 In this fourth module on monolayers, we today look into condensed and liquid expanded monolayers and move forward on the phase transformations or phase transitions that I had commented upon earlier. But before we go any further, let us have a quick recapitulation of the pressures, surface pressures. In absence of any repulsive pressure, the total pressure will be this pi k plus pi s and pi k will be given by k t by a minus a 0, pi s is given by minus 400 m by 8 to the power 3 by 2, where the area is supposed to exceed about 100 angstrom square. And we could for cohesive films, write the equation of state of this kind pi plus 400 m by 8 to the power 3 by 2 into a minus a 0 equal to k t. And one could obtain the pi s by considering the total pressure at air liquid against oil liquid interfaces, where the cohesion will be omitted. The last topic we had seen was charged films for which the Davis equation gives you the repulse energy from this expression on the right hand side of equation 7. Pi r is now 6.1 c i to the power half cos hyperbolic sin hyperbolic inverse of 134 by a c i to the power 1 by 2 minus 1, which I had asked you to simplify under conditions where a c i to the power half becomes less than 38. That would give you this approximation. So, when you combine this approximation 8 for 7 with equation 2, we would obtain the surface equation of state for charged films as follows pi a minus a 0, which is basically pi k plus pi s plus pi r times a minus a 0 equal to 3 k t minus 400 m by a to the power 3 by 2 into a minus a 0 minus 6.1 c i to the power 1 by 2 a minus a 0 minus 2 k 2 2 k t a 0 by a. At oil water interface, because the oil can get in between the chains which have cohesion acting amongst them, spacing them apart that would lead to dropping of the second term altogether at oil water interface. And this equation has been tested for a number of surfactants both on electrolytes as well as in absence of any electrolyte like salt. So, we now move on to the discussion on the liquid expanded films and condensed films. Let us understand that if the hydrocarbon chains have a strong cohesion and this would usually be true if there is no electrical repulsion and the number of CH 2 groups is in excess of about 10, then the pi versus a curve will be similar to what we get for oleic acid. And these liquid expanded films have cohesion which is nearly constant for a range of areas ok. This will become clearer as we go further. So, suffices it to say here that when you think of liquid expanded films, you could first presume strong cohesion, reasonably strong cohesion, absence of electrical repulsion and nearly constant cohesion over a considerable range of areas. And the equation of state would follow from these three equations 10, 11 and 12, pi k is pi minus pi s no repulsion, pi k times a minus a 0 is k t. So, when you substitute for pi k from equation 10, we get pi minus pi s into a minus a 0 equal to k t. So, the liquid expanded films are represented by this surface equation of state pi minus pi s a minus a 0 equal to k t. We could take some examples of hydrocarbon chains. If we have hydrocarbon chains of let us say C 10, then pi s will be about minus 10 dynes per centimeter. This is easy to remember C 22, hydrocarbon chain has pi s equal to minus 22 dynes per centimeter. With these number of carbon atoms, cohesion is sufficiently strong to give you liquid expanded nature on the force area curve. However, if there is unsaturation in the chain, then we understand that there would be a certain amount of repulsion present which will not permit the chains to pack too closely, which means pi s value the cohesive pressure will be somewhat reduced. Now, you can think of the next scenario. If we look at much greater areas and higher temperatures, then the films become gaseous. There will be a gradual variation, more gradual variation of pi versus a. Condensed films on the other hand would be represented by an example like steric acid. If we have materials like steric acid, the films form with very strong cohesion. The cohesive pressure pi s now may be of the order of minus 50 dynes per centimeter to minus 100 dynes per centimeter. That is a large magnitude of the cohesive pressure. With such high cohesive pressures, if we expand the film such that each individual molecule has an area at the interface, phase greater than the limiting area is 0, then it leads to that breakup of the film into islands. We had mentioned this earlier also and cohesion dominates in these islands. But what happens if you look at the areas less than a 0? You could visualize a surface film which is condensed and on top of which you have sprinkled fine ignited talc particles for visualization. And now, you try to move the barriers closer together such that the area per molecule is less than a 0. If you try to pack in this film to areas per molecule less than a 0, the surface pressure climbs up very steeply. That is behavior like a solid. So, condensed films are like solid monolayers with a very steep pi versus a curve for areas less than a 0. Now, it is here that I want you to look at some numbers. On that steep pi versus a curve, if we think of a gentle pressure of 20 dynes per centimeter, surface pressure of 20 dynes per centimeter acting on the molecules which are about 20 angstroms. The film is about 20 angstroms thick and if you apply a pressure of 20 dynes per centimeter, it would translate into an equivalent of 10 to the power 8 dynes per centimeter square and that you know is about 100 atmospheres. So, you see here, one of the confirmations of what I said earlier, the small looking values in the context of surface actually translate into rather large numbers. Here it is an open system. Think of conducting a reaction which requires about 100 atmospheres. So, material of construction, your fabrication will have to take care of those high pressures. Here, apparently very gentle changes in surface pressures allow you to obtain equivalents of those high bulk values. We will see more of this later. So, one possibility is that you might be able to obtain conditions in the surfaces or interfaces which are equivalent to rather severe conditions in bulk terms, but from operational point of view, there is a great deal of ease. It is an open system. We next look at values of A0. Some typical values of the limiting area for single molecules, we will see for a range of them. Some simple molecules will correspond to limiting area equal to about the cross sectional area. If it is just a molecule anchored vertically at the interface or surface, the cross sectional area itself will be the limiting area. But with more complex molecules like for example, diastres, the diastres will be anchored onto the surface in a flat position because of the polar groups. And if we take even more complex molecules like some of the proteins, then the situation is even more different. So, let us look at some magnitudes of these limiting areas. For straight chain acids on water, A0 is about 20.5 angstrom square as expected. Consider similar molecules, but on dilute HCl, A0 is now 25.1 angstrom square. We have these straight chain acids on an electrolyte now, which will mean there is an electrical energy associated with the monolayer which creates that repulsion. So, it is harder now to bring these charged acid groups closer together when they are placed on electrolyte. So, the pi versus A curve will become steeper at larger areas. For saturated acids esters, the value is about 22 angstrom squares. For primary alcohols, 21.6 long chain phenols, about 24.0. A biological relevance for cholesterol is probably reflected in the limiting area which is much larger, 40.8 angstrom square. For lecithins, about 50 angstrom square. And for di-loryl oxalate, di-dacyl adipate, di-octylsubacate and di-butyl thapsate, the values are in the range 90 to 190 angstrom square. And for all proteins, we change the units. It is more convenient to talk in terms of milligram of protein and the limiting areas of the order of about a meter square per milligram. A naught is that limiting area which you cannot reduce any further because you have already reduced from those exteriors like Rayleigh's exteriors. You have already compressed the surface molecules to a very close packed state. Now, the areas cannot diminish below whatever is minimum. It is the area per chain in the surface. You remember the excluded volume equivalent I talked about? You take a gas and you compress it. If you compress it, the gas molecules will come together in close packed state. You cannot decrease the volume further. So, that is the correction which the Van der Waals equation makes for the excluded volume. Similarly, you cannot decrease the area. According to that equation pi equal to kT, if you keep on increasing pi, A will keep decreasing, but does not happen in practice. In practice, you cannot decrease the area below some minimum area. And therefore, we have to write that as pi A minus A 0 equal to kT. And any attempt to compress it below A 0 will make pressure climb very steeply. So, that is the limiting area we are talking of. You cannot get these molecules closer together. For simple molecules like these state chain acids, it will be cross sectional area itself of molecule. But if the same molecules are placed on an acidic solution, then because of the interaction of the electrolyte and the charge head groups, you will find that it is now not possible to bring them even as close as the geometrical area. The repulsion becomes very large and so on. So, with other molecules, the areas we see are larger like diesters we talked of. They will have much larger areas here, because it is not now a chain which is vertically oriented. It will be two polar groups which are attaching the molecule now flat. Obviously, the areas now you can get as minimum areas will be much larger. And for a different reason for complex molecules like proteins, we do not talk of per molecule now. We talk of how much area per milligram, because the molecular weight itself at times might not be known. We will see later that it is possible to infer molecular weight from measurements of the force area curve or the surface equation of state. I think I agree with Richard Feynman here with Gibbons thought that the power instruction is seldom of much efficacy except in those rare dispositions where it is superfluous. We return to the equilibrium in films. So, with insoluble films at air-water surface, we may have systems which will be so coherent that on expansion of the available area, we have these discrete islands left on water surface. And how do these islands behave? They could be solid like or they could be liquid expanded in nature. But what is in between those islands? That is the question that we raise next. You start with the monolayer, close packed and then expand the area. If you provide too large an area, you find that this breaks up into islands. Each island will have depending on the cohesion either condensed behavior or liquid expanded behavior, solid like and liquid like behavior. But is there nothing in between the islands or do we have another kind of surface state for matter which might be existing in the area between the islands? It turns out that between the islands, there is this two dimensional gases film. So, when the monolayer has been expanded, it has split up into either condensed or liquid expanded islands. And the space in between the islands is occupied by matter, surface phase which exhibits the gaseous film behavior. Now, this is the pivotal point for a discussion. We have either solid islands in equilibrium with each other through a gaseous surface film or liquid expanded islands with a two dimensional gaseous monolayer in between. Is this just a visualization or can we quantify properties associated with the two dimensional gaseous film? We look into this. Upon raising the temperature, the coherent films undergo expansion. First noted by Labrust. If we have a solid coherent film, condensed film, on raising the temperature, it melts into liquid expanded film. As you raise the temperature, we have equivalent of melting. And in condensed films of aliphatic materials, the arrangement of chains is roughly parallel as in solid acids. On raising the temperature, these parallel chains in the condensed films would transform into liquid expanded state which will be more like chains are coiled and interlocked, not exactly parallel like in solid acids. So, this is one thing you should visualize. Upon raising the temperature, we go from the parallel films parallel molecules to the interlocked coiled molecules. And they form a kind of hydrocarbon layer and below that there are the polar head groups immersed in water. So, in a sense in liquid expanded films, you see this structure of two layers. The top hydrocarbon layer where the chains are interlocked and the polar head groups which are anchored in aqueous phase below. This is what laid Langmuir to call such liquid expanded films as duplex films. The equation of state for such films we have seen already is pi minus pi s times a minus a 0 equal to kt. Liquid expanded films have this surface equation of state. What about the gaseous film? When you provide sufficient expansions, then both condensed as well as liquid expanded films which are unable to cover the entire surface remain in equilibrium with a 2D vapor film. Do we have numbers to characterize this two dimensional gaseous or vapor film? People have made measurements and we could term the surface pressures corresponding to the vapor film or the gaseous film in between islands as surface equilibrium vapor pressures. Evidently, they look small. We have some data here for tri desilic acid. The surface vapor pressure at 14 and half degree centigrade is about 0.3 dynes per centimeter. Miristic acid 0.19 dynes per centimeter. Penta desilic acid 0.11 dynes per centimeter. Pharmatic acid much lower 0.04 dynes per centimeter. All of these are a tenth or hundredth of dynes per centimeter and may appear small, but once again I remind you that properties associated with surface, phase may have small magnitudes in the 2D terms, but in translated 3D terms those become large. So, these small looking vapor pressures for the surface phase when expressed in dynes per centimeters will easily translate into several atmospheres for bulk phase. Interestingly, these bulk equivalents are not much smaller than the critical pressures of the liquid. So, we continue with this argument that we might have much gentler systems to handle in the 2 dimensional terms allowing almost similar conditions as in bulk phase terms. So, these vapor pressures are only slightly lower than the critical pressures of bulk liquids. Next we talk about this transition a little further. When we look at this transition of a monolayer from one state to another, it would depend among other things on temperature. So, temperature will of course, lead to this transition. Other factors should be the 2 dimensional pressure, the surface pressure and the third which may be less obvious to you at this point would be the extent of interaction among the polar head groups with the substrate. If we have let us say acid groups on aqueous solution, then it may be important that the pH is a relevant parameter. Depending on pH, the interaction between the polar head groups and the substrate will be a factor governing the transition of monolayer from one state to another. You may also relate this to the magnitude of A0 that we had seen in this table when we have a long chain fatty acid, stray chain fatty acid on water. A0 was lower on HCl solution that was higher. That is the result of the interaction between the polar head group and the substrate underneath. You could think of more clever things here. It is possible to think of products which might change their properties depending on the pH of the solution underneath. What might be insoluble monolayer under acidic pH could be made a soluble monolayer under alkaline pH. So, those kind of product designs are possible where the properties of the system change depending on the altered interaction between the polar head groups and the properties of the substrate underneath. So, if you look at the force area curves from experiments, the data suggests that we have separate states of matter in monolayers and these have to be regarded as separate phases which can come to equilibrium with another one another. And if that is the case then one would suspect that some kind of phase rule should be applicable. Wherever we have this phase transitions between different phases, we should have a associated phase rule. Now, there is a certain phase rule deduced by Chris which shows that if you have single component system and you have a phase transition, then there must be an invariant point similar to melting point or freezing point or boiling point. The melting point will accompany phase transition between a solid and liquid. A boiling point is the invariant point between the transition between the liquid and paper or liquid and gas. Similarly, here in a single component system, the simplest phase transition follows Chris' phase rule and there is a predicted invariant point for two surface phases when they are present in equilibrium. We call such phase transitions first kind of phase transitions, vapor to liquid or vapor to solid transitions. These are the first kind transitions. But like in bulk phases, we might have other more complex phase transitions. We will just dwell upon a couple of them. At times against area when you plot the surface pressures, you find discontinuities. And discontinuities in the pi A curves show existence of other kinds of phase transitions. Second order phase transitions now do not correspond to an invariant point, but these occur over a range of temperatures. And even higher order phase transitions are possible. These have been detected from data which show discontinuities in several things, surface incompressibilities, surface potentials or surface viscosities. And what is the reason for these higher order phase transitions? That is the next question. Is the increase in the degrees of freedom which are permitted to the hydrocarbon chains? It is a progressive increase in the degrees of freedom which leads to these higher order phase transitions over range of temperatures. But the question may crop up again. Is this only an argument or can we have a quantitative substantiation? To that end, I would present the following facts to you. The thermodynamic quantities which are associated with phase changes could be actually calculated. One could use an equation similar to Clapeyron Clausius equation which is d ln p by dt equal to delta Hv by RT square. And using this kind of relation it should be possible to calculate the heat of surface evaporation when you have a liquid expanded monolayer transforming into the 2D vapor. So, if you take the 3 acids that we talked of here earlier, tri-dacillic, meristic and pentadacillic, the surface vapor pressures are 0.3, 0.19 and 0.11 dynes per centimeter. The external data show from use of Clausius-Clapeyron kind of relation, the heat of surface evaporation to be about 2000 calories per mole, 3200 calories per mole and 9500 calories per mole. These are values which characterize the transition from liquid to vapor. So, it is possible to carry out that analogy quantitatively for phase transitions inside the surface phase. When we talked about without it like higher order system there only exists for more than one component. Those are possibilities, but you might have complex molecules which have a gradual, because of their structure they allow for gradual increase in the degrees of freedom. One component for simple system, first one first order transition and the higher order transitions could be more components and single component, but complex systems. So, these numbers are real you can have the heats of surface evaporation predicted from or deduced from experimental data on surface pressure changes. We could take this analysis to a different level. We could think of latent heat of spreading, think of let us say solid-pometic acid when it spreads on water. So, latent heat of spreading can be calculated from the temperature cohesion of spreading pressure. Just as here we have the first derivative of pressure with temperature that is the temperature cohesion. Similarly, for the surface or spreading pressures we can have the temperature cohesion and based on that we could calculate this latent heat of spreading. It turns out to be 8 kilo calories per mole for the solid-pometic acid, solid-pometic acid to the monolayer. Whereas, if you start with a liquid lens of ametic acid the latent heat of spreading is minus 4.9 kilo calories per mole. Think of these numbers solid to the monolayer 8 kilo calories per mole, the liquid lens to monolayer minus 4.9 kilo calories per mole. Difference between these values is a measure of heat of fusion of palmitic acid or we take it to another level where we look at the entropy changes. So, the corresponding entropy changes are from solid to the vapor 28.9 entropy units from liquid lens to the vapor 2D vapor minus 9.5 entropy units. What does this suggest? It suggests what is written over there. The monolayer is definitely disoriented relative to the solid, but more oriented compared to the bulk liquid. This is yet another example where a surface phase or interfacial state of matter due to a certain degree of orientation gives you properties in between the solid and the liquid. So, the monolayer of palmitic acid will be intermediate between a solid-pometic acid crystal and the liquid lens. The entropy changes also suggest that for spreading of acids the films behave as if they are immobile. The entropy of the film is accounted for by not the two-dimensional translation, but by rotation of these molecules about their long axis. That is what is the reason behind entropies which are observed. Next, go into the films of polymers. I had mentioned in passing while talking about the limiting areas for proteins or complex polymers. We talk of the limiting area in terms of meter square per milligram and I just mentioned that with some polymers we may not even know the molecular weight. The question is can we use the force area curve data for inferring molecular weight. Let me just prompt your thinking here. Both for natural as well as synthetic polymers we often observe spreading rapidly at surfaces and these films generally show cohesion. It could be intermolecular cohesion among different molecules or intramolecular cohesion that is within the molecule itself sometimes both intermolecular and intramolecular cohesion. Now if we measure the surface pressures at different areas given areas we expect it would be greater at the oil water surface compared to air water surface. Is the reason clear? Would you anticipate surface pressures to be greater at oil water interface as against air water interface? Why? May be just draw a sketch in your notebooks for a monolayer at air water and oil water surface and try to think of the force area curve. Make a rough sketch of force area curve keeping the equations in mind and therefore, within this lecture itself I referred to this point pi s right. So, when we looked at the total pressure pi s pi k plus pi s plus pi r we argued that at oil water interface pi s will be 0. Cohesion now is eliminated right and pi s is a negative part. So, it is clear that the surface pressure measured would be higher at the oil water interface for any given area clear. So, we could actually look at the data force area curves for example, hemoglobin monolayer spread on 10th of a normal aqueous HCl at an interface with air and with cyclohexane. The pi values depend on one thing the molecular weight and intramolecular forces and intermolecular forces all these will matter right. If we think of such molecules if the molecular weight is higher then it will be harder to get them out of the interface into the bulk. So, it will be more difficult to dissolve the larger molecular weight film molecules right. For many hydrocarbon residues which are oriented away from water the total energy of desorption can be very high, but is it impossible to achieve their desorption. We had seen the third factor the interaction among the film molecules and the substrate. So, even if otherwise these molecules are difficult to dissolve either on account of their molecular weight or their hydrophobic character overall hydrophobic character. It might be possible to tune the interaction with the substrate such that that desorption becomes possible. It is indeed possible to think of molecules like polyacrylic acid which are adsorbed at the surface of water. It is possible to think of desorbing them by altering the pH. If you make the water underneath alkaline then there is a lot greater electrical interaction between the head groups of polyacrylic acid and the electrolyte underneath and you can have these polymer molecules becoming soluble. In fact, this was one particular idea which we had investigated for making a new class of inks which could be made soluble by altering the interaction energy between the adsorbed molecules and the substrate underneath. We actually have equivalent of a polyacrylic acid mimicking the role of a resin in an ink giving you an ink system which could become washable using instead of water that is used in printing in offset printing. You could use alkaline dilute alkaline solutions and the inks which are otherwise hydrophobic could be rendered hydrophilic because the polyacrylic acid then reacts with hydroxide produces a salt that changes the interfacial tension in such a manner that it is possible for these surface molecules to become desorbable. Therefore, they become soluble. It is here that I would like to leave you with the last thought and we will pick up from here question of molecular weight. How do we use the data on pi versus A for monolayers which are made of polymers and we have only this pi versus A data. We do not know the molecular weight. Your task is to figure out how you could use the pi versus A data to find out what is the molecular weight of the polymer. I would like you to do this before the class on Wednesday and anyway we will resume from here. Suffice it to say here that measurements of force area occurs or surface equation of state for such monolayers can be a useful way of determining the molecular weight. You know number of ways in which molecular weights of polymers could be determined. You could use the intrinsic viscosities and so on, but here we are talking of a surface phenomenon, measurements of surface pressure versus area and the picture which is known to you how the molecules are anchored at the surface. So, treat the system to be simple vertically oriented molecules forming the monolayer and your job is to find out the molecular weight. We will take up from here next time. Thank you.