1 00:00:10,470 --> 00:00:12,470 PROFESSOR: So let's look at a little more detail 2 00:00:12,470 --> 00:00:16,160 at the equilibrium size of respiratory droplets 3 00:00:16,160 --> 00:00:18,950 that are emitted during breathing, 4 00:00:18,950 --> 00:00:22,030 or coughing, or speaking. 5 00:00:22,030 --> 00:00:26,070 And the key idea is that these droplets are not pure liquid. 6 00:00:26,070 --> 00:00:29,440 As explained in the wells curve, pure droplets 7 00:00:29,440 --> 00:00:32,049 that are small enough will shrink completely 8 00:00:32,049 --> 00:00:33,280 and evaporate. 9 00:00:33,280 --> 00:00:36,220 However, these droplets contain a significant amount 10 00:00:36,220 --> 00:00:39,630 of solutes. 11 00:00:39,630 --> 00:00:43,380 And those solutes, in the case of mucus coming from your lungs 12 00:00:43,380 --> 00:00:46,830 or from your vocal cords, your nasal pharynx, 13 00:00:46,830 --> 00:00:51,990 are full of proteins and other macromolecules, carbohydrates. 14 00:00:51,990 --> 00:00:55,770 And also there are always in bodily fluids plenty 15 00:00:55,770 --> 00:00:58,980 of dissolved salts such as sodium and chloride or calcium 16 00:00:58,980 --> 00:01:01,620 or potassium ions. 17 00:01:01,620 --> 00:01:04,170 Also, in saliva, many of these species are present, 18 00:01:04,170 --> 00:01:07,980 although it's not quite as thick of a liquid. 19 00:01:07,980 --> 00:01:11,970 And of course, virions as well will find themselves in here, 20 00:01:11,970 --> 00:01:14,010 and it also constitutes solutes. 21 00:01:14,010 --> 00:01:17,289 So the idea is that we don't just have a pure liquid. 22 00:01:17,289 --> 00:01:23,100 So there is some initial volume fraction 23 00:01:23,100 --> 00:01:26,110 of solute in the liquid. 24 00:01:34,170 --> 00:01:37,300 And in addition to that, most of these liquids -- 25 00:01:40,860 --> 00:01:43,289 sorry, the solutes I should say are 26 00:01:43,289 --> 00:01:51,650 charged and thus hygroscopic. 27 00:01:51,650 --> 00:01:54,650 What that means is that, of course, the salt, 28 00:01:54,650 --> 00:01:59,080 those ions are literally charged species, but also the proteins 29 00:01:59,080 --> 00:02:00,500 and other macromolecules that many 30 00:02:00,500 --> 00:02:07,100 charge residues and sites along the molecule, which 31 00:02:07,100 --> 00:02:08,270 attract water. 32 00:02:08,270 --> 00:02:10,460 And essentially, there is a layer 33 00:02:10,460 --> 00:02:14,100 of bound water solvating all of these species I just mentioned, 34 00:02:14,100 --> 00:02:14,960 including the virus. 35 00:02:14,960 --> 00:02:18,860 So I'll just sketch that there's lots 36 00:02:18,860 --> 00:02:24,210 of bound water, which is surrounding each species, 37 00:02:24,210 --> 00:02:25,220 including the virus. 38 00:02:25,220 --> 00:02:27,860 There's essentially a layer of water 39 00:02:27,860 --> 00:02:33,510 mostly around these molecules here and other species. 40 00:02:33,510 --> 00:02:38,220 So it's solutes plus the bound water 41 00:02:38,220 --> 00:02:43,190 that's coming from solvation of these molecules in liquid. 42 00:02:43,190 --> 00:02:44,950 So that water is pretty firmly attached. 43 00:02:44,950 --> 00:02:48,300 And even if you dry the material, a lot of that water 44 00:02:48,300 --> 00:02:49,260 will still be left over. 45 00:02:49,260 --> 00:02:53,960 It takes a significant amount of energy to remove it. 46 00:02:53,960 --> 00:02:56,140 And so if we think that -- 47 00:02:56,140 --> 00:03:00,280 if we describe there is initial volume of the droplet V_O, 48 00:03:00,280 --> 00:03:08,270 and a radius R_0, so let's say it's initially 49 00:03:08,270 --> 00:03:11,150 a circular droplet, there is an initial amount 50 00:03:11,150 --> 00:03:16,550 of solid, V_s, which is phi_s^0 times V_0. 51 00:03:16,550 --> 00:03:19,410 So there's a certain amount of solutes in there which cannot 52 00:03:19,410 --> 00:03:19,910 be removed. 53 00:03:19,910 --> 00:03:23,540 So the water can evaporate, but the solutes will not. 54 00:03:23,540 --> 00:03:25,130 So let's think a little bit about what 55 00:03:25,130 --> 00:03:27,110 the consequences of that are. 56 00:03:27,110 --> 00:03:29,430 So let me do a brief derivation here 57 00:03:29,430 --> 00:03:32,480 for looking at the thermodynamics of this system. 58 00:03:32,480 --> 00:03:35,120 And the key idea is just to get to the final result. 59 00:03:35,120 --> 00:03:37,630 I don't want to dwell on the details of thermodynamics. 60 00:03:37,630 --> 00:03:42,200 But an important concept here is the relative humidity 61 00:03:42,200 --> 00:03:42,829 of the air. 62 00:03:46,770 --> 00:03:48,070 So there's moisture in the air. 63 00:03:48,070 --> 00:03:49,650 There's water vapor. 64 00:03:49,650 --> 00:03:52,640 And it's at a certain level. 65 00:03:52,640 --> 00:03:59,640 So we often write that rh for relative humidity. 66 00:03:59,640 --> 00:04:03,870 And that can be defined as the concentration of vapor, 67 00:04:03,870 --> 00:04:09,830 water vapor in the air relative to the vapor concentration 68 00:04:09,830 --> 00:04:13,350 that would be in equilibrium with pure liquid. 69 00:04:13,350 --> 00:04:17,700 So when the concentration water vapor gets high enough, 70 00:04:17,700 --> 00:04:20,750 eventually you start to nucleate water droplets. 71 00:04:20,750 --> 00:04:23,150 And you start to have condensation water. 72 00:04:23,150 --> 00:04:26,760 That's essentially how rain forms from the clouds. 73 00:04:26,760 --> 00:04:28,260 So that's that ratio. 74 00:04:28,260 --> 00:04:29,720 So relative humidity is telling you 75 00:04:29,720 --> 00:04:32,750 how close you are to basically having, water liquid water come 76 00:04:32,750 --> 00:04:33,690 out of the air. 77 00:04:33,690 --> 00:04:37,550 OK, now, the relative humidity also tells us 78 00:04:37,550 --> 00:04:42,830 something about how far you are from that phase transition 79 00:04:42,830 --> 00:04:44,090 point. 80 00:04:44,090 --> 00:04:45,950 And there's a very simple approximation. 81 00:04:45,950 --> 00:04:48,620 I'll put approximate here. 82 00:04:48,620 --> 00:04:53,000 We can also write that this is -- scales with, and is -- 83 00:04:53,000 --> 00:04:56,150 can be in fact close to the liquid volume 84 00:04:56,150 --> 00:04:59,570 fraction in equilibrium inside the drop. 85 00:04:59,570 --> 00:05:02,630 So that'll be the water volume fraction of water liquid 86 00:05:02,630 --> 00:05:04,160 inside the droplet. 87 00:05:04,160 --> 00:05:06,810 At least you can see here in this relationship when this 88 00:05:06,810 --> 00:05:08,480 volume fraction is one -- in other words, 89 00:05:08,480 --> 00:05:09,920 we have pure water -- 90 00:05:09,920 --> 00:05:12,520 then the relative humidity is 100%. 91 00:05:12,520 --> 00:05:16,990 OK, and on the other hand, when you have, let's say, only 50% 92 00:05:16,990 --> 00:05:21,160 water, over here, that's like having relative humidity 50%. 93 00:05:21,160 --> 00:05:23,530 This can be derived by more careful consideration 94 00:05:23,530 --> 00:05:26,500 of the ideal entropy of mixing where essentially 95 00:05:26,500 --> 00:05:28,600 this term here is take into account the excluded 96 00:05:28,600 --> 00:05:34,600 volume and the fact that all the sites in this droplet 97 00:05:34,600 --> 00:05:36,070 are not available for the water. 98 00:05:36,070 --> 00:05:38,740 So they're being excluded by all the solutes and the bound water 99 00:05:38,740 --> 00:05:39,909 that are present. 100 00:05:39,909 --> 00:05:45,610 And similarly, we have a buildup of free energy 101 00:05:45,610 --> 00:05:47,870 in the bulk as well. 102 00:05:47,870 --> 00:05:50,140 So basically, this comes from some thermodynamic 103 00:05:50,140 --> 00:05:52,870 considerations of equilibrium between water vapor 104 00:05:52,870 --> 00:05:54,950 and water liquid. 105 00:05:54,950 --> 00:05:59,460 We can write this as 1 minus the volume fraction 106 00:05:59,460 --> 00:06:02,530 in equilibrium of the solid. 107 00:06:02,530 --> 00:06:08,460 OK, now the thing is that we can now write this. 108 00:06:08,460 --> 00:06:14,900 So if we multiply through, we can write this as 1 minus the -- 109 00:06:14,900 --> 00:06:17,360 so what is the volume -- so when we get to equilibrium, 110 00:06:17,360 --> 00:06:19,160 this droplet is going to change its shape. 111 00:06:19,160 --> 00:06:20,740 It's going to reach a new shape, which 112 00:06:20,740 --> 00:06:24,230 we're going to calculate our new volume, V_equilibrium. 113 00:06:24,230 --> 00:06:27,210 And so what this would would be V_solid, 114 00:06:27,210 --> 00:06:32,909 which is phi_s^0*V_0 divided by V_equilibrium. 115 00:06:32,909 --> 00:06:35,450 So it's going to be new volume, V_equilibrium, which 116 00:06:35,450 --> 00:06:37,040 will be achieved then. 117 00:06:37,040 --> 00:06:40,210 And then we'll end up with the equilibrium volume fraction. 118 00:06:40,210 --> 00:06:42,960 So if I take these equations here, 119 00:06:42,960 --> 00:06:46,770 and I solve, I get a fundamental result, which 120 00:06:46,770 --> 00:06:53,430 is that the equilibrium volume of the droplet 121 00:06:53,430 --> 00:06:58,080 relative to the initial volume is equal to, 122 00:06:58,080 --> 00:07:01,290 well, we have put this on the other side. 123 00:07:01,290 --> 00:07:02,840 That'll be (1-RH). 124 00:07:02,840 --> 00:07:04,200 And we divide that out. 125 00:07:04,200 --> 00:07:07,760 And we find that it's the initial volume fractions 126 00:07:07,760 --> 00:07:10,710 solutes divided by (1-RH). 127 00:07:13,290 --> 00:07:17,130 That is our key result. And let's plot what this looks 128 00:07:17,130 --> 00:07:18,300 like. 129 00:07:18,300 --> 00:07:27,690 So if we prop the relative humidity 130 00:07:27,690 --> 00:07:35,130 on the horizontal axis, from 0 to 100%, so at 100%, 131 00:07:35,130 --> 00:07:37,570 the water vapor is saturating the air. 132 00:07:37,570 --> 00:07:40,270 And you would start to nucleate and condensed water 133 00:07:40,270 --> 00:07:41,460 liquid from that. 134 00:07:41,460 --> 00:07:43,450 At 0, the air is completely dry. 135 00:07:43,450 --> 00:07:46,570 And there is essentially no water vapor present. 136 00:07:46,570 --> 00:07:47,409 So that's the range. 137 00:07:47,409 --> 00:07:50,140 And typical comfortable rooms have a relative humidity 138 00:07:50,140 --> 00:07:51,730 around 50%. 139 00:07:51,730 --> 00:07:53,810 This is a typical number. 140 00:07:53,810 --> 00:07:59,290 And let's plot on this axis the equilibrium volume. 141 00:07:59,290 --> 00:08:01,480 It could also be the equilibrium radius 142 00:08:01,480 --> 00:08:04,420 because I should say that if there are spheres, 143 00:08:04,420 --> 00:08:10,990 that this is also equal to R_equilibrium divided by (R_0)^3. 144 00:08:10,990 --> 00:08:13,060 So I can also take a cube root of this. 145 00:08:13,060 --> 00:08:14,750 And I would have the ratio of radii. 146 00:08:14,750 --> 00:08:17,040 So we would know if we started a certain radius, what's 147 00:08:17,040 --> 00:08:17,810 the final radius. 148 00:08:17,810 --> 00:08:19,540 OK, so we can talk about volume. 149 00:08:19,540 --> 00:08:21,380 We can talk about radius. 150 00:08:21,380 --> 00:08:26,220 So here is the initial size of the drop. 151 00:08:26,220 --> 00:08:29,740 And somewhere down here is V_s, 152 00:08:29,740 --> 00:08:36,970 which is the solute volume, which is phi_s^0*V_0. 153 00:08:36,970 --> 00:08:39,440 Now what is this value? 154 00:08:39,440 --> 00:08:41,390 It depends on the kind of liquid. 155 00:08:41,390 --> 00:08:44,770 So saliva is mostly water with some salt 156 00:08:44,770 --> 00:08:46,900 and a few other molecules. 157 00:08:46,900 --> 00:08:53,110 But in saliva, the volume fraction phi_s^0 158 00:08:53,110 --> 00:08:59,750 is 0.5% in saliva. 159 00:08:59,750 --> 00:09:01,780 OK, so that's just gives you a sense. 160 00:09:01,780 --> 00:09:04,830 So this is quite far down, right? 161 00:09:04,830 --> 00:09:08,220 But then, if you look in mucus, it 162 00:09:08,220 --> 00:09:10,080 depends which mucus you're talking about. 163 00:09:10,080 --> 00:09:13,620 But the mucus that comes from the lungs or from the pharynx, 164 00:09:13,620 --> 00:09:14,670 it can vary. 165 00:09:14,670 --> 00:09:17,400 But what has also been measured in droplets 166 00:09:17,400 --> 00:09:21,000 that are emitted by breathing is that this can range anywhere 167 00:09:21,000 --> 00:09:23,100 from 5% to 10%. 168 00:09:23,100 --> 00:09:27,600 So a fairly significant amount of the volume of the droplet 169 00:09:27,600 --> 00:09:30,430 is containing all these molecules and the bound water 170 00:09:30,430 --> 00:09:30,930 around that. 171 00:09:30,930 --> 00:09:32,860 Now we know that because mucus is very sticky. 172 00:09:32,860 --> 00:09:34,290 It's a non-Newtonian fluid. 173 00:09:34,290 --> 00:09:37,230 It doesn't maintain a nice round shape even. 174 00:09:37,230 --> 00:09:39,060 It can have a regular shape. 175 00:09:39,060 --> 00:09:39,930 It flows slowly. 176 00:09:39,930 --> 00:09:42,830 It has a high viscosity. 177 00:09:42,830 --> 00:09:44,720 And that's because it has a large amount 178 00:09:44,720 --> 00:09:48,460 of these hygroscopic solutes. 179 00:09:48,460 --> 00:09:50,990 So mucus might be a little bit higher up. 180 00:09:50,990 --> 00:09:56,800 But in any case, what you then find is 181 00:09:56,800 --> 00:10:01,240 we can sketch different regions of this plot now. 182 00:10:01,240 --> 00:10:06,040 So this curve this formula drive here, 183 00:10:06,040 --> 00:10:09,460 when the relative humidity is zero, we start here. 184 00:10:09,460 --> 00:10:12,010 So that's saying when there's no water in the air, 185 00:10:12,010 --> 00:10:13,720 you completely dry the droplet. 186 00:10:13,720 --> 00:10:16,810 And you're left just with the molecules, the solid molecules, 187 00:10:16,810 --> 00:10:19,300 and possibly the bound water around it, 188 00:10:19,300 --> 00:10:21,790 depending on how dry the air actually is. 189 00:10:21,790 --> 00:10:26,000 And then it rises up and blows up at 100%. 190 00:10:26,000 --> 00:10:29,260 So when you get to 100%, then droplets 191 00:10:29,260 --> 00:10:31,660 are getting really large. 192 00:10:31,660 --> 00:10:33,250 And if you actually hit 100%, then you 193 00:10:33,250 --> 00:10:35,050 can't really speak of an equilibrium size 194 00:10:35,050 --> 00:10:38,770 because you'll just start to get lots and lots of water. 195 00:10:38,770 --> 00:10:40,270 So that's that limit. 196 00:10:40,270 --> 00:10:44,140 And so now we can look at three different regimes 197 00:10:44,140 --> 00:10:47,300 of the kinds of droplets that we'd expect to see. 198 00:10:47,300 --> 00:10:52,420 So down here at 0% or close to zero, 199 00:10:52,420 --> 00:10:59,770 we have a dried droplet nuclei as they're called 200 00:10:59,770 --> 00:11:02,320 in the public health field. 201 00:11:02,320 --> 00:11:05,350 These respiratory aerosols, if they completely dry out, 202 00:11:05,350 --> 00:11:07,600 and you're left with just these solutes, then that's 203 00:11:07,600 --> 00:11:08,640 called a droplet nucleus. 204 00:11:08,640 --> 00:11:10,100 So it doesn't necessarily mean it's 205 00:11:10,100 --> 00:11:11,710 a nucleus for phase transformation 206 00:11:11,710 --> 00:11:15,100 as we use that term in, say, engineering or in physics. 207 00:11:15,100 --> 00:11:18,190 But it's really just refers to the core 208 00:11:18,190 --> 00:11:21,190 of just the hydrated solutes. 209 00:11:21,190 --> 00:11:25,530 So if I could sketch what that looks like, that would be-- 210 00:11:25,530 --> 00:11:27,160 for example, all those molecules I just 211 00:11:27,160 --> 00:11:29,830 sketched there might be condensed 212 00:11:29,830 --> 00:11:31,930 into some little blob, which, by the way, 213 00:11:31,930 --> 00:11:34,450 could include a virion. 214 00:11:34,450 --> 00:11:36,880 In fact, it could even be just one virion 215 00:11:36,880 --> 00:11:39,260 if that were all that were in there. 216 00:11:39,260 --> 00:11:43,570 And you would have a little bit of bound water around it. 217 00:11:43,570 --> 00:11:48,610 But you essentially have a dried up blob of just the solutes. 218 00:11:48,610 --> 00:11:50,560 OK, and so that -- and then, of course, 219 00:11:50,560 --> 00:11:54,730 the smallest volume you can get is just the initial solute 220 00:11:54,730 --> 00:11:58,690 volume that you started with, plus the bound water. 221 00:11:58,690 --> 00:12:03,130 On the other end, if we are near 100% relative humidity, then 222 00:12:03,130 --> 00:12:06,100 the fact that these are hygroscopic solutes, which 223 00:12:06,100 --> 00:12:09,400 like to have water near them, will form as a nucleation site 224 00:12:09,400 --> 00:12:11,440 to actually cause more and more water 225 00:12:11,440 --> 00:12:14,560 to grow and be absorbed into this droplet. 226 00:12:14,560 --> 00:12:17,710 And not only do the droplets not shrink, 227 00:12:17,710 --> 00:12:21,970 as predicted by the Wells curve for a pure liquid, 228 00:12:21,970 --> 00:12:23,480 but they can actually grow. 229 00:12:23,480 --> 00:12:25,880 So if the size here is small enough to begin with, 230 00:12:25,880 --> 00:12:29,190 let's say it were a several-micron droplet to begin with, 231 00:12:29,190 --> 00:12:30,940 but it contains a lot of solutes, and we're 232 00:12:30,940 --> 00:12:34,480 at very high humidity, actually the particle can grow. 233 00:12:34,480 --> 00:12:40,350 So over here, we could end up with an even larger droplet 234 00:12:40,350 --> 00:12:43,210 than we started with where now because the humidity is 235 00:12:43,210 --> 00:12:45,630 so high, and we have the same number of molecules in there 236 00:12:45,630 --> 00:12:51,280 that I sketched before, that's more dilute now. 237 00:12:51,280 --> 00:12:54,660 And there's maybe a virus or a virion here and there. 238 00:12:54,660 --> 00:12:58,680 And of course, there's also some salt. 239 00:12:58,680 --> 00:13:00,640 But basically, the droplet is growing. 240 00:13:00,640 --> 00:13:11,290 So here we have hygroscopic growth 241 00:13:11,290 --> 00:13:18,500 and also we have what's called deliquescence, which 242 00:13:18,500 --> 00:13:21,950 refers to water that's absorbing around these salt molecules 243 00:13:21,950 --> 00:13:24,410 and even causing some other molecules or charges 244 00:13:24,410 --> 00:13:26,800 on these macromolecules to dissolve into solution 245 00:13:26,800 --> 00:13:28,220 because it's more and more water present 246 00:13:28,220 --> 00:13:30,450 and it can solvate more species. 247 00:13:30,450 --> 00:13:33,260 And so whereas hygroscopic growth 248 00:13:33,260 --> 00:13:35,720 refers to water being absorbed into a more 249 00:13:35,720 --> 00:13:37,340 solid-like framework, you can also 250 00:13:37,340 --> 00:13:40,910 be generating more aqueous solution, which is deliquescence. 251 00:13:40,910 --> 00:13:42,980 So basically, the droplet can actually grow. 252 00:13:42,980 --> 00:13:46,790 And that would be like when you're here, let's just say. 253 00:13:46,790 --> 00:13:49,680 And this might be when you're here. 254 00:13:49,680 --> 00:13:52,530 And then, of course, when you're at 50% relative humidity, 255 00:13:52,530 --> 00:13:54,360 you can see the droplet has shrunken 256 00:13:54,360 --> 00:13:58,230 but not all the way down to the initial solute volume fraction, 257 00:13:58,230 --> 00:13:59,760 but something larger. 258 00:13:59,760 --> 00:14:02,400 And in fact, if the relative humidity is 50%, 259 00:14:02,400 --> 00:14:05,650 you end up at exactly twice the solid volume fraction. 260 00:14:05,650 --> 00:14:09,070 So if the solid volume fraction of mucus is 10%, 261 00:14:09,070 --> 00:14:10,950 you may end up with a droplet that 262 00:14:10,950 --> 00:14:12,550 is maybe 20% of the volume. 263 00:14:12,550 --> 00:14:15,900 So maybe it looks something like this. 264 00:14:15,900 --> 00:14:21,200 OK, and so we have a little bit of shrinking going on. 265 00:14:21,200 --> 00:14:24,260 And maybe there's even a virus in there as well. 266 00:14:24,260 --> 00:14:26,870 But there is still plenty of water. 267 00:14:26,870 --> 00:14:31,040 And so you can see also now the value of having solutes 268 00:14:31,040 --> 00:14:35,810 in mucus in terms of making the virions more viable and more 269 00:14:35,810 --> 00:14:38,960 easily transmittable because they hold onto the water. 270 00:14:38,960 --> 00:14:41,820 So that the virion is in a stable environment. 271 00:14:41,820 --> 00:14:44,990 So that when it ends up being inhaled into someone else's 272 00:14:44,990 --> 00:14:47,600 lungs that it can then more easily 273 00:14:47,600 --> 00:14:52,220 diffuse out of that region and infect the host cells. 274 00:14:52,220 --> 00:14:55,010 In contrast, if you have a nearly pure liquid 275 00:14:55,010 --> 00:14:58,290 that the virion is in, let's say pure water or even saliva, 276 00:14:58,290 --> 00:15:00,620 which is actually mostly water, then 277 00:15:00,620 --> 00:15:04,010 the droplet will shrink by a factor of 100. 278 00:15:04,010 --> 00:15:07,520 And it might be just literally a virion with a [couple of 279 00:15:07,520 --> 00:15:10,270 ions] just enveloped 280 00:15:10,270 --> 00:15:11,390 with a tiny bit of water. 281 00:15:11,390 --> 00:15:16,640 And maybe that, in some cases, would be not as viable 282 00:15:16,640 --> 00:15:18,860 of a situation for the virions. 283 00:15:18,860 --> 00:15:21,170 So basically the mucus fragments are 284 00:15:21,170 --> 00:15:25,460 likely to be the more common source of the aerosols that 285 00:15:25,460 --> 00:15:29,440 will stay in the air and remain infectious.