 In this last module which is 7th for this semester, we will be trying to become ambitious for covering the entire space of dispersed systems in just about 40 minutes and the topics I intend to touch upon are outlined there colloids, aerosols, emulsions, foams, coagulation and one of the most important theories for coagulation, Smolikowski's theory. I did mention a bit about dispersed systems in one of the earlier lectures. However, we have not had a chance to look into how the dispersed systems owe their thermodynamic nature, how partly the intrinsic thermodynamic nature could be modified and in those systems where the effect of interfacial energy is too overwhelming how we could analyze the breakdown of dispersed systems. That is what we will try to look into today. Let me begin with the slides we might interchange with the visualizer. First some generic remarks about dispersed systems. First point to note here is that the dispersed systems are thermodynamically unstable. Why does this happen? As the name suggests we have one of the phases dispersed in another perhaps at very small dimensions and if we have particles or droplets present within the continuous phase of the dispersed phase, we expect the interfacial energy to be very high. That is the genesis of the unstable nature of the system, the very high interfacial energy associated with the large interfacial area in these dispersed systems. It is a thermodynamic tendency for a system with high interfacial energy to respond to lower that high energy. That could be done by spontaneous coagulation or coalescence or aggregation of these droplets or particles forming larger clumps or coalesced droplets which then will have lesser specific interfacial energy. That is one way the system could respond to the unstable nature. So, it is the interfacial free energy which is large and positive which leads to this kind of behavior and we call such systems as liophobic dispersions. By that we mean that the droplets would aggregate or coalesce rather than remain in contact with the continuous phase or solvent. So, it is the tendency to coalesce or aggregate and minimize the contact with the continuous phase or the solvent phase. We did refer to colloids very early on when we talked about fluid bounded persistent structures, submicroscopically small in at least one dimension and here we come to that dimension. We will call a dispersed system as a colloidal solution if that one dimension is less than about 10 raise to minus 7 centimeters to 10 raise to minus 4 centimeters. If the particles are smaller than this range we will call them colloidal. Next question will arise as to how these liophobic colloids can be prepared. There are two classes of making these colloidal dispersions or solutions. First we have the class of methods which are the dispersion which referred to one of the phases being toned in another phase making smaller and smaller fragments of this particular phase which will end up as a dispersed phase. The other one is condensation as you are familiar with condensation it would involve super saturation of a phase with a solute. As a result of this super saturation we will have precipitation into very small particles or droplets leading to the colloidal dispersion by the condensation method. It is here that it might be worthwhile to know that in condensation which is akin to crystallization the super saturation can itself lead to two different ways of forming these submicroscopically small particles. One could think of nucleation here. We may have ultra high super saturations leading to spontaneous formation of molecular clusters which are few thousands or few tens of thousands of molecules forming aggregates as a result of thermodynamic fluctuation. Such kind of clusters would always form when we have very high super saturation. What fate do these clusters or nuclei suffer? There are two possibilities just a similar kind of fluctuation which has brought these large number of molecules together forming the nucleus subsequent fluctuation could take them apart. So we might have this dynamic formation and disintegration of nuclei. However, when you do the analysis based on total energy for such clusters it turns out that there is a critical radius of a cluster below which these clusters will be necessarily unstable. They may form but by the similar fluctuation as is creating those nuclei or sub-nuclea size particles those will disintegrate also. But if the particles have these nuclei have critical size or higher then they will grow into larger particles droplets crystals whatever we have. Right. So we have the surface energy and then we have the volumetric the bulk term corresponding to energy. So when we take this together it is possible to show that there is a critical size of nucleus below which the clusters would be unstable. This is purely talking about super saturation and generally very high super saturation driven nuclea formation or nucleation. But most realistic systems as you would anticipate would not be so pure. They would always be some kind of sites for nucleation very fine dust particles for example. Those fine heterogenities could be the preferred site for condensation or precipitation of the solute which is coming from this super saturated surrounding solution. So if we have those extraneous particles serving as a shortcut to nucleation we have heterogeneous nucleation as opposed to very clean systems where the nucleic will have to exceed the critical size and that will require very high super saturations. The heterogeneous nucleation is relatively easier at lesser super saturation driving force we will have the condensation bringing about formation of these colloidal particles. That is precisely the reason why when it comes to the precipitation leading to rain we have the role for dust particles or in artificial rain we seed the clouds with nucleic silver iodide particles for example or in crystallization you have the seed crystals added which serves as sites for nucleation. In any case this is the other method where the super saturation driven formation of colloidal suspensions of particle solutions could occur. The lyophobic colloids with concentrations of particles in excess of over 10 to the power 10 per cc would tend to coagulate rapidly. The particles see each other the colloid and if they have very high surface energy this is one sure way of removing that unstable character particles could come together join together so that the effective area goes down the energy goes down. So it is here that you can think that lyophobic colloids with high very high surface energy for particles would tend to have every collision fruitful every collision results into clumping together or aggregation of particles. This is how the colloids would break down but having said that yeah. I said there are two ways in which you can have the colloidal particles forming in condensation itself. In condensation you could have one kind of system which is relatively rare very pure system there is only the super saturated solution. It could be the water vapor in air free of dust particles or it could be super saturated solution in a crystallization system without any seed crystals or roughness on the surface because those particles if they are present they are sites of lower free energy. So the system goes by relieving the super saturation by depositing this solute which may be the vapor or the solute in the super saturated mother liquor it is now deposited on to these particles. So if the particles are present it is a heterogeneous nucleation. If there are no extraneous particles thus particles or roughness on the surface or seed crystals then we have only the solvent and the solute and if such a system has to form nuclei spontaneously in these very clean systems you need very very high super saturation. In crystallization literature you would come across super saturation to the tune of say about 1500 percent over saturation you require that high super saturations in order to have any crystal nuclei forming and that is homogenous nucleation. So homogenous nucleation is in a very clean system heterogeneous nucleation where there are some seed nucleicides to build up the nuclei. Now we are returning to something which is colloidal particles which have very high energy we might only have the colloidal particles of the dispersed phase and the continuous phase and these particles they are moving about they may be large compared to let us say molecular dimensions or the molecular free path but they are still not so large as to be in relatively static positions. Now what I am referring to is probably the dimensions of particles which might exhibit Brownian kind of movement. The zigzag motion which is the result of non equilibrium conditions in the continuous phase. The particles are small enough to be able to show the non equilibrium effects but they are large compared to molecular dimensions or mean free paths. Now it may take a different level of understanding this if such particles are there they are moving about it is interesting that the hypothesis that such particles such small particles capable of displaying Brownian movement might exhibit a property similar to what we see in diffusion. How these particles move from one region to another might show a similarity with diffusion. Obviously we are not dealing with molecular system but much larger particles. So the characteristic transport cohesion is the Brownian diffusion cohesion. We will see later how to predict or estimate that Brownian diffusion cohesion given the size of particles. So you can visualize on one hand the kinetic molecular theory for molecules. Molecules are getting bombarded by other molecules. They move about in random chaotic fashion and on the other hand you have these colloidal particles much larger than molecular dimensions or mean free paths but they are also slightly more sluggish leaf but moving randomly and chaotic and the two have a certain analogy which is of diffusional character ok. So these particles which are moving around if they happen to collide against each other they will tend to stick to each other. There by they will lower the energy. We might otherwise say that these particles are droplets which are getting bombarded against each other they get adhere to each other. You might have adhesion of these particles. The moment they come together they clump together and the chances of such a thing happening are obviously high if the particle concentrations are large. So that is why this number was there. If we have number of particles per cc in excess of 10 to the power 10 we might have such frequent collisions leading to the clumping or agglomeration or aggregation of particles. Now for such a system which is called as rapidly coagulating system a certain theory was advanced by Smolikowski in 1916 to be able to estimate how the number of particles in the system would change with time. And it turns out that simple theory is in pretty good agreement with experiment. We will shortly see what that theory is. I will try to simplify the picture as much as possible but we can alongside look at the more rigorous derivation also. Before we do that let us think of the practical part of it. If we have such rapid coagulation systems in practice you may not always want. After all the dispersions may have to be processed over certain lengths of time. You might not want these dispersions to break down very quickly and many products will require the dispersions to be metastable as stable as you can practically have. So often we need this stabilization that could be done either through electrical factors or through solvation barrier. Electrical factors you can imagine if let us say the colloidal particles carry surface charge. Typically very small droplets of particles may carry negative charges and if they are suspended in the medium then it means that even though particles come very close to each other the electrostatic charges would repel them. So that is a stabilizing factor. If you want to destabilize you should be aiming at removing that charge or alternatively we might have solvation barriers. We might deliberately coat these particles or droplets with amphipelic molecules like surfactants. Therefore there is a barrier the energy is lowered and if the particles have to bring the native surfaces together they will have to rip off these surrounding surfactant layers which will not be easy. So there are those kind of barriers possible. So through such expedience it is possible to stabilize dispersions even when the concentrations are in excess of about 10 to the power 14 particles per cc. So we can make such dispersions colloidal dispersions last for long periods. If we achieve such stabilization it means now very few of the collisions will be fruitful or effective in reducing the energy whatever residual energy in the surfaces and that will amount to something like 1 in 10 to the power 6 collisions or 1 in about 10 raise to 8 collisions might actually succeed in coagulation. So it is possible to tailor make the stabilizing factor such that you reduce the coagulation frequency or efficiency by a very significant factor and such very stable dispersions are quite common in practice. Some of the dispersions could be made stable for up to several years. Having said that and having earlier mentioned that the mists or smogs aerosols could be made harder to disperse if tarry material is present we cannot actually stabilize mists or smoke that much. Forms on the other hand can be stabilized as I mentioned earlier by having factors like these electrostatic repulsions and you can preserve soap films for several years. Yeah, they will coagulate you might be able to delay it a bit but not so much you cannot make. What is the limit of the coagulation? You just have to look at what is the time it takes for the smog to last even in the most polluted cities of the order of few hours at the most. We cannot however use this Smolikowski's theory for coagulation of foam because we have very high volume fraction of gas. In typical foam you may have 97 percent or higher percentage of volume of gas disperse in liquid. In those systems the stability depends on other factors the drainage of liquid or thinning of foam films and other mechanical factor and often it is required to break the colloidal dispersions. It will help here how the colloids are stabilized. Once we know what is the stabilizing factor we can take counter measures against the stabilizing factors. If it is the charge you might think of neutralizing the charge by introducing let us say in a negative charge stabilized colloidal system you can introduce maybe cationics or factors where the adsorbing component will carry positive charge. So it will annihilate the negative charges once the particles become neutral then it may be easier to break down those kind of suspensions or emulsions. So in the remaining time let us focus on the collision rates in disperse systems or Smolikowski's theory. It is here that I would like you to think of the system as having all initial population of particles which we will call primary particles. Primary particles are the particles that we start with. For simplicity we may say all those particles that we have in the beginning are of the same size. It is possible to build up more rigorous versions of this theory allowing for the polydispersity of particles but let us say to get the key concepts and approach right we think of all particles to be of same size and they are moving around with some characteristic transport cohesion the diffusion cohesion D like the Brownian diffusion cohesion. With the same units as of diffusion cohesion molecular diffusion cohesion this will relate the flux of the particles to their gradient through this Brownian diffusion cohesion. We may think of the primary particles number concentration as being represented by the symbol n subscript 1 n 1 is the number of primary particles per cc and they will come one other parameter called collision radius of particles capital R or what is stated originally by Smolikowski the radius of sphere of coagulation I need to explain this a bit. Let us say you have certain primary particles we have these particles of same size all moving around randomly and over period of time will have reduction in their number and so on. If the number of primary particles per volume is n 1 that will be function of time and we might be interested in how this total number of particles within a system is changing with time. Now in Smolikowski's theory what is visualized is the following Smolikowski hypothesized the following approach that you can take any one of the particles primary particles and regard it as stationary and around this particle he said there is a sphere of coagulation or sphere of influence with radius which is capital R. Now that is a hypothetical envelope around the particle he says if any other particle crosses this sphere of coagulation it will get bonded to this particular particle under reference. So you can now think that in this population we have one particle arbitrarily chosen to be stationary and around that there is a whole population of these other primary particles they are all moving about around this arbitrarily chosen particle we have a certain sphere of influence or coagulation such that any other particle which crosses the boundary of this sphere will get bonded to this. In other words in this case it is simple another particle will cross this when their centers differ by the sum of their radii or the particles touch each other right every contact between particles is fruitful leading to aggregation. So this sphere of coagulation is a concept around the sphere of coagulation till infinity you have all the particles if this sphere is acting as something which is able to make particles vanish we could say the number part concentration of particles at this boundary will be always 0 anything which comes to that point we will get immediately removed from the system. So the free primary particles number concentration on this spherical envelope will be 0 far away it will be whatever is initial concentration supposing initial concentration is n 0 that will be the n 1 0 that is the initial concentration far away here the concentration is 0. Now you have two ways of looking at it regardless is speaking you take the spherical coordinates and find out what is the profile of concentration which makes it vary from n 1 0 at infinity at the radial coordinate of infinity to 0 at this sphere radius or at r. Then you can actually find out what that concentration profile is you can try calculate the flux and then you multiply that flux by the area of the sphere and get the flux that will give you a measure of what will be the collisional frequency of like particles primary particles then we should be able to relate it to how the total number is changing with time or maybe even simpler and I will just see this idea you could take take the film theoretic picture you can think around this sphere a film across which there is a steady state kind of transport and you are using a pseudo steady state kind of approximation the flux is given by the film theoretic picture and again the total area of the sphere times that flux will give you the rate of disappearance of particles primary particles alright. So, based on these approaches we could consider the equations here. So, let us look at this minus d n 1 by d t that is the rate of change of number of primary particles n 1 per volume that will be equal to 8 pi d r n 1 square and this equation presumes that every collision is effective in removing two primary particles from the system if their centers approach to within the collision radius capital r, but in our approach here we are concerned with total number of particles in the system and each collision in effect will remove only one particle because before collision there are two like particles after collision there is one particle. So, in effect we are removed only one particle from the total number. So, we change our notation here a bit moving from the primary particles we are now n the total number of particles in the system that is d n by d t this should be minus d n by d t equal to 4 pi d r n square integration of this equation leads to 1 by n equal to 4 pi d r small t plus a constant at time 0 the number of particles per c c is n 0. So, that constant is 1 by n 0 substituting that we get 1 by n minus 1 by n 0 is equal to 4 pi d r t and this equation this result has been checked against experiments many experiments including steric acid smoke coagulation. So, there may be one way of checking it you could plot 1 by n versus t that should be a straight line with a constant slope of this 4 pi d r or the other way would be that you look at a time when the total number of particles changes reduces from n 0 to half that value. So, at t equal to t half n will become equal to n 0 by 2 this should read as n 0 by 2 at t equal to 0 n is n 0 at t equal to t half n is n 0 by 2 correspondingly the t half will come out to be equal to 1 by 4 pi d r n 0. So, you could check at what time the initial total number of particles per c c reduces to half the original value n 0 by 2. As for second order chemical reaction we see that this t half is inversely proportional to the initial number of particles total number of particles present n 0. The diffusion cohesion here for the particles this is not molecular diffusion cohesion may be regarded as Brownian diffusion cohesion for that dimension of particles is given by the equation given by Einstein and that is k t by 6 pi eta a where k is Boltzmann constant t is absolute temperature eta is the viscosity of the medium the continuous phase or solvent whatever we have and this small a is the radius of the particle. So, if you look at d estimated from here and then plotted for this steric acid smoke coagulation is 1 by n versus t you get a pretty good agreement between the prediction and the experiment. There is only one limitation for this diffusion cohesion relation the particles that you are calculating diffusion cohesion for should be large compared to the mean free path of molecules. We could think of next emulsions. Emulsions are made by subjecting large oil droplets to high shear and you split this into smaller daughter droplets. If you are thinking of oil being dispersed in water you can think of water being vigorously mixed with a mechanical agitator and you add large oil drops to it which when the enter water are subjected to turbulent shear which will tend to break these droplets into smaller ones. Supposing that in the initial concentration we only look at the dispersed phase and the continuous phase. The dispersed phase is oil, continuous phase is water. Oil droplets subjected to turbulent shear conditions will break into smaller droplets. There is nothing to stabilize. So, you can think of this droplet being bombarded by turbulent eddies and then breaking into these ones. Intermediate picture could be maybe something like this where the instability takes the droplets into separate droplets, but the rivers can also happen. When you have these daughter droplets or other droplets which are floating around by the similar eddies movement random chaotic movement they may actually bump against each other. If you look at the rivers it might actually be these droplets are coming closer together. This is oil. There is a water film in between. If these approach each other and squeeze out the water in between they may actually coalesce. So, this part is the reversible part, reverse part. So, you might have shear breaking it into smaller droplets and random eddies bringing together these droplets and squeezing the film of continuous phase will cause the mergers or coalescences or coagulation. So, one of the simplest models for emulsion formation was based on this picture. So, I like you to think about a few things here. The first is if we have to write how the number of emulsion droplets dispersed phase droplets will change with time. Based on this mechanism how would we formulate the equation? So, you are already saying that it is to be treated like a reaction. The forward reaction is breakage, the backward reaction is coalescence and the coalescence is the backward reaction or the reverse reaction. Now, if you choose say the number of dispersed phase droplets as N and you want to write N droplets per volume of the continuous phase, N droplets of dispersed phase per volume of continuous phase. How these are changing with time? You want to write dN by dt. How would we write this? Or on similar terms minus dN by dt? How the number of droplets of dispersed phase per volume of continuous phase will change with time? We go by the reaction. The forward reaction is corresponding to breakage and let us say we are looking at only the binary breakage or coalescence. If a droplet breaks, it breaks into two droplets. If droplets have to coalesce two droplets combined together and form one droplet, that is the simplest picture we keep. Then for the breakage we might say there is a breakage rate constant Kb and for coalescence we may say a coalescence rate constant Kc. Now, in terms of Kb and Kc how would we write this minus dN by dt? Kb into N, Kb into N minus Kc into N square. . . . . . . . . . . . . So, you will write this as if the breakage is there, only breakage is there number of particles will increase. If there is coalescence, if there is only coalescence and no breakage, the number of particles will decrease. If the two are present together, we will have this sign correct. Then at t equal to 0, you may have the same N equal to N 0 and therefore, we can integrate this and obtain the expression for the number of emulsion droplets changing your time. You can obtain an index of stability half life for the number of droplets to come down to N 0 by 2, if you start with initial number N 0. This is one way in which you can approach the emulsions breakage and coalescence. There are certain two phase systems containing oil and water which are stabilized with the help of a surfactant and perhaps a co-surfactant such that you move from the unstable region of dispersions to a thermodynamically stable dispersion that is the micro emulsion. So, we will not worry about that. Those have many applications. Next is foam. Foam or froth will have very little liquid separating the bubbles. So, in foam the bubbles are actually polyhedral if we roughly represent them by the gas pockets here and the liquid film here. Then as a result of gravitational drainage, the liquid drains from within the liquid films bringing these phases closer together. If they are stabilized by anionic or cationic surfactants, there will be some electrostatic repulsion when these phases start seeing each other. But this is only an isolated bubble. In a polyhedral foam, you have large number of such bubbles in proximity and these films have a certain interesting ways to combine with other bubbles. These gas pockets are for different bubbles. Two bubbles are sharing this liquid film. If you take the polyhedral foam bubble, then one of the approximations or idealizations could be regular pentagonal dodecahedra. What it means is, each flat film is regarded as pentagonal of equal sides regular and each bubble has 12 such phases. So, the common foam that you see with large bubbles, they are polyhedral. If the bubbles are very small, they will be spherical. So, the foam is one which is polyhedral and spherical bubbles are in froth. Some people reserve the word froth if there are particles within the liquid. So, that if you let a layer of froth vaporize, it will leave a thin film of solids at the bottom. That probably has to do with the froth flotation, which is a much older process compared to other applications of foam. Now, if we have this structural idealization that each polyhedral foam bubble is like a dodecahedron, each with 12 pentagonal phases, each phase with the same length of side, you have one visualization. You take many such polyhedral together, dodecahedrons together. If they are all of same size and you put them together, there is one conceptual catch that is you are not able to fill space completely. If you take cubes, you can fill space completely because they will be exactly matching phases, leaving no gap. But in this dodecahedron, there are certain gaps which are not filled. They amount about 3 percent. So, it is presumable that those voids are filled up with liquid forming the films. Now, each phase has those 3 sides. Those sides in real foam are not line segments, but they have certain thickness. They are called plateau borders. They have a cross section which is curvilinear. If you take a cross section, this is how plateau border will look. This is a part of one film. This is part of another film. This is a part of third film. So, if you call them F1, F2 and F3, 3 films are meeting together in this edge or the side, which is like a curvilinear pipe all made of liquid, no sides, no surface, no wall. Then we have liquid here and a little consideration here from the Laplace Young equation will tell you that the pressure at the center and pressure in the film in the flat region when compared with the pressure in the center of the bubble will tell you something interesting about drainage. Think of this. Supposing you call this P1, this as P2 and this as P3, we want to know how these pressures are related to each other such that that analysis can tell us how liquid will flow within the plateau border. Now you can think of let us say first comparing P1 and P3. P1-P3 will be given by Laplace Young equation as 2 gamma by R between 1 and 3. But what is the radius of curvature for this flat region? Radius of curvature is the radius of that sphere of which this will be the surface. For a sphere to have flat surface, the radius will have to be infinity, is not it? So, that means this is infinity. If this is infinity, 2 gamma, a finite quantity by infinity will be 0 which means P1 is equal to P3. If similarly, you consider P1 and P2, P1-P2, it will be 2 gamma by R12. Radius of curvature for this surface is some finite radius R12. So, this is a positive quantity which means P1 is greater than P2, but P1 is equal to P3. P2 is equal to P1. So, P2 is greater than P3 which means the pressure here is higher than the pressure here and the liquid will flow from a higher pressure region to lower pressure region which means liquid goes from the center of the plate of order. You have P1 equal to P3, P1 greater than P2. So, P3 is greater than P2 which means the liquid flows from P3 to P2, reverse of what I am saying which means the liquid is flowing from the flat region of the film towards the center of the plate of orders. All these sides are connected, all the plate of orders are connected, they form a network. So, everywhere the liquid is flowing from the center of the film towards the center of the plate of order and through the plate of order network the liquid is flowing downward alright which means this should be possible to visualize. If you were to take a height of film of some surfactant solution, aqueous solution and in the same solution it dissolves a dye for visualization like a red dye and you put the drop at the top what would you get to see. The color of this drop will let you visualize how that dye solution is trickling down. If this theory is correct it shows that the liquid film must thin and liquid should be dumped into the plate of order. If you add such a drop of liquid at the top even if it were to be placed at the center of the plate of order it would not be able to go into the flat region of the films. It will be drained off through the network of plate of orders. So, you will be able to see actually this red drop traveling down through the network of plate of orders. The films will remain colorless. You might object to this by saying that the films are very thin. So, even if they are colored you might not be able to see. But actually even if you make that observation carefully you would see that the flow is through the plate of orders whereas the center of the film which is at higher pressure compared to center of plate of order will not accept liquid. So, the drainage occurs through the network of plate of orders. But the film drainage does occur and these films thin down over period of time and they may become so thin that any extraneous vibration may be able to break this foam film. And you know in actual foam in practice we have all bubbles not of same size they are of different sizes. So, the smaller bubbles will have a higher pressure excess pressure. So, the gas will diffuse from smaller bubbles into larger bubbles. And there is another way in which the foam breakdown occurs without any rupture of films. So, these are the two mechanisms we will stop here for today.