[MUSIC] Dear students in today's lecture we will discuss the Fick's diffusion laws or the quantitative description of diffusion. So from last lecture, we have defined the diffusion as the mass transport by atomic motion. So essentially there are two ways to describe diffusion. The first approach is so-called phenomenal logical theory of diffusion which correlates the atomic or molecular flux, which are the number of items or molecules passing through unit area per unit time. So correlate the atomic or molecular flux with the thermodynamic driving force That drives the diffusion. So this is so called phenomenological theory. And the second approach is to study the atomic mechanism of diffusion. In particular, how do the items that move from one equilibrium condition to another to carry out this diffusion process. Now we start from our discussion on diffusion through this phenomenological approach, and we'll talk about the Fick's first law first. So, the Fick's first law, describes the correlation between your diffusion flux J, with respect to your concentration gradient, d rho over dx. So suppose you have a small piece of material with the thickness of dx, and the concentration on the left hand side of the material is rho 1, the concentration on the right hand is rho 2. And if there are diffusion flux flowing across this small dx thickness and then fix propose that diffusion flux will be equal to a minus sign times a term called diffusivity times the concentration gradient d rho over dx. So j is the flux which have a unit of number of atoms or molecules per unit area per unit time. And d is the diffusivity have this unit, which describes how capable or how fast an item can diffuse. And row is the concentration gradient having this number of items per unit wallum and d rho over dx is the concentration gradient. And this Fick's first law is a purely empirical law. So you have a minus sign here, which means the diffusion is driven by concentration gradient and will happen in the direction of reducing the concentration gradient. So this is your Fick's first law which describes the correlation between your concentration and your diffusion flux. So here we show an example. So this is an example or a classical experiment done in the 1950s published in this classical paper. So, suppose you have a thin shell, where the internal radius is r2 and the external radius is r1. And you have a high temperature carbon item of high concentration as well inside the cylinder. And then these high density or high concentration carbon atoms want to diffuse from the internal part of the cylinder to the external part side. So, this will be a classical diffusion problem and there are a few assumptions. The first assumption is you have this lens or height of your cylinder much larger than the wall thickness of your shell. So then the question is at steady state, how to calculate the diffusion flux. So the term steady state, probably you have already met in your college physics which means if you have a physical term of, there's a question mark, well, corresponds to any physical term, you are time derivative of any physical term will be 0. I under steady state, right? So nothing is changing under steady state over time. So if you write down this diffusion flux j, you can write down the J as a function of your diffusivity and your concentration gradient. So this is your definition, or this is just your Fick's first law. And if you define the total number of carbon items traveling across the thin shell is Q and if you divide the total number by the cross sectional area of the shell, which is two pi r times L and also you divided this by your time you just to generate the flux as well. So if you equal these two together, you will generate this equation. And because you're under steady state, your q over t is a constant. So q over t is a constant. So your diffusivity times the concentration gradient where it's a modified concentration gradient which is equal to d rho over dlnr will be a constant. So, in practice such an experiment is for the measurement of your diffusivity d. So, in particular, after diffusion you can cut the shell into small pieces. And then you just measure the carbon concentration in each of these small pieces. As long as your pieces are sufficiently small, you can generate a concentration profile as a function of lowering lnr, your sink shell. And with this profile and with this keeps as a constant, you can then calculate the diffusivity of carbon atoms in the shell. So, essentially, you can calculate or you can obtain your of your diffusion coefficient or diffusivity from this concentration profile. So, as the product of your concentration gradient and your diffusivity is a constant. So if your concentration or modified concentration gradient is a constant which means the slope of this concentration profile is a constant, you will have a constant diffusivity as well. However, what actually measured in the scientists original paper is that your concentration versus lnr is not linear which means your diffusivity is concentration dependent. And in particular if you have a higher concentration, you will have a higher diffusivity as well. So, this is how you measure the diffusivity under steady state condition. Now we move on to the Fick's second law. So previously, the Fick's first law doesn't have a time term there. So, how about the situations where time is a parameter in the diffusion problem? So this is what described by Fick's second law. So essentially the derivation of the Fick's second law, the derivation of the second law is the combination of your Fick's first law and mass conservation. So suppose you have this piece of material and you are considering a small thickness of dx here and the flux flowing in the dx is j1, the flux flowing out of the dx is j2 and the position here is x the position here is x + dx. And then you can derive the diffusion equations by combining Fick's first law and mass conservation. So essentially the processes like this. So, you will know by mass concentration the items or molecules accumulated in a small dx wallum will be equal to the items flow in or diffused in minus items diffused out, okay? So mathematically it means this. Okay, the change in your concentration in your dx times the wallum of your unit, so is a cross section area times dx will be equal to the flux in minus flux out, which is delta j minus, right? Times the cross section area, times the time interval. So if you derive this equation, and do some mathematical treatment, you will know that it's essentially just partial concentration over partial time equals minus partial flux over partial position. So this is essentially your Fick's second law. And if you put your Fick's first law for the expression of flux, you can generate this equation where you have your partial concentration over partial time equals this relatively complex term here, and you have a diffusivity in the bracket. So to summarize a little bit, your Fick's first law can describe any situation time dependent or time independent. However, if you are, Situation is time dependent, you can only use the combination of your Fick's first law and Fick's second law. So here we show the equation for Fick's second law. So we can see that if your diffusivity is independent of concentration, from the previous equation you can take the diffusivity out. So essentially you have this partial concentration over partial time equals the diffusivity time the secondary derivative of your concentration over position. So this is the case where you have your diffusivity independent of concentration. And in general in the three dimensional case, you draw or you can conclude with this equation, right? You have x, y, and z coordinates. So essentially, we have some final comments on the Fick's second law and Fick's first law. So for the Fick's first law is to reduce the concentration gradient, right? Is driven by the concentration gradient and diffuse will take place in order to reduce the concentration gradient. For Fick's second law if you really learn your college mathematics, well you will know that the second order derivative corresponds to the curvature of your curve, right? So, essentially for a Fick's second law you are trying to or the diffusion is trying to reduce the curvature in a concentration profile. So today's lecture we have talked about the Fick's two diffusion laws. Thank you very much. [MUSIC]
