 OK, so I'm going to be telling you about soft matter physics. And before I get started, I want to also say that throughout this talk, feel free to stop and ask me questions. I'm not in a hurry. We're going to easily finish by 10 o'clock. So feel free to ask questions if anything is confusing in the middle of the talk. All right. So soft matter, what are soft materials? Soft materials are pretty common. So food is a soft material. Toothpaste, shampoo, a lot of consumer products would be considered soft materials. Standpiles, so if you're walking along a beach, or if you have foam, like when you're shaving, if you have shaving cream foam, all of these are soft materials. It's somewhat intermediate between liquids and solids. And they're often mixtures of different components. And it's the mixtures, the fact that you're mixing together some simple ingredients, but then the mixture is complicated. That's what makes it an interesting material. So why should we study soft materials? There's going to be a lot of cool physics, which is what I'm going to be focusing on this talk, but I want to give you some sense of the big picture as well. So I'm a physicist, and I work in an academic laboratory. And I tend to do very simple model system experiments and so forth. But there is a lot of relevance for the real world. So food, as I mentioned, is a soft material. We've been eating a lot of soft materials for lunch and dinner. And so for example, suppose you want to make a healthier food product or maybe something that is cheaper, so you want to change the ingredients of the food product, but you want the texture to be the same. If you're going to have gluten-free pasta, you want it to be the same texture as regular pasta. So how do you engineer the properties of the material with different ingredients to give you the same textures? So a chemist might tell me what kind of chemicals to put in the food, or some food chemist would tell me what kind of chemicals. I would say if you're going to put in those chemicals, they have a different density or a different viscosity or something that's different about them. And so here's how to modify the food so that it has the same texture and the same appearance as what you would normally expect. Also improving shelf life. There's a lot of food products that when they go bad, they're not actually chemically bad for you. They're not full of contaminants or not something that would harm you to eat it. It's just that the texture changes. And at that point, the food just no longer doesn't taste any good or doesn't seem right. So people have to throw out food not because it's chemically bad, but because the texture has changed. So there's a way to make that shelf life longer for the texture that could be useful. The other big category of soft materials is biology. So we're all squishy materials. If you think about what fraction of the human body is water? Was it like 90? OK, so we're 90% water, but yet somehow we're not puddles on the ground. And this is good, right? So why is that? What are the mechanical properties of cells? What is it about the stuff inside our cells that gives us some texture and gives us some stiffness so that we're not just liquid? Clearly, the 10% of us that's not water is a pretty important 10%. So again, you'd like to understand this. And then maybe when cells go bad, you'd like to understand the soft mechanics of that as well. But again, I really want to focus on the physics of all this in the talk. So there might be some applications near the end of the talk, but for right now I may be focusing more on the basic physics. So sometimes I call this squishy physics, but the technical term for this field is often soft condensed matter. And my definition of what this is, this is a study of soft systems. And there's a lot of examples. I'm going to define what some of these examples are. So these are, right now, just think of them as words that you may or may not know. But they're just a variety of kinds of materials. Sometimes they're called complex fluids. And the key question is relating the microscopic properties to the macroscopic properties. So if you have food products, the textures might be developed on a scale of micrometers. But then what you care about is how they spread, when you spread butter across bread, or how they feel in your mouth, macroscopic things. Or skin lotions. If you're rubbing cream on your arm, you want it to feel right. That's a macroscopic property. But that depends on the details that are going on at the micron scale. Usually not the atomic scale, but usually the micron scale. So my favorite example, because a lot of my research relates to is collides. So collides are small, solid particles in a liquid. So paint is a collide. When you paint a wall, you want it to be a liquid. But then when the liquid evaporates, you want the particle stuck on the wall to give you the color. So that paint is a collide. Blood is a collide. Toothpaste. So by small, they're typically nanometers to micron scale. You want to think of them as spherical objects. You don't want to think about them as chemicals or molecules. You want them to be spheres. We're physicists, right? We like spheres. I'm a physicist. I like spheres. Thermal energy is important. They do Brownian motion. So the energy scale that's often relevant, you think about, is KT. And they're nice because you can see them with a microscope. They move on human time scales. You can often do video microscopy. You can do tabletop science. So they're a nice thing to start with. They're a nice thing to study. And the most basic thing to start with is Brownian motion and diffusion. So how do these things move around due to thermal energy? So they're not moving ballistically. They're not moving with constant velocity. They're just jiggling around. I'm going to show you moving a second. And you have the mean square displacement, delta R squared. It's now going linearly with the lag time delta T. So as it diffuses, it spreads out. But it kind of spreads out the size of a diffusing. Blob spreads out as a square root of time. It's spreading out slower. And the diffusion coefficients related to the thermal temperature, this is kT, thermal energy, the viscosity eta and the particle radius A. And just for fun, I'm going to show you a movie here. So this is a real time movie of micron radius particles that are just doing Brownian motion in a liquid, taken with microscopy. This is in the middle of a sample. So sometimes they go in and out of focus. And in just a second, we're going to switch to a dense review. So this is more like a paste, something that would be more like toothpaste, where they're kind of crowded in. But they still do Brownian motion. They still jiggle around. They can still rearrange. So this is what Brownian motion looks like under a microscope. I forget, Pietro, have you been doing Brownian motion in your hands? OK, good. And so some of you have seen Brownian motion already last week in Pietro's session. Great. Just to briefly say how you would calculate a mean square displacement, you look at a particle trajectory. You could maybe measure with particle tracking or measure it some other way. You look at displacement, so you subtract the motion at one time later from the time now. And then you would square this and average this over all particles in all time, just to say how you would measure this kind of thing. All right. The next bit of physics I want to teach you is about sedimentation. So the particles are often not the same density as the liquid. In fact, it's hard to make them exactly the same density. So at best, you can make it close to the same density. So they like to fall. Or maybe they're lighter than liquid, and then they like to float up. But there's also a drag force that prevents them from going infinitely fast. Or they don't accelerate. They reach a constant velocity. This is the Stokes drag force. So again, it depends on the viscosity. It depends on the particle radius A. And then it depends on the velocity. So the faster they move, the more the drag force. Of course, what's making them go up or down is the buoyant force, the gravitational force. So either they're floating up because they're lighter than the liquid, or they're floating down. And so it depends on the density difference. It depends on the volume. So the volume of the particle is 4 third pi A cubed. And then times g. That seems good. Particle falls down. Stokes drag force. All right. Now I know it's Monday morning, and I know I've just kind of started the talk. But now it's time for you to do some work. I've got some problems I want you to think about. Actually, I want you to do the problems to be more specific. So I'd like you to balance the drag force and the gravitational force and figure out how fast do particles sediment? How fast do particles sediment? I want you to separately balance out the gravitational potential energy, mgh, with kt. So when you find the particles sitting at the bottom, they also still have thermal energy. So there's a scale height. So the density varies with height. This is very similar to the atmosphere, right? The atmospheric pressure is biggest down here at sea level. And it gets smaller pressure as you go up due to kt of energy. So I want you to do the same thing for collage. So how high can they go by balancing out kt of energy to mgh? And the last thing is when you're diffusing, I want you to solve using plugging in for d. When is the time, typically, that particles have moved their own size, so a, or I guess plugging in for a squared? OK, go. You are allowed to talk with your neighbor, especially if you think that they have something useful to say about this. It seems OK. I'm right now. Not right now. If you have any questions or need any assistance, raise your hand. Or if you get an answer, raise your hand. I want formulas. You want just formulas. I want formulas. I'll give you numbers in a minute. No questions yet? Or answers? OK. Yeah, just to be clear, you want to be looking for your answers in terms of a. So you don't want d in your answer. You just want a. Or a and other factors like the viscosity and so forth. I think it's right, though. Do you need maybe one more minute? Who wants me to keep going and who wants one more minute? One more minute? A couple of people are nodding. All right. Sorry, I'm making you work on Monday morning. No, no, no, no, no. Do you have some answers? No? Do you have answers? Yeah, that's right. So just solve that for, just solve for b, yeah. All right, let's look at some answers. OK, so for the first one, you're balancing forces. And this is a formula for the sedimentation velocity. The key thing is that it goes as a squared. So if you got to the sedimentation velocity when it's something times a squared, you should feel good. If you didn't get the two 9s, you probably just made some algebra mistake. You'd get it right if you had more time. But it should go as a squared. For the scale height, it goes as inverse a cubed. So you've got kt on the top. You've got an mg on the bottom. And that ends up going as inverse with a cubed. And for the mean squared displacement formula, you're just plugging in a squared for delta x squared. And then this one goes as a cubed because you get an extra factor of a from the d. So there's an a squared on the top from the mean squared displacement. And there's an extra a from the d. So the key thing here is to think about how these scale with a and what that means. So the three factors, the two 9s and the pi's and so forth, are not really the key thing about this calculation. It's about the scaling with a. So let's look at the meaning of these answers. So the sedimentation velocity goes as delta rho a squared g. What this means is that small particles sediment slowly. Big particles go quick. Like if you jump into a swimming pool, you're going to sink to the bottom more quickly because you're big. But if you're a small particle, you feel more drag force. So then you go slowly. If you want to centrifuge small things like viruses or something, you have to go really, really big centrifuges. You have to use a centrifuge to increase g. And it's hard because the sedimentation velocity goes as a squared. g is linear. So you have to go 100 times more in the centrifuge for something that's 10 times smaller. So small stuff is hard to sediment. The scale height means that there's a huge size dependence on gravity. So large particles sink really, really to the bottom and they don't distribute themselves throughout the system. Smaller things can distribute themselves more equally. So large stuff really, really feels the influence of gravity. And again, you can think about the atmosphere. This is why the atmosphere does not change dramatically from the bottom of this room to the top of this room. Because gas molecules are really small. But if you went up several to the top of a mountain, you'd start feeling the atmosphere is changing. For larger particles than gas molecules, this is happening over a much shorter length scale. And the diffusion time scale goes as a cube. So this means that small particles move fast. Large things don't diffuse very much. Again, this is why you're not diffusing right now in this room because you're large particles, but small things can diffuse their own size. They really explore their environment very rapidly. OK, let's put some numbers on this. Let's consider two examples. So these are both typical kinds of examples studied by labs around the world. One would be polystyrene particles. This is the same plastic that's in styrofoam. So if we had micron sized particles in water, and then another would be a system that we study in my lab a lot, which is polymethyl methacrylate. That's the same plastic that's plexiglass or lusite. So here we can make the solvent much more close to the density of the particles to try and turn off gravity. So the sedimentation velocity for polystyrene is slow. A tenth of a micron per second, but for the density match system it's even slower. So a nanometer per second, really, really slow. The scale height is two microns for polystyrene particles. So that would mean that if you had a sample, like a microscope chamber with polystyrene particles in it, you would expect to see them near the bottom of the slide. Maybe a micron or two off, but you wouldn't see that many that would be 100 microns away from the bottom of the slide. They'd mostly be near the bottom. The other particles, the density match ones, can go 200 microns, so in a thin sample chamber, you'd see them evenly spread throughout the sample chamber. But if I handed you a jar of these, you'd say, OK, they're going to be mostly on the bottom, because they just don't have enough energy to be up at the top of the jar so easily. And then the diffusion time scale is about the same for both of these, because it doesn't depend on the density. It just depends on the viscosity. All right, let's now look at the effective size. So if I look at just polystyrene particles and compare the small ones I just talked about to large ones, and by large, let's just say 100 microns. So you could see these with your eye, but not easily. These aren't huge, 100 microns, but a lot of these depend on A to some higher power. So the sedimentation velocity now is millimeters per second. If you had a jar of these, and you shook them up, you could see them sinking slowly in the jar, and after a minute or so, they'd be at the bottom of the jar. So that's pretty fast velocity. The scale height is two picometers. Once they're on the bottom of the jar, they're going to stay there until you stir them up. They're not going to go anywhere due to thermal energy. So thermal energy is a certain amount, but it's not enough to lift these particles up off the bottom of the jar. They're really at the bottom, and they're going to stay there unless you do something to lift them up. And diffusion takes now, it goes to A cubed. So when the particle size goes up by 100, the diffusion time scale goes up by a million. And so it takes nine days for them to diffuse their own size. So diffusion is off. These things don't diffuse. So this is usually what you think of as a not-colortal particle. It's something that is extremely heavy in terms of this kind of reasoning, and something that just doesn't diffuse anymore. So KT is no longer helping us do anything interesting. And this is what makes large particles distinctly different physics from small particles, among other things. OK, and again, the point of this exercise was that just knowing a few of these really simple formulas, you can work at the scaling with the particle size and start thinking through some of the physics of this. That's really an approach that's often very useful in soft matter, is to think through just the scaling with particle size. Any questions so far? Everyone's OK? So to sum up on collides, the important point is that the small size of these things is important. The scaling with A is really the main message I'm trying to tell you, not the exact formulas, but the scaling. Large particles, so anything much larger than 10 microns, is not really thermal. You no longer think about diffusion or KT of energy once you get very large. So that's why the boulders outside are much different than collides. And the other thing is that, in some sense, this means that this particle size gets really, really small. You can stop thinking about gravity. You no longer need to worry about gravity if you're talking about things that are nanometer sized. One time, somebody was telling me about a simulation they had done where they were doing molecular dynamics of atoms in a material, and they were claiming that gravity was doing something. I'm just like, no, that can't be true. Because when you get really, really small, gravity cannot be playing a role because the thermal energy is so much more important than the gravitational energy, for example. So this really means that the smaller it gets, the more you can just stop thinking about gravity, and that's a very useful simplification. OK, that's all I have to say about collides. Everybody seems OK. Let's move on to emulsions. So emulsions are what we're using in our hands-on module. So half of you have seen this, and half of you are seeing it right now. These are liquid droplets in another liquid, like oil and water, and you have to have a soap of some sort, a surfactant molecule that coats these droplets, that prevents the droplets from coalescing and coming together and turning into bigger droplets. So examples are things like butter and mayonnaise, a lot of food products, skin lotions are usually emulsions, and the samples in our hands-on module. So the surfactants are these molecules like this. They have a part that likes to be in water and a part, a tail that likes to be in oil. So these molecules like to coat the boundaries so they can have the part that wants to be in water, be in water, do the part be in oil. This makes the surfactants very happy. And that means that the oil droplets, when they come together, they bump into their surfactants. They don't end up coalescing as much. So what makes these different from collides are that these are now softer things. They can deform. And you have a surface tension now. And the surface tension makes all the difference. All right, how many people feel they understand surface tension really well? One, maybe. So what is surface tension? Surface tension, the simplest way to think about this is it's an energy cost to have an interface. So if you have a droplet of radius a, then the surface area is proportional to a squared. And so the energy that that surface requires is gamma times a squared, with factors of pi and so forth. So for example, then, if you're going to make an emulsion, so you've got some oil and some water and some surfactant, what you could do is you could just take it and shake it really, really hard. Then you're putting in energy to try and break up these droplets. Again, half of you have done this in our module already. So what does that energy do? Well, it goes to make the surface. So if you think about how much surface you have to make, if you've got a volume v of the emulsion, you can roughly guess at the number of droplets, because each droplet has a volume related to a cubed. So the number of droplets is v divided by a cubed. Then the energy of the surface goes as a squared. So the total area you're making is the volume, or it's the number of droplets times a squared, or the volume divided by a. So the amount of energy you have to put into shaking the sample goes as v times gamma divided by a. The smaller you want to make the droplets, the more energy you have to put in. So if you're trying to make some kind of skin cream in industry that is made out of oil and water, you have to have a mixer. And this tells you how much energy you have to expend on your mixing process to make the sample with droplets of smaller sizes. So small sizes require you to mix that much harder. It's going to be that much more power to run your factory. Does this make sense? And you can change this by changing the surface tension. So you can certainly try to make it easier by adding in more surfactants or surfactants that modify the surface tension. But the bottom line is if you want to make really small droplets, you've got to work harder, because that requires more energy just to make the surface. You don't get surface for free. All right. This is now a question which I'm not going to make you work through. You've already worked hard once. I'm going to talk you through this one. This is a very straightforward question. If we had water leaking from above in this room and it was coming down to the ceiling tiles, at what size were the droplets when they drip off the ceiling? This is a really critical question if we start getting leaked from upstairs. We need to know this, right? How big are these droplets? So what you can do is you can think of, as it's coming off the ceiling, let's just approximate everything as a sphere, because we're a physicist or I'm a physicist. That's what I do. Everything's a sphere. So figure out the change in potential energy, gravitational energy, for a spherical droplet to descend from the ceiling. If it can drop a distance a below the ceiling, then it might have a chance to actually drop off and then go the rest of the way. But the first thing it's going to do is form a bump that comes down by distance a. But to do that, cost surface energy. So if you just have water as a flat layer, you've minimized the surface energy. But as it falls down, you're making more surface. So that's a cost. You get a reward for going down, but you pay this cost. So you can try and balance these things out and ask where does the reward and the cost equal each other. I'm going to do this, but does this make sense as an idea for how we're going to solve the problem? Okay. So the change in gravitational potential energy, so it's going to be MGA from gravitational potential energy. The M depends on the volume. So this goes as a to the fourth, droplet size to the fourth power. So you get a cubed from the volume and then you get an extra a because that's how far we really want it to go is that distance. For surface tension, this is just the surface of the spherical droplet. So this is four pi a squared. So this goes as a squared. So we want to check these things and we want to compare them. So one of these goes as a to the fourth, one goes as a squared. So the ratio of these goes as a squared which means that if a is really, really small then this ratio gets smaller. So then to make any bump of a really small size is just really costly in terms of surface tension. It doesn't give you enough reward in terms of gravitational potential energy so you don't make small droplets. If you make really big droplets, if I tried to dump a bucket of water out gravitational energy really controls it so everything falls. It's right when they're in balance is with the size of the droplet. So we can do this with numbers. So here's the exact formulas. We equate, if we set this ratio equal to one we can say what's the size so that this ratio is equal to one and plugging in the actual surface tension between water and air you get a droplet size of about four millimeters in radius. So about eight millimeters in diameter. So if there's a catastrophic leak of water from upstairs and starts dripping to the ceiling you're gonna get hit with droplets of about this size. And this should kind of make sense, right? If you've ever seen water dripping off things this is about the size that you get. This is also the same kind of physics that tells you that the bugs that can walk on water are roughly millimeter in size and you, which are much larger cannot walk on water because the water doesn't care about surface energy. The water just says if you walk on it there's a huge reward in terms of gravitational energy for sinking through the water. Okay. Any questions about this solution? I am gonna be posting a copy of these notes on the Google drive. So you'll have all this and you can go back through it and do more detail if you need it. Okay. So implications for emulsion droplets they like to be round because they minimize surface energy and that's more and more to the smaller you are. The smaller you are the more surface tension becomes important. So in this kind of image of the emulsion the small droplets those are the ones that are the really circular ones and the large droplets are the ones that are kind of squishy and being deformed because then other forces can be more significant. But the smaller you are the more round you're gonna be. And you can modify this argument with surfactants which modify the surface tension but really it's the size that makes a bigger difference than the surface tension. Okay. One more material, foams. So foams are basically emulsions but rather than having oil in the droplets you're gonna have air. You still need soap to make a foam. So this is just soapy water and air and we added some red dye to make it look interesting but it's basically just the air bubbles. All right. So foams are straightforward in that sense. Let me talk now about one more property of these kind of materials. So toothpaste is a paste of collides and then we have shaving cream here which is a foam. Whoa, no, go back. So these things are soft and squishy. So the technical term for this is viscoelastic. So here we are, we're making a pile of toothpaste on the table and again you cannot do this with water. If I cannot make a pile of water on the table. If I squeeze harder it goes out faster but it depends on how hard I'm pushing. With a foam we've got soapy water and air and again both these things are fluids but you can make a pile of shaving cream on the table and it holds its own weight and you can't do this with either soapy water or air by themselves. The other thing though is that under some force these will flow pretty easily. So they're not very hard solids. They squish out very nicely when you apply force to them. So you can start asking questions about how much force does it take to do that. Not much, yeah. So why are these things squishy at all? Why do they have any kind of elastic properties? So suppose you have say an emulsion or a foam and you try and shear. You're trying to move the sample back and forth. You're trying to push the droplets. You're trying to push the sample. You squish the droplets. You deform the droplets and they're no longer spherical and that costs you surface energy and that gives you an elastic response. So then these droplets would like to go back to their original shape. This stores energy and that gives you the elasticity. So this is why a foam can hold up its own weight because a foam in order to sink requires these droplets to change their shape and they don't wanna do that because of the surface energy. So they stay stiff in a pile. If you do move around anyway, you've got a fluid back here. So they're all in a fluid, this blue stuff. So you also get a viscous response. So these materials have both elastic properties and viscous properties and it can depend on the frequency and the amplitude. It depends on how you're forcing them but you can get both elastic and viscous. All right, any questions about this? Yeah. If they're small enough, they're brown in motion. The movies I was showing you and the pictures I showed you were mostly larger. They were maybe like 100 microns. So there's no brown in motion. They're like the 100 micron polystyrene particle. If you make an emulsion droplet that's a micron size, then you definitely have brown in motion. So yeah, you can have it but you've gotta work hard to make your small emulsion droplets. Any other questions? Okay. So we're right on schedule for R1 to B. We're now at the most important part of the talk. Since this is Monday morning and I know that you had a long weekend and you were probably doing things that were making you get tired and so forth, I thought I would end the talk with like 20 minutes of something that is a critical, crucial importance to many of you in this room. We're gonna talk about the physics of beer foam. So now you've learned a little bit about soft materials. Let's look at an application which is beer foam. And you have to understand, I don't actually drink beer. This is my friend's beer I'm holding. I didn't even drink any of that. But still it's an interesting topic so I'm gonna talk, and it relates to soft materials and it is Monday morning so I don't wanna spend the whole talk making you do algebra problems. Let's talk about the physics of beer foam. So you have foam on a beer and I wanna talk first where the bubbles come from and you all know that beer is carbonated. What can I tell you beyond just that the beer is carbonated? But let me actually tell you why the beer is carbonated. Does anyone know why the beer is carbonated? It's not the same reason as Coke. Coke they especially add carbonation to. With beer, the process that makes beer uses yeast to digest sugars and other things and the yeast release carbon dioxide and that's what carbonates the beer. So carbonation and beer is part of the process. It's not done by us because we wanna have a carbonated. It's actually that was what the product is, it's carbonated because of the yeast. All right, so you have to have surfactants and I told you before that surfactants are silk molecules and that does not sound like something you want in your beer but another kind of surfactant is proteins. And so the proteins can do the same business. These are kind of like more like model surfactants. Again, they've got a head part that likes to be in the water or the beer. The tail part likes to be in the air. For beer, it's proteins. Where do the proteins come from? Any guess where the proteins come from? Well, I guess we add organic ingredients so your hops or barley, whatever there's ingredients. Also there's yeast and yeast has protein in it. So some of the bits of yeast are the proteins and so forth as well as other constituents. So you have to have the stuff to make the foam in the beer. And in fact, there's no surfactants in champagne or in soft drinks like Coke. This is why you don't get foam, like a head of foam on a champagne glass or on a bottle of Coke or a cup of Coke. Root beer, I'll bet they add in something that gives you the foam. Yeah, root beer must be obviously has to have something else in there to give you the foam. So, but this is a particularly important distinction with champagne is that there's no surfactants in champagne. All right, so the carbon dioxide comes from the fermentation process. That's what initially gives you the carbonation. You can also, if you want to, you can add more carbonation later if you want a really extra carbonated beer. And then you keep the beer under pressure to keep the carbon dioxide dissolved in the beer. You want it to come out only when you're pouring it. So when you pour the beer, you get mechanical agitation. Your beer's flashing around, so that's gonna release some of the amount of bubbles. But then extra bubbles are released and they just come out of the solution when the beer is in the glass. So why is that? You have to have microscopic pockets of air already present in your beer glass to give you more bubbles coming up. This is a picture from an article about beer bubbles in the American Journal of Physics. So what happens is that if you have perfectly clean glass, you don't get very many bubbles coming up out of the glass because glass is really, really smooth. But when you clean glass, you normally wipe it dry with a towel. And the towel leaves tiny, tiny cellulose fibers behind on your glass surface. And those fibers have tracked air pockets and the air pockets are where you grow new bubbles coming from. Because again, remember it takes energy to make surface, right? And there's some energy given because the carbon dioxide wants to get out of the beer but it helps to have a starter bubble. So a little fiber like this will nucleate the growth of more bubbles coming up out of your beer. And that really has to do with how you clean the glass and just the cellulose fibers just from drying the glasses quite often. And then when the bubbles are rising up, they've got surfactants on them, which coat the bubble. That makes bubbles rise slower. So then more carbon dioxide can come into the bubble and it gets bigger. So this is also important. And this is something that if you look at the bubbles rising in beer, they rise slower than they do in coke. Again, because of the surfactants. Okay. Any questions about this? Yeah. It's mostly just surface tension. So there's viscous forces because of the beer. So there's the hydrodynamic interactions but otherwise the two bubbles just, they've got their surface energy so if they get crowded together, there's a penalty they pay for deforming them from spheres. There's a little bit of attraction but not very much. They kind of just, they're being pushed out because of gravity. And when they get to the top, they're kind of crowding because they all want to go up because of gravity but there's a limit because to go up even more, they have to deform and that causes surface energy. Yeah, yeah, these are fragile but that's you squeezing it. You're big. You're bigger than a bubble. Actually, I can go back to my phone picture. Yeah, if you look at, nope. If you look at this picture, you can see this process. So the bubbles at the bottom are round because they're having a weak gravitational force but up here to get higher you have to kind of be pushed together more and so you're deforming but you're paying a surface tension price at that point. To be no longer spherical costs you energy. So you have to have something else going on to make you be not round. Yeah, Mark. Is there a limit to how big the bubble can grow? I think in principle there's not really a limit but if you've got a huge bubble rising you might think that it's gonna start being sheared apart due to the viscous forces from the surrounding liquid. So yeah, usually there's something else that cuts you off if you start making something really, really big. While they get bigger there's dissolved carbon dioxide in the liquid and that gives you a reward for going into the bubble. And you can think of it also as the reward is how much carbon dioxide you have in the bubble and that goes as the volume, a cubed. The surface penalty that you pay for making the bubble bigger only goes as a squared. So if your bubble's big enough it wants to grow because it definitely rewards you more so than it costs you. So again, it gets back to the scaling. Reward goes as a cubed. Penalty goes as a squared. It's the same thing as nucleation. Fusion, that's prevented to some extent by the surfactant. No, the problem is that you've got these surfactant molecules and they interfere with each other. And that is a strong enough interference that they can't get close enough to say oh actually we'd prefer to merge. There's an energy barrier to merge in. Yeah, actually you can get diffusion between bubbles and that can be, that actually causes the foam to course in. So yeah, diffusion can give you, but it's hard to get them to coalesce because of the energy barrier surfactants. But you're right, diffusion gets another way to get change bubble. Yeah, that's a good question. So it could be a size effect but what I've read is that it's also a surface reality effect. It's because of the surfactants coating the bubble that that slows it down a little bit. That it's a little bit more like a rigid capsule than like a free slip band definition on the bubble. So I'm told that the surfactants are a bigger effect but you are absolutely right. There's also be a size effect. But yeah, there can be a lot of different contributions. So the nucleation of the bubbles relates to these fibers. So that's certainly in some sense an environmental factor. If you had other things in the glass, a lot of the solubility of carbon dioxide could depend on temperature. It could depend on other composition factors in the beer. Yeah, there's a lot of extra factors that can go into this. And a lot of times the beer companies think about these factors quite a lot. I told you about these cellulose fibers. You can also make special beer mugs that have extra ways to capture air pockets that are more better at making foam heads. So there's special beer glasses you can do which would be an environmental factor. Okay, we've got a few minutes left. I'm gonna talk about one more beer topic which is nitrogenated beer. So a lot of stout beers are foamed with nitrogen, dissolved nitrogen rather than carbon dioxide. The one on the left here is made with carbon dioxide. The one on the right is with nitrogen. So you can see the head of foam is taller. It's a little bit thicker head of foam. It's different, right? So what's going on with this? The real answer is I'm not 100% sure but I have some guesses or some knowledge I've got from some places. So it adds pressure which changes how it dispenses. The nitrogen is less soluble than carbon dioxide so it changes the nature of the bubbles and in the end you get smaller bubbles because of this. So the bubbles are smaller and that changes the properties of the foam because we've been looking at all throughout this talk how things depend on A squared or A cubed or things like that. So you know that size makes a difference. So smaller bubbles are stiffer. They're less easy to deform so that gives you a thicker head of foam. It doesn't nucleate the bubbles as easily. So these cellulose fibers that make bubbles it's harder to get that process to work with nitrogenated beers. You have to take special measures to get the foam. And if you have a question about the widget in Guinness cans ask me about it after the talk. I've got a slide on that but I don't want to talk about it right now. But the main message is special measures are needed. And then the internet says that if you use argon you get the same kind of effect as if you get nitrogen. And if you can't trust the internet, who can you trust? So that's probably true. When you pour a stout beer the bubbles sometimes are observed to go down. How many people have heard of this? I've seen it. You've seen it. This is another video from another physics article. So this is a beer being poured. You can see that on the right-hand side there's stuff going down. That's bubbles that you're seeing. And they're going down. On the left-hand side they're going up like they should be. So on the right-hand side they're going down. Let me show one more time. You're not seeing. OK, so come up afterwards and you can see it. Or you can talk to Brian who's seen it in real life. So yeah, this is hard to see and that's why I tried to put on a black background. The reason why they go down is that the beer is flowing downward as it's going in the glass. It's circulating around. And some of the beer is flowing down. And then it gets back to the scaling, which is why this is in here. So the drag force for these Reynolds numbers, these kind of problems, goes as a squared. The buoyant weight goes as a cubed. And I already just told you that the bubbles are smaller. So smaller bubbles have less of a buoyant force pulling them up. They're overwhelmed by the drag force pulling them down to go with the beer that's flowing in the glass. That's why the bubbles can go down while they're being poured in. It just gets back to the drag force is winning out because of the scaling argument. And this doesn't happen in regular beer because you get the larger bubbles. Finally, let's consider drainage. This is a movie on the left of a real beer over, I think, like maybe 20 or 30 minutes. And you can see that the beer foam drains. So apparently there's beer in the foam. That makes sense. There's some liquid between the droplets, between the bubbles. And that liquid is beer. So when it drains out, you get more beer in the bottom of your glass and less foam. And then this is the picture I showed you before. So you can kind of see that the red stuff is the liquid. So there is red stuff. There is liquid in the foam. So where is it? So let me give you a couple of words about the structure of foam. So you have bubbles. When two bubbles are pressed together, you have a face. And then between three bubbles, like three bubbles, you get these things called channels. And then where the channels come together, you get nodes. But the liquid is mostly in these channels. So you've got three bubbles coming together. And then basically these red things in here, those are the channels between three bubbles. So that's where all the beer is being stored in the foam head is in these channels. The faces are kind of pressed up against each other. There's no beer between them, but the channels. So liquid's draining through the channels. Here's a microscope we took of some channels in a, actually it's an emulsion, but it's the same. So there's some foam, so it's flowing through these things. And these are like pipes, right? This is like some kind of complicated, self-assembled network of pipes that your beer is draining through. So we want to understand about these pipes. And the question is, are they rigid pipes or slippery pipes? I mean, these aren't real pipes. They're made out of bubbles. And bubbles seem slipperier than pipes, right? So if they were rigid pipes, if they were working like pipes in our house, you would expect that the speed is fastest in the center and goes to a no-slit boundary condition at the boundaries. I know some chunk of you do fluid mechanics in your PhD work or whatever, so that you understand this. If they're slippery pipes, then you'd expect that the speed is the same everywhere, that you're just going to have everything flowing like a plug. And then the drainage would be faster. All right, which do you think this is? How many people think that beer foam is a rigid pipe? Raise your hand? Raise your hand. How many people wish I was handing out free samples of beer right now? All right, so we did some experiments on this. And I will confess right away, we only did a couple experiments with real beer. We mostly used model systems. We made foams with water or protein or small molecules. We added particles, and we visualized them with a microscope. So here's a movie we took on the left. And what you can see is that in the center of the channel, the particles are going quickly. At the edges of the channel, they're going slowly. This is more like a rigid pipe. And this is with proteins, like what you get in beer. Here's our velocity profile. So it's really going as, you know, obviously it's not as nice as a cylindrical pipe. It's got a lot of details here. But it's really going to zero at the edges of the channel and as fast as in the center. So the speed is slowed down because the walls are like rigid pipes with these proteins. And drainage is slower. We did some experiments with some smaller soap molecules. So SDS, a very common physics soap molecule. And then here we saw that there's a slippery boundary and we got more like a plug flow where the flow was, so the drainage is faster if you use smaller surfactant molecules. But the point is that in beer, you've got these big proteins left over from the yeast and the hops and everything. The proteins slowed down the drainage. And this is good. Guinness beer adds proteins to make the drainage even slower because they want a long lasting head of foam. The idea is that if you see this, you'd be like, oh yeah, I'll pay more for that. And that's good for Guinness if you pay more. So Guinness actually engineers their foam by adding in more proteins. It doesn't affect the flavor. It just affects the appearance in the glass and you'll pay more for it. And this is why they do it. So they use nitrogenated beer, which are smaller bubbles. So the channels are also smaller. So you get slower drainage because of that. And then you've got the slower drainage because of the surfactants. So to end the talk, the whole reason that we're talking about beer foam at all is because the whole process of making beer adds carbonation. This is different from wine and so forth. Beer just naturally is carbonated. And then you've got the surfactants. So now people are actually putting in thought process about how can we make the foam look better? How can we make it better for the consumer? So now beer is greatly engineered on the things that we think beer should have, such as foam. So take home messages are nitrogen is good and proteins are good. And drainage is bad, especially if you're trying to look for a nice looking head of foam. All right, thank you very much.