[MUSIC] Hello everyone, I'm Surya Kalidindi. And I'm the instructor for the Materials Data Science and Informatics class. Today's lesson will be on Digital Representation of Material Structure. In this lesson we hope to understand what is meant by the terms local states, local state spaces, and microstructure function. After we understand the basic ideas, we'll learn about the discretized representation of the microstructure function which perhaps is the most practically useful form of the microstructure function. Let's start with the word local state. The local state is the word used to define all the attributes we need to identify in order to specify each microscale constituent in the internal structure of microstructure. Let's work through this through an example. For example, here you have an optical micrograph of a 2-phase microstructure from a steel. In this particular microstructure, you can easily identify two differently shaded regions. With some experience and background in metallurgy, you'll understand that one of these shaded region is ferrite. The darkly shaded is ferrite and the other region is austenite. [COUGH] Now in order to uniquely identify that this is ferrite, in addition to understanding that it is a particular thermodynamic structure, we also need to identify chemical composition. For example, you might need to know the percentage carbon concentration. And there may be other chemical species of interest. And there may also other features of interest, such as crystal orientations and so on, so forth. So this list can be fairly large. We don't know exactly how many attributes we might need. Nevertheless, once we identify all the attributes we need to clearly mark our tag, the state, the local state in every voxel, we are now ready to define the local state. So in this particular example, if we say the local state is identified as a combination of the phase identifier, rho, in which in this particular case, maybe rho = 1 is ferrite. And maybe rho = 2 is austenite. And then in addition to identifying the phase, we want to identify the chemical composition. In this case, this might be percentage carbon, or it could be some other elemental species of interest. This combination, the combination of rho and ci, is what we are going to call as local state. And that has a simple symbol h. The local state has the single symbol h in the rest of the slides. Now that we have defined the local state, we can move on to the local state space. The term space is a mathematical term used to define the complete set of all possible values that the elements of the set might take. In our case, we are interested in local state space. So we are interested in the complete set of all theoretically possible local states one might expect in studying a particular material. On the right here, you see a bunch of potential micrographs of interest in studying steels. And in this case there are a large number of potential local states that one might encounter. And for each local state, as we discussed in the previous slide, there may be some chemical compositions of interest. The combination of all possible combinations, all possible combinations of local state and chemical composition, is the local state space defined by the H. Just to remind you again. The local state space, in fact, in our definition includes all local states that may not even be present in a given sample of a selected material system. In other words, we want to be conservative. And we want to include in the local state space all potential states that you might encounter, even if you don't happen to encounter them in that particular micrograph. Now that we have defined local states and local state spaces we are ready to define a microstructure function. Any function needs to have a set of inputs and an output. So one can think of a microstructure as a function where the spatial coordinate and time coordinates are inputs, so x and t are inputs. And the local state is the output. So for example, on the micrograph on the right here, we might point to a particular location. And at that particular location, you know what the value of x is, and time, of course. Because you're looking at it in an instant in time. And you can say that that particular space or location is occupied by a particular local state. In this micrograph there are only two local states. One is shown white and the other is shown black. So h would take on the value of either, point to either black or white. And if you do that for every spatial position x and time t, you have the microstructure function. Now that's a fairly complicated microstructure function. In reality, in practice, what happens is that you only get a discretized description of the microstructure. What that means is if we now expand this small region here, you'll see that the information, the raw data, is presented in a voxellized fashion. So the space is divided into voxels and you only have one measurement in each voxel. So what you are measuring is really over a finite volume and also a finite time interval. The resolution, of course, is set by the instrument. So the size of the voxel or the time interval over which you're measuring is set by how we're measuring or what equipment you are using. The most important factor though, the most important point to remember is that what you're measuring is only an average measure, Over the finite probe volume and the finite time step. What this then raises is the fact that the local state you might find in any particular voxel or at any particular time step may not be unique. In other words, there may not be a single value of h at any particular spatial location x and time instant t. So this raises a problem and we need to fix that problem. And the easiest way to fix the problem is to think in terms of a microstructure function defined as m(h, x, t). Or if you don't want the time dependence, you can simply think of it as h, x. In this description, you're not specifically stating what local state exists at every spatial location, as we did in the previous slide. But instead, we now are defining the probability of finding a particular local state of interest at a particular location x at a particular time t. Because you're looking at the probability density and the probabilities, you can actually talk about multiple local states coexisting in each spatial voxel. So that takes care of the deterministic description. So now we can actually have a statistical description. However, the other problem with the probability density definition is that probability density definition becomes a little difficult to handle. So in order to understand the difference between probability density and probabilities, one needs to remember that if you have discrete outcomes, you can only define probabilities. And the simple example is tossing a coin. So it can only be head or tails, there's nothing in between. However, when have continuous outcomes, for example, if on the micrograph on the right, you have grayscales all the way going from 0 to 1 in a continuous fashion, then you cannot really talk in terms of probabilities. You can only talk in terms of probability densities. And what we're going to do here, we're going to simply avoid this problem and simply talk in terms of probabilities themselves by discretizing the microstructure. And this is okay because most experimental methods provide discrete microstructures anyway. And looking at probabilities is perfectly fine for our purpose, because most experiments actually produce only discretized information. So we're really only interested in the discrete version of the microstructure function, anyway. So how does that look? The discretized version of the microstructure function is schematically shown here on the left. In this case, we take the microstructure spatial domain and cut it up into voxels. In this case we have 16 voxels. And the way we are labeling the voxels, we're using two indices to label each voxel, s1 and s2. And you can follow the labeling scheme. And once we have a labeling scheme for each voxel, we're deciding what are the possible states we can place in each of these voxels. And in this case, we decided that there are two possible phases. And the two phases can be identified as either a white phase or a gray phase. And they're simply assigned to indices, n = 1 and n = 2. Now that we have established this convention, one can then assign the microstructure function, a discretized microstructure function, simply by following the conventions already presented. So for example, if you look at the microstructure function in cell 1,2, which would be here, cell 1,2, the volume fraction of white phase in cell 1,2 is 1, which is 100%. Volume fraction of the gray phase in cell 1,2 is 0 because there's no gray phase in the cell 1,2. Similarly, you can see that the microstructure function for cell 2,1 is described by these two variables. And all it is saying is the volume fraction of the grey phase in cell 2,1 is 100% and the volume fraction of the white phase in 2,1 is 0. So that's, in essence, my description of the descretized microstructure function. One other point is that these particular examples we have shown so far are what are called as eigen microstructures. Because we're allowing the microstructure function, the discretized microstructure function, to take values of only 0 or 1. In principle, they can take any value between 0 and 1. But so far we haven't introduced any that have values between 0 and 1. So the examples we have seen so far are would be called as eigen microstructures. And by definition, any time you assign a value between 0 and 1 it would become a non-eigen microstructure. So in summary, what we have done so far is to understand that microstructure functions are better defined as using the symbol m(h, x) and that we really want only the discretized version of this microstructure function. Because that's what is recoverable experimentally. And this discretized function has two indices. The index in the subscript we're going reserve to label all the spatial cells involved. And the index in the superscript we're going to reserve to index all the local states possible. And the physical interpretation of m s n, which we're going to encounter again and again in the rest of this class, the physical interpretation is that it represents the volume fraction of all local states from bin n. And this bin n responds to local states, Local state space. The volume fraction of all the local states that correspond to this bin in the spatial bin s. So it is simply has an interpretation as a simple probability. In that sense the values of m s n have to lie between 0 and 1 because probabilities have to be between 0 and 100%. And moreover, if you sum up the values of m s n over all the possible local states for any spatial bin s, they have to add up to 1. Because you have to find one of the possible local states within that cell s. That's why it's very important that when we define the local state space we are very comprehensive and include all possible local states. In summary, from this lesson we learned that microstructure functions are central to stochastic representation of the material structure. And that discrete representations allow practically useful forms of the microstructure function. Thank you. [MUSIC]