1 00:00:16,015 --> 00:00:20,086 As I mentioned last Friday, there is one more topic 2 00:00:20,086 --> 00:00:23,456 that I want to cover. 3 00:00:23,456 --> 00:00:26,092 And that is the topic of diffusion. 4 00:00:26,092 --> 00:00:31,965 And so today, I'm going to talk about diffusion, which 5 00:00:31,965 --> 00:00:37,470 will be in the context of what diffusion depends on 6 00:00:37,470 --> 00:00:39,973 and how do we describe it. 7 00:00:39,973 --> 00:00:41,674 And in particular, there's two laws, 8 00:00:41,674 --> 00:00:43,777 Fick's first law and Fick's second law, 9 00:00:43,777 --> 00:00:44,911 that I want to talk about. 10 00:00:44,911 --> 00:00:46,846 And I want to talk about how this 11 00:00:46,846 --> 00:00:48,848 looks in the context of some of the chemistry 12 00:00:48,848 --> 00:00:51,551 that we've learned, some of the things that we've learned. 13 00:00:51,551 --> 00:00:54,921 So diffusion, this is kind of a standard picture 14 00:00:54,921 --> 00:00:57,157 people show for diffusion, although really, this 15 00:00:57,157 --> 00:01:02,462 is probably mostly convection, which is the movement. 16 00:01:02,462 --> 00:01:05,397 It's not diffusion. 17 00:01:05,397 --> 00:01:09,269 But diffusion kind of has taken on this broader meaning. 18 00:01:09,269 --> 00:01:12,138 And so what do we mean by diffusion? 19 00:01:12,138 --> 00:01:18,912 Well, diffusion comes from the Latin word diffundere. 20 00:01:18,912 --> 00:01:21,381 And that means to spread out. 21 00:01:25,418 --> 00:01:30,289 But in this class, we're going to start 22 00:01:30,289 --> 00:01:35,962 by thinking about it as a particular movement 23 00:01:35,962 --> 00:01:37,564 down a concentration gradient. 24 00:01:37,564 --> 00:01:39,265 And that is what's happening here. 25 00:01:39,265 --> 00:01:42,202 You see there's a very concentrated droplet 26 00:01:42,202 --> 00:01:43,402 of dye that you put in there. 27 00:01:43,402 --> 00:01:46,239 And then it's less concentrated out here. 28 00:01:46,239 --> 00:01:47,607 You probably don't have any. 29 00:01:47,607 --> 00:01:49,676 So the concentration of the dye out there is zero. 30 00:01:49,676 --> 00:01:51,444 And in here it's very high. 31 00:01:51,444 --> 00:01:56,348 And so it wants to go down a concentration gradient. 32 00:01:56,348 --> 00:02:03,556 So that's movement down a concentration gradient. 33 00:02:10,396 --> 00:02:10,896 OK. 34 00:02:14,267 --> 00:02:16,069 How do we describe this movement? 35 00:02:16,069 --> 00:02:21,207 And it really goes back to experiments 36 00:02:21,207 --> 00:02:22,742 that Robert Brown did. 37 00:02:22,742 --> 00:02:26,679 And this is the original paper that he wrote way back 38 00:02:26,679 --> 00:02:29,048 in the early 1800s. 39 00:02:29,048 --> 00:02:32,352 I love the title of this paper, "A Brief Account 40 00:02:32,352 --> 00:02:36,289 of Microscopical Observations Made Specifically 41 00:02:36,289 --> 00:02:39,659 in the Months of June, July, and August 1827 on the Particles 42 00:02:39,659 --> 00:02:41,594 Contained in the Pollen and Plants 43 00:02:41,594 --> 00:02:44,597 and on the General Existence of Active Molecules in Organic 44 00:02:44,597 --> 00:02:46,699 and Inorganic Bodies." 45 00:02:46,699 --> 00:02:50,970 What a nice, detailed title. 46 00:02:50,970 --> 00:02:53,339 That was Brown. 47 00:02:53,339 --> 00:02:56,609 So he started with pollen. 48 00:02:56,609 --> 00:02:57,477 And he looked at it. 49 00:02:57,477 --> 00:02:58,478 And it looked like this. 50 00:02:58,478 --> 00:03:00,146 [MUSIC PLAYING] 51 00:03:00,146 --> 00:03:01,281 Oh, that's very loud. 52 00:03:06,786 --> 00:03:09,189 There wasn't supposed to be music. 53 00:03:09,189 --> 00:03:12,458 So that's what he saw, except this isn't what he saw. 54 00:03:12,458 --> 00:03:15,929 This is what Koshu Endo randomly posted online 55 00:03:15,929 --> 00:03:17,664 and I'm showing you. 56 00:03:17,664 --> 00:03:20,133 But if you look at pollen in a microscope, 57 00:03:20,133 --> 00:03:22,735 and you're Robert Brown back in the early 1800s, 58 00:03:22,735 --> 00:03:24,837 you'd see this. 59 00:03:24,837 --> 00:03:28,541 So he thought, aha, OK, there's life. 60 00:03:28,541 --> 00:03:31,844 This is must have to do with biology. 61 00:03:31,844 --> 00:03:33,112 And things are alive. 62 00:03:33,112 --> 00:03:34,314 And so they move. 63 00:03:34,314 --> 00:03:36,249 And it must be something like that. 64 00:03:36,249 --> 00:03:38,284 And so he put other things in. 65 00:03:38,284 --> 00:03:41,988 And they did the same thing, they jiggled around. 66 00:03:41,988 --> 00:03:43,856 And then he said, well, OK, what about 67 00:03:43,856 --> 00:03:48,328 if I put something that was once alive but is not alive anymore. 68 00:03:48,328 --> 00:03:50,196 And it did the same thing. 69 00:03:50,196 --> 00:03:51,331 He said, OK, wait a second. 70 00:03:51,331 --> 00:03:54,701 Maybe there's something to do with having ever been alive, 71 00:03:54,701 --> 00:03:57,503 the spirit of the particles. 72 00:03:57,503 --> 00:03:59,239 And he said, OK, the ultimate spirit, I'm 73 00:03:59,239 --> 00:04:00,707 going to break some glass-- 74 00:04:00,707 --> 00:04:03,576 which is what he did, I love that. 75 00:04:03,576 --> 00:04:04,978 I like breaking glass too. 76 00:04:04,978 --> 00:04:06,512 And so he broke the glass. 77 00:04:06,512 --> 00:04:08,047 And he took little pieces of glass 78 00:04:08,047 --> 00:04:09,382 and put it under his microscope. 79 00:04:09,382 --> 00:04:10,383 They did the same thing. 80 00:04:10,383 --> 00:04:13,353 They all did the same thing. 81 00:04:13,353 --> 00:04:15,088 So they were really perplexed. 82 00:04:15,088 --> 00:04:17,223 What was going on? 83 00:04:17,223 --> 00:04:20,125 This is the microscopic basis. 84 00:04:20,125 --> 00:04:22,295 This is now called Brownian motion. 85 00:04:22,295 --> 00:04:28,635 It's random motion due to thermal energy and collisions. 86 00:04:28,635 --> 00:04:34,440 And it's the connection of this microscopic random motion 87 00:04:34,440 --> 00:04:38,878 due to thermal energy and macroscopic observables that 88 00:04:38,878 --> 00:04:44,384 led to a big advance in our understanding of diffusion. 89 00:04:44,384 --> 00:04:49,589 And so the question of how far took 80 years. 90 00:04:49,589 --> 00:04:51,124 So one of the first questions, you 91 00:04:51,124 --> 00:04:53,558 say, how far do these things go. 92 00:04:53,558 --> 00:04:54,527 How far are they going? 93 00:04:54,527 --> 00:04:55,695 They're randomly moving around. 94 00:04:55,695 --> 00:04:56,195 I got that. 95 00:04:56,195 --> 00:04:57,530 I see it in the microscope. 96 00:04:57,530 --> 00:04:59,165 How far do they go? 97 00:04:59,165 --> 00:04:59,832 So how far? 98 00:05:02,969 --> 00:05:06,673 Well, that was a question that you 99 00:05:06,673 --> 00:05:13,646 can answer if you can get the mean squared displacement. 100 00:05:13,646 --> 00:05:14,180 I don't know. 101 00:05:14,180 --> 00:05:15,481 I say tomato. 102 00:05:15,481 --> 00:05:24,724 And so displacement, it's OK, mean squared displacement. 103 00:05:24,724 --> 00:05:32,031 And it really came from Einstein in 1905. 104 00:05:32,031 --> 00:05:35,468 Actually, this is what Einstein did for his dissertation, 105 00:05:35,468 --> 00:05:37,770 his Ph.D. work was on this topic. 106 00:05:37,770 --> 00:05:40,640 It was on understanding how to go 107 00:05:40,640 --> 00:05:45,745 from these kind of microscopic random fluctuations 108 00:05:45,745 --> 00:05:48,881 to some macroscopic observables. 109 00:05:48,881 --> 00:05:51,084 And so, the mean squared displacement, well, 110 00:05:51,084 --> 00:05:57,090 that's just the average of some measurement of some distance 111 00:05:57,090 --> 00:05:58,224 the particles go. 112 00:05:58,224 --> 00:06:02,261 So if you go back to the dye, there it is. 113 00:06:02,261 --> 00:06:05,064 But now it's in atom form or molecule form. 114 00:06:05,064 --> 00:06:07,033 And it's diffusing around, diffusing around. 115 00:06:07,033 --> 00:06:10,670 And OK, you say, well, how far did it go. 116 00:06:10,670 --> 00:06:12,839 That would be the circle. 117 00:06:12,839 --> 00:06:16,642 And that's something that Einstein tackled in his Ph.D. 118 00:06:16,642 --> 00:06:20,947 and found that the mean squared displacement is-- 119 00:06:20,947 --> 00:06:26,152 well, let's just say it's equal to 6 times some constant times 120 00:06:26,152 --> 00:06:28,488 time. 121 00:06:28,488 --> 00:06:33,092 This is in three dimensions. 122 00:06:33,092 --> 00:06:35,728 If you're in two dimensions, it's 4. 123 00:06:35,728 --> 00:06:38,865 If you're in one dimension, it's 2. 124 00:06:38,865 --> 00:06:40,133 But the main point isn't that. 125 00:06:40,133 --> 00:06:43,636 The main point is that the mean squared displacement 126 00:06:43,636 --> 00:06:48,174 is a function of some constant times time. 127 00:06:48,174 --> 00:06:50,443 And so if you just wait long enough 128 00:06:50,443 --> 00:06:52,812 and you know this constant, then you 129 00:06:52,812 --> 00:06:55,114 can figure out how far things diffuse. 130 00:06:55,114 --> 00:06:57,150 That was pretty cool. 131 00:06:57,150 --> 00:06:58,017 That was pretty cool. 132 00:06:58,017 --> 00:06:59,986 But you see, now, what we're going 133 00:06:59,986 --> 00:07:04,457 to be more interested in in this class is not how 134 00:07:04,457 --> 00:07:06,692 far but how fast. 135 00:07:06,692 --> 00:07:09,896 We're interested in how fast. 136 00:07:09,896 --> 00:07:14,901 And again, these are questions that Brown couldn't answer. 137 00:07:14,901 --> 00:07:17,870 Brown observed these things and wrote about them. 138 00:07:17,870 --> 00:07:19,639 And that's why it's called Brownian motion. 139 00:07:19,639 --> 00:07:22,141 But it was later that people said, well, 140 00:07:22,141 --> 00:07:24,110 let's try to think about how to write this down 141 00:07:24,110 --> 00:07:27,380 and come up with theories that explain it. 142 00:07:27,380 --> 00:07:31,784 And so for how fast, now we have the flux. 143 00:07:31,784 --> 00:07:38,191 So we have the flux or the diffusion flux J. 144 00:07:38,191 --> 00:07:42,695 And what we want is a way of describing how much. 145 00:07:42,695 --> 00:07:52,572 So it's an amount of something, the amount 146 00:07:52,572 --> 00:07:54,440 of the substance per area. 147 00:07:54,440 --> 00:07:55,641 You're going to normalize it. 148 00:07:55,641 --> 00:07:57,743 So you say, I'm going to take this sheet. 149 00:07:57,743 --> 00:07:59,245 And I'm going to say, how much of it 150 00:07:59,245 --> 00:08:03,950 is crossing this sheet area per time. 151 00:08:03,950 --> 00:08:18,030 So amount of a substance per area per time, that's the flux. 152 00:08:18,030 --> 00:08:21,434 So it could be grams. 153 00:08:21,434 --> 00:08:26,873 The amount could be grams, mass, kilograms, micrograms, 154 00:08:26,873 --> 00:08:28,841 nanograms. 155 00:08:28,841 --> 00:08:32,544 It could be number of atoms, moles. 156 00:08:32,544 --> 00:08:34,447 That's also an amount, moles. 157 00:08:34,447 --> 00:08:36,849 So we just say amount, it's general. 158 00:08:36,849 --> 00:08:45,958 Grams, moles, et cetera, n number of atoms. 159 00:08:45,958 --> 00:08:47,960 That's what moles tells us. 160 00:08:47,960 --> 00:08:49,495 Grams, et cetera-- 161 00:08:49,495 --> 00:08:50,963 The time could also be-- 162 00:08:50,963 --> 00:08:52,999 well, the area could be centimeter squared, 163 00:08:52,999 --> 00:08:53,933 meters squared. 164 00:08:53,933 --> 00:08:54,867 That's an area. 165 00:08:54,867 --> 00:08:57,003 And then the time is usually seconds. 166 00:08:57,003 --> 00:09:00,172 Usually, it doesn't have to be. 167 00:09:00,172 --> 00:09:02,975 And so this is what Fick's law gives us. 168 00:09:02,975 --> 00:09:07,680 So this was first established. 169 00:09:07,680 --> 00:09:10,983 How do you get a dependence of this flux? 170 00:09:10,983 --> 00:09:12,685 How do you get a dependence of this flux? 171 00:09:12,685 --> 00:09:14,954 And again, we've got this constant in there. 172 00:09:14,954 --> 00:09:18,424 And Fick's first law is all about that. 173 00:09:18,424 --> 00:09:26,065 Fick's first law says that this flux is proportional to-- 174 00:09:26,065 --> 00:09:28,134 and I'm just going to write it like this for now-- 175 00:09:28,134 --> 00:09:32,772 concentration gradient. 176 00:09:37,543 --> 00:09:43,583 So that's C divided by the change in X. C 177 00:09:43,583 --> 00:09:45,351 is concentration here. 178 00:09:45,351 --> 00:09:47,386 We'll draw an arrow. 179 00:09:47,386 --> 00:09:49,288 Concentration is C. 180 00:09:49,288 --> 00:09:51,591 Now, we could use brackets. 181 00:09:51,591 --> 00:09:55,027 Bracket A, concentration of A, mass per liter, 182 00:09:55,027 --> 00:09:56,529 something like that. 183 00:09:56,529 --> 00:10:00,533 I'll use the letter C, delta C over delta X. 184 00:10:00,533 --> 00:10:02,068 So that's the concentration gradient. 185 00:10:05,538 --> 00:10:08,341 Oh, by the way this wasn't as far. 186 00:10:08,341 --> 00:10:09,809 It didn't take as long as Einstein. 187 00:10:09,809 --> 00:10:15,448 But this was 1855. 188 00:10:15,448 --> 00:10:17,650 OK, good. 189 00:10:17,650 --> 00:10:20,853 So Fick came along maybe 30 years later. 190 00:10:20,853 --> 00:10:23,356 And he said, well, I want to know how fast things are going. 191 00:10:23,356 --> 00:10:25,758 I want to understand how to describe 192 00:10:25,758 --> 00:10:30,096 the flux of these randomly moving things. 193 00:10:30,096 --> 00:10:34,133 And I notice that it depends on the concentration difference. 194 00:10:34,133 --> 00:10:40,106 And in particular, I like it when diffusion is positive. 195 00:10:40,106 --> 00:10:45,444 So J is equal to minus D times dc/dx. 196 00:10:45,444 --> 00:10:47,880 And that is Fick's first law. 197 00:10:47,880 --> 00:10:49,949 Gesundheit. 198 00:10:49,949 --> 00:10:50,915 And D is a constant. 199 00:10:50,915 --> 00:10:52,018 D is this constant again. 200 00:10:52,018 --> 00:10:54,153 So D is the diffusion constant. 201 00:10:54,153 --> 00:10:58,357 We'll talk about this, diffusion constant. 202 00:11:01,093 --> 00:11:02,962 There's the concentration gradient. 203 00:11:02,962 --> 00:11:05,931 And we got a minus sign. 204 00:11:05,931 --> 00:11:10,836 Notice, movement down a concentration gradient. 205 00:11:10,836 --> 00:11:12,271 But if I didn't have a minus sign, 206 00:11:12,271 --> 00:11:14,440 then I wouldn't have a positive flux. 207 00:11:14,440 --> 00:11:17,443 Then the droplet wouldn't spread out. 208 00:11:17,443 --> 00:11:19,245 That wouldn't be good. 209 00:11:19,245 --> 00:11:21,414 So we're going from high concentration 210 00:11:21,414 --> 00:11:22,548 to low concentration. 211 00:11:22,548 --> 00:11:25,384 When I've got a bigger change in the concentration, 212 00:11:25,384 --> 00:11:27,119 I've got faster diffusion. 213 00:11:27,119 --> 00:11:29,455 That's what Fick's first law tells us. 214 00:11:29,455 --> 00:11:31,924 OK, good. 215 00:11:31,924 --> 00:11:38,097 Very importantly, Fick's first law means it's steady state. 216 00:11:38,097 --> 00:11:41,367 Notice, there is no t here. 217 00:11:41,367 --> 00:11:43,469 There's no time. 218 00:11:43,469 --> 00:11:45,438 For that, we have to wait for the second law. 219 00:11:45,438 --> 00:11:51,744 So steady state diffusion, and that 220 00:11:51,744 --> 00:11:56,215 means not a function of time. 221 00:12:01,954 --> 00:12:07,226 The kinds of problems where Fick's first law matters is-- 222 00:12:07,226 --> 00:12:09,695 and I know some of you have probably seen Fick's first law. 223 00:12:09,695 --> 00:12:10,196 I get that. 224 00:12:10,196 --> 00:12:15,267 But in a little bit, we're going to connect it to crystals. 225 00:12:15,267 --> 00:12:19,338 But the kind of diffusion where the first floor matters 226 00:12:19,338 --> 00:12:23,943 is the kind where you're holding concentrations equal. 227 00:12:23,943 --> 00:12:26,846 Let's say you're holding a concentration gradient, 228 00:12:26,846 --> 00:12:29,315 and it's not changing. 229 00:12:29,315 --> 00:12:31,751 And so you're wondering, well, if I 230 00:12:31,751 --> 00:12:35,020 hold a concentration at one thing here and at another thing 231 00:12:35,020 --> 00:12:37,690 here, how fast do things diffuse. 232 00:12:37,690 --> 00:12:39,992 That's what Fick's law tells us. 233 00:12:39,992 --> 00:12:43,195 And so let's do an example. 234 00:12:43,195 --> 00:12:43,996 Here's my example. 235 00:12:43,996 --> 00:12:50,336 I found these gloves, because I was looking for copolymers. 236 00:12:50,336 --> 00:12:55,274 Here's a copolymer that doesn't have the nitrile groups. 237 00:12:55,274 --> 00:12:57,409 Remember, the nitrile groups last Friday 238 00:12:57,409 --> 00:13:00,279 that were so important when we were doing our polymer design. 239 00:13:00,279 --> 00:13:02,081 This doesn't have it. 240 00:13:02,081 --> 00:13:03,649 This is butyl rubber. 241 00:13:03,649 --> 00:13:04,250 OK, fine. 242 00:13:04,250 --> 00:13:05,684 So there's the copolymer. 243 00:13:05,684 --> 00:13:08,420 It's copolymerized with these two groups. 244 00:13:08,420 --> 00:13:12,725 But now, it's got those groups, because it gave me 245 00:13:12,725 --> 00:13:15,461 the right flexibility and some of the other factors 246 00:13:15,461 --> 00:13:16,862 that I want in the glove. 247 00:13:16,862 --> 00:13:19,665 But is it strong enough chemically? 248 00:13:19,665 --> 00:13:21,967 And that's a very important question. 249 00:13:21,967 --> 00:13:26,539 We talked about this last Friday that these chemical groups 250 00:13:26,539 --> 00:13:27,907 change the mechanical properties. 251 00:13:27,907 --> 00:13:31,343 They also use the chemical resilience. 252 00:13:31,343 --> 00:13:32,845 And so here I have it. 253 00:13:32,845 --> 00:13:34,680 I have a toxin. 254 00:13:34,680 --> 00:13:36,982 This is why we wear gloves, because we 255 00:13:36,982 --> 00:13:42,154 don't want to get methyl chloride into our skin. 256 00:13:42,154 --> 00:13:45,925 Methyl chloride is toxic at 50 parts per million. 257 00:13:45,925 --> 00:13:47,993 Keep it out. 258 00:13:47,993 --> 00:13:49,361 Be very careful. 259 00:13:49,361 --> 00:13:51,197 So I don't know. 260 00:13:51,197 --> 00:13:54,133 I look up online and say, well, OK, butyl rubber. 261 00:13:54,133 --> 00:13:55,201 These feel comfortable. 262 00:13:55,201 --> 00:13:56,135 But how are they? 263 00:13:56,135 --> 00:13:59,905 OK, with methyl chloride, which is a toxin, it's a molecule, 264 00:13:59,905 --> 00:14:02,708 it has a diffusion constant of 1 times 10 265 00:14:02,708 --> 00:14:05,144 to the minus 6 centimeters squared per second. 266 00:14:05,144 --> 00:14:08,581 The gloves are 0.04 centimeters thick. 267 00:14:08,581 --> 00:14:11,717 And I'm working with paint remover. 268 00:14:11,717 --> 00:14:12,952 I'm working with paint room. 269 00:14:12,952 --> 00:14:14,353 The concentration of paint remover 270 00:14:14,353 --> 00:14:16,455 is a gram per centimeter cubed. 271 00:14:16,455 --> 00:14:17,990 Am I safe? 272 00:14:17,990 --> 00:14:21,260 How much am I going to be exposed to this? 273 00:14:21,260 --> 00:14:23,429 At my skin, I'm starting out with zero. 274 00:14:23,429 --> 00:14:25,464 What's the flux through the glove? 275 00:14:25,464 --> 00:14:26,565 Can I stay safe? 276 00:14:26,565 --> 00:14:30,469 Well, first of all, one thing that we can do 277 00:14:30,469 --> 00:14:31,971 is we can look at-- 278 00:14:31,971 --> 00:14:33,305 just to make sure we understand. 279 00:14:33,305 --> 00:14:35,841 So there's the diffusion constant. 280 00:14:35,841 --> 00:14:39,044 Why does it have those units? 281 00:14:39,044 --> 00:14:40,379 You can see this. 282 00:14:40,379 --> 00:14:47,453 The units of J are an amount-- 283 00:14:47,453 --> 00:14:49,288 this is what we already said-- 284 00:14:49,288 --> 00:14:55,361 divided by an area times a time. 285 00:14:55,361 --> 00:15:01,300 So if the units of C are an amount-- 286 00:15:01,300 --> 00:15:03,402 OK, I stayed with grams-- 287 00:15:03,402 --> 00:15:10,242 over a volume, then you can see right away that diffusion 288 00:15:10,242 --> 00:15:18,350 must be area over time. 289 00:15:18,350 --> 00:15:21,320 It has to be. 290 00:15:21,320 --> 00:15:26,325 If it's not, then the units don't work. 291 00:15:26,325 --> 00:15:28,861 I've got concentration over a length. 292 00:15:28,861 --> 00:15:32,765 This is a 1 D. I'm going across some membrane 293 00:15:32,765 --> 00:15:35,634 maybe or some area. 294 00:15:35,634 --> 00:15:38,704 This is a membrane, the glove. 295 00:15:38,704 --> 00:15:40,973 And so you can think about this for yourself. 296 00:15:40,973 --> 00:15:45,678 The units have to be area over time. 297 00:15:45,678 --> 00:15:47,846 Is it area over time? 298 00:15:47,846 --> 00:15:49,982 Centimeters squared per second. 299 00:15:49,982 --> 00:15:51,016 They were, OK. 300 00:15:51,016 --> 00:15:53,052 So that worked. 301 00:15:53,052 --> 00:15:59,758 Now, another thing is, before we do the math, 302 00:15:59,758 --> 00:16:02,728 why does this happen anyway? 303 00:16:02,728 --> 00:16:04,863 Why is there diffusion? 304 00:16:04,863 --> 00:16:09,635 And it's actually quite straightforward to understand, 305 00:16:09,635 --> 00:16:13,839 even when you go all the way back to Brown's experiments 306 00:16:13,839 --> 00:16:14,540 with pollen. 307 00:16:14,540 --> 00:16:17,509 You say, well, things are moving randomly. 308 00:16:17,509 --> 00:16:20,012 Why is there now a direction? 309 00:16:20,012 --> 00:16:21,747 Why is there now a direction? 310 00:16:21,747 --> 00:16:22,715 How is that possible? 311 00:16:22,715 --> 00:16:28,587 How can there be a direction if it's all random? 312 00:16:28,587 --> 00:16:30,622 How can there be a concentration gradient? 313 00:16:30,622 --> 00:16:32,558 And then you can get that, because if you 314 00:16:32,558 --> 00:16:34,560 have some high concentration-- 315 00:16:34,560 --> 00:16:39,331 Let's suppose that this is a high concentration, 316 00:16:39,331 --> 00:16:42,368 and this is a low concentration. 317 00:16:42,368 --> 00:16:46,138 And you've got the Brownian stuff going on. 318 00:16:46,138 --> 00:16:49,942 It's all in something. 319 00:16:49,942 --> 00:16:51,243 There's Brownian motion. 320 00:16:51,243 --> 00:16:54,646 Whatever containers these are, whatever is in whatever, 321 00:16:54,646 --> 00:16:56,949 it's got Brownian motion. 322 00:16:56,949 --> 00:16:59,485 And so that means that, because it's random, 323 00:16:59,485 --> 00:17:02,688 it's equally likely to go that way as it is to go that way. 324 00:17:06,058 --> 00:17:07,826 But see, at the high concentration, 325 00:17:07,826 --> 00:17:11,096 I've got more particles. 326 00:17:11,096 --> 00:17:13,799 You can see that, because I drew bigger, longer arrows, 327 00:17:13,799 --> 00:17:17,236 so it's obvious, more particles, because it's 328 00:17:17,236 --> 00:17:18,103 more concentration. 329 00:17:18,103 --> 00:17:19,438 There's just a lot more of them. 330 00:17:19,438 --> 00:17:22,608 So there's more going across. 331 00:17:22,608 --> 00:17:27,246 If I tried to draw a boundary here or boundaries inside here, 332 00:17:27,246 --> 00:17:29,615 and I look at those as my areas. 333 00:17:29,615 --> 00:17:30,983 There's a high concentration. 334 00:17:30,983 --> 00:17:32,551 There's a lot of stuff going randomly 335 00:17:32,551 --> 00:17:33,685 back and forth across them. 336 00:17:33,685 --> 00:17:36,722 Down here, you can do the same thing. 337 00:17:36,722 --> 00:17:38,157 But there's a lower concentration, 338 00:17:38,157 --> 00:17:39,925 so there's less stuff going back and forth 339 00:17:39,925 --> 00:17:41,260 across those barriers. 340 00:17:41,260 --> 00:17:43,762 And now I bring them together. 341 00:17:43,762 --> 00:17:44,963 I bring them together. 342 00:17:44,963 --> 00:17:52,771 I bring the methyl chloride to the glove, to my hand. 343 00:17:52,771 --> 00:17:56,341 Or I bring something with a high concentration next to something 344 00:17:56,341 --> 00:17:57,743 with a low concentration. 345 00:17:57,743 --> 00:18:02,381 And I keep the concentrations of the endpoints fixed. 346 00:18:02,381 --> 00:18:03,949 I hold these fixed. 347 00:18:03,949 --> 00:18:06,251 Fick's law tells me now how this happens. 348 00:18:06,251 --> 00:18:09,521 And you know that if I bring them together now, well, look, 349 00:18:09,521 --> 00:18:12,024 this was going just a little bit back and forth. 350 00:18:12,024 --> 00:18:14,193 And this was going a lot bit back and forth. 351 00:18:14,193 --> 00:18:18,197 And so there's going to be a net diffusion that way. 352 00:18:18,197 --> 00:18:20,365 Right there at the edge, you can see it. 353 00:18:20,365 --> 00:18:24,103 More goes to the right than goes to the left. 354 00:18:24,103 --> 00:18:28,774 And so this starts to create a diffusion gradient. 355 00:18:28,774 --> 00:18:30,275 That'll create a diffusion gradient. 356 00:18:33,412 --> 00:18:34,379 So that's why it works. 357 00:18:34,379 --> 00:18:38,050 That's why concentration gradients cause diffusion. 358 00:18:38,050 --> 00:18:41,753 And Fick's first law tells us how. 359 00:18:41,753 --> 00:18:44,857 OK, well, if we go to the question-- 360 00:18:44,857 --> 00:18:46,959 so let's see-- so the J-- 361 00:18:46,959 --> 00:18:49,528 so now I've got my glove. 362 00:18:49,528 --> 00:18:56,235 And so the J is equal minus 1.1 times 10 363 00:18:56,235 --> 00:19:02,207 to the minus 6 centimeters squared per second times-- 364 00:19:02,207 --> 00:19:04,143 and this is going to be-- 365 00:19:04,143 --> 00:19:07,146 OK, I started with what? 366 00:19:07,146 --> 00:19:11,049 I started with 1 gram right per centimeter cubed. 367 00:19:11,049 --> 00:19:14,553 And on the skin I've got 0. 368 00:19:14,553 --> 00:19:19,758 So it's going to be time 0 minus 1 gram per centimeter cubed. 369 00:19:19,758 --> 00:19:24,163 And that's going to be divided by the distance 0.04 370 00:19:24,163 --> 00:19:26,064 centimeters. 371 00:19:26,064 --> 00:19:34,606 And this goes to something like 2.5 times 10 372 00:19:34,606 --> 00:19:41,513 to the minus 5 grams per centimeter squared second. 373 00:19:41,513 --> 00:19:42,881 That's the flux. 374 00:19:42,881 --> 00:19:45,684 That's what Fick's law tells us, because my skin is good enough. 375 00:19:45,684 --> 00:19:48,654 My body is good enough that any time a toxin, a methyl chloride 376 00:19:48,654 --> 00:19:51,990 molecule touches it, the skin just takes it 377 00:19:51,990 --> 00:19:56,361 away and poisons me. 378 00:19:56,361 --> 00:19:57,863 But that's important. 379 00:19:57,863 --> 00:20:00,265 It's not good, but it's important, 380 00:20:00,265 --> 00:20:04,336 because it keeps this concentration fixed. 381 00:20:04,336 --> 00:20:06,338 Otherwise, this would build up. 382 00:20:06,338 --> 00:20:08,140 And it would be changing. 383 00:20:08,140 --> 00:20:10,742 So you can't use Fick's law. 384 00:20:10,742 --> 00:20:11,243 No. 385 00:20:11,243 --> 00:20:15,280 Fick's law applies when these two concentrations are fixed, 386 00:20:15,280 --> 00:20:17,749 and there's not a time dependence. 387 00:20:17,749 --> 00:20:19,117 Instead, there's a rate. 388 00:20:19,117 --> 00:20:23,455 There's a flux that you're after, J. 389 00:20:23,455 --> 00:20:26,592 And so now you say, well, can I handle this. 390 00:20:26,592 --> 00:20:27,659 Can I handle this? 391 00:20:27,659 --> 00:20:31,430 Work it out, 50 parts per million. 392 00:20:31,430 --> 00:20:33,131 I should have used the nitrile. 393 00:20:33,131 --> 00:20:35,167 Should have used the nitrile copolymer. 394 00:20:35,167 --> 00:20:37,436 Go back and make new polymers. 395 00:20:37,436 --> 00:20:40,906 Maybe you need much, much better chemical resilience, 396 00:20:40,906 --> 00:20:45,210 which might lower this diffusion constant. 397 00:20:45,210 --> 00:20:47,913 This is serious stuff. 398 00:20:47,913 --> 00:20:49,348 This is serious stuff. 399 00:20:49,348 --> 00:20:51,516 And it's so serious that I want to talk about eggs. 400 00:20:56,888 --> 00:21:00,125 The rubber glove stops this toxin. 401 00:21:00,125 --> 00:21:01,627 Well, it slowed it down. 402 00:21:01,627 --> 00:21:04,029 Did it slow down enough? 403 00:21:04,029 --> 00:21:05,764 Use Fick's first law. 404 00:21:05,764 --> 00:21:10,102 But you can make membranes that are actually very selective. 405 00:21:10,102 --> 00:21:11,136 It's the coolest thing. 406 00:21:11,136 --> 00:21:14,039 You make a membrane that only allows water 407 00:21:14,039 --> 00:21:21,146 to go through but not salt. That's a selective membrane. 408 00:21:21,146 --> 00:21:22,914 And that's called osmosis when you can 409 00:21:22,914 --> 00:21:25,550 allow one fluid to go through. 410 00:21:25,550 --> 00:21:28,453 It's semi-permeable, permeable to one type of fluid 411 00:21:28,453 --> 00:21:29,121 but not another. 412 00:21:29,121 --> 00:21:30,689 And the egg is such a great example, 413 00:21:30,689 --> 00:21:31,890 I had to show you this video. 414 00:21:31,890 --> 00:21:35,861 I made this years ago in another class, 415 00:21:35,861 --> 00:21:37,696 because I wanted to show it. 416 00:21:37,696 --> 00:21:39,398 This is what you can do with eggs. 417 00:21:39,398 --> 00:21:40,866 They're so cool. 418 00:21:40,866 --> 00:21:41,500 Here's a video. 419 00:21:41,500 --> 00:21:42,734 So you soak them in vinegar. 420 00:21:42,734 --> 00:21:45,137 OK, it's even got directions and everything. 421 00:21:45,137 --> 00:21:46,471 We don't need any volume. 422 00:21:46,471 --> 00:21:47,239 And there they are. 423 00:21:47,239 --> 00:21:48,473 OK, you soak them in vinegar. 424 00:21:48,473 --> 00:21:50,842 Now, the vinegar dissolves the calcium carbonate. 425 00:21:50,842 --> 00:21:52,444 You know this. 426 00:21:52,444 --> 00:21:54,613 You've done these reactions. 427 00:21:54,613 --> 00:21:55,280 So there you go. 428 00:21:55,280 --> 00:21:56,214 You soak them in vinegar. 429 00:21:56,214 --> 00:21:58,083 Leave it overnight, I'd say, is probably best. 430 00:21:58,083 --> 00:21:59,318 Then you peel off the vinegar. 431 00:21:59,318 --> 00:22:01,286 And look, there's a membrane. 432 00:22:01,286 --> 00:22:03,088 The egg has a really cool membrane. 433 00:22:03,088 --> 00:22:03,855 This isn't cooked. 434 00:22:03,855 --> 00:22:05,457 This is just a raw egg. 435 00:22:05,457 --> 00:22:08,393 But now I add a bunch of syrup. 436 00:22:08,393 --> 00:22:11,496 Now, this is almost pure sugar. 437 00:22:11,496 --> 00:22:15,267 So the concentration gradient is enormous, 438 00:22:15,267 --> 00:22:17,803 because the liquid inside the egg 439 00:22:17,803 --> 00:22:19,438 doesn't have any sugar in it. 440 00:22:19,438 --> 00:22:22,741 But outside, the liquid has huge amounts of sugar in it. 441 00:22:22,741 --> 00:22:25,210 So the water inside the egg wants to diffuse out. 442 00:22:25,210 --> 00:22:28,146 Eight hours, well, it's kind of enough. 443 00:22:28,146 --> 00:22:31,083 18 hours, you pick that membrane up. 444 00:22:31,083 --> 00:22:33,218 It's a really cool membrane, the egg membrane. 445 00:22:33,218 --> 00:22:34,686 It's a semi-permeable membrane. 446 00:22:34,686 --> 00:22:35,754 And it's hard to see here. 447 00:22:35,754 --> 00:22:38,223 But you see, all the water is kind of gone. 448 00:22:38,223 --> 00:22:40,492 It's totally deflated but completely stable, 449 00:22:40,492 --> 00:22:42,427 because the membrane's that strong. 450 00:22:42,427 --> 00:22:44,529 The water inside has left. 451 00:22:44,529 --> 00:22:46,698 And then if you get some green dye 452 00:22:46,698 --> 00:22:48,900 and you put it back into some green water, 453 00:22:48,900 --> 00:22:51,370 then you can make green eggs. 454 00:22:51,370 --> 00:22:53,572 And you can make green ham that way. 455 00:22:53,572 --> 00:22:54,373 And that's osmosis. 456 00:22:57,676 --> 00:23:00,946 But it's semi permeable. 457 00:23:00,946 --> 00:23:02,848 You've got to give the credit right. 458 00:23:02,848 --> 00:23:05,417 You got to give the credit right in these things. 459 00:23:05,417 --> 00:23:08,353 This was years ago when I thought I 460 00:23:08,353 --> 00:23:13,859 had a career in that direction. 461 00:23:13,859 --> 00:23:17,129 So the egg is just such a cool membrane to play with. 462 00:23:17,129 --> 00:23:19,264 And I though, most of us can access an egg. 463 00:23:19,264 --> 00:23:20,632 If you want to play with a really 464 00:23:20,632 --> 00:23:23,835 cool semi-permeable membrane, that is cool. 465 00:23:23,835 --> 00:23:28,106 And you really will see Fick's first law at play. 466 00:23:28,106 --> 00:23:31,843 You will see it, and you will feel it. 467 00:23:31,843 --> 00:23:35,213 Now, there's one thing I want to comment 468 00:23:35,213 --> 00:23:36,715 on about all this stuff. 469 00:23:36,715 --> 00:23:41,820 And we are not going to learn the details of entropy 470 00:23:41,820 --> 00:23:42,487 in this class. 471 00:23:42,487 --> 00:23:47,259 That is really something for a thermodynamics class. 472 00:23:47,259 --> 00:23:49,261 But I got to just correct something, 473 00:23:49,261 --> 00:23:54,032 because it's in your textbook, because a lot of this diffusion 474 00:23:54,032 --> 00:23:57,402 stuff, and in fact dissolution-- remember we 475 00:23:57,402 --> 00:23:58,837 talked about salt dissolving? 476 00:23:58,837 --> 00:24:02,140 We did whole lectures on this, the dissolution. 477 00:24:02,140 --> 00:24:05,677 Remember, these all have to do with the total energy 478 00:24:05,677 --> 00:24:06,311 of the system. 479 00:24:09,347 --> 00:24:13,518 But we really focus in this class on one part of that, 480 00:24:13,518 --> 00:24:17,556 on one part, which is called the enthalpy, the bonding, 481 00:24:17,556 --> 00:24:18,757 the potential of the bonding. 482 00:24:18,757 --> 00:24:23,395 But there's a whole other part that's related to the entropy. 483 00:24:23,395 --> 00:24:26,264 And we're not going to go into this in detail, like I said. 484 00:24:26,264 --> 00:24:27,999 You won't be tested on this. 485 00:24:27,999 --> 00:24:30,168 But I got to bring it up. 486 00:24:30,168 --> 00:24:34,105 So entropy can be really important. 487 00:24:34,105 --> 00:24:35,273 Why does something dissolve? 488 00:24:35,273 --> 00:24:38,643 It may be dominated by entropy. 489 00:24:38,643 --> 00:24:41,246 It may be dominated by entropy. 490 00:24:41,246 --> 00:24:43,482 That goes into the total energy still. 491 00:24:43,482 --> 00:24:48,386 So we still have our happy place, lower energy better. 492 00:24:48,386 --> 00:24:51,890 But as you learn more about what that energy entails, 493 00:24:51,890 --> 00:24:55,861 you've got to start including entropy in the future. 494 00:24:55,861 --> 00:24:59,898 So Averill says, "...for now, we can state that entropy is 495 00:24:59,898 --> 00:25:02,534 a thermodynamic property of all substances that is proportional 496 00:25:02,534 --> 00:25:05,470 to their degree disorder." 497 00:25:05,470 --> 00:25:07,839 No! 498 00:25:07,839 --> 00:25:10,675 That's not true. 499 00:25:10,675 --> 00:25:12,744 That is not true. 500 00:25:12,744 --> 00:25:14,513 So I have a question. 501 00:25:14,513 --> 00:25:17,182 So you see this all the time. 502 00:25:17,182 --> 00:25:20,352 Entropy, higher entropy, more disorder. 503 00:25:20,352 --> 00:25:23,989 My room is messy, it's got a high entropy. 504 00:25:23,989 --> 00:25:27,959 So which of these has a higher entropy? 505 00:25:27,959 --> 00:25:33,398 Which one of these has a higher entropy, and which is lower? 506 00:25:33,398 --> 00:25:35,033 This is a very simple grid. 507 00:25:35,033 --> 00:25:37,435 They've got the same exact number. 508 00:25:37,435 --> 00:25:41,473 How many think that the one on the right is-- 509 00:25:41,473 --> 00:25:43,108 it's the smoother one, right? 510 00:25:43,108 --> 00:25:47,178 How many think that this is the lower entropy system? 511 00:25:47,178 --> 00:25:48,613 I've set this up. 512 00:25:48,613 --> 00:25:51,283 I've set this up, because you know that's not. 513 00:25:51,283 --> 00:25:53,051 But why? 514 00:25:53,051 --> 00:25:55,754 The reason is the real reason for what entropy. 515 00:25:55,754 --> 00:26:00,225 It is not simply, because in this-- these 516 00:26:00,225 --> 00:26:02,727 are computerized experiments. 517 00:26:02,727 --> 00:26:06,698 So we told a computer to pick randomly a place 518 00:26:06,698 --> 00:26:09,200 on a grid a certain number of times and then fill it in. 519 00:26:09,200 --> 00:26:12,103 But in this case, you restricted it. 520 00:26:12,103 --> 00:26:12,837 You said, no. 521 00:26:12,837 --> 00:26:16,207 I can't have any two squares next to each other. 522 00:26:16,207 --> 00:26:18,977 And in this case, you simply did it randomly. 523 00:26:18,977 --> 00:26:21,212 So this looks smoother. 524 00:26:21,212 --> 00:26:25,116 This looks less disordered. 525 00:26:25,116 --> 00:26:27,752 But this has a higher entropy. 526 00:26:27,752 --> 00:26:28,320 Why? 527 00:26:28,320 --> 00:26:30,622 Because you have fewer options. 528 00:26:30,622 --> 00:26:32,891 And that is what entropy is about. 529 00:26:32,891 --> 00:26:37,896 Entropy is about the number of possibilities that you have. 530 00:26:37,896 --> 00:26:41,366 Now, that might look more disordered, it might not. 531 00:26:41,366 --> 00:26:44,569 But under the hood of entropy is possibilities. 532 00:26:44,569 --> 00:26:48,974 It's number of states, of accessible states. 533 00:26:48,974 --> 00:26:54,613 A salt atom in the crystal has very few possibilities. 534 00:26:54,613 --> 00:26:56,915 If it were a perfect crystal, one, 535 00:26:56,915 --> 00:27:00,285 it's got to be in the lattice. 536 00:27:00,285 --> 00:27:02,153 But once it goes into the liquid, 537 00:27:02,153 --> 00:27:03,655 it's got many more possibilities. 538 00:27:03,655 --> 00:27:07,025 It's those numbers of states that matter. 539 00:27:07,025 --> 00:27:09,828 That's why its entropy goes way, way up. 540 00:27:09,828 --> 00:27:11,429 It's the number of possibilities, 541 00:27:11,429 --> 00:27:13,598 the number of possible states. 542 00:27:13,598 --> 00:27:14,432 OK, good. 543 00:27:14,432 --> 00:27:16,267 That was my aside on entropy. 544 00:27:16,267 --> 00:27:20,839 Now, we talked about Fick's first law. 545 00:27:20,839 --> 00:27:22,007 We talked about diffusion. 546 00:27:22,007 --> 00:27:22,841 We looked at drops. 547 00:27:22,841 --> 00:27:26,578 And we made sure the gloves we choose are going to be safe. 548 00:27:26,578 --> 00:27:28,313 What about solids? 549 00:27:28,313 --> 00:27:29,047 What about solids? 550 00:27:29,047 --> 00:27:32,250 Well, things diffuse around in solids too. 551 00:27:32,250 --> 00:27:35,253 Here's a red atom that is going to diffuse. 552 00:27:35,253 --> 00:27:37,489 Watch this super high-tech video. 553 00:27:37,489 --> 00:27:38,256 How does it do it? 554 00:27:38,256 --> 00:27:39,658 A vacancy comes in. 555 00:27:39,658 --> 00:27:43,828 Look at that vacancy, moving around, moving around, bam! 556 00:27:43,828 --> 00:27:47,632 That atom moved, it moved, self-diffusion. 557 00:27:47,632 --> 00:27:49,167 It moved. 558 00:27:49,167 --> 00:27:52,003 It's an atom of the same type that 559 00:27:52,003 --> 00:27:55,473 was able to move in the lattice because a vacancy diffused 560 00:27:55,473 --> 00:27:56,941 through right. 561 00:27:56,941 --> 00:27:58,576 And so there's the beginning. 562 00:27:58,576 --> 00:27:59,411 And there's the end. 563 00:27:59,411 --> 00:28:01,312 You see it moved over by one position. 564 00:28:01,312 --> 00:28:03,048 But now we're all thinking. 565 00:28:03,048 --> 00:28:05,050 We're all thinking, and we're going back 566 00:28:05,050 --> 00:28:09,521 to our days of, well, OK, hold on. 567 00:28:09,521 --> 00:28:11,589 What really happened? 568 00:28:11,589 --> 00:28:14,225 So here's the vacancy on one side. 569 00:28:14,225 --> 00:28:15,927 Here, it went over to this side. 570 00:28:15,927 --> 00:28:18,697 There's the atom going through. 571 00:28:18,697 --> 00:28:20,031 So the vacancy moved that way. 572 00:28:20,031 --> 00:28:21,032 The atom moved that way. 573 00:28:21,032 --> 00:28:26,805 But as you know, that atom had to experience some barrier. 574 00:28:26,805 --> 00:28:30,208 So it had to get over some activation. 575 00:28:30,208 --> 00:28:31,676 It had to get over some activation. 576 00:28:31,676 --> 00:28:34,245 And that really helps us answer a question that 577 00:28:34,245 --> 00:28:37,582 has been burning in our minds this whole lecture, which is, 578 00:28:37,582 --> 00:28:41,720 what else does flux depend on. 579 00:28:41,720 --> 00:28:45,123 We saw the concentration gradient thing. 580 00:28:45,123 --> 00:28:47,258 We looked at it in very simple terms. 581 00:28:47,258 --> 00:28:48,193 We understood it. 582 00:28:48,193 --> 00:28:51,096 But what about D itself? 583 00:28:51,096 --> 00:28:54,299 Well, when you see something that's an activated process, 584 00:28:54,299 --> 00:28:57,769 there's only one person that ever comes to mind, right? 585 00:28:57,769 --> 00:29:02,207 First name S, last name A. And it's 586 00:29:02,207 --> 00:29:06,044 Svante, because you know that if it's an activated process, 587 00:29:06,044 --> 00:29:08,213 then there's some pre-exponential factor 588 00:29:08,213 --> 00:29:12,150 times the activation energy divided by k B T 589 00:29:12,150 --> 00:29:14,119 if it's for a single atom. 590 00:29:14,119 --> 00:29:17,422 Or if it's a mole, you use R. 591 00:29:17,422 --> 00:29:20,992 There's your activation energy right there. 592 00:29:20,992 --> 00:29:23,228 But so now, we can start connecting this. 593 00:29:23,228 --> 00:29:27,065 So one thing you know about that diffusion constant is it 594 00:29:27,065 --> 00:29:30,168 will have a temperature dependence. 595 00:29:30,168 --> 00:29:35,440 And you also know now that if you plot the log of it versus 1 596 00:29:35,440 --> 00:29:38,343 over T, you will get that. 597 00:29:38,343 --> 00:29:39,010 Oh, we're back. 598 00:29:39,010 --> 00:29:41,379 We're in our happy place with this. 599 00:29:41,379 --> 00:29:43,915 It's Svante. 600 00:29:43,915 --> 00:29:50,221 So this is an activated process for an atom diffusing. 601 00:29:50,221 --> 00:29:54,926 These kind of atoms diffusing through their own lattice, 602 00:29:54,926 --> 00:29:56,895 remember, these have high barriers. 603 00:29:56,895 --> 00:29:58,263 This might be kind of hard to do, 604 00:29:58,263 --> 00:30:00,198 but it's still going to happen. 605 00:30:00,198 --> 00:30:01,299 It's still going to happen. 606 00:30:01,299 --> 00:30:02,934 You might also get something like this. 607 00:30:02,934 --> 00:30:05,036 And this goes back to our discussions 608 00:30:05,036 --> 00:30:07,071 of defects in crystals. 609 00:30:07,071 --> 00:30:12,744 You might get an interstitial diffusion. 610 00:30:12,744 --> 00:30:15,747 So this is diffusion. 611 00:30:15,747 --> 00:30:18,583 This is the topic that we're covering. 612 00:30:18,583 --> 00:30:21,519 But now we can relate the equation 613 00:30:21,519 --> 00:30:26,257 that governs diffusion, the D, to the crystal 614 00:30:26,257 --> 00:30:29,661 and to the chemistry, because what is the barrier that that 615 00:30:29,661 --> 00:30:31,095 feels when it goes through. 616 00:30:31,095 --> 00:30:36,167 So remember, an interstitial, like carbon and iron 617 00:30:36,167 --> 00:30:37,468 to make steel. 618 00:30:37,468 --> 00:30:42,540 That would be an interstitial defect or an interstitial atom 619 00:30:42,540 --> 00:30:46,678 that you've added to the system, to the iron lattice. 620 00:30:46,678 --> 00:30:48,479 And how does it move? 621 00:30:48,479 --> 00:30:50,481 Well, that's absolutely critical, 622 00:30:50,481 --> 00:30:52,417 because it tells you how to make it. 623 00:30:52,417 --> 00:30:54,552 And we'll get to an example of that. 624 00:30:54,552 --> 00:30:58,756 So if I look now at these plots-- 625 00:30:58,756 --> 00:31:01,626 and these are plots I think I showed you back when we talked 626 00:31:01,626 --> 00:31:03,828 about interstitial defects. 627 00:31:03,828 --> 00:31:06,998 You can really connect it to what we've learned. 628 00:31:06,998 --> 00:31:11,102 So just to start, these are the log diffusion coefficients. 629 00:31:11,102 --> 00:31:12,570 So that's the diffusion coefficient 630 00:31:12,570 --> 00:31:14,539 that we're talking about right there. 631 00:31:14,539 --> 00:31:18,009 Versus 1,000 over T-- so it's 1 over T scaled by 1,000. 632 00:31:18,009 --> 00:31:24,215 And these are those beautiful Svante Arrhenius dependencies. 633 00:31:24,215 --> 00:31:26,251 And these are experimental measurements. 634 00:31:26,251 --> 00:31:30,054 And look, here's hydrogen in BCC iron. 635 00:31:30,054 --> 00:31:31,689 Here's hydrogen in FCC iron. 636 00:31:31,689 --> 00:31:36,394 Now, notice, at the same temperature, 637 00:31:36,394 --> 00:31:37,862 there's a huge difference. 638 00:31:37,862 --> 00:31:42,300 And then, I have FCC up here, and then I've 639 00:31:42,300 --> 00:31:44,035 got BCC over there. 640 00:31:44,035 --> 00:31:44,535 Why? 641 00:31:44,535 --> 00:31:47,171 Well, those are the stable phases at those temperatures. 642 00:31:47,171 --> 00:31:49,374 But notice that when they're at the same temperature, 643 00:31:49,374 --> 00:31:50,275 there's a huge difference. 644 00:31:50,275 --> 00:31:50,775 Why? 645 00:31:50,775 --> 00:31:52,844 Because you know from the crystal 646 00:31:52,844 --> 00:31:56,981 structure that there's more room. 647 00:31:56,981 --> 00:32:03,154 So the activation barrier will be lower in BCC. 648 00:32:03,154 --> 00:32:04,522 Those voids are bigger. 649 00:32:04,522 --> 00:32:05,290 There's more space. 650 00:32:05,290 --> 00:32:07,992 It's not as well packed. 651 00:32:07,992 --> 00:32:10,862 We've covered that. 652 00:32:10,862 --> 00:32:11,562 And look at this. 653 00:32:11,562 --> 00:32:13,231 So there's hydrogen, it's really small. 654 00:32:13,231 --> 00:32:18,102 There's iron, iron in BCC Fe compared to hydrogen in BCC Fe. 655 00:32:18,102 --> 00:32:19,837 Look at those orders of magnitude. 656 00:32:19,837 --> 00:32:22,707 So there's an iron atom diffusing in its own solid. 657 00:32:22,707 --> 00:32:23,775 And here's hydrogen atoms. 658 00:32:23,775 --> 00:32:26,077 You can see there's carbon and so forth. 659 00:32:26,077 --> 00:32:29,213 So this is the picture that comes back. 660 00:32:29,213 --> 00:32:31,382 If we want to think about diffusion in solids, 661 00:32:31,382 --> 00:32:34,485 we can connect it to the packing and crystal structures, 662 00:32:34,485 --> 00:32:37,288 because we know that these atoms, these interstitial atoms 663 00:32:37,288 --> 00:32:41,526 are going to diffuse through the interstitial space. 664 00:32:41,526 --> 00:32:43,027 What is the interstitial space? 665 00:32:43,027 --> 00:32:48,766 Well here, as just and example, I showed just BCC and FCC. 666 00:32:48,766 --> 00:32:49,500 Here they are. 667 00:32:49,500 --> 00:32:51,102 Here's is an octahedral site. 668 00:32:51,102 --> 00:32:53,071 This is BCC down here. 669 00:32:53,071 --> 00:32:58,443 So notice, the shaded in atoms are the lattice sites, BCC, 670 00:32:58,443 --> 00:33:01,012 at the corners and in the middle. 671 00:33:01,012 --> 00:33:05,350 And now, you see, the red atom is in the middle. 672 00:33:05,350 --> 00:33:06,651 Gesundheit. 673 00:33:06,651 --> 00:33:08,119 Let's say the red atom was here. 674 00:33:08,119 --> 00:33:09,153 And it's moving to there. 675 00:33:09,153 --> 00:33:10,655 And now it's in the middle. 676 00:33:10,655 --> 00:33:13,157 So it might take that path as it diffuses. 677 00:33:13,157 --> 00:33:15,793 It's going to find the voids. 678 00:33:15,793 --> 00:33:18,129 Diffusing atoms find the voids. 679 00:33:18,129 --> 00:33:20,365 And so if you have something that's trying to diffuse, 680 00:33:20,365 --> 00:33:22,000 it's going to want to find these voids. 681 00:33:22,000 --> 00:33:25,503 This is an octahedral site, because this is an octahedron. 682 00:33:25,503 --> 00:33:28,039 And you can have one of those in an FCC crystals as well. 683 00:33:28,039 --> 00:33:29,540 You can have an octahedral side. 684 00:33:29,540 --> 00:33:33,478 So in FCC, you've got lattice sites in the face 685 00:33:33,478 --> 00:33:34,912 but not in the center. 686 00:33:34,912 --> 00:33:38,750 And so its octahedral site is here. 687 00:33:38,750 --> 00:33:39,951 You've got tetrahedral sites. 688 00:33:39,951 --> 00:33:44,088 Those are other voids in the crystals. 689 00:33:44,088 --> 00:33:47,392 We touched on this a little bit when we talked about defects 690 00:33:47,392 --> 00:33:48,026 in crystals. 691 00:33:48,026 --> 00:33:50,528 But now I want to relate it to diffusion, 692 00:33:50,528 --> 00:34:00,238 because how a crystal has voids is critical to D, how something 693 00:34:00,238 --> 00:34:01,906 diffuses in it, whether that something 694 00:34:01,906 --> 00:34:03,307 is something in the crystal itself 695 00:34:03,307 --> 00:34:06,411 or something you've added to it. 696 00:34:06,411 --> 00:34:10,915 And how it has voids depends on the bonding, the chemistry, 697 00:34:10,915 --> 00:34:13,618 and the crystal structure. 698 00:34:13,618 --> 00:34:15,553 So you could answer questions like this. 699 00:34:15,553 --> 00:34:22,326 These are some just fairly straightforward questions 700 00:34:22,326 --> 00:34:23,226 that you could answer. 701 00:34:26,030 --> 00:34:30,234 What if I had carbon-- this is what I just showed you-- 702 00:34:30,234 --> 00:34:33,704 in alpha or gamma iron? 703 00:34:33,704 --> 00:34:35,940 Well, so I'll just write the answer. 704 00:34:35,940 --> 00:34:41,179 It's alpha ion, because it's an open-- 705 00:34:41,179 --> 00:34:42,647 I don't need to calculate stuff. 706 00:34:45,183 --> 00:34:48,018 Which one is BCC, which one is FCC? 707 00:34:48,018 --> 00:34:51,522 And it's alpha, because it's an open BCC structure. 708 00:34:51,522 --> 00:34:56,393 And so the diffusion barriers are going to be lower. 709 00:34:56,393 --> 00:34:59,964 Iron in alpha iron at 500 or 900? 710 00:34:59,964 --> 00:35:02,700 Well, now I've got a temperature question. 711 00:35:02,700 --> 00:35:09,073 I know D is going to be higher, so 900 degrees C 712 00:35:09,073 --> 00:35:12,477 because of influence of temperature. 713 00:35:12,477 --> 00:35:16,747 Iron or hydrogen in alpha iron. 714 00:35:16,747 --> 00:35:18,382 It's like what we just talked about. 715 00:35:18,382 --> 00:35:20,885 So this would be particle size. 716 00:35:20,885 --> 00:35:22,787 H is smaller. 717 00:35:26,557 --> 00:35:30,828 These are conceptual questions. 718 00:35:30,828 --> 00:35:31,896 What's the fourth one? 719 00:35:31,896 --> 00:35:37,001 Silicon vacancy in silicon or helium in silicon. 720 00:35:37,001 --> 00:35:40,304 Well, for a vacancy to move, you need 721 00:35:40,304 --> 00:35:42,607 those kind of same atoms move. 722 00:35:42,607 --> 00:35:46,944 And if a same moving through those sites, that's harder. 723 00:35:46,944 --> 00:35:51,449 That's harder than if a small interstitial like hydrogen 724 00:35:51,449 --> 00:35:54,719 is moving through those voids. 725 00:35:54,719 --> 00:35:56,654 That's why, when you look at this, 726 00:35:56,654 --> 00:36:00,491 iron is orders and orders of magnitude slower. 727 00:36:00,491 --> 00:36:02,260 The D is orders and orders of magnitude 728 00:36:02,260 --> 00:36:07,498 slower than carbon or hydrogen. And so that's 729 00:36:07,498 --> 00:36:12,703 going to be He, because the vacancy diffusion is so much 730 00:36:12,703 --> 00:36:13,538 slower. 731 00:36:13,538 --> 00:36:16,240 And then five, Mg, this is a good one, 732 00:36:16,240 --> 00:36:20,545 because hydrogen in platinum or hydrogen in magnesium. 733 00:36:20,545 --> 00:36:23,181 Well, magnesium has a larger atomic radius. 734 00:36:31,656 --> 00:36:32,523 And you look it up. 735 00:36:32,523 --> 00:36:34,759 And you say, they've got the same close-packed crystal 736 00:36:34,759 --> 00:36:35,960 structures. 737 00:36:35,960 --> 00:36:39,664 But one of them has a larger radius. 738 00:36:39,664 --> 00:36:40,932 What does that mean? 739 00:36:40,932 --> 00:36:45,169 Well, it says something about the voids as well. 740 00:36:45,169 --> 00:36:48,306 The voids are going to be larger. 741 00:36:48,306 --> 00:36:51,309 Remember, this is happening through the voids. 742 00:36:51,309 --> 00:36:52,843 This is happening through the voids. 743 00:36:52,843 --> 00:36:56,881 This is how batteries are engineered. 744 00:36:56,881 --> 00:37:02,420 This is why we have a revolution in batteries. 745 00:37:02,420 --> 00:37:04,622 So this is a couple of years old. 746 00:37:04,622 --> 00:37:06,424 There so many great review articles 747 00:37:06,424 --> 00:37:08,226 on battery technologies. 748 00:37:12,964 --> 00:37:13,831 This is the cathode. 749 00:37:13,831 --> 00:37:15,066 By the way, what is a battery? 750 00:37:15,066 --> 00:37:20,137 Well, a lithium battery is basically two sponges. 751 00:37:20,137 --> 00:37:24,508 And one of them is usually carbon. 752 00:37:24,508 --> 00:37:26,043 And when you're running your battery, 753 00:37:26,043 --> 00:37:28,879 lithium ions are going in. 754 00:37:28,879 --> 00:37:31,549 And the carbon is so strong and able 755 00:37:31,549 --> 00:37:35,586 to hold those lithium in and not break down. 756 00:37:35,586 --> 00:37:41,792 So we often use carbon on the anode side. 757 00:37:41,792 --> 00:37:44,195 And then when we charge it up, we 758 00:37:44,195 --> 00:37:46,030 push them back over to where? 759 00:37:46,030 --> 00:37:47,498 Where they came from. 760 00:37:47,498 --> 00:37:50,935 That's the cathode, which is a lithium compound. 761 00:37:50,935 --> 00:37:55,539 But it has to be a material that can hold lithium 762 00:37:55,539 --> 00:37:58,109 in it, hopefully at very high concentration, 763 00:37:58,109 --> 00:38:03,781 but also be OK to lose it and not completely collapse. 764 00:38:03,781 --> 00:38:06,984 And so you've got lithium ion phosphate, 765 00:38:06,984 --> 00:38:08,352 those are these ones. 766 00:38:08,352 --> 00:38:10,221 You've got manganates. 767 00:38:10,221 --> 00:38:13,557 You've got cobalts, cobalt oxide materials. 768 00:38:13,557 --> 00:38:17,495 There are so many battery materials. 769 00:38:17,495 --> 00:38:19,530 They all work the same way. 770 00:38:19,530 --> 00:38:21,799 If it's a lithium battery, the lithium 771 00:38:21,799 --> 00:38:25,403 is going to leave some crystal that has lithium in it. 772 00:38:25,403 --> 00:38:27,305 But the crystal has to stay. 773 00:38:27,305 --> 00:38:30,241 So it's like a sponge that loses something from it 774 00:38:30,241 --> 00:38:31,242 but stays intact. 775 00:38:31,242 --> 00:38:33,844 And that's something goes over to the anode. 776 00:38:33,844 --> 00:38:35,746 It gets soaked up there. 777 00:38:35,746 --> 00:38:39,050 And you can recharge, discharge, recharge, 778 00:38:39,050 --> 00:38:42,853 as you're just pushing these lithiums back and forth. 779 00:38:42,853 --> 00:38:45,790 Oh, but you're pushing them back and forth. 780 00:38:45,790 --> 00:38:48,092 You are diffusing them. 781 00:38:48,092 --> 00:38:49,694 That's what you're doing. 782 00:38:49,694 --> 00:38:52,363 You are diffusing them through whatever structure 783 00:38:52,363 --> 00:38:54,832 you're putting them into. 784 00:38:54,832 --> 00:38:57,868 So if it's the cathode, and you're 785 00:38:57,868 --> 00:39:01,739 trying to get the lithium to come back in or to go back out, 786 00:39:01,739 --> 00:39:05,376 then you can imagine that the barriers and the dimensionality 787 00:39:05,376 --> 00:39:07,645 itself are crucial. 788 00:39:07,645 --> 00:39:10,281 What is an activation energy for a lithium 789 00:39:10,281 --> 00:39:13,784 atom in a given material? 790 00:39:13,784 --> 00:39:15,653 And does it take a 1 D path? 791 00:39:15,653 --> 00:39:19,457 It turns out these can be extremely efficient. 792 00:39:19,457 --> 00:39:20,057 Look at those. 793 00:39:20,057 --> 00:39:21,592 They're practically holes, they're 794 00:39:21,592 --> 00:39:23,327 lines in the lithium ion phosphate. 795 00:39:23,327 --> 00:39:25,396 It's a really nice cathode material, 796 00:39:25,396 --> 00:39:26,831 because you've got these channels. 797 00:39:26,831 --> 00:39:30,968 These high conduction channels, these highways. 798 00:39:30,968 --> 00:39:34,071 But then you say, well, but maybe I need 3D voids. 799 00:39:34,071 --> 00:39:35,473 Maybe it shouldn't have channels. 800 00:39:35,473 --> 00:39:40,111 It should have voids, so it can choose more paths. 801 00:39:40,111 --> 00:39:41,245 Maybe we have planes. 802 00:39:41,245 --> 00:39:43,647 Maybe it's a layered structure where they have these 2D 803 00:39:43,647 --> 00:39:45,282 planes they can travel through. 804 00:39:45,282 --> 00:39:46,250 What's the best? 805 00:39:46,250 --> 00:39:50,121 Well, it's not trivial, because it's 806 00:39:50,121 --> 00:39:51,822 one of these, as usual, constrained 807 00:39:51,822 --> 00:39:54,692 optimization problems, where, when you make the pathway 808 00:39:54,692 --> 00:39:58,329 faster, you might need to have more voids where 809 00:39:58,329 --> 00:40:00,531 you'll lose material. 810 00:40:00,531 --> 00:40:03,234 Or maybe you make the material unstable. 811 00:40:03,234 --> 00:40:06,036 So you want it to have a really high density. 812 00:40:08,572 --> 00:40:10,775 The lithium diffusion and a battery material 813 00:40:10,775 --> 00:40:12,143 is related to the charge time. 814 00:40:15,479 --> 00:40:18,149 So here's another paper from a few years ago. 815 00:40:18,149 --> 00:40:22,153 If you plot materials-- now, here's the specific power. 816 00:40:22,153 --> 00:40:25,456 And so that's how much power, watts per weight. 817 00:40:25,456 --> 00:40:26,991 Here's the watt hours. 818 00:40:26,991 --> 00:40:30,194 So that's the energy per weight. 819 00:40:30,194 --> 00:40:31,996 And this is the charge rate. 820 00:40:31,996 --> 00:40:33,397 This is the charge rate. 821 00:40:33,397 --> 00:40:37,201 And notice, so you want it to be very, very fast-charging. 822 00:40:37,201 --> 00:40:38,436 And these are good materials. 823 00:40:38,436 --> 00:40:42,540 There's the lead-acid battery that's still in your car. 824 00:40:42,540 --> 00:40:45,142 These are nickel metal hydrides and so forth, 825 00:40:45,142 --> 00:40:46,177 other technologies. 826 00:40:46,177 --> 00:40:49,146 Here are the lithium ion batteries. 827 00:40:49,146 --> 00:40:50,548 This is what's in your cell phone. 828 00:40:50,548 --> 00:40:55,786 Notice something, as I get up to faster charge rates, 829 00:40:55,786 --> 00:41:02,493 more open voids in material, I lose energy density. 830 00:41:02,493 --> 00:41:04,195 I go this way. 831 00:41:04,195 --> 00:41:06,063 I don't want to go that way. 832 00:41:06,063 --> 00:41:08,365 I want to go this way. 833 00:41:08,365 --> 00:41:11,969 Constrained optimization, these are the things. 834 00:41:11,969 --> 00:41:14,972 What goes into solving these problems 835 00:41:14,972 --> 00:41:19,510 is calculating diffusion barriers and energy densities 836 00:41:19,510 --> 00:41:21,045 and all the other things that matter. 837 00:41:21,045 --> 00:41:23,481 But if you want to target fast charging, 838 00:41:23,481 --> 00:41:26,417 you've got to know the diffusion barriers. 839 00:41:26,417 --> 00:41:29,787 One of the number one challenges in battery design 840 00:41:29,787 --> 00:41:34,425 is to keep that high while continuing to push up that way. 841 00:41:34,425 --> 00:41:37,795 Polymers, you could imagine that a polymer is slower. 842 00:41:37,795 --> 00:41:39,897 It makes sense, right? 843 00:41:39,897 --> 00:41:41,799 Because it's harder-- as we talked 844 00:41:41,799 --> 00:41:43,300 about when we talked about polymers, 845 00:41:43,300 --> 00:41:46,704 it's harder to make crystalline polymers, 846 00:41:46,704 --> 00:41:50,674 to keep a polymer from having some amorphous region. 847 00:41:50,674 --> 00:41:54,144 After all, you've got these 100,000-unit long strands 848 00:41:54,144 --> 00:41:56,113 of spaghetti. 849 00:41:56,113 --> 00:41:58,249 So it kind of makes sense that the diffusion 850 00:41:58,249 --> 00:42:00,284 is going to be lower in a polymer 851 00:42:00,284 --> 00:42:03,587 than in one of these crystals, like an olivine, 852 00:42:03,587 --> 00:42:05,022 lithium ion phosphate. 853 00:42:05,022 --> 00:42:06,023 So that makes sense. 854 00:42:06,023 --> 00:42:09,760 But polymers would be great, because they're lighter. 855 00:42:09,760 --> 00:42:12,596 And they can be flexible. 856 00:42:12,596 --> 00:42:14,765 And so we like polymer batteries a lot. 857 00:42:14,765 --> 00:42:17,768 We want them for many applications. 858 00:42:17,768 --> 00:42:20,137 So these are the kinds of tradeoffs that you think about. 859 00:42:20,137 --> 00:42:23,240 And D is right there in the center, because none of us 860 00:42:23,240 --> 00:42:25,609 want a battery that takes two days to charge. 861 00:42:28,212 --> 00:42:33,651 And now we get to the next part, which I'll just introduce, 862 00:42:33,651 --> 00:42:37,187 which is, now we're going to think about time. 863 00:42:37,187 --> 00:42:39,790 And I want to set this problem up. 864 00:42:39,790 --> 00:42:42,626 And then we'll solve it on Wednesday and then 865 00:42:42,626 --> 00:42:44,395 and then finish. 866 00:42:44,395 --> 00:42:46,664 But I want to set it up, because the next thing that's 867 00:42:46,664 --> 00:42:49,466 related to diffusion is time. 868 00:42:49,466 --> 00:42:51,035 And I want to give you an example 869 00:42:51,035 --> 00:42:53,304 of how that's so important with something 870 00:42:53,304 --> 00:42:55,940 called case hardening. 871 00:42:55,940 --> 00:42:57,641 So if you buy a gear-- 872 00:42:57,641 --> 00:43:00,477 and you do buy gears all the time. 873 00:43:00,477 --> 00:43:03,280 Whether you like it or not or know it or not, lots of stuff 874 00:43:03,280 --> 00:43:05,950 has gears in it. 875 00:43:05,950 --> 00:43:07,518 And you look at the gear closely, 876 00:43:07,518 --> 00:43:09,186 it would look different on the outside. 877 00:43:09,186 --> 00:43:12,356 If you do a cross-section, most gears 878 00:43:12,356 --> 00:43:15,326 are going to look different on the outside. 879 00:43:15,326 --> 00:43:18,529 And the reason has to do with exactly what we just 880 00:43:18,529 --> 00:43:20,030 talked about. 881 00:43:20,030 --> 00:43:23,834 If this is a piece of iron, pure iron, 882 00:43:23,834 --> 00:43:25,836 it's going to be too soft. 883 00:43:25,836 --> 00:43:27,037 So you're now turning this. 884 00:43:27,037 --> 00:43:31,742 And it's got to have mechanical integrity. 885 00:43:31,742 --> 00:43:35,613 And so you've got to make it harder. 886 00:43:35,613 --> 00:43:38,282 But the thing is, if you put it in and it's pure iron, 887 00:43:38,282 --> 00:43:38,949 it's too soft. 888 00:43:38,949 --> 00:43:41,719 But now if you make it this really ultra-high strength, 889 00:43:41,719 --> 00:43:46,490 carbon-infused steel, it gets too brittle. 890 00:43:46,490 --> 00:43:47,858 It gets too brittle. 891 00:43:47,858 --> 00:43:49,760 And so you need a balance. 892 00:43:49,760 --> 00:43:53,397 And one of the ways to balance the mechanical strength 893 00:43:53,397 --> 00:43:57,067 of something like a gear or many, many other materials 894 00:43:57,067 --> 00:43:58,969 is to harden just the outside. 895 00:43:58,969 --> 00:44:00,304 That's cold case hardening. 896 00:44:00,304 --> 00:44:02,506 The case, it's the case. 897 00:44:02,506 --> 00:44:03,641 How do you do it? 898 00:44:03,641 --> 00:44:05,576 Well, you get carbon. 899 00:44:05,576 --> 00:44:09,613 Remember, carbon going into the iron makes it a lot harder, 900 00:44:09,613 --> 00:44:11,015 but it also makes it more brittle. 901 00:44:11,015 --> 00:44:14,084 So if you made the whole gear the same amount 902 00:44:14,084 --> 00:44:17,154 of carbon infused in iron, is going to be too brittle. 903 00:44:17,154 --> 00:44:19,723 If you don't put any on, it's not mechanically strong enough. 904 00:44:19,723 --> 00:44:23,260 So what you do is you very, very delicately 905 00:44:23,260 --> 00:44:27,665 expose this piece of iron to carbon. 906 00:44:27,665 --> 00:44:30,567 You harden it, but just on the outside. 907 00:44:30,567 --> 00:44:33,003 Now, how does the carbon get in? 908 00:44:33,003 --> 00:44:34,171 Diffusion. 909 00:44:34,171 --> 00:44:37,941 It gets into the material through diffusion. 910 00:44:37,941 --> 00:44:41,445 So then the question at your gear-making plant 911 00:44:41,445 --> 00:44:48,152 that you have to know is, what temperature 912 00:44:48,152 --> 00:44:53,557 and how long should I expose this gear to how much carbon. 913 00:44:53,557 --> 00:44:55,526 You've got concentration. 914 00:44:55,526 --> 00:44:56,794 You've got temperature. 915 00:44:56,794 --> 00:44:58,729 And you've got time. 916 00:44:58,729 --> 00:45:01,598 And if you know that, you can actually 917 00:45:01,598 --> 00:45:03,867 process this material in just the way you want. 918 00:45:03,867 --> 00:45:06,203 And you have to know those parameters. 919 00:45:06,203 --> 00:45:10,107 And that is Fick's second law. 920 00:45:10,107 --> 00:45:14,244 So these are steel pipes. 921 00:45:14,244 --> 00:45:16,947 And they are going to be case hardened. 922 00:45:16,947 --> 00:45:17,781 So what's happening? 923 00:45:17,781 --> 00:45:20,451 You've got an iron with a certain amount 924 00:45:20,451 --> 00:45:22,519 of carbon content. 925 00:45:22,519 --> 00:45:26,056 But now, I need to add more carbon but just 926 00:45:26,056 --> 00:45:31,895 the right amount to the exterior of those steel pipes. 927 00:45:31,895 --> 00:45:33,130 How long should I do it? 928 00:45:33,130 --> 00:45:36,100 And that is Fick's second law, which I'll write down. 929 00:45:38,669 --> 00:45:40,571 So how do we figure that out? 930 00:45:40,571 --> 00:45:45,075 Well, we use the time dependent version of Fick's law. 931 00:45:45,075 --> 00:45:57,254 And Fick's second law says that the change in concentration 932 00:45:57,254 --> 00:46:02,960 with time is equal to the diffusion constant times 933 00:46:02,960 --> 00:46:09,099 the second derivative of the concentration with position. 934 00:46:09,099 --> 00:46:10,534 That's Fick's second law. 935 00:46:10,534 --> 00:46:17,207 And so this is non-steady state, which means-- 936 00:46:17,207 --> 00:46:18,642 and we're not going to derive this. 937 00:46:18,642 --> 00:46:21,411 But I do want you to know that this is the Fick's 938 00:46:21,411 --> 00:46:23,680 second equation and how you might apply it, 939 00:46:23,680 --> 00:46:26,216 which is why we'll do a problem with it. 940 00:46:26,216 --> 00:46:28,085 But it's non-steady state. 941 00:46:28,085 --> 00:46:34,057 And that means that the concentration 942 00:46:34,057 --> 00:46:35,459 that you're going to get is going 943 00:46:35,459 --> 00:46:39,730 to be a function of position and time. 944 00:46:39,730 --> 00:46:42,599 It's going to be a function of both. 945 00:46:42,599 --> 00:46:45,002 So now, I can answer the question. 946 00:46:45,002 --> 00:46:52,409 Remember before, I had to keep the concentrations fixed 947 00:46:52,409 --> 00:46:54,945 to apply Fick's first law. 948 00:46:54,945 --> 00:46:57,047 Now I can answer the question. 949 00:46:57,047 --> 00:47:01,485 Well, OK, if I have certain parameters, 950 00:47:01,485 --> 00:47:04,555 what is the concentration going to be. 951 00:47:04,555 --> 00:47:06,089 What is the concentration going to be 952 00:47:06,089 --> 00:47:08,525 at some time and some place? 953 00:47:08,525 --> 00:47:11,228 And that's exactly the question I have to answer. 954 00:47:11,228 --> 00:47:16,533 How much carbon am I going to have at what 955 00:47:16,533 --> 00:47:20,704 position in the case hardening? 956 00:47:20,704 --> 00:47:23,407 In what position of the pipe exterior, 957 00:47:23,407 --> 00:47:25,642 how much carbon am I going to have at what time? 958 00:47:25,642 --> 00:47:27,911 How long should I leave it in the oven? 959 00:47:27,911 --> 00:47:30,080 That's the question I'm trying to answer. 960 00:47:30,080 --> 00:47:32,749 And by the way, what temperature? 961 00:47:32,749 --> 00:47:37,988 And so the thing that Fick's second law helps us understand 962 00:47:37,988 --> 00:47:40,791 is, if I start with some C S that's 963 00:47:40,791 --> 00:47:48,665 the starting concentration, and I have a starting concentration 964 00:47:48,665 --> 00:47:56,240 C 0, and I expose at the surface some higher concentration-- 965 00:47:56,240 --> 00:47:59,209 remember, before, I kept these fixed. 966 00:47:59,209 --> 00:47:59,977 I kept these fixed. 967 00:47:59,977 --> 00:48:01,245 And I got a flux. 968 00:48:01,245 --> 00:48:03,580 Now, I'm going to be able to get this. 969 00:48:03,580 --> 00:48:09,052 So this line here is time equals 0. 970 00:48:09,052 --> 00:48:14,558 But now I can actually get this profile. 971 00:48:14,558 --> 00:48:24,101 Time 2, time 3, that's the position through some surface. 972 00:48:24,101 --> 00:48:27,004 So this is what Fick's second law tells us. 973 00:48:27,004 --> 00:48:31,575 It tells us these profiles, the concentration in the material. 974 00:48:31,575 --> 00:48:36,313 And it just simply is a time-dependent version 975 00:48:36,313 --> 00:48:37,080 of diffusion. 976 00:48:37,080 --> 00:48:38,548 OK, what we're going to do is we're 977 00:48:38,548 --> 00:48:40,117 going to pick up with this. 978 00:48:40,117 --> 00:48:42,486 We'll do this example of case hardening. 979 00:48:42,486 --> 00:48:44,922 We'll do a few more comments on the class. 980 00:48:44,922 --> 00:48:47,658 I'll tell you a little bit about the final. 981 00:48:47,658 --> 00:48:51,528 And please bring your laptops on Wednesday for the last 15 982 00:48:51,528 --> 00:48:52,029 minutes. 983 00:48:52,029 --> 00:48:55,499 We'll reserve for you to do evaluations.