WEBVTT 00:00:00.030 --> 00:00:02.400 The following content is provided under a Creative 00:00:02.400 --> 00:00:03.830 Commons license. 00:00:03.830 --> 00:00:06.850 Your support will help MIT OpenCourseWare continue to 00:00:06.850 --> 00:00:10.510 offer high-quality educational resources for free. 00:00:10.510 --> 00:00:13.390 To make a donation or view additional materials from 00:00:13.390 --> 00:00:17.490 hundreds of MIT courses, visit MIT OpenCourseWare at 00:00:17.490 --> 00:00:18.740 ocw.mit.edu. 00:00:21.470 --> 00:00:22.300 PROFESSOR: OK. 00:00:22.300 --> 00:00:24.260 OK, settle down. 00:00:24.260 --> 00:00:25.710 Settle down. 00:00:25.710 --> 00:00:28.860 Let's get started. 00:00:28.860 --> 00:00:31.740 So last day we started talking about diffusion. 00:00:31.740 --> 00:00:37.540 Diffusion is a solid-state mass transport by random 00:00:37.540 --> 00:00:38.420 atomic motion. 00:00:38.420 --> 00:00:41.150 I made the first point last day and today I'll make the 00:00:41.150 --> 00:00:41.820 second point. 00:00:41.820 --> 00:00:45.350 We'll see why it's random atomic motion. 00:00:45.350 --> 00:00:47.530 We reasoned that we could describe 00:00:47.530 --> 00:00:48.920 the ingress of material. 00:00:48.920 --> 00:00:51.655 Here I'm showing a concentration profile. 00:00:51.655 --> 00:00:56.310 A profile is nothing more than a plot of something as a 00:00:56.310 --> 00:00:57.880 function of position. 00:00:57.880 --> 00:01:00.870 So, this is looking at, for example, 00:01:00.870 --> 00:01:03.330 the doping of a wafer. 00:01:03.330 --> 00:01:05.740 This is the free surface at x equals 0. 00:01:05.740 --> 00:01:08.780 As we go to the right, it represents depth. 00:01:08.780 --> 00:01:13.080 And we fix the concentration at the surface and the 00:01:13.080 --> 00:01:18.470 material tails off to zero deep inside the specimen. 00:01:18.470 --> 00:01:23.010 Fick in 1855 annunciated the law of diffusion, shown here, 00:01:23.010 --> 00:01:27.590 that says that the rate of ingress expressed by the flux 00:01:27.590 --> 00:01:31.740 is proportional to the instant gradient in concentration. 00:01:31.740 --> 00:01:33.730 And the constant proportionality is the 00:01:33.730 --> 00:01:34.680 diffusion coefficient. 00:01:34.680 --> 00:01:36.400 This is where all the material science lies. 00:01:36.400 --> 00:01:38.120 It's inside the D. 00:01:38.120 --> 00:01:40.450 That's what we're going to talk about more today. 00:01:40.450 --> 00:01:44.480 And so we see things tail off and what we want to do is 00:01:44.480 --> 00:01:46.905 figure out what the mathematics are that'll allow 00:01:46.905 --> 00:01:51.355 us to render this thing into a functional representation. 00:01:51.355 --> 00:01:54.410 Because all we know right now is that we've got something 00:01:54.410 --> 00:01:55.480 tailing off. 00:01:55.480 --> 00:01:56.140 Ok? 00:01:56.140 --> 00:01:59.580 And so this is the gradient multiplied by the diffusion 00:01:59.580 --> 00:02:02.230 coefficient and you can see that the red line 00:02:02.230 --> 00:02:03.850 represents the flux. 00:02:03.850 --> 00:02:07.170 The gradient is maximum at the surface, so the flux is 00:02:07.170 --> 00:02:08.770 maximum at the surface. 00:02:08.770 --> 00:02:12.300 And as we move farther and farther into the piece, the 00:02:12.300 --> 00:02:15.680 gradient in concentration becomes less steep, which 00:02:15.680 --> 00:02:19.140 means the absolute magnitude of the flux becomes less 00:02:19.140 --> 00:02:23.780 steep, and we go deep inside where there is no doping yet. 00:02:23.780 --> 00:02:29.920 The variation of concentration is constantly 0, so the slope 00:02:29.920 --> 00:02:31.840 is zero, which means the flux is zero. 00:02:31.840 --> 00:02:34.380 So the two sort of track, one and the other. 00:02:34.380 --> 00:02:36.930 And we recognize these are negative slopes since the 00:02:36.930 --> 00:02:38.150 minus sign is here. 00:02:38.150 --> 00:02:39.630 Even though we have a negative slope, we 00:02:39.630 --> 00:02:41.560 have a positive ingress. 00:02:41.560 --> 00:02:44.930 And then the other thing that we realized was that the 00:02:44.930 --> 00:02:48.500 diffusion coefficient has a temperature dependence. 00:02:48.500 --> 00:02:51.130 And the temperature dependence looks like this. 00:02:51.130 --> 00:02:54.280 It's an exponential of some kind of a characteristic 00:02:54.280 --> 00:02:57.320 energy divided by the product of -- this is the gas 00:02:57.320 --> 00:03:00.050 constant, this is not the Rydberg constant. 00:03:00.050 --> 00:03:03.980 The gas constant is simply the Boltzmann constant per mole. 00:03:03.980 --> 00:03:06.130 So if you take the Boltzmann constant, multiply it by the 00:03:06.130 --> 00:03:08.160 Avogadro number, you get this. 00:03:08.160 --> 00:03:11.210 But if you see something times T, you know it's got to be 00:03:11.210 --> 00:03:13.880 either gas constant or Boltzmann constant. 00:03:13.880 --> 00:03:18.360 This two-letter character string has to have something 00:03:18.360 --> 00:03:20.970 akin to a Boltzmann constant, otherwise you don't end up 00:03:20.970 --> 00:03:22.250 with an energy term. 00:03:22.250 --> 00:03:24.510 And this in the top is a barrier energy. 00:03:24.510 --> 00:03:28.160 It's a barrier energy to diffusion and out here we have 00:03:28.160 --> 00:03:29.310 the pre-exponential. 00:03:29.310 --> 00:03:32.515 Unfortunately we didn't give it the letter F in honor of 00:03:32.515 --> 00:03:33.860 Fick, it's just D naught. 00:03:33.860 --> 00:03:35.330 And so there it is. 00:03:35.330 --> 00:03:39.200 So if we take this equation and we plot it, natural log of 00:03:39.200 --> 00:03:43.580 d versus 1 over T, that'll linearize the equation, and we 00:03:43.580 --> 00:03:46.130 end up with something that gives us a straight line. 00:03:46.130 --> 00:03:51.320 And the slope is minus Q over R, where this Q is some kind 00:03:51.320 --> 00:03:52.980 of a barrier energy. 00:03:52.980 --> 00:03:57.810 And the R is the gas constant or the Boltzmann constant. 00:03:57.810 --> 00:04:02.320 So this is the gas constant. 00:04:02.320 --> 00:04:03.960 It's on your table of constants. 00:04:03.960 --> 00:04:11.360 It's got a value in SI units of 8.314 joules per mole 00:04:11.360 --> 00:04:13.360 Kelvin, OK? 00:04:13.360 --> 00:04:16.740 Just the Boltzmann times Avogadro number. 00:04:16.740 --> 00:04:20.730 So now I want to invite you to join me in some pattern 00:04:20.730 --> 00:04:21.860 recognition. 00:04:21.860 --> 00:04:27.510 If I give you the temperature dependence of a quantity that 00:04:27.510 --> 00:04:30.850 looks like this , in other words, the logarithm of that 00:04:30.850 --> 00:04:34.300 physical parameter versus 1 over T, gives 00:04:34.300 --> 00:04:35.700 you a straight line. 00:04:35.700 --> 00:04:39.640 You know from the work that we did on chemical kinetics, this 00:04:39.640 --> 00:04:42.370 is the representation of something that is 00:04:42.370 --> 00:04:43.860 an activated process. 00:04:43.860 --> 00:04:46.650 Remember the box that would fall on its side, and it had 00:04:46.650 --> 00:04:48.020 to go up on its corner? 00:04:48.020 --> 00:04:51.250 So we had some physical sense of what activation is. 00:04:51.250 --> 00:04:55.620 Now this has a log, something versus 1 over T dependence. 00:04:55.620 --> 00:05:00.550 This represents some kind of an activation energy. 00:05:00.550 --> 00:05:01.790 An activation energy. 00:05:01.790 --> 00:05:05.810 But what's the activated process that's going on here? 00:05:05.810 --> 00:05:09.800 So that's where I want to take you into the atomistics. 00:05:09.800 --> 00:05:13.890 So here's a cartoon taken from one of the readings. 00:05:13.890 --> 00:05:18.700 And it shows a set of atoms here and the atom next to the 00:05:18.700 --> 00:05:21.350 vacancy wants to move into the vacancy. 00:05:21.350 --> 00:05:24.130 And you can see the artist's rendition is showing that in 00:05:24.130 --> 00:05:27.190 order to get from the current position into the vacant 00:05:27.190 --> 00:05:31.300 position, it's got to squeeze through this narrow channel. 00:05:31.300 --> 00:05:34.650 And it takes energy to move through that channel. 00:05:34.650 --> 00:05:37.010 And there's an activation barrier 00:05:37.010 --> 00:05:38.330 associated with that motion. 00:05:38.330 --> 00:05:40.040 Can you see when the atom is sitting right 00:05:40.040 --> 00:05:41.550 at the saddle point? 00:05:41.550 --> 00:05:44.250 The system is highly activated because it's pushing out 00:05:44.250 --> 00:05:47.170 against those atoms. And then finally it falls into the new 00:05:47.170 --> 00:05:49.090 slot and the energy drops. 00:05:49.090 --> 00:05:52.560 So the activation energy is associated with moving through 00:05:52.560 --> 00:05:53.760 this saddle point. 00:05:53.760 --> 00:05:57.440 So what really happens here, I'll tell you right now. 00:05:57.440 --> 00:06:00.950 First of all, we have to recognize that the system is 00:06:00.950 --> 00:06:05.030 depicted there, because it's a freeze frame, but we know that 00:06:05.030 --> 00:06:08.150 above 0 Kelvin, everything is in motion. 00:06:08.150 --> 00:06:12.230 So the only way we can rationalize what's going on is 00:06:12.230 --> 00:06:14.310 to, first of all, recognize that we 00:06:14.310 --> 00:06:16.960 have a pulsating lattice. 00:06:16.960 --> 00:06:18.590 We have a pulsating lattice. 00:06:18.590 --> 00:06:20.410 Everything's vibrating. 00:06:20.410 --> 00:06:21.750 And what's the heartbeat? 00:06:21.750 --> 00:06:22.900 What's the idle speed? 00:06:22.900 --> 00:06:24.220 I've told you this before. 00:06:24.220 --> 00:06:27.070 Everything that's going on in this room has an idle speed, a 00:06:27.070 --> 00:06:31.190 heartbeat, of 10 trillion times a second. 00:06:31.190 --> 00:06:33.360 That's the Debye frequency, OK? 00:06:33.360 --> 00:06:36.630 The idle speed or the heartbeat. 00:06:36.630 --> 00:06:40.140 The heartbeat is called the Debye frequency. 00:06:40.140 --> 00:06:45.470 Here's frequency, lowercase nu, in honor of Peter Debye. 00:06:45.470 --> 00:06:47.310 And it's on the order of about 10 to the 00:06:47.310 --> 00:06:49.270 thirteenth per second. 00:06:49.270 --> 00:06:50.720 10 to the thirteenth Hertz. 00:06:50.720 --> 00:06:52.670 That's 10 trillion. 00:06:52.670 --> 00:06:56.300 So 10 trillion times a second atoms are going like this. 00:06:56.300 --> 00:06:57.430 Pulsating. 00:06:57.430 --> 00:06:58.380 All right? 00:06:58.380 --> 00:07:02.500 So what has to happen for that operation to occur? 00:07:02.500 --> 00:07:06.950 Well, I've got an atom here, vacancy next door, but because 00:07:06.950 --> 00:07:11.960 it's close packed the atoms on either side close in. 00:07:11.960 --> 00:07:14.230 There's no way this can squeeze through. 00:07:14.230 --> 00:07:17.360 But, imagine, 10 trillion times a second. 00:07:17.360 --> 00:07:21.610 If I'm lucky enough to see the moment when the atom above 00:07:21.610 --> 00:07:26.240 pulsates up, the atom below pulsates down, a channel can 00:07:26.240 --> 00:07:28.950 open up and then this thing can squirt through. 00:07:28.950 --> 00:07:30.930 Now, that doesn't happen every time. 00:07:30.930 --> 00:07:33.620 If it did, history would have ended. 00:07:33.620 --> 00:07:37.260 At 10 trillion times a second, it's all over for everybody. 00:07:37.260 --> 00:07:37.790 Right? 00:07:37.790 --> 00:07:39.390 So what happens? 00:07:39.390 --> 00:07:42.080 What's the frequency at which we will get this unique 00:07:42.080 --> 00:07:43.290 combination? 00:07:43.290 --> 00:07:47.050 It turns out that frequency, noted gamma, is called the 00:07:47.050 --> 00:07:49.230 jump frequency. 00:07:49.230 --> 00:07:52.550 That's the jump frequency. 00:07:52.550 --> 00:07:55.260 And the jump frequency is on the order of about 10 to the 00:07:55.260 --> 00:07:56.980 eighth Hertz. 00:07:56.980 --> 00:08:01.530 10 to the eighth Hertz, which is about 100 million. 00:08:01.530 --> 00:08:05.690 So if you take the ratio of the Debye frequency to the 00:08:05.690 --> 00:08:10.300 jump frequency, you see you get a value of about one try 00:08:10.300 --> 00:08:14.940 in about 10 to the fifth is successful. 00:08:14.940 --> 00:08:17.980 One try in about 10 to the fifth is successful. 00:08:17.980 --> 00:08:20.320 And that explains what's going on here. 00:08:20.320 --> 00:08:24.430 So, what now does that mean in terms of Q? 00:08:24.430 --> 00:08:25.900 What's the breakdown of Q? 00:08:25.900 --> 00:08:28.940 Well, clearly, Q, the activation energy for 00:08:28.940 --> 00:08:33.220 diffusion, must involve both the energy to 00:08:33.220 --> 00:08:35.170 form the vacancy -- 00:08:35.170 --> 00:08:37.720 I'm going to call it delta Hv, and that's the energy for 00:08:37.720 --> 00:08:38.940 vacancy formation -- 00:08:38.940 --> 00:08:42.300 if you don't have vacancies, you cannot have motion -- 00:08:42.300 --> 00:08:45.750 plus the delta H sub m which is the 00:08:45.750 --> 00:08:49.450 energy for atom migration. 00:08:49.450 --> 00:08:51.425 So you have to form the vacancies and then you've got 00:08:51.425 --> 00:08:53.910 to squeeze through those saddle points. 00:08:53.910 --> 00:08:56.330 So that's how we get a sense of it. 00:08:56.330 --> 00:08:59.250 And I think I've got another slide. 00:08:59.250 --> 00:09:02.340 OK, so they're I've simply put the markings on for you so you 00:09:02.340 --> 00:09:05.670 can see the atoms above and below jumping 00:09:05.670 --> 00:09:06.540 in the right direction. 00:09:06.540 --> 00:09:10.370 That allows the atom on the left to squirt to the right. 00:09:10.370 --> 00:09:16.090 Now, this is also related to energetics. 00:09:16.090 --> 00:09:19.300 It's related to energetics because the energy to form a 00:09:19.300 --> 00:09:22.920 vacancy and the energy to squeeze through those saddle 00:09:22.920 --> 00:09:25.450 points must be related to the binding energy. 00:09:25.450 --> 00:09:27.880 And what else is related to binding energy? 00:09:27.880 --> 00:09:28.770 Melting point. 00:09:28.770 --> 00:09:32.790 So, this is an elegant plot that simply shows that the 00:09:32.790 --> 00:09:36.540 energy -- these are all FCC metals: lead, aluminum, 00:09:36.540 --> 00:09:38.760 silver, gold, copper, and iron. 00:09:38.760 --> 00:09:42.380 They're all FCC on this chart here. 00:09:42.380 --> 00:09:44.620 Iron has both BCC and FCC, but this is 00:09:44.620 --> 00:09:46.600 comparing apples to apples. 00:09:46.600 --> 00:09:50.060 And you can see that there's lead melts at 327, aluminum at 00:09:50.060 --> 00:09:53.360 660, silver 980, all the way up. 00:09:53.360 --> 00:09:54.560 Iron 1535. 00:09:54.560 --> 00:09:57.780 And you can see that the activation energy for -- 00:09:57.780 --> 00:09:59.300 that's this Q-value -- 00:09:59.300 --> 00:10:02.610 the activation energy for diffusion scales 00:10:02.610 --> 00:10:03.590 with melting point. 00:10:03.590 --> 00:10:07.330 Again, an indication that this analysis makes sense. 00:10:07.330 --> 00:10:10.060 And then, here's the other one, the interstitial. 00:10:10.060 --> 00:10:12.670 If you have interstitial diffusion you've got plenty of 00:10:12.670 --> 00:10:16.260 interstitials, so you don't have to pay the penalty to 00:10:16.260 --> 00:10:20.340 create the interstitial, but it's much, much more difficult 00:10:20.340 --> 00:10:22.390 to squeeze through the saddle point. 00:10:22.390 --> 00:10:25.810 So, it turns out that activation energy for 00:10:25.810 --> 00:10:28.100 diffusion by interstitial means is still 00:10:28.100 --> 00:10:29.980 fairly highly activated. 00:10:29.980 --> 00:10:32.160 So this is in the case of substitutional 00:10:32.160 --> 00:10:33.860 systems, all right? 00:10:33.860 --> 00:10:37.090 This is for substitutional. 00:10:37.090 --> 00:10:40.220 Substitutional atoms. What do I mean by that? 00:10:40.220 --> 00:10:42.980 An atom that sits on a lattice site. 00:10:42.980 --> 00:10:45.290 If an atom sits on a lattice site, it has to jump to 00:10:45.290 --> 00:10:47.510 another lattice site, which means there needs to be a 00:10:47.510 --> 00:10:48.960 vacant lattice site. 00:10:48.960 --> 00:10:50.660 On the other hand, if we're talking about interstitial 00:10:50.660 --> 00:10:52.380 special atoms -- 00:10:52.380 --> 00:10:55.720 not substitutional atoms, interstitial atoms -- 00:10:55.720 --> 00:10:59.130 the assumption is that there's plenty of interstitial sites. 00:10:59.130 --> 00:11:01.890 There are very few systems that I can think of where the 00:11:01.890 --> 00:11:04.980 interstitial sites are so heavily occupied that this 00:11:04.980 --> 00:11:06.380 assumption is invalid. 00:11:06.380 --> 00:11:07.860 So in that case, you don't have to 00:11:07.860 --> 00:11:09.920 pay for vacancy formation. 00:11:09.920 --> 00:11:13.400 It's simply equal to the enthalpy, which we know is 00:11:13.400 --> 00:11:15.310 analogous to the energy for a condensed 00:11:15.310 --> 00:11:18.420 system of atom migration. 00:11:18.420 --> 00:11:21.440 Atom migration gives you the whole story and it goes back 00:11:21.440 --> 00:11:24.480 to the same picture, and so on. 00:11:24.480 --> 00:11:27.480 Now, I said it's random walks, so let's look at another one. 00:11:27.480 --> 00:11:30.400 So, this is an interesting example. 00:11:30.400 --> 00:11:32.890 This is self-diffusion in cobalt. 00:11:32.890 --> 00:11:35.120 Cobalt has a number of isotopes. 00:11:35.120 --> 00:11:37.780 One of them is 59 and one of them is 60. 00:11:37.780 --> 00:11:40.380 60 is a radioisotope. 00:11:40.380 --> 00:11:45.400 Cobalt 60 is a radioisotope and it's used in a variety of 00:11:45.400 --> 00:11:46.670 technical endeavors. 00:11:46.670 --> 00:11:50.600 And so what we've got here is a sandwich of cobalt 59 -- 00:11:50.600 --> 00:11:53.310 cobalt 60 and cobalt 59. 00:11:53.310 --> 00:11:57.320 So even though cobalt 60 has radioactivity and it's got a 00:11:57.320 --> 00:12:01.670 different number of neutrons in the nucleus, chemically it 00:12:01.670 --> 00:12:05.750 is identical to cobalt 59, just as carbon 14 is 00:12:05.750 --> 00:12:07.850 chemically identical to carbon 12. 00:12:07.850 --> 00:12:10.800 So there's no concentration gradient. 00:12:10.800 --> 00:12:12.830 Fick's Law says it's supposed to go by 00:12:12.830 --> 00:12:13.860 concentration gradient. 00:12:13.860 --> 00:12:17.400 The concentration gradient is zero across that piece. 00:12:17.400 --> 00:12:20.540 And yet if you wait for long enough, you will 00:12:20.540 --> 00:12:21.570 start to see -- 00:12:21.570 --> 00:12:25.970 the dark atoms here are meant to represent cobalt 60 in the 00:12:25.970 --> 00:12:26.850 sandwich -- 00:12:26.850 --> 00:12:31.560 after some period of time, the cobalt 60's will move outward. 00:12:31.560 --> 00:12:34.980 And if you wait long enough, the cobalt 60 concentration 00:12:34.980 --> 00:12:38.530 will be uniform throughout the specimen. 00:12:38.530 --> 00:12:39.930 So what's going on? 00:12:39.930 --> 00:12:42.390 There's no chemical concentration gradient, and 00:12:42.390 --> 00:12:45.730 yet over time, the cobalt 60 spreads out. 00:12:45.730 --> 00:12:48.530 And the answer is this is happening by random walk. 00:12:48.530 --> 00:12:51.030 We've got the pulsating lattice. 00:12:51.030 --> 00:12:52.710 Pulsating lattice. 00:12:52.710 --> 00:12:55.220 10 trillion times a second those atoms are vibrating and 00:12:55.220 --> 00:12:56.440 there's going to be some vacancy. 00:12:56.440 --> 00:12:59.050 All of a sudden an atom falls into a vacancy. 00:12:59.050 --> 00:13:01.610 And the mathematics of this -- look at this, this is no 00:13:01.610 --> 00:13:04.570 accident -- this curve here represents something that 00:13:04.570 --> 00:13:06.760 looks like a Gaussian distribution. 00:13:06.760 --> 00:13:10.230 At any given time, the amount of cobalt 60 that's veered 00:13:10.230 --> 00:13:14.210 from the center band can be portrayed by the Gaussian 00:13:14.210 --> 00:13:16.020 distribution, the bell curve. 00:13:16.020 --> 00:13:18.450 And what's the physical model for the bell curve? 00:13:18.450 --> 00:13:20.110 It's the drunken sailor. 00:13:20.110 --> 00:13:21.530 Or I guess I have to be PC. 00:13:21.530 --> 00:13:23.370 I'm not going to just pick on sailors. 00:13:23.370 --> 00:13:25.650 Maybe it's the drunken cougar, the soccer mom 00:13:25.650 --> 00:13:26.460 that went out drinking. 00:13:26.460 --> 00:13:31.110 OK, so what happens with the drunken cougar, she comes to 00:13:31.110 --> 00:13:34.950 the door, and she starts to walk, all right? 00:13:34.950 --> 00:13:38.200 And then she walks one way or the other way, until she 00:13:38.200 --> 00:13:40.200 finally falls. 00:13:40.200 --> 00:13:42.510 And if you go through the number of steps that it'll 00:13:42.510 --> 00:13:46.310 take before the drunken sailor falls, you get this 00:13:46.310 --> 00:13:47.120 distribution. 00:13:47.120 --> 00:13:48.980 It's an old physics problem. 00:13:48.980 --> 00:13:50.540 The drunken sailor problem. 00:13:50.540 --> 00:13:52.880 And it's a normal distribution. 00:13:52.880 --> 00:13:55.930 And that normal distribution tells you that everything's 00:13:55.930 --> 00:13:58.730 going about by random jumping. 00:13:58.730 --> 00:14:02.740 Random jumping, according to this, all right? 00:14:02.740 --> 00:14:05.450 So that's why we say that diffusion is 00:14:05.450 --> 00:14:07.150 a random walk problem. 00:14:07.150 --> 00:14:10.110 Before I turn this off, I want you to take a look carefully. 00:14:10.110 --> 00:14:13.345 What's wrong with this textbook cartoon? 00:14:17.770 --> 00:14:19.480 Any ideas? 00:14:19.480 --> 00:14:21.260 It's a substitutional system, right? 00:14:21.260 --> 00:14:23.250 It's cobalt and cobalt. 00:14:23.250 --> 00:14:25.860 This is absolutely impossible. 00:14:25.860 --> 00:14:27.840 What's wrong with it? 00:14:27.840 --> 00:14:29.450 There are no vacancies. 00:14:29.450 --> 00:14:30.580 There are no vacancies. 00:14:30.580 --> 00:14:32.020 It's impossible. 00:14:32.020 --> 00:14:33.460 There's no vacancies to jump into. 00:14:33.460 --> 00:14:36.980 This thing is jam-packed, so it's physically unrealistic 00:14:36.980 --> 00:14:38.390 from a thermodynamic standpoint. 00:14:38.390 --> 00:14:41.460 There has to be some finite population of vacancies. 00:14:41.460 --> 00:14:45.320 Given this system, there's no way that the black atom could 00:14:45.320 --> 00:14:47.860 jump into the neighboring site, because you'd have to 00:14:47.860 --> 00:14:52.700 have the entire system open up, and the chances of that 00:14:52.700 --> 00:14:54.170 happening are vanishingly small. 00:14:54.170 --> 00:14:57.340 So it just shows you, just because something costs 150 00:14:57.340 --> 00:15:00.120 bucks and is pressed between two hard covers doesn't mean 00:15:00.120 --> 00:15:01.160 it's full of truth, OK? 00:15:01.160 --> 00:15:03.430 Be careful here when you look at this. 00:15:03.430 --> 00:15:05.280 So, again, the artist needed to be guided. 00:15:05.280 --> 00:15:07.610 We need to put a few of these things out, just to give 00:15:07.610 --> 00:15:09.920 symbolic recognition to the fact that we need 00:15:09.920 --> 00:15:12.220 to have some vacancies. 00:15:12.220 --> 00:15:12.665 OK. 00:15:12.665 --> 00:15:15.150 All right, let's keep going. 00:15:15.150 --> 00:15:15.670 Let's keep going. 00:15:15.670 --> 00:15:16.670 Here is some data. 00:15:16.670 --> 00:15:17.670 Here is some data. 00:15:17.670 --> 00:15:18.410 Various data. 00:15:18.410 --> 00:15:19.490 This is diffusion. 00:15:19.490 --> 00:15:22.280 This is the logarithm of the diffusion coefficient 00:15:22.280 --> 00:15:24.050 versus 1 over T. 00:15:24.050 --> 00:15:25.050 So let's look at the top. 00:15:25.050 --> 00:15:27.430 This is hydrogen in iron. 00:15:27.430 --> 00:15:29.940 So temperature increases from right to left. 00:15:29.940 --> 00:15:31.320 So, go up, up, up, up, up. 00:15:31.320 --> 00:15:36.200 And roughly 900 degrees Centigrade, iron changes from 00:15:36.200 --> 00:15:38.040 BCC to FCC. 00:15:38.040 --> 00:15:40.790 And you know from our previous unit, the number of nearest 00:15:40.790 --> 00:15:44.110 neighbors in FCC is greater, so there's less 00:15:44.110 --> 00:15:46.110 void fraction, right? 00:15:46.110 --> 00:15:49.840 So that means that it's much more difficult to move. 00:15:49.840 --> 00:15:54.160 And you can see, first of all, the order of magnitude drop in 00:15:54.160 --> 00:15:56.860 the diffusion coefficient, and also, it's subtle, but the 00:15:56.860 --> 00:15:59.660 slope is steeper, because the activation energy for 00:15:59.660 --> 00:16:02.600 diffusion is steeper in a closer-packed system. 00:16:02.600 --> 00:16:04.570 This is carbon in iron. 00:16:04.570 --> 00:16:05.880 Same thing. 00:16:05.880 --> 00:16:09.560 Relatively gentle slope at the same temperature, there's a 00:16:09.560 --> 00:16:14.190 phase change to FCC and it takes much, much more energy 00:16:14.190 --> 00:16:16.870 per unit increase in temperature to get the 00:16:16.870 --> 00:16:19.265 comparable increase in the value of diffusion 00:16:19.265 --> 00:16:19.680 coefficient. 00:16:19.680 --> 00:16:21.950 This is iron, self-diffusion in iron. 00:16:21.950 --> 00:16:24.680 So this would be radio tracer iron in iron. 00:16:24.680 --> 00:16:25.620 Very much lower. 00:16:25.620 --> 00:16:28.640 You see these values are up around 10 to the minus 5, 10 00:16:28.640 --> 00:16:29.310 to the minus 6. 00:16:29.310 --> 00:16:32.340 This is down to 10 to the minus 11, 10 to the minus 12 00:16:32.340 --> 00:16:33.950 centimeters squared per second. 00:16:33.950 --> 00:16:38.180 Iron diffusion changes at the transition temperature. 00:16:38.180 --> 00:16:40.480 Now iron going through FCC iron. 00:16:40.480 --> 00:16:41.330 Here's the last one. 00:16:41.330 --> 00:16:41.680 Look at this one. 00:16:41.680 --> 00:16:43.790 This is carbon in graphite. 00:16:43.790 --> 00:16:45.620 So how does carbon -- suppose I had some 00:16:45.620 --> 00:16:48.410 carbon 14 in graphite. 00:16:48.410 --> 00:16:49.670 What's the bonding in graphite? 00:16:55.100 --> 00:16:56.180 Silence. 00:16:56.180 --> 00:16:57.310 It's covalent. 00:16:57.310 --> 00:16:58.740 It's sp2 hybridized. 00:16:58.740 --> 00:17:00.860 It's really strong bonds. 00:17:00.860 --> 00:17:02.440 The activation energy -- 00:17:02.440 --> 00:17:04.900 you know, forget just jumping through that saddle point -- 00:17:04.900 --> 00:17:06.550 you've got to break those covalent bonds. 00:17:06.550 --> 00:17:09.140 It's very, very high energy to get carbon to 00:17:09.140 --> 00:17:09.950 move through graphite. 00:17:09.950 --> 00:17:12.780 And so you see very, very low values and very, very steep 00:17:12.780 --> 00:17:14.030 temperature dependence. 00:17:16.190 --> 00:17:18.320 Now, there's another way to think about this. 00:17:18.320 --> 00:17:20.980 Again, this is now looking at the atomistics. 00:17:20.980 --> 00:17:25.125 So this is the same material trying to indicate that if you 00:17:25.125 --> 00:17:29.310 use Fick's Law, moving in from the surface into the bulk, we 00:17:29.310 --> 00:17:30.410 have a certain front. 00:17:30.410 --> 00:17:33.480 So this is some isoconcentration line. 00:17:33.480 --> 00:17:39.300 So we could take this value off of the graph arbitrarily. 00:17:39.300 --> 00:17:41.820 So they're saying c greater than c i. 00:17:41.820 --> 00:17:43.620 Because it trails off to zero, right? 00:17:43.620 --> 00:17:44.590 It's asymptotic. 00:17:44.590 --> 00:17:46.450 There's no sharp cut-off here. 00:17:46.450 --> 00:17:47.560 It trails off. 00:17:47.560 --> 00:17:50.550 So what they're doing in that graph is arbitrarily saying, 00:17:50.550 --> 00:17:53.200 let's call this ci. 00:17:53.200 --> 00:17:57.330 And where's the front at concentration ci, given a 00:17:57.330 --> 00:18:00.050 constant value of cs at the surface? 00:18:00.050 --> 00:18:03.900 And you can see that there's a near-constant rate of advance 00:18:03.900 --> 00:18:04.960 through the bulk. 00:18:04.960 --> 00:18:07.590 But along the grain boundary you go much, much farther in 00:18:07.590 --> 00:18:08.830 the same amount of time. 00:18:08.830 --> 00:18:09.730 Why? 00:18:09.730 --> 00:18:12.770 Because the atoms in the grain boundary are not bounded the 00:18:12.770 --> 00:18:14.500 same way as the atoms in the bulk. 00:18:14.500 --> 00:18:17.080 There's this gap, this misalignment. 00:18:17.080 --> 00:18:20.585 And so, it's easier to make that jump, and you can see you 00:18:20.585 --> 00:18:22.410 get much, much more advancement 00:18:22.410 --> 00:18:23.790 along the grain boundary. 00:18:23.790 --> 00:18:25.110 And this is a crack. 00:18:25.110 --> 00:18:27.690 And this material covers the surface. 00:18:27.690 --> 00:18:30.030 You get surface diffusion. 00:18:30.030 --> 00:18:34.140 Surface diffusion is very important in many processes, 00:18:34.140 --> 00:18:37.120 and it's very fast, because on a surface, you've got no atoms 00:18:37.120 --> 00:18:38.640 on the top. 00:18:38.640 --> 00:18:40.770 It's almost liquid-like, isn't it? 00:18:40.770 --> 00:18:43.300 You've got atoms to the bottom, but you've got no 00:18:43.300 --> 00:18:47.110 atoms to the top, so you can move unconstrained. 00:18:47.110 --> 00:18:48.360 Go back to this one. 00:18:51.320 --> 00:18:52.710 Imagine here. 00:18:52.710 --> 00:18:55.820 Imagine if you didn't have any atoms on the top, how easy it 00:18:55.820 --> 00:18:57.695 would be to make that jump. 00:18:57.695 --> 00:19:00.660 All right, let's go and see what the data are. 00:19:00.660 --> 00:19:02.930 Nothing like data. 00:19:02.930 --> 00:19:04.380 And I got data for you. 00:19:04.380 --> 00:19:05.780 Let's look. 00:19:05.780 --> 00:19:07.030 Click, click, click. 00:19:09.540 --> 00:19:10.280 All right. 00:19:10.280 --> 00:19:11.280 Here's the data. 00:19:11.280 --> 00:19:14.220 This is all for the diffusion of silver. 00:19:14.220 --> 00:19:19.110 So I'm going to use this value, this schematic, and I'm 00:19:19.110 --> 00:19:21.370 going to give you the value for silver diffusion in the 00:19:21.370 --> 00:19:25.240 bulk, silver diffusion in the grain boundary, and silver 00:19:25.240 --> 00:19:26.890 diffusion along the surface. 00:19:26.890 --> 00:19:28.090 So these are the data. 00:19:28.090 --> 00:19:31.220 So this is log diffusion coefficient versus 1 over T. 00:19:31.220 --> 00:19:33.140 Only, I hate this graph, because 00:19:33.140 --> 00:19:35.340 somebody got really wimpy. 00:19:35.340 --> 00:19:36.210 You know, it bothers them. 00:19:36.210 --> 00:19:39.000 You see, when you plot something like this, you end 00:19:39.000 --> 00:19:44.030 up with, if 1 over T increases from left to right, this 00:19:44.030 --> 00:19:45.960 really means that high temperature 00:19:45.960 --> 00:19:47.250 is over here, right? 00:19:47.250 --> 00:19:50.430 1 over T increases this way, which means temperature is 00:19:50.430 --> 00:19:51.890 increasing this way. 00:19:51.890 --> 00:19:55.090 So somebody, some weenie, got all nervous here and put the 00:19:55.090 --> 00:19:57.070 numbers backwards, you see? 00:19:57.070 --> 00:20:00.280 That way, the high temperature is over here, but it looks 00:20:00.280 --> 00:20:03.150 stupid, because when I see an activation plot, I want a 00:20:03.150 --> 00:20:03.930 negative slope. 00:20:03.930 --> 00:20:06.450 It's non-physical to have a positive slope. 00:20:06.450 --> 00:20:09.090 So I condemned this and I turned it around. 00:20:09.090 --> 00:20:12.900 So now let's look at it the right way. 00:20:12.900 --> 00:20:17.360 So this is log d versus 1 over T, and this is the bulk, 00:20:17.360 --> 00:20:18.640 lattice or bulk. 00:20:18.640 --> 00:20:22.010 This is grain boundary and this is surface diffusion, all 00:20:22.010 --> 00:20:24.100 in solid silver. 00:20:24.100 --> 00:20:25.620 OK, look at this. 00:20:25.620 --> 00:20:28.700 First of all -- 00:20:28.700 --> 00:20:31.090 I'm going to jump from here -- one, two, three, four. 00:20:31.090 --> 00:20:34.970 Four orders of magnitude faster diffusion along a grain 00:20:34.970 --> 00:20:37.700 boundary, which makes sense, because it's less constrained. 00:20:37.700 --> 00:20:40.100 And now let's go from grain boundary at constant 00:20:40.100 --> 00:20:41.930 temperature up to surface. 00:20:41.930 --> 00:20:43.960 One, two, three. 00:20:43.960 --> 00:20:47.140 Three orders of magnitude along the surface. 00:20:47.140 --> 00:20:49.920 So you can see the effects. 00:20:49.920 --> 00:20:53.750 And there's the melting point of silver, right here. 00:20:53.750 --> 00:20:56.070 So at the melting point of silver, the diffusion 00:20:56.070 --> 00:20:58.650 coefficient is about 10 to the minus 8 centimeters squared 00:20:58.650 --> 00:21:01.220 per second, which is the diffusion coefficient of 00:21:01.220 --> 00:21:03.600 virtually every metal at its melting point. 00:21:03.600 --> 00:21:04.930 So I happen to know that number. 00:21:04.930 --> 00:21:07.640 10 to the minus 8 centimeters squared per second. 00:21:07.640 --> 00:21:09.840 Not that I know it as a function of temperature, but I 00:21:09.840 --> 00:21:11.450 know it can't be any faster than that. 00:21:14.390 --> 00:21:17.610 So I said, this is near liquid-like behavior, what's 00:21:17.610 --> 00:21:19.000 the next thing to look for? 00:21:19.000 --> 00:21:21.910 Let's get some data from real liquids and see if this hunch 00:21:21.910 --> 00:21:22.580 is correct. 00:21:22.580 --> 00:21:23.850 What's the number here? 00:21:23.850 --> 00:21:26.660 10 to the minus 5 to 10 to the minus 4 centimeters squared 00:21:26.660 --> 00:21:27.480 per second. 00:21:27.480 --> 00:21:29.980 This is diffusion in molten ferrous alloys. 00:21:29.980 --> 00:21:32.430 This is manganese in molten iron. 00:21:32.430 --> 00:21:34.800 10 to the minus 5, 10 to the minus 4. 00:21:34.800 --> 00:21:36.100 Bingo. 00:21:36.100 --> 00:21:39.410 It really does behave like a liquid, in terms of diffusion. 00:21:39.410 --> 00:21:43.520 Surface diffusion is very fast, very fast. And it's all 00:21:43.520 --> 00:21:47.980 explained by this simple model of atom jumping. 00:21:47.980 --> 00:21:48.760 And I continued it. 00:21:48.760 --> 00:21:50.210 Here's data from glasses. 00:21:50.210 --> 00:21:51.550 We studied glasses. 00:21:51.550 --> 00:21:53.440 So here's a suite of different glasses. 00:21:53.440 --> 00:21:54.610 Log k. 00:21:54.610 --> 00:21:57.080 This is permeability, which is related to diffusion 00:21:57.080 --> 00:21:57.780 coefficient. 00:21:57.780 --> 00:21:59.850 There's a gas constant in there, but forget about it. 00:21:59.850 --> 00:22:02.920 So this is essentially log of the diffusion coefficient 00:22:02.920 --> 00:22:04.370 versus 1 over T. 00:22:04.370 --> 00:22:05.970 We've got a family of lines. 00:22:05.970 --> 00:22:07.530 And what's the difference here? 00:22:07.530 --> 00:22:09.630 This is pure fused silica up here. 00:22:09.630 --> 00:22:10.960 SiO2. 00:22:10.960 --> 00:22:13.970 There's borosilicate, there's soda lime, and 00:22:13.970 --> 00:22:15.070 there's lead borate. 00:22:15.070 --> 00:22:15.900 So what's happening? 00:22:15.900 --> 00:22:18.820 We're adding more and more modifier as we 00:22:18.820 --> 00:22:20.810 go from top to bottom. 00:22:20.810 --> 00:22:24.230 And as we add modifier, we break the length of the 00:22:24.230 --> 00:22:27.990 silicate network, which means it packs tighter and tighter, 00:22:27.990 --> 00:22:30.650 and as you get tighter and tighter packing, the diffusion 00:22:30.650 --> 00:22:32.170 coefficient gets lower and lower. 00:22:32.170 --> 00:22:33.050 So train your eye. 00:22:33.050 --> 00:22:35.920 Let's pick 200 degrees Centigrade, right here. 00:22:35.920 --> 00:22:40.230 So at constant temperature, the more modifier you put in, 00:22:40.230 --> 00:22:43.050 the lower the diffusion coefficient because things are 00:22:43.050 --> 00:22:44.330 more tightly packed. 00:22:44.330 --> 00:22:47.090 It's like walking down the infinite corridor in the 00:22:47.090 --> 00:22:50.370 middle of the hour, versus walking down on the hour. 00:22:50.370 --> 00:22:52.340 On the hour, there's too many people. 00:22:52.340 --> 00:22:53.610 It's tighter packing. 00:22:53.610 --> 00:22:55.590 And if you're the least bit civilized, you're going to 00:22:55.590 --> 00:22:58.380 have to move a little bit slower, whereas, if you walk 00:22:58.380 --> 00:23:02.270 down the infinite at around now, no problem. 00:23:02.270 --> 00:23:05.000 It's liquid-like behavior. 00:23:05.000 --> 00:23:07.670 And now this is the isotherm at 300 hundred degrees C. 00:23:07.670 --> 00:23:10.090 This is from some old Corning literature. 00:23:10.090 --> 00:23:12.060 Corning used to make glass, at one time. 00:23:12.060 --> 00:23:15.200 And this is the logarithm of diffusivity. 00:23:15.200 --> 00:23:21.030 And this is now the set of silica B203 and P205, and 00:23:21.030 --> 00:23:23.270 these are all network formers, right? 00:23:23.270 --> 00:23:25.280 They all form covalent bonds. 00:23:25.280 --> 00:23:30.300 So the more silicate, borate, phosphate you put in, the more 00:23:30.300 --> 00:23:33.360 stretched out becomes the network. 00:23:33.360 --> 00:23:34.570 And sure enough, look up here. 00:23:34.570 --> 00:23:37.890 See they didn't have names for their glasses like the Malibu 00:23:37.890 --> 00:23:39.390 or the Corvette or something. 00:23:39.390 --> 00:23:41.950 They called their glasses by four-digit numbers. 00:23:41.950 --> 00:23:45.560 So someone would say, oh, this customer needs, oh, I'd sell 00:23:45.560 --> 00:23:46.385 them a 7040. 00:23:46.385 --> 00:23:46.990 All right? 00:23:46.990 --> 00:23:49.030 Because it has a certain mix of properties. 00:23:49.030 --> 00:23:51.780 So forget about what those numbers are, just know that 00:23:51.780 --> 00:23:55.540 this axis indicates that as you move up, up, up, up, up, 00:23:55.540 --> 00:23:59.200 the higher diffusivity is associated with a much more 00:23:59.200 --> 00:24:00.740 open structure. 00:24:00.740 --> 00:24:04.270 So, again, a link between atomistic behavior and 00:24:04.270 --> 00:24:05.690 atomistic structure. 00:24:05.690 --> 00:24:08.560 It's all there. 00:24:08.560 --> 00:24:09.870 OK. 00:24:09.870 --> 00:24:12.780 So let's leave that up there while we go 00:24:12.780 --> 00:24:13.640 a little bit forward. 00:24:13.640 --> 00:24:17.340 OK, so now, so we've been down buried at the atomistic level. 00:24:17.340 --> 00:24:20.050 Now let's jump up to the continuum level. 00:24:20.050 --> 00:24:22.610 We'll go back to Fick's first law. 00:24:22.610 --> 00:24:25.670 And I want to do something a little more quantitative. 00:24:25.670 --> 00:24:30.200 So I want to look at diffusion across a permeable membrane. 00:24:30.200 --> 00:24:35.250 So this is gas through a membrane. 00:24:35.250 --> 00:24:36.600 Gas through a membrane. 00:24:36.600 --> 00:24:38.230 And the membrane is shown here. 00:24:38.230 --> 00:24:40.190 This is the walls of the membrane. 00:24:40.190 --> 00:24:42.980 Membrane has thickness L. 00:24:42.980 --> 00:24:44.770 Membrane of thickness L. 00:24:44.770 --> 00:24:45.630 In fact, we'll give it a name. 00:24:45.630 --> 00:24:48.565 Let's call it membrane. 00:24:48.565 --> 00:24:52.280 All right, so here's the membrane of thickness L, and 00:24:52.280 --> 00:24:55.720 I'm going to put gas on the left at P1, and it's a 00:24:55.720 --> 00:24:56.850 constant P1. 00:24:56.850 --> 00:24:57.960 So we keep -- 00:24:57.960 --> 00:25:00.300 we've got a ballast here, so even though something starts 00:25:00.300 --> 00:25:03.340 diffusing, the amount that we lose doesn't affect the 00:25:03.340 --> 00:25:05.020 constant pressure over here. 00:25:05.020 --> 00:25:09.440 And on the right side, I'm going to put a second 00:25:09.440 --> 00:25:10.890 pressure, P2. 00:25:10.890 --> 00:25:15.120 And just for argument's sake, I'm going to say that P2 is 00:25:15.120 --> 00:25:16.210 less than P1. 00:25:16.210 --> 00:25:19.470 So that means we're going to end up with mass transport 00:25:19.470 --> 00:25:21.050 from left to right. 00:25:21.050 --> 00:25:25.690 And so, I want to ask, what's this going to look like if we 00:25:25.690 --> 00:25:27.010 use Fick's Law? 00:25:27.010 --> 00:25:31.700 And so beneath I'll make a plot, a concentration profile, 00:25:31.700 --> 00:25:35.230 from x equals 0 to x equals L. 00:25:35.230 --> 00:25:39.670 And I can morph this into concentration. 00:25:39.670 --> 00:25:43.860 We know the gas law is PV equals nRT. 00:25:43.860 --> 00:25:45.320 This is the gas law. 00:25:45.320 --> 00:25:47.750 P is pressure. 00:25:47.750 --> 00:25:49.460 P is pressure. 00:25:49.460 --> 00:25:51.570 V you know is volume. 00:25:51.570 --> 00:25:53.200 But I'm going to write all these out, because some of 00:25:53.200 --> 00:25:55.730 these letters are used so many times that we don't know, is 00:25:55.730 --> 00:25:56.990 this the Rydberg constant? 00:25:56.990 --> 00:26:00.000 No, it's the gas constant that you've seen 00:26:00.000 --> 00:26:00.920 on the first board. 00:26:00.920 --> 00:26:04.690 It's the Boltzmann constant times the Avogadro number. 00:26:04.690 --> 00:26:08.380 T is absolute temperature, temperature in Kelvins. 00:26:08.380 --> 00:26:13.060 And n is not the quantum number, it's mole number. 00:26:13.060 --> 00:26:14.520 Mole number. 00:26:14.520 --> 00:26:16.500 PV equals nRT. 00:26:16.500 --> 00:26:22.710 And we know that concentration of i is mole 00:26:22.710 --> 00:26:24.270 number of i over V. 00:26:24.270 --> 00:26:26.040 Both of those appear in here. 00:26:26.040 --> 00:26:28.680 So if you take what's underneath there, and I'll 00:26:28.680 --> 00:26:32.210 expose it in a second, you'll convince yourself that Ci 00:26:32.210 --> 00:26:35.920 equals Pi divided by RT. 00:26:35.920 --> 00:26:38.820 And that's how you get that permeability thing to turn 00:26:38.820 --> 00:26:40.290 around as well. 00:26:40.290 --> 00:26:46.320 OK, so, I started with gas pressure, but I'm now going to 00:26:46.320 --> 00:26:48.070 write this in terms of concentration. 00:26:48.070 --> 00:26:55.030 So I'm going to morph P1 into C1 and over here, P2 into C2. 00:26:55.030 --> 00:26:57.120 And I want to plot the profile. 00:26:57.120 --> 00:26:58.930 So what's the profile look like? 00:26:58.930 --> 00:27:02.120 I'm going to plot the profile at steady state. 00:27:02.120 --> 00:27:04.020 At steady state. 00:27:04.020 --> 00:27:06.650 At steady state, after I've waited a while, I have a 00:27:06.650 --> 00:27:10.320 straight-line profile across here, at steady state. 00:27:10.320 --> 00:27:12.370 It's pinned at the two ends. 00:27:12.370 --> 00:27:14.240 It's pinned at the two ends and varies 00:27:14.240 --> 00:27:16.100 continuously across here. 00:27:16.100 --> 00:27:22.440 And you can see from Fick's Law, that if you have a 00:27:22.440 --> 00:27:29.270 situation in which the gradient is invariant across 00:27:29.270 --> 00:27:32.250 the piece, the gradient is -- 00:27:32.250 --> 00:27:37.190 dc by dx is the same here, as it is here, as it is here, as 00:27:37.190 --> 00:27:44.180 it is here. dc by dx is invariant, right? 00:27:44.180 --> 00:27:45.020 It doesn't change. 00:27:45.020 --> 00:27:46.330 This is a straight line. 00:27:46.330 --> 00:27:49.220 Well, if dc by dx is invariant, and d is a 00:27:49.220 --> 00:27:52.240 constant, then J is invariant, which just means 00:27:52.240 --> 00:27:53.740 no sources are sinks. 00:27:53.740 --> 00:27:56.590 All of the material that goes in, must come out. 00:27:56.590 --> 00:28:01.450 This means that J is not a function of time. 00:28:01.450 --> 00:28:02.700 J is invariant. 00:28:05.500 --> 00:28:08.220 J is not a function of time. 00:28:08.220 --> 00:28:09.960 I come back here ten minutes later, I 00:28:09.960 --> 00:28:11.100 have the same profile. 00:28:11.100 --> 00:28:13.360 That's what steady state means. 00:28:13.360 --> 00:28:16.450 That's what study state means. 00:28:16.450 --> 00:28:18.920 OK, and then, of course, if I wanted to be a little bit 00:28:18.920 --> 00:28:21.890 pedantic, I could say, in certain systems, the diffusion 00:28:21.890 --> 00:28:24.720 coefficient is a function of concentration. 00:28:24.720 --> 00:28:27.630 So, when the diffusion coefficient is a function of 00:28:27.630 --> 00:28:30.670 concentration, then even at steady state, you might have 00:28:30.670 --> 00:28:32.900 some bowing of the profile, but that's a 00:28:32.900 --> 00:28:34.710 little bit more advanced. 00:28:34.710 --> 00:28:35.620 OK. 00:28:35.620 --> 00:28:38.900 So, I can describe everything that's going on here. 00:28:38.900 --> 00:28:44.130 If I wanted to know how much material leaves from inside 00:28:44.130 --> 00:28:48.290 over a certain period of time, then I can simply say that 00:28:48.290 --> 00:28:55.120 total material lost is given by Fick's first law when you 00:28:55.120 --> 00:28:56.490 have steady state. 00:28:56.490 --> 00:29:00.040 It's simply going to equal the product of the flux, which is 00:29:00.040 --> 00:29:03.240 mass per unit area per unit time times the 00:29:03.240 --> 00:29:06.450 area times the time. 00:29:06.450 --> 00:29:10.940 And that's sort of analogous to Faraday's Law, right? 00:29:10.940 --> 00:29:17.530 If I wanted to look at how much electrical charge I get 00:29:17.530 --> 00:29:21.120 for a certain period of time with current passing, I could 00:29:21.120 --> 00:29:24.000 say that the total charge is the product of the current 00:29:24.000 --> 00:29:24.860 times the time. 00:29:24.860 --> 00:29:26.350 You know this relationship. 00:29:26.350 --> 00:29:30.970 Well, can you see that J times A is analogous to the current? 00:29:30.970 --> 00:29:32.960 It's the mass current. 00:29:32.960 --> 00:29:34.150 It's mass flow rate. 00:29:34.150 --> 00:29:37.730 Mass flow rate per unit area times area gives me something 00:29:37.730 --> 00:29:39.750 that's analogous to the current, and the time is the 00:29:39.750 --> 00:29:42.090 time, this is the total material lost, this is the 00:29:42.090 --> 00:29:43.590 total electrical charge. 00:29:43.590 --> 00:29:46.830 These two are the comparable equations. 00:29:46.830 --> 00:29:47.670 OK. 00:29:47.670 --> 00:29:50.260 Now, I want to go to the more sophisticated question, and 00:29:50.260 --> 00:29:53.240 that is, what happens before steady state? 00:29:53.240 --> 00:29:55.110 Before steady state is achieved? 00:29:55.110 --> 00:29:59.110 Suppose at time zero, the membrane has no gas in it. 00:29:59.110 --> 00:30:02.070 And at time zero, I inflate the left side. 00:30:02.070 --> 00:30:03.900 And just to keep things simple, I'm going to 00:30:03.900 --> 00:30:05.430 set c2 equals 0. 00:30:05.430 --> 00:30:06.440 It's a linear equation. 00:30:06.440 --> 00:30:09.330 You can always add it, but I just want to keep it simpler 00:30:09.330 --> 00:30:10.240 for the analysis. 00:30:10.240 --> 00:30:24.620 So, suppose at t equals 0, we inflate to c1 on the left side 00:30:24.620 --> 00:30:26.420 of the empty membrane. 00:30:26.420 --> 00:30:27.670 There's nothing in the membrane. 00:30:32.470 --> 00:30:33.950 So, let's see what happens. 00:30:33.950 --> 00:30:35.770 First, I'm going to solve the problem graphically. 00:30:35.770 --> 00:30:37.170 Because we know what's going to happen. 00:30:37.170 --> 00:30:38.960 I'm going to have to do the mathematics. 00:30:38.960 --> 00:30:41.440 So, here's what we expect. 00:30:41.440 --> 00:30:46.100 This is going from 0 to L. 00:30:46.100 --> 00:30:48.490 And we've got nothing in the membrane on both sides, so 00:30:48.490 --> 00:30:49.430 that's good. 00:30:49.430 --> 00:30:54.320 And then, at time zero, I inflate to c1. 00:30:54.320 --> 00:30:56.500 There's nothing in here. 00:30:56.500 --> 00:30:57.990 So, what do I have? 00:30:57.990 --> 00:31:00.420 Well, I know if I wait long enough, it's eventually going 00:31:00.420 --> 00:31:01.730 to look like this. 00:31:01.730 --> 00:31:04.570 Because that's the steady state solution. 00:31:04.570 --> 00:31:07.510 I've put here c2 equals 0, for simplicity, but 00:31:07.510 --> 00:31:08.300 we can change it. 00:31:08.300 --> 00:31:09.430 It's a linear equation. 00:31:09.430 --> 00:31:10.850 We can just add. 00:31:10.850 --> 00:31:14.580 So, what happens at time 0 plus? 00:31:14.580 --> 00:31:17.740 I start getting diffusion in from left to right, following 00:31:17.740 --> 00:31:18.310 Fick's Law. 00:31:18.310 --> 00:31:20.730 So it's going to look like this, isn't it? 00:31:20.730 --> 00:31:24.560 This is the approach to steady state. 00:31:24.560 --> 00:31:27.320 Some books will call it nonsteady state or unsteady. 00:31:27.320 --> 00:31:30.500 Unsteady to me means unsteady, so I refuse to use those 00:31:30.500 --> 00:31:32.870 terms. I call this the transient. 00:31:32.870 --> 00:31:34.060 That's the literary term. 00:31:34.060 --> 00:31:35.980 The transient. 00:31:35.980 --> 00:31:38.350 We're talking about the approach to steady state. 00:31:44.080 --> 00:31:46.520 And in some systems, you never get to steady state. 00:31:46.520 --> 00:31:47.780 But that's OK. 00:31:47.780 --> 00:31:50.080 So here's what it's going to look like at some very, very 00:31:50.080 --> 00:31:51.055 short time afterwards. 00:31:51.055 --> 00:31:55.170 It's going to look like that, only somewhat modified for 00:31:55.170 --> 00:31:55.980 this geometry. 00:31:55.980 --> 00:31:58.020 So it's going to look like this, isn't it? 00:31:58.020 --> 00:32:01.640 It's pinned at c1, and it's pinned at 0, and it varies 00:32:01.640 --> 00:32:02.860 continuously. 00:32:02.860 --> 00:32:06.050 So this is at time t 1. 00:32:06.050 --> 00:32:08.780 What happens at time t 2? 00:32:08.780 --> 00:32:12.405 At time t 2, it advances a little bit farther. 00:32:16.680 --> 00:32:20.990 But in all instances, the slope and the shape is given 00:32:20.990 --> 00:32:23.210 by this, Fick's first law. 00:32:23.210 --> 00:32:26.360 So this is at time t 2. 00:32:26.360 --> 00:32:27.600 And let's do one more. 00:32:27.600 --> 00:32:28.150 Three's a charm. 00:32:28.150 --> 00:32:30.790 So, let's take a green one. 00:32:30.790 --> 00:32:35.170 All right, so here's at t 3, still pinned at c 1 and still 00:32:35.170 --> 00:32:37.960 pinned at 0. 00:32:37.960 --> 00:32:40.400 So this is at t3. 00:32:40.400 --> 00:32:45.280 And then finally, this is steady state. 00:32:45.280 --> 00:32:48.890 So if we park at any place -- 00:32:48.890 --> 00:32:50.140 let's park at x1. 00:32:52.430 --> 00:32:53.650 Watch here. 00:32:53.650 --> 00:32:56.700 Can you see that the slope at t1 is gentle? 00:32:56.700 --> 00:33:00.060 The slope at t2 is steeper than t1. 00:33:00.060 --> 00:33:03.130 The slope at t3 is steeper than t2, and this is the 00:33:03.130 --> 00:33:07.990 highest. So, we go from nothing, to gradual, gradual, 00:33:07.990 --> 00:33:09.710 gradual, until we finally reach steady 00:33:09.710 --> 00:33:11.070 state, and we stop. 00:33:11.070 --> 00:33:13.590 And then at that point, we just continue to have to mass 00:33:13.590 --> 00:33:14.100 flowing through. 00:33:14.100 --> 00:33:17.600 And look, the concentration is zero, but the flux is not 00:33:17.600 --> 00:33:20.740 zero, because the flux isn't related to the concentration. 00:33:20.740 --> 00:33:22.820 It's related to the concentration gradient. 00:33:22.820 --> 00:33:24.960 Zero concentration, but non-zero 00:33:24.960 --> 00:33:26.000 concentration gradient. 00:33:26.000 --> 00:33:28.800 In fact, just for grins and chuckles, I could plot the 00:33:28.800 --> 00:33:30.440 ascent of this. 00:33:30.440 --> 00:33:35.060 I could plot time at x equals x1. 00:33:35.060 --> 00:33:37.710 I could plot J. 00:33:37.710 --> 00:33:38.610 And what's J? 00:33:38.610 --> 00:33:40.420 At time zero, it's zero. 00:33:40.420 --> 00:33:43.430 Then it's small, then it's larger, larger, larger 00:33:43.430 --> 00:33:44.360 asymptotically. 00:33:44.360 --> 00:33:46.730 And this is J of steady state. 00:33:46.730 --> 00:33:52.160 And this is t1, t2, t3. 00:33:52.160 --> 00:33:56.350 That's the approach to steady state. 00:33:56.350 --> 00:33:58.750 And you might say, well, this is kind of 00:33:58.750 --> 00:34:00.200 a professor's pedantry. 00:34:00.200 --> 00:34:01.330 It's not! 00:34:01.330 --> 00:34:03.660 This is doping of semiconductors. 00:34:03.660 --> 00:34:06.160 You don't run processes at equilibrium. 00:34:06.160 --> 00:34:08.520 You run them far from equilibrium. 00:34:08.520 --> 00:34:11.990 So in the transient mode, doping, outgassing, drying, 00:34:11.990 --> 00:34:12.850 what have you. 00:34:12.850 --> 00:34:15.370 So, now I want to give you the shape of the 00:34:15.370 --> 00:34:16.770 concentration profile. 00:34:16.770 --> 00:34:18.170 So how do I do that? 00:34:18.170 --> 00:34:27.890 OK, so the shape of the transient profile is not given 00:34:27.890 --> 00:34:28.720 by Fick's Law. 00:34:28.720 --> 00:34:31.720 The Fick's Law just gives me the gradient. 00:34:31.720 --> 00:34:33.660 In other words, here's what I want. 00:34:33.660 --> 00:34:38.500 I want C as a function of x at any given time. 00:34:38.500 --> 00:34:39.860 That's what I want. 00:34:39.860 --> 00:34:41.360 I want to be able to plot the profile. 00:34:41.360 --> 00:34:44.890 So, profile is C of x, but C of x varies with time. 00:34:44.890 --> 00:34:47.440 So, I need C of x at different times. 00:34:47.440 --> 00:34:48.470 What gives me that? 00:34:48.470 --> 00:34:52.310 What gives me that is Fick's second law. 00:34:52.310 --> 00:34:55.980 Go to FSL, Fick's second law. 00:34:55.980 --> 00:34:58.320 Now, Fick's second law, I'm just going to put it up here. 00:34:58.320 --> 00:34:59.920 I'm going to show you the solution. 00:34:59.920 --> 00:35:01.730 I don't expect you to solve it. 00:35:01.730 --> 00:35:05.530 I would, you were all required to have differential equations 00:35:05.530 --> 00:35:06.990 as a pre-req, but you don't. 00:35:06.990 --> 00:35:09.690 So, I'm just going to put it up here so that you see it. 00:35:09.690 --> 00:35:12.160 It's a partial differential equation, and 00:35:12.160 --> 00:35:13.490 it looks like this. 00:35:13.490 --> 00:35:15.740 And it's really beautiful. 00:35:15.740 --> 00:35:19.290 It's got a beautiful symmetry and the fonts look great, so 00:35:19.290 --> 00:35:20.480 that's why we put it up here. 00:35:20.480 --> 00:35:24.500 So, this is the partial in time goes as the double 00:35:24.500 --> 00:35:27.390 derivative in space, because we need a two-variable 00:35:27.390 --> 00:35:28.080 function, right? 00:35:28.080 --> 00:35:29.540 We want x and t. 00:35:29.540 --> 00:35:34.070 And this assumes that the diffusion coefficient is not a 00:35:34.070 --> 00:35:35.380 function of concentration. 00:35:35.380 --> 00:35:37.510 If it is, the equation is messier. 00:35:37.510 --> 00:35:39.860 This assumes a constant diffusion coefficient. 00:35:39.860 --> 00:35:42.590 So, this, as you can see, is a partial differential equation, 00:35:42.590 --> 00:35:45.610 which is a bummer, because it's hard to solve, all right? 00:35:45.610 --> 00:35:47.350 But, it's linear. 00:35:47.350 --> 00:35:50.080 It's a linear partial differential equation, which 00:35:50.080 --> 00:35:51.560 means the solutions are additive. 00:35:51.560 --> 00:35:54.300 We know this already from LCAOMO. 00:35:54.300 --> 00:35:57.300 So, I can handle, because I can solve it once and then I 00:35:57.300 --> 00:36:02.580 can give you ways to patch it, to make it useful. 00:36:02.580 --> 00:36:06.200 So, the solution. 00:36:06.200 --> 00:36:09.300 First of all, how do I specify this fully? 00:36:09.300 --> 00:36:10.550 Every time you differentiate -- 00:36:10.550 --> 00:36:12.240 I'm going to give you some math here, real math 00:36:12.240 --> 00:36:13.260 that you can use. 00:36:13.260 --> 00:36:15.880 When you differentiate, you throw away information, right? 00:36:15.880 --> 00:36:17.960 The derivative of a constant is zero. 00:36:17.960 --> 00:36:19.380 So, what was the value of the constant? 00:36:19.380 --> 00:36:20.330 It's gone. 00:36:20.330 --> 00:36:21.160 You lost it. 00:36:21.160 --> 00:36:22.490 So you have to bring it back. 00:36:22.490 --> 00:36:25.780 So when I double derivative this, I need two pieces of 00:36:25.780 --> 00:36:27.345 spacial information. 00:36:27.345 --> 00:36:30.040 And when I take a derivative with respect to time, I need 00:36:30.040 --> 00:36:32.560 one piece of temporal information. 00:36:32.560 --> 00:36:33.670 That's math. 00:36:33.670 --> 00:36:36.110 All the other lemmas and postulates, forget them. 00:36:36.110 --> 00:36:37.590 This is how you use it, all right? 00:36:37.590 --> 00:36:39.860 So, I need two pieces of spatial information and one 00:36:39.860 --> 00:36:41.620 piece of temporal information. 00:36:41.620 --> 00:36:45.150 So these specify our boundary conditions, right? 00:36:45.150 --> 00:36:46.450 So, what are our boundary conditions. 00:36:46.450 --> 00:36:49.920 So, I will need the concentration of 00:36:49.920 --> 00:36:51.710 all x at time 0. 00:36:51.710 --> 00:36:53.140 That's this. 00:36:53.140 --> 00:36:54.400 The answer to this question. 00:36:54.400 --> 00:36:58.180 And what we're going to do is say that it's a constant. 00:36:58.180 --> 00:37:00.350 I don't know what the constant is, it could be zero, it could 00:37:00.350 --> 00:37:02.800 be any arbitrary number, but I'm going to give you the 00:37:02.800 --> 00:37:05.340 solution for the situation in which the initial 00:37:05.340 --> 00:37:08.650 concentration is a constant. 00:37:08.650 --> 00:37:10.700 So, you could look at this and say, well, what happens if 00:37:10.700 --> 00:37:11.790 it's a non-zero constant? 00:37:11.790 --> 00:37:13.010 You ready for linearity? 00:37:13.010 --> 00:37:13.870 Watch this. 00:37:13.870 --> 00:37:15.530 This is how cool linearity is. 00:37:15.530 --> 00:37:17.230 You see this set of solutions? 00:37:17.230 --> 00:37:20.400 This is the set of solutions for C equals 0. 00:37:20.400 --> 00:37:23.030 What if C equals some non-zero value? 00:37:23.030 --> 00:37:26.210 I just shift the origin down to here to the non-zero value 00:37:26.210 --> 00:37:27.440 and everything sits there. 00:37:27.440 --> 00:37:29.390 That's what superposition gives you. 00:37:29.390 --> 00:37:32.570 That whole set of solutions just gets jacked up by the 00:37:32.570 --> 00:37:34.580 value of C naught. 00:37:34.580 --> 00:37:38.680 If C naught is 0, this slams down onto the x-axis. 00:37:38.680 --> 00:37:40.660 That's real math. 00:37:40.660 --> 00:37:42.360 You're the master. 00:37:42.360 --> 00:37:47.620 The math is the slave. Never let math enslave you. 00:37:47.620 --> 00:37:48.510 I won't. 00:37:48.510 --> 00:37:49.470 I refuse. 00:37:49.470 --> 00:37:52.840 So that's how you make C equals C naught work. 00:37:52.840 --> 00:37:57.190 And then you have to peg the concentration at the surface. 00:37:57.190 --> 00:37:59.830 I can give you examples where that's not the situation, 00:37:59.830 --> 00:38:02.340 where you have a variable concentration at the surface. 00:38:02.340 --> 00:38:04.550 The solution I'm going to give you won't work for those. 00:38:04.550 --> 00:38:07.370 So, x equals 0 is the surface. 00:38:07.370 --> 00:38:10.680 So, at x equals 0 at all time, I have some fixed value, which 00:38:10.680 --> 00:38:12.550 I'm going to just call C surface. 00:38:12.550 --> 00:38:14.600 And I need a third boundary condition. 00:38:14.600 --> 00:38:16.100 And the third boundary condition is a 00:38:16.100 --> 00:38:17.390 mathematical trick. 00:38:17.390 --> 00:38:20.540 We're going to say that this only works at short times. 00:38:20.540 --> 00:38:24.310 It's called the short time. 00:38:24.310 --> 00:38:26.560 And at short times, I'm going to say that -- 00:38:26.560 --> 00:38:29.850 from the perspective of the diffusion experiment -- 00:38:29.850 --> 00:38:33.390 at very, very short times, this sample appears to be 00:38:33.390 --> 00:38:35.120 infinitely long. 00:38:35.120 --> 00:38:38.220 Because the amount of material that goes in doesn't hit the 00:38:38.220 --> 00:38:39.290 other side. 00:38:39.290 --> 00:38:42.160 So, at very, very short times, we can pretend that it's 00:38:42.160 --> 00:38:43.090 semi-infinite. 00:38:43.090 --> 00:38:46.000 And so then we can write that the concentration -- 00:38:46.000 --> 00:38:52.130 this is the short time, which means infinite size, and 00:38:52.130 --> 00:38:53.740 you'll see that the two are related -- 00:38:53.740 --> 00:38:57.520 that the concentration at infinity, then, doesn't change 00:38:57.520 --> 00:38:58.620 from the initial value. 00:38:58.620 --> 00:39:02.200 The concentration at infinity for all time must equal the 00:39:02.200 --> 00:39:03.770 initial value C naught. 00:39:03.770 --> 00:39:06.250 And so if you put these three boundary conditions into that 00:39:06.250 --> 00:39:09.600 equation, you end up with this. 00:39:09.600 --> 00:39:10.340 This is the thing. 00:39:10.340 --> 00:39:16.140 You get C minus Cs over C naught minus Cs. 00:39:16.140 --> 00:39:17.880 This is surface concentration. 00:39:17.880 --> 00:39:19.320 This is initial concentration. 00:39:19.320 --> 00:39:20.560 This is C. 00:39:20.560 --> 00:39:23.780 It varies by this amount. 00:39:23.780 --> 00:39:28.520 Error function of x over 2 times the square root of Dt. 00:39:28.520 --> 00:39:31.960 So, you tell me the time and I can tell you the relationship 00:39:31.960 --> 00:39:34.500 between concentration and position. 00:39:34.500 --> 00:39:38.320 Now I've got to tell you what the error function is. 00:39:38.320 --> 00:39:39.940 Here's the error function. 00:39:39.940 --> 00:39:42.900 This is this problem before. 00:39:42.900 --> 00:39:44.490 This is the solution I'm going to show you. 00:39:44.490 --> 00:39:46.550 Here's Cs. 00:39:46.550 --> 00:39:47.905 Here's C naught. 00:39:47.905 --> 00:39:49.410 The most general case. 00:39:49.410 --> 00:39:51.470 C naught could be zero. 00:39:51.470 --> 00:39:55.370 And this curve here is given by this equation, OK? 00:39:55.370 --> 00:39:57.740 And this is an example of doping, isn't it? 00:39:57.740 --> 00:39:59.680 This is how doping would work. 00:39:59.680 --> 00:40:04.110 And this is the case where C s is greater than C naught. 00:40:04.110 --> 00:40:06.630 If we turn it around and we make the surface concentration 00:40:06.630 --> 00:40:09.270 less than the initial concentration, we'll get the 00:40:09.270 --> 00:40:12.306 complementary situation. 00:40:12.306 --> 00:40:14.800 The complementary situation looks like this. 00:40:17.340 --> 00:40:21.360 So in this case, C naught is up here, Cs is down here, and 00:40:21.360 --> 00:40:22.610 we end up with this. 00:40:25.160 --> 00:40:28.660 So now matter is going out, because Cs 00:40:28.660 --> 00:40:29.760 is less than C naught. 00:40:29.760 --> 00:40:31.140 So this is -- 00:40:31.140 --> 00:40:34.760 what do you want to call it -- this is effusion, this is some 00:40:34.760 --> 00:40:36.920 kind of a drying process, outgassing. 00:40:36.920 --> 00:40:38.340 And this is doping. 00:40:38.340 --> 00:40:39.685 And what's the shape of this curve? 00:40:39.685 --> 00:40:40.830 The shape of this curve. 00:40:40.830 --> 00:40:42.130 ERF. 00:40:42.130 --> 00:40:43.300 That's what math is. 00:40:43.300 --> 00:40:45.700 You find a mathematical function that 00:40:45.700 --> 00:40:47.660 templates for the curve. 00:40:47.660 --> 00:40:50.800 And ERF, I'm going to show you, is related to random walk 00:40:50.800 --> 00:40:51.410 statistics. 00:40:51.410 --> 00:40:54.970 So, it's physically validated. 00:40:54.970 --> 00:40:56.205 It's the best fit. 00:40:56.205 --> 00:40:59.080 I mean, heck, if you find anything that is pinned at two 00:40:59.080 --> 00:41:02.260 ends and has curvature, you'll get a reasonable fit, but 00:41:02.260 --> 00:41:03.880 what's the right fit? 00:41:03.880 --> 00:41:04.980 The right fit is this. 00:41:04.980 --> 00:41:07.220 The error function looks like this. 00:41:07.220 --> 00:41:08.370 Here's the error function. 00:41:08.370 --> 00:41:09.800 Error function. 00:41:09.800 --> 00:41:13.600 I'm going to plot ERF of z as a function of z. 00:41:13.600 --> 00:41:16.570 And it goes from 0 to 1. 00:41:21.120 --> 00:41:22.300 It looks like this. 00:41:22.300 --> 00:41:23.640 All right? 00:41:23.640 --> 00:41:26.260 And here's an error function table. 00:41:26.260 --> 00:41:27.730 Those are exact values. 00:41:27.730 --> 00:41:30.700 And it turns out that ERF is pretty much linear up 00:41:30.700 --> 00:41:32.830 to about point 6. 00:41:32.830 --> 00:41:34.100 Up to about point 6. 00:41:34.100 --> 00:41:39.950 ERF of point 6 not to less than about 1% is point 6. 00:41:39.950 --> 00:41:42.180 And then as you go farther and farther out, and eventually, 00:41:42.180 --> 00:41:45.240 ERF of infinity equals 1. 00:41:45.240 --> 00:41:49.200 But you don't even need tables for 0 up to point 6. 00:41:49.200 --> 00:41:52.150 And you can push it, go to point 6, 5, if you want. 00:41:52.150 --> 00:41:52.790 Yeah, let's do it. 00:41:52.790 --> 00:41:53.980 Let's push harder. 00:41:53.980 --> 00:41:55.110 6, 5. 00:41:55.110 --> 00:41:55.750 It's linear. 00:41:55.750 --> 00:41:57.360 You don't need tables. 00:41:57.360 --> 00:41:57.880 Right? 00:41:57.880 --> 00:42:01.460 And so ERF of 0 equals 0. 00:42:01.460 --> 00:42:03.350 Oh, let me give you the function. 00:42:03.350 --> 00:42:06.450 You know how you can define sine? 00:42:06.450 --> 00:42:09.940 Sine is that integral of something over 1 minus blah, 00:42:09.940 --> 00:42:10.820 blah, blah? 00:42:10.820 --> 00:42:13.090 We've got a definition for this one, too. 00:42:13.090 --> 00:42:14.320 It's cool. 00:42:14.320 --> 00:42:14.960 It's cool. 00:42:14.960 --> 00:42:17.170 Here's what the error function looks like. 00:42:17.170 --> 00:42:24.810 ERF of z equals the integral from 0 to z of e to the minus 00:42:24.810 --> 00:42:26.840 u squared du. 00:42:26.840 --> 00:42:29.770 You might say, why is he showing us all of this? 00:42:29.770 --> 00:42:31.860 Because he wants to torment you, that's why. 00:42:31.860 --> 00:42:33.590 No, it's because I'm trying to teach you something. 00:42:33.590 --> 00:42:35.090 What's e to the minus u squared? 00:42:35.090 --> 00:42:37.140 Actually, I think I can squeeze it in right there. 00:42:37.140 --> 00:42:40.290 What's e to the minus u squared du? 00:42:40.290 --> 00:42:43.640 This function here, that's the bell curve, isn't it? e to the 00:42:43.640 --> 00:42:44.550 minus u squared. 00:42:44.550 --> 00:42:46.470 That's this thing. 00:42:46.470 --> 00:42:47.430 It's symmetric, right? 00:42:47.430 --> 00:42:49.850 It's a minus u squared, so it doesn't matter if u is 00:42:49.850 --> 00:42:51.080 positive or negative. 00:42:51.080 --> 00:42:55.040 And e to the 0 was 1, and then back and forth. 00:42:55.040 --> 00:42:58.700 And so what this is doing, is it's integrating from zero out 00:42:58.700 --> 00:43:00.600 to some value. 00:43:00.600 --> 00:43:01.180 That's all. 00:43:01.180 --> 00:43:02.330 That's the random walk. 00:43:02.330 --> 00:43:03.460 This is the drunken sailor. 00:43:03.460 --> 00:43:05.700 How far does the drunken sailor go? 00:43:05.700 --> 00:43:07.830 That's the shape of this curve, the integral. 00:43:07.830 --> 00:43:11.310 And we'd like it to be so that if you integrate from zero to 00:43:11.310 --> 00:43:14.040 infinity, the area will be one. 00:43:14.040 --> 00:43:18.830 It turns out that this area here is root pi over 2. 00:43:18.830 --> 00:43:22.120 So, therefore, to normalize, we put 2 over root pi. 00:43:22.120 --> 00:43:23.915 That's how you get the error function. 00:43:23.915 --> 00:43:24.650 All right? 00:43:24.650 --> 00:43:27.300 So, the integral from zero to zero is zero, and the integral 00:43:27.300 --> 00:43:31.280 from 0 to 1 is going to be, whatever it is, point 8, 4, et 00:43:31.280 --> 00:43:32.200 cetera, et cetera. 00:43:32.200 --> 00:43:33.550 So, that's it. 00:43:33.550 --> 00:43:34.010 That's it. 00:43:34.010 --> 00:43:38.350 Now we have the functional relationship to describe any 00:43:38.350 --> 00:43:40.610 of these curves. 00:43:40.610 --> 00:43:43.180 And you'll get some practice in doing that. 00:43:43.180 --> 00:43:44.730 And it's a lot of fun. 00:43:44.730 --> 00:43:47.380 Because now you can talk about how long it's going to take to 00:43:47.380 --> 00:43:49.050 dope something, and so on. 00:43:49.050 --> 00:43:49.960 Outgassing. 00:43:49.960 --> 00:43:52.240 And now I'm going to show you something really cool, and I'm 00:43:52.240 --> 00:43:53.010 not kidding you. 00:43:53.010 --> 00:43:54.010 This has served me well. 00:43:54.010 --> 00:43:56.490 I've been here 30 some odd years, and I've been in 00:43:56.490 --> 00:44:00.070 consulting situations where I can, in my head, figure out 00:44:00.070 --> 00:44:02.830 order of magnitude of what it's going to take to run a 00:44:02.830 --> 00:44:03.950 diffusion problem. 00:44:03.950 --> 00:44:10.080 Because you've got, up here, C minus Cs over C naught minus 00:44:10.080 --> 00:44:17.930 Cs equals ERF error function of x over 2 roots of Dt. 00:44:17.930 --> 00:44:21.100 So, this runs from 0 to 1. 00:44:21.100 --> 00:44:22.160 The right side. 00:44:22.160 --> 00:44:24.940 When I went to school, the left side of the equation had 00:44:24.940 --> 00:44:27.320 to do the same thing as the right side of the equation. 00:44:27.320 --> 00:44:30.040 So if the right side of the equation runs from 0 to 1, the 00:44:30.040 --> 00:44:32.040 left side of the equation runs from 0 to 1. 00:44:32.040 --> 00:44:34.250 So, let's pick a -- are you ready -- average 00:44:34.250 --> 00:44:35.940 value from 0 to 1. 00:44:35.940 --> 00:44:37.270 What would you pick? 00:44:37.270 --> 00:44:39.990 I'd pick point 5. 00:44:39.990 --> 00:44:42.440 Now, I don't have to go to the error function tables because 00:44:42.440 --> 00:44:44.420 point 5 is less than point 6. 00:44:44.420 --> 00:44:49.540 So ERF of x over 2 root Dt is essentially equal to x over 2 00:44:49.540 --> 00:44:52.010 roots of Dt. 00:44:52.010 --> 00:44:56.550 So, now I've got one half equals one half x over root 00:44:56.550 --> 00:45:00.645 Dt, from which I can say that x is approximately equal to 00:45:00.645 --> 00:45:02.120 the square root of Dt. 00:45:02.120 --> 00:45:04.390 And I can be sitting in a meeting and someone says, gee, 00:45:04.390 --> 00:45:05.490 I don't know how long it's going to take. 00:45:05.490 --> 00:45:08.440 This thing's going to go down about 100 microns. 00:45:08.440 --> 00:45:10.040 And how long is it going to take? 00:45:10.040 --> 00:45:13.100 So, I pick a number, like, OK, if this is 100 microns, and 00:45:13.100 --> 00:45:15.930 this thing here is 10 to the minus 8, now you just take the 00:45:15.930 --> 00:45:18.110 square root of that, and tell them, oh, it's going to take 00:45:18.110 --> 00:45:19.790 so many minutes, and they go, huh? 00:45:19.790 --> 00:45:20.780 How did you get that? 00:45:20.780 --> 00:45:22.130 It's right there. 00:45:22.130 --> 00:45:24.220 And I'm not trying to tell them a number for three 00:45:24.220 --> 00:45:25.410 significant figures. 00:45:25.410 --> 00:45:27.580 They just want to know is it going to take a minute, an 00:45:27.580 --> 00:45:30.580 hour, a day, or are we out of business? 00:45:30.580 --> 00:45:33.840 And you can make that calculation with this. 00:45:33.840 --> 00:45:36.210 This is so powerful. 00:45:36.210 --> 00:45:39.100 And everything I told you about diffusion applies to 00:45:39.100 --> 00:45:40.540 conductive heat transport. 00:45:40.540 --> 00:45:42.480 You can do the same thing for heat transfer, figure out how 00:45:42.480 --> 00:45:45.050 long it's going to take for a wave to go in. 00:45:45.050 --> 00:45:47.220 This is really powerful stuff. 00:45:47.220 --> 00:45:48.390 Anyway. 00:45:48.390 --> 00:45:50.920 Let's move on. 00:45:50.920 --> 00:45:53.650 Well, there's our pal. 00:45:53.650 --> 00:45:56.030 OK, so I was going to give you the last element today. 00:45:56.030 --> 00:45:56.830 The last element. 00:45:56.830 --> 00:45:59.050 Remember, we talked about clean air. 00:45:59.050 --> 00:46:02.720 So, I'm showing you the catalytic converter. 00:46:02.720 --> 00:46:05.410 That's this thing here, coated with platinum, 00:46:05.410 --> 00:46:08.300 rhodium, and so on. 00:46:08.300 --> 00:46:11.510 And I'm showing you the electronic control module. 00:46:11.510 --> 00:46:13.330 That's the CPU. 00:46:13.330 --> 00:46:16.400 It's a little bit toned down from a Pentium, but it 00:46:16.400 --> 00:46:19.130 certainly started from one of these units. 00:46:19.130 --> 00:46:22.200 So, we've studied all of this. 00:46:22.200 --> 00:46:23.840 Right? 00:46:23.840 --> 00:46:24.810 OK. 00:46:24.810 --> 00:46:28.240 And, so now, I want to talk about this piece here, the 00:46:28.240 --> 00:46:30.240 oxygen sensor. why do we need an oxygen sensor? 00:46:30.240 --> 00:46:33.750 Remember, the last day I told you that to get rid of NO, the 00:46:33.750 --> 00:46:38.180 NOx, you have to reduce, And to get rid of CO and unburned 00:46:38.180 --> 00:46:40.970 hydrocarbons, you have to oxidize. 00:46:40.970 --> 00:46:42.850 Well, you have to pick. 00:46:42.850 --> 00:46:45.180 You can't have two different atmospheres in the same place 00:46:45.180 --> 00:46:46.250 at the same time. 00:46:46.250 --> 00:46:48.320 Turns out there's a lucky sweet spot. 00:46:48.320 --> 00:46:51.210 Can you see that the conversion efficiency is high 00:46:51.210 --> 00:46:55.200 for both reduction and oxidation reactions if you peg 00:46:55.200 --> 00:46:58.490 the air-to-fuel ratio at 14.6. 00:46:58.490 --> 00:47:00.520 So you can't just set the carburetor or the fuel 00:47:00.520 --> 00:47:03.570 injectors at some arbitrary ratio, because as you change 00:47:03.570 --> 00:47:05.320 humidity, as you change temperature -- what happens 00:47:05.320 --> 00:47:06.790 when the temperature goes down? 00:47:06.790 --> 00:47:08.480 How much oxygen is there in the air? 00:47:08.480 --> 00:47:12.070 It's still 20%, but PV equals nRT. 00:47:12.070 --> 00:47:14.580 You ever try barbecuing when it's -- 00:47:14.580 --> 00:47:16.090 I mean, I don't know, maybe you're not from the Northeast. 00:47:16.090 --> 00:47:19.250 But if you want to barbecue when it's 0 Fahrenheit? 00:47:19.250 --> 00:47:22.230 Do you think you get a hotter fire or a colder fire? 00:47:22.230 --> 00:47:24.640 How much oxygen is there per unit volume in cold 00:47:24.640 --> 00:47:26.570 air versus warm air? 00:47:26.570 --> 00:47:27.730 Think about it. 00:47:27.730 --> 00:47:28.930 It'll be on the next exam. 00:47:28.930 --> 00:47:29.620 I'm just kidding. 00:47:29.620 --> 00:47:29.880 OK. 00:47:29.880 --> 00:47:34.490 So, anyway, here's the sweet spot that gives you both 00:47:34.490 --> 00:47:36.190 oxidation and reduction. 00:47:36.190 --> 00:47:38.290 And so we need a feedback loop. 00:47:38.290 --> 00:47:40.940 That feedback loop is provided by the oxygen sensor. 00:47:40.940 --> 00:47:44.800 Dave, would you mind cutting to the document camera? 00:47:44.800 --> 00:47:49.020 So this is an oxygen sensor that's used on automobiles. 00:47:49.020 --> 00:47:51.000 Here's the plug. 00:47:51.000 --> 00:47:52.750 This goes into the pin set. 00:47:52.750 --> 00:47:55.410 And the oxygen sensor is inside this housing. 00:47:55.410 --> 00:47:56.660 This housing, you can see -- 00:48:00.310 --> 00:48:02.020 OK, what I'm going to show you is what's 00:48:02.020 --> 00:48:03.100 going on inside here. 00:48:03.100 --> 00:48:05.210 This has got all kinds of protective stuff on it so that 00:48:05.210 --> 00:48:07.890 it doesn't get dented during installation and so on. 00:48:07.890 --> 00:48:12.230 But it sits inside the exhaust train. 00:48:12.230 --> 00:48:13.600 It sits inside the exhaust train. 00:48:13.600 --> 00:48:18.030 Now, may we go back to the slides, please? 00:48:18.030 --> 00:48:21.260 Every car that's running a catalytic converter has to 00:48:21.260 --> 00:48:24.980 have one of those in order to control, right here, to make 00:48:24.980 --> 00:48:26.650 sure the air-to-fuel ratio is optimum. 00:48:26.650 --> 00:48:29.370 Otherwise, the catalytic converter is either not doing 00:48:29.370 --> 00:48:32.510 a good job on NOx, or it's not doing a good job on 00:48:32.510 --> 00:48:37.770 hydrocarbons and whatever the other thing is. 00:48:37.770 --> 00:48:39.100 Anyway, so here's what it looks like. 00:48:39.100 --> 00:48:40.310 So, this is what's inside. 00:48:40.310 --> 00:48:43.240 It's a closed, one-end tube made of zirconia. 00:48:43.240 --> 00:48:44.690 Zirconium oxide. 00:48:44.690 --> 00:48:47.840 And zirconium oxide conducts oxide ions. 00:48:47.840 --> 00:48:49.590 And there are platinum electrodes on the 00:48:49.590 --> 00:48:50.280 front and the back. 00:48:50.280 --> 00:48:52.380 And what I showed you was from this side. 00:48:52.380 --> 00:48:54.930 And this is the exhaust gas going along here, and this is 00:48:54.930 --> 00:48:57.580 the edge of your exhaust train. 00:48:57.580 --> 00:49:01.200 And two wires coming off of that sensor go to the CPU. 00:49:01.200 --> 00:49:05.380 The CPU measures voltage and then from there, it regulates 00:49:05.380 --> 00:49:08.040 the air-to-fuel mix to the engine, to the exhaust gas, 00:49:08.040 --> 00:49:09.900 and around and around and around we go, so that when 00:49:09.900 --> 00:49:12.480 exhaust gas gets to the catalytic convert, it is 00:49:12.480 --> 00:49:13.890 optimally being converted. 00:49:17.280 --> 00:49:20.130 So, we want to get fast response. 00:49:20.130 --> 00:49:21.650 We don't want slow response. 00:49:21.650 --> 00:49:24.670 And it's a solid-state sensor and solid-state diffusion is 00:49:24.670 --> 00:49:26.700 slow, but that's we have to rely upon. 00:49:26.700 --> 00:49:29.510 So we add a dopant. 00:49:29.510 --> 00:49:33.020 We add a dopant to zirconia to stabilize its crystal 00:49:33.020 --> 00:49:36.350 structure and create more oxygen vacancies to get faster 00:49:36.350 --> 00:49:38.590 diffusion and a shorter response time. 00:49:38.590 --> 00:49:41.320 So what we add is calcia, one of the candidates, and it 00:49:41.320 --> 00:49:45.660 creates oxygen vacancies, and these oxygen vacancies allow 00:49:45.660 --> 00:49:48.500 us to get more rapid response time. 00:49:48.500 --> 00:49:52.020 So, these oxygen vacancies compensate for the charge 00:49:52.020 --> 00:49:54.090 imbalance, thanks to the addition of calcia. 00:49:54.090 --> 00:49:55.720 So, it's an engineered material. 00:49:55.720 --> 00:49:58.040 Without this it doesn't work, all right? 00:49:58.040 --> 00:50:01.120 And so, now what I've shown you is that on the basis of 00:50:01.120 --> 00:50:05.980 this oxygen sensor, we can monitor the exhaust gas flow 00:50:05.980 --> 00:50:08.380 so as to optimize the conversion here in the 00:50:08.380 --> 00:50:12.870 catalytic converter and do so by sending information to the 00:50:12.870 --> 00:50:14.390 CPU on the car. 00:50:14.390 --> 00:50:18.130 And the number one consumer of CPU's is the automobile 00:50:18.130 --> 00:50:21.440 industry, because every car has at least one, if not 00:50:21.440 --> 00:50:22.950 multiple, CPU's. 00:50:22.950 --> 00:50:26.460 So this is a good example of how understanding the lessons 00:50:26.460 --> 00:50:29.920 of solid state chemistry can allow us to build fuel 00:50:29.920 --> 00:50:35.020 efficient automobiles that have minimum toxic emissions 00:50:35.020 --> 00:50:36.840 to the environment. 00:50:36.840 --> 00:50:39.480 All right, we'll see you on Monday.