In this lesson we're going to be looking at Fick's Second Law, and we're going to apply the law to the solution of a particular problem called the problem of the Thin Film. Let's examine how Fick's Second Law was developed. First, we're going to look at a volume element, and that volume element is going to be described in terms of the width of the element times the cross sectional area. So we're looking at a region which is a uniform cross section. We're going to be examining the flux that comes into this system, and the flux that leaves the system. And what we can do is we can write a governing equation that describes the time dependent change in composition of that volume element with respect to the flux coming in and going out. Now, what we'll do is rearrange these terms and when we have rearranged these terms, what we wind up with is the changing composition is directly related to the change influx with respect to the region over which the diffusion processes occurring. And notice because we are diffusing down the flux gradient, what we see is that changing concentration has a negative sign in front of it. Now we know that we can write an expression for the actual flux in terms of the expression for Fick's First Law. So now what we can do is we can substitute and now we have an equation that relates the change in composition with respect to time on the left hand side of the equation and the diffusivity on the right hand side along with the second derivative of the change in composition with respect to position. So with all this information, we're now able to look at a process which is not necessarily a time independent process. It could be very well time dependent. Now let's examine the thin film. So we're going to look at two different materials, material A and B. And what we're doing is taking two semi-infinite slabs of A, and between them what we're going to do is to put a small amount of B that turns out to be completely soluble in A. So consequently what we expect to see happen is, as the diffusion process goes on, B moves into A, and it moves there in both directions from positive x to negative x. So, here is our picture of what happens when we put these two together and we put the thin film in-between. And it may be in order for us to see some reasonable diffusion we might to make sure that we prepare the surfaces that are in contact with one another to allow the diffusion process to occur readily without the influence of an oxide film presenting preventing the diffusion process. So here we are our picture at some time, t is equal to zero. We put this whole thing in a furnace and as a result of putting it in a furnace for a certain period of time, what we find is, there is a distribution of B that goes into the left and a distribution of B that goes into the right. And what we're doing is along this Y axis of the structure, we're plotting the composition. Now, what you can see here is the behavior looks to be Gaussian and you would expect it to be that way, in that the diffusion process is an exponential process. So, we see this behavior, and this is what we anticipate. So, I'm going to take a simple exponential function. C is the exponential of minus x squared, and I have graphed that function. And then what I'm going to do is I'm going to come along here and I'm going to take local derivatives as indicated by those red lines. If I take each one of those points where I have taken the first derivative, I have the first derivative then plotted as a function of composition to the right. So I have the actual first derivative plotted next to the composition as a function of distance. This slide now contains three pieces of information. The first is the Gaussian behavior which is the composition that we are as a function of position that we begin with. Then the next graph is the derivative of that function. And then the third graph that is superimposed is that of the associated second derivative. And if you go back in your review and think about what the meaning of the second derivative is, it's actually describing what the concavity of that particular function is that we are looking at. So, the second derivative with respect to x tells us the concavity. Now, between the points that are indicated by the blue arrow. What we're going to see is the composition in this region winds up decreasing with time. And if we look at what happens on the outside of those points, what we're going to see is this is the region where the composition increases with respect to time. And those two red arrows that appear on the screen now represent the region where the concavity of the function is changing. And the portions between the central region, where the function is decreasing, the concavity is negative and over to the left of those points, and to the right of those points we're seeing that the concavity is positive. And hence the composition is going to be increasing with time. Now if we go back with respect to examining Fick's Second Law, and we think about it in terms of the concavity of the composition function, what we see is, here's Fick's equation. And when the concavity is negative, we're seeing that the composition is decreasing with time. When the concavity is positive, what we're seeing is that the composition is increasing with time. Now, the point at which we have a change in concavity can also be used to identify the position as the diffusion process continues. So, it can tell us about the extent of the fusion process. And we can see that more quickly by looking at a series of plots where we have looked at different times, and what you can see is that the function is spreading. You can see how the portions where we have concave down, what we're seeing is a decrease with respect to temperature and in terms of composition. When the function is concave up, what we're seeing is an increase. We see those points where we change our concavity from negative to positive. What we're seeing is those points are moving with respect to time. So we can follow the process of diffusion that's occurring in this thin film. Now the actual solution to this including all of the boundary conditions of this particular function is given in this slide, and what we see is we can describe the change in composition not only with position, but with respect to time. So we have two variables here, and this is then a partial differential equation. The term in the pre-exponential tells us about the geometry of the process, what's the cross sectional area. It also describes for us how much mass we have in the material. And then in the exponential portion X represents the location where we are examining the composition and D and t have their normal meanings, D being the diffusivity and t representing time. So what we're going to do then is to look at this equation and see how we might be able to modify this equation to describe another type of diffusion that involves slightly different boundary conditions than the thin film. Thank you.