 visualization using our imagination to something very mundane, stuff we use in daily life. So I call this surfactant dynamics complexity in a Petri dish, this part of the course you will be doing experiment, not theoretical experiments, not experimental theory you will be doing a real experiment and analyzing the data. So I will give the conceptual lecture now and then after lunch we will meet up the hill in the M lab, the multidisciplinary lab and we will do the experiments. All of you will be divided into groups of three and I will show the groupings at the very end and if you do not see your name there please come up and we will add you to the groupings and then we will take it from there. So you have had coffee, let us start with the joke. Anybody know how we count in physics, how do we count in physics? No idea? 1, 2, infinity. 1, 2, 3. 1, 2, infinity. Yes close enough, 1, 2, 3, infinity and that is because we know the answer to the one body problem thanks to Newton. We know how to solve the two body problem by taking the reduced mass, for the three body problem Poincare showed us formally that there is no closed form analytical solution, so we go to infinity and take statistical averages. So there is a very humorous way of explaining what Jorge explained yesterday that for the few degrees of freedom we have the Hamiltonian description or the dynamical systems description and in the limit of very large degrees of freedom we have the statistical mechanical description but the real world we live in, the messy real world sits somewhere in between and that is the world I study. So and it is sometimes complex, sometimes complicated we have to figure out. So before I proceed I just want to give you some of you students have asked me what do I work on. So I do not have a fixed area of research. I work in statistical hydrodynamics, my PhD was in turbulence, I work in interfacial merengoni driven flows which will be what your experiment will be on. Nowadays I study statistics in spectra of solar place which is related to turbulence and in future I will be studying evaporation and condensation phenomena and then there is quantitative life sciences, there was no method to this madness because biology life sciences is too broad and I am not a trained life scientist. So I always tagged along with smart people who had a crisp sharp question in biology and if I could contribute something useful to it then I could get my name on the paper with them and it has taken me about ten years to form my own ideas and now I am just beginning to embark on my own experiments which is on mudskipper locomotion and now we are starting two experiments on the colony behavior of garden eels. Go look up on YouTube what garden eels are, we find them in Okinawa and then redback spiders and ant interactions, so predator-prey interactions so we do experiments in the lab, we bring in spiders and send ants as food and watch them devour the ants. So the next is something I started after I moved to Japan, what I loosely called statistical physics of energy and sustainability, primarily applying statistical mechanics fluctuations to renewable energy and finally something I started doing in my post-doc days which was granular media or amorphous media. I have, I feel like I have understood what I had set out to understand in granular jamming that was the topic I studied, now I am starting to explore what is known as triboelectric granular media that is granular media that is charged and this is a difficult problem and if you have, if you are curious you can visit my group webpage you will find more details or come talk to me and so that is as much as I can tell you about my introduction so when people ask me what do you study which field of physics I tell them doesn't matter physics, nowadays I also say I study condensed matter biology, experimental theory, I am not a trained theorist, okay, so brief preliminaries, so why, why care to do an experiment in a predominantly theoretical course because the relationship between theory and experiment is symbiotic, what do I mean by that, a good experimentalist when trying to test or validate a theory must know the limits of the theory, must understand what were the assumptions that went into developing of the theory, what are the parameter ranges under which the theory works and design the experiment appropriately, otherwise it's going to give you an answer for sure but it's not going to make much sense to with regards to testing the theory that you had set up out in the first place, likewise for a good theorist to come up with a theory it has to be experimentally testable which means they have to make, it has to have predictive value in testing some aspect of your theory and how do you go about doing that, you have to design your theory that can be tested using some observables, so if you cannot do that, I've since I lack originality I borrow from Ludwig Wittgenstein who in his tractatus logico philosophicus, Proposition 7, the English translation of the German statement is where of one cannot speak, there of one must be silent, so if a theory cannot be tested experimentally then it may be excellent mathematics but how do we know it is the right theory, so now we get started with the brief historical background of the experiment you're going to do, it goes back to 1490s, the first people to study surface tension in the form of capillary action, you all must have done some experiments on capillarity as when you were in school, if you put water in a container and put a tube then the water rises up the tube, so this is basically capillary action, the first known records of studies in capillary action date back to Leonardo da Vinci and Niccolò Aggiunti, not far from here, Firenze, and then there are records of by Nelly who was I think Galileo student who actually took the notes of Aggiunti's observations on capillary action, the first systematic measurements were conducted around 1709 by Sir Francis Hawksby, is that a name familiar to any of you, the reason I call this a brief review of a tortuous history is for two reasons, one surface tension phenomena are easy to observe in daily life, but they are conceptually, historically they have been conceptually difficult to understand and experimentally they are notorious for being difficult to quantitatively study, the first systematic measurements were done by Sir Francis Hawksby who is not known to you, but he was a remarkable person, he was the son of a draper, went into draping business as an assistant to his brother in the city of London and then midway through his life he switched to a career making scientific instruments and then he was hired by Sir Isaac Newton to develop experimental demonstrations for the Royal Society, he performed all the duties of a curator, experimental curator for the Royal Society, but he was never given the title nor was he ever given a salary and we have forgotten him, in the year 2010 the Royal Society recognized his contributions to science and they set up the Hawksby medal or the Hawksby prize which is given to the unsung heroes of science. So, the study of interfacial science, surface science of which surface tension is the most fundamental concept stands on the shoulders of some very humble giants, whose we do not know many of them. Then based on Sir Francis Hawksby's work James Juryn comes up with Juryn's law of capillary rise which is the example I just gave you which basically says that the height of the capillary rise, the liquid rising of the water column, the capillary column is inversely proportional to the radius. James Juryn was the son of a diar and he won a scholarship and went to Trinity College, Cambridge and that is where he figured this out. Then the concept of surface tension, does anybody know Andreas von Senna? He defined the concept of surface tension, in what year? 1751 we did not even know stuff was made of atoms or molecules, the atomic hypothesis was not even figured out by them. Then come the first two famous people young and Laplace in 1805 and 1806 they come up with the first quantitative equation that we know as the young Laplace equation. Sir Thomas Young wrote it down in verbose English Laplace wrote down concisely as an equation and they both had a fight about who was right for the next 24 years until Carl Friedrich Gauss came along in 1830 to say guys stop it you both are right. What Laplace said is the same as what Sir Thomas Young said. Gauss is the prince of mathematicians and he said he's given the title for no ordinary reason. It took him to equate the ideas of young and Laplace. Up until then we were talking about statics that is surface tension as an equilibrium concept where everything is static time does not enter. For the first time in the year 1855 James Thompson who was the brother of Lord Kelvin he comes up with this paper in the philosophical magazine called tears of wine, tears of strong wine I think. And so we were discussing some of us about it yesterday evening while holding our glasses of wine and that was not explained until 14 years later by Carlo Marangoni of the Marangoni effect. I'll tell you a bit about him shortly and then it goes on 1869 we have van der Men's Brugge explaining camphor board dynamics. Van der Men's Brugge was scientific assistant and son-in-law of Lato of the Rayleigh-Plato instability. The first reliable measurement of surface tension was not taken up until the year 1881 and the work itself was published in 1891 by a young lady of 18 years by the name of Agnes Pockels. She was born not far from here in Venice and she could never enter the university. In her times women were not allowed in the university that is Agnes Pockels and this is her first paper published in the year 1891 in the journal Nature. So in her time women were not allowed in the university so her brother who was a student at the University of Goettingen used to smuggle books from the library so that she could study at home and using that she realized that nobody had conducted a reliable experimental measurement of the surface tension. So she devised a simple apparatus at home using tin and a button and a needle and she made the first reliable measurement of surface tension of water and she sends it as a letter in German to Lord Rayleigh in England who takes help from his wife to translate it and sends it to Nature and has it published as it is. And this is the modern-day schematic of the apparatus she made which was known as the Pockels scale in her time which was improved upon by Sir Irving Langmuir at the GE Labs in Schenectady in upstate New York working with another young lady his assistant Catherine Blodgett for which he won the Nobel Prize. So commentators have said and I quote when Langmuir received the Nobel Prize for chemistry in 1932 for his work in investigating monolayers on solids and on liquids part of his achievement was founded on original experiments first made with a button and a thin tray by a young lady of 18 who had had no formal scientific training. So this field interfacial science or surface science stands on the shoulders of all these unsung heroes all these gentle or humble giants. So what is the basic mechanism of surface tension? We all have studied surface tension we know that where there is a surface there is an energy or a tension involved. It happens because we know all matter is made of elementary particles atoms which come come together through some form of bonds to form molecules. So if you think of this drop of liquid as comprised of several molecules of water they all have an interaction of bond energy that is taking place between these individual molecules that is in the bulk. So if you have a molecule in the bulk and it looks around itself along every direction in the solid angle it is going to feel the presence of a neighboring molecule through this bond energy. But what about the molecules that are at the surface? They are not going to see a molecule on the other side of the medium that is air in our example and there is a bond missing. So that excess energy is what manifests itself at the surface and macroscopically that microscopic excess bond energy macroscopic macroscopically shows up as what we know as surface tension. Surface tension is represented by either the symbol gamma or omega I normally use gamma it has units of force per unit length or energy per unit area. Now what happens when we introduce a drop of oil or soap on the surface of water? Soap is made up of tiny molecules that look like this they have what we call a polar head or hydrophilic head. So the head likes water so it sticks itself into the water and hydrophobic tails which stick their tail out of the water and they coat the surface. So when you introduce a little bit of soap very few molecules initially the surface tension is not going to change but as the number of those molecules increases and they crowd the surface the surface tension falls to a new value. So this for example the surface tension of water is 72 dynes per centimeter or million newtons per meter and when you introduce soap it falls to something lower around 28 or 32 dynes per centimeter. So the mechanism by which it goes so here you are looking at a plot graph of surface tension versus the concentration of the soap molecules. Initially there is not much happening you see the surface tension is more or less constant as the number as you start introducing the soap molecules as the concentration increases it falls rapidly until it reaches a point. Today we call this as a CMC point but long ago it was called the Pockels point because Agnes Pockels figured it out for the first time and it is called the CMC point CMC stands for critical micellar concentration what is micellar. So when you have concentration that has saturated the surface the excess molecules they form spheres of these micelles. So they basically come together like this. So all this is water but this part is packed close together so that water cannot penetrate but sorry I have drawn it in reverse this is wrong and so forth. So the tails are shielded from water and only the polar heads are exposed to water in the bulk. So whatever excess concentration of the soap molecules is there it goes into the bulk of water and they form these spherical balls that we call micelles. So that is the critical concentration at which the surface is now saturated and from there you get start getting micelles. This is known as the Pockels point because Agnes Pockels was the first person to determine this and from there she did a calculation which was also done by Lord Rehler separately. The calculation was that both of them independently figured out that once the surface is saturated with this concentration there is assumed that it is a monomolecular layer. It is a layer of a single row of molecules or single layer of molecules of the soap the excess is going into the bulk. So they know the amount of soap they introduced onto the surface and from there they could estimate what would be the size of a single molecule. So Lord Rehler came up with the estimate of 10 angstroms Agnes Pockels came up with the estimate of 12 angstroms but close enough and that was the first time anybody had figured out if the atomic hypothesis is correct what might be the size of these atoms and molecules. This was in the 1890s. Now all this is equilibrium so everything I have explained so far about the surface tension changing as a function of the soap concentration. What I mean by that is we introduce a little bit of soap on the surface of water and then we wait the molecules will dance about and then they will settle down and then we make the measurement of surface tension. Then we change the concentration a little more and then we wait until everything equilibrates and then we make the measurement of surface. So the surface tension values we are talking about are equilibrium surface tension values. Time is not a quantity that enters at this point. So will there be any case when sharp change in concentration happens surface tension happens sharp change. This is considered a sharpen of change. This is a very small region. So it is continuous change. It is continuous it is not discontinuous it is continuous no there is there is no discontinuous transition. Unless you introduce a high enough concentration at the get go you are not going to get a discontinuous transition. But typically the way these experiments are done these measurements are done. They have a trough what is known as a long Langmuir trough which was originally the pockel scale and they have a divider. So they move it to some portion. So let me draw it conceptually here and let us suppose that this region has soaked. Now for this area you have introduced a known quantity of the molecules and you get a certain value of surface tension. Now if you move this you have increased the area but you have kept the total number of molecules fixed. So the concentration goes down or you can do it vice versa which is you start with a large area with a known concentration and then start packing it. Either way since this is going to happen continuously the change is not going to be discontinuous. There is a point between 1 and 2 where surface tension starts to decrease. So what happens at that point and is it not worthy of some name? Maybe but people study that region as well because these are the intermediate regions where surface tension value has an equilibrium value but it is not the saturated value but it does not have a name to the best of my knowledge. What is happening between 1 and 2 that makes this change? The concentration is increasing enough that the surface tension will start falling. Okay so this has to do with the interaction between the molecules. Any other question? Okay. Please feel free to ask questions because we will be proceeding at a fast clip and there is lot to cover before we get you to do experiments. So we were discussing the equilibrium picture so far. What happens in non-equilibrium situations? So this video was taken in 2016 in Venice and you look at the soap bubbles. We know from elementary surface tension concepts that we study in school that a soap bubble takes a spherical shape because it is the minimal surface that bubble can assume but if you look at these bubbles they are not spherical. They are changing shape constantly. Why do you think that is happening? Because they have a large enough size that the air currents that are blowing around the bubble are deforming it momentarily. What do you think is happening in the bubble itself, the thin film of soap that is covering this bubble during that period? Any idea? Gravity is also pulling them down. Yes. And making more saturated in the bottom and then that is why also it breaks. It breaks but I am just talking about the deformation of the shape. So it is changing surface because it is doing a constant minimization of energy because there are so many forces acting. Okay. Fair enough. There is a force that is acting on the surface of the bubble due to the air blowing and that is stretching the surface and it is trying to optimize or minimize the energy within the new forces that are now present. But what is happening in the thin film of soap itself? You notice this is something like a dynamical case of the Langmuir trough. This surface area is suddenly changing. So what is happening when the surface area suddenly changes? So what happens is, let us say this is, that is the bubble and we have soap molecules. The soap molecules are what are imparting stability to this film and you have them on both sides. This is cross section of a bubble. So think of it as a large spherical bubble. I have just made a cut and I am looking at, so it could go down like that. But even within the bubble, I told you they are these micelles. Right? They are sitting all over here. So when the surface area is suddenly increased, these micelles come up and occupy the empty spots. So the surface tension takes some time to reach the equilibrium value again. So in that instant when the soap bubble is being stretched and these micelles are moving up to populate the empty space on the surface, there is a non-equilibrium condition. So there time enters. Now we are not dealing with equilibrium or statics and our understanding of what happens in these transients or where dynamic centers was explained by these two gentlemen. James Thompson as I mentioned was brother of Lord Kelvin. He was a very interesting character. He was an engineer, got his PhD from the University of Glasgow. He held the Regius chair at the University of Glasgow. Regius in Latin means royal. The chair was endowed by the royal family of England. And the previous holder of the chair, he was a successor of William Rankine who is a famous engineer and physicist. Rankine numbers and all good are named after him. And he introduced many neologisms to scientific English. The term torque, triple point, radian, they all go back to him. So he had a paper he wrote in 1855, I think we will see in the next slide, called the tears of strong wine. And the paper starts, this morning as I sat in bed staring at my glass of wine, I noticed this curious phenomenon which I shall describe. And you wonder why was he having wine in the morning in bed? We don't know. Anyway, but this tears of wine is a dynamical effect in surface tension. And he explained it qualitatively, James Thompson. It was quantitatively explained by Carlo Marangoni who got his PhD in 1865 at the University of Pavia. And he published his thesis as a paper in 1869 for which the effect is known as the Marangoni effect. Another unsung hero of science, he spent his time as a high school physics teacher for 45 years in forensic. And the title of his PhD thesis, if I am not wrong is the spreading of water droplets, expansion of water droplets. So tears of strong wine, philosophical magazine 1855. So that is the effect. So if you take wine with high alcohol content and you just swig it and you look at it, this is spread up 12 times. They form these legs. So these are also known as legs of wine. This was made with Nebbiolo grapes, high alcohol content. So you will never see wine the same way ever again. Now we are going to switch gears. Yes. Why it is happening? Good. So it is happening because let us assume that is a caricature of a wine glass and you have and that is your red wine. If you zoom in on this region, you see what is what we call the meniscus which is due to the capillary effect. What happens in this region is that the wine is made of a mixture of alcohol, water and some other colloidal particles that are coming from grapes that are imparting the color. Forget the colloidal mess. You have think of it as a mixture of water and alcohol. Alcohol evaporates faster than water. So water I told you has a surface tension of 72 dynes per centimeter. It is one of the highest surface tension values. The only other liquid I am aware of with even higher surface tension value is mercury. So water has high surface tension. So when the alcohol evaporates quicker than water, you have water left behind here, sorry here and alcohol has evaporated. So the surface tension has increased here. Now you have a higher surface tension perhaps closer to 72 dynes per centimeter over there and you have a lower surface tension here which is or even here which is perhaps let us say 30 dynes per centimeter. What did I tell you surface tension has units of force per unit length, right? So Newton's per meter, sorry not F over M. So if it is force per unit length and if I take the difference in surface tension value over this distance between this point and this point that comes out to 72 minus I have just assumed I have made up a number. It is going to be lower 30. How much is 72 minus 30? 30. 40, 50, 60, 70, 72, 42. Sorry I can only do symbolic math. I am not good at arithmetic. I failed in math 3, 3 years in a row before I got a respectable 69 percent in 10th class in high school. Okay. And we divide by the distance over which this difference exists. Let us say it is of the order of 1 millimeter or 1 centimeter. So 42 dynes divided by per centimeter divided by 1 centimeter. What do we have now? This has units of force per unit length. I have taken if I take the limit of delta L going to 0, I will get what is that? That is a surface tension gradient, right? Sorry. If surface tension has units of force per unit length, surface tension gradient has units of what? Force per, force divided by length squared. That is a stress, right? So this is and that is why we call it Marangoni stress after Carlo Marangoni. So the surface tension gradient causes the flow of more wine up the lip of this wine glass to replenish, to reduce the surface tension because the surface tension has to be minimized. It has to reach its equilibrium again. So when wine is transported up, then again alcohol evaporates, but the water is being dumped there. It is being left behind. So the water starts forming droplets and then it falls, which we call tears of wine. Okay. Make sense? All right. So far we have looked at surface tension and surface tension gradient. Now let us introduce some particles in it. Now what happens when you introduce a spherical particle at the air-water interface? Just as with glass. So if I take a glass of water and if I blow up, I get this. That is the capillary rise, the capillary column, right? So if instead I take, if instead I take a dish with water and I have a particle, what happens then? Even there the capillary effect is at work. The water rises to form a meniscus. So the water rises to form a meniscus if the particle is hydrophilic, which means it likes water. If the particle is hydrophobic, you see the opposite effect. So if the particle is hydrophobic, it dips down. That is what I show through the schematic here. If you have hydrophobic particles at the air-water interface, the water dips down. And if the particles are hydrophilic, the water rises up. That is all what the textbooks would tell you. There is one other effect that happens. I am going to blow this up now just to, and then the water rises to form a meniscus. And it also forms a thin lubrication film. That is something that the specialists would know, but the textbooks don't state because it's of little significance in daily life. But it has a curious effect if you bring these particles together somehow like the barrier in the Pockels scale or the Langmuir trough. If you start packing these particles, because of this lubrication film, if these particles come in contact with each other, they are going to slide over each other and stack up out of plane in the third dimension. But if you have hydrophobic particles where the meniscus dips down, there is no lubrication film. What happens when you move the barrier and they come and touch each other? Yeah, that's what happens. So they resist each other and they start buckling out of plane. But they buckle out of plane collectively. They don't buckle out of plane. It's like not popping out like popcorn. They behave like a two-dimensional solid because of this capillary interaction between particles. Mind you, these particles are like powder, granular powder, which don't interact with each other if you keep them in a bottle. But when you sprinkle them on the surface of water, because of this meniscus, this hydrophobic interaction, they interact with each other like a solid, a two-dimensional solid. And if you keep moving this boundary that's trying to compress these particles, they'll buckle out of plane with a wavelength. And that can be calculated mathematically from elasticity theory. And it was done by Dominic Vela as a professor at University of Oxford in 2004. So now we are slowly entering into the realm of the experimental conduct. So the setup that I'm going to describe for what I did maybe 10 years ago, 12 years ago, and what you're going to do today, you'll have a flat light tablet, an illumination source on which you'll place a petri dish, a circular dish filled with water. You can do this experiment at your kitchen countertop. That's where we did the experiments when we first did it. Take a dish, fill it with water, and take a needle with surfactant, which would be soap, sprinkle the surface of water with the hydrophobic particles or hydrophilic particles, and you'll see the two different effects. So what happens? Let us conceptually imagine what's going to happen. Forget the particles for an instant. You take the needle, touch the surface, and the soap will spread to reduce the surface tension. So for a split second, the soap is spreading due to the Marangoni effect, due to surface tension gradients. But if I want to visualize it, I'm going to sprinkle particles. But the particles are going to behave differently depending upon whether they are hydrophobic particles that hate water or hydrophilic particles that like water. So let's see what happens. So you're looking at... So this was the experimental schematic, and there was a camera from up above looking down. So you're going to see a circle with the particles floating on the air-water interface. Any questions? All good so far? So that is what you're seeing, and you have a needle coming down. There's a drop of soap, not soap actually, oleic acid, which is a particular surfactant. Anybody know what oleic acid is? Anybody know olive oil? Anybody know extra virgin olive oil? Oleic acid is extra, extra, extra virgin olive oil. It's a very pure form of olive oil. So when the needle touches the surface, this movie was taken at 600 frames per second using a high-speed camera, but we'll do it with our smartphones. Our phones are smart. Mine wasn't. So that's how it spreads. So as it spreads, it is pushing those particles out radially, and the particles, because these are hydrophilic particles, and they have this lubrication film, they can slide on top of each other, and they're forming a thin band. Now let us ask, what happens if I repeat this experiment with hydrophobic particles? Hydrophobic particles, I told you, behave like a two-dimensional elastic sheet or a solid. So here is the same experiment, but now I have replaced the hydrophilic particles with hydrophobic particles, and I repeat the experiment again at 600 frames per second. Instead, this is what happens. You get this star-like pattern, and that is happening because the solid is fracturing. It is forming cracks. These particles which flow like salt or pepper, when put on water, they behave like a solid together, not the individual particle, but all the particles as an aggregate behave like a solid. So let us describe the experiment and the analysis. So his question was, do these fracture patterns have a fractal property to it? It's a very good question. I left it out of my talk, but I'll explain it towards the end of my talk. Please ask the question again towards the end of the talk, because I know I will forget. See, I'm just not a professor. I'm an absent-minded one too. So normally, when you want to do these experiments in the lab, you have to follow some careful procedures. Why? As I told you, water is material with one of the highest surface tension values, which means it can very easily get contaminated. That means its surface tension value can change very easily, even with dirt particles in the air that land on the surface of water will change its surface tension value, which is why it was so difficult for many people to measure it reliably in the first place until this young lady came along, Agnes Pockels, and made the first reliable determination. So we filled distilled water in a clean petri dish of some diameter. What is clean? We acid wash, then water rinse, then bake it dry, and then subject it to UV treatment. In fact, even this is not the best way to do it. We have to treat it with plasma to get rid of organic impurities, which basically oil. Then you place the dish on a light tablet, and we record the transmitted light. And we had the camera from above. And then we throw in hydrophobic particles, which are basically hollow glass spheres coated with Teflon. For your purposes today, you're going to use pepper. Pepper is hydrophobic. And that was the diameter I used, 50 microns plus or minus 10 microns, and the density is 0.25 grams per centimeter cube, which means they naturally float on the surface. They're not going to sink. These particles were washed with ethanol, and then water, and then baked dry, which is to make sure they're also clean. They're not changing the surface tension of water when they land on the surface of water. And they are allowed to naturally puff into air and naturally allowed sediment onto the surface. We cannot control the packing fraction. Packing fraction is basically the total area occupied by the particles divided by the total area of the air-water interface itself. It's just a ratio. Then we take a clean needle, which is flame-clean. So we insert it in flame to get rid of any impurity. Then we dip it in oleic acid. Or in our case, it will be soap. And then we touch the surface and then record the dynamics with a fast camera. How do we do the image analysis? So before the experimental run, we took a background snapshot of the water-filled Petri dish. And then we subtract the background from all the experimental images we get. That way, we can get rid of any inhomogeneity in the lighting. So it's what we call flattening the image. So the image now looks as its own negative. We see the particles as grayscale particles in a black dark background. Then this is just fancy image analysis. Forget about it. And then we exploit the symmetry in the problem. So the problem has a circular symmetry. We are introducing the soap at the center. So when we do the image analysis, we do an azimuthal scan. That means I record what is the packing fraction along this radial line. And I repeat it for many such lines and divide by the total number of lines to get the average, the azimuthally averaged radial packing fraction. That means I will get a packing fraction. So what this notation denotes is, I am averaging it, sorry, 5. I am averaging it along the azimuthal angle. And I am getting the packing fraction, local packing fraction at this, this radial distance r, this radial distance r. And I am getting it along this average, averaged along this azimuth for every time instant. That means for every frame of the image. And that is what you will also do. Now, is it necessary to do a line instead of doing it in circles? No, you will get the same error. There is both methods work. So there is nothing holy about the way I have done it. I am not lazy. I am just an effort minimalist. I did it this way. If you know better, take the Petri dish. And if that is, you can do a circular scan. That is also okay. Remember, you are taking digital images, which means there are pixels. So you are not going to get a continuous line, a straight line or a continuous circle. It will be pixelated. That means there are errors. The error is going to be the same, whether you do it along a line or along a circle. It just adds up to the same. So the more lines you take, the more error you minimize up to a point or vice versa, the more closer the circles are. So when you do that, what I am showing you now is the ratio of this azimuthally averaged radial packing fraction as a function of the radial distance divided by what I call phi RCP, random close packing. I will come to it in a moment. But let us ask ourselves, how does this plot behave in time? So this is the time t equals zero before I introduce the oleic acid. So I see some fluctuations in the concentration or the packing fraction. Now at t equals zero when the oleic acid hits it, there is a shock going. And that was basically the surfactant that was spreading and pushing the particles and they were cracking, they were forming fractures. But before they could form fractures, they had to form a solid. That means they have to pack, they have to jam. So in granular jamming, there is a density or a packing fraction at which the jamming is supposed to occur if your particles are not, if they do not have friction. And that is known as random close packed density. Hydrophobic. The hydrophilic wouldn't be very different. What would be different would be that you see that this wave that is going from left to right, once it reaches a height, it just maintains that height. For hydrophilic, that height would keep increasing because the particles are stacking up on top of each other. Otherwise the dynamics will be the same. Any other questions? That is coming from two sources. Good question. Yes, sorry. So you wanted to know why is there an increase in the, in here on the extreme left corner. That is coming from two sources. It is a spurious effect. One is, is the needle. And two, when the circle is small, the area, the error is much larger. So at very small radii, you are going to get very huge errors. So up to that distance, you are not going to get any reliable readout. Yes? If we have come this far, we have already, so you wanted to know if there is a way to measure the shock velocity. So yes, there is, but the velocity is not very useful here. So there is, we can measure it. But the remember that this is not a steady state phenomenon. It is a transient. And it is occurring in a radially divergent geometry. So the velocity is not going to have much significance. But as I explained in the preliminary slide where I said a good theorist always tries to design a theory where you can measure an experimental quantity that, where a prediction is made on an experimental observable or a physical observable that is experimentally accessible. So yes, velocity is accessible, but that is not the useful quantity to come from theory. Another quantity is, and we will come to that. So what are the features of this plot? Now I am showing you the same movie, but I am showing you snapshots of it, superimposed on top of each other at different instance of time. So this is, so this is at very early time. You see lot of jagged features. This is all the error. So we discard that. And then you notice that the peak of that wave is increasing over some time, up to some time that I call p star. At that instant it has saturated because these particles have jammed to form a solid. And now they are starting to fracture. Fracture can only happen in solids. They cannot happen in liquids because yeah. So basically these particles have to slam into each of the first and form a solid before they can fail. So some of the quantities we defined. So now I am answering the question that you raised. Can we measure the velocity? Yes, we can measure the velocity, but there are other quantities that we can pull out from this quantitative data that is more useful. We can pull out the radial distance from the point of the tip out to the trailing edge of this wave, the leading edge. So trailing edge is r sub t. The leading edge is r sub l. And the width of that wave is w. So how do we start making sense of this? So I told you about the Merangoni effect or the Merangoni stress. So using this concept of surface tension gradient, a bunch of people had made a prediction. The first one was made in 1969 by J. A. Fay. The paper was titled Oil on the Sea. But the most rigorous theoretical treatment comes from this paper by Olivia Jensen in Journal of Fluid Mechanics in 1995. So he started, now here I am getting you into the conceptual aspects of the theory, how it was constructed and why it was important for me to understand what were the assumptions that went into it. He made the assumptions that he has an infinite point source of surfactant being introduced at t equals 0. And he is asking how is this spreading? So now if you want to physically visualize what might be happening at the surface as a surfactant is spreading, how are we doing on the time? Oh we are okay. So we need blue chalk. So let's say this is the surface of water. And I have surfactant spreading. So this is a surfactant. It's going left to right. So presumably my needle with drop of surfactant was somewhere here. As this is spreading it is shearing the fluid below. So it's disturbing the fluid and it is entraining what we technically call a boundary layer. This type of boundary layer is called Blasius boundary layer. In fluid mechanics the way it is solved is you assume you have an infinite flat plate being moved on the surface of water and how it shears the fluid below. So now you have two forces. The first force or stress, Marangoni stress, is coming from surface tension. That is being balanced by the viscosity of the water below. So the two are competing against each other. So the velocity as the gentleman asked what can we measure the velocity? The velocity of this spreading surfactant is determined by the surface tension difference which is provided in the force and the viscosity which is trying to stop it from spreading. So if we were to repeat this experiment on honey instead of water, this surfactant would spread much slower. So we have done this experiment on corn syrup, not honey. And it takes several seconds for it to go. So the question or rather the prediction, the theoretical prediction that was made by Olivia Jensen and a few people before him was by balancing the spreading force that is the Marangoni stress with the viscous drag or the viscous force, you get a relation for, so if this is your dish and you have an infinite point source, there are no infinities in the real world. So how am I going to cook up an infinite point source in the experiment? If my drop of surfactant has so many molecules of the surfactant, the oleic acid molecules, that it is going to completely saturate the surface and the excess is going into the bulk when this whole experiment is over, then I can approximate it as an infinite point source for the duration of that experiment. And our calculations showed that, remember I showed you this CMC, the critical micellar concentration that is the concentration at which any excess molecules will go into the bulk and form these spheres micelles. The concentration of the in the droplet of oleic acid was 20,000 times CMC. So for all practical intents and purposes, it was infinite point source. So as it spreads, so the surfactant is spreading across, I will call this the radial distance of the surfactant from the point source at time t. So you will get many such circles at earlier times and later times and that is what is being tracked by this wave that is going from left to right in that video I showed you. Now that is expected to go as some constant K times some constant which is, which has quantities like the surface tension difference squared, delta gamma squared is surface tension difference squared. So the Marangoni stress is coming from there divided by mu which is the dynamical viscosity, rho is the density times t to the 3 force. So this was dimensional analysis that yielded when you balance the surface tension force or Marangoni stress with the viscous force. So you get the surface tension difference which is the forcing and the viscosity mu that is providing the drag, the resistance to flow times t to the 3 force. So from dimensional analysis you will get something this radial distance if you can measure it will go as measurement time to the 3 force. So it is not an integer, it is a fractional number or it is an interesting fraction, so rational fraction right can be measured. So we can do this. Now another assumption he made was this time to the 3 fourth relation. So that is a prediction and it can be experimentally measured. So instead of velocity which we can measure he did not have a prediction about the velocity, he had a measurement prediction about the position, the radial, the position of the surfactant radially going outwards as a function of time right. But the other assumption he made was that this will happen for a deep fluid layer. What do we mean by a deep fluid layer? Remember I told you about this blazius boundary layer. So this keeps growing as this keeps moving to the right it will keep growing like this. At some point it is going to hit the bottom of the dish. The moment it hits the bottom of the dish the dynamics of this surfactant will change. So the time, so r sub s going as t to the 3 fourth holds as long as this boundary layer has not grown so deep that it is hitting the bottom of the dish. So now we have to ask ourselves is my experimental design in the deep fluid layer limit or the shallow fluid layer limit. How do we do that? So there is a time scale. The question is how long before the boundary layer hits the bottom of my Petri dish? So h 0 is the fluid depth in my experiment that is 1 centimeter divided by, so h 0 squared divided by nu nu is what we call kinematic viscosity. So it is the same as dynamical viscosity nu is basically nu over rho. So kinematic viscosity is the usual viscosity value you get divided by the density of the fluid in our case water. That gives you a time that is the time it takes for the boundary layer to hit the bottom of the Petri dish if you have a layer of 1 centimeter deep water in the dish and that time comes to about 100 seconds. Our experiment was over in the order of one second. So our experiment is never in that shallow layer limit it is always in the deep fluid layer limit. So we should expect to see this t to the three fourth scaling people had seen it. So what Dominic Weller and co-workers had found was these cracks that are proceeding they themselves actually go as time to the three fourths the length of the crack. Why? Because as the surfactant is spreading and it is creating these cracks it is invading into these cracks. So if you just track the length of the crack as a function of time you are going to get by proxy the radial distance to which the surfactant has spread. From our experiments which you saw the leading edge is going as time to the three fourth the trailing edge is also going as time to the three fourth and the width is also going as time to the three fourth. So this is plotted in log log scale is anybody here unfamiliar with log log scales or power laws? Yes no no is a perfectly reasonable answer do you know what a power law is anybody who does not know what a power law is it is okay not to know okay good I did not know what a power law is when I was born I learned okay. Let us say we have some quantity x of t or let us say f of x some function f of x and let us say f of x is going as a times x to some alpha and alpha is not an integer it is a fraction like time to the three fourths that is called a power law. Now if I want to look at a power law on a plot how do I look at it why do I see a power law as a straight line in this plot you notice that the gradations are not equidistant I have plotted this in log scale so the x axis and the y axis are in log scale what y is that if I take the log of the left hand side and the right hand side I can write the right hand side as log a plus alpha log x right what does this equation look like y equals mx plus c it is like a straight line but in log scale so either I can take the log of this quantity and plot it in linear linear axis or I can just plot this quantity as it is in a log log axis so I plot it in log log axis so y axis is in log x axis is in log so it comes up as a straight line and a slope is the exponent alpha so this is the most important plot for you people you have to do this experiment pull this quantity out and show that it goes as a power law if it goes as time to the three fourths you get full marks chances are you won't and it's for no fault of yours let me explain why we'll get into that now but before I do that let's discuss a little bit about the formation of this disordered solid and its failure because there was this question so remember I told you that the wave that is proceeding from the tip as it goes out radially outwards the wave keeps rising and then it saturates at some height at some time t star okay once that initial transient is that initial transient is the time over which a solid is an annular solid is formed right let me go back and show you the video again so you see this dark annulus that is the solid part so this solid has to form first as a surfactant is pushing the particles circularly outwards and only when that solid has formed can it crack or fail so okay good so so the question was in this dark band which part is the r sub s r sub s is over here so think of if you want to imagine what is happening at the in the plane think of it like a snow shovel snowmobile so you are shoveling snow so as the surfactant is spreading think of it as a snow shovel and it is scooping snow ahead of it or the particles ahead of it but this the snow particles are weird they can't pile up on top of each other so they they jam into they crash into each other and they become solid but there is more stress piling up from the surfactant and the crack so the surfactant does not invade into here it is along this region yes much smoother that's because the hydrophilic particles have this thin lubrication there so the particles can slide on top of each other so they don't form a solid any other questions what exactly are you calling cracks these jagged triangular things they keep growing okay and the question was do they have a fractal structure who asked the question yes we'll come to that soon okay so how shall we think about the formation and failure of the solid so the experimental observation is the fracture onset coincides with the time t star when the peak of the side or the wave starts saturating that means the solid has formed at that instant and the fracture has formed what is it telling us that the stresses are already large enough but you can't have a crack unless there is a solid so the solid has to form first so by t star the solid has formed as soon as it has formed all stress is relieved by undergoing fracture the funny aspect of the system is the same cause that leads to the formation of the solid is also leading to its failure okay and all the stress is relieved in a single instant at time t star that gives us an idea the question is how many cracks do we get so this was an answer question that was solved in an unrelated context by Sir Neville Mott he never it this was a puzzle he never even published it so he asked the question if I have a bar of solid of length l and width w that is under stress how many cracks do I need to relieve the stress across this entire beam at the same instant you need an integer number of l over w cracks that was the answer why because you will need to nucleate cracks at one width in order to relieve the stress this can be done in a couple of ways there is what we call a hoop stress calculation I will not go into it because then you would have to go in deep into elasticity theory which is outside the scope of what you're going to work on the your experiment stops with this effectant forcing basically this plot I'm just giving you the rest of the stuff because you did notice that there were cracks forming it's always fun to destroy stuff right okay so there's a simple way to go ahead this is already assuming that all the cracks would have the same width no no the question is posed differently from what a standard elasticity or a fracture dynamics question would be he said I have already stressed the bar how many cracks do I need to relieve the stress everywhere at the same time so he's not asking how much stress does the crack relieve which is the usual question asked he's asking how many cracks do I need to get rid of all the stress but assuming that all the cracks are going to be the same or they are allowed to be different I don't know what you mean by same and different so if the crack is like like a triangular opening right so that this triangle can have different angle so is it assuming that this he makes no such assumptions he's not going into the geometry of the problem he's just asking how many cracks he's not asking what type of crack I think it's what you what you raise is a relevant point but it is not necessary to answer that question so uh do people use this experiment as a some kind of proxy to study the crack in solid in like 2d dimension something like that also think of it you all know about numerical simulations right fracture mechanics is a very difficult problem so think of this as an analog simulation not a digital simulation to study fracture mechanics in two dimensions people do use the system now to study to learn more about fracture mechanics through this prototype because you can visualize the shock you can visualize the crack you can control the forcing by controlling the surface tension of the surfactant you're introducing so you can you can play games you can even slow down the dynamics by increasing the viscosity of the fluid below you can replace the disordered solid with colloidal particles which will give you a crystalline solid then the dynamics will be different because then the fracture or failure will occur across along what we call topological defects so there's an entire body of literature that gets that that kicks in if you are getting to the elasticity in the fracture mechanics but we are limiting ourselves to the surfactant dynamics have you studied the height of the like the free surface surface or whatever area like how does if i see it yeah that it deforms very slightly at the very early instant when we introduce the drop of surfactant and it travels really fast at almost one meter per second it is what we call the shear wave or the s wave you know earthquakes unfortunately right so the so when when two plates rupture and there is a wave that travels that's a shear wave he has a question please i have a question about the the region that we are going to calculate so in the case of a idophobic swifata we are going to have many cracks so uh how do you consider those cracks to measure the are you going to consider those cracks to measure your region or you are just going to get a hit of them so when you do the averaging as you mentally you're going to average over the cracks so think of it this way let's say right so this is the particulate region this is the surfactant region if you do an average here this is a particulate region this is a particle free region there's a particle region this is a particle free region so they're going to average cancel each other so it's okay there are also tricks if you start with if you sprinkle fewer particles too few particles visualizing it is difficult if you put too many particles you're going to get very jagged cracks there is a sweet spot somewhere in between in between these two limits so this is the part that Edison referred to as one percent inspiration and 99 percent perspiration so figuring out these empirical details of an experiment is the 99 percent perspiration any other questions or shall we proceed okay so we have n number of cracks and it seems to remain nearly constant as this whole dynamics takes place over the duration of the experiment which is roughly typically about 600 milliseconds and if we take the mott calculation the number of cracks is length divided by w we can take the radial distance hang on yes so the question he asked is is there a way to not reach the saturation now that would violate the theory that we are trying to test right Olivia Jensen developed a theory under the assumption of an infinite point source of surfactant so when I say I have in excess of 20 000 cmc I'm approximating it as infinite point source but you're asking what happens if I don't have an infinite point source I have if I have a finite point source and that was another paper I published with some friends a few years ago what happens when I change the surface tension right that is another way of posing the same question you have posed the cracks don't go all the way you you can go from basically all right I'll just pull out the paper it's easier so you get a family of cracks depending upon the amount of surfactant you introduce you can get three cracks you can get four cracks you can get five cracks there you can start with two cracks or just a line you can get three or four five so as you increase the surface tension the stress buildup is higher as a result you need more cracks to relieve it but the argument I'm applying in my case does not apply here here you have started with a solid that is you had a high enough particle fraction to begin with in my case you are starting with lower particle fraction so that you have to first form a solid and then it has to fail so two situations are different but the question of what happens when you have a dilute concentration below cmc is also studied any other questions okay so we can translate Sir Neville Mott's result that the number of cracks is length divided by width in a straightforward manner in our case it's not a bar it is an annulus so I take 2 pi r star divided by w star and I get number of cracks but this is going to fail in some limiting cases we can go a little further and invoke mass conservation that is all the particles that are in the annulus the dark band have come from the particle free hole in bit that is residing inside now so if we apply mass conservation and do a little bit of approximation basically out of the quadratic term you can change Mott's result which for our case gave 2 pi r star by w star will give you 4 pi times the random close packing density divided by the initial packing fraction now we have an experimental way if we can change the number change the initial fraction of particles at the air water interface we can change the number of cracks so we have a testable idea so we did a simple theory using mass conservation it I wouldn't call it a theory a simple calculation using mass conservation which gave us a testable thing it allows us to test is Mott's idea applicable to us so if we go there and test it so the dark black circles are the total number of cracks we measured from experiment if I look at r star and w star which I get from the experiment and ask how many cracks do I get I get the squares the red square and if I apply the mass conservation argument that is a dashed line you notice the dashed line is not working very well at low packing fractions why because I don't have enough particles for in that limit for this idea to work because if I have a low enough pack initial particle concentration then the surfactant has to spread out to a larger radial distance before it forms an annulus of solid so it takes longer and longer because I have fewer particles right okay yeah she has a question I'm done with my lecture that is the experiment you will be doing what is w star and how can we measure in our experiment in the patriot dish I doubt you can measure it in your experiment and you don't need to measure it so w is the width of the wave and w star is the width of the wave at the time t star when the packing fraction saturates so at that peak what was w the width of the wave at that at the time that the wave peak saturated you won't see that wave peak and you will see how your experiment will look because see you're doing the same experiment but not under these rigorous experimental conditions right we are going to simplify things for your experiment and so I'm going to show you how your experiment is going to look we did this experiment with Maria Liz and Luis Garcia yesterday at the m lab upstairs and that is how it looks so you take a petri dish you sprinkle pepper flakes so I told you you can do this in your kitchen take a dish or mineral water sprinkle pepper flakes take a needle dip it in dish dishwashing soap and touch it and that's what happens so this was taken with an iPhone in slow motion so it's going at 240 frames per second so you don't see the tannulus you don't see the dark band and that is because I have much larger particles here the pepper flakes are much larger you're not going to see all that part so I'm not expecting you to actually I ask you not to try and measure r star w star because those quantities are meaningless here just measure the just do the image analysis to see if you get the r sub s goes as t to the three fourths now I told you you're not going to see r sub s goes go as t to the three fourths why pepper flakes have oil in them so when you introduce pepper on surface you notice that it immediately spreads out it's almost like an explosive dispersion and there's a paper from 2010 or 2011 in proceedings of the national academy of sciences that explains this that initial explosive dispersion once that is done now the surface of water already has oil from the pepper flakes in it that means the surface tension of water has already gone down it has contaminated you haven't washed these pepper flakes with ethanol and and baked it and dried it and then did the experiment but it gives you an advantage because the surface tension of water has gone down but not down enough that it is lesser than the surface tension of soap if when you introduce the needle with dipped in soap the surface tension difference is not 30 minus 70 or 72 minus 30 it is going to be maybe 40 minus 30 that means the force marangoni stress is the magnitude is smaller that means the dynamics have slowed down that means you can imagine it with a smartphone so all the smartness did not come from the phone I had some contribution to make as well okay but this is what we expect to for each group to have by the end of today that kind of a video okay so what are you going to do that is your experimental protocol pull out your phones take a photo because I've already given you the homework assignment for tonight once you have your video on your phone you you have to go online I'm assuming you all have laptops or a tablet you can go online look for a free movie to image converter so you'll get an .mov or .mp4 format video convert it into a sequence of images and then we'll meet tomorrow at 11 in the informatics lab and we'll start the data analysis we'll start pulling out numbers from your images so before we leave these are the groups we have 18 groups so we have a very busy afternoon at least I have a very busy afternoon if your name is not on this list as I suspect one or two might be missing please come see me I'm going to assign you to a group because at least one the last group has only two people two members now what are we going to do we are going to so because I'm aware there was at least one student from China and I haven't come across a Chinese name in this list so I know at least one person is missing one person's name is missing so come see me we'll we'll include you in one of the groups you are going to work in groups of three you will participate in collecting the data you will participate in brainstorming together to figure out what is happening how to analyze it and how to pull out numbers from it and then towards the end of this course your exam is basically a 15 minute presentation to convince me that you've got a power law why won't you get a power law because as I told you the pepper flakes have oil that means the water surface is contaminated so you're not going to get t to the three fourths but you will get t to the half or t to the point six or t to the point six five or t to the point five five that is good enough for me because we are doing an experiment in this age using what pepper flakes water and smartphones and not large hadron colliders and quantum what's scanning tunneling microscopes so and where are we going to meet we are going to meet in m lab so for those of you who don't know you see one one is where adiatico guest house is if so if you go if you come down where the bus stops and you make a U turn you see stairs going up up the hill go up the stairs so these are the so where where one is over there the stairs go up this way and this is the strada costiera here be careful as you cross and then you take stairs and that is m lab or if you feel like having a walk in the park you can take the stairs up here and enter through the park and come this way so you come out like this and there's a road there that takes you to m lab my honest suggestion don't try it take these steps this is quicker do your experiment and then take a walk in the park i'll tell you why because you get to visit the castello miramare right it's open it has nice gardens you've done hard work you've collected your first experimental data now it's time to relax all right enjoy your lunch