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 a composition 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 stuff (like particles) you try to disperse Case 2. Interdiffusion (the carburization of steel):doping of steel with carbon Situation a): Doping with fixed amount of dopant Consider a thin layer of B deposited onto A, through annealing at high temperature, we will be able to get the concentration profile at different times, from there then we can determine the diffusion coefficient, D as determined by this diffusion kinetics equation, the concentration profile of carbon as various times will be like this The above diffusion is one-direction (0 →+∞). But if we extends it to two-way, from -∞ to +∞ (like a