菲克第二定律.doc

Lecture 4: Diffusion: Fick’s second law
Today’s topics
• Learn how to deduce the Fick’s second law, and understand the basic meaning, in
comparison to the first law.
• Learn how to apply the second law in several practical cases, including
homogenization, interdiffusion in carburization of steel, where diffusion plays dominant
role.
Continued from last lecture, we will learn how to deduce the Fick’s second law, and
understand the meanings when applied to some practical cases.
Let’s consider a case like this

We can define the local concentration and diffusion flux (through a unit area) at position “x”
as:
So, Fick’s first law can be considered as a specific (simplified) format of the second law when
applied to a steady state.
Now, let’s consider two real practical cases, and see how to solve the Fick’s second law in
these specific cases.
Case 1. Homogenization: (non-uniform →uniform)
Consider position profile as superimposed sinusoidal variation as shown below, where
the solid line represents the initial concentration profile (at t=0), and the dashed line
represents the profile after time τ.
It is an exponential decay, the longer the wavelength (l), the longer the relaxation time (τ),
then the slower decay. Short wavelength dies fast. That’s why shaking always helps speed up
the dispersion, because it enables wide spreading (smaller l) of the