WEBVTT 00:00:00.000 --> 00:00:12.050 align:middle line:90% 00:00:12.050 --> 00:00:15.590 align:middle line:84% RAFAEL JARAMILLO: So let's talk about phase coexistence. 00:00:15.590 --> 00:00:22.830 align:middle line:90% 00:00:22.830 --> 00:00:32.680 align:middle line:84% Phase coexistence in a common tangent construction. 00:00:32.680 --> 00:00:39.050 align:middle line:90% 00:00:39.050 --> 00:00:41.690 align:middle line:84% Right, so what's the situation here? 00:00:41.690 --> 00:00:42.485 align:middle line:90% We have two phases. 00:00:42.485 --> 00:00:46.830 align:middle line:90% 00:00:46.830 --> 00:00:50.910 align:middle line:90% We'll just label them 1 and 2. 00:00:50.910 --> 00:01:13.050 align:middle line:84% And they can freely exchange components A and B. 00:01:13.050 --> 00:01:16.860 align:middle line:90% So two phases, two components. 00:01:16.860 --> 00:01:28.430 align:middle line:84% And we know that phased coexistence at equilibrium 00:01:28.430 --> 00:01:37.570 align:middle line:84% requires that the Gibbs free energy is stationary. 00:01:37.570 --> 00:01:41.980 align:middle line:90% So here's our little picture. 00:01:41.980 --> 00:01:52.550 align:middle line:84% We have phase A, phase one, phase two. 00:01:52.550 --> 00:01:54.965 align:middle line:90% And let's say A is-- 00:01:54.965 --> 00:02:00.940 align:middle line:90% 00:02:00.940 --> 00:02:04.860 align:middle line:90% it can be red squares. 00:02:04.860 --> 00:02:09.840 align:middle line:90% And B can be green circles. 00:02:09.840 --> 00:02:13.800 align:middle line:84% And we know that this boundary is open. 00:02:13.800 --> 00:02:17.980 align:middle line:84% If this boundary is open, then what? 00:02:17.980 --> 00:02:23.343 align:middle line:84% Components can freely jump between phases. 00:02:23.343 --> 00:02:24.760 align:middle line:84% That's something that can happen-- 00:02:24.760 --> 00:02:28.520 align:middle line:84% it's an unconstrained, internal process. 00:02:28.520 --> 00:02:32.550 align:middle line:84% We're not going to stop that from happening. 00:02:32.550 --> 00:02:37.960 align:middle line:84% So given that these matter exchanges 00:02:37.960 --> 00:02:44.820 align:middle line:84% can happen, how do we determine the equilibrium condition? 00:02:44.820 --> 00:02:58.750 align:middle line:84% And I say, these fluctuations must leave 00:02:58.750 --> 00:03:06.090 align:middle line:90% G stationary at equilibrium. 00:03:06.090 --> 00:03:09.130 align:middle line:90% 00:03:09.130 --> 00:03:12.980 align:middle line:84% OK, so this is now zooming in at the molecular level. 00:03:12.980 --> 00:03:16.240 align:middle line:84% And I think people understand that if you have a glass of ice 00:03:16.240 --> 00:03:18.190 align:middle line:84% water, there's a boundary between the ice 00:03:18.190 --> 00:03:20.860 align:middle line:84% and the water, the liquid water and the solid water. 00:03:20.860 --> 00:03:26.080 align:middle line:84% And ice and water molecules will be jumping back and forth 00:03:26.080 --> 00:03:28.780 align:middle line:84% between the two phases at that boundary. 00:03:28.780 --> 00:03:32.300 align:middle line:84% On a molecular scale, and on a molecular timescale. 00:03:32.300 --> 00:03:38.530 align:middle line:84% So what does it mean for that process to leave G stationary? 00:03:38.530 --> 00:03:40.960 align:middle line:84% That would be our equilibrium condition. 00:03:40.960 --> 00:03:43.160 align:middle line:84% So we can get into the math with that. 00:03:43.160 --> 00:03:47.440 align:middle line:90% We say, the change of G-- 00:03:47.440 --> 00:03:50.265 align:middle line:84% now we're going to be at fixed temperature and pressure. 00:03:50.265 --> 00:03:51.890 align:middle line:84% So we're not going to worry about that. 00:03:51.890 --> 00:03:53.800 align:middle line:84% And we know the total differential 00:03:53.800 --> 00:03:56.740 align:middle line:84% of Gibbs free energy from weeks past. 00:03:56.740 --> 00:03:59.400 align:middle line:90% 00:03:59.400 --> 00:04:05.350 align:middle line:84% It's a sum over components and phases, chemical potential 00:04:05.350 --> 00:04:12.770 align:middle line:84% of each component in each phase times the change 00:04:12.770 --> 00:04:16.950 align:middle line:84% in mole number of each component in each phase. 00:04:16.950 --> 00:04:22.920 align:middle line:90% So K going-- let's see A, B-- 00:04:22.920 --> 00:04:24.090 align:middle line:90% these are components. 00:04:24.090 --> 00:04:28.700 align:middle line:90% 00:04:28.700 --> 00:04:36.515 align:middle line:84% I going one, two, these are phases. 00:04:36.515 --> 00:04:42.000 align:middle line:90% 00:04:42.000 --> 00:04:46.680 align:middle line:90% So let's write this out. 00:04:46.680 --> 00:04:55.530 align:middle line:90% This goes mu, A, 1 d mu-- 00:04:55.530 --> 00:05:00.250 align:middle line:90% d n, sorry, let me start over. 00:05:00.250 --> 00:05:28.450 align:middle line:84% Mu A 1 d n a 1 plus mu B 1 dn B 1 plus mu a 2 dn a 2 00:05:28.450 --> 00:05:36.370 align:middle line:90% plus mu B 2 dn B 2. 00:05:36.370 --> 00:05:38.005 align:middle line:90% OK, so we're written that out. 00:05:38.005 --> 00:05:39.880 align:middle line:84% Now, we're going to use conservation of mass. 00:05:39.880 --> 00:05:44.950 align:middle line:90% 00:05:44.950 --> 00:05:48.580 align:middle line:84% And conservation of mass tells me 00:05:48.580 --> 00:05:52.150 align:middle line:84% that I can have molecules or atoms jumping back and forth, 00:05:52.150 --> 00:05:54.400 align:middle line:84% but I can't have them just disappearing. 00:05:54.400 --> 00:05:55.900 align:middle line:90% Or just appearing. 00:05:55.900 --> 00:05:58.190 align:middle line:90% I've got to have mass conserved. 00:05:58.190 --> 00:06:08.980 align:middle line:84% So conservation of mass tells me that dn a 1 equals minus dn a2. 00:06:08.980 --> 00:06:11.560 align:middle line:90% 00:06:11.560 --> 00:06:17.320 align:middle line:84% And dn b1 equals minus the dn b2. 00:06:17.320 --> 00:06:20.200 align:middle line:84% So two equations, and so we're going 00:06:20.200 --> 00:06:22.510 align:middle line:84% to reduce our number of independent variables 00:06:22.510 --> 00:06:26.010 align:middle line:90% from four to two. 00:06:26.010 --> 00:06:27.560 align:middle line:90% So I'm going to reduce-- 00:06:27.560 --> 00:06:29.090 align:middle line:90% I'm going to do that. 00:06:29.090 --> 00:06:31.640 align:middle line:90% And I get this-- 00:06:31.640 --> 00:07:02.430 align:middle line:84% mu a 1 minus mu a 2 dn a 1 plus mu b 1 minus mu b 2 dn b 1. 00:07:02.430 --> 00:07:06.120 align:middle line:84% So I have four independent variables, 00:07:06.120 --> 00:07:08.310 align:middle line:84% I applied two constraints, I end up 00:07:08.310 --> 00:07:12.250 align:middle line:90% with two independent variables. 00:07:12.250 --> 00:07:13.960 align:middle line:84% This has a familiar form for us-- these 00:07:13.960 --> 00:07:27.400 align:middle line:84% here, these here, are the unconstrained, independent 00:07:27.400 --> 00:07:28.240 align:middle line:90% variables. 00:07:28.240 --> 00:07:36.850 align:middle line:90% 00:07:36.850 --> 00:07:41.590 align:middle line:84% Those are the unconstrained, independent variables. 00:07:41.590 --> 00:07:48.540 align:middle line:84% And what do we call the multipliers 00:07:48.540 --> 00:07:51.325 align:middle line:84% in front of the independent variables in this differential 00:07:51.325 --> 00:07:51.825 align:middle line:90% form? 00:07:51.825 --> 00:07:55.010 align:middle line:90% 00:07:55.010 --> 00:07:58.650 align:middle line:84% What do we call this thing and this thing? 00:07:58.650 --> 00:08:01.330 align:middle line:84% This is a differentiable form for a dependent variable, 00:08:01.330 --> 00:08:04.530 align:middle line:84% and how it varies with two independent variables. 00:08:04.530 --> 00:08:05.220 align:middle line:90% And we have-- 00:08:05.220 --> 00:08:09.778 align:middle line:90% 00:08:09.778 --> 00:08:11.400 align:middle line:90% STUDENT: It's the coefficient. 00:08:11.400 --> 00:08:13.020 align:middle line:84% RAFAEL JARAMILLO: Their coefficients, thank you. 00:08:13.020 --> 00:08:14.853 align:middle line:84% I didn't notice who that was, but thank you. 00:08:14.853 --> 00:08:20.850 align:middle line:90% 00:08:20.850 --> 00:08:23.070 align:middle line:90% Right, coefficients. 00:08:23.070 --> 00:08:24.060 align:middle line:90% Good. 00:08:24.060 --> 00:08:26.820 align:middle line:84% So this is a form that we are familiar with. 00:08:26.820 --> 00:08:29.730 align:middle line:90% 00:08:29.730 --> 00:08:33.450 align:middle line:84% And we know from when we've solved equilibrium conditions 00:08:33.450 --> 00:08:44.560 align:middle line:84% before in this class, that d g p t equals 0 requires 00:08:44.560 --> 00:08:48.510 align:middle line:90% coefficients to be 0. 00:08:48.510 --> 00:08:50.530 align:middle line:84% That's our equilibrium condition. 00:08:50.530 --> 00:08:53.070 align:middle line:84% And in this case, what that means-- 00:08:53.070 --> 00:08:58.170 align:middle line:84% the chemical potential of A is the same in both phases. 00:08:58.170 --> 00:09:08.160 align:middle line:84% And the chemical potential of B is the same in both phases. 00:09:08.160 --> 00:09:24.020 align:middle line:84% When that happens, phases can exchange mass freely 00:09:24.020 --> 00:09:29.070 align:middle line:90% without changing-- 00:09:29.070 --> 00:09:33.560 align:middle line:90% 00:09:33.560 --> 00:09:37.550 align:middle line:90% overall Gibbs free energy. 00:09:37.550 --> 00:09:41.240 align:middle line:84% Or in other words, Gibbs free energy is stationary. 00:09:41.240 --> 00:09:48.080 align:middle line:90% 00:09:48.080 --> 00:09:49.850 align:middle line:90% So the math is the same. 00:09:49.850 --> 00:09:51.350 align:middle line:84% There's a lot of pattern recognition 00:09:51.350 --> 00:09:53.570 align:middle line:90% that kicks in, in this class. 00:09:53.570 --> 00:09:58.560 align:middle line:84% The math is the same formally, as we've seen before. 00:09:58.560 --> 00:10:01.120 align:middle line:84% We're just exploring different cases of this. 00:10:01.120 --> 00:10:04.610 align:middle line:84% And so right, chemical potential is the same. 00:10:04.610 --> 00:10:08.068 align:middle line:84% So these particles, they can move back and forth. 00:10:08.068 --> 00:10:10.360 align:middle line:84% They can change their phase, but the chemical potential 00:10:10.360 --> 00:10:11.990 align:middle line:90% remain unchanged. 00:10:11.990 --> 00:10:16.910 align:middle line:84% So the overall Gibbs free energy will remain unchanged. 00:10:16.910 --> 00:10:20.870 align:middle line:90% Now, how do we analyze that? 00:10:20.870 --> 00:10:36.110 align:middle line:84% Right, in general, in general, each phase 00:10:36.110 --> 00:10:49.120 align:middle line:84% has its own solution model, and partial molar properties 00:10:49.120 --> 00:10:49.630 align:middle line:90% of mixing. 00:10:49.630 --> 00:11:01.440 align:middle line:90% 00:11:01.440 --> 00:11:02.970 align:middle line:90% So what do I mean by that? 00:11:02.970 --> 00:11:11.670 align:middle line:84% Mu a in phase one equals mu a in its reference state, 00:11:11.670 --> 00:11:19.310 align:middle line:84% plus delta mu a of mixing in phase one. 00:11:19.310 --> 00:11:26.480 align:middle line:84% Mu a in phase two equals mu a in its reference state, 00:11:26.480 --> 00:11:34.830 align:middle line:84% plus delta mu a in phase two mixing. 00:11:34.830 --> 00:11:38.000 align:middle line:84% And I can write the same for the other component-- 00:11:38.000 --> 00:11:44.940 align:middle line:84% mu B in phase one, equals mu B naught 00:11:44.940 --> 00:11:48.210 align:middle line:84% plus the change in chemical potential of B 00:11:48.210 --> 00:11:53.260 align:middle line:90% due to mixing in phase one. 00:11:53.260 --> 00:11:56.740 align:middle line:84% Chemical potential of B in phase two 00:11:56.740 --> 00:12:01.840 align:middle line:84% equals the reference state plus the change 00:12:01.840 --> 00:12:07.650 align:middle line:84% in chemical potential of B due to mixing, phase two. 00:12:07.650 --> 00:12:10.160 align:middle line:90% 00:12:10.160 --> 00:12:12.215 align:middle line:90% Why did I write it in this way? 00:12:12.215 --> 00:12:15.180 align:middle line:90% 00:12:15.180 --> 00:12:19.900 align:middle line:84% I wrote it in this way because we see 00:12:19.900 --> 00:12:21.460 align:middle line:90% that these pure components-- 00:12:21.460 --> 00:12:28.200 align:middle line:90% 00:12:28.200 --> 00:12:44.920 align:middle line:84% pure component A, and pure component B-- 00:12:44.920 --> 00:12:49.290 align:middle line:84% those terms are the same in this equation as this equation. 00:12:49.290 --> 00:12:51.610 align:middle line:84% Or they're the same in this equation as this equation. 00:12:51.610 --> 00:12:54.900 align:middle line:84% It's the chemical potential of A in its pure state. 00:12:54.900 --> 00:12:58.095 align:middle line:90% 00:12:58.095 --> 00:13:00.220 align:middle line:84% It's the chemical potential of A in its pure state. 00:13:00.220 --> 00:13:02.680 align:middle line:84% It doesn't care what kind of mixture 00:13:02.680 --> 00:13:04.730 align:middle line:90% you're going to make out of it. 00:13:04.730 --> 00:13:06.730 align:middle line:84% This is like ingredients on the shelf-- 00:13:06.730 --> 00:13:09.610 align:middle line:84% if you've got sugar on the shelf, that's pure sugar, 00:13:09.610 --> 00:13:11.110 align:middle line:84% it's thermodynamic properties don't 00:13:11.110 --> 00:13:12.985 align:middle line:84% care whether you're going to go make cupcakes 00:13:12.985 --> 00:13:14.920 align:middle line:84% or you're going to make syrup out of it. 00:13:14.920 --> 00:13:16.210 align:middle line:90% It's just pure sugar. 00:13:16.210 --> 00:13:18.400 align:middle line:84% So its reference state is unchanged, 00:13:18.400 --> 00:13:20.950 align:middle line:84% doesn't depend on what you're about to mix it into. 00:13:20.950 --> 00:13:24.130 align:middle line:90% 00:13:24.130 --> 00:13:29.350 align:middle line:84% So in words, the reference states-- 00:13:29.350 --> 00:13:40.200 align:middle line:90% 00:13:40.200 --> 00:13:47.870 align:middle line:84% the reference states are fixed, and the mu k naught 00:13:47.870 --> 00:13:54.205 align:middle line:90% don't depend on mixing. 00:13:54.205 --> 00:13:56.855 align:middle line:90% 00:13:56.855 --> 00:13:58.230 align:middle line:84% The properties of the ingredients 00:13:58.230 --> 00:13:59.397 align:middle line:90% don't depend on your recipe. 00:13:59.397 --> 00:14:01.870 align:middle line:90% 00:14:01.870 --> 00:14:13.680 align:middle line:84% So if we have mu a 1 equals mu a 2 as our equilibrium condition, 00:14:13.680 --> 00:14:24.510 align:middle line:84% this becomes delta mu a of mixing 1 equals delta mu a 00:14:24.510 --> 00:14:27.900 align:middle line:90% of mixing 2. 00:14:27.900 --> 00:14:42.520 align:middle line:84% And similarly, mu B 1 equals mu B 2 delta mu B of mixing, 00:14:42.520 --> 00:14:51.080 align:middle line:90% 1 equals delta mu B of mixing 2. 00:14:51.080 --> 00:15:04.960 align:middle line:84% And these are our two-phase co-existence equilibrium 00:15:04.960 --> 00:15:05.860 align:middle line:90% conditions. 00:15:05.860 --> 00:15:10.470 align:middle line:90% 00:15:10.470 --> 00:15:12.540 align:middle line:84% Those are our equilibrium conditions. 00:15:12.540 --> 00:15:14.040 align:middle line:84% So now, we're going to use something 00:15:14.040 --> 00:15:16.830 align:middle line:84% which we introduced maybe a week or more ago, 00:15:16.830 --> 00:15:19.690 align:middle line:90% which is the graphical-- 00:15:19.690 --> 00:15:30.360 align:middle line:90% 00:15:30.360 --> 00:15:34.230 align:middle line:84% the graphical solution for parsing all the properties. 00:15:34.230 --> 00:15:37.530 align:middle line:84% And to remind you what that was, we had-- 00:15:37.530 --> 00:15:43.630 align:middle line:90% 00:15:43.630 --> 00:15:45.370 align:middle line:84% now, specific to the case of Gibbs, 00:15:45.370 --> 00:15:47.530 align:middle line:84% because that's pretty much the only way 00:15:47.530 --> 00:15:49.810 align:middle line:90% we're going to be using this-- 00:15:49.810 --> 00:15:58.420 align:middle line:84% we had here Gibbs free energy, and we had this concept 00:15:58.420 --> 00:16:01.550 align:middle line:90% of a solution model. 00:16:01.550 --> 00:16:03.790 align:middle line:84% So let me just draw a section of a solution model. 00:16:03.790 --> 00:16:07.760 align:middle line:90% 00:16:07.760 --> 00:16:10.400 align:middle line:84% Draw the whole thing, because I don't need the whole thing. 00:16:10.400 --> 00:16:15.560 align:middle line:84% And let's imagine some system composition here, 00:16:15.560 --> 00:16:20.660 align:middle line:90% that would be x 2-- 00:16:20.660 --> 00:16:23.420 align:middle line:90% that's x 2. 00:16:23.420 --> 00:16:25.530 align:middle line:84% The way that we found the partial molar Gibbs 00:16:25.530 --> 00:16:34.510 align:middle line:84% free energy of mixing was we drew the tangent 00:16:34.510 --> 00:16:41.100 align:middle line:84% to the solution model at that composition. 00:16:41.100 --> 00:16:44.440 align:middle line:84% And the intercepts were the partial molar properties 00:16:44.440 --> 00:16:44.940 align:middle line:90% of mixing. 00:16:44.940 --> 00:16:57.430 align:middle line:84% So this intercept here is mu 2 at x 2. 00:16:57.430 --> 00:17:05.440 align:middle line:84% And this intercept here is mu 1 at x 1. 00:17:05.440 --> 00:17:11.079 align:middle line:84% So I hope you do remember this-- if you don't, just rewind 00:17:11.079 --> 00:17:12.530 align:middle line:90% for a couple of lectures. 00:17:12.530 --> 00:17:14.420 align:middle line:84% So we had this graphical solution. 00:17:14.420 --> 00:17:16.960 align:middle line:84% Now, we introduce this for one solution model. 00:17:16.960 --> 00:17:20.040 align:middle line:90% 00:17:20.040 --> 00:17:24.200 align:middle line:84% So what we're going to do now is use this tangent construction 00:17:24.200 --> 00:17:28.339 align:middle line:84% for the case of two different phases. 00:17:28.339 --> 00:17:38.650 align:middle line:84% This becomes a common tangent construction. 00:17:38.650 --> 00:17:41.350 align:middle line:84% It's called-- or if you like, you just call it a condition-- 00:17:41.350 --> 00:17:44.850 align:middle line:90% 00:17:44.850 --> 00:17:52.260 align:middle line:90% for two-phase equilibrium. 00:17:52.260 --> 00:17:53.570 align:middle line:90% So how does that look? 00:17:53.570 --> 00:18:09.710 align:middle line:90% 00:18:09.710 --> 00:18:11.050 align:middle line:90% Well, let's drive one phase. 00:18:11.050 --> 00:18:16.640 align:middle line:90% 00:18:16.640 --> 00:18:19.340 align:middle line:84% There's part of a solution model for one phase. 00:18:19.340 --> 00:18:21.285 align:middle line:90% Let's draw another phase. 00:18:21.285 --> 00:18:23.410 align:middle line:84% There's part of a solution model for another phase. 00:18:23.410 --> 00:18:27.320 align:middle line:90% 00:18:27.320 --> 00:18:29.345 align:middle line:84% And now, I'm going to look for a common tangent. 00:18:29.345 --> 00:18:34.310 align:middle line:90% 00:18:34.310 --> 00:18:36.110 align:middle line:84% Let's say that I drew that accurately, 00:18:36.110 --> 00:18:39.710 align:middle line:84% and we have a tangent, which is common to both curves. 00:18:39.710 --> 00:18:45.020 align:middle line:90% 00:18:45.020 --> 00:18:49.525 align:middle line:84% This is phase one, this is phase two. 00:18:49.525 --> 00:18:54.490 align:middle line:90% 00:18:54.490 --> 00:18:58.210 align:middle line:84% But say, now using our notation from the previous slides, 00:18:58.210 --> 00:19:05.270 align:middle line:84% let's say this is composition of component B and phase one. 00:19:05.270 --> 00:19:14.510 align:middle line:84% This is composition of component B in phase two. 00:19:14.510 --> 00:19:21.730 align:middle line:84% These intercepts are the chemical potentials 00:19:21.730 --> 00:19:42.480 align:middle line:84% B and A. So what we've shown is a common tangent, which 00:19:42.480 --> 00:19:58.490 align:middle line:84% is satisfied at compositions xb 1 and xb 2. 00:19:58.490 --> 00:20:00.710 align:middle line:84% Right, it's only satisfied with these compositions. 00:20:00.710 --> 00:20:03.350 align:middle line:84% If you look at these curves, the way I've drawn them, 00:20:03.350 --> 00:20:05.990 align:middle line:90% there's no other common tangent. 00:20:05.990 --> 00:20:11.070 align:middle line:84% It's only those two particular compositions. 00:20:11.070 --> 00:20:13.350 align:middle line:84% And for those two particular compositions, 00:20:13.350 --> 00:20:19.860 align:middle line:84% common tangent satisfied composition, 00:20:19.860 --> 00:20:27.710 align:middle line:84% that ensures that the chemical potential mu a 1 equals 00:20:27.710 --> 00:20:36.590 align:middle line:90% mu a 2 mu B 1 equals mu B 2. 00:20:36.590 --> 00:20:39.590 align:middle line:84% This is the equilibrium condition. 00:20:39.590 --> 00:20:46.200 align:middle line:90% 00:20:46.200 --> 00:20:50.280 align:middle line:84% So a common tangent on a free energy composition diagram 00:20:50.280 --> 00:20:53.340 align:middle line:84% is the same as the equilibrium condition. 00:20:53.340 --> 00:20:56.500 align:middle line:90% 00:20:56.500 --> 00:20:58.390 align:middle line:84% When you can draw the common tangent, 00:20:58.390 --> 00:21:01.330 align:middle line:84% then you can have two-phase equilibrium. 00:21:01.330 --> 00:21:03.460 align:middle line:84% And only at those compositions that 00:21:03.460 --> 00:21:05.900 align:middle line:90% satisfy the common tangent. 00:21:05.900 --> 00:21:08.750 align:middle line:84% If there's no common tangent you can draw, 00:21:08.750 --> 00:21:10.760 align:middle line:84% then there's no two-phase equilibrium possible. 00:21:10.760 --> 00:21:16.930 align:middle line:90% 00:21:16.930 --> 00:21:19.272 align:middle line:84% So we're knitting things together here 00:21:19.272 --> 00:21:20.355 align:middle line:90% across different lectures. 00:21:20.355 --> 00:21:28.660 align:middle line:90% 00:21:28.660 --> 00:21:31.540 align:middle line:84% I'm going to spend the rest of the class 00:21:31.540 --> 00:21:35.090 align:middle line:84% on how this works in the case of spinodal systems. 00:21:35.090 --> 00:21:38.050 align:middle line:84% So before I make it a little bit more specific 00:21:38.050 --> 00:21:42.810 align:middle line:84% to spinodal systems, are there any questions on the concept, 00:21:42.810 --> 00:21:45.210 align:middle line:90% or on the math of this, so far? 00:21:45.210 --> 00:21:46.670 align:middle line:90% OK. 00:21:46.670 --> 00:21:48.170 align:middle line:90% So let's move on, then. 00:21:48.170 --> 00:21:50.530 align:middle line:90% 00:21:50.530 --> 00:21:53.030 align:middle line:84% OK, so now, we're going to make this a little more specific. 00:21:53.030 --> 00:22:00.570 align:middle line:84% We're going to talk about an example of common tangent 00:22:00.570 --> 00:22:01.380 align:middle line:90% construction. 00:22:01.380 --> 00:22:11.750 align:middle line:90% 00:22:11.750 --> 00:22:15.280 align:middle line:90% And that is spinodal systems. 00:22:15.280 --> 00:22:17.110 align:middle line:84% So something a little bit familiar to us. 00:22:17.110 --> 00:22:22.660 align:middle line:90% 00:22:22.660 --> 00:22:30.530 align:middle line:84% So here's the general view of a free energy composition 00:22:30.530 --> 00:22:33.365 align:middle line:90% diagram for a spinodal system. 00:22:33.365 --> 00:22:40.870 align:middle line:90% 00:22:40.870 --> 00:22:47.810 align:middle line:84% This is going to be delta G of mixing, 00:22:47.810 --> 00:22:54.380 align:middle line:84% and let's draw a solution model for a phase that exhibits 00:22:54.380 --> 00:22:55.890 align:middle line:90% spinodal decomposition. 00:22:55.890 --> 00:22:59.685 align:middle line:90% So let's draw something like-- 00:22:59.685 --> 00:23:02.280 align:middle line:90% that works. 00:23:02.280 --> 00:23:06.578 align:middle line:84% You've been playing with stuff, something like this in PSETs 00:23:06.578 --> 00:23:07.620 align:middle line:90% and in previous lectures. 00:23:07.620 --> 00:23:10.890 align:middle line:84% And so this has a common tangent. 00:23:10.890 --> 00:23:11.610 align:middle line:90% Let me draw it. 00:23:11.610 --> 00:23:18.120 align:middle line:90% 00:23:18.120 --> 00:23:21.910 align:middle line:84% OK, so it has a common tangent, let's say, here, and let's say, 00:23:21.910 --> 00:23:22.410 align:middle line:90% here. 00:23:22.410 --> 00:23:25.670 align:middle line:90% 00:23:25.670 --> 00:23:27.620 align:middle line:90% Just eyeballing this, obviously. 00:23:27.620 --> 00:23:33.290 align:middle line:84% And so let's note the composition of the system when 00:23:33.290 --> 00:23:35.960 align:middle line:90% it satisfies common tangent. 00:23:35.960 --> 00:23:47.020 align:middle line:84% Let's say, we have x b, we'll call that phase one, x b, 00:23:47.020 --> 00:23:51.300 align:middle line:90% we'll call that phase two. 00:23:51.300 --> 00:23:55.200 align:middle line:84% The 0 of the free energy mixing is there. 00:23:55.200 --> 00:24:04.860 align:middle line:84% And these intercepts are delta mu b of mixing, 00:24:04.860 --> 00:24:14.340 align:middle line:90% and delta mu a of mixing. 00:24:14.340 --> 00:24:16.050 align:middle line:84% And so what's happened in this system? 00:24:16.050 --> 00:24:18.040 align:middle line:90% What happens at equilibrium? 00:24:18.040 --> 00:24:19.650 align:middle line:90% The system has phase-separated. 00:24:19.650 --> 00:24:27.250 align:middle line:84% It has spontaneously unmixed into two different phases, 00:24:27.250 --> 00:24:28.660 align:middle line:90% with what? 00:24:28.660 --> 00:24:29.980 align:middle line:90% These two compositions. 00:24:29.980 --> 00:24:46.070 align:middle line:84% So you could say, this is a composition rich in component 00:24:46.070 --> 00:24:47.720 align:middle line:90% a. 00:24:47.720 --> 00:24:51.620 align:middle line:90% So that composition were a-rich. 00:24:51.620 --> 00:25:00.490 align:middle line:84% And here's a composition rich in component. 00:25:00.490 --> 00:25:05.880 align:middle line:90% 00:25:05.880 --> 00:25:08.250 align:middle line:84% And we saw some examples of this on the slides 00:25:08.250 --> 00:25:12.640 align:middle line:90% from Monday's lecture. 00:25:12.640 --> 00:25:14.760 align:middle line:84% So there's the common tangent construction-- 00:25:14.760 --> 00:25:15.600 align:middle line:90% [INAUDIBLE] system. 00:25:15.600 --> 00:25:19.740 align:middle line:90% 00:25:19.740 --> 00:25:35.920 align:middle line:84% When the common tangent is possible, 00:25:35.920 --> 00:25:58.870 align:middle line:84% the free energy of two-phase system is lower then that 00:25:58.870 --> 00:26:07.890 align:middle line:84% of fully mixed one phase system-- 00:26:07.890 --> 00:26:09.480 align:middle line:84% as you would expect for equilibrium. 00:26:09.480 --> 00:26:12.585 align:middle line:90% It's got the lowest free energy. 00:26:12.585 --> 00:26:14.460 align:middle line:84% I'm not going to prove this to you right now, 00:26:14.460 --> 00:26:17.160 align:middle line:84% but I'm going to show you graphically how it comes about. 00:26:17.160 --> 00:26:23.580 align:middle line:90% 00:26:23.580 --> 00:26:33.000 align:middle line:84% Here, again, is a free energy composition diagram. 00:26:33.000 --> 00:26:35.780 align:middle line:84% And let me say, my solution model looks kind of like this-- 00:26:35.780 --> 00:26:36.830 align:middle line:90% [INAUDIBLE] here. 00:26:36.830 --> 00:26:40.830 align:middle line:90% 00:26:40.830 --> 00:26:42.730 align:middle line:90% I'd say it looks like that. 00:26:42.730 --> 00:26:50.100 align:middle line:84% And here is my common tangent, points of tangency. 00:26:50.100 --> 00:26:52.770 align:middle line:90% 00:26:52.770 --> 00:26:58.050 align:middle line:84% And my composition is here at those points 00:26:58.050 --> 00:26:59.130 align:middle line:90% of common tangency. 00:26:59.130 --> 00:27:07.120 align:middle line:84% And let's imagine an overall system composition here. 00:27:07.120 --> 00:27:10.360 align:middle line:84% Let's imagine an overall system composition there. 00:27:10.360 --> 00:27:17.880 align:middle line:84% It can be shown that relative to a fully mixed system, 00:27:17.880 --> 00:27:25.360 align:middle line:84% the free energy can be decreased by exactly this amount, 00:27:25.360 --> 00:27:30.010 align:middle line:84% by splitting into two different phases. 00:27:30.010 --> 00:27:33.420 align:middle line:84% So spontaneous un-mixing lowers the free energy 00:27:33.420 --> 00:27:36.510 align:middle line:84% by exactly the amount that I drew here on the plot. 00:27:36.510 --> 00:27:41.230 align:middle line:90% 00:27:41.230 --> 00:27:44.310 align:middle line:84% So the single phase, the single phase, 00:27:44.310 --> 00:27:46.860 align:middle line:84% the Gibbs free energy of the single phase system 00:27:46.860 --> 00:27:49.650 align:middle line:84% equals the Gibbs free energy of the components 00:27:49.650 --> 00:27:51.390 align:middle line:90% in their pure state. 00:27:51.390 --> 00:27:56.430 align:middle line:84% Plus just the value of this curve evaluated 00:27:56.430 --> 00:28:08.070 align:middle line:84% at the overall composition, plus delta G mixed evaluated, 00:28:08.070 --> 00:28:10.830 align:middle line:90% at overall x b. 00:28:10.830 --> 00:28:18.800 align:middle line:84% And two phase, two phase, the Gibbs free energy 00:28:18.800 --> 00:28:23.180 align:middle line:84% is, again, the Gibbs free energy of the components unmixed, 00:28:23.180 --> 00:28:30.905 align:middle line:84% plus the Gibbs free energy of each of those two phases-- 00:28:30.905 --> 00:28:34.490 align:middle line:90% 00:28:34.490 --> 00:28:34.990 align:middle line:90% sorry. 00:28:34.990 --> 00:28:56.550 align:middle line:90% 00:28:56.550 --> 00:29:01.320 align:middle line:84% Right, so single phase, it's just the reference state 00:29:01.320 --> 00:29:03.900 align:middle line:84% plus the evaluation of this solution 00:29:03.900 --> 00:29:05.580 align:middle line:90% model at that composition. 00:29:05.580 --> 00:29:07.920 align:middle line:84% Two phases, it's the reference state, 00:29:07.920 --> 00:29:10.440 align:middle line:84% plus the evaluation of the solution model at these two 00:29:10.440 --> 00:29:16.620 align:middle line:84% compositions, weighted by the phase fractions. 00:29:16.620 --> 00:29:25.700 align:middle line:84% So this here's a solution model, solution model. 00:29:25.700 --> 00:29:31.490 align:middle line:90% This is solution model. 00:29:31.490 --> 00:29:37.750 align:middle line:90% And these are phase fractions. 00:29:37.750 --> 00:29:42.610 align:middle line:90% 00:29:42.610 --> 00:29:49.230 align:middle line:84% OK, so you're working some of these concepts on the problem 00:29:49.230 --> 00:29:50.910 align:middle line:90% set. 00:29:50.910 --> 00:29:53.100 align:middle line:84% All right, so now we're going to continue 00:29:53.100 --> 00:29:55.330 align:middle line:90% exploring the spinodal case. 00:29:55.330 --> 00:30:11.920 align:middle line:84% The common tangents-- they define the tie lines 00:30:11.920 --> 00:30:13.150 align:middle line:90% on a binary phase diagram. 00:30:13.150 --> 00:30:23.190 align:middle line:90% 00:30:23.190 --> 00:30:32.140 align:middle line:84% So again, for the example of spinodal system, 00:30:32.140 --> 00:30:33.425 align:middle line:90% example of a spinodal system-- 00:30:33.425 --> 00:30:43.890 align:middle line:90% 00:30:43.890 --> 00:30:45.615 align:middle line:90% here, I'll draw a phase diagram. 00:30:45.615 --> 00:30:53.590 align:middle line:84% So we have temperature, composition, 00:30:53.590 --> 00:30:57.000 align:middle line:84% and we have a region of spontaneous-- or I should say, 00:30:57.000 --> 00:31:00.010 align:middle line:90% region of phase separation. 00:31:00.010 --> 00:31:02.640 align:middle line:84% And let's see, we have the highest temperature here, which 00:31:02.640 --> 00:31:06.930 align:middle line:90% is sometimes marked as star-- 00:31:06.930 --> 00:31:08.260 align:middle line:90% different notations there. 00:31:08.260 --> 00:31:09.135 align:middle line:90% I'll just say T star. 00:31:09.135 --> 00:31:14.590 align:middle line:90% 00:31:14.590 --> 00:31:16.180 align:middle line:90% And we have these tie lines-- 00:31:16.180 --> 00:31:18.280 align:middle line:84% these tie lines define the compositions 00:31:18.280 --> 00:31:22.020 align:middle line:84% that can coexist at equilibrium with each other. 00:31:22.020 --> 00:31:25.690 align:middle line:90% 00:31:25.690 --> 00:31:28.460 align:middle line:84% So in a system like this, mixing-- 00:31:28.460 --> 00:31:32.790 align:middle line:90% 00:31:32.790 --> 00:31:35.445 align:middle line:90% mixing favored at high temp. 00:31:35.445 --> 00:31:39.930 align:middle line:90% 00:31:39.930 --> 00:31:48.970 align:middle line:84% All right, in fact, it's fully immiscible at high temp. 00:31:48.970 --> 00:31:51.580 align:middle line:90% 00:31:51.580 --> 00:31:59.900 align:middle line:84% At low temp, the system spontaneously unmixes. 00:31:59.900 --> 00:32:07.580 align:middle line:90% 00:32:07.580 --> 00:32:13.520 align:middle line:84% Spontaneously unmixes, and the tie lines 00:32:13.520 --> 00:32:24.390 align:middle line:84% that are drawn in the two-phase region, 00:32:24.390 --> 00:32:37.591 align:middle line:84% connect compositions that co-exist at equilibrium. 00:32:37.591 --> 00:32:45.460 align:middle line:90% 00:32:45.460 --> 00:32:47.190 align:middle line:84% And so this is a reminder, basically, 00:32:47.190 --> 00:32:49.890 align:middle line:90% of what tie lines are. 00:32:49.890 --> 00:32:54.780 align:middle line:84% How do the common tangents define the tie lines? 00:32:54.780 --> 00:32:57.510 align:middle line:84% They do it through the free energy composition diagrams. 00:32:57.510 --> 00:33:22.830 align:middle line:84% So phase diagram emerges from temp dependence of free energy 00:33:22.830 --> 00:33:23.895 align:middle line:90% composition diagrams. 00:33:23.895 --> 00:33:31.570 align:middle line:90% 00:33:31.570 --> 00:33:34.280 align:middle line:84% So I'm going to draw some snapshots here 00:33:34.280 --> 00:33:35.940 align:middle line:84% of how that happens, and then we're 00:33:35.940 --> 00:33:38.273 align:middle line:84% going to spend a lot of time in the next couple of weeks 00:33:38.273 --> 00:33:39.570 align:middle line:90% on this. 00:33:39.570 --> 00:33:47.250 align:middle line:84% So let's say, low temp, T1, let me draw a free energy 00:33:47.250 --> 00:33:51.950 align:middle line:84% composition diagram for this spinodal system 00:33:51.950 --> 00:33:53.540 align:middle line:90% at low temperature. 00:33:53.540 --> 00:34:01.250 align:middle line:84% At low temperature, T1, delta G of mixing 00:34:01.250 --> 00:34:02.920 align:middle line:90% is basically positive-- 00:34:02.920 --> 00:34:04.130 align:middle line:90% that's why we have-- 00:34:04.130 --> 00:34:08.360 align:middle line:90% 00:34:08.360 --> 00:34:09.514 align:middle line:90% let me draw it like this. 00:34:09.514 --> 00:34:12.690 align:middle line:90% 00:34:12.690 --> 00:34:13.550 align:middle line:90% This is an example. 00:34:13.550 --> 00:34:15.665 align:middle line:90% It could look like this. 00:34:15.665 --> 00:34:20.500 align:middle line:84% This system has a common tangent, right? 00:34:20.500 --> 00:34:26.650 align:middle line:90% 00:34:26.650 --> 00:34:29.350 align:middle line:84% So now, right below this, we're going 00:34:29.350 --> 00:34:30.860 align:middle line:90% to draw the phase diagram. 00:34:30.860 --> 00:34:41.260 align:middle line:90% 00:34:41.260 --> 00:34:43.520 align:middle line:90% There's going to be temperature. 00:34:43.520 --> 00:34:47.739 align:middle line:84% I'm going to drop down those compositions. 00:34:47.739 --> 00:34:52.080 align:middle line:90% 00:34:52.080 --> 00:34:56.780 align:middle line:90% This is x b. 00:34:56.780 --> 00:35:01.380 align:middle line:84% What I've just learned from this complement tangent, 00:35:01.380 --> 00:35:09.530 align:middle line:84% is that these two compositions can coexist at equilibrium 00:35:09.530 --> 00:35:14.430 align:middle line:90% at temperature T1. 00:35:14.430 --> 00:35:17.207 align:middle line:84% So those are two points on the phase diagram. 00:35:17.207 --> 00:35:19.540 align:middle line:84% And I know what's going to happen if I vary temperature. 00:35:19.540 --> 00:35:22.950 align:middle line:84% So I'm just going to draw those as two points on little line 00:35:22.950 --> 00:35:29.000 align:middle line:84% segments, and I'm going to connect them with a tie line. 00:35:29.000 --> 00:35:32.530 align:middle line:90% What about medium temp? 00:35:32.530 --> 00:35:39.660 align:middle line:90% 00:35:39.660 --> 00:35:41.410 align:middle line:90% What about medium temp? 00:35:41.410 --> 00:35:45.130 align:middle line:84% So as I raise the temperature, my free energy composition 00:35:45.130 --> 00:35:47.117 align:middle line:90% diagram will change. 00:35:47.117 --> 00:35:49.450 align:middle line:84% As I raise the temperature, mixing becomes more favored. 00:35:49.450 --> 00:36:05.050 align:middle line:90% 00:36:05.050 --> 00:36:09.160 align:middle line:84% So now, my 0 of delta G of mixing will be there. 00:36:09.160 --> 00:36:11.695 align:middle line:84% And I'll draw it maybe like this. 00:36:11.695 --> 00:36:20.900 align:middle line:90% 00:36:20.900 --> 00:36:24.470 align:middle line:84% I've got a common tangent, right? 00:36:24.470 --> 00:36:26.900 align:middle line:90% There and there. 00:36:26.900 --> 00:36:29.830 align:middle line:90% 00:36:29.830 --> 00:36:32.065 align:middle line:84% Let's draw the corresponding phase diagram. 00:36:32.065 --> 00:36:40.670 align:middle line:90% 00:36:40.670 --> 00:36:42.514 align:middle line:90% And this is x b. 00:36:42.514 --> 00:36:46.470 align:middle line:90% 00:36:46.470 --> 00:36:53.280 align:middle line:90% At T1, I had coexistence at-- 00:36:53.280 --> 00:36:58.490 align:middle line:84% let's say, here, and here, being a little bit rough with that. 00:36:58.490 --> 00:37:01.860 align:middle line:90% So we had a tie line here. 00:37:01.860 --> 00:37:05.010 align:middle line:84% Now, at T2, I've got coexistence. 00:37:05.010 --> 00:37:11.950 align:middle line:90% 00:37:11.950 --> 00:37:13.990 align:middle line:90% OK, let's say this is T2. 00:37:13.990 --> 00:37:18.055 align:middle line:84% Let's say my coexistence is here and here. 00:37:18.055 --> 00:37:20.970 align:middle line:90% 00:37:20.970 --> 00:37:24.060 align:middle line:90% All right, I'm calling this T2. 00:37:24.060 --> 00:37:27.625 align:middle line:90% And I've got a tie line there. 00:37:27.625 --> 00:37:30.000 align:middle line:84% And now, I'm only drawing discrete points in temperature, 00:37:30.000 --> 00:37:33.520 align:middle line:84% but you can imagine this now starts to define 00:37:33.520 --> 00:37:37.860 align:middle line:90% a family of tie lines. 00:37:37.860 --> 00:37:40.920 align:middle line:84% So I've shown you how the tie lines at T1 and T2 00:37:40.920 --> 00:37:45.240 align:middle line:84% emerge, but you can imagine that as I sweep the temperature, 00:37:45.240 --> 00:37:46.890 align:middle line:90% I have a family of tie lines. 00:37:46.890 --> 00:37:51.570 align:middle line:84% And this starts to sweep out the two-phase region. 00:37:51.570 --> 00:37:55.207 align:middle line:90% 00:37:55.207 --> 00:37:56.165 align:middle line:90% And then, a high temp-- 00:37:56.165 --> 00:38:00.500 align:middle line:90% 00:38:00.500 --> 00:38:05.190 align:middle line:90% T3-- we see it again. 00:38:05.190 --> 00:38:16.612 align:middle line:90% 00:38:16.612 --> 00:38:19.070 align:middle line:84% At high temp, I know that my free energy mixing will pretty 00:38:19.070 --> 00:38:23.630 align:middle line:90% much be strictly negative. 00:38:23.630 --> 00:38:26.340 align:middle line:84% So I'm going to draw the 0 way up there. 00:38:26.340 --> 00:38:31.810 align:middle line:84% And I can say, my solution model might look like that. 00:38:31.810 --> 00:38:35.500 align:middle line:84% So no common tangents, no inflection points. 00:38:35.500 --> 00:38:39.370 align:middle line:84% Strictly positive curvature throughout. 00:38:39.370 --> 00:38:42.150 align:middle line:84% And so that section of my phase diagram 00:38:42.150 --> 00:38:45.725 align:middle line:90% will show full miscibility. 00:38:45.725 --> 00:38:50.850 align:middle line:90% 00:38:50.850 --> 00:38:54.840 align:middle line:90% Here's x and b. 00:38:54.840 --> 00:38:59.300 align:middle line:90% Here's T1, here is T2. 00:38:59.300 --> 00:39:03.810 align:middle line:84% And let's say this up here is T3. 00:39:03.810 --> 00:39:05.343 align:middle line:84% I'm just going to sketch this now. 00:39:05.343 --> 00:39:06.260 align:middle line:90% We'll see if we have-- 00:39:06.260 --> 00:39:11.260 align:middle line:90% 00:39:11.260 --> 00:39:12.080 align:middle line:90% all right. 00:39:12.080 --> 00:39:16.855 align:middle line:84% So now, my region of two-phase region has closed. 00:39:16.855 --> 00:39:23.500 align:middle line:90% 00:39:23.500 --> 00:39:28.130 align:middle line:84% And I have finished drawing my spinodal phase diagram. 00:39:28.130 --> 00:39:33.520 align:middle line:84% So the way I did this here is I did a very, very slow motion 00:39:33.520 --> 00:39:35.650 align:middle line:90% animation for you. 00:39:35.650 --> 00:39:37.990 align:middle line:84% At each temperature, you have a free energy composition 00:39:37.990 --> 00:39:40.190 align:middle line:90% diagram. 00:39:40.190 --> 00:39:42.980 align:middle line:84% At each temperature, that free energy composition diagram 00:39:42.980 --> 00:39:44.720 align:middle line:84% determines for you whether or not 00:39:44.720 --> 00:39:46.610 align:middle line:84% there are any tie lines in the phase diagram. 00:39:46.610 --> 00:39:50.540 align:middle line:84% It tells you about the phase stability and equilibrium 00:39:50.540 --> 00:39:54.320 align:middle line:84% as a function of composition at that temperature. 00:39:54.320 --> 00:39:55.100 align:middle line:90% 00:39:55.100 --> 00:39:57.950 align:middle line:84% And so as you tune the temperatures and knob, 00:39:57.950 --> 00:40:00.500 align:middle line:84% you sweep through a series of free energy composition 00:40:00.500 --> 00:40:02.870 align:middle line:90% diagrams. 00:40:02.870 --> 00:40:04.820 align:middle line:84% And you gradually build up the phase diagram. 00:40:04.820 --> 00:40:08.650 align:middle line:90% 00:40:08.650 --> 00:40:12.210 align:middle line:84% So you can imagine a flipbook style version of this. 00:40:12.210 --> 00:40:14.130 align:middle line:84% The textbook has some nice examples 00:40:14.130 --> 00:40:17.040 align:middle line:84% of this-- of these free energy composition diagrams giving 00:40:17.040 --> 00:40:18.105 align:middle line:90% rise to phase diagrams. 00:40:18.105 --> 00:40:20.620 align:middle line:90% 00:40:20.620 --> 00:40:22.810 align:middle line:84% These single temperature snapshots. 00:40:22.810 --> 00:40:24.340 align:middle line:84% And in the weeks ahead, we're also 00:40:24.340 --> 00:40:27.400 align:middle line:84% going to play with an online kind of animated 00:40:27.400 --> 00:40:31.980 align:middle line:84% version of this, which we've coded up for use in this class. 00:40:31.980 --> 00:40:35.052 align:middle line:84% OK, that's where I want to leave it for today. 00:40:35.052 --> 00:40:36.760 align:middle line:84% We're finishing about five minutes early, 00:40:36.760 --> 00:40:39.170 align:middle line:84% so we have plenty of time for questions. 00:40:39.170 --> 00:40:41.653 align:middle line:90% I just saw there was a chat-- 00:40:41.653 --> 00:40:43.070 align:middle line:84% does a common tangent construction 00:40:43.070 --> 00:40:44.510 align:middle line:84% help us to say anything about the kinetics 00:40:44.510 --> 00:40:45.830 align:middle line:90% of the process of segregation? 00:40:45.830 --> 00:40:46.640 align:middle line:90% No, it doesn't. 00:40:46.640 --> 00:40:48.990 align:middle line:84% This doesn't tell us anything about kinetics. 00:40:48.990 --> 00:40:50.913 align:middle line:84% So this tells us about the equilibrium, 00:40:50.913 --> 00:40:52.580 align:middle line:84% the common tangent construction tells us 00:40:52.580 --> 00:40:57.230 align:middle line:84% about the equilibrium configuration of the system. 00:40:57.230 --> 00:41:01.280 align:middle line:84% Now, we can infer a lot of things about kinetics 00:41:01.280 --> 00:41:04.860 align:middle line:84% from the phase diagrams, and we can also 00:41:04.860 --> 00:41:07.320 align:middle line:84% infer things about kinetics from the driving 00:41:07.320 --> 00:41:10.380 align:middle line:90% force for phase segregation. 00:41:10.380 --> 00:41:12.950 align:middle line:90% 00:41:12.950 --> 00:41:14.930 align:middle line:84% It goes beyond the scope of the class, 00:41:14.930 --> 00:41:17.240 align:middle line:84% but kinetics is really interesting. 00:41:17.240 --> 00:41:19.670 align:middle line:84% And it starts from this point-- kinetics studies start 00:41:19.670 --> 00:41:21.470 align:middle line:90% from phase diagrams like this. 00:41:21.470 --> 00:41:25.400 align:middle line:84% And a tendency of a system to spontaneously unmix-- 00:41:25.400 --> 00:41:31.770 align:middle line:84% that is, a tendency for pattern formation. 00:41:31.770 --> 00:41:34.403 align:middle line:84% And what patterns actually form depends on the kinetics-- 00:41:34.403 --> 00:41:35.820 align:middle line:84% depends on diffusion coefficients, 00:41:35.820 --> 00:41:38.610 align:middle line:90% depends on temperature. 00:41:38.610 --> 00:41:44.600 align:middle line:84% And that's where this curriculum takes off in the fall. 00:41:44.600 --> 00:41:47.000 align:middle line:90%