1 00:00:03,010 --> 00:00:06,520 So I would like to tell you about friction today. 2 00:00:06,520 --> 00:00:08,910 Friction is a very mysterious subject, 3 00:00:08,910 --> 00:00:13,460 because it has such simple rules at the macroscopic scale, 4 00:00:13,460 --> 00:00:15,560 and yet-- as I will show you and tell you 5 00:00:15,560 --> 00:00:17,850 a little bit about today-- very different rules 6 00:00:17,850 --> 00:00:19,750 at the nanoscopic scale. 7 00:00:19,750 --> 00:00:34,125 So I would like to tell you about friction at the nano 8 00:00:34,125 --> 00:00:34,625 scale. 9 00:00:38,340 --> 00:00:41,090 So at the macro scale, there are classic rules that 10 00:00:41,090 --> 00:00:43,380 were discovered very early on. 11 00:00:43,380 --> 00:00:49,020 For macroscopic objects, it was actually 12 00:00:49,020 --> 00:00:53,290 Leonardo da Vinci who found the first laws 13 00:00:53,290 --> 00:00:56,010 of so-called "dry" friction. 14 00:00:56,010 --> 00:01:00,460 So this was da Vinci as early as in the 15th century. 15 00:01:03,880 --> 00:01:09,810 And basically, he discovered that friction depends 16 00:01:09,810 --> 00:01:14,730 on the force, on the weight of the object on the surface, 17 00:01:14,730 --> 00:01:17,150 and that there's a certain coefficient that 18 00:01:17,150 --> 00:01:20,750 depends on the material. 19 00:01:20,750 --> 00:01:23,810 Da Vinci did not publish his notes. 20 00:01:23,810 --> 00:01:27,060 They were just hidden in his notebooks for a long time. 21 00:01:27,060 --> 00:01:33,100 And Amontons in the 17th century rediscovered the very same laws 22 00:01:33,100 --> 00:01:34,680 that da Vinci had found. 23 00:01:37,259 --> 00:01:41,490 And these are two very simple laws. 24 00:01:41,490 --> 00:01:44,509 The first law is that if you have an object sliding 25 00:01:44,509 --> 00:01:50,270 on a surface-- so there's a force that you are pulling 26 00:01:50,270 --> 00:01:54,080 on the object, and there's the friction force acting 27 00:01:54,080 --> 00:01:56,780 on it in the opposite direction-- then 28 00:01:56,780 --> 00:02:00,980 Amontons found that the friction force is 29 00:02:00,980 --> 00:02:04,990 proportional to the normal force. 30 00:02:04,990 --> 00:02:07,158 So the normal force is the weight of the object. 31 00:02:11,370 --> 00:02:13,030 So he found that the friction force 32 00:02:13,030 --> 00:02:14,710 as a function of normal force is just 33 00:02:14,710 --> 00:02:16,430 the simple interdependence. 34 00:02:16,430 --> 00:02:19,230 So if you make the object twice as heavy, 35 00:02:19,230 --> 00:02:23,310 the friction force will be twice as large. 36 00:02:23,310 --> 00:02:25,220 This makes, maybe, sense. 37 00:02:25,220 --> 00:02:28,430 It agrees with our intuition that maybe an object 38 00:02:28,430 --> 00:02:32,200 twice as heavy should have a force twice as large. 39 00:02:32,200 --> 00:02:37,540 The second law of friction is much more surprising-- namely, 40 00:02:37,540 --> 00:02:43,100 that if you have two objects of the same weight-- 41 00:02:43,100 --> 00:02:50,150 so you have the same normal force-- but very different 42 00:02:50,150 --> 00:02:54,640 areas of contact, then the friction force is the same. 43 00:02:54,640 --> 00:02:57,340 So if you have a friction force for this object-- 44 00:02:57,340 --> 00:02:59,110 let me call it friction force one-- 45 00:02:59,110 --> 00:03:01,420 and the friction force of this object-- friction 46 00:03:01,420 --> 00:03:06,350 force two-- then the friction forces are the same. 47 00:03:06,350 --> 00:03:08,290 One equals friction force two. 48 00:03:08,290 --> 00:03:10,490 So surprisingly, the friction force 49 00:03:10,490 --> 00:03:12,370 is independent of contact area. 50 00:03:23,340 --> 00:03:24,870 This seems very counter-intuitive, 51 00:03:24,870 --> 00:03:27,950 because if you think of friction as some kind of sticking, 52 00:03:27,950 --> 00:03:31,520 some kind of effect where two surfaces rub together, 53 00:03:31,520 --> 00:03:34,180 then we might naively assume that the larger the surface, 54 00:03:34,180 --> 00:03:37,400 the larger the friction force. 55 00:03:37,400 --> 00:03:40,720 So one maybe intuition of why this might not 56 00:03:40,720 --> 00:03:42,640 be the case, or one possible explanation, 57 00:03:42,640 --> 00:03:45,310 is that surfaces always have roughness. 58 00:03:45,310 --> 00:03:47,920 And so the actual contact area between two 59 00:03:47,920 --> 00:03:52,180 surfaces-- so if I really draw the surface of the object 60 00:03:52,180 --> 00:03:55,140 here-- then the actual contact area between two surfaces 61 00:03:55,140 --> 00:03:59,520 might not be the observed area of the macroscopic object. 62 00:03:59,520 --> 00:04:03,190 And so this may be one way to explain why the friction is 63 00:04:03,190 --> 00:04:05,450 independent of the area, because it might only 64 00:04:05,450 --> 00:04:09,892 be proportional to the actual contact area of the system. 65 00:04:09,892 --> 00:04:11,600 We have already described two of the laws 66 00:04:11,600 --> 00:04:13,280 of macroscopic friction. 67 00:04:13,280 --> 00:04:18,750 There's a third law that was actually discovered by Coulomb. 68 00:04:18,750 --> 00:04:22,860 And again, this is a little bit surprising result-- namely, 69 00:04:22,860 --> 00:04:27,480 that for this kind of friction, the dry friction, 70 00:04:27,480 --> 00:04:29,480 it is independent of the velocity of the object. 71 00:04:29,480 --> 00:04:31,146 So this is, for instance, something that 72 00:04:31,146 --> 00:04:32,540 is not true for air resistance. 73 00:04:32,540 --> 00:04:35,200 If you drive a car, the faster you go, 74 00:04:35,200 --> 00:04:37,790 the more friction there is, the more air resistance there is. 75 00:04:37,790 --> 00:04:41,280 But if you have two surfaces sliding on top of each other, 76 00:04:41,280 --> 00:04:44,640 then Coulomb found that the friction force 77 00:04:44,640 --> 00:04:45,765 is independent of velocity. 78 00:04:58,590 --> 00:05:01,150 Again, this is something quite counter-- 79 00:05:01,150 --> 00:05:03,290 not quite, but somewhat counter-intuitive. 80 00:05:03,290 --> 00:05:05,820 And as I will show you a little bit later, 81 00:05:05,820 --> 00:05:08,330 it actually is no longer true when 82 00:05:08,330 --> 00:05:10,680 you look at the nano scale. 83 00:05:10,680 --> 00:05:18,020 Coulomb was also the first one to distinguish static friction, 84 00:05:18,020 --> 00:05:21,770 which is the force with which an object resists 85 00:05:21,770 --> 00:05:23,252 motion for a while. 86 00:05:23,252 --> 00:05:25,460 You make the force larger and larger, so for a while, 87 00:05:25,460 --> 00:05:26,910 the object will stick. 88 00:05:26,910 --> 00:05:28,580 And then it will start sliding. 89 00:05:28,580 --> 00:05:31,460 So he distinguished static friction from dynamic friction. 90 00:05:37,500 --> 00:05:39,610 And because we have this law that 91 00:05:39,610 --> 00:05:41,310 says that the frictional force is 92 00:05:41,310 --> 00:05:42,730 proportional to the normal force, 93 00:05:42,730 --> 00:05:48,800 I will remind you of the first law-- the friction force being 94 00:05:48,800 --> 00:05:51,350 proportional to the normal force, 95 00:05:51,350 --> 00:05:54,645 like so-- we can define a slope. 96 00:05:58,110 --> 00:06:00,220 So we can write the friction force 97 00:06:00,220 --> 00:06:04,540 as some slope, mu, times the normal force. 98 00:06:04,540 --> 00:06:06,970 These are both forces, so they have the same units, 99 00:06:06,970 --> 00:06:11,721 which means that the so-called coefficient of friction-- which 100 00:06:11,721 --> 00:06:13,220 could be therefore static frictional 101 00:06:13,220 --> 00:06:16,920 or for dynamic friction, for kinetic friction-- 102 00:06:16,920 --> 00:06:19,210 this coefficient is just a number. 103 00:06:19,210 --> 00:06:30,070 And typically, this number is between 0.3 and 0.6. 104 00:06:30,070 --> 00:06:32,100 This makes a lot of sense. 105 00:06:32,100 --> 00:06:35,350 The friction force is not quite as strong as the normal force. 106 00:06:35,350 --> 00:06:39,520 So the friction force with which an object resists motion 107 00:06:39,520 --> 00:06:41,790 is not quite as large as the normal force 108 00:06:41,790 --> 00:06:44,990 with which it pushes down on a certain surface. 109 00:06:44,990 --> 00:06:48,510 However, surprisingly, there are materials where 110 00:06:48,510 --> 00:06:50,380 mu is actually larger than 1. 111 00:06:50,380 --> 00:06:55,470 It's possible, for instance, for rubber. 112 00:06:55,470 --> 00:06:59,420 Which means that an object can be harder to pull sideways 113 00:06:59,420 --> 00:07:04,300 than actually-- it can be harder to pull an object sideways 114 00:07:04,300 --> 00:07:07,270 than to actually lift it from the surface. 115 00:07:07,270 --> 00:07:10,980 So this is a surprising result. 116 00:07:10,980 --> 00:07:15,270 Now, it turns out that despite these very simple laws, 117 00:07:15,270 --> 00:07:18,520 our fundamental understanding of friction is rather limited. 118 00:07:18,520 --> 00:07:20,420 So for instance, nobody in the world 119 00:07:20,420 --> 00:07:23,250 can predict this coefficient mu of friction. 120 00:07:23,250 --> 00:07:26,670 Nobody in the world can, from microscopic or principle 121 00:07:26,670 --> 00:07:30,120 say-- being told what the two surfaces are, 122 00:07:30,120 --> 00:07:34,170 say plastic or wood-- what is the friction coefficient. 123 00:07:34,170 --> 00:07:35,610 Can you calculate it for me? 124 00:07:35,610 --> 00:07:36,990 We cannot predict it. 125 00:07:36,990 --> 00:07:39,630 We can only measure it, and then tabulate it-- make a table-- 126 00:07:39,630 --> 00:07:42,490 and say this kind of object on this kind of surface 127 00:07:42,490 --> 00:07:45,920 has this kind of coefficient for friction. 128 00:07:45,920 --> 00:07:48,890 So that's one, if you want, failure of physics. 129 00:07:48,890 --> 00:07:52,730 How is it possible that we can't describe, and quantitatively 130 00:07:52,730 --> 00:07:54,950 describe, something as simple as that? 131 00:07:54,950 --> 00:07:57,100 Also, in many instances, we would 132 00:07:57,100 --> 00:07:59,120 like to change the coefficient to friction. 133 00:07:59,120 --> 00:08:01,490 Typically, we would like to reduce friction, 134 00:08:01,490 --> 00:08:06,000 because friction is something where we dissipate energy. 135 00:08:06,000 --> 00:08:15,650 It is estimated that as much as 3% of the nation's GDP 136 00:08:15,650 --> 00:08:24,260 is "wasted" on friction. 137 00:08:24,260 --> 00:08:29,810 So friction destroys energy in the amount of billions 138 00:08:29,810 --> 00:08:33,340 and billions of dollars-- cars driving on streets, 139 00:08:33,340 --> 00:08:35,400 machines working, and so on. 140 00:08:35,400 --> 00:08:37,308 Sometimes friction is, of course, desirable. 141 00:08:37,308 --> 00:08:39,240 If you want to stop your car on the road, 142 00:08:39,240 --> 00:08:41,600 you want friction to actually work. 143 00:08:41,600 --> 00:08:44,760 You wouldn't like to do away with it completely. 144 00:08:44,760 --> 00:08:49,270 But in many instances, it's just a source of energy loss. 145 00:08:49,270 --> 00:08:53,010 So if we could even make a small change in the friction, 146 00:08:53,010 --> 00:08:55,770 this would be not only important for science, 147 00:08:55,770 --> 00:08:59,330 but it might also be very important for technology. 148 00:08:59,330 --> 00:09:01,550 So here, I've told you about friction 149 00:09:01,550 --> 00:09:04,390 of macroscopic objects. 150 00:09:04,390 --> 00:09:07,890 Now people-- chemists, physicists, other scientists, 151 00:09:07,890 --> 00:09:11,110 engineers-- are able to build smaller and smaller machines. 152 00:09:11,110 --> 00:09:14,230 And there's tremendous interest to build nanoscopic machines-- 153 00:09:14,230 --> 00:09:18,210 machines that are maybe only a few molecules or a small number 154 00:09:18,210 --> 00:09:20,930 of molecules or atoms wide and big. 155 00:09:20,930 --> 00:09:22,430 And there's been tremendous success. 156 00:09:22,430 --> 00:09:27,210 People have built nano pumps-- tiny pumps-- even a tiny car, 157 00:09:27,210 --> 00:09:29,810 consisting of a relatively small number of molecules 158 00:09:29,810 --> 00:09:32,830 that can move on the surface if you inject the electrons. 159 00:09:32,830 --> 00:09:38,390 So as I would like now to tell you about how bad friction 160 00:09:38,390 --> 00:09:39,550 is at the nano scale. 161 00:09:39,550 --> 00:09:42,700 So let's ask the following question-- 162 00:09:42,700 --> 00:09:47,812 how bad is friction at the nano scale? 163 00:09:51,910 --> 00:09:54,090 Your general intuition might be that friction 164 00:09:54,090 --> 00:09:58,330 gets worse at the nano scale, simply because the bulk-- 165 00:09:58,330 --> 00:10:00,940 the volume of an object-- is what 166 00:10:00,940 --> 00:10:03,320 determines how many atoms are in it, 167 00:10:03,320 --> 00:10:05,410 whereas friction is a surface effect. 168 00:10:05,410 --> 00:10:08,020 And because the volume increases more quickly 169 00:10:08,020 --> 00:10:10,610 with the size of the object then does the surface, 170 00:10:10,610 --> 00:10:12,730 you might think that probably, there's 171 00:10:12,730 --> 00:10:14,780 a bigger problem with friction at the nano scale 172 00:10:14,780 --> 00:10:17,930 than there is for macroscopic objects. 173 00:10:17,930 --> 00:10:20,170 To illustrate how bad this really is, 174 00:10:20,170 --> 00:10:22,635 let's consider a tire of a car. 175 00:10:25,490 --> 00:10:29,830 So this tire is rolling as your car is driving. 176 00:10:29,830 --> 00:10:31,990 And we can try to estimate what happens 177 00:10:31,990 --> 00:10:34,970 to the wear of this tire. 178 00:10:34,970 --> 00:10:42,100 So let's say that this tire maybe wears off. 179 00:10:42,100 --> 00:10:46,830 Wear is on the order of maybe, let's say, 180 00:10:46,830 --> 00:10:58,870 10 millimeters every, maybe, 50,000 kilometers. 181 00:10:58,870 --> 00:11:02,790 So that means we take off a number of atomic layers 182 00:11:02,790 --> 00:11:05,640 here every 50,000 kilometers. 183 00:11:05,640 --> 00:11:08,510 Now, how many atomic layers is 10 millimeter? 184 00:11:08,510 --> 00:11:15,140 If we say that one atomic layer is 185 00:11:15,140 --> 00:11:18,330 on the order of, let's call it 1 Angstrom-- which is 10 186 00:11:18,330 --> 00:11:24,660 to the minus 10 meters-- then 10 millimeters 187 00:11:24,660 --> 00:11:28,870 is 10 to the minus 2 meters. 188 00:11:28,870 --> 00:11:31,180 So that is equal-- since one layer 189 00:11:31,180 --> 00:11:34,280 is 10 to minus 10 meters-- that is equal to 10 190 00:11:34,280 --> 00:11:36,180 to the 8 atomic layers. 191 00:11:42,270 --> 00:11:51,260 We can also ask ourself, how far has the tire been moving? 192 00:11:51,260 --> 00:11:56,260 So if we assume that the circumference of the tire-- so 193 00:11:56,260 --> 00:11:58,480 this is the radius of the tire. 194 00:11:58,480 --> 00:12:01,010 Let's guess that maybe the circumference of the tire here, 195 00:12:01,010 --> 00:12:06,810 which is given by 2 pi r, is on the order of 1 meter. 196 00:12:06,810 --> 00:12:12,000 Then this means that 50,000 kilometers-- or 5 times 10 197 00:12:12,000 --> 00:12:20,300 to the 7 meters, which is 50,000 kilometers-- 198 00:12:20,300 --> 00:12:27,728 is basically 5 times 10 to the 7 revolutions. 199 00:12:32,970 --> 00:12:37,800 So in 50 million revolutions-- 5 times 10 to the 7 revolutions-- 200 00:12:37,800 --> 00:12:40,690 we lose 10 to the 8 atomic layers. 201 00:12:40,690 --> 00:12:46,760 So it comes out that we lose one to two atomic layers 202 00:12:46,760 --> 00:12:47,385 per revolution. 203 00:12:51,880 --> 00:12:54,180 If we had used maybe a size of an atom 204 00:12:54,180 --> 00:12:55,730 more closely as two Angstroms, we 205 00:12:55,730 --> 00:12:57,930 would come out at about one atomic layer. 206 00:12:57,930 --> 00:13:00,100 So this is a remarkable effect. 207 00:13:00,100 --> 00:13:04,160 Even when we drive our car, each time the tire rolls around, 208 00:13:04,160 --> 00:13:06,289 we leave one atomic layer on the street. 209 00:13:06,289 --> 00:13:08,580 For a macroscopic object, as we see, it doesn't matter. 210 00:13:08,580 --> 00:13:11,540 We can drive a very large macroscopic distance 211 00:13:11,540 --> 00:13:14,330 before we wear off a few millimeters of the tire. 212 00:13:14,330 --> 00:13:16,930 But if the tire of the car was itself 213 00:13:16,930 --> 00:13:19,570 only a few atomic layers big, then you 214 00:13:19,570 --> 00:13:22,070 see we would have a huge problem. 215 00:13:22,070 --> 00:13:24,610 So this is why people are interested in understanding, 216 00:13:24,610 --> 00:13:29,140 and maybe manipulating, friction at the nano scale. 217 00:13:29,140 --> 00:13:32,360 So what does friction at the nano scale really look like? 218 00:13:32,360 --> 00:13:34,730 And why are people interested in it 219 00:13:34,730 --> 00:13:40,180 after so many centuries of first studying friction? 220 00:13:40,180 --> 00:13:43,410 One reason is that now one can do experiments 221 00:13:43,410 --> 00:13:44,348 at the nano scale. 222 00:13:52,540 --> 00:13:55,190 And the new tool that has enabled that 223 00:13:55,190 --> 00:13:57,668 is what is called an atomic force microscope. 224 00:14:02,670 --> 00:14:04,520 What is an atomic force microscope? 225 00:14:04,520 --> 00:14:07,310 Well, it's essentially a very, very sharp tip-- 226 00:14:07,310 --> 00:14:08,480 atomically sharp tip. 227 00:14:08,480 --> 00:14:12,550 So you have a surface here, which consists of atoms. 228 00:14:15,590 --> 00:14:16,901 And these are individual atoms. 229 00:14:19,830 --> 00:14:22,040 And you're trying to study the friction. 230 00:14:22,040 --> 00:14:26,400 And now what you do is, you bring an object here shaped 231 00:14:26,400 --> 00:14:29,850 in the form of the tip, which also consists of atoms. 232 00:14:29,850 --> 00:14:32,500 And the object is so sharp that you have, 233 00:14:32,500 --> 00:14:36,690 ideally, just one atom near the surface-- one 234 00:14:36,690 --> 00:14:39,480 or more atoms near the surface. 235 00:14:39,480 --> 00:14:41,960 So this is what an atomic force microscope is. 236 00:14:41,960 --> 00:14:45,720 You can pull on this with some force. 237 00:14:45,720 --> 00:14:48,910 And then you can try to measure the force of resistance-- 238 00:14:48,910 --> 00:14:54,810 the friction force that is due to the interaction 239 00:14:54,810 --> 00:14:57,940 between the atom near the surface and the atoms 240 00:14:57,940 --> 00:14:59,660 forming the surface. 241 00:14:59,660 --> 00:15:02,060 So we might call this the substrate. 242 00:15:05,060 --> 00:15:07,460 And this would be the moving object. 243 00:15:12,240 --> 00:15:14,560 And we can either study static friction 244 00:15:14,560 --> 00:15:17,760 by pulling the objects, but not with a force large enough 245 00:15:17,760 --> 00:15:22,060 to move, or maybe we can apply a larger force 246 00:15:22,060 --> 00:15:24,460 so the object starts moving at a certain velocity v, 247 00:15:24,460 --> 00:15:27,330 and we can measure friction. 248 00:15:27,330 --> 00:15:29,020 Now this, as simple as it may seem, 249 00:15:29,020 --> 00:15:31,430 is still too complicated for physicists. 250 00:15:31,430 --> 00:15:34,610 So we try to make the model even simpler. 251 00:15:34,610 --> 00:15:36,500 So how do physicists do this? 252 00:15:39,310 --> 00:15:44,580 Well, they say that the surface still consists of atoms, 253 00:15:44,580 --> 00:15:47,490 but now we're mostly interested what happens to this tip. 254 00:15:47,490 --> 00:15:49,450 We're going to say, well, somehow, 255 00:15:49,450 --> 00:15:52,700 what matters is that this atom is bound to the tip. 256 00:15:52,700 --> 00:15:56,430 So maybe we can model it in the following way. 257 00:15:56,430 --> 00:16:00,200 We have here the macroscopic part of the tip. 258 00:16:00,200 --> 00:16:03,480 And then we're going to say that the surface atom is really 259 00:16:03,480 --> 00:16:06,570 bound in some way to the tip. 260 00:16:06,570 --> 00:16:10,060 And the simplest way that we can imagine the surface atom 261 00:16:10,060 --> 00:16:15,070 to be bound to the tip is via some spring 262 00:16:15,070 --> 00:16:17,960 with some specific spring constant. 263 00:16:17,960 --> 00:16:20,020 So basically, the one atom that makes 264 00:16:20,020 --> 00:16:24,350 the contact with the surface is bound via spring 265 00:16:24,350 --> 00:16:26,510 to the rest of a tip that is moving-- 266 00:16:26,510 --> 00:16:28,480 to the rest of the moving object. 267 00:16:28,480 --> 00:16:32,374 So this is how we view the tip as physicists, to make a model. 268 00:16:32,374 --> 00:16:34,540 And then how do we view the surface to make a model? 269 00:16:34,540 --> 00:16:36,770 Well, we say this is really a periodic arrangement 270 00:16:36,770 --> 00:16:37,750 of the atoms. 271 00:16:37,750 --> 00:16:41,240 So maybe it makes sense to kind of assume 272 00:16:41,240 --> 00:16:44,740 that the associated potential is also 273 00:16:44,740 --> 00:16:47,690 periodic at the atomic scale. 274 00:16:47,690 --> 00:16:51,560 So this is the potential as a function of position. 275 00:16:51,560 --> 00:17:06,760 So basically, we model friction as spring 276 00:17:06,760 --> 00:17:08,429 plus periodic potential. 277 00:17:15,108 --> 00:17:17,400 This is a very simple model. 278 00:17:17,400 --> 00:17:21,550 And it was first introduced now about 90 years ago. 279 00:17:21,550 --> 00:17:28,170 So by Prandtl-- by a German physicist 280 00:17:28,170 --> 00:17:32,400 called Prandtl-- and independently 281 00:17:32,400 --> 00:17:35,910 by another physicist called Tomlinson. 282 00:17:35,910 --> 00:17:39,070 And this is called the Prandtl-Tomlinson model. 283 00:17:39,070 --> 00:17:42,450 And they discovered it, or introduced it, 284 00:17:42,450 --> 00:17:46,577 in 1928 and 1929. 285 00:17:46,577 --> 00:17:48,410 And it turns out that this very simple model 286 00:17:48,410 --> 00:17:50,710 of a spring and the periodic potential 287 00:17:50,710 --> 00:17:55,090 captures much of the essence of nano friction. 288 00:17:55,090 --> 00:17:56,970 So as physicists, now we have a spring 289 00:17:56,970 --> 00:17:58,660 and the periodic potential. 290 00:17:58,660 --> 00:18:00,430 How do we think about this? 291 00:18:00,430 --> 00:18:04,900 Well, one way is to think about the energy in the system. 292 00:18:04,900 --> 00:18:08,900 So we can plot the energy as a function of position. 293 00:18:08,900 --> 00:18:14,720 And now we have our periodic potential, like so. 294 00:18:14,720 --> 00:18:16,440 And how do we model the spring? 295 00:18:16,440 --> 00:18:19,150 Well, this is going to be the potential energy, v. How 296 00:18:19,150 --> 00:18:20,360 do we model the spring? 297 00:18:20,360 --> 00:18:23,290 Well, the spring has a linear force. 298 00:18:23,290 --> 00:18:25,400 The force is proportional to the displacement, 299 00:18:25,400 --> 00:18:28,500 which means the potential is quadratic in this placement. 300 00:18:28,500 --> 00:18:31,510 So this is the potential for spring. 301 00:18:31,510 --> 00:18:33,200 This is what the spring does, and this 302 00:18:33,200 --> 00:18:34,571 is what the substrate does. 303 00:18:37,130 --> 00:18:41,160 And what we need to do is, we need to add these two together. 304 00:18:41,160 --> 00:18:43,780 So what that means qualitatively is 305 00:18:43,780 --> 00:18:46,040 that the total potential of the system 306 00:18:46,040 --> 00:18:47,832 might look something like this. 307 00:18:54,740 --> 00:18:57,710 So as we now translate the spring 308 00:18:57,710 --> 00:19:03,930 across the surface with some velocity, 309 00:19:03,930 --> 00:19:06,390 then you can see that this addition 310 00:19:06,390 --> 00:19:11,200 between the fixed substrate potential-- this one is fixed-- 311 00:19:11,200 --> 00:19:14,120 and this spring potential is moving. 312 00:19:14,120 --> 00:19:17,200 You can see that this will lead to a time-varying potential 313 00:19:17,200 --> 00:19:18,520 for the object. 314 00:19:18,520 --> 00:19:21,620 So let's look now, in a simulation, what that time 315 00:19:21,620 --> 00:19:22,750 variation might look like. 316 00:19:26,050 --> 00:19:28,600 So what you see here is a combination 317 00:19:28,600 --> 00:19:31,192 of a spring and the periodic potential. 318 00:19:31,192 --> 00:19:32,650 And the periodic potential has been 319 00:19:32,650 --> 00:19:34,370 chosen a slightly different strength 320 00:19:34,370 --> 00:19:35,660 than I'm showing you here. 321 00:19:35,660 --> 00:19:38,670 It has been chosen weaker so that instead 322 00:19:38,670 --> 00:19:41,340 of many minima in this total potential, 323 00:19:41,340 --> 00:19:42,840 there are only two minima. 324 00:19:42,840 --> 00:19:44,610 And what you see happening in this system 325 00:19:44,610 --> 00:19:47,610 is that, as the particle is moving, 326 00:19:47,610 --> 00:19:51,900 the spring tries to pull it across a maximum 327 00:19:51,900 --> 00:19:53,470 of the periodic potential. 328 00:19:53,470 --> 00:19:55,820 At some point, the atom is released, 329 00:19:55,820 --> 00:19:58,860 and it releases energy that is taken up 330 00:19:58,860 --> 00:20:01,500 as heat by the substrate. 331 00:20:01,500 --> 00:20:05,840 And then, the object is pulled towards the next minimum. 332 00:20:05,840 --> 00:20:07,810 The minimum disappears slowly. 333 00:20:07,810 --> 00:20:09,940 The object is released again, et cetera. 334 00:20:09,940 --> 00:20:12,330 So basically, in this model, we can understand friction 335 00:20:12,330 --> 00:20:15,140 as the external force pulling the object 336 00:20:15,140 --> 00:20:18,120 over successive maxima. 337 00:20:18,120 --> 00:20:19,160 So that's nice. 338 00:20:19,160 --> 00:20:21,930 In this model we can understand why there is heat generated. 339 00:20:21,930 --> 00:20:25,290 There's always heat generated when the atom loses 340 00:20:25,290 --> 00:20:27,970 this extra energy, because it is stuck 341 00:20:27,970 --> 00:20:31,340 for a while in a minimum of the potential, which 342 00:20:31,340 --> 00:20:33,510 is not the absolute minimum of the potential. 343 00:20:33,510 --> 00:20:35,130 And then the moment it's released, 344 00:20:35,130 --> 00:20:38,510 it releases kinetic energy that is converted into heat. 345 00:20:38,510 --> 00:20:42,490 Now we can have a different situation. 346 00:20:42,490 --> 00:20:49,295 So this was for strong or moderately strong potential. 347 00:20:59,150 --> 00:21:01,710 So basically, this was a situation 348 00:21:01,710 --> 00:21:06,780 where there are just two minima in the potential, like so. 349 00:21:06,780 --> 00:21:10,380 We can make the potential a little bit weaker. 350 00:21:10,380 --> 00:21:14,700 And in that case, the curvature of this potential-- 351 00:21:14,700 --> 00:21:18,440 of the periodic potential-- the curvature in one direction 352 00:21:18,440 --> 00:21:21,090 might not be enough to overcome the opposite curvature 353 00:21:21,090 --> 00:21:22,050 of the spring. 354 00:21:22,050 --> 00:21:24,760 So in this case, we can end up with a potential just slightly 355 00:21:24,760 --> 00:21:27,935 distorted, but we have only one minimum. 356 00:21:33,020 --> 00:21:34,730 So in the next movie, I will show you 357 00:21:34,730 --> 00:21:37,590 what happens when we have just one minimum. 358 00:21:37,590 --> 00:21:39,730 In that case, you can see that, because there 359 00:21:39,730 --> 00:21:44,670 is no second minimum in the system, the object-- 360 00:21:44,670 --> 00:21:47,170 in this case, the tip-- follows quite smoothly 361 00:21:47,170 --> 00:21:48,760 the minimum of the potential. 362 00:21:48,760 --> 00:21:50,420 And no energy is released. 363 00:21:50,420 --> 00:21:52,630 No heat is released in the problem. 364 00:21:52,630 --> 00:21:54,800 This model-- this Prandtl-Tomlinson model-- 365 00:21:54,800 --> 00:21:55,960 is interesting. 366 00:21:55,960 --> 00:22:02,220 If in this model, we plot the friction force 367 00:22:02,220 --> 00:22:09,790 versus the corrugation of the potential corrugation-- 368 00:22:09,790 --> 00:22:12,440 so we'll call this the corrugation of the potential, 369 00:22:12,440 --> 00:22:17,180 u, which is proportional to the normal force 370 00:22:17,180 --> 00:22:20,950 for a macroscopic object-- then what we find is, 371 00:22:20,950 --> 00:22:23,020 yes, we find a linear dependence. 372 00:22:23,020 --> 00:22:25,360 So this is, if you want, the corrugation 373 00:22:25,360 --> 00:22:27,560 or the normal force. 374 00:22:27,560 --> 00:22:30,280 What we find is a linear dependence, but only 375 00:22:30,280 --> 00:22:32,890 above a certain critical value. 376 00:22:32,890 --> 00:22:38,350 So the friction force actually looks like this. 377 00:22:38,350 --> 00:22:41,760 It's 0 until the potential becomes strong enough. 378 00:22:41,760 --> 00:22:44,490 Basically, you can think of this curvature becoming larger 379 00:22:44,490 --> 00:22:46,070 than that curvature. 380 00:22:46,070 --> 00:22:48,830 And then, the friction force sets in. 381 00:22:48,830 --> 00:22:50,333 So this is at the nano scale. 382 00:22:52,960 --> 00:22:56,180 And our simple macroscopic friction law 383 00:22:56,180 --> 00:23:01,190 would have predicted something of the same slope 384 00:23:01,190 --> 00:23:04,170 at the macro scale-- something of the same slope 385 00:23:04,170 --> 00:23:05,880 with a [INAUDIBLE] offset. 386 00:23:05,880 --> 00:23:08,244 So we can kind of see that, at the nano scale, 387 00:23:08,244 --> 00:23:10,160 the friction is a little bit more complicated. 388 00:23:10,160 --> 00:23:12,770 There's a region where there's no friction whatsoever. 389 00:23:12,770 --> 00:23:15,530 But then it increases linearly with the normal force. 390 00:23:15,530 --> 00:23:19,250 So you can imagine that if I go to macroscopic normal loads, 391 00:23:19,250 --> 00:23:21,840 then the difference between these curves, at least 392 00:23:21,840 --> 00:23:25,376 fractionally, will be quite small. 393 00:23:25,376 --> 00:23:27,880 The difference between these curves, at least fractionally, 394 00:23:27,880 --> 00:23:29,170 will be quite small. 395 00:23:29,170 --> 00:23:33,340 And so we can see how the law of the nano scale, 396 00:23:33,340 --> 00:23:36,320 explained by this very simple Prandtl-Tomlinson model, 397 00:23:36,320 --> 00:23:38,570 approaches the law at the macro scale. 398 00:23:38,570 --> 00:23:41,730 So far, I have told you about something very simple-- 399 00:23:41,730 --> 00:23:46,972 namely when the contact is a single atom. 400 00:23:50,310 --> 00:23:52,550 And even making this very simple approximation, 401 00:23:52,550 --> 00:23:55,190 we can already understand why the friction 402 00:23:55,190 --> 00:23:57,930 force is approximately proportional 403 00:23:57,930 --> 00:23:59,890 to the normal force. 404 00:23:59,890 --> 00:24:03,420 Now in real life, probably there is more than one atom 405 00:24:03,420 --> 00:24:05,010 touching the surface. 406 00:24:05,010 --> 00:24:15,740 So let's consider a contact area where several atoms-- maybe 407 00:24:15,740 --> 00:24:18,950 a long chain, maybe just a few atoms, in the case of nanoscale 408 00:24:18,950 --> 00:24:22,110 probably just a few atoms-- several atoms will be making up 409 00:24:22,110 --> 00:24:23,370 the contact. 410 00:24:23,370 --> 00:24:27,350 So instead of considering this situation 411 00:24:27,350 --> 00:24:33,140 with one atom on a spring, which is the single atom case, 412 00:24:33,140 --> 00:24:38,340 now let's consider a situation where several atoms make up 413 00:24:38,340 --> 00:24:39,810 the contact. 414 00:24:39,810 --> 00:24:43,300 So again, here we have the substrate 415 00:24:43,300 --> 00:24:46,430 with our periodic potential that is ultimately coming 416 00:24:46,430 --> 00:24:48,050 from the individual atoms. 417 00:24:48,050 --> 00:24:53,946 But now we will consider more than one atom making up 418 00:24:53,946 --> 00:24:54,445 the surface. 419 00:24:58,710 --> 00:25:05,010 So in this case, how do we model the system? 420 00:25:05,010 --> 00:25:08,670 Well, we think that these atoms are still connected by springs 421 00:25:08,670 --> 00:25:09,985 to the macroscopic object. 422 00:25:12,630 --> 00:25:17,280 But typically, these atoms will also have forces between them. 423 00:25:17,280 --> 00:25:20,330 So this is my physicist's model of what 424 00:25:20,330 --> 00:25:23,530 happens when more than one atom touches the surface. 425 00:25:23,530 --> 00:25:25,690 Now these are only masses of springs 426 00:25:25,690 --> 00:25:27,530 and some simple periodic potential, 427 00:25:27,530 --> 00:25:30,880 and yet the situation is quite complex, and in many ways, 428 00:25:30,880 --> 00:25:32,690 counter-intuitive. 429 00:25:32,690 --> 00:25:36,440 And the first thing that changes compared to the single atom 430 00:25:36,440 --> 00:25:39,780 is that now I have the distance between the atoms 431 00:25:39,780 --> 00:25:41,140 as a parameter. 432 00:25:41,140 --> 00:25:46,370 So if I label the period of the periodic potential 433 00:25:46,370 --> 00:25:53,470 as a, and maybe the period of my object of the distance 434 00:25:53,470 --> 00:25:56,816 between the atoms in the object as d, 435 00:25:56,816 --> 00:25:59,730 then I can have different situations. 436 00:25:59,730 --> 00:26:07,520 So one very simple case is when d 437 00:26:07,520 --> 00:26:10,360 is equal to a-- when the two periods are the same-- 438 00:26:10,360 --> 00:26:14,630 or in general, when d is a multiple integer of a, 439 00:26:14,630 --> 00:26:15,743 and n is an integer. 440 00:26:20,000 --> 00:26:22,280 So let's see what happens in this case. 441 00:26:22,280 --> 00:26:25,390 Very naively, all the atoms are doing the same thing 442 00:26:25,390 --> 00:26:27,090 relative to the surface. 443 00:26:27,090 --> 00:26:29,150 So you might expect them, maybe, to move 444 00:26:29,150 --> 00:26:30,710 in exactly the same way. 445 00:26:30,710 --> 00:26:32,200 They will all move together. 446 00:26:32,200 --> 00:26:35,220 And these springs between them will not stretch. 447 00:26:35,220 --> 00:26:38,020 In this case, I can forget about the springs 448 00:26:38,020 --> 00:26:41,880 between the atoms in this simple case 449 00:26:41,880 --> 00:26:45,190 that we'll call commensurate. 450 00:26:45,190 --> 00:26:49,710 Basically, when the period of the object 451 00:26:49,710 --> 00:26:51,440 matches the period of the substrate-- 452 00:26:51,440 --> 00:26:57,770 the commensurate case-- then the atoms are 453 00:26:57,770 --> 00:26:59,870 at equivalent positions throughout the substrate 454 00:26:59,870 --> 00:27:02,560 period, the lattice of the springs between them 455 00:27:02,560 --> 00:27:04,270 will not stretch. 456 00:27:04,270 --> 00:27:06,870 And it's as if they are not there. 457 00:27:07,730 --> 00:27:09,300 So in the commensurate case, what 458 00:27:09,300 --> 00:27:11,920 you can see here is, as the atoms are pulled 459 00:27:11,920 --> 00:27:14,620 across the periodic potential, because 460 00:27:14,620 --> 00:27:17,750 of this commensurability condition where d is equal to a 461 00:27:17,750 --> 00:27:19,570 or a multiple integer of a-- basically, 462 00:27:19,570 --> 00:27:21,830 two periods are matched-- all the atoms 463 00:27:21,830 --> 00:27:23,580 are doing the same thing at the same time. 464 00:27:23,580 --> 00:27:25,440 So they get pulled, pulled, pulled. 465 00:27:25,440 --> 00:27:30,960 They first stick, and then they all slip at the same time. 466 00:27:30,960 --> 00:27:33,380 And you can imagine-- and you can kind of see visually 467 00:27:33,380 --> 00:27:36,310 in this case-- that the friction is 468 00:27:36,310 --> 00:27:39,940 the same as the single atom friction multiplied simply 469 00:27:39,940 --> 00:27:40,890 by n. 470 00:27:40,890 --> 00:27:43,070 So this is just the single atom friction 471 00:27:43,070 --> 00:27:45,560 that we have seen before multiplied 472 00:27:45,560 --> 00:27:48,242 by just the number of atoms that makes up the contact area. 473 00:27:49,640 --> 00:27:54,080 Now there's a different case, which 474 00:27:54,080 --> 00:28:04,960 we might call incommensurate, where d is not equal to a, 475 00:28:04,960 --> 00:28:07,900 and d is not an integer multiple of a. 476 00:28:07,900 --> 00:28:13,380 So maybe d could be 1.5 a, or 2/3 of a, or some other number. 477 00:28:13,380 --> 00:28:15,310 And what is the most incommensurate case 478 00:28:15,310 --> 00:28:16,630 that we can imagine? 479 00:28:16,630 --> 00:28:22,070 Well, the most incommensurate case for d and a 480 00:28:22,070 --> 00:28:25,716 is that the ratio of d and a is an irrational number. 481 00:28:31,190 --> 00:28:32,940 So an irrational number would be something 482 00:28:32,940 --> 00:28:37,510 like square root of 2, or for our purposes, 483 00:28:37,510 --> 00:28:39,930 it turns out that the most irrational number 484 00:28:39,930 --> 00:28:46,030 in a mathematical sense is 1/2 squared of 5 plus 1. 485 00:28:46,030 --> 00:28:47,560 This is the so-called Golden Ratio. 486 00:28:50,860 --> 00:28:54,530 The ancient Greeks believed that it had magical properties. 487 00:28:54,530 --> 00:28:57,670 For instance, when they built the temples, the two 488 00:28:57,670 --> 00:29:00,710 sides of the temples-- the two sides of the rectangle-- 489 00:29:00,710 --> 00:29:03,830 were related by this very strange number, 490 00:29:03,830 --> 00:29:05,430 the so-called Golden Ratio. 491 00:29:05,430 --> 00:29:08,956 It is assumed to be aesthetically very pleasing. 492 00:29:08,956 --> 00:29:10,830 Now mathematically speaking, the Golden Ratio 493 00:29:10,830 --> 00:29:15,750 is very interesting, because it is the most irrational number, 494 00:29:15,750 --> 00:29:20,560 in some sense, that you can devise. 495 00:29:20,560 --> 00:29:23,770 Now we can choose such an incommensurate ratio 496 00:29:23,770 --> 00:29:26,783 of d over a, and see what happens in this case. 497 00:29:27,850 --> 00:29:30,640 So you can see here that, in the case 498 00:29:30,640 --> 00:29:34,480 of an incommensurate ratio, the behavior of the transport-- 499 00:29:34,480 --> 00:29:36,230 the behavior of the motion of the object-- 500 00:29:36,230 --> 00:29:37,710 changes dramatically. 501 00:29:37,710 --> 00:29:40,320 Instead of all the atoms sticking and slipping together, 502 00:29:40,320 --> 00:29:42,570 like we had in the commensurate case, 503 00:29:42,570 --> 00:29:45,040 now the atoms move one by one. 504 00:29:45,040 --> 00:29:48,180 Basically, the first atom moves over the barrier, the spring 505 00:29:48,180 --> 00:29:53,500 stretches, which facilitates the motion 506 00:29:53,500 --> 00:29:56,020 of the next atom over the barrier, the next atom, 507 00:29:56,020 --> 00:29:56,960 and so on. 508 00:29:56,960 --> 00:30:00,590 So these atom chains move like a caterpillar. 509 00:30:00,590 --> 00:30:02,390 They essentially move one by one. 510 00:30:02,390 --> 00:30:05,011 In technical language, we call these things kinks. 511 00:30:05,011 --> 00:30:07,010 You can see that they are periodic compressions. 512 00:30:07,010 --> 00:30:09,140 They are compressions in the chain 513 00:30:09,140 --> 00:30:10,840 and stretches in the chain. 514 00:30:10,840 --> 00:30:13,700 And so these atoms move like a caterpillar. 515 00:30:13,700 --> 00:30:17,370 And now an interesting fact is that in this case, 516 00:30:17,370 --> 00:30:19,353 friction is dramatically reduced. 517 00:30:29,300 --> 00:30:31,640 And depending on which regime you are, 518 00:30:31,640 --> 00:30:37,040 the friction can even disappear altogether at the nano scale. 519 00:30:37,040 --> 00:30:41,680 The reduction was so large that some people have even 520 00:30:41,680 --> 00:30:46,020 coined this with the word superlubricity. 521 00:30:48,610 --> 00:30:51,440 Interestingly enough, even though this is a very simple 522 00:30:51,440 --> 00:30:55,570 system, it's just, if you want, balls and springs, 523 00:30:55,570 --> 00:31:00,760 this superlubricity was only discovered in the late 1980s, 524 00:31:00,760 --> 00:31:03,130 even though we know all the physics 525 00:31:03,130 --> 00:31:05,170 for more than a century. 526 00:31:05,170 --> 00:31:08,580 So you can see just a tiny, tiny rearrangement of the atoms-- 527 00:31:08,580 --> 00:31:11,010 not to be any more commensurate with the lattice, 528 00:31:11,010 --> 00:31:13,810 but to be incommensurate in terms 529 00:31:13,810 --> 00:31:16,180 of an irrational number-- can change friction 530 00:31:16,180 --> 00:31:19,560 properties dramatically. 531 00:31:19,560 --> 00:31:21,900 In fact, there was a French scientist 532 00:31:21,900 --> 00:31:24,900 called Aubry that first pointed out 533 00:31:24,900 --> 00:31:27,690 that there's a very interesting transition that happens 534 00:31:27,690 --> 00:31:30,000 in this system of a periodic potential 535 00:31:30,000 --> 00:31:34,160 and atoms connected with springs when you choose, as the ratio 536 00:31:34,160 --> 00:31:38,470 between these two length scales, the Golden Ratio. 537 00:31:38,470 --> 00:31:42,900 And this is called the Aubry Transition. 538 00:31:42,900 --> 00:31:48,100 So far, we have considered only, if you 539 00:31:48,100 --> 00:31:51,830 want, the mechanics of the motion, which 540 00:31:51,830 --> 00:31:54,710 is equivalent to saying that we have assumed that the system is 541 00:31:54,710 --> 00:31:56,360 at temperature T equal 0. 542 00:32:01,190 --> 00:32:04,430 Basically, the atoms remains at the minimum. 543 00:32:04,430 --> 00:32:07,610 It doesn't wiggle around because of temperature, 544 00:32:07,610 --> 00:32:10,910 and we have derived the friction in this limit. 545 00:32:10,910 --> 00:32:18,830 Now what happens for finite temperature-- 546 00:32:18,830 --> 00:32:22,550 by finite, meaning a temperature that is larger than 0. 547 00:32:22,550 --> 00:32:25,970 Well, if you think back of our simple single-atom model 548 00:32:25,970 --> 00:32:28,980 of friction, where maybe an atom is stuck here 549 00:32:28,980 --> 00:32:31,940 as the potential is moving with some velocity v, 550 00:32:31,940 --> 00:32:34,550 then in the 0 temperature limit, this atom 551 00:32:34,550 --> 00:32:40,420 would only be released at the time when this minimum actually 552 00:32:40,420 --> 00:32:41,220 disappears. 553 00:32:41,220 --> 00:32:43,740 And then it would have a high energy 554 00:32:43,740 --> 00:32:46,530 and it would dissipate this energy 555 00:32:46,530 --> 00:32:48,530 to be cooled to the next minimum. 556 00:32:48,530 --> 00:32:51,220 However, if we have a finite temperature T-- 557 00:32:51,220 --> 00:32:53,520 some temperature scale T-- that means 558 00:32:53,520 --> 00:32:57,810 that the energy of the atom in the potential is not 0, 559 00:32:57,810 --> 00:32:59,050 the kinetic energy. 560 00:32:59,050 --> 00:33:02,200 But the atom has a smeared-out kinetic energy 561 00:33:02,200 --> 00:33:06,050 and potential energy that is like so. 562 00:33:06,050 --> 00:33:09,460 So what that means is that the atom can [INAUDIBLE] hop over 563 00:33:09,460 --> 00:33:12,820 the barrier and find the new minimum, without actually 564 00:33:12,820 --> 00:33:15,820 having to be pulled over this maximum. 565 00:33:15,820 --> 00:33:19,230 So what we expect is that temperature effect 566 00:33:19,230 --> 00:33:22,540 might reduce friction. 567 00:33:22,540 --> 00:33:30,222 So people call this thermolubricity-- the effect 568 00:33:30,222 --> 00:33:31,680 that when you heat up the surface-- 569 00:33:31,680 --> 00:33:34,263 and in many instances, you would have to substantially heat up 570 00:33:34,263 --> 00:33:36,720 the surface-- then friction is reduced, 571 00:33:36,720 --> 00:33:39,730 because the atom can find the new minimum of the potential 572 00:33:39,730 --> 00:33:43,070 without actually having to be pulled over the barrier. 573 00:33:43,070 --> 00:33:44,990 So let's see in a small simulation what 574 00:33:44,990 --> 00:33:45,890 that might look like. 575 00:33:46,389 --> 00:33:49,070 What you see here is now that the atom 576 00:33:49,070 --> 00:33:53,047 has a chance of hopping between the two minima, back and forth. 577 00:33:53,047 --> 00:33:54,630 And it has a certain probability to be 578 00:33:54,630 --> 00:33:56,580 found at either one of the two minima, 579 00:33:56,580 --> 00:33:59,840 which is indicated by the size of the red circle, 580 00:33:59,840 --> 00:34:00,860 in this case. 581 00:34:00,860 --> 00:34:03,580 So you can see that when temperature is present, 582 00:34:03,580 --> 00:34:06,720 then the atom can follow the minimum locally-- 583 00:34:06,720 --> 00:34:08,770 the absolute global minimum-- simply because it 584 00:34:08,770 --> 00:34:11,050 can hop back and forth between the barrier 585 00:34:11,050 --> 00:34:15,340 without actually experiencing much friction. 586 00:34:15,340 --> 00:34:18,270 So in particular in the limit when the temperature is very, 587 00:34:18,270 --> 00:34:20,370 very high, or the velocity of the atom 588 00:34:20,370 --> 00:34:24,310 is very, very low, then the atom with hop many times back 589 00:34:24,310 --> 00:34:26,010 and forth between the two minima. 590 00:34:26,010 --> 00:34:28,500 And the distribution between the two minima 591 00:34:28,500 --> 00:34:31,739 would be simply given by the Maxwell-Boltzmann distribution, 592 00:34:31,739 --> 00:34:34,790 which means that the atom will be predominantly found 593 00:34:34,790 --> 00:34:35,969 in the global minimum. 594 00:34:35,969 --> 00:34:38,360 And they'll just be following the potential along 595 00:34:38,360 --> 00:34:46,199 and the friction in this system will be quite small. 596 00:34:46,199 --> 00:34:51,340 So in this case, if we do a measurement of friction 597 00:34:51,340 --> 00:34:56,010 versus velocity in a simulator-- that we do, 598 00:34:56,010 --> 00:34:59,020 then this is what the result might look like, 599 00:34:59,020 --> 00:35:01,220 or what the result does look like. 600 00:35:01,220 --> 00:35:03,390 We see here different ranges of friction. 601 00:35:03,390 --> 00:35:06,350 We see a range where the friction does not 602 00:35:06,350 --> 00:35:09,360 depend on velocity at all. 603 00:35:09,360 --> 00:35:10,540 Friction is 0. 604 00:35:10,540 --> 00:35:12,880 Then there's a range where the friction increases 605 00:35:12,880 --> 00:35:15,490 with velocity, but only very weakly logarithmically. 606 00:35:15,490 --> 00:35:18,790 Please notice that this is a logarithmic scale 607 00:35:18,790 --> 00:35:20,190 for the velocity. 608 00:35:20,190 --> 00:35:25,170 So the velocity here changes over five orders of magnitude. 609 00:35:25,170 --> 00:35:28,280 So friction increases with increasing velocity. 610 00:35:28,280 --> 00:35:30,110 Then there's a range of frictions 611 00:35:30,110 --> 00:35:32,090 actually independent of velocity, 612 00:35:32,090 --> 00:35:35,320 just like the simple macroscopic law would predict. 613 00:35:35,320 --> 00:35:38,062 And then it turns out when you move the atom very, very fast, 614 00:35:38,062 --> 00:35:39,520 then it basically doesn't have time 615 00:35:39,520 --> 00:35:41,090 to dissipate all the energy. 616 00:35:41,090 --> 00:35:43,020 And friction is reduced again, effectively, 617 00:35:43,020 --> 00:35:44,690 because the atom is hotter. 618 00:35:44,690 --> 00:35:49,620 So we can see that the simple macroscopic law of friction 619 00:35:49,620 --> 00:35:52,770 only applies in a region, which in this experiment 620 00:35:52,770 --> 00:35:58,210 was relatively narrow, of relatively low temperatures. 621 00:35:58,210 --> 00:36:01,750 So let's summarize what happens to the loss at the macro scale. 622 00:36:06,230 --> 00:36:07,910 So the first law at the macro scale 623 00:36:07,910 --> 00:36:13,740 was that the friction force was proportional 624 00:36:13,740 --> 00:36:15,750 to the normal force. 625 00:36:15,750 --> 00:36:19,730 And we found that at the nano scale, actually what happens 626 00:36:19,730 --> 00:36:22,760 is a displaced curve, where there 627 00:36:22,760 --> 00:36:24,380 is no friction in a certain region, 628 00:36:24,380 --> 00:36:29,080 and then the friction follows parallel to the macro results. 629 00:36:29,080 --> 00:36:31,690 So this would be the actual friction at the nano scale. 630 00:36:31,690 --> 00:36:35,090 And we see that it's a fairly good approximation 631 00:36:35,090 --> 00:36:38,330 to the macroscopic friction, but with a small offset. 632 00:36:38,330 --> 00:36:44,440 Our second law of friction was that friction 633 00:36:44,440 --> 00:36:57,060 is independent of surface area, of contact area-- 634 00:36:57,060 --> 00:37:01,420 basically, the idea that a high object 635 00:37:01,420 --> 00:37:03,210 and a flat object of the same mass 636 00:37:03,210 --> 00:37:04,830 experience the same friction. 637 00:37:04,830 --> 00:37:08,360 And we see that at the nano scale, 638 00:37:08,360 --> 00:37:12,730 it's replaced by a much more complicated law that 639 00:37:12,730 --> 00:37:18,162 depends on the arrangement of the atoms, or on, if you want, 640 00:37:18,162 --> 00:37:18,870 commensurability. 641 00:37:24,590 --> 00:37:26,330 so we have non-equivalent arrangements 642 00:37:26,330 --> 00:37:28,090 of the atoms that can either lead 643 00:37:28,090 --> 00:37:29,830 to large or small friction. 644 00:37:29,830 --> 00:37:32,410 And somehow, the macroscopic law is some kind 645 00:37:32,410 --> 00:37:36,340 of average over these behaviors, if we allow for randomness 646 00:37:36,340 --> 00:37:38,370 in the positions of the surfaces. 647 00:37:38,370 --> 00:37:43,990 And finally, our third law-- that at the macroscopic scale, 648 00:37:43,990 --> 00:37:54,400 friction is independent of velocity-- 649 00:37:54,400 --> 00:38:03,370 we found at the nano scale that this can be true, 650 00:38:03,370 --> 00:38:19,730 but it is generally true only in some finite temperature range-- 651 00:38:19,730 --> 00:38:23,910 namely, that if you move the object very, very slowly, 652 00:38:23,910 --> 00:38:27,910 then thermal excitations allow you to always find 653 00:38:27,910 --> 00:38:31,790 the global minimum, and friction disappears-- the effect 654 00:38:31,790 --> 00:38:34,610 called thermolubricity. 655 00:38:34,610 --> 00:38:39,580 So one can see that there are nice connections 656 00:38:39,580 --> 00:38:41,570 between the nano scale and the macro scale. 657 00:38:41,570 --> 00:38:44,630 But many open questions remain, in particular, 658 00:38:44,630 --> 00:38:48,030 concerning the point two, which is 659 00:38:48,030 --> 00:38:50,670 that the friction is independent of the contact area. 660 00:38:50,670 --> 00:38:52,120 What does the contact area really 661 00:38:52,120 --> 00:38:56,090 look like for two macroscopic objects? 662 00:38:56,090 --> 00:39:00,490 And how is it that this independence on the surface 663 00:39:00,490 --> 00:39:03,540 area-- or at least on the apparent contact area-- 664 00:39:03,540 --> 00:39:06,160 actually arises from the properties of friction 665 00:39:06,160 --> 00:39:08,310 at the nano scale?