D (or known function of C).
- Gas permeation through membranes.
- Corrosion and oxidation.
- Doping of semiconductors.
- Diffusion in batteries and fuel cells.
The law generalises to J = –D · ∇C, where ∇C is the concentration gradient vector. This is used in 2D and 3D diffusion problems.