WEBVTT 00:00:00.000 --> 00:00:01.972 align:middle line:90% [SQUEAKING] 00:00:01.972 --> 00:00:03.944 align:middle line:90% [RUSTLING] 00:00:03.944 --> 00:00:07.395 align:middle line:90% [CLICKING] 00:00:07.395 --> 00:00:12.070 align:middle line:90% 00:00:12.070 --> 00:00:13.820 align:middle line:84% RAFAEL JARAMILLO: Let's get right into it. 00:00:13.820 --> 00:00:24.460 align:middle line:84% We are-- I'm going to overview here thermodynamics 00:00:24.460 --> 00:00:25.562 align:middle line:90% of binary phase diagrams. 00:00:25.562 --> 00:00:26.770 align:middle line:90% That's what we're working on. 00:00:26.770 --> 00:00:37.100 align:middle line:90% 00:00:37.100 --> 00:00:49.180 align:middle line:84% So we've done notation and bookkeeping, notation 00:00:49.180 --> 00:00:51.253 align:middle line:90% and bookkeeping. 00:00:51.253 --> 00:00:52.795 align:middle line:84% We've talked about solution modeling. 00:00:52.795 --> 00:00:59.870 align:middle line:90% 00:00:59.870 --> 00:01:04.610 align:middle line:84% And we've talked about spinodal systems in some detail-- 00:01:04.610 --> 00:01:10.390 align:middle line:90% 00:01:10.390 --> 00:01:12.300 align:middle line:90% spinodal systems. 00:01:12.300 --> 00:01:23.130 align:middle line:84% And this is spontaneous unmixing, so phase separation, 00:01:23.130 --> 00:01:31.310 align:middle line:84% when both phases are in the same-- 00:01:31.310 --> 00:01:37.870 align:middle line:90% 00:01:37.870 --> 00:01:40.290 align:middle line:84% so this is-- we haven't made too much of this yet. 00:01:40.290 --> 00:01:43.380 align:middle line:84% But it's been underlying our talk about the spinodal, which 00:01:43.380 --> 00:01:45.750 align:middle line:90% is that-- 00:01:45.750 --> 00:01:47.640 align:middle line:90% this red pen is dying. 00:01:47.640 --> 00:01:52.560 align:middle line:84% Life is too short to keep dying sharpies on your desk. 00:01:52.560 --> 00:01:56.760 align:middle line:84% So spinodal systems are spontaneously 00:01:56.760 --> 00:01:59.430 align:middle line:84% unmixed when both phases are in the same structure. 00:01:59.430 --> 00:02:00.930 align:middle line:84% We haven't made a lot of this yet, 00:02:00.930 --> 00:02:05.490 align:middle line:84% but it's been an important point that all the spinodal systems, 00:02:05.490 --> 00:02:06.790 align:middle line:90% it's the same structure. 00:02:06.790 --> 00:02:12.490 align:middle line:84% You have a chromium-rich bcc and an iron-rich bcc, 00:02:12.490 --> 00:02:15.450 align:middle line:84% or an ethanol-rich liquid and dodecane-rich liquid, 00:02:15.450 --> 00:02:16.990 align:middle line:90% or so forth and so on. 00:02:16.990 --> 00:02:19.420 align:middle line:90% But the structure is the same. 00:02:19.420 --> 00:02:22.230 align:middle line:84% And so this is our story thus far. 00:02:22.230 --> 00:02:29.900 align:middle line:90% 00:02:29.900 --> 00:02:33.320 align:middle line:84% And what we're going to do is we're 00:02:33.320 --> 00:02:39.950 align:middle line:84% going to now generalize this to analyze what happens when you 00:02:39.950 --> 00:02:43.940 align:middle line:84% have phase separation between two phases that 00:02:43.940 --> 00:02:46.840 align:middle line:90% have different structures. 00:02:46.840 --> 00:02:51.830 align:middle line:84% And so when you had two phases that had the same structure, 00:02:51.830 --> 00:02:57.745 align:middle line:84% you only need one solution model. 00:02:57.745 --> 00:03:03.570 align:middle line:90% 00:03:03.570 --> 00:03:18.960 align:middle line:84% You only need one solution model to analyze the free energy, 00:03:18.960 --> 00:03:21.930 align:middle line:84% to analyze the free energy composition plots. 00:03:21.930 --> 00:03:24.450 align:middle line:84% So today we're going to start introducing 00:03:24.450 --> 00:03:27.840 align:middle line:84% more complexity, which is having multiple solution models. 00:03:27.840 --> 00:03:32.930 align:middle line:90% 00:03:32.930 --> 00:03:43.540 align:middle line:84% And more generally, the pure components-- 00:03:43.540 --> 00:04:03.040 align:middle line:90% 00:04:03.040 --> 00:04:04.080 align:middle line:90% What do I mean? 00:04:04.080 --> 00:04:06.450 align:middle line:84% Pure components may be in different structures. 00:04:06.450 --> 00:04:08.550 align:middle line:84% Let me illustrate this with a simple example. 00:04:08.550 --> 00:04:16.130 align:middle line:90% 00:04:16.130 --> 00:04:18.750 align:middle line:90% Let's do silicon germanium. 00:04:18.750 --> 00:04:19.829 align:middle line:90% It's one of my favorites. 00:04:19.829 --> 00:04:29.070 align:middle line:90% 00:04:29.070 --> 00:04:31.890 align:middle line:84% Does anyone remember, who has a higher melting point, 00:04:31.890 --> 00:04:33.600 align:middle line:90% silicon or germanium? 00:04:33.600 --> 00:04:39.090 align:middle line:90% 00:04:39.090 --> 00:04:39.930 align:middle line:90% Give you a hint. 00:04:39.930 --> 00:04:41.790 align:middle line:90% STUDENT: Is it germanium? 00:04:41.790 --> 00:04:43.530 align:middle line:84% RAFAEL JARAMILLO: No, it's silicon. 00:04:43.530 --> 00:04:46.890 align:middle line:90% Lead is very soft, very soft. 00:04:46.890 --> 00:04:47.910 align:middle line:90% We know that lead melts. 00:04:47.910 --> 00:04:50.340 align:middle line:84% We saw it run off the roof of the Notre Dame 00:04:50.340 --> 00:04:54.180 align:middle line:90% a couple of years ago in Paris-- 00:04:54.180 --> 00:04:57.690 align:middle line:84% then tin, then germanium, then silicon. 00:04:57.690 --> 00:04:59.910 align:middle line:84% Carbon, it doesn't even melt. It's 00:04:59.910 --> 00:05:01.990 align:middle line:90% [INAUDIBLE] at 3,700 kelvin. 00:05:01.990 --> 00:05:05.160 align:middle line:84% So as you go up here, you have stronger and stronger covalent 00:05:05.160 --> 00:05:07.710 align:middle line:84% bonds, meaning higher and higher melting points. 00:05:07.710 --> 00:05:10.500 align:middle line:84% So silicon has a high melting point than germanium. 00:05:10.500 --> 00:05:12.270 align:middle line:90% Thank you, mousepad. 00:05:12.270 --> 00:05:15.210 align:middle line:84% All right, so here's what we're going to have is 00:05:15.210 --> 00:05:18.435 align:middle line:84% we're going to have a lens diagram, which looks like this. 00:05:18.435 --> 00:05:21.990 align:middle line:90% 00:05:21.990 --> 00:05:28.410 align:middle line:84% Here's the melting point of germanium. 00:05:28.410 --> 00:05:33.280 align:middle line:84% And here's the melting point of silicon. 00:05:33.280 --> 00:05:35.335 align:middle line:84% And, of course, this is a two-phase region. 00:05:35.335 --> 00:05:37.960 align:middle line:90% 00:05:37.960 --> 00:05:41.500 align:middle line:84% As we saw in the last lecture, this is a fully miscible system 00:05:41.500 --> 00:05:44.770 align:middle line:84% with a fully miscible liquid, and a fully miscible 00:05:44.770 --> 00:05:47.650 align:middle line:90% diamond-structured solid-- 00:05:47.650 --> 00:05:54.183 align:middle line:84% sometimes just use alpha for the ground state solid structure-- 00:05:54.183 --> 00:05:55.225 align:middle line:90% so fully miscible system. 00:05:55.225 --> 00:05:58.810 align:middle line:90% 00:05:58.810 --> 00:06:02.350 align:middle line:84% So if you're below the melting point of germanium, 00:06:02.350 --> 00:06:05.350 align:middle line:84% you only need one solution model to analyze this whole phase 00:06:05.350 --> 00:06:06.550 align:middle line:90% diagram. 00:06:06.550 --> 00:06:11.020 align:middle line:84% That is the solution model for diamond-structured silicon 00:06:11.020 --> 00:06:12.398 align:middle line:90% germanium. 00:06:12.398 --> 00:06:14.440 align:middle line:84% And if you're above the melting point of silicon, 00:06:14.440 --> 00:06:16.450 align:middle line:84% you also only need one solution model 00:06:16.450 --> 00:06:18.040 align:middle line:90% to analyze the whole system. 00:06:18.040 --> 00:06:22.210 align:middle line:84% That is the liquid-phase solution model. 00:06:22.210 --> 00:06:26.980 align:middle line:84% But if you are in between these two temperatures, 00:06:26.980 --> 00:06:30.490 align:middle line:84% you need two different models because the endpoints are 00:06:30.490 --> 00:06:33.420 align:middle line:90% at two different structures. 00:06:33.420 --> 00:06:46.150 align:middle line:84% So if you have a temperature, which is intermediate, 00:06:46.150 --> 00:06:49.945 align:middle line:84% we need two different solution models to analyze the system. 00:06:49.945 --> 00:06:52.960 align:middle line:90% 00:06:52.960 --> 00:06:55.380 align:middle line:84% So let's look what that looks like. 00:06:55.380 --> 00:06:59.220 align:middle line:90% 00:06:59.220 --> 00:07:07.953 align:middle line:84% So for temperature less than the melting point of germanium, 00:07:07.953 --> 00:07:09.870 align:middle line:84% we can draw a free energy composition diagram. 00:07:09.870 --> 00:07:17.670 align:middle line:90% 00:07:17.670 --> 00:07:20.190 align:middle line:84% And let's just say it looks something like this. 00:07:20.190 --> 00:07:27.270 align:middle line:90% 00:07:27.270 --> 00:07:28.775 align:middle line:90% This is going to be x. 00:07:28.775 --> 00:07:31.730 align:middle line:90% 00:07:31.730 --> 00:07:33.380 align:middle line:84% And let's just say it looks like this. 00:07:33.380 --> 00:07:39.520 align:middle line:84% For some , temperature, low temperature it's fully 00:07:39.520 --> 00:07:40.390 align:middle line:90% miscible. 00:07:40.390 --> 00:07:43.370 align:middle line:84% All the solutions, solid solutions, are stable. 00:07:43.370 --> 00:07:45.850 align:middle line:84% So we have that positive curvature. 00:07:45.850 --> 00:07:49.590 align:middle line:84% And I'm going to label that alpha because that is 00:07:49.590 --> 00:07:52.980 align:middle line:90% the alpha-phase solution model. 00:07:52.980 --> 00:07:53.480 align:middle line:90% Here. 00:07:53.480 --> 00:07:57.270 align:middle line:90% 00:07:57.270 --> 00:08:03.450 align:middle line:84% For temperature greater than the melting point of silicon, 00:08:03.450 --> 00:08:05.610 align:middle line:84% I'm again going to have a relatively simple looking 00:08:05.610 --> 00:08:06.960 align:middle line:90% for energy composition diagram. 00:08:06.960 --> 00:08:16.830 align:middle line:90% 00:08:16.830 --> 00:08:19.850 align:middle line:90% The liquid is fully miscilbe. 00:08:19.850 --> 00:08:24.550 align:middle line:84% So we expected downwards curving for energy composition diagram. 00:08:24.550 --> 00:08:27.310 align:middle line:84% And I'm going to label that liquid. 00:08:27.310 --> 00:08:30.060 align:middle line:90% 00:08:30.060 --> 00:08:33.740 align:middle line:84% But things get a little bit more interesting 00:08:33.740 --> 00:08:35.689 align:middle line:90% when we are in between. 00:08:35.689 --> 00:08:44.400 align:middle line:90% 00:08:44.400 --> 00:08:48.610 align:middle line:84% When we were in between and we have different reference 00:08:48.610 --> 00:08:54.110 align:middle line:84% states, that is the pure components 00:08:54.110 --> 00:08:56.787 align:middle line:90% are in different structures-- 00:08:56.787 --> 00:08:58.370 align:middle line:84% I'm just going to go ahead and draw it 00:08:58.370 --> 00:08:59.900 align:middle line:84% and we could talk about why it is. 00:08:59.900 --> 00:09:02.720 align:middle line:90% So let's see. 00:09:02.720 --> 00:09:06.030 align:middle line:84% This is the silicon axis, the x silicon axis. 00:09:06.030 --> 00:09:08.300 align:middle line:84% So when I'm on the left-hand side 00:09:08.300 --> 00:09:10.700 align:middle line:84% of this free energy composition diagram, what 00:09:10.700 --> 00:09:11.720 align:middle line:90% is my composition? 00:09:11.720 --> 00:09:17.410 align:middle line:84% What is my system made out of when I'm along this axis? 00:09:17.410 --> 00:09:22.670 align:middle line:90% 00:09:22.670 --> 00:09:26.640 align:middle line:84% This is the silicon opposition axis. 00:09:26.640 --> 00:09:28.915 align:middle line:84% STUDENT: From the left it's germanium and on the right 00:09:28.915 --> 00:09:29.457 align:middle line:90% it's silicon. 00:09:29.457 --> 00:09:30.873 align:middle line:84% RAFAEL JARAMILLO: Right, OK, good. 00:09:30.873 --> 00:09:32.820 align:middle line:84% So left is germanium, right is silicon. 00:09:32.820 --> 00:09:38.370 align:middle line:84% So what is the reference state of germanium 00:09:38.370 --> 00:09:40.500 align:middle line:90% in this temperature range? 00:09:40.500 --> 00:09:45.530 align:middle line:84% That is, what state do you find pure germanium 00:09:45.530 --> 00:09:48.470 align:middle line:90% at this temperature? 00:09:48.470 --> 00:09:51.740 align:middle line:84% And it's unstated that the pressure is 1 atmosphere 00:09:51.740 --> 00:09:54.510 align:middle line:90% unless stated otherwise. 00:09:54.510 --> 00:09:56.150 align:middle line:90% So what is the reference state? 00:09:56.150 --> 00:09:58.010 align:middle line:84% What is my state here of germanium? 00:09:58.010 --> 00:10:02.280 align:middle line:90% 00:10:02.280 --> 00:10:05.220 align:middle line:84% Or I should say, what's the phase of germanium 00:10:05.220 --> 00:10:06.510 align:middle line:90% in this temperature range? 00:10:06.510 --> 00:10:11.660 align:middle line:90% 00:10:11.660 --> 00:10:12.608 align:middle line:90% STUDENT: Liquid. 00:10:12.608 --> 00:10:13.650 align:middle line:90% RAFAEL JARAMILLO: Liquid. 00:10:13.650 --> 00:10:15.770 align:middle line:84% So my liquid-phase solution model 00:10:15.770 --> 00:10:19.090 align:middle line:84% is going to start from zero because the free energy 00:10:19.090 --> 00:10:24.302 align:middle line:84% of turning pure germanium into a liquid is zero. 00:10:24.302 --> 00:10:25.260 align:middle line:90% It already is a liquid. 00:10:25.260 --> 00:10:27.900 align:middle line:90% 00:10:27.900 --> 00:10:39.430 align:middle line:84% However, this is going to zoom up above zero for pure silicon. 00:10:39.430 --> 00:10:43.890 align:middle line:90% 00:10:43.890 --> 00:10:46.770 align:middle line:90% Pure silicon is solid alpha. 00:10:46.770 --> 00:10:49.455 align:middle line:84% So the solid alpha solution model starts at zero. 00:10:49.455 --> 00:10:55.490 align:middle line:90% 00:10:55.490 --> 00:11:00.230 align:middle line:84% And that is going to zoom up above the zero 00:11:00.230 --> 00:11:03.780 align:middle line:90% axis for germanium. 00:11:03.780 --> 00:11:05.250 align:middle line:90% What is that telling us? 00:11:05.250 --> 00:11:10.490 align:middle line:84% That's telling us that if you have pure germanium, 00:11:10.490 --> 00:11:16.170 align:middle line:84% you have to input free energy to turn it into a solid. 00:11:16.170 --> 00:11:18.450 align:middle line:84% Or if you have pure silicon, you have to input 00:11:18.450 --> 00:11:22.590 align:middle line:84% free energy to turn it into a liquid 00:11:22.590 --> 00:11:26.077 align:middle line:84% because those are not equilibrium states. 00:11:26.077 --> 00:11:27.660 align:middle line:84% So we're going to read the equilibrium 00:11:27.660 --> 00:11:31.710 align:middle line:84% states of the pure components from the solution model 00:11:31.710 --> 00:11:35.430 align:middle line:84% that intercepts zero on a free energy and mixing plot. 00:11:35.430 --> 00:11:36.610 align:middle line:90% And what do I have here? 00:11:36.610 --> 00:11:37.610 align:middle line:90% I have a common tangent. 00:11:37.610 --> 00:11:43.960 align:middle line:90% 00:11:43.960 --> 00:11:46.510 align:middle line:84% And I expect to have a common tangent because I 00:11:46.510 --> 00:11:48.340 align:middle line:90% have a two-phase regime. 00:11:48.340 --> 00:11:53.250 align:middle line:84% For these temperatures, I have a two-phase region. 00:11:53.250 --> 00:11:57.060 align:middle line:84% So we're going to be analyzing this in quite a bit of detail. 00:11:57.060 --> 00:11:58.245 align:middle line:90% Let's consider the process. 00:11:58.245 --> 00:12:34.410 align:middle line:90% 00:12:34.410 --> 00:12:36.720 align:middle line:84% All right, let's consider the process 00:12:36.720 --> 00:12:39.620 align:middle line:84% of actually making a solid solution 00:12:39.620 --> 00:12:42.370 align:middle line:84% at some intermediate temperature. 00:12:42.370 --> 00:12:44.920 align:middle line:84% And this is not going to be the process that you 00:12:44.920 --> 00:12:46.150 align:middle line:90% would use in the real world. 00:12:46.150 --> 00:12:47.620 align:middle line:84% This is going to be thermodynamics 00:12:47.620 --> 00:12:51.250 align:middle line:84% where we can take any process we like and imagine 00:12:51.250 --> 00:12:54.250 align:middle line:84% a reversible process and calculate changes 00:12:54.250 --> 00:12:55.250 align:middle line:90% in state functions. 00:12:55.250 --> 00:12:57.940 align:middle line:84% So what I'm going to first do, so I have-- 00:12:57.940 --> 00:13:01.900 align:middle line:84% what are my starting-- what are my starting materials? 00:13:01.900 --> 00:13:02.990 align:middle line:90% Again, liquid. 00:13:02.990 --> 00:13:15.010 align:middle line:84% So I have here a beaker, a beaker of liquid germanium. 00:13:15.010 --> 00:13:18.470 align:middle line:90% That is not so clear. 00:13:18.470 --> 00:13:20.870 align:middle line:84% Here's a beaker of liquid germanium. 00:13:20.870 --> 00:13:35.650 align:middle line:84% And I have, let's say, a crucible of solid silicon. 00:13:35.650 --> 00:13:36.940 align:middle line:90% That's my starting point. 00:13:36.940 --> 00:13:39.370 align:middle line:90% I'm at this high temperature. 00:13:39.370 --> 00:13:42.400 align:middle line:84% My pure components are liquid germanium and solid silicon. 00:13:42.400 --> 00:13:47.770 align:middle line:84% And let's imagine a process that converts this starting system 00:13:47.770 --> 00:13:51.110 align:middle line:84% into an alpha-phase solid solution. 00:13:51.110 --> 00:13:52.540 align:middle line:90% So here's my process. 00:13:52.540 --> 00:14:04.055 align:middle line:84% First, I'm going to convert the germanium from liquid to alpha 00:14:04.055 --> 00:14:04.555 align:middle line:90% phase. 00:14:04.555 --> 00:14:09.350 align:middle line:90% 00:14:09.350 --> 00:14:10.727 align:middle line:84% Maybe you can do this in reality. 00:14:10.727 --> 00:14:12.560 align:middle line:84% But you can definitely do this in your mind. 00:14:12.560 --> 00:14:16.700 align:middle line:84% You can definitely do this when you do a calculation. 00:14:16.700 --> 00:14:18.730 align:middle line:84% So in your mind, you're going to take 00:14:18.730 --> 00:14:20.230 align:middle line:90% some pair of atomic tweezers. 00:14:20.230 --> 00:14:21.910 align:middle line:84% And you're going to pick up each and every atom. 00:14:21.910 --> 00:14:24.035 align:middle line:84% And you're going to force it into the diamond-cubic 00:14:24.035 --> 00:14:28.080 align:middle line:90% structure and hold them there. 00:14:28.080 --> 00:14:29.750 align:middle line:84% This is called a reference state change. 00:14:29.750 --> 00:14:42.950 align:middle line:90% 00:14:42.950 --> 00:14:45.140 align:middle line:84% reference date change, it's sort of the topic 00:14:45.140 --> 00:14:46.580 align:middle line:90% of today's lectures. 00:14:46.580 --> 00:14:49.250 align:middle line:84% And the second thing I'm going to do 00:14:49.250 --> 00:15:00.780 align:middle line:84% is I'm going to mix solid silicon and solid germanium 00:15:00.780 --> 00:15:04.440 align:middle line:90% in the alpha structure. 00:15:04.440 --> 00:15:07.940 align:middle line:90% 00:15:07.940 --> 00:15:09.470 align:middle line:90% I'm going to mix them. 00:15:09.470 --> 00:15:21.820 align:middle line:84% This is described by solution model, which we have already 00:15:21.820 --> 00:15:22.840 align:middle line:90% spent some time with. 00:15:22.840 --> 00:15:26.165 align:middle line:90% 00:15:26.165 --> 00:15:29.120 align:middle line:84% All right, so we have two steps here, two conceptual steps. 00:15:29.120 --> 00:15:31.257 align:middle line:84% The second step we spent time with you're 00:15:31.257 --> 00:15:32.590 align:middle line:90% at least somewhat familiar with. 00:15:32.590 --> 00:15:33.730 align:middle line:90% The first step is new. 00:15:33.730 --> 00:15:36.250 align:middle line:90% 00:15:36.250 --> 00:15:38.657 align:middle line:84% Reference state change, this is new. 00:15:38.657 --> 00:15:42.790 align:middle line:90% 00:15:42.790 --> 00:15:47.340 align:middle line:84% And if I want to do this, if I have liquid germanium, 00:15:47.340 --> 00:15:49.980 align:middle line:84% and I imagine a process that converts it 00:15:49.980 --> 00:15:53.460 align:middle line:84% from liquid to alpha phase, is it 00:15:53.460 --> 00:15:56.755 align:middle line:84% going to go up in Gibbs free energy or down in Gibbs 00:15:56.755 --> 00:15:57.255 align:middle line:90% free energy? 00:15:57.255 --> 00:16:00.900 align:middle line:90% 00:16:00.900 --> 00:16:03.360 align:middle line:84% STUDENT: Up in Gibbs free energy. 00:16:03.360 --> 00:16:05.130 align:middle line:84% RAFAEL JARAMILLO: Up, I think you said up. 00:16:05.130 --> 00:16:09.860 align:middle line:84% Let's say you said up because up is right. 00:16:09.860 --> 00:16:12.900 align:middle line:84% At this temperature, the germanium is liquid. 00:16:12.900 --> 00:16:17.560 align:middle line:84% So nature is telling you, I am happiest as a liquid. 00:16:17.560 --> 00:16:21.870 align:middle line:84% My Gibbs free energy is lowest when I'm a liquid. 00:16:21.870 --> 00:16:23.280 align:middle line:90% That's the reference state. 00:16:23.280 --> 00:16:25.930 align:middle line:84% That's what equilibrium is in nature. 00:16:25.930 --> 00:16:28.890 align:middle line:84% So if you're going to force it into a different state 00:16:28.890 --> 00:16:32.550 align:middle line:84% that you don't find at equilibrium, by definition 00:16:32.550 --> 00:16:34.830 align:middle line:84% you're going up in Gibbs free energy 00:16:34.830 --> 00:16:36.780 align:middle line:84% because at fixed temperature and pressure, 00:16:36.780 --> 00:16:40.120 align:middle line:84% equilibrium is a state of lowest Gibbs free energy. 00:16:40.120 --> 00:16:41.350 align:middle line:90% So thank you for that. 00:16:41.350 --> 00:16:44.160 align:middle line:84% And if you look at our free energy composition diagram 00:16:44.160 --> 00:16:48.640 align:middle line:84% here, for pure germanium, we're going 00:16:48.640 --> 00:16:53.502 align:middle line:84% to imagine there's an intercept of this curve. 00:16:53.502 --> 00:16:55.210 align:middle line:84% And we'll be drawing these more and more. 00:16:55.210 --> 00:16:59.573 align:middle line:90% 00:16:59.573 --> 00:17:00.990 align:middle line:84% This is a gamma [INAUDIBLE] alpha. 00:17:00.990 --> 00:17:03.540 align:middle line:90% 00:17:03.540 --> 00:17:04.439 align:middle line:90% This was a liquid. 00:17:04.439 --> 00:17:07.560 align:middle line:90% 00:17:07.560 --> 00:17:10.140 align:middle line:84% And on this plot, which is a plot with units of joules 00:17:10.140 --> 00:17:12.150 align:middle line:84% per moles, a Gibbs free energy plot, 00:17:12.150 --> 00:17:16.339 align:middle line:84% there's some Gibbs free energy you need to put in. 00:17:16.339 --> 00:17:20.420 align:middle line:84% You need to pay some Gibbs free energy to convert the system 00:17:20.420 --> 00:17:24.910 align:middle line:90% from liquid into alpha phase. 00:17:24.910 --> 00:17:26.710 align:middle line:84% And, again, with a few exceptions, 00:17:26.710 --> 00:17:31.060 align:middle line:84% this is not how you actually do it in the lab or the factory. 00:17:31.060 --> 00:17:35.800 align:middle line:84% But it is the way you analyze it and get the right answer. 00:17:35.800 --> 00:17:39.220 align:middle line:84% So let's talk about how much free energy 00:17:39.220 --> 00:17:44.470 align:middle line:84% is required to effect these reference state changes. 00:17:44.470 --> 00:17:46.427 align:middle line:84% STUDENT: I have think quick question actually, 00:17:46.427 --> 00:17:47.010 align:middle line:90% if that's all. 00:17:47.010 --> 00:17:48.100 align:middle line:90% RAFAEL JARAMILLO: Yeah. 00:17:48.100 --> 00:17:53.260 align:middle line:84% STUDENT: So is the reason that the Gibbs free energy goes up, 00:17:53.260 --> 00:17:54.670 align:middle line:90% is that because of-- 00:17:54.670 --> 00:17:58.740 align:middle line:90% that the delta S is changing? 00:17:58.740 --> 00:18:08.630 align:middle line:84% RAFAEL JARAMILLO: Well, delta G equals delta H minus T delta 00:18:08.630 --> 00:18:11.450 align:middle line:90% S for isothermal processes. 00:18:11.450 --> 00:18:14.232 align:middle line:90% This is sort of-- 00:18:14.232 --> 00:18:16.690 align:middle line:84% I want to bring you a question all the way back to the baby 00:18:16.690 --> 00:18:18.070 align:middle line:90% book. 00:18:18.070 --> 00:18:22.180 align:middle line:84% The whole class is about balancing delta H and delta S. 00:18:22.180 --> 00:18:24.880 align:middle line:84% And so, in general, you don't know. 00:18:24.880 --> 00:18:27.980 align:middle line:84% If you have some transformation that raises Gibbs free energy, 00:18:27.980 --> 00:18:29.710 align:middle line:84% it could be an enthalpy-driven thing, 00:18:29.710 --> 00:18:31.780 align:middle line:84% or an entropy-driven thing, or both. 00:18:31.780 --> 00:18:33.040 align:middle line:90% In general, you don't know. 00:18:33.040 --> 00:18:36.670 align:middle line:84% Now, in this case, if we take liquid germanium 00:18:36.670 --> 00:18:38.710 align:middle line:84% and we're turning it into solid germanium, 00:18:38.710 --> 00:18:43.453 align:middle line:84% what do you think the change in entropy is? 00:18:43.453 --> 00:18:44.870 align:middle line:84% STUDENT: I think it would decrease 00:18:44.870 --> 00:18:46.108 align:middle line:90% just because it's a solid. 00:18:46.108 --> 00:18:47.900 align:middle line:84% RAFAEL JARAMILLO: So let's write that down. 00:18:47.900 --> 00:18:56.760 align:middle line:84% Delta H liquid to alpha, and delta S liquid to alpha. 00:18:56.760 --> 00:18:58.370 align:middle line:90% So delta H liquid to alpha-- 00:18:58.370 --> 00:19:01.040 align:middle line:84% delta S, sorry, delta S is less than zero. 00:19:01.040 --> 00:19:03.500 align:middle line:84% That's what you just said, that the entropy decreases. 00:19:03.500 --> 00:19:04.670 align:middle line:90% And that's exactly right. 00:19:04.670 --> 00:19:05.720 align:middle line:90% The liquids are more mixed up. 00:19:05.720 --> 00:19:06.720 align:middle line:90% They're more disordered. 00:19:06.720 --> 00:19:09.388 align:middle line:90% They have higher entropy. 00:19:09.388 --> 00:19:10.680 align:middle line:90% Let's just finish the argument. 00:19:10.680 --> 00:19:11.472 align:middle line:90% What about delta H? 00:19:11.472 --> 00:19:16.680 align:middle line:84% Is delta H bigger than zero or smaller than zero? 00:19:16.680 --> 00:19:21.000 align:middle line:84% STUDENT: Is it like an exothermic reaction, 00:19:21.000 --> 00:19:23.898 align:middle line:84% going from liquid to some sort of solid? 00:19:23.898 --> 00:19:25.440 align:middle line:84% RAFAEL JARAMILLO: It's good thinking. 00:19:25.440 --> 00:19:27.630 align:middle line:84% So remember, high-temperature phases 00:19:27.630 --> 00:19:29.130 align:middle line:90% are higher enthalpy phases. 00:19:29.130 --> 00:19:31.260 align:middle line:84% The liquid is the higher temperature phase. 00:19:31.260 --> 00:19:33.540 align:middle line:84% The solid is the lower temperature phase. 00:19:33.540 --> 00:19:35.123 align:middle line:90% So it's exothermic. 00:19:35.123 --> 00:19:36.540 align:middle line:84% Another way to thinking of this is 00:19:36.540 --> 00:19:38.790 align:middle line:84% there are heats of crystallization. 00:19:38.790 --> 00:19:40.860 align:middle line:84% There are heats of crystallization Or heats 00:19:40.860 --> 00:19:44.640 align:middle line:90% of condensation if it's-- 00:19:44.640 --> 00:19:46.680 align:middle line:84% maybe you're thinking about climate systems. 00:19:46.680 --> 00:19:49.180 align:middle line:84% And I'm looking at the window thinking about climate systems 00:19:49.180 --> 00:19:51.857 align:middle line:84% and thinking about phase changes in water. 00:19:51.857 --> 00:19:53.190 align:middle line:90% But, yeah, you're exactly right. 00:19:53.190 --> 00:19:54.810 align:middle line:84% It's going from a high-temperature phase 00:19:54.810 --> 00:19:55.935 align:middle line:90% to a low-temperature phase. 00:19:55.935 --> 00:19:59.280 align:middle line:84% So it's necessarily lowering the enthalpy 00:19:59.280 --> 00:20:02.110 align:middle line:90% and also lowering the entropy. 00:20:02.110 --> 00:20:04.845 align:middle line:84% So we have this thing which is negative, 00:20:04.845 --> 00:20:06.220 align:middle line:84% and this thing which is negative, 00:20:06.220 --> 00:20:07.780 align:middle line:90% so this thing which is positive. 00:20:07.780 --> 00:20:10.590 align:middle line:84% So you don't necessarily know whether delta G will 00:20:10.590 --> 00:20:13.440 align:middle line:84% be positive or negative if this is all the information you're 00:20:13.440 --> 00:20:14.220 align:middle line:90% given. 00:20:14.220 --> 00:20:17.040 align:middle line:84% But in this case, I'm telling you 00:20:17.040 --> 00:20:19.860 align:middle line:84% the temperature is above the melting point of germanium. 00:20:19.860 --> 00:20:22.710 align:middle line:84% So you know that the Gibbs free energy, change 00:20:22.710 --> 00:20:26.360 align:middle line:84% has to be positive because if it were negative, 00:20:26.360 --> 00:20:28.730 align:middle line:84% this wouldn't be the reference state. 00:20:28.730 --> 00:20:30.730 align:middle line:84% But let's analyze this graphically a little bit, 00:20:30.730 --> 00:20:31.720 align:middle line:90% is where we're heading. 00:20:31.720 --> 00:20:32.710 align:middle line:90% Does that address your question? 00:20:32.710 --> 00:20:34.210 align:middle line:90% I didn't see you ask that, but-- 00:20:34.210 --> 00:20:36.790 align:middle line:84% STUDENT: Yeah, that makes sense, thank you. 00:20:36.790 --> 00:20:39.400 align:middle line:84% RAFAEL JARAMILLO: I love the questions and interruptions. 00:20:39.400 --> 00:20:41.570 align:middle line:84% It's hard to really have a back and forth over zoom. 00:20:41.570 --> 00:20:44.110 align:middle line:84% I know that, but the more the better. 00:20:44.110 --> 00:20:46.780 align:middle line:84% So let's talk about the temperature dependence-- 00:20:46.780 --> 00:20:56.610 align:middle line:90% 00:20:56.610 --> 00:21:04.690 align:middle line:84% temperature dependence of the free energy 00:21:04.690 --> 00:21:06.970 align:middle line:90% of the pure components. 00:21:06.970 --> 00:21:11.270 align:middle line:90% 00:21:11.270 --> 00:21:18.530 align:middle line:84% Everything that we learned from unary phase 00:21:18.530 --> 00:21:19.970 align:middle line:84% diagrams we're going to need here. 00:21:19.970 --> 00:21:24.650 align:middle line:84% We're going to need to use all that stuff in order to build up 00:21:24.650 --> 00:21:26.400 align:middle line:90% binary phase diagrams. 00:21:26.400 --> 00:21:29.765 align:middle line:84% So let's draw that for germanium and silicon. 00:21:29.765 --> 00:21:33.950 align:middle line:90% 00:21:33.950 --> 00:21:37.520 align:middle line:84% This now is going to be temperature. 00:21:37.520 --> 00:21:40.920 align:middle line:84% And this is going to be Gibbs free energy. 00:21:40.920 --> 00:21:47.250 align:middle line:84% And this is going to be for pure germanium-- 00:21:47.250 --> 00:21:48.553 align:middle line:90% pure germanium. 00:21:48.553 --> 00:21:50.970 align:middle line:84% I'm going to draw the Gibbs free energy versus temperature 00:21:50.970 --> 00:21:54.120 align:middle line:84% curves for the two phases in question here, 00:21:54.120 --> 00:21:57.690 align:middle line:90% alpha and liquid. 00:21:57.690 --> 00:22:01.410 align:middle line:84% OK, first off, let's recall, Gibbs free energy 00:22:01.410 --> 00:22:03.000 align:middle line:90% versus temperature, is this-- 00:22:03.000 --> 00:22:05.370 align:middle line:84% have a positive slope or does it have a negative slope? 00:22:05.370 --> 00:22:17.112 align:middle line:90% 00:22:17.112 --> 00:22:18.195 align:middle line:90% STUDENT: A negative slope. 00:22:18.195 --> 00:22:19.653 align:middle line:84% RAFAEL JARAMILLO: A negative slope. 00:22:19.653 --> 00:22:22.450 align:middle line:84% And the curvature is negative as well. 00:22:22.450 --> 00:22:28.980 align:middle line:84% So let's draw the Gibbs energy for alpha phase. 00:22:28.980 --> 00:22:32.470 align:middle line:84% Let's just say it looks like that. 00:22:32.470 --> 00:22:34.840 align:middle line:84% All right, now I want to draw it for liquid phase. 00:22:34.840 --> 00:22:38.440 align:middle line:84% How should the Gibbs free energy for liquid phase look? 00:22:38.440 --> 00:22:41.680 align:middle line:90% By the way, let's just mark-- 00:22:41.680 --> 00:22:43.200 align:middle line:84% let's mark the melting point here. 00:22:43.200 --> 00:22:46.730 align:middle line:90% 00:22:46.730 --> 00:22:47.750 align:middle line:90% It's kind of a hint. 00:22:47.750 --> 00:22:50.300 align:middle line:84% How should the Gibbs free energy per mole of the liquid phase 00:22:50.300 --> 00:22:50.800 align:middle line:90% look? 00:22:50.800 --> 00:22:58.357 align:middle line:90% 00:22:58.357 --> 00:23:00.190 align:middle line:84% STUDENT: Should it also be negatively sloped 00:23:00.190 --> 00:23:01.782 align:middle line:84% and cross through the alpha phase? 00:23:01.782 --> 00:23:03.490 align:middle line:84% RAFAEL JARAMILLO: Cross through, perfect. 00:23:03.490 --> 00:23:04.907 align:middle line:84% So it's going to more or less look 00:23:04.907 --> 00:23:08.320 align:middle line:84% like this, except there's going to be some important features. 00:23:08.320 --> 00:23:12.430 align:middle line:84% It has to cross the alpha phase where they coexist. 00:23:12.430 --> 00:23:15.100 align:middle line:84% Where those phases can coexist, their Gibbs free energy 00:23:15.100 --> 00:23:16.060 align:middle line:90% has to be equal. 00:23:16.060 --> 00:23:19.240 align:middle line:84% We know that from the unary phase diagram module. 00:23:19.240 --> 00:23:22.180 align:middle line:90% So it's got to cross that. 00:23:22.180 --> 00:23:25.360 align:middle line:84% At high temperature, does it lie below or above the solid? 00:23:25.360 --> 00:23:30.848 align:middle line:90% 00:23:30.848 --> 00:23:32.140 align:middle line:90% STUDENT: Below the [INAUDIBLE]. 00:23:32.140 --> 00:23:32.765 align:middle line:90% STUDENT: Below. 00:23:32.765 --> 00:23:34.130 align:middle line:90% RAFAEL JARAMILLO: Right, great. 00:23:34.130 --> 00:23:35.060 align:middle line:90% So here we have it. 00:23:35.060 --> 00:23:38.980 align:middle line:90% 00:23:38.980 --> 00:23:39.790 align:middle line:90% How's that? 00:23:39.790 --> 00:23:41.500 align:middle line:90% That looks good. 00:23:41.500 --> 00:23:44.590 align:middle line:84% This is completely consistent with what we 00:23:44.590 --> 00:23:46.470 align:middle line:90% know about the phase diagram. 00:23:46.470 --> 00:23:53.890 align:middle line:84% At high temperature, the liquid phase is the equilibrium phase. 00:23:53.890 --> 00:23:55.540 align:middle line:90% It has lower Gibbs free energy. 00:23:55.540 --> 00:23:58.720 align:middle line:84% At low temperature, the solid phase is the equilibrium phase. 00:23:58.720 --> 00:24:00.310 align:middle line:90% It has lower Gibbs free energy. 00:24:00.310 --> 00:24:02.830 align:middle line:84% At the coexisting temperature, they cross. 00:24:02.830 --> 00:24:05.140 align:middle line:84% And they have the same Gibbs free energy. 00:24:05.140 --> 00:24:12.670 align:middle line:84% And now we can see graphically the measure 00:24:12.670 --> 00:24:20.900 align:middle line:84% of what we need to pay to transform germanium 00:24:20.900 --> 00:24:27.060 align:middle line:90% from liquid to solid. 00:24:27.060 --> 00:24:32.750 align:middle line:84% And we can see that this measure flips sign 00:24:32.750 --> 00:24:34.880 align:middle line:84% once you cross through the coexistence point. 00:24:34.880 --> 00:24:40.490 align:middle line:90% 00:24:40.490 --> 00:24:43.600 align:middle line:84% So this is the energy you need to pay for that reference state 00:24:43.600 --> 00:24:44.860 align:middle line:90% change. 00:24:44.860 --> 00:24:47.560 align:middle line:84% Let me draw something similar for pure silicon, which 00:24:47.560 --> 00:24:55.900 align:middle line:84% will basically look the same except I'll 00:24:55.900 --> 00:24:59.260 align:middle line:90% put the melting point higher. 00:24:59.260 --> 00:25:01.660 align:middle line:84% That's make believe because I haven't really labeled 00:25:01.660 --> 00:25:02.650 align:middle line:90% the axes, but. 00:25:02.650 --> 00:25:30.020 align:middle line:90% 00:25:30.020 --> 00:25:42.440 align:middle line:84% Liquid, alpha, and we have, again, a measure delta G-- 00:25:42.440 --> 00:25:47.210 align:middle line:84% this is silicon-- liquid to alpha. 00:25:47.210 --> 00:25:50.240 align:middle line:90% 00:25:50.240 --> 00:25:53.570 align:middle line:84% And it's sign flips as we go through the melting point. 00:25:53.570 --> 00:25:56.880 align:middle line:90% 00:25:56.880 --> 00:26:01.070 align:middle line:84% So there's a reason why we spent time on these sorts of plots. 00:26:01.070 --> 00:26:03.530 align:middle line:84% I know we plotted S and we plotted H. Now we're 00:26:03.530 --> 00:26:04.310 align:middle line:90% plotting G. 00:26:04.310 --> 00:26:06.560 align:middle line:84% There's a reason we spent time on these sorts of plots 00:26:06.560 --> 00:26:09.020 align:middle line:84% back when we were doing unary phase diagrams. 00:26:09.020 --> 00:26:11.720 align:middle line:84% It's because you need to have these in the back of your mind 00:26:11.720 --> 00:26:13.940 align:middle line:84% as you're building binary phase diagrams. 00:26:13.940 --> 00:26:15.490 align:middle line:84% You need to understand this stuff is 00:26:15.490 --> 00:26:18.860 align:middle line:84% the data input that goes into binary phase diagrams. 00:26:18.860 --> 00:26:22.000 align:middle line:90% 00:26:22.000 --> 00:26:25.210 align:middle line:84% Or as I've mentioned before, if you look at a binary phase 00:26:25.210 --> 00:26:31.930 align:middle line:84% diagram, the unary phase diagrams 00:26:31.930 --> 00:26:33.220 align:middle line:90% are kind of hidden here. 00:26:33.220 --> 00:26:36.970 align:middle line:84% That is an isobaric slice of the unary phase 00:26:36.970 --> 00:26:38.500 align:middle line:90% diagram of germanium. 00:26:38.500 --> 00:26:41.380 align:middle line:84% And that is an isobaric slice of the unary phase 00:26:41.380 --> 00:26:44.390 align:middle line:90% diagram of silicon. 00:26:44.390 --> 00:26:47.470 align:middle line:84% So this stuff comes-- this stuff is essential. 00:26:47.470 --> 00:26:57.170 align:middle line:90% 00:26:57.170 --> 00:26:58.970 align:middle line:84% OK, so how do we account for this? 00:26:58.970 --> 00:27:01.775 align:middle line:90% 00:27:01.775 --> 00:27:02.900 align:middle line:90% How do we account for this? 00:27:02.900 --> 00:27:09.490 align:middle line:90% 00:27:09.490 --> 00:27:17.187 align:middle line:84% Accounting for delta G of some component k-- 00:27:17.187 --> 00:27:19.270 align:middle line:84% now I'm going to make this a little more general-- 00:27:19.270 --> 00:27:22.600 align:middle line:84% any transformation between any two phases alpha to beta. 00:27:22.600 --> 00:27:24.790 align:middle line:84% And that, of course, is a function of temperature 00:27:24.790 --> 00:27:26.950 align:middle line:90% that is temperature dependent. 00:27:26.950 --> 00:27:36.250 align:middle line:84% And as we've just written down, this is delta H, alpha to beta. 00:27:36.250 --> 00:27:39.070 align:middle line:84% And that thing is temperature dependent 00:27:39.070 --> 00:27:45.610 align:middle line:90% minus T delta S alpha to beta. 00:27:45.610 --> 00:27:48.679 align:middle line:84% And that thing, of course, is also temperature dependent. 00:27:48.679 --> 00:28:03.360 align:middle line:90% 00:28:03.360 --> 00:28:06.270 align:middle line:84% It does both, maybe temperature dependent. 00:28:06.270 --> 00:28:19.200 align:middle line:84% All right, at the alpha/beta coexistence temp, 00:28:19.200 --> 00:28:22.460 align:middle line:84% at their equilibrium coexistence temp, 00:28:22.460 --> 00:28:37.360 align:middle line:84% we'll call that T alpha, beta, at that temp, delta G 00:28:37.360 --> 00:28:40.241 align:middle line:84% equals zero because they're in equilibrium. 00:28:40.241 --> 00:28:43.400 align:middle line:90% 00:28:43.400 --> 00:28:47.740 align:middle line:84% Often, we will make an assumption. 00:28:47.740 --> 00:28:50.230 align:middle line:84% And I want you to recall back to some problem sets and such 00:28:50.230 --> 00:28:51.010 align:middle line:90% where we did this. 00:28:51.010 --> 00:28:54.820 align:middle line:84% If the transformation heat capacity, 00:28:54.820 --> 00:28:57.340 align:middle line:84% that is the difference in heat capacity between the two 00:28:57.340 --> 00:29:19.330 align:middle line:84% phases, can be ignored, then delta H and delta S 00:29:19.330 --> 00:29:27.930 align:middle line:84% are approximately temp independent. 00:29:27.930 --> 00:29:33.170 align:middle line:90% 00:29:33.170 --> 00:29:35.650 align:middle line:84% So, again, this is stuff from unary phase diagrams 00:29:35.650 --> 00:29:36.730 align:middle line:90% we're recalling. 00:29:36.730 --> 00:29:38.890 align:middle line:84% In the case that the heat capacity difference 00:29:38.890 --> 00:29:41.980 align:middle line:84% is negligible, then you have approximately temperature 00:29:41.980 --> 00:29:44.720 align:middle line:90% independence of those things. 00:29:44.720 --> 00:29:52.990 align:middle line:84% And when that is the case, your expression 00:29:52.990 --> 00:29:59.050 align:middle line:84% for the temperature dependence of the Gibbs free energy 00:29:59.050 --> 00:29:59.620 align:middle line:90% simplifies. 00:29:59.620 --> 00:30:01.000 align:middle line:84% And you should convince yourself. 00:30:01.000 --> 00:30:04.885 align:middle line:84% Work this on the side after class-- 00:30:04.885 --> 00:30:07.885 align:middle line:84% a few lines of algebra, five minutes of your time. 00:30:07.885 --> 00:30:16.500 align:middle line:90% 00:30:16.500 --> 00:30:25.660 align:middle line:84% All right, so that is a simplification. 00:30:25.660 --> 00:30:35.090 align:middle line:84% This thing is linear in T. And what do you need? 00:30:35.090 --> 00:30:37.550 align:middle line:84% What data do you need to evaluate this? 00:30:37.550 --> 00:30:58.530 align:middle line:84% You need transition temp and the entropy of the transformation. 00:30:58.530 --> 00:31:02.030 align:middle line:84% And that's the sort of data that-- 00:31:02.030 --> 00:31:04.530 align:middle line:84% as a reminder, that's the sort of data you get in databases. 00:31:04.530 --> 00:31:13.040 align:middle line:84% So here's, for instance, phase transformations 00:31:13.040 --> 00:31:13.790 align:middle line:90% of the elements. 00:31:13.790 --> 00:31:15.920 align:middle line:84% Here's a bunch of phase transformations 00:31:15.920 --> 00:31:17.570 align:middle line:90% of pure materials. 00:31:17.570 --> 00:31:19.882 align:middle line:84% And normally you get these triples. 00:31:19.882 --> 00:31:20.840 align:middle line:90% You get a temperature-- 00:31:20.840 --> 00:31:23.540 align:middle line:84% I know it's blurry, but I'm looking 00:31:23.540 --> 00:31:25.850 align:middle line:84% at the book, Appendix C. There's a temperature 00:31:25.850 --> 00:31:28.730 align:middle line:84% of the transformation, a delta S of the transformation, 00:31:28.730 --> 00:31:30.870 align:middle line:84% and a delta H of the transformation. 00:31:30.870 --> 00:31:34.760 align:middle line:84% So this is the data that you find in databases. 00:31:34.760 --> 00:31:39.760 align:middle line:90% 00:31:39.760 --> 00:31:42.400 align:middle line:90% OK, so let's keep moving. 00:31:42.400 --> 00:31:45.620 align:middle line:90% 00:31:45.620 --> 00:31:47.720 align:middle line:84% We're talking about two-phase equilibrium. 00:31:47.720 --> 00:31:55.880 align:middle line:84% We're talking about coexistence, solving 00:31:55.880 --> 00:32:01.510 align:middle line:90% for two-phase coexistence. 00:32:01.510 --> 00:32:06.370 align:middle line:90% 00:32:06.370 --> 00:32:07.490 align:middle line:90% Let's make this specific. 00:32:07.490 --> 00:32:15.930 align:middle line:84% Let's go back to the case of silicon germanium 00:32:15.930 --> 00:32:17.790 align:middle line:84% where we know we have two phase coexistence 00:32:17.790 --> 00:32:19.960 align:middle line:90% in between the melting temps. 00:32:19.960 --> 00:32:24.280 align:middle line:84% And so we're going to use our two-phase coexistence 00:32:24.280 --> 00:32:27.700 align:middle line:90% conditions from previous. 00:32:27.700 --> 00:32:28.810 align:middle line:90% What's that? 00:32:28.810 --> 00:32:31.010 align:middle line:90% Chemical potentials are equal. 00:32:31.010 --> 00:32:34.130 align:middle line:84% Chemical potential of silicon and the alpha phase 00:32:34.130 --> 00:32:39.330 align:middle line:84% equals the chemical potential of silicon in the liquid phase. 00:32:39.330 --> 00:32:43.120 align:middle line:90% 00:32:43.120 --> 00:32:47.620 align:middle line:84% And that's going to become the following. 00:32:47.620 --> 00:32:50.020 align:middle line:84% The chemical potential of silicon 00:32:50.020 --> 00:32:51.385 align:middle line:90% in its reference state-- 00:32:51.385 --> 00:32:54.007 align:middle line:90% 00:32:54.007 --> 00:32:55.590 align:middle line:84% what's the reference state of silicon, 00:32:55.590 --> 00:32:59.400 align:middle line:84% somebody, at this temperature, below the melting point 00:32:59.400 --> 00:33:00.810 align:middle line:90% of silicon? 00:33:00.810 --> 00:33:01.702 align:middle line:90% STUDENT: Solid. 00:33:01.702 --> 00:33:03.160 align:middle line:84% RAFAEL JARAMILLO: Solid, thank you. 00:33:03.160 --> 00:33:05.430 align:middle line:84% So the chemical potential of silicon in its reference 00:33:05.430 --> 00:33:18.130 align:middle line:84% state, which is a solid, plus the mixing term 00:33:18.130 --> 00:33:19.255 align:middle line:90% in the solid phase-- 00:33:19.255 --> 00:33:24.040 align:middle line:90% 00:33:24.040 --> 00:33:32.690 align:middle line:90% that, that's that. 00:33:32.690 --> 00:33:34.440 align:middle line:84% Now we're going to do the right-hand side. 00:33:34.440 --> 00:33:37.710 align:middle line:84% The right-hand side, we have chemical potential of silicon 00:33:37.710 --> 00:33:40.530 align:middle line:84% in its reference state, which is solid. 00:33:40.530 --> 00:33:42.450 align:middle line:84% Now we need to pay some free energy. 00:33:42.450 --> 00:33:50.420 align:middle line:84% We need to get silicon from solid to liquid. 00:33:50.420 --> 00:33:59.315 align:middle line:84% And then we need to mix in the liquid phase. 00:33:59.315 --> 00:34:11.580 align:middle line:90% 00:34:11.580 --> 00:34:23.010 align:middle line:84% So mu silicon zero equals reference state, 00:34:23.010 --> 00:34:32.650 align:middle line:84% reference state for pure silicon, which here is alpha-- 00:34:32.650 --> 00:34:40.230 align:middle line:90% delta mu silicon alpha mixed. 00:34:40.230 --> 00:34:41.300 align:middle line:90% What's that? 00:34:41.300 --> 00:34:51.710 align:middle line:84% Solution model for alpha phase solid solution. 00:34:51.710 --> 00:34:57.390 align:middle line:90% 00:34:57.390 --> 00:35:03.310 align:middle line:84% Delta mu silicon mixed in the liquid, that 00:35:03.310 --> 00:35:15.250 align:middle line:84% comes from a solution model for liquid phase solution. 00:35:15.250 --> 00:35:23.040 align:middle line:84% And delta mu silicon alpha to liquid, 00:35:23.040 --> 00:35:24.885 align:middle line:84% that's called a reference state change. 00:35:24.885 --> 00:35:32.960 align:middle line:90% 00:35:32.960 --> 00:35:35.410 align:middle line:84% So you're all going to be great accountants when 00:35:35.410 --> 00:35:37.160 align:middle line:84% you're done with this class because that's 00:35:37.160 --> 00:35:38.540 align:middle line:90% what we're doing here. 00:35:38.540 --> 00:35:41.030 align:middle line:84% It's just accounting, accounting, accounting. 00:35:41.030 --> 00:35:43.700 align:middle line:84% Let's do the same thing for the germanium. 00:35:43.700 --> 00:35:45.230 align:middle line:84% The other equilibrium condition is 00:35:45.230 --> 00:35:50.900 align:middle line:84% that the chemical potential of germanium in the alpha phase 00:35:50.900 --> 00:35:54.647 align:middle line:84% is the chemical potential of germanium in liquid phase. 00:35:54.647 --> 00:35:56.480 align:middle line:84% So this is a two-phase equilibrium condition 00:35:56.480 --> 00:35:58.760 align:middle line:84% that we've derived very recently. 00:35:58.760 --> 00:36:03.290 align:middle line:84% And for this case, you end up with chemical potential 00:36:03.290 --> 00:36:05.510 align:middle line:84% of germanium in its reference state. 00:36:05.510 --> 00:36:08.590 align:middle line:84% Its reference state is the liquid. 00:36:08.590 --> 00:36:12.810 align:middle line:84% So we need to pay some free energy to get germanium 00:36:12.810 --> 00:36:16.050 align:middle line:90% from liquid to the alpha phase. 00:36:16.050 --> 00:36:26.230 align:middle line:84% And then we're going to mix germanium in the alpha phase. 00:36:26.230 --> 00:36:31.553 align:middle line:84% And that is going to be equal to chemical potential of germanium 00:36:31.553 --> 00:36:33.220 align:middle line:84% in its reference state, which is liquid. 00:36:33.220 --> 00:36:41.640 align:middle line:84% And so now all we need to do is make the liquid solution. 00:36:41.640 --> 00:36:53.240 align:middle line:84% So, again, we have the liquid stuff and the solid stuff. 00:36:53.240 --> 00:37:00.990 align:middle line:90% 00:37:00.990 --> 00:37:04.520 align:middle line:84% So I'll be really explicit here, mu germanium zero 00:37:04.520 --> 00:37:15.330 align:middle line:84% equals reference state of pure germanium, 00:37:15.330 --> 00:37:20.970 align:middle line:90% which here is the liquid phase. 00:37:20.970 --> 00:37:29.340 align:middle line:84% Delta mu germanium mix in the liquid 00:37:29.340 --> 00:37:37.305 align:middle line:84% comes from the solution model for liquid-phase solution. 00:37:37.305 --> 00:37:42.180 align:middle line:90% 00:37:42.180 --> 00:37:53.760 align:middle line:84% Delta mu germanium mix alpha comes from the solution model 00:37:53.760 --> 00:37:57.645 align:middle line:90% for alpha-phase solid solution. 00:37:57.645 --> 00:38:01.230 align:middle line:90% 00:38:01.230 --> 00:38:08.490 align:middle line:84% And delta mu germanium liquid to alpha 00:38:08.490 --> 00:38:09.690 align:middle line:90% is a reference state change. 00:38:09.690 --> 00:38:22.650 align:middle line:90% 00:38:22.650 --> 00:38:24.720 align:middle line:84% So we're really, we're bringing out all the stuff 00:38:24.720 --> 00:38:27.090 align:middle line:84% that we've done to this point in the semester. 00:38:27.090 --> 00:38:29.250 align:middle line:84% Everything we've done is needed here. 00:38:29.250 --> 00:38:33.450 align:middle line:84% We need to be able to calculate the temperature dependence. 00:38:33.450 --> 00:38:36.090 align:middle line:84% We need to be able to identify our reference state. 00:38:36.090 --> 00:38:39.610 align:middle line:84% So we need to be able to read unary phase diagrams. 00:38:39.610 --> 00:38:43.540 align:middle line:84% We need to be able to analyze solution models, solution 00:38:43.540 --> 00:38:44.170 align:middle line:90% models. 00:38:44.170 --> 00:38:45.370 align:middle line:84% And you're going to have different solution 00:38:45.370 --> 00:38:46.537 align:middle line:90% models for different phases. 00:38:46.537 --> 00:38:48.370 align:middle line:84% So we need to analyze solution models. 00:38:48.370 --> 00:38:51.730 align:middle line:84% We need to be able to calculate Gibbs free energy 00:38:51.730 --> 00:38:56.635 align:middle line:84% changes for pure materials going through transformations. 00:38:56.635 --> 00:38:58.510 align:middle line:84% And in order to do that, we need to calculate 00:38:58.510 --> 00:39:01.000 align:middle line:84% temperature dependence of enthalpy change transformations 00:39:01.000 --> 00:39:02.680 align:middle line:90% and entropy transformations. 00:39:02.680 --> 00:39:05.650 align:middle line:84% And in order to do that, we need data for temperature dependence 00:39:05.650 --> 00:39:07.690 align:middle line:90% of heat capacity. 00:39:07.690 --> 00:39:10.480 align:middle line:84% So it's like, it's all coming together 00:39:10.480 --> 00:39:14.650 align:middle line:84% in this process of making binary phase diagrams. 00:39:14.650 --> 00:39:17.080 align:middle line:84% So I want to say one final thing, and then pause, 00:39:17.080 --> 00:39:19.110 align:middle line:84% and have a brief discussion about the p-set 00:39:19.110 --> 00:39:20.300 align:middle line:90% and take questions. 00:39:20.300 --> 00:39:28.180 align:middle line:84% So if both solutions-- we have two solution models here, 00:39:28.180 --> 00:39:29.410 align:middle line:90% alpha and liquid. 00:39:29.410 --> 00:39:32.275 align:middle line:84% If both solutions behave ideally-- 00:39:32.275 --> 00:39:35.500 align:middle line:90% 00:39:35.500 --> 00:39:37.590 align:middle line:84% which is not realistic, but it's a nice model. 00:39:37.590 --> 00:39:44.840 align:middle line:84% If both solutions behave ideally and if the heat capacity 00:39:44.840 --> 00:39:53.150 align:middle line:84% difference is approximately zero for both pure components-- 00:39:53.150 --> 00:39:55.920 align:middle line:90% 00:39:55.920 --> 00:39:56.865 align:middle line:90% so a lot of ifs here-- 00:39:56.865 --> 00:40:00.420 align:middle line:90% 00:40:00.420 --> 00:40:04.470 align:middle line:84% then the coexistence conditions-- 00:40:04.470 --> 00:40:09.460 align:middle line:90% 00:40:09.460 --> 00:40:15.175 align:middle line:84% phase coexistence conditions being mu k alpha equals mu k-- 00:40:15.175 --> 00:40:17.753 align:middle line:90% 00:40:17.753 --> 00:40:19.170 align:middle line:84% well, let me make this not liquid. 00:40:19.170 --> 00:40:22.050 align:middle line:84% Let me make it beta so it's a little more general. 00:40:22.050 --> 00:40:29.150 align:middle line:84% That's coexistence conditions-- can be solved explicitly. 00:40:29.150 --> 00:40:32.300 align:middle line:90% 00:40:32.300 --> 00:40:36.200 align:middle line:84% And you have closed-form equations, 00:40:36.200 --> 00:40:40.290 align:middle line:84% functions for the resulting phase diagram. 00:40:40.290 --> 00:40:42.360 align:middle line:84% So the math there is a little bit complicated. 00:40:42.360 --> 00:40:44.090 align:middle line:84% We're not going to do it in this class, 00:40:44.090 --> 00:40:53.540 align:middle line:84% but section 10.2.1, figure 10.22. 00:40:53.540 --> 00:40:57.500 align:middle line:90% So DeHoff does this. 00:40:57.500 --> 00:41:12.450 align:middle line:84% It's kind of interesting-- figure 10.22. 00:41:12.450 --> 00:41:16.838 align:middle line:84% So if those conditions are met, then you 00:41:16.838 --> 00:41:17.880 align:middle line:90% can have different types. 00:41:17.880 --> 00:41:19.755 align:middle line:84% All different phase diagrams that can emerge. 00:41:19.755 --> 00:41:23.220 align:middle line:84% And this is an example of the different types of binary phase 00:41:23.220 --> 00:41:25.750 align:middle line:90% diagrams that you can get. 00:41:25.750 --> 00:41:28.470 align:middle line:84% And so these are all two-phase systems. 00:41:28.470 --> 00:41:31.350 align:middle line:84% It's alpha and liquid, alpha and liquid, alpha and liquid, alpha 00:41:31.350 --> 00:41:31.862 align:middle line:90% and liquid. 00:41:31.862 --> 00:41:33.570 align:middle line:84% It could have been called alpha and beta. 00:41:33.570 --> 00:41:35.128 align:middle line:90% It doesn't matter. 00:41:35.128 --> 00:41:37.170 align:middle line:84% And you can get different shapes, different types 00:41:37.170 --> 00:41:38.850 align:middle line:90% of lens diagrams there. 00:41:38.850 --> 00:41:40.210 align:middle line:90% But that's not very general. 00:41:40.210 --> 00:41:43.230 align:middle line:84% It doesn't look like what nature provides us. 00:41:43.230 --> 00:41:51.510 align:middle line:84% In all other cases, we use computers. 00:41:51.510 --> 00:42:01.058 align:middle line:90% 00:42:01.058 --> 00:42:03.600 align:middle line:84% So in all other cases, we let the computer do the work for us 00:42:03.600 --> 00:42:06.580 align:middle line:84% because it becomes unrealistic to do it ourselves. 00:42:06.580 --> 00:42:09.600 align:middle line:84% And so we use the computer to evaluate these equilibrium 00:42:09.600 --> 00:42:13.020 align:middle line:84% conditions, or it's saying the same thing 00:42:13.020 --> 00:42:15.870 align:middle line:84% to say we use a computer to evaluate the common tangent 00:42:15.870 --> 00:42:21.460 align:middle line:84% conditions and draw the phase diagrams for us. 00:42:21.460 --> 00:42:23.440 align:middle line:84% So in the days and weeks ahead, we're 00:42:23.440 --> 00:42:25.360 align:middle line:84% going to be using CALPHAD more and more. 00:42:25.360 --> 00:42:27.460 align:middle line:84% We're going to be using Thermo-Calc more and more. 00:42:27.460 --> 00:42:29.043 align:middle line:84% We're going to use some other software 00:42:29.043 --> 00:42:31.840 align:middle line:84% that we wrote to play with these free energy composition 00:42:31.840 --> 00:42:32.680 align:middle line:90% diagrams. 00:42:32.680 --> 00:42:36.250 align:middle line:84% We'll have a guest lecture by Professor Greg Olson who 00:42:36.250 --> 00:42:40.900 align:middle line:84% has made a very successful career, built largely 00:42:40.900 --> 00:42:45.830 align:middle line:84% from data-driven predictions of high-performance materials 00:42:45.830 --> 00:42:49.360 align:middle line:90% using CALPHAD tools. 00:42:49.360 --> 00:42:51.420 align:middle line:90% So that's where we're heading. 00:42:51.420 --> 00:42:54.000 align:middle line:90%