What is Fick's Second Law?
Fick's Second Law of Diffusion describes how the concentration of a substance changes with time in a given volume. Unlike the First Law, which deals with steady-state diffusion (where the concentration gradient does not change), the Second Law is used for non-steady-state systems. In these scenarios, the concentration at any specific point changes as the diffusion process progresses.
The Formula
The partial differential equation for Fick's Second Law is: ∂C/∂t = D(∂²C/∂x²). For the specific case of diffusion into a semi-infinite solid with constant surface concentration, the solution used by this calculator is:
(Cx - C0) / (Cs - C0) = 1 - erf(x / (2 * sqrt(Dt)))
How to Use This Calculator
To use the Fick's Second Law Calculator, follow these steps:
- Surface Concentration (Cs): Enter the concentration of the diffusing species at the very edge of the material.
- Initial Concentration (C0): Enter the uniform concentration throughout the material before diffusion begins.
- Diffusion Coefficient (D): This is a material-specific property (usually in m²/s) that depends on temperature.
- Distance (x): The depth into the material where you want to find the concentration.
- Time (t): Total duration the diffusion process has been occurring.
Practical Applications
Fick's Second Law is fundamental in materials science and engineering. It is used to determine the necessary time for heat treatment processes like gas carburizing of steel, where carbon is diffused into the surface to increase hardness. It also plays a vital role in semiconductor fabrication for doping silicon wafers and in pharmacology to model how drugs permeate through biological tissues over time.
Frequently Asked Questions
What is the Error Function (erf)?
The error function is a mathematical function that occurs frequently in probability, statistics, and partial differential equations. In diffusion, it represents the profile of the concentration gradient as it smooths out over time.
Why is it called "non-steady state"?
It is non-steady because the concentration at a specific depth x is not constant; it increases or decreases as time passes until the entire system reaches equilibrium.