Lecture 36

[MUSIC] [MUSIC] We will start lecture 36. We have seen over the last few classes that if del h mix is ​​a non-zero value either positive or negative then the δ logic mix also changes and its shape changes. H mix becomes positive then if the temperature of δ logic mix decreases then we see that the shape changes and two minima are coming ok and from those two minima we understand a concept of miscibility gap and we also saw this in last class if the δ logic mix changes and I know G2 which is a solution free energy will also change, those two downward points will be found, the minima will be found, because Dell G2 Mix is also changing and G2 is G1 plus Dell G2 Mix, so G2 is also changing and We can also take the concept of visibility gap from G2 and see that if the temperature changes, if we go to high temperature, G2 will be continuous downward concave, but if we go to low temperature, G2 will reach those two minima points and there we can apply the phase equilibrium concept. And we will see that a single phase will break into two phases and those two phases will be in equilibrium and we have also seen that in the phase diagram it means temperature composition that there is a dome that gives an upward dome but within that dome there will be two phases and the two extreme ends indicate the position of those two phases. It will be done but within that dome but whenever any composition is taken from single phase high temperature, when it comes to that dome, two phases will be broken, so we have already seen that, we will discuss more about this. Coming means the critical temperature, the dome is slowly starting to form. OK, I will try to calculate that point and also see how the solution phase energy is changing. We have been able to calculate and show how many examples of practical examples can be given with the Gaussian chemical method. Today's topic is but we will continue with the same topic. We'll talk a little bit more about the energy and miscibility gap, so what we've seen is a binary solution, a binary solution, and there are A and B elements, N atoms, Nb atoms, and adding that up gives Avogadro's number of atoms. Avogadro Number of Atoms Atoms And if I take the mole fraction exp equal to one it is one mole mole solution and is t constant rather I will say that t is fixed and at different temperature we will see the free energy plot pressure but always in the atmosphere is fixed temperature I see at different temperature But whenever I plot the free energy of pressure at different temperatures, but the pressure is always at another atmosphere, we see in the regular solution approach that del G mix is ​​equal to del h mix minus t del s mix and another thing we see is del h mix. Equal to the interaction parameter x is xp and what we saw in the last class is we assumed this to be a constant del h mix and how does it affect the del h mix at different temperatures what is the effect on that shape from ok load to high temperature We saw that reading and that's where the concept of the miscibility gap comes from, which is a constant DelH mix but the DelH mix can change in different systems. There will continue to be changes and there we will get that critical value of Dell H mix there but my phase separation or miscibility gap will start to appear and if we look at the temperature aspect of constant Delch mix there will also be a critical temperature there will be my Del H phase separation or The miscibility gap will start to appear, so this point is nothing. The extreme point of the head of the dom is where we see if we can change the temperature before that. So we'll see this time changing the Dell H mix and the same temperature we'll see how the Dell mix changes so let's look at that first we'll see the effect of change in Dell H mix so let's see that before that I'll do a treatment here. This way we can write del h mix equal to rt alpha x a x b here alpha is a function that is called alpha equal to interaction parameter omega divided by rt ​​so again i can change δ log g mix here i am saying if i complete Let's write the expression rt α x a xb + rt x a ln xa + xb ln xb where δ is the mixture of - r x ln x a plus xpl n xp So this time I'm going to keep changing this alpha value at the same temperature let's see what the plot pattern is. It is available here, I will not show the minus t del s mix in the plot, it will not be at a constant place because the temperature is fixed, so if I look at a plot, I will show a schematic that is this, let's say minus t del s If mix since temperature is kept constant then its position will be fixed I mean only position that will change is δH mix I del H mix I show with blue color del H mix first like this first zero can be zero then small del H mix can large slowly del h mix increasing δ l h mix is ​​increasing this is del h mix increasing that means we can also say that since X A X B is taking a particular composition at a point I'm taking it at this point I'm taking the valence at this composition there I just have If I see both r and t are constant that means alpha is a particular constant value I see here alpha will be equal to zero slowly if del x mix increases that means alpha will also increase so that means i will see del g mix but this also remains fixed these particles At a particular temperature, at an alpha value, I also have this del H mix too fixed, only changing, that means del g mix is del g mix, let's see what happens at high temperature, let's see what happens at low temperature. I will not draw only δL H mix rather I will see how δL H mix will change based on the alpha parameter and how δL G mix will change with respect. With the blue color I will plot the Dell H mix, with the red color I will plot the Dell H mix with the red green color, I will plot the Dell H mix when I take the Dell H and at the particular temperature. Either here alpha is alpha is zero ok that means del h mix is ​​also zero so that means del h mix is ​​equal to zero and another thing from here I can get is if I simplify here I get alpha x a xp plus x l n x A plus X B L N XP is equal to delta G mix by RT so I can also plot this delta G mix by RT so if I plot this then I can also plot the del H mix I can also plot this mine divided by RT here I Let's write alpha x a x b equal to del h mix by RT so I'll plot this so we 'll work with a normalized value like this and this when we're taking alpha zero my delgy mix will come out like this delgy mix is ​​actually del g mix Plotting the B&T because we're plotting this here and plotting this so now if I keep increasing the alpha the alpha starts to increase symmetrically like this comes Delgy Mix at the same temperature its temperature and pressure constant the Delgy Mix will also rise up the Dell gimmicks Let's see if this is called alpha equal to two I can do this alpha equal to let's one here if I immediately put the value of alpha equal to zero in this case alpha equal to 1 and this is Dell G mix by RT so that's it. Then when my alpha becomes two it means it will continue to increase and if radar alpha is two then alpha becomes two so alpha is equal to two if alpha is two I will see that the plot of Dell G Mix by RT will stand like this and a plateau will be found. OK, this is Dell G mix by RT when alpha is equal to 20. Now if I increase the alpha a little more and I draw the radar alpha a little lower, then the advantage will be alpha equal to 20. Now if I increase the alpha more, let's make it alpha 30. Whenever Alpha 22 comes, a plateau will be available. We will discuss why the plateau is coming later. As soon as Alpha 30 is reached, there will be a pattern here. The blue plots that happened are all del H mix divided by RT OK so that means I rationalized my RT by RT I rationalized del H mix and del G mix there I first saw that I had a common tangent It turns out OK and there, whenever I get to the common tangent, my two compositions break up my single face into two compositions, and if I then increase the alpha, I increase the alpha, equal to three points, five, let's say, Del H. Mix by RT, too. Steep upward and then this too but this time if I see this value this value this value this value this value at the midpoint I've put two a little bit apart just to illustrate these two midpoints mean when XB will be in 05 composition and I'm not writing that separately but it's So the xp will vary and it's x equals two and it's x equals two. These two pure sides are binary because whenever these two are equal, that means my plot will be like this. And when I draw the common tangent. Here also a common tangent I can see composition two on the right means when the XP increases from point five then that equilibrium point will move to the right and when the XP less 05 then the equilibrium point will move it to the left now and then if we can increase alpha a little more alpha a little more and here Since these two arms are equal, these two arms are equal, this arm of this armature is equal, then what will happen to me at the mid point will touch zero, del g mix by RT, and here another reason to divide by RT is because the temperature is fixed constant because RT is is a constant value and then if alpha can be increased a little more alpha if we can increase alpha a little more let's say alpha here is 40 then my Dell G mix will move up and it will become like this and there will also be a common tangent so this composition and this composition I mean, if I change the alpha too, then it was also seen how the plots of Dell H Mix by RT constant temperature, Dell H Mix by RT and Dell G Mix by RT will change, so here too, there is a critical alpha. Value is so alpha alpha critical then but face separation will start to appear or miscibility gap will start to appear just appear and there I this is critical limit means where else what will just appear and if alpha even slightly less than alpha critical then the radar increases. So my alpha critical increases immediately but my phase separation or miscibility gap will start appearing in the phase diagram so ok now we can change the critical value of alpha and at the same time we also see as a function of temperature alpha in fixed condition means alpha when we It's fixed I mean Dell H Mixta fixed but here Alpha if you think I should fix Alpha there I'm fixing Dell H Mixta because T is changing so to keep my Dell H Mixta fixed Alpha will also change but if I Dell If we keep the H mix fixed as a function of temperature then we will see that the miscibility gap is coming and that is but we have reflected it in the face diagram so we can see if we can change that critical temperature but at least one thing is understood that my miscibility gap To appear I have to go lower or increase alpha value so in temperature fix condition then my miscibility gap can come that means I have two conditions for miscibility gap one temperature sud be low at constant del h mix and second alpha sud B high at constant temperature so here we plot the temperature and composition on our phase diagram so when the temperature here has an alpha critical value there will also be a temperature critical value where from where we go to a little lower temperature we can see the miscibility gap. That critical temperature can be calculated ok, to do that we need to understand a little diagram and do a little calculation. Phase diagram, this is my critical point and here X is equal to one, X is equal to one, we will calculate this point and this is the miscibility gap. That is the matter of DELGI MIX is LG MIX equal to Omega X A XP plus RTX L N X A plus XP L N XP so I take fast derivative with respect to XP partial should be taken XP equal to extend a little X A X I can write this as x b one minus x b x b plus r t one minus x b l n one minus x b plus x b l n x b because x a plus x b is equal to one we already know so if i xp Respect if we take a partial then I get 1 - to xp plus rt you can try this it's going to be ln xb - ln x a ok now if I do the second derivative one minus x p and here it will be one minus x a plus because Here X A is one minus X PR and if we take the third derivative it will be R T one minus X A squared minus one minus X B squared. First, I took a plot like this. I'm not writing, it will vary the XP here and it's pure A and it's pure B. I'll go to infinity here, it'll go to infinite slope, so the next plot will come. And this point the slope is infinite we will start a little right to left because data management from infinite slope will be difficult what we will do from here let's start from here a little off composition we see that slowly here slowly the slope is increasing here the maximum is again It's starting to decrease so the radar slope is decreasing here I'm wrong I said slope is decreasing from here if we see the slope is decreasing now it's increasing then again the slope is increasing and everything but positive slope I'll get the pattern like this Minimum slope here I'll get a plot like this ok this is Delta G Mix The second derivative and the third derivative will be zero zero. Now, if we look at the third derivative, let's start from a finite slope from this point. Let's do it with green let's do it with green let's do it with green I'm going to get one like this and this is the third derivative right and this is and when does this pattern come this is always the delgy mix is concave downward this is the delgy mix can come When alpha less gives alpha critical or temperature more gives temperature critical if you look carefully what happens at higher temperature the value of t del s mix is greater than the value of del h mix by magnitude so what happens is the whole curve is down downwards that one A change in slope will not come in delji mix so it will come in a condition like this. Now what can be the second condition when alpha equal to alpha critical and temperature equal to temperature critical temperature critical so then what will be the plot delji mix plot I will see here a plateau will come. Where just starting there will be a plateau so this is the plot of delgy mix now I will start doing the derivatives how will the fast derivative come here also start from infinity and here a plateau is coming ok here in the plateau I see the slope is zero so it is coming to me One like this will go flat and one like this will increase so one like this will come from there I'll slope it again this time I'll see when I do the slope of it since it's infinite at the extreme end I'll come a little off composition from that I'll come to a finite slope so I'll see when I come to a finite slope Slowly the slope is decreasing here here it's going to start increasing here zero slope I'm going to be like this it's going to be flat again it's going to start increasing fine so we're going to get a pattern like this so we're going to get a pattern like this now this is xp now let's look at the third derivative ok here's a mess you guys Correct it is third derivative ok not square third so third derivative again here comes this point from a finite slope here the slope will slowly decrease to constant and then increase again so we will get something like this now it looks like when I come to the critical point there is a thing at the critical point It is understood that this becomes equal to zero for t equal to three critical and alpha equal to alpha critical. Now let's see if I give alpha greater, alpha critical and t less give t critical, what kind of plot will be there. This will appear, face separation will appear, there will be a pattern like that of Delgy Mix, if I show the pattern of Delgy Mix like this, then I will make it again all this is Delgy Mix, as I did the previous ones, I will do the fast second and third derivative of it. There will be a pattern like this, first the negative infinity slope is zero here here the slope is zero here also the slope is zero those three points we already extend it after extending it here the slope is coming like this then slowly at this point the strange thing is here the slope is slowly decreasing decreasing again decreasing decreasing Here it will start to increase again like this the comma will increase so here will come here a plot like this will come a plot like this will come a plot like this here zero slope here zero slope this too zero slope then the next one will be ok it will start from my slightly off composition because Here infinite slope at the extreme end it's going to be like this here then these two points zero slope ok these two points will cross my diagram plot with these two points my plot will be like this you'll get a similar plot like this ok and third If I do the derivative, it's definitely zero. It's zero points. You can put zero points on all of them. So here's the plot. Here's the slope at the midpoint. Again, I'm going to get something like this. And this is ok. So I can see that when critical means this is critical. Critical t or alpha there if I look at the second derivative if I equate it to equal zero I already saw that equals two plus RT rather x b plus one x we saw it and I see that when it comes to the critical point in this condition you have all these graphs If you observe at this point I have it should be zero and since my plot of free energy delta mix is ​​symmetric then what can I do I can take a condition exp equal to x equal half if I take 05 composition then from here I get two omega Equal to RT for RT if I put these values then I get Omega equal to two to RT so and from there I can do t critical times t critical because it's a little bit critical equal to omega by 2 and so now what can alpha be I know alpha Equal to Omega RT rather than Omega BI RT because I know Delta H Mix Equal to RT Alpha X A X P Again it's Omega X A X B that means Alpha is Omega BY RT If I put it here it means I Alpha equals to rather put here alpha equals to t critical and omega by 2 and from there I got 2 means t is critical by 2 and alpha critical is 2 this time I I mean here I did this plot this is the phrase diagram I have shown the plot where I can bar this critical point and if the temperature goes below it it will start to break. Both compositions are fine. Another thing here is an important thing to say. Look at this plot, I have two points coming from Delgy Mix, these two points are coming like this, these two points are coming like this. Either there is a decomposition, it is called spinodal decomposition. Okay, between these two points, spinodal decomposition is spinodal decomposition, but we will not discuss that, so these two points are always. In the middle but the spinoral will lie ok if I look at this diagram here in the phase diagram I will get the spinoral point like this always it will be in the middle and the spinoral point will always be at t critical but will merge so my spinoral will be the line at those points where my second derivative is coming to zero If I continue to connect the two points, we can draw a line in the middle, which is the face miscibility gap. This gap is very important in the face transformation. Here is the spinodal decomposition. Fine A is where the two points are coming from when we look at them. If I keep plotting those two points where the second derivative goes to zero, if I keep reducing it, I will see that these two points will continue to get wider and wider, and I will also see that these two points of zero will start to widen as the temperature decreases. And so those loci, but we've plotted here a zone of spinodal decomposition in the miscibility gap, we'll discuss this later if there's a course on face transformation. When H mix is ​​different value a del H mix value can be positive negative can be zero and negative I have seen that my delgy mix plot is always downwards but whenever del h mix becomes positive I have a miscibility gap. will appear and how is that miscibility gap coming and what is the relation of the face diagram of phase transformation and how we are plotting the miscibility gap from there we can calculate the critical temperature value that is the apex point from there we can change a single face into two phases We've seen those that can be broken down and we've also seen how the alpha parameter changes and how we get to the critical point. Alpha is equal to two then this plateau will come, I discussed earlier, so this time it was understood why this plateau is coming to alpha equal to two and it is proved that by doing these derivatives, we proved that fine ok this is the miscibility gap and with it But the possibility of spinodal decomposition is very important in phase transformations and is seen in engineering alloys, and we can get many different properties from it, so our discussion of this miscibility gap and sphenol decomposition is very important to have our ideas, but that's not part of this course. We will not discuss that, we have seen how del H mix varies the delgy mix its plot pattern and from there we know a lot about miscibility gap spinodal decomposition gap and how it is plotted on face face diagram and also How to calculate those critical values ​​are all understood ok you have to look it up a bit better if not but it's a bit hard to understand you have to practice on your own I would say you all practice and slowly draw on your own you will see that You'll get a pattern like this and from there you'll get a pretty good idea of ​​the face transformation face diagram. Okay, we'll end this class here. Gonna go slow and try to see how the different phase diagram patterns form from there we'll end here for now thanks [MUSIC] [MUSIC] [MUSIC]