Variables
| Symbol | Quantity | Unit |
|---|---|---|
| dC/dt | Rate of change of concentration with time | |
| D | Diffusion coefficient | m2/s |
| d2C/dx2 | Second spatial derivative of concentration |
What it means
Fick’s second law is the differential equation that describes how the concentration C changes with time t and position x during non‑steady‑state diffusion: ∂C/∂t = D ∂²C/∂x². It is derived from Fick’s first law and the conservation of mass. Solving this equation with appropriate initial and boundary conditions (e.g., error function solutions) allows prediction of concentration profiles after a given time. This is crucial for designing diffusion treatments, such as case hardening (carburising, nitriding), homogenisation of alloys, and dopant diffusion in semiconductors. It is also used in modelling corrosion, diffusion in polymers, and biological systems. Understanding Fick’s second law is essential for materials engineers and chemical engineers to control processing parameters and achieve desired compositions.
Worked example
Fick's Second Law – Two Examples
Real‑World| Parameter | Value |
|---|---|
| D | 1×10⁻¹¹ m²/s |
| t | 2 hours = 7200 s |
| Parameter | Value |
|---|---|
| D | 1×10⁻¹² m²/s |
| t | 4 hours = 14400 s |
Common mistakes
- Fick’s second law: ∂C/∂t = D·∂²C/∂x² – describes non‑steady‑state (transient) diffusion.
- Partial derivatives: Use partial differential equations – not ordinary derivatives.
- D: Assume D is constant (independent of concentration) – otherwise, the equation becomes non‑linear.
- Boundary conditions: Solutions depend on initial and boundary conditions – choose the appropriate solution (e.g., error function for infinite source).
- Units: C in mol/m³, t in s, x in m, D in m²/s.
Applications
Fick's second law (∂C/∂t = D·∂²C/∂x²) describes transient diffusion, enabling the prediction of concentration changes over time. It is used to model diffusion in non‑steady‑state conditions, such as during heat treatment, carburising, and homogenisation of alloys. Engineers use it to determine the diffusion time required to achieve a desired case depth, to simulate the concentration profile, and to design schedules for heat treatments. It is also applied in the study of corrosion, sintering, and material degradation. By solving Fick's second law with appropriate boundary conditions, materials engineers can control diffusion processes to enhance surface properties and overall performance.
- Carburising and nitriding process design for case hardening
- Homogenisation heat treatments for cast alloys
- Diffusion in semiconductors and thin‑film deposition
- Hydration and drying of polymers and ceramics
- Prediction of diffusion‑controlled phase transformations
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