In this lesson, we're going to introduce the Johnson, Mehl and Avrami Equation, or sometimes referred to as the JMA equation. The JMA equation is a very powerful tool to help us model the process of a phase transformation. And we're going to see several examples of this as we go through the next several lessons, but first let's begin with the basic idea of what we mean by the JMA equation. If we have a process by which something is disappearing, and as a result of the disappearance of that one phase, another phase is beginning to grow, what we can do is we can follow the transformation process by looking at the fraction transformed as a function of the log of time. And we're using the log of time because these processes are going to be diffusion processes. They're going to be be controlled by diffusion. And hence, because of the exponential nature, we are going to be looking at the log of time. So we are plotting the log of time along the x, and the y axis is fraction transformed. For example, if we are studying the crystallization behavior of an amorphous material, we would be plotting the fraction of material that has crystalized during a particular time period. So, the way we look at the Johnson, Mehl, Avrami equation is up on the screen. What we see and why it's so important is that there are basically only two parameters that are necessary in order to fit the data, if the data behave as we see on the screen. So this is referred to, this particular behavior is referred to, as a sigmoidal curve, and what it does is it follows it's behavior. Now for example if you take a look at what happens as time goes to zero, the value of x then goes to zero because what we have is one minus one. On the other hand, as the amount of time continues on, and t becomes very large, then what will happen is this term will go to zero, and the fraction transform is now equal to x. And the general behavior of what usually happens during a phase change is it starts out slow, it then begins to accelerate in its process, and then eventually it begins to slow down. And we'll see more a more physical interpretation of these behaviors in a bit. So, the two values that we're looking at are k and n, and they're two variables. Now, what I can do, and the way I've written this equation, there are other ways that you can write the equation. But I'm choosing to write the equation where the product of kt is inside so that they are raised to the n power. And what that means then is that k actually can be a considered a rate constant. So, when we look at the beginning of the JMA curve, what we see is a part that we refer to as nucleation. To there's a certain period of time for the grains to nucleate, for the grains to begin to form, and then we have a particular point that we say start. The reason we fix a start point on here, as opposed to trying to draw the line so that the nucleation goes to zero, is the fact that the function is behaving in asymptotic way. So what we want to do is start when we can see a notable change in the structure. And then what we have is at the end, the finish time, and what's happening at that particular point is that again we have an asymptotic behavior of the transform function. And what we want to do is be able to terminate that reasonably, so we'll call the termination the finish. Now, a lot of times the start and finish will be basically determined by how easy it is for you to identify while you're making measurements, the start and the finish. Okay, but once you've done that then you can go ahead and you can assign the values for the start and the finish on the diagram. When you look at the curve, the other thing that happens is, the curve is beginning to slow down at longer periods of time. And I've written up there the word impingement. And basically what that means is, as the crystallization amount increases, we're beginning to slowly run out of space. And the growing grains begin to physically impinge on one another. So the process does begin to slow down. So now we look at the Johnson Mehl Avrami equation. And I've just reorganized it. And so now what I have is, I take that function, take the log of both sides of the equation. And I get the log of the quantity one minus x. And that's equal to minus kt raised to the n power. Now what I'm going to do is in effect multiply that equation by n minus one, and this is what I get. So i take the reciprocal of the one minus x, and that's going to be one over one minus x, and that's going to be equal to kt. Now if we take the logarithm one more time, what we're going to wind up with is this function will ultimately become linear, and it's linear as a result of that second logarithm that we're taking. So if we plot the log of the log of one over the fraction transformed subtracted from one. What I'm going to get is the behavior that I'm looking for. And that's going to be a linear behavior. And I'm going to be able to extract based upon the plot of the log log of one over one minus x. I'll be able to determine what the value of n is. If we look at a variety of different temperatures, for example, I've given T2 and T1. So we do the transformations, we define the start, and we define the finish points of the transformation at the two different temperatures. Then what we do is, we go to the second temperature, and we do essentially the same thing. Determine the start and the finish. Then what we wind up doing, is to take those points that we determine using the JMA equation, and we identify what we refer to as the start point and we refer to it as the finish point. And we have for the second set of data at temperature T2, we have the blue points. And this curve, this full black curve, for the start and the finish or in this case, 1% and 99%, those curves are determined by running a whole series of different temperatures. So we can fill in the data between what we have at T1 and above T1, and what we have at T2 and below T2. So consequently, we're able to produce the curve, and the curve appears as it does in the visual. Thus, the bottom curve represents something that we refer to as an isothermal transformation curve. We do experiments at constant temperature, and we plot those. And that plot tells us the progression of the transformation as a function of temperature. So we're going to be using these isothermal transformation curves in subsequent lessons where we can begin to talk about the different types of phase transformations that are important to the material scientist. Thank you.