1 00:00:00,030 --> 00:00:02,400 The following content is provided under a Creative 2 00:00:02,400 --> 00:00:03,830 Commons license. 3 00:00:03,830 --> 00:00:06,850 Your support will help MIT OpenCourseWare continue to 4 00:00:06,850 --> 00:00:10,510 offer high-quality educational resources for free. 5 00:00:10,510 --> 00:00:13,390 To make a donation or view additional materials from 6 00:00:13,390 --> 00:00:17,490 hundreds of MIT courses, visit MIT OpenCourseWare at 7 00:00:17,490 --> 00:00:18,740 ocw.mit.edu. 8 00:00:21,470 --> 00:00:22,300 PROFESSOR: OK. 9 00:00:22,300 --> 00:00:24,260 OK, settle down. 10 00:00:24,260 --> 00:00:25,710 Settle down. 11 00:00:25,710 --> 00:00:28,860 Let's get started. 12 00:00:28,860 --> 00:00:31,740 So last day we started talking about diffusion. 13 00:00:31,740 --> 00:00:37,540 Diffusion is a solid-state mass transport by random 14 00:00:37,540 --> 00:00:38,420 atomic motion. 15 00:00:38,420 --> 00:00:41,150 I made the first point last day and today I'll make the 16 00:00:41,150 --> 00:00:41,820 second point. 17 00:00:41,820 --> 00:00:45,350 We'll see why it's random atomic motion. 18 00:00:45,350 --> 00:00:47,530 We reasoned that we could describe 19 00:00:47,530 --> 00:00:48,920 the ingress of material. 20 00:00:48,920 --> 00:00:51,655 Here I'm showing a concentration profile. 21 00:00:51,655 --> 00:00:56,310 A profile is nothing more than a plot of something as a 22 00:00:56,310 --> 00:00:57,880 function of position. 23 00:00:57,880 --> 00:01:00,870 So, this is looking at, for example, 24 00:01:00,870 --> 00:01:03,330 the doping of a wafer. 25 00:01:03,330 --> 00:01:05,740 This is the free surface at x equals 0. 26 00:01:05,740 --> 00:01:08,780 As we go to the right, it represents depth. 27 00:01:08,780 --> 00:01:13,080 And we fix the concentration at the surface and the 28 00:01:13,080 --> 00:01:18,470 material tails off to zero deep inside the specimen. 29 00:01:18,470 --> 00:01:23,010 Fick in 1855 annunciated the law of diffusion, shown here, 30 00:01:23,010 --> 00:01:27,590 that says that the rate of ingress expressed by the flux 31 00:01:27,590 --> 00:01:31,740 is proportional to the instant gradient in concentration. 32 00:01:31,740 --> 00:01:33,730 And the constant proportionality is the 33 00:01:33,730 --> 00:01:34,680 diffusion coefficient. 34 00:01:34,680 --> 00:01:36,400 This is where all the material science lies. 35 00:01:36,400 --> 00:01:38,120 It's inside the D. 36 00:01:38,120 --> 00:01:40,450 That's what we're going to talk about more today. 37 00:01:40,450 --> 00:01:44,480 And so we see things tail off and what we want to do is 38 00:01:44,480 --> 00:01:46,905 figure out what the mathematics are that'll allow 39 00:01:46,905 --> 00:01:51,355 us to render this thing into a functional representation. 40 00:01:51,355 --> 00:01:54,410 Because all we know right now is that we've got something 41 00:01:54,410 --> 00:01:55,480 tailing off. 42 00:01:55,480 --> 00:01:56,140 Ok? 43 00:01:56,140 --> 00:01:59,580 And so this is the gradient multiplied by the diffusion 44 00:01:59,580 --> 00:02:02,230 coefficient and you can see that the red line 45 00:02:02,230 --> 00:02:03,850 represents the flux. 46 00:02:03,850 --> 00:02:07,170 The gradient is maximum at the surface, so the flux is 47 00:02:07,170 --> 00:02:08,770 maximum at the surface. 48 00:02:08,770 --> 00:02:12,300 And as we move farther and farther into the piece, the 49 00:02:12,300 --> 00:02:15,680 gradient in concentration becomes less steep, which 50 00:02:15,680 --> 00:02:19,140 means the absolute magnitude of the flux becomes less 51 00:02:19,140 --> 00:02:23,780 steep, and we go deep inside where there is no doping yet. 52 00:02:23,780 --> 00:02:29,920 The variation of concentration is constantly 0, so the slope 53 00:02:29,920 --> 00:02:31,840 is zero, which means the flux is zero. 54 00:02:31,840 --> 00:02:34,380 So the two sort of track, one and the other. 55 00:02:34,380 --> 00:02:36,930 And we recognize these are negative slopes since the 56 00:02:36,930 --> 00:02:38,150 minus sign is here. 57 00:02:38,150 --> 00:02:39,630 Even though we have a negative slope, we 58 00:02:39,630 --> 00:02:41,560 have a positive ingress. 59 00:02:41,560 --> 00:02:44,930 And then the other thing that we realized was that the 60 00:02:44,930 --> 00:02:48,500 diffusion coefficient has a temperature dependence. 61 00:02:48,500 --> 00:02:51,130 And the temperature dependence looks like this. 62 00:02:51,130 --> 00:02:54,280 It's an exponential of some kind of a characteristic 63 00:02:54,280 --> 00:02:57,320 energy divided by the product of -- this is the gas 64 00:02:57,320 --> 00:03:00,050 constant, this is not the Rydberg constant. 65 00:03:00,050 --> 00:03:03,980 The gas constant is simply the Boltzmann constant per mole. 66 00:03:03,980 --> 00:03:06,130 So if you take the Boltzmann constant, multiply it by the 67 00:03:06,130 --> 00:03:08,160 Avogadro number, you get this. 68 00:03:08,160 --> 00:03:11,210 But if you see something times T, you know it's got to be 69 00:03:11,210 --> 00:03:13,880 either gas constant or Boltzmann constant. 70 00:03:13,880 --> 00:03:18,360 This two-letter character string has to have something 71 00:03:18,360 --> 00:03:20,970 akin to a Boltzmann constant, otherwise you don't end up 72 00:03:20,970 --> 00:03:22,250 with an energy term. 73 00:03:22,250 --> 00:03:24,510 And this in the top is a barrier energy. 74 00:03:24,510 --> 00:03:28,160 It's a barrier energy to diffusion and out here we have 75 00:03:28,160 --> 00:03:29,310 the pre-exponential. 76 00:03:29,310 --> 00:03:32,515 Unfortunately we didn't give it the letter F in honor of 77 00:03:32,515 --> 00:03:33,860 Fick, it's just D naught. 78 00:03:33,860 --> 00:03:35,330 And so there it is. 79 00:03:35,330 --> 00:03:39,200 So if we take this equation and we plot it, natural log of 80 00:03:39,200 --> 00:03:43,580 d versus 1 over T, that'll linearize the equation, and we 81 00:03:43,580 --> 00:03:46,130 end up with something that gives us a straight line. 82 00:03:46,130 --> 00:03:51,320 And the slope is minus Q over R, where this Q is some kind 83 00:03:51,320 --> 00:03:52,980 of a barrier energy. 84 00:03:52,980 --> 00:03:57,810 And the R is the gas constant or the Boltzmann constant. 85 00:03:57,810 --> 00:04:02,320 So this is the gas constant. 86 00:04:02,320 --> 00:04:03,960 It's on your table of constants. 87 00:04:03,960 --> 00:04:11,360 It's got a value in SI units of 8.314 joules per mole 88 00:04:11,360 --> 00:04:13,360 Kelvin, OK? 89 00:04:13,360 --> 00:04:16,740 Just the Boltzmann times Avogadro number. 90 00:04:16,740 --> 00:04:20,730 So now I want to invite you to join me in some pattern 91 00:04:20,730 --> 00:04:21,860 recognition. 92 00:04:21,860 --> 00:04:27,510 If I give you the temperature dependence of a quantity that 93 00:04:27,510 --> 00:04:30,850 looks like this , in other words, the logarithm of that 94 00:04:30,850 --> 00:04:34,300 physical parameter versus 1 over T, gives 95 00:04:34,300 --> 00:04:35,700 you a straight line. 96 00:04:35,700 --> 00:04:39,640 You know from the work that we did on chemical kinetics, this 97 00:04:39,640 --> 00:04:42,370 is the representation of something that is 98 00:04:42,370 --> 00:04:43,860 an activated process. 99 00:04:43,860 --> 00:04:46,650 Remember the box that would fall on its side, and it had 100 00:04:46,650 --> 00:04:48,020 to go up on its corner? 101 00:04:48,020 --> 00:04:51,250 So we had some physical sense of what activation is. 102 00:04:51,250 --> 00:04:55,620 Now this has a log, something versus 1 over T dependence. 103 00:04:55,620 --> 00:05:00,550 This represents some kind of an activation energy. 104 00:05:00,550 --> 00:05:01,790 An activation energy. 105 00:05:01,790 --> 00:05:05,810 But what's the activated process that's going on here? 106 00:05:05,810 --> 00:05:09,800 So that's where I want to take you into the atomistics. 107 00:05:09,800 --> 00:05:13,890 So here's a cartoon taken from one of the readings. 108 00:05:13,890 --> 00:05:18,700 And it shows a set of atoms here and the atom next to the 109 00:05:18,700 --> 00:05:21,350 vacancy wants to move into the vacancy. 110 00:05:21,350 --> 00:05:24,130 And you can see the artist's rendition is showing that in 111 00:05:24,130 --> 00:05:27,190 order to get from the current position into the vacant 112 00:05:27,190 --> 00:05:31,300 position, it's got to squeeze through this narrow channel. 113 00:05:31,300 --> 00:05:34,650 And it takes energy to move through that channel. 114 00:05:34,650 --> 00:05:37,010 And there's an activation barrier 115 00:05:37,010 --> 00:05:38,330 associated with that motion. 116 00:05:38,330 --> 00:05:40,040 Can you see when the atom is sitting right 117 00:05:40,040 --> 00:05:41,550 at the saddle point? 118 00:05:41,550 --> 00:05:44,250 The system is highly activated because it's pushing out 119 00:05:44,250 --> 00:05:47,170 against those atoms. And then finally it falls into the new 120 00:05:47,170 --> 00:05:49,090 slot and the energy drops. 121 00:05:49,090 --> 00:05:52,560 So the activation energy is associated with moving through 122 00:05:52,560 --> 00:05:53,760 this saddle point. 123 00:05:53,760 --> 00:05:57,440 So what really happens here, I'll tell you right now. 124 00:05:57,440 --> 00:06:00,950 First of all, we have to recognize that the system is 125 00:06:00,950 --> 00:06:05,030 depicted there, because it's a freeze frame, but we know that 126 00:06:05,030 --> 00:06:08,150 above 0 Kelvin, everything is in motion. 127 00:06:08,150 --> 00:06:12,230 So the only way we can rationalize what's going on is 128 00:06:12,230 --> 00:06:14,310 to, first of all, recognize that we 129 00:06:14,310 --> 00:06:16,960 have a pulsating lattice. 130 00:06:16,960 --> 00:06:18,590 We have a pulsating lattice. 131 00:06:18,590 --> 00:06:20,410 Everything's vibrating. 132 00:06:20,410 --> 00:06:21,750 And what's the heartbeat? 133 00:06:21,750 --> 00:06:22,900 What's the idle speed? 134 00:06:22,900 --> 00:06:24,220 I've told you this before. 135 00:06:24,220 --> 00:06:27,070 Everything that's going on in this room has an idle speed, a 136 00:06:27,070 --> 00:06:31,190 heartbeat, of 10 trillion times a second. 137 00:06:31,190 --> 00:06:33,360 That's the Debye frequency, OK? 138 00:06:33,360 --> 00:06:36,630 The idle speed or the heartbeat. 139 00:06:36,630 --> 00:06:40,140 The heartbeat is called the Debye frequency. 140 00:06:40,140 --> 00:06:45,470 Here's frequency, lowercase nu, in honor of Peter Debye. 141 00:06:45,470 --> 00:06:47,310 And it's on the order of about 10 to the 142 00:06:47,310 --> 00:06:49,270 thirteenth per second. 143 00:06:49,270 --> 00:06:50,720 10 to the thirteenth Hertz. 144 00:06:50,720 --> 00:06:52,670 That's 10 trillion. 145 00:06:52,670 --> 00:06:56,300 So 10 trillion times a second atoms are going like this. 146 00:06:56,300 --> 00:06:57,430 Pulsating. 147 00:06:57,430 --> 00:06:58,380 All right? 148 00:06:58,380 --> 00:07:02,500 So what has to happen for that operation to occur? 149 00:07:02,500 --> 00:07:06,950 Well, I've got an atom here, vacancy next door, but because 150 00:07:06,950 --> 00:07:11,960 it's close packed the atoms on either side close in. 151 00:07:11,960 --> 00:07:14,230 There's no way this can squeeze through. 152 00:07:14,230 --> 00:07:17,360 But, imagine, 10 trillion times a second. 153 00:07:17,360 --> 00:07:21,610 If I'm lucky enough to see the moment when the atom above 154 00:07:21,610 --> 00:07:26,240 pulsates up, the atom below pulsates down, a channel can 155 00:07:26,240 --> 00:07:28,950 open up and then this thing can squirt through. 156 00:07:28,950 --> 00:07:30,930 Now, that doesn't happen every time. 157 00:07:30,930 --> 00:07:33,620 If it did, history would have ended. 158 00:07:33,620 --> 00:07:37,260 At 10 trillion times a second, it's all over for everybody. 159 00:07:37,260 --> 00:07:37,790 Right? 160 00:07:37,790 --> 00:07:39,390 So what happens? 161 00:07:39,390 --> 00:07:42,080 What's the frequency at which we will get this unique 162 00:07:42,080 --> 00:07:43,290 combination? 163 00:07:43,290 --> 00:07:47,050 It turns out that frequency, noted gamma, is called the 164 00:07:47,050 --> 00:07:49,230 jump frequency. 165 00:07:49,230 --> 00:07:52,550 That's the jump frequency. 166 00:07:52,550 --> 00:07:55,260 And the jump frequency is on the order of about 10 to the 167 00:07:55,260 --> 00:07:56,980 eighth Hertz. 168 00:07:56,980 --> 00:08:01,530 10 to the eighth Hertz, which is about 100 million. 169 00:08:01,530 --> 00:08:05,690 So if you take the ratio of the Debye frequency to the 170 00:08:05,690 --> 00:08:10,300 jump frequency, you see you get a value of about one try 171 00:08:10,300 --> 00:08:14,940 in about 10 to the fifth is successful. 172 00:08:14,940 --> 00:08:17,980 One try in about 10 to the fifth is successful. 173 00:08:17,980 --> 00:08:20,320 And that explains what's going on here. 174 00:08:20,320 --> 00:08:24,430 So, what now does that mean in terms of Q? 175 00:08:24,430 --> 00:08:25,900 What's the breakdown of Q? 176 00:08:25,900 --> 00:08:28,940 Well, clearly, Q, the activation energy for 177 00:08:28,940 --> 00:08:33,220 diffusion, must involve both the energy to 178 00:08:33,220 --> 00:08:35,170 form the vacancy -- 179 00:08:35,170 --> 00:08:37,720 I'm going to call it delta Hv, and that's the energy for 180 00:08:37,720 --> 00:08:38,940 vacancy formation -- 181 00:08:38,940 --> 00:08:42,300 if you don't have vacancies, you cannot have motion -- 182 00:08:42,300 --> 00:08:45,750 plus the delta H sub m which is the 183 00:08:45,750 --> 00:08:49,450 energy for atom migration. 184 00:08:49,450 --> 00:08:51,425 So you have to form the vacancies and then you've got 185 00:08:51,425 --> 00:08:53,910 to squeeze through those saddle points. 186 00:08:53,910 --> 00:08:56,330 So that's how we get a sense of it. 187 00:08:56,330 --> 00:08:59,250 And I think I've got another slide. 188 00:08:59,250 --> 00:09:02,340 OK, so they're I've simply put the markings on for you so you 189 00:09:02,340 --> 00:09:05,670 can see the atoms above and below jumping 190 00:09:05,670 --> 00:09:06,540 in the right direction. 191 00:09:06,540 --> 00:09:10,370 That allows the atom on the left to squirt to the right. 192 00:09:10,370 --> 00:09:16,090 Now, this is also related to energetics. 193 00:09:16,090 --> 00:09:19,300 It's related to energetics because the energy to form a 194 00:09:19,300 --> 00:09:22,920 vacancy and the energy to squeeze through those saddle 195 00:09:22,920 --> 00:09:25,450 points must be related to the binding energy. 196 00:09:25,450 --> 00:09:27,880 And what else is related to binding energy? 197 00:09:27,880 --> 00:09:28,770 Melting point. 198 00:09:28,770 --> 00:09:32,790 So, this is an elegant plot that simply shows that the 199 00:09:32,790 --> 00:09:36,540 energy -- these are all FCC metals: lead, aluminum, 200 00:09:36,540 --> 00:09:38,760 silver, gold, copper, and iron. 201 00:09:38,760 --> 00:09:42,380 They're all FCC on this chart here. 202 00:09:42,380 --> 00:09:44,620 Iron has both BCC and FCC, but this is 203 00:09:44,620 --> 00:09:46,600 comparing apples to apples. 204 00:09:46,600 --> 00:09:50,060 And you can see that there's lead melts at 327, aluminum at 205 00:09:50,060 --> 00:09:53,360 660, silver 980, all the way up. 206 00:09:53,360 --> 00:09:54,560 Iron 1535. 207 00:09:54,560 --> 00:09:57,780 And you can see that the activation energy for -- 208 00:09:57,780 --> 00:09:59,300 that's this Q-value -- 209 00:09:59,300 --> 00:10:02,610 the activation energy for diffusion scales 210 00:10:02,610 --> 00:10:03,590 with melting point. 211 00:10:03,590 --> 00:10:07,330 Again, an indication that this analysis makes sense. 212 00:10:07,330 --> 00:10:10,060 And then, here's the other one, the interstitial. 213 00:10:10,060 --> 00:10:12,670 If you have interstitial diffusion you've got plenty of 214 00:10:12,670 --> 00:10:16,260 interstitials, so you don't have to pay the penalty to 215 00:10:16,260 --> 00:10:20,340 create the interstitial, but it's much, much more difficult 216 00:10:20,340 --> 00:10:22,390 to squeeze through the saddle point. 217 00:10:22,390 --> 00:10:25,810 So, it turns out that activation energy for 218 00:10:25,810 --> 00:10:28,100 diffusion by interstitial means is still 219 00:10:28,100 --> 00:10:29,980 fairly highly activated. 220 00:10:29,980 --> 00:10:32,160 So this is in the case of substitutional 221 00:10:32,160 --> 00:10:33,860 systems, all right? 222 00:10:33,860 --> 00:10:37,090 This is for substitutional. 223 00:10:37,090 --> 00:10:40,220 Substitutional atoms. What do I mean by that? 224 00:10:40,220 --> 00:10:42,980 An atom that sits on a lattice site. 225 00:10:42,980 --> 00:10:45,290 If an atom sits on a lattice site, it has to jump to 226 00:10:45,290 --> 00:10:47,510 another lattice site, which means there needs to be a 227 00:10:47,510 --> 00:10:48,960 vacant lattice site. 228 00:10:48,960 --> 00:10:50,660 On the other hand, if we're talking about interstitial 229 00:10:50,660 --> 00:10:52,380 special atoms -- 230 00:10:52,380 --> 00:10:55,720 not substitutional atoms, interstitial atoms -- 231 00:10:55,720 --> 00:10:59,130 the assumption is that there's plenty of interstitial sites. 232 00:10:59,130 --> 00:11:01,890 There are very few systems that I can think of where the 233 00:11:01,890 --> 00:11:04,980 interstitial sites are so heavily occupied that this 234 00:11:04,980 --> 00:11:06,380 assumption is invalid. 235 00:11:06,380 --> 00:11:07,860 So in that case, you don't have to 236 00:11:07,860 --> 00:11:09,920 pay for vacancy formation. 237 00:11:09,920 --> 00:11:13,400 It's simply equal to the enthalpy, which we know is 238 00:11:13,400 --> 00:11:15,310 analogous to the energy for a condensed 239 00:11:15,310 --> 00:11:18,420 system of atom migration. 240 00:11:18,420 --> 00:11:21,440 Atom migration gives you the whole story and it goes back 241 00:11:21,440 --> 00:11:24,480 to the same picture, and so on. 242 00:11:24,480 --> 00:11:27,480 Now, I said it's random walks, so let's look at another one. 243 00:11:27,480 --> 00:11:30,400 So, this is an interesting example. 244 00:11:30,400 --> 00:11:32,890 This is self-diffusion in cobalt. 245 00:11:32,890 --> 00:11:35,120 Cobalt has a number of isotopes. 246 00:11:35,120 --> 00:11:37,780 One of them is 59 and one of them is 60. 247 00:11:37,780 --> 00:11:40,380 60 is a radioisotope. 248 00:11:40,380 --> 00:11:45,400 Cobalt 60 is a radioisotope and it's used in a variety of 249 00:11:45,400 --> 00:11:46,670 technical endeavors. 250 00:11:46,670 --> 00:11:50,600 And so what we've got here is a sandwich of cobalt 59 -- 251 00:11:50,600 --> 00:11:53,310 cobalt 60 and cobalt 59. 252 00:11:53,310 --> 00:11:57,320 So even though cobalt 60 has radioactivity and it's got a 253 00:11:57,320 --> 00:12:01,670 different number of neutrons in the nucleus, chemically it 254 00:12:01,670 --> 00:12:05,750 is identical to cobalt 59, just as carbon 14 is 255 00:12:05,750 --> 00:12:07,850 chemically identical to carbon 12. 256 00:12:07,850 --> 00:12:10,800 So there's no concentration gradient. 257 00:12:10,800 --> 00:12:12,830 Fick's Law says it's supposed to go by 258 00:12:12,830 --> 00:12:13,860 concentration gradient. 259 00:12:13,860 --> 00:12:17,400 The concentration gradient is zero across that piece. 260 00:12:17,400 --> 00:12:20,540 And yet if you wait for long enough, you will 261 00:12:20,540 --> 00:12:21,570 start to see -- 262 00:12:21,570 --> 00:12:25,970 the dark atoms here are meant to represent cobalt 60 in the 263 00:12:25,970 --> 00:12:26,850 sandwich -- 264 00:12:26,850 --> 00:12:31,560 after some period of time, the cobalt 60's will move outward. 265 00:12:31,560 --> 00:12:34,980 And if you wait long enough, the cobalt 60 concentration 266 00:12:34,980 --> 00:12:38,530 will be uniform throughout the specimen. 267 00:12:38,530 --> 00:12:39,930 So what's going on? 268 00:12:39,930 --> 00:12:42,390 There's no chemical concentration gradient, and 269 00:12:42,390 --> 00:12:45,730 yet over time, the cobalt 60 spreads out. 270 00:12:45,730 --> 00:12:48,530 And the answer is this is happening by random walk. 271 00:12:48,530 --> 00:12:51,030 We've got the pulsating lattice. 272 00:12:51,030 --> 00:12:52,710 Pulsating lattice. 273 00:12:52,710 --> 00:12:55,220 10 trillion times a second those atoms are vibrating and 274 00:12:55,220 --> 00:12:56,440 there's going to be some vacancy. 275 00:12:56,440 --> 00:12:59,050 All of a sudden an atom falls into a vacancy. 276 00:12:59,050 --> 00:13:01,610 And the mathematics of this -- look at this, this is no 277 00:13:01,610 --> 00:13:04,570 accident -- this curve here represents something that 278 00:13:04,570 --> 00:13:06,760 looks like a Gaussian distribution. 279 00:13:06,760 --> 00:13:10,230 At any given time, the amount of cobalt 60 that's veered 280 00:13:10,230 --> 00:13:14,210 from the center band can be portrayed by the Gaussian 281 00:13:14,210 --> 00:13:16,020 distribution, the bell curve. 282 00:13:16,020 --> 00:13:18,450 And what's the physical model for the bell curve? 283 00:13:18,450 --> 00:13:20,110 It's the drunken sailor. 284 00:13:20,110 --> 00:13:21,530 Or I guess I have to be PC. 285 00:13:21,530 --> 00:13:23,370 I'm not going to just pick on sailors. 286 00:13:23,370 --> 00:13:25,650 Maybe it's the drunken cougar, the soccer mom 287 00:13:25,650 --> 00:13:26,460 that went out drinking. 288 00:13:26,460 --> 00:13:31,110 OK, so what happens with the drunken cougar, she comes to 289 00:13:31,110 --> 00:13:34,950 the door, and she starts to walk, all right? 290 00:13:34,950 --> 00:13:38,200 And then she walks one way or the other way, until she 291 00:13:38,200 --> 00:13:40,200 finally falls. 292 00:13:40,200 --> 00:13:42,510 And if you go through the number of steps that it'll 293 00:13:42,510 --> 00:13:46,310 take before the drunken sailor falls, you get this 294 00:13:46,310 --> 00:13:47,120 distribution. 295 00:13:47,120 --> 00:13:48,980 It's an old physics problem. 296 00:13:48,980 --> 00:13:50,540 The drunken sailor problem. 297 00:13:50,540 --> 00:13:52,880 And it's a normal distribution. 298 00:13:52,880 --> 00:13:55,930 And that normal distribution tells you that everything's 299 00:13:55,930 --> 00:13:58,730 going about by random jumping. 300 00:13:58,730 --> 00:14:02,740 Random jumping, according to this, all right? 301 00:14:02,740 --> 00:14:05,450 So that's why we say that diffusion is 302 00:14:05,450 --> 00:14:07,150 a random walk problem. 303 00:14:07,150 --> 00:14:10,110 Before I turn this off, I want you to take a look carefully. 304 00:14:10,110 --> 00:14:13,345 What's wrong with this textbook cartoon? 305 00:14:17,770 --> 00:14:19,480 Any ideas? 306 00:14:19,480 --> 00:14:21,260 It's a substitutional system, right? 307 00:14:21,260 --> 00:14:23,250 It's cobalt and cobalt. 308 00:14:23,250 --> 00:14:25,860 This is absolutely impossible. 309 00:14:25,860 --> 00:14:27,840 What's wrong with it? 310 00:14:27,840 --> 00:14:29,450 There are no vacancies. 311 00:14:29,450 --> 00:14:30,580 There are no vacancies. 312 00:14:30,580 --> 00:14:32,020 It's impossible. 313 00:14:32,020 --> 00:14:33,460 There's no vacancies to jump into. 314 00:14:33,460 --> 00:14:36,980 This thing is jam-packed, so it's physically unrealistic 315 00:14:36,980 --> 00:14:38,390 from a thermodynamic standpoint. 316 00:14:38,390 --> 00:14:41,460 There has to be some finite population of vacancies. 317 00:14:41,460 --> 00:14:45,320 Given this system, there's no way that the black atom could 318 00:14:45,320 --> 00:14:47,860 jump into the neighboring site, because you'd have to 319 00:14:47,860 --> 00:14:52,700 have the entire system open up, and the chances of that 320 00:14:52,700 --> 00:14:54,170 happening are vanishingly small. 321 00:14:54,170 --> 00:14:57,340 So it just shows you, just because something costs 150 322 00:14:57,340 --> 00:15:00,120 bucks and is pressed between two hard covers doesn't mean 323 00:15:00,120 --> 00:15:01,160 it's full of truth, OK? 324 00:15:01,160 --> 00:15:03,430 Be careful here when you look at this. 325 00:15:03,430 --> 00:15:05,280 So, again, the artist needed to be guided. 326 00:15:05,280 --> 00:15:07,610 We need to put a few of these things out, just to give 327 00:15:07,610 --> 00:15:09,920 symbolic recognition to the fact that we need 328 00:15:09,920 --> 00:15:12,220 to have some vacancies. 329 00:15:12,220 --> 00:15:12,665 OK. 330 00:15:12,665 --> 00:15:15,150 All right, let's keep going. 331 00:15:15,150 --> 00:15:15,670 Let's keep going. 332 00:15:15,670 --> 00:15:16,670 Here is some data. 333 00:15:16,670 --> 00:15:17,670 Here is some data. 334 00:15:17,670 --> 00:15:18,410 Various data. 335 00:15:18,410 --> 00:15:19,490 This is diffusion. 336 00:15:19,490 --> 00:15:22,280 This is the logarithm of the diffusion coefficient 337 00:15:22,280 --> 00:15:24,050 versus 1 over T. 338 00:15:24,050 --> 00:15:25,050 So let's look at the top. 339 00:15:25,050 --> 00:15:27,430 This is hydrogen in iron. 340 00:15:27,430 --> 00:15:29,940 So temperature increases from right to left. 341 00:15:29,940 --> 00:15:31,320 So, go up, up, up, up, up. 342 00:15:31,320 --> 00:15:36,200 And roughly 900 degrees Centigrade, iron changes from 343 00:15:36,200 --> 00:15:38,040 BCC to FCC. 344 00:15:38,040 --> 00:15:40,790 And you know from our previous unit, the number of nearest 345 00:15:40,790 --> 00:15:44,110 neighbors in FCC is greater, so there's less 346 00:15:44,110 --> 00:15:46,110 void fraction, right? 347 00:15:46,110 --> 00:15:49,840 So that means that it's much more difficult to move. 348 00:15:49,840 --> 00:15:54,160 And you can see, first of all, the order of magnitude drop in 349 00:15:54,160 --> 00:15:56,860 the diffusion coefficient, and also, it's subtle, but the 350 00:15:56,860 --> 00:15:59,660 slope is steeper, because the activation energy for 351 00:15:59,660 --> 00:16:02,600 diffusion is steeper in a closer-packed system. 352 00:16:02,600 --> 00:16:04,570 This is carbon in iron. 353 00:16:04,570 --> 00:16:05,880 Same thing. 354 00:16:05,880 --> 00:16:09,560 Relatively gentle slope at the same temperature, there's a 355 00:16:09,560 --> 00:16:14,190 phase change to FCC and it takes much, much more energy 356 00:16:14,190 --> 00:16:16,870 per unit increase in temperature to get the 357 00:16:16,870 --> 00:16:19,265 comparable increase in the value of diffusion 358 00:16:19,265 --> 00:16:19,680 coefficient. 359 00:16:19,680 --> 00:16:21,950 This is iron, self-diffusion in iron. 360 00:16:21,950 --> 00:16:24,680 So this would be radio tracer iron in iron. 361 00:16:24,680 --> 00:16:25,620 Very much lower. 362 00:16:25,620 --> 00:16:28,640 You see these values are up around 10 to the minus 5, 10 363 00:16:28,640 --> 00:16:29,310 to the minus 6. 364 00:16:29,310 --> 00:16:32,340 This is down to 10 to the minus 11, 10 to the minus 12 365 00:16:32,340 --> 00:16:33,950 centimeters squared per second. 366 00:16:33,950 --> 00:16:38,180 Iron diffusion changes at the transition temperature. 367 00:16:38,180 --> 00:16:40,480 Now iron going through FCC iron. 368 00:16:40,480 --> 00:16:41,330 Here's the last one. 369 00:16:41,330 --> 00:16:41,680 Look at this one. 370 00:16:41,680 --> 00:16:43,790 This is carbon in graphite. 371 00:16:43,790 --> 00:16:45,620 So how does carbon -- suppose I had some 372 00:16:45,620 --> 00:16:48,410 carbon 14 in graphite. 373 00:16:48,410 --> 00:16:49,670 What's the bonding in graphite? 374 00:16:55,100 --> 00:16:56,180 Silence. 375 00:16:56,180 --> 00:16:57,310 It's covalent. 376 00:16:57,310 --> 00:16:58,740 It's sp2 hybridized. 377 00:16:58,740 --> 00:17:00,860 It's really strong bonds. 378 00:17:00,860 --> 00:17:02,440 The activation energy -- 379 00:17:02,440 --> 00:17:04,900 you know, forget just jumping through that saddle point -- 380 00:17:04,900 --> 00:17:06,550 you've got to break those covalent bonds. 381 00:17:06,550 --> 00:17:09,140 It's very, very high energy to get carbon to 382 00:17:09,140 --> 00:17:09,950 move through graphite. 383 00:17:09,950 --> 00:17:12,780 And so you see very, very low values and very, very steep 384 00:17:12,780 --> 00:17:14,030 temperature dependence. 385 00:17:16,190 --> 00:17:18,320 Now, there's another way to think about this. 386 00:17:18,320 --> 00:17:20,980 Again, this is now looking at the atomistics. 387 00:17:20,980 --> 00:17:25,125 So this is the same material trying to indicate that if you 388 00:17:25,125 --> 00:17:29,310 use Fick's Law, moving in from the surface into the bulk, we 389 00:17:29,310 --> 00:17:30,410 have a certain front. 390 00:17:30,410 --> 00:17:33,480 So this is some isoconcentration line. 391 00:17:33,480 --> 00:17:39,300 So we could take this value off of the graph arbitrarily. 392 00:17:39,300 --> 00:17:41,820 So they're saying c greater than c i. 393 00:17:41,820 --> 00:17:43,620 Because it trails off to zero, right? 394 00:17:43,620 --> 00:17:44,590 It's asymptotic. 395 00:17:44,590 --> 00:17:46,450 There's no sharp cut-off here. 396 00:17:46,450 --> 00:17:47,560 It trails off. 397 00:17:47,560 --> 00:17:50,550 So what they're doing in that graph is arbitrarily saying, 398 00:17:50,550 --> 00:17:53,200 let's call this ci. 399 00:17:53,200 --> 00:17:57,330 And where's the front at concentration ci, given a 400 00:17:57,330 --> 00:18:00,050 constant value of cs at the surface? 401 00:18:00,050 --> 00:18:03,900 And you can see that there's a near-constant rate of advance 402 00:18:03,900 --> 00:18:04,960 through the bulk. 403 00:18:04,960 --> 00:18:07,590 But along the grain boundary you go much, much farther in 404 00:18:07,590 --> 00:18:08,830 the same amount of time. 405 00:18:08,830 --> 00:18:09,730 Why? 406 00:18:09,730 --> 00:18:12,770 Because the atoms in the grain boundary are not bounded the 407 00:18:12,770 --> 00:18:14,500 same way as the atoms in the bulk. 408 00:18:14,500 --> 00:18:17,080 There's this gap, this misalignment. 409 00:18:17,080 --> 00:18:20,585 And so, it's easier to make that jump, and you can see you 410 00:18:20,585 --> 00:18:22,410 get much, much more advancement 411 00:18:22,410 --> 00:18:23,790 along the grain boundary. 412 00:18:23,790 --> 00:18:25,110 And this is a crack. 413 00:18:25,110 --> 00:18:27,690 And this material covers the surface. 414 00:18:27,690 --> 00:18:30,030 You get surface diffusion. 415 00:18:30,030 --> 00:18:34,140 Surface diffusion is very important in many processes, 416 00:18:34,140 --> 00:18:37,120 and it's very fast, because on a surface, you've got no atoms 417 00:18:37,120 --> 00:18:38,640 on the top. 418 00:18:38,640 --> 00:18:40,770 It's almost liquid-like, isn't it? 419 00:18:40,770 --> 00:18:43,300 You've got atoms to the bottom, but you've got no 420 00:18:43,300 --> 00:18:47,110 atoms to the top, so you can move unconstrained. 421 00:18:47,110 --> 00:18:48,360 Go back to this one. 422 00:18:51,320 --> 00:18:52,710 Imagine here. 423 00:18:52,710 --> 00:18:55,820 Imagine if you didn't have any atoms on the top, how easy it 424 00:18:55,820 --> 00:18:57,695 would be to make that jump. 425 00:18:57,695 --> 00:19:00,660 All right, let's go and see what the data are. 426 00:19:00,660 --> 00:19:02,930 Nothing like data. 427 00:19:02,930 --> 00:19:04,380 And I got data for you. 428 00:19:04,380 --> 00:19:05,780 Let's look. 429 00:19:05,780 --> 00:19:07,030 Click, click, click. 430 00:19:09,540 --> 00:19:10,280 All right. 431 00:19:10,280 --> 00:19:11,280 Here's the data. 432 00:19:11,280 --> 00:19:14,220 This is all for the diffusion of silver. 433 00:19:14,220 --> 00:19:19,110 So I'm going to use this value, this schematic, and I'm 434 00:19:19,110 --> 00:19:21,370 going to give you the value for silver diffusion in the 435 00:19:21,370 --> 00:19:25,240 bulk, silver diffusion in the grain boundary, and silver 436 00:19:25,240 --> 00:19:26,890 diffusion along the surface. 437 00:19:26,890 --> 00:19:28,090 So these are the data. 438 00:19:28,090 --> 00:19:31,220 So this is log diffusion coefficient versus 1 over T. 439 00:19:31,220 --> 00:19:33,140 Only, I hate this graph, because 440 00:19:33,140 --> 00:19:35,340 somebody got really wimpy. 441 00:19:35,340 --> 00:19:36,210 You know, it bothers them. 442 00:19:36,210 --> 00:19:39,000 You see, when you plot something like this, you end 443 00:19:39,000 --> 00:19:44,030 up with, if 1 over T increases from left to right, this 444 00:19:44,030 --> 00:19:45,960 really means that high temperature 445 00:19:45,960 --> 00:19:47,250 is over here, right? 446 00:19:47,250 --> 00:19:50,430 1 over T increases this way, which means temperature is 447 00:19:50,430 --> 00:19:51,890 increasing this way. 448 00:19:51,890 --> 00:19:55,090 So somebody, some weenie, got all nervous here and put the 449 00:19:55,090 --> 00:19:57,070 numbers backwards, you see? 450 00:19:57,070 --> 00:20:00,280 That way, the high temperature is over here, but it looks 451 00:20:00,280 --> 00:20:03,150 stupid, because when I see an activation plot, I want a 452 00:20:03,150 --> 00:20:03,930 negative slope. 453 00:20:03,930 --> 00:20:06,450 It's non-physical to have a positive slope. 454 00:20:06,450 --> 00:20:09,090 So I condemned this and I turned it around. 455 00:20:09,090 --> 00:20:12,900 So now let's look at it the right way. 456 00:20:12,900 --> 00:20:17,360 So this is log d versus 1 over T, and this is the bulk, 457 00:20:17,360 --> 00:20:18,640 lattice or bulk. 458 00:20:18,640 --> 00:20:22,010 This is grain boundary and this is surface diffusion, all 459 00:20:22,010 --> 00:20:24,100 in solid silver. 460 00:20:24,100 --> 00:20:25,620 OK, look at this. 461 00:20:25,620 --> 00:20:28,700 First of all -- 462 00:20:28,700 --> 00:20:31,090 I'm going to jump from here -- one, two, three, four. 463 00:20:31,090 --> 00:20:34,970 Four orders of magnitude faster diffusion along a grain 464 00:20:34,970 --> 00:20:37,700 boundary, which makes sense, because it's less constrained. 465 00:20:37,700 --> 00:20:40,100 And now let's go from grain boundary at constant 466 00:20:40,100 --> 00:20:41,930 temperature up to surface. 467 00:20:41,930 --> 00:20:43,960 One, two, three. 468 00:20:43,960 --> 00:20:47,140 Three orders of magnitude along the surface. 469 00:20:47,140 --> 00:20:49,920 So you can see the effects. 470 00:20:49,920 --> 00:20:53,750 And there's the melting point of silver, right here. 471 00:20:53,750 --> 00:20:56,070 So at the melting point of silver, the diffusion 472 00:20:56,070 --> 00:20:58,650 coefficient is about 10 to the minus 8 centimeters squared 473 00:20:58,650 --> 00:21:01,220 per second, which is the diffusion coefficient of 474 00:21:01,220 --> 00:21:03,600 virtually every metal at its melting point. 475 00:21:03,600 --> 00:21:04,930 So I happen to know that number. 476 00:21:04,930 --> 00:21:07,640 10 to the minus 8 centimeters squared per second. 477 00:21:07,640 --> 00:21:09,840 Not that I know it as a function of temperature, but I 478 00:21:09,840 --> 00:21:11,450 know it can't be any faster than that. 479 00:21:14,390 --> 00:21:17,610 So I said, this is near liquid-like behavior, what's 480 00:21:17,610 --> 00:21:19,000 the next thing to look for? 481 00:21:19,000 --> 00:21:21,910 Let's get some data from real liquids and see if this hunch 482 00:21:21,910 --> 00:21:22,580 is correct. 483 00:21:22,580 --> 00:21:23,850 What's the number here? 484 00:21:23,850 --> 00:21:26,660 10 to the minus 5 to 10 to the minus 4 centimeters squared 485 00:21:26,660 --> 00:21:27,480 per second. 486 00:21:27,480 --> 00:21:29,980 This is diffusion in molten ferrous alloys. 487 00:21:29,980 --> 00:21:32,430 This is manganese in molten iron. 488 00:21:32,430 --> 00:21:34,800 10 to the minus 5, 10 to the minus 4. 489 00:21:34,800 --> 00:21:36,100 Bingo. 490 00:21:36,100 --> 00:21:39,410 It really does behave like a liquid, in terms of diffusion. 491 00:21:39,410 --> 00:21:43,520 Surface diffusion is very fast, very fast. And it's all 492 00:21:43,520 --> 00:21:47,980 explained by this simple model of atom jumping. 493 00:21:47,980 --> 00:21:48,760 And I continued it. 494 00:21:48,760 --> 00:21:50,210 Here's data from glasses. 495 00:21:50,210 --> 00:21:51,550 We studied glasses. 496 00:21:51,550 --> 00:21:53,440 So here's a suite of different glasses. 497 00:21:53,440 --> 00:21:54,610 Log k. 498 00:21:54,610 --> 00:21:57,080 This is permeability, which is related to diffusion 499 00:21:57,080 --> 00:21:57,780 coefficient. 500 00:21:57,780 --> 00:21:59,850 There's a gas constant in there, but forget about it. 501 00:21:59,850 --> 00:22:02,920 So this is essentially log of the diffusion coefficient 502 00:22:02,920 --> 00:22:04,370 versus 1 over T. 503 00:22:04,370 --> 00:22:05,970 We've got a family of lines. 504 00:22:05,970 --> 00:22:07,530 And what's the difference here? 505 00:22:07,530 --> 00:22:09,630 This is pure fused silica up here. 506 00:22:09,630 --> 00:22:10,960 SiO2. 507 00:22:10,960 --> 00:22:13,970 There's borosilicate, there's soda lime, and 508 00:22:13,970 --> 00:22:15,070 there's lead borate. 509 00:22:15,070 --> 00:22:15,900 So what's happening? 510 00:22:15,900 --> 00:22:18,820 We're adding more and more modifier as we 511 00:22:18,820 --> 00:22:20,810 go from top to bottom. 512 00:22:20,810 --> 00:22:24,230 And as we add modifier, we break the length of the 513 00:22:24,230 --> 00:22:27,990 silicate network, which means it packs tighter and tighter, 514 00:22:27,990 --> 00:22:30,650 and as you get tighter and tighter packing, the diffusion 515 00:22:30,650 --> 00:22:32,170 coefficient gets lower and lower. 516 00:22:32,170 --> 00:22:33,050 So train your eye. 517 00:22:33,050 --> 00:22:35,920 Let's pick 200 degrees Centigrade, right here. 518 00:22:35,920 --> 00:22:40,230 So at constant temperature, the more modifier you put in, 519 00:22:40,230 --> 00:22:43,050 the lower the diffusion coefficient because things are 520 00:22:43,050 --> 00:22:44,330 more tightly packed. 521 00:22:44,330 --> 00:22:47,090 It's like walking down the infinite corridor in the 522 00:22:47,090 --> 00:22:50,370 middle of the hour, versus walking down on the hour. 523 00:22:50,370 --> 00:22:52,340 On the hour, there's too many people. 524 00:22:52,340 --> 00:22:53,610 It's tighter packing. 525 00:22:53,610 --> 00:22:55,590 And if you're the least bit civilized, you're going to 526 00:22:55,590 --> 00:22:58,380 have to move a little bit slower, whereas, if you walk 527 00:22:58,380 --> 00:23:02,270 down the infinite at around now, no problem. 528 00:23:02,270 --> 00:23:05,000 It's liquid-like behavior. 529 00:23:05,000 --> 00:23:07,670 And now this is the isotherm at 300 hundred degrees C. 530 00:23:07,670 --> 00:23:10,090 This is from some old Corning literature. 531 00:23:10,090 --> 00:23:12,060 Corning used to make glass, at one time. 532 00:23:12,060 --> 00:23:15,200 And this is the logarithm of diffusivity. 533 00:23:15,200 --> 00:23:21,030 And this is now the set of silica B203 and P205, and 534 00:23:21,030 --> 00:23:23,270 these are all network formers, right? 535 00:23:23,270 --> 00:23:25,280 They all form covalent bonds. 536 00:23:25,280 --> 00:23:30,300 So the more silicate, borate, phosphate you put in, the more 537 00:23:30,300 --> 00:23:33,360 stretched out becomes the network. 538 00:23:33,360 --> 00:23:34,570 And sure enough, look up here. 539 00:23:34,570 --> 00:23:37,890 See they didn't have names for their glasses like the Malibu 540 00:23:37,890 --> 00:23:39,390 or the Corvette or something. 541 00:23:39,390 --> 00:23:41,950 They called their glasses by four-digit numbers. 542 00:23:41,950 --> 00:23:45,560 So someone would say, oh, this customer needs, oh, I'd sell 543 00:23:45,560 --> 00:23:46,385 them a 7040. 544 00:23:46,385 --> 00:23:46,990 All right? 545 00:23:46,990 --> 00:23:49,030 Because it has a certain mix of properties. 546 00:23:49,030 --> 00:23:51,780 So forget about what those numbers are, just know that 547 00:23:51,780 --> 00:23:55,540 this axis indicates that as you move up, up, up, up, up, 548 00:23:55,540 --> 00:23:59,200 the higher diffusivity is associated with a much more 549 00:23:59,200 --> 00:24:00,740 open structure. 550 00:24:00,740 --> 00:24:04,270 So, again, a link between atomistic behavior and 551 00:24:04,270 --> 00:24:05,690 atomistic structure. 552 00:24:05,690 --> 00:24:08,560 It's all there. 553 00:24:08,560 --> 00:24:09,870 OK. 554 00:24:09,870 --> 00:24:12,780 So let's leave that up there while we go 555 00:24:12,780 --> 00:24:13,640 a little bit forward. 556 00:24:13,640 --> 00:24:17,340 OK, so now, so we've been down buried at the atomistic level. 557 00:24:17,340 --> 00:24:20,050 Now let's jump up to the continuum level. 558 00:24:20,050 --> 00:24:22,610 We'll go back to Fick's first law. 559 00:24:22,610 --> 00:24:25,670 And I want to do something a little more quantitative. 560 00:24:25,670 --> 00:24:30,200 So I want to look at diffusion across a permeable membrane. 561 00:24:30,200 --> 00:24:35,250 So this is gas through a membrane. 562 00:24:35,250 --> 00:24:36,600 Gas through a membrane. 563 00:24:36,600 --> 00:24:38,230 And the membrane is shown here. 564 00:24:38,230 --> 00:24:40,190 This is the walls of the membrane. 565 00:24:40,190 --> 00:24:42,980 Membrane has thickness L. 566 00:24:42,980 --> 00:24:44,770 Membrane of thickness L. 567 00:24:44,770 --> 00:24:45,630 In fact, we'll give it a name. 568 00:24:45,630 --> 00:24:48,565 Let's call it membrane. 569 00:24:48,565 --> 00:24:52,280 All right, so here's the membrane of thickness L, and 570 00:24:52,280 --> 00:24:55,720 I'm going to put gas on the left at P1, and it's a 571 00:24:55,720 --> 00:24:56,850 constant P1. 572 00:24:56,850 --> 00:24:57,960 So we keep -- 573 00:24:57,960 --> 00:25:00,300 we've got a ballast here, so even though something starts 574 00:25:00,300 --> 00:25:03,340 diffusing, the amount that we lose doesn't affect the 575 00:25:03,340 --> 00:25:05,020 constant pressure over here. 576 00:25:05,020 --> 00:25:09,440 And on the right side, I'm going to put a second 577 00:25:09,440 --> 00:25:10,890 pressure, P2. 578 00:25:10,890 --> 00:25:15,120 And just for argument's sake, I'm going to say that P2 is 579 00:25:15,120 --> 00:25:16,210 less than P1. 580 00:25:16,210 --> 00:25:19,470 So that means we're going to end up with mass transport 581 00:25:19,470 --> 00:25:21,050 from left to right. 582 00:25:21,050 --> 00:25:25,690 And so, I want to ask, what's this going to look like if we 583 00:25:25,690 --> 00:25:27,010 use Fick's Law? 584 00:25:27,010 --> 00:25:31,700 And so beneath I'll make a plot, a concentration profile, 585 00:25:31,700 --> 00:25:35,230 from x equals 0 to x equals L. 586 00:25:35,230 --> 00:25:39,670 And I can morph this into concentration. 587 00:25:39,670 --> 00:25:43,860 We know the gas law is PV equals nRT. 588 00:25:43,860 --> 00:25:45,320 This is the gas law. 589 00:25:45,320 --> 00:25:47,750 P is pressure. 590 00:25:47,750 --> 00:25:49,460 P is pressure. 591 00:25:49,460 --> 00:25:51,570 V you know is volume. 592 00:25:51,570 --> 00:25:53,200 But I'm going to write all these out, because some of 593 00:25:53,200 --> 00:25:55,730 these letters are used so many times that we don't know, is 594 00:25:55,730 --> 00:25:56,990 this the Rydberg constant? 595 00:25:56,990 --> 00:26:00,000 No, it's the gas constant that you've seen 596 00:26:00,000 --> 00:26:00,920 on the first board. 597 00:26:00,920 --> 00:26:04,690 It's the Boltzmann constant times the Avogadro number. 598 00:26:04,690 --> 00:26:08,380 T is absolute temperature, temperature in Kelvins. 599 00:26:08,380 --> 00:26:13,060 And n is not the quantum number, it's mole number. 600 00:26:13,060 --> 00:26:14,520 Mole number. 601 00:26:14,520 --> 00:26:16,500 PV equals nRT. 602 00:26:16,500 --> 00:26:22,710 And we know that concentration of i is mole 603 00:26:22,710 --> 00:26:24,270 number of i over V. 604 00:26:24,270 --> 00:26:26,040 Both of those appear in here. 605 00:26:26,040 --> 00:26:28,680 So if you take what's underneath there, and I'll 606 00:26:28,680 --> 00:26:32,210 expose it in a second, you'll convince yourself that Ci 607 00:26:32,210 --> 00:26:35,920 equals Pi divided by RT. 608 00:26:35,920 --> 00:26:38,820 And that's how you get that permeability thing to turn 609 00:26:38,820 --> 00:26:40,290 around as well. 610 00:26:40,290 --> 00:26:46,320 OK, so, I started with gas pressure, but I'm now going to 611 00:26:46,320 --> 00:26:48,070 write this in terms of concentration. 612 00:26:48,070 --> 00:26:55,030 So I'm going to morph P1 into C1 and over here, P2 into C2. 613 00:26:55,030 --> 00:26:57,120 And I want to plot the profile. 614 00:26:57,120 --> 00:26:58,930 So what's the profile look like? 615 00:26:58,930 --> 00:27:02,120 I'm going to plot the profile at steady state. 616 00:27:02,120 --> 00:27:04,020 At steady state. 617 00:27:04,020 --> 00:27:06,650 At steady state, after I've waited a while, I have a 618 00:27:06,650 --> 00:27:10,320 straight-line profile across here, at steady state. 619 00:27:10,320 --> 00:27:12,370 It's pinned at the two ends. 620 00:27:12,370 --> 00:27:14,240 It's pinned at the two ends and varies 621 00:27:14,240 --> 00:27:16,100 continuously across here. 622 00:27:16,100 --> 00:27:22,440 And you can see from Fick's Law, that if you have a 623 00:27:22,440 --> 00:27:29,270 situation in which the gradient is invariant across 624 00:27:29,270 --> 00:27:32,250 the piece, the gradient is -- 625 00:27:32,250 --> 00:27:37,190 dc by dx is the same here, as it is here, as it is here, as 626 00:27:37,190 --> 00:27:44,180 it is here. dc by dx is invariant, right? 627 00:27:44,180 --> 00:27:45,020 It doesn't change. 628 00:27:45,020 --> 00:27:46,330 This is a straight line. 629 00:27:46,330 --> 00:27:49,220 Well, if dc by dx is invariant, and d is a 630 00:27:49,220 --> 00:27:52,240 constant, then J is invariant, which just means 631 00:27:52,240 --> 00:27:53,740 no sources are sinks. 632 00:27:53,740 --> 00:27:56,590 All of the material that goes in, must come out. 633 00:27:56,590 --> 00:28:01,450 This means that J is not a function of time. 634 00:28:01,450 --> 00:28:02,700 J is invariant. 635 00:28:05,500 --> 00:28:08,220 J is not a function of time. 636 00:28:08,220 --> 00:28:09,960 I come back here ten minutes later, I 637 00:28:09,960 --> 00:28:11,100 have the same profile. 638 00:28:11,100 --> 00:28:13,360 That's what steady state means. 639 00:28:13,360 --> 00:28:16,450 That's what study state means. 640 00:28:16,450 --> 00:28:18,920 OK, and then, of course, if I wanted to be a little bit 641 00:28:18,920 --> 00:28:21,890 pedantic, I could say, in certain systems, the diffusion 642 00:28:21,890 --> 00:28:24,720 coefficient is a function of concentration. 643 00:28:24,720 --> 00:28:27,630 So, when the diffusion coefficient is a function of 644 00:28:27,630 --> 00:28:30,670 concentration, then even at steady state, you might have 645 00:28:30,670 --> 00:28:32,900 some bowing of the profile, but that's a 646 00:28:32,900 --> 00:28:34,710 little bit more advanced. 647 00:28:34,710 --> 00:28:35,620 OK. 648 00:28:35,620 --> 00:28:38,900 So, I can describe everything that's going on here. 649 00:28:38,900 --> 00:28:44,130 If I wanted to know how much material leaves from inside 650 00:28:44,130 --> 00:28:48,290 over a certain period of time, then I can simply say that 651 00:28:48,290 --> 00:28:55,120 total material lost is given by Fick's first law when you 652 00:28:55,120 --> 00:28:56,490 have steady state. 653 00:28:56,490 --> 00:29:00,040 It's simply going to equal the product of the flux, which is 654 00:29:00,040 --> 00:29:03,240 mass per unit area per unit time times the 655 00:29:03,240 --> 00:29:06,450 area times the time. 656 00:29:06,450 --> 00:29:10,940 And that's sort of analogous to Faraday's Law, right? 657 00:29:10,940 --> 00:29:17,530 If I wanted to look at how much electrical charge I get 658 00:29:17,530 --> 00:29:21,120 for a certain period of time with current passing, I could 659 00:29:21,120 --> 00:29:24,000 say that the total charge is the product of the current 660 00:29:24,000 --> 00:29:24,860 times the time. 661 00:29:24,860 --> 00:29:26,350 You know this relationship. 662 00:29:26,350 --> 00:29:30,970 Well, can you see that J times A is analogous to the current? 663 00:29:30,970 --> 00:29:32,960 It's the mass current. 664 00:29:32,960 --> 00:29:34,150 It's mass flow rate. 665 00:29:34,150 --> 00:29:37,730 Mass flow rate per unit area times area gives me something 666 00:29:37,730 --> 00:29:39,750 that's analogous to the current, and the time is the 667 00:29:39,750 --> 00:29:42,090 time, this is the total material lost, this is the 668 00:29:42,090 --> 00:29:43,590 total electrical charge. 669 00:29:43,590 --> 00:29:46,830 These two are the comparable equations. 670 00:29:46,830 --> 00:29:47,670 OK. 671 00:29:47,670 --> 00:29:50,260 Now, I want to go to the more sophisticated question, and 672 00:29:50,260 --> 00:29:53,240 that is, what happens before steady state? 673 00:29:53,240 --> 00:29:55,110 Before steady state is achieved? 674 00:29:55,110 --> 00:29:59,110 Suppose at time zero, the membrane has no gas in it. 675 00:29:59,110 --> 00:30:02,070 And at time zero, I inflate the left side. 676 00:30:02,070 --> 00:30:03,900 And just to keep things simple, I'm going to 677 00:30:03,900 --> 00:30:05,430 set c2 equals 0. 678 00:30:05,430 --> 00:30:06,440 It's a linear equation. 679 00:30:06,440 --> 00:30:09,330 You can always add it, but I just want to keep it simpler 680 00:30:09,330 --> 00:30:10,240 for the analysis. 681 00:30:10,240 --> 00:30:24,620 So, suppose at t equals 0, we inflate to c1 on the left side 682 00:30:24,620 --> 00:30:26,420 of the empty membrane. 683 00:30:26,420 --> 00:30:27,670 There's nothing in the membrane. 684 00:30:32,470 --> 00:30:33,950 So, let's see what happens. 685 00:30:33,950 --> 00:30:35,770 First, I'm going to solve the problem graphically. 686 00:30:35,770 --> 00:30:37,170 Because we know what's going to happen. 687 00:30:37,170 --> 00:30:38,960 I'm going to have to do the mathematics. 688 00:30:38,960 --> 00:30:41,440 So, here's what we expect. 689 00:30:41,440 --> 00:30:46,100 This is going from 0 to L. 690 00:30:46,100 --> 00:30:48,490 And we've got nothing in the membrane on both sides, so 691 00:30:48,490 --> 00:30:49,430 that's good. 692 00:30:49,430 --> 00:30:54,320 And then, at time zero, I inflate to c1. 693 00:30:54,320 --> 00:30:56,500 There's nothing in here. 694 00:30:56,500 --> 00:30:57,990 So, what do I have? 695 00:30:57,990 --> 00:31:00,420 Well, I know if I wait long enough, it's eventually going 696 00:31:00,420 --> 00:31:01,730 to look like this. 697 00:31:01,730 --> 00:31:04,570 Because that's the steady state solution. 698 00:31:04,570 --> 00:31:07,510 I've put here c2 equals 0, for simplicity, but 699 00:31:07,510 --> 00:31:08,300 we can change it. 700 00:31:08,300 --> 00:31:09,430 It's a linear equation. 701 00:31:09,430 --> 00:31:10,850 We can just add. 702 00:31:10,850 --> 00:31:14,580 So, what happens at time 0 plus? 703 00:31:14,580 --> 00:31:17,740 I start getting diffusion in from left to right, following 704 00:31:17,740 --> 00:31:18,310 Fick's Law. 705 00:31:18,310 --> 00:31:20,730 So it's going to look like this, isn't it? 706 00:31:20,730 --> 00:31:24,560 This is the approach to steady state. 707 00:31:24,560 --> 00:31:27,320 Some books will call it nonsteady state or unsteady. 708 00:31:27,320 --> 00:31:30,500 Unsteady to me means unsteady, so I refuse to use those 709 00:31:30,500 --> 00:31:32,870 terms. I call this the transient. 710 00:31:32,870 --> 00:31:34,060 That's the literary term. 711 00:31:34,060 --> 00:31:35,980 The transient. 712 00:31:35,980 --> 00:31:38,350 We're talking about the approach to steady state. 713 00:31:44,080 --> 00:31:46,520 And in some systems, you never get to steady state. 714 00:31:46,520 --> 00:31:47,780 But that's OK. 715 00:31:47,780 --> 00:31:50,080 So here's what it's going to look like at some very, very 716 00:31:50,080 --> 00:31:51,055 short time afterwards. 717 00:31:51,055 --> 00:31:55,170 It's going to look like that, only somewhat modified for 718 00:31:55,170 --> 00:31:55,980 this geometry. 719 00:31:55,980 --> 00:31:58,020 So it's going to look like this, isn't it? 720 00:31:58,020 --> 00:32:01,640 It's pinned at c1, and it's pinned at 0, and it varies 721 00:32:01,640 --> 00:32:02,860 continuously. 722 00:32:02,860 --> 00:32:06,050 So this is at time t 1. 723 00:32:06,050 --> 00:32:08,780 What happens at time t 2? 724 00:32:08,780 --> 00:32:12,405 At time t 2, it advances a little bit farther. 725 00:32:16,680 --> 00:32:20,990 But in all instances, the slope and the shape is given 726 00:32:20,990 --> 00:32:23,210 by this, Fick's first law. 727 00:32:23,210 --> 00:32:26,360 So this is at time t 2. 728 00:32:26,360 --> 00:32:27,600 And let's do one more. 729 00:32:27,600 --> 00:32:28,150 Three's a charm. 730 00:32:28,150 --> 00:32:30,790 So, let's take a green one. 731 00:32:30,790 --> 00:32:35,170 All right, so here's at t 3, still pinned at c 1 and still 732 00:32:35,170 --> 00:32:37,960 pinned at 0. 733 00:32:37,960 --> 00:32:40,400 So this is at t3. 734 00:32:40,400 --> 00:32:45,280 And then finally, this is steady state. 735 00:32:45,280 --> 00:32:48,890 So if we park at any place -- 736 00:32:48,890 --> 00:32:50,140 let's park at x1. 737 00:32:52,430 --> 00:32:53,650 Watch here. 738 00:32:53,650 --> 00:32:56,700 Can you see that the slope at t1 is gentle? 739 00:32:56,700 --> 00:33:00,060 The slope at t2 is steeper than t1. 740 00:33:00,060 --> 00:33:03,130 The slope at t3 is steeper than t2, and this is the 741 00:33:03,130 --> 00:33:07,990 highest. So, we go from nothing, to gradual, gradual, 742 00:33:07,990 --> 00:33:09,710 gradual, until we finally reach steady 743 00:33:09,710 --> 00:33:11,070 state, and we stop. 744 00:33:11,070 --> 00:33:13,590 And then at that point, we just continue to have to mass 745 00:33:13,590 --> 00:33:14,100 flowing through. 746 00:33:14,100 --> 00:33:17,600 And look, the concentration is zero, but the flux is not 747 00:33:17,600 --> 00:33:20,740 zero, because the flux isn't related to the concentration. 748 00:33:20,740 --> 00:33:22,820 It's related to the concentration gradient. 749 00:33:22,820 --> 00:33:24,960 Zero concentration, but non-zero 750 00:33:24,960 --> 00:33:26,000 concentration gradient. 751 00:33:26,000 --> 00:33:28,800 In fact, just for grins and chuckles, I could plot the 752 00:33:28,800 --> 00:33:30,440 ascent of this. 753 00:33:30,440 --> 00:33:35,060 I could plot time at x equals x1. 754 00:33:35,060 --> 00:33:37,710 I could plot J. 755 00:33:37,710 --> 00:33:38,610 And what's J? 756 00:33:38,610 --> 00:33:40,420 At time zero, it's zero. 757 00:33:40,420 --> 00:33:43,430 Then it's small, then it's larger, larger, larger 758 00:33:43,430 --> 00:33:44,360 asymptotically. 759 00:33:44,360 --> 00:33:46,730 And this is J of steady state. 760 00:33:46,730 --> 00:33:52,160 And this is t1, t2, t3. 761 00:33:52,160 --> 00:33:56,350 That's the approach to steady state. 762 00:33:56,350 --> 00:33:58,750 And you might say, well, this is kind of 763 00:33:58,750 --> 00:34:00,200 a professor's pedantry. 764 00:34:00,200 --> 00:34:01,330 It's not! 765 00:34:01,330 --> 00:34:03,660 This is doping of semiconductors. 766 00:34:03,660 --> 00:34:06,160 You don't run processes at equilibrium. 767 00:34:06,160 --> 00:34:08,520 You run them far from equilibrium. 768 00:34:08,520 --> 00:34:11,990 So in the transient mode, doping, outgassing, drying, 769 00:34:11,990 --> 00:34:12,850 what have you. 770 00:34:12,850 --> 00:34:15,370 So, now I want to give you the shape of the 771 00:34:15,370 --> 00:34:16,770 concentration profile. 772 00:34:16,770 --> 00:34:18,170 So how do I do that? 773 00:34:18,170 --> 00:34:27,890 OK, so the shape of the transient profile is not given 774 00:34:27,890 --> 00:34:28,720 by Fick's Law. 775 00:34:28,720 --> 00:34:31,720 The Fick's Law just gives me the gradient. 776 00:34:31,720 --> 00:34:33,660 In other words, here's what I want. 777 00:34:33,660 --> 00:34:38,500 I want C as a function of x at any given time. 778 00:34:38,500 --> 00:34:39,860 That's what I want. 779 00:34:39,860 --> 00:34:41,360 I want to be able to plot the profile. 780 00:34:41,360 --> 00:34:44,890 So, profile is C of x, but C of x varies with time. 781 00:34:44,890 --> 00:34:47,440 So, I need C of x at different times. 782 00:34:47,440 --> 00:34:48,470 What gives me that? 783 00:34:48,470 --> 00:34:52,310 What gives me that is Fick's second law. 784 00:34:52,310 --> 00:34:55,980 Go to FSL, Fick's second law. 785 00:34:55,980 --> 00:34:58,320 Now, Fick's second law, I'm just going to put it up here. 786 00:34:58,320 --> 00:34:59,920 I'm going to show you the solution. 787 00:34:59,920 --> 00:35:01,730 I don't expect you to solve it. 788 00:35:01,730 --> 00:35:05,530 I would, you were all required to have differential equations 789 00:35:05,530 --> 00:35:06,990 as a pre-req, but you don't. 790 00:35:06,990 --> 00:35:09,690 So, I'm just going to put it up here so that you see it. 791 00:35:09,690 --> 00:35:12,160 It's a partial differential equation, and 792 00:35:12,160 --> 00:35:13,490 it looks like this. 793 00:35:13,490 --> 00:35:15,740 And it's really beautiful. 794 00:35:15,740 --> 00:35:19,290 It's got a beautiful symmetry and the fonts look great, so 795 00:35:19,290 --> 00:35:20,480 that's why we put it up here. 796 00:35:20,480 --> 00:35:24,500 So, this is the partial in time goes as the double 797 00:35:24,500 --> 00:35:27,390 derivative in space, because we need a two-variable 798 00:35:27,390 --> 00:35:28,080 function, right? 799 00:35:28,080 --> 00:35:29,540 We want x and t. 800 00:35:29,540 --> 00:35:34,070 And this assumes that the diffusion coefficient is not a 801 00:35:34,070 --> 00:35:35,380 function of concentration. 802 00:35:35,380 --> 00:35:37,510 If it is, the equation is messier. 803 00:35:37,510 --> 00:35:39,860 This assumes a constant diffusion coefficient. 804 00:35:39,860 --> 00:35:42,590 So, this, as you can see, is a partial differential equation, 805 00:35:42,590 --> 00:35:45,610 which is a bummer, because it's hard to solve, all right? 806 00:35:45,610 --> 00:35:47,350 But, it's linear. 807 00:35:47,350 --> 00:35:50,080 It's a linear partial differential equation, which 808 00:35:50,080 --> 00:35:51,560 means the solutions are additive. 809 00:35:51,560 --> 00:35:54,300 We know this already from LCAOMO. 810 00:35:54,300 --> 00:35:57,300 So, I can handle, because I can solve it once and then I 811 00:35:57,300 --> 00:36:02,580 can give you ways to patch it, to make it useful. 812 00:36:02,580 --> 00:36:06,200 So, the solution. 813 00:36:06,200 --> 00:36:09,300 First of all, how do I specify this fully? 814 00:36:09,300 --> 00:36:10,550 Every time you differentiate -- 815 00:36:10,550 --> 00:36:12,240 I'm going to give you some math here, real math 816 00:36:12,240 --> 00:36:13,260 that you can use. 817 00:36:13,260 --> 00:36:15,880 When you differentiate, you throw away information, right? 818 00:36:15,880 --> 00:36:17,960 The derivative of a constant is zero. 819 00:36:17,960 --> 00:36:19,380 So, what was the value of the constant? 820 00:36:19,380 --> 00:36:20,330 It's gone. 821 00:36:20,330 --> 00:36:21,160 You lost it. 822 00:36:21,160 --> 00:36:22,490 So you have to bring it back. 823 00:36:22,490 --> 00:36:25,780 So when I double derivative this, I need two pieces of 824 00:36:25,780 --> 00:36:27,345 spacial information. 825 00:36:27,345 --> 00:36:30,040 And when I take a derivative with respect to time, I need 826 00:36:30,040 --> 00:36:32,560 one piece of temporal information. 827 00:36:32,560 --> 00:36:33,670 That's math. 828 00:36:33,670 --> 00:36:36,110 All the other lemmas and postulates, forget them. 829 00:36:36,110 --> 00:36:37,590 This is how you use it, all right? 830 00:36:37,590 --> 00:36:39,860 So, I need two pieces of spatial information and one 831 00:36:39,860 --> 00:36:41,620 piece of temporal information. 832 00:36:41,620 --> 00:36:45,150 So these specify our boundary conditions, right? 833 00:36:45,150 --> 00:36:46,450 So, what are our boundary conditions. 834 00:36:46,450 --> 00:36:49,920 So, I will need the concentration of 835 00:36:49,920 --> 00:36:51,710 all x at time 0. 836 00:36:51,710 --> 00:36:53,140 That's this. 837 00:36:53,140 --> 00:36:54,400 The answer to this question. 838 00:36:54,400 --> 00:36:58,180 And what we're going to do is say that it's a constant. 839 00:36:58,180 --> 00:37:00,350 I don't know what the constant is, it could be zero, it could 840 00:37:00,350 --> 00:37:02,800 be any arbitrary number, but I'm going to give you the 841 00:37:02,800 --> 00:37:05,340 solution for the situation in which the initial 842 00:37:05,340 --> 00:37:08,650 concentration is a constant. 843 00:37:08,650 --> 00:37:10,700 So, you could look at this and say, well, what happens if 844 00:37:10,700 --> 00:37:11,790 it's a non-zero constant? 845 00:37:11,790 --> 00:37:13,010 You ready for linearity? 846 00:37:13,010 --> 00:37:13,870 Watch this. 847 00:37:13,870 --> 00:37:15,530 This is how cool linearity is. 848 00:37:15,530 --> 00:37:17,230 You see this set of solutions? 849 00:37:17,230 --> 00:37:20,400 This is the set of solutions for C equals 0. 850 00:37:20,400 --> 00:37:23,030 What if C equals some non-zero value? 851 00:37:23,030 --> 00:37:26,210 I just shift the origin down to here to the non-zero value 852 00:37:26,210 --> 00:37:27,440 and everything sits there. 853 00:37:27,440 --> 00:37:29,390 That's what superposition gives you. 854 00:37:29,390 --> 00:37:32,570 That whole set of solutions just gets jacked up by the 855 00:37:32,570 --> 00:37:34,580 value of C naught. 856 00:37:34,580 --> 00:37:38,680 If C naught is 0, this slams down onto the x-axis. 857 00:37:38,680 --> 00:37:40,660 That's real math. 858 00:37:40,660 --> 00:37:42,360 You're the master. 859 00:37:42,360 --> 00:37:47,620 The math is the slave. Never let math enslave you. 860 00:37:47,620 --> 00:37:48,510 I won't. 861 00:37:48,510 --> 00:37:49,470 I refuse. 862 00:37:49,470 --> 00:37:52,840 So that's how you make C equals C naught work. 863 00:37:52,840 --> 00:37:57,190 And then you have to peg the concentration at the surface. 864 00:37:57,190 --> 00:37:59,830 I can give you examples where that's not the situation, 865 00:37:59,830 --> 00:38:02,340 where you have a variable concentration at the surface. 866 00:38:02,340 --> 00:38:04,550 The solution I'm going to give you won't work for those. 867 00:38:04,550 --> 00:38:07,370 So, x equals 0 is the surface. 868 00:38:07,370 --> 00:38:10,680 So, at x equals 0 at all time, I have some fixed value, which 869 00:38:10,680 --> 00:38:12,550 I'm going to just call C surface. 870 00:38:12,550 --> 00:38:14,600 And I need a third boundary condition. 871 00:38:14,600 --> 00:38:16,100 And the third boundary condition is a 872 00:38:16,100 --> 00:38:17,390 mathematical trick. 873 00:38:17,390 --> 00:38:20,540 We're going to say that this only works at short times. 874 00:38:20,540 --> 00:38:24,310 It's called the short time. 875 00:38:24,310 --> 00:38:26,560 And at short times, I'm going to say that -- 876 00:38:26,560 --> 00:38:29,850 from the perspective of the diffusion experiment -- 877 00:38:29,850 --> 00:38:33,390 at very, very short times, this sample appears to be 878 00:38:33,390 --> 00:38:35,120 infinitely long. 879 00:38:35,120 --> 00:38:38,220 Because the amount of material that goes in doesn't hit the 880 00:38:38,220 --> 00:38:39,290 other side. 881 00:38:39,290 --> 00:38:42,160 So, at very, very short times, we can pretend that it's 882 00:38:42,160 --> 00:38:43,090 semi-infinite. 883 00:38:43,090 --> 00:38:46,000 And so then we can write that the concentration -- 884 00:38:46,000 --> 00:38:52,130 this is the short time, which means infinite size, and 885 00:38:52,130 --> 00:38:53,740 you'll see that the two are related -- 886 00:38:53,740 --> 00:38:57,520 that the concentration at infinity, then, doesn't change 887 00:38:57,520 --> 00:38:58,620 from the initial value. 888 00:38:58,620 --> 00:39:02,200 The concentration at infinity for all time must equal the 889 00:39:02,200 --> 00:39:03,770 initial value C naught. 890 00:39:03,770 --> 00:39:06,250 And so if you put these three boundary conditions into that 891 00:39:06,250 --> 00:39:09,600 equation, you end up with this. 892 00:39:09,600 --> 00:39:10,340 This is the thing. 893 00:39:10,340 --> 00:39:16,140 You get C minus Cs over C naught minus Cs. 894 00:39:16,140 --> 00:39:17,880 This is surface concentration. 895 00:39:17,880 --> 00:39:19,320 This is initial concentration. 896 00:39:19,320 --> 00:39:20,560 This is C. 897 00:39:20,560 --> 00:39:23,780 It varies by this amount. 898 00:39:23,780 --> 00:39:28,520 Error function of x over 2 times the square root of Dt. 899 00:39:28,520 --> 00:39:31,960 So, you tell me the time and I can tell you the relationship 900 00:39:31,960 --> 00:39:34,500 between concentration and position. 901 00:39:34,500 --> 00:39:38,320 Now I've got to tell you what the error function is. 902 00:39:38,320 --> 00:39:39,940 Here's the error function. 903 00:39:39,940 --> 00:39:42,900 This is this problem before. 904 00:39:42,900 --> 00:39:44,490 This is the solution I'm going to show you. 905 00:39:44,490 --> 00:39:46,550 Here's Cs. 906 00:39:46,550 --> 00:39:47,905 Here's C naught. 907 00:39:47,905 --> 00:39:49,410 The most general case. 908 00:39:49,410 --> 00:39:51,470 C naught could be zero. 909 00:39:51,470 --> 00:39:55,370 And this curve here is given by this equation, OK? 910 00:39:55,370 --> 00:39:57,740 And this is an example of doping, isn't it? 911 00:39:57,740 --> 00:39:59,680 This is how doping would work. 912 00:39:59,680 --> 00:40:04,110 And this is the case where C s is greater than C naught. 913 00:40:04,110 --> 00:40:06,630 If we turn it around and we make the surface concentration 914 00:40:06,630 --> 00:40:09,270 less than the initial concentration, we'll get the 915 00:40:09,270 --> 00:40:12,306 complementary situation. 916 00:40:12,306 --> 00:40:14,800 The complementary situation looks like this. 917 00:40:17,340 --> 00:40:21,360 So in this case, C naught is up here, Cs is down here, and 918 00:40:21,360 --> 00:40:22,610 we end up with this. 919 00:40:25,160 --> 00:40:28,660 So now matter is going out, because Cs 920 00:40:28,660 --> 00:40:29,760 is less than C naught. 921 00:40:29,760 --> 00:40:31,140 So this is -- 922 00:40:31,140 --> 00:40:34,760 what do you want to call it -- this is effusion, this is some 923 00:40:34,760 --> 00:40:36,920 kind of a drying process, outgassing. 924 00:40:36,920 --> 00:40:38,340 And this is doping. 925 00:40:38,340 --> 00:40:39,685 And what's the shape of this curve? 926 00:40:39,685 --> 00:40:40,830 The shape of this curve. 927 00:40:40,830 --> 00:40:42,130 ERF. 928 00:40:42,130 --> 00:40:43,300 That's what math is. 929 00:40:43,300 --> 00:40:45,700 You find a mathematical function that 930 00:40:45,700 --> 00:40:47,660 templates for the curve. 931 00:40:47,660 --> 00:40:50,800 And ERF, I'm going to show you, is related to random walk 932 00:40:50,800 --> 00:40:51,410 statistics. 933 00:40:51,410 --> 00:40:54,970 So, it's physically validated. 934 00:40:54,970 --> 00:40:56,205 It's the best fit. 935 00:40:56,205 --> 00:40:59,080 I mean, heck, if you find anything that is pinned at two 936 00:40:59,080 --> 00:41:02,260 ends and has curvature, you'll get a reasonable fit, but 937 00:41:02,260 --> 00:41:03,880 what's the right fit? 938 00:41:03,880 --> 00:41:04,980 The right fit is this. 939 00:41:04,980 --> 00:41:07,220 The error function looks like this. 940 00:41:07,220 --> 00:41:08,370 Here's the error function. 941 00:41:08,370 --> 00:41:09,800 Error function. 942 00:41:09,800 --> 00:41:13,600 I'm going to plot ERF of z as a function of z. 943 00:41:13,600 --> 00:41:16,570 And it goes from 0 to 1. 944 00:41:21,120 --> 00:41:22,300 It looks like this. 945 00:41:22,300 --> 00:41:23,640 All right? 946 00:41:23,640 --> 00:41:26,260 And here's an error function table. 947 00:41:26,260 --> 00:41:27,730 Those are exact values. 948 00:41:27,730 --> 00:41:30,700 And it turns out that ERF is pretty much linear up 949 00:41:30,700 --> 00:41:32,830 to about point 6. 950 00:41:32,830 --> 00:41:34,100 Up to about point 6. 951 00:41:34,100 --> 00:41:39,950 ERF of point 6 not to less than about 1% is point 6. 952 00:41:39,950 --> 00:41:42,180 And then as you go farther and farther out, and eventually, 953 00:41:42,180 --> 00:41:45,240 ERF of infinity equals 1. 954 00:41:45,240 --> 00:41:49,200 But you don't even need tables for 0 up to point 6. 955 00:41:49,200 --> 00:41:52,150 And you can push it, go to point 6, 5, if you want. 956 00:41:52,150 --> 00:41:52,790 Yeah, let's do it. 957 00:41:52,790 --> 00:41:53,980 Let's push harder. 958 00:41:53,980 --> 00:41:55,110 6, 5. 959 00:41:55,110 --> 00:41:55,750 It's linear. 960 00:41:55,750 --> 00:41:57,360 You don't need tables. 961 00:41:57,360 --> 00:41:57,880 Right? 962 00:41:57,880 --> 00:42:01,460 And so ERF of 0 equals 0. 963 00:42:01,460 --> 00:42:03,350 Oh, let me give you the function. 964 00:42:03,350 --> 00:42:06,450 You know how you can define sine? 965 00:42:06,450 --> 00:42:09,940 Sine is that integral of something over 1 minus blah, 966 00:42:09,940 --> 00:42:10,820 blah, blah? 967 00:42:10,820 --> 00:42:13,090 We've got a definition for this one, too. 968 00:42:13,090 --> 00:42:14,320 It's cool. 969 00:42:14,320 --> 00:42:14,960 It's cool. 970 00:42:14,960 --> 00:42:17,170 Here's what the error function looks like. 971 00:42:17,170 --> 00:42:24,810 ERF of z equals the integral from 0 to z of e to the minus 972 00:42:24,810 --> 00:42:26,840 u squared du. 973 00:42:26,840 --> 00:42:29,770 You might say, why is he showing us all of this? 974 00:42:29,770 --> 00:42:31,860 Because he wants to torment you, that's why. 975 00:42:31,860 --> 00:42:33,590 No, it's because I'm trying to teach you something. 976 00:42:33,590 --> 00:42:35,090 What's e to the minus u squared? 977 00:42:35,090 --> 00:42:37,140 Actually, I think I can squeeze it in right there. 978 00:42:37,140 --> 00:42:40,290 What's e to the minus u squared du? 979 00:42:40,290 --> 00:42:43,640 This function here, that's the bell curve, isn't it? e to the 980 00:42:43,640 --> 00:42:44,550 minus u squared. 981 00:42:44,550 --> 00:42:46,470 That's this thing. 982 00:42:46,470 --> 00:42:47,430 It's symmetric, right? 983 00:42:47,430 --> 00:42:49,850 It's a minus u squared, so it doesn't matter if u is 984 00:42:49,850 --> 00:42:51,080 positive or negative. 985 00:42:51,080 --> 00:42:55,040 And e to the 0 was 1, and then back and forth. 986 00:42:55,040 --> 00:42:58,700 And so what this is doing, is it's integrating from zero out 987 00:42:58,700 --> 00:43:00,600 to some value. 988 00:43:00,600 --> 00:43:01,180 That's all. 989 00:43:01,180 --> 00:43:02,330 That's the random walk. 990 00:43:02,330 --> 00:43:03,460 This is the drunken sailor. 991 00:43:03,460 --> 00:43:05,700 How far does the drunken sailor go? 992 00:43:05,700 --> 00:43:07,830 That's the shape of this curve, the integral. 993 00:43:07,830 --> 00:43:11,310 And we'd like it to be so that if you integrate from zero to 994 00:43:11,310 --> 00:43:14,040 infinity, the area will be one. 995 00:43:14,040 --> 00:43:18,830 It turns out that this area here is root pi over 2. 996 00:43:18,830 --> 00:43:22,120 So, therefore, to normalize, we put 2 over root pi. 997 00:43:22,120 --> 00:43:23,915 That's how you get the error function. 998 00:43:23,915 --> 00:43:24,650 All right? 999 00:43:24,650 --> 00:43:27,300 So, the integral from zero to zero is zero, and the integral 1000 00:43:27,300 --> 00:43:31,280 from 0 to 1 is going to be, whatever it is, point 8, 4, et 1001 00:43:31,280 --> 00:43:32,200 cetera, et cetera. 1002 00:43:32,200 --> 00:43:33,550 So, that's it. 1003 00:43:33,550 --> 00:43:34,010 That's it. 1004 00:43:34,010 --> 00:43:38,350 Now we have the functional relationship to describe any 1005 00:43:38,350 --> 00:43:40,610 of these curves. 1006 00:43:40,610 --> 00:43:43,180 And you'll get some practice in doing that. 1007 00:43:43,180 --> 00:43:44,730 And it's a lot of fun. 1008 00:43:44,730 --> 00:43:47,380 Because now you can talk about how long it's going to take to 1009 00:43:47,380 --> 00:43:49,050 dope something, and so on. 1010 00:43:49,050 --> 00:43:49,960 Outgassing. 1011 00:43:49,960 --> 00:43:52,240 And now I'm going to show you something really cool, and I'm 1012 00:43:52,240 --> 00:43:53,010 not kidding you. 1013 00:43:53,010 --> 00:43:54,010 This has served me well. 1014 00:43:54,010 --> 00:43:56,490 I've been here 30 some odd years, and I've been in 1015 00:43:56,490 --> 00:44:00,070 consulting situations where I can, in my head, figure out 1016 00:44:00,070 --> 00:44:02,830 order of magnitude of what it's going to take to run a 1017 00:44:02,830 --> 00:44:03,950 diffusion problem. 1018 00:44:03,950 --> 00:44:10,080 Because you've got, up here, C minus Cs over C naught minus 1019 00:44:10,080 --> 00:44:17,930 Cs equals ERF error function of x over 2 roots of Dt. 1020 00:44:17,930 --> 00:44:21,100 So, this runs from 0 to 1. 1021 00:44:21,100 --> 00:44:22,160 The right side. 1022 00:44:22,160 --> 00:44:24,940 When I went to school, the left side of the equation had 1023 00:44:24,940 --> 00:44:27,320 to do the same thing as the right side of the equation. 1024 00:44:27,320 --> 00:44:30,040 So if the right side of the equation runs from 0 to 1, the 1025 00:44:30,040 --> 00:44:32,040 left side of the equation runs from 0 to 1. 1026 00:44:32,040 --> 00:44:34,250 So, let's pick a -- are you ready -- average 1027 00:44:34,250 --> 00:44:35,940 value from 0 to 1. 1028 00:44:35,940 --> 00:44:37,270 What would you pick? 1029 00:44:37,270 --> 00:44:39,990 I'd pick point 5. 1030 00:44:39,990 --> 00:44:42,440 Now, I don't have to go to the error function tables because 1031 00:44:42,440 --> 00:44:44,420 point 5 is less than point 6. 1032 00:44:44,420 --> 00:44:49,540 So ERF of x over 2 root Dt is essentially equal to x over 2 1033 00:44:49,540 --> 00:44:52,010 roots of Dt. 1034 00:44:52,010 --> 00:44:56,550 So, now I've got one half equals one half x over root 1035 00:44:56,550 --> 00:45:00,645 Dt, from which I can say that x is approximately equal to 1036 00:45:00,645 --> 00:45:02,120 the square root of Dt. 1037 00:45:02,120 --> 00:45:04,390 And I can be sitting in a meeting and someone says, gee, 1038 00:45:04,390 --> 00:45:05,490 I don't know how long it's going to take. 1039 00:45:05,490 --> 00:45:08,440 This thing's going to go down about 100 microns. 1040 00:45:08,440 --> 00:45:10,040 And how long is it going to take? 1041 00:45:10,040 --> 00:45:13,100 So, I pick a number, like, OK, if this is 100 microns, and 1042 00:45:13,100 --> 00:45:15,930 this thing here is 10 to the minus 8, now you just take the 1043 00:45:15,930 --> 00:45:18,110 square root of that, and tell them, oh, it's going to take 1044 00:45:18,110 --> 00:45:19,790 so many minutes, and they go, huh? 1045 00:45:19,790 --> 00:45:20,780 How did you get that? 1046 00:45:20,780 --> 00:45:22,130 It's right there. 1047 00:45:22,130 --> 00:45:24,220 And I'm not trying to tell them a number for three 1048 00:45:24,220 --> 00:45:25,410 significant figures. 1049 00:45:25,410 --> 00:45:27,580 They just want to know is it going to take a minute, an 1050 00:45:27,580 --> 00:45:30,580 hour, a day, or are we out of business? 1051 00:45:30,580 --> 00:45:33,840 And you can make that calculation with this. 1052 00:45:33,840 --> 00:45:36,210 This is so powerful. 1053 00:45:36,210 --> 00:45:39,100 And everything I told you about diffusion applies to 1054 00:45:39,100 --> 00:45:40,540 conductive heat transport. 1055 00:45:40,540 --> 00:45:42,480 You can do the same thing for heat transfer, figure out how 1056 00:45:42,480 --> 00:45:45,050 long it's going to take for a wave to go in. 1057 00:45:45,050 --> 00:45:47,220 This is really powerful stuff. 1058 00:45:47,220 --> 00:45:48,390 Anyway. 1059 00:45:48,390 --> 00:45:50,920 Let's move on. 1060 00:45:50,920 --> 00:45:53,650 Well, there's our pal. 1061 00:45:53,650 --> 00:45:56,030 OK, so I was going to give you the last element today. 1062 00:45:56,030 --> 00:45:56,830 The last element. 1063 00:45:56,830 --> 00:45:59,050 Remember, we talked about clean air. 1064 00:45:59,050 --> 00:46:02,720 So, I'm showing you the catalytic converter. 1065 00:46:02,720 --> 00:46:05,410 That's this thing here, coated with platinum, 1066 00:46:05,410 --> 00:46:08,300 rhodium, and so on. 1067 00:46:08,300 --> 00:46:11,510 And I'm showing you the electronic control module. 1068 00:46:11,510 --> 00:46:13,330 That's the CPU. 1069 00:46:13,330 --> 00:46:16,400 It's a little bit toned down from a Pentium, but it 1070 00:46:16,400 --> 00:46:19,130 certainly started from one of these units. 1071 00:46:19,130 --> 00:46:22,200 So, we've studied all of this. 1072 00:46:22,200 --> 00:46:23,840 Right? 1073 00:46:23,840 --> 00:46:24,810 OK. 1074 00:46:24,810 --> 00:46:28,240 And, so now, I want to talk about this piece here, the 1075 00:46:28,240 --> 00:46:30,240 oxygen sensor. why do we need an oxygen sensor? 1076 00:46:30,240 --> 00:46:33,750 Remember, the last day I told you that to get rid of NO, the 1077 00:46:33,750 --> 00:46:38,180 NOx, you have to reduce, And to get rid of CO and unburned 1078 00:46:38,180 --> 00:46:40,970 hydrocarbons, you have to oxidize. 1079 00:46:40,970 --> 00:46:42,850 Well, you have to pick. 1080 00:46:42,850 --> 00:46:45,180 You can't have two different atmospheres in the same place 1081 00:46:45,180 --> 00:46:46,250 at the same time. 1082 00:46:46,250 --> 00:46:48,320 Turns out there's a lucky sweet spot. 1083 00:46:48,320 --> 00:46:51,210 Can you see that the conversion efficiency is high 1084 00:46:51,210 --> 00:46:55,200 for both reduction and oxidation reactions if you peg 1085 00:46:55,200 --> 00:46:58,490 the air-to-fuel ratio at 14.6. 1086 00:46:58,490 --> 00:47:00,520 So you can't just set the carburetor or the fuel 1087 00:47:00,520 --> 00:47:03,570 injectors at some arbitrary ratio, because as you change 1088 00:47:03,570 --> 00:47:05,320 humidity, as you change temperature -- what happens 1089 00:47:05,320 --> 00:47:06,790 when the temperature goes down? 1090 00:47:06,790 --> 00:47:08,480 How much oxygen is there in the air? 1091 00:47:08,480 --> 00:47:12,070 It's still 20%, but PV equals nRT. 1092 00:47:12,070 --> 00:47:14,580 You ever try barbecuing when it's -- 1093 00:47:14,580 --> 00:47:16,090 I mean, I don't know, maybe you're not from the Northeast. 1094 00:47:16,090 --> 00:47:19,250 But if you want to barbecue when it's 0 Fahrenheit? 1095 00:47:19,250 --> 00:47:22,230 Do you think you get a hotter fire or a colder fire? 1096 00:47:22,230 --> 00:47:24,640 How much oxygen is there per unit volume in cold 1097 00:47:24,640 --> 00:47:26,570 air versus warm air? 1098 00:47:26,570 --> 00:47:27,730 Think about it. 1099 00:47:27,730 --> 00:47:28,930 It'll be on the next exam. 1100 00:47:28,930 --> 00:47:29,620 I'm just kidding. 1101 00:47:29,620 --> 00:47:29,880 OK. 1102 00:47:29,880 --> 00:47:34,490 So, anyway, here's the sweet spot that gives you both 1103 00:47:34,490 --> 00:47:36,190 oxidation and reduction. 1104 00:47:36,190 --> 00:47:38,290 And so we need a feedback loop. 1105 00:47:38,290 --> 00:47:40,940 That feedback loop is provided by the oxygen sensor. 1106 00:47:40,940 --> 00:47:44,800 Dave, would you mind cutting to the document camera? 1107 00:47:44,800 --> 00:47:49,020 So this is an oxygen sensor that's used on automobiles. 1108 00:47:49,020 --> 00:47:51,000 Here's the plug. 1109 00:47:51,000 --> 00:47:52,750 This goes into the pin set. 1110 00:47:52,750 --> 00:47:55,410 And the oxygen sensor is inside this housing. 1111 00:47:55,410 --> 00:47:56,660 This housing, you can see -- 1112 00:48:00,310 --> 00:48:02,020 OK, what I'm going to show you is what's 1113 00:48:02,020 --> 00:48:03,100 going on inside here. 1114 00:48:03,100 --> 00:48:05,210 This has got all kinds of protective stuff on it so that 1115 00:48:05,210 --> 00:48:07,890 it doesn't get dented during installation and so on. 1116 00:48:07,890 --> 00:48:12,230 But it sits inside the exhaust train. 1117 00:48:12,230 --> 00:48:13,600 It sits inside the exhaust train. 1118 00:48:13,600 --> 00:48:18,030 Now, may we go back to the slides, please? 1119 00:48:18,030 --> 00:48:21,260 Every car that's running a catalytic converter has to 1120 00:48:21,260 --> 00:48:24,980 have one of those in order to control, right here, to make 1121 00:48:24,980 --> 00:48:26,650 sure the air-to-fuel ratio is optimum. 1122 00:48:26,650 --> 00:48:29,370 Otherwise, the catalytic converter is either not doing 1123 00:48:29,370 --> 00:48:32,510 a good job on NOx, or it's not doing a good job on 1124 00:48:32,510 --> 00:48:37,770 hydrocarbons and whatever the other thing is. 1125 00:48:37,770 --> 00:48:39,100 Anyway, so here's what it looks like. 1126 00:48:39,100 --> 00:48:40,310 So, this is what's inside. 1127 00:48:40,310 --> 00:48:43,240 It's a closed, one-end tube made of zirconia. 1128 00:48:43,240 --> 00:48:44,690 Zirconium oxide. 1129 00:48:44,690 --> 00:48:47,840 And zirconium oxide conducts oxide ions. 1130 00:48:47,840 --> 00:48:49,590 And there are platinum electrodes on the 1131 00:48:49,590 --> 00:48:50,280 front and the back. 1132 00:48:50,280 --> 00:48:52,380 And what I showed you was from this side. 1133 00:48:52,380 --> 00:48:54,930 And this is the exhaust gas going along here, and this is 1134 00:48:54,930 --> 00:48:57,580 the edge of your exhaust train. 1135 00:48:57,580 --> 00:49:01,200 And two wires coming off of that sensor go to the CPU. 1136 00:49:01,200 --> 00:49:05,380 The CPU measures voltage and then from there, it regulates 1137 00:49:05,380 --> 00:49:08,040 the air-to-fuel mix to the engine, to the exhaust gas, 1138 00:49:08,040 --> 00:49:09,900 and around and around and around we go, so that when 1139 00:49:09,900 --> 00:49:12,480 exhaust gas gets to the catalytic convert, it is 1140 00:49:12,480 --> 00:49:13,890 optimally being converted. 1141 00:49:17,280 --> 00:49:20,130 So, we want to get fast response. 1142 00:49:20,130 --> 00:49:21,650 We don't want slow response. 1143 00:49:21,650 --> 00:49:24,670 And it's a solid-state sensor and solid-state diffusion is 1144 00:49:24,670 --> 00:49:26,700 slow, but that's we have to rely upon. 1145 00:49:26,700 --> 00:49:29,510 So we add a dopant. 1146 00:49:29,510 --> 00:49:33,020 We add a dopant to zirconia to stabilize its crystal 1147 00:49:33,020 --> 00:49:36,350 structure and create more oxygen vacancies to get faster 1148 00:49:36,350 --> 00:49:38,590 diffusion and a shorter response time. 1149 00:49:38,590 --> 00:49:41,320 So what we add is calcia, one of the candidates, and it 1150 00:49:41,320 --> 00:49:45,660 creates oxygen vacancies, and these oxygen vacancies allow 1151 00:49:45,660 --> 00:49:48,500 us to get more rapid response time. 1152 00:49:48,500 --> 00:49:52,020 So, these oxygen vacancies compensate for the charge 1153 00:49:52,020 --> 00:49:54,090 imbalance, thanks to the addition of calcia. 1154 00:49:54,090 --> 00:49:55,720 So, it's an engineered material. 1155 00:49:55,720 --> 00:49:58,040 Without this it doesn't work, all right? 1156 00:49:58,040 --> 00:50:01,120 And so, now what I've shown you is that on the basis of 1157 00:50:01,120 --> 00:50:05,980 this oxygen sensor, we can monitor the exhaust gas flow 1158 00:50:05,980 --> 00:50:08,380 so as to optimize the conversion here in the 1159 00:50:08,380 --> 00:50:12,870 catalytic converter and do so by sending information to the 1160 00:50:12,870 --> 00:50:14,390 CPU on the car. 1161 00:50:14,390 --> 00:50:18,130 And the number one consumer of CPU's is the automobile 1162 00:50:18,130 --> 00:50:21,440 industry, because every car has at least one, if not 1163 00:50:21,440 --> 00:50:22,950 multiple, CPU's. 1164 00:50:22,950 --> 00:50:26,460 So this is a good example of how understanding the lessons 1165 00:50:26,460 --> 00:50:29,920 of solid state chemistry can allow us to build fuel 1166 00:50:29,920 --> 00:50:35,020 efficient automobiles that have minimum toxic emissions 1167 00:50:35,020 --> 00:50:36,840 to the environment. 1168 00:50:36,840 --> 00:50:39,480 All right, we'll see you on Monday.