 We move on to now lecture 12 in which we will continue to discuss the spreading in view of the contact angles, work of adhesion and a new phenomenon called debating. Continuing from where we left, we have for theta equal to 0 degrees for surface with exposed polar groups which will be hydrophilic and with theta equal to 110 degrees for the part of surface which is exposed to hydrocarbon tails, we will have cos theta equal to f 1 minus 0.34 into 1 minus f 1. We can go back to this equation last equation theta 1 is 0 and theta 2 is 110 degrees. So, when you plug plug in these values, we get cos theta prime simply written here as cos theta as f 1 minus 0.34 1 minus f 1. If we consider some certain solid acids like C 16 acid or C 16 alcohol, then experimentally it turns out that theta can be between 100 degrees and 50 degrees depending on how we cut the crystal. This we would like to understand a little bit now. If only hydrocarbon chains are exposed through the cleats cleft surface, then that will give you a theta prime of 110 degrees. Whereas, if you have only about 70 percent of surface exposed to carboxylic groups which is which will be the hydrophilic groups, the theta will come down drastically to about 50 degrees. So, this provides for another variation in measurements that one comes in practice. These would have been difficult to understand otherwise, where it not for the fact that we have now a theory for composite surfaces with different contact angles for microscopic regions. We next consider the spreading cohesions of liquids and solids. By definition, we write in terms of the free energies S S A as F S A S minus gamma L A plus F S L S. In your imagination, you would see that we have replaced the surface tension of the solid by the free energy F S A S. And we replaced the interfacial tension between solid and liquid gamma S L by F S L S. S A and L like usual represent the solid, air and liquid respectively. And if we consider a situation where a liquid placed on solid could maintain an equilibrium contact angle, we should be able to write the following equation. F S A S is equal to F S L S plus gamma L A cos theta. Mentally, you can run the parallel equations in terms of the surface tensions understanding that for solids it is preferable to talk of the surface free energies. So, gamma S A is replaced by F S A S and gamma S L by F S L S. Now, substituting for this surface energy of solid per area F S A S into the previous expression for the spreading cohesion, we have the bracket is same minus gamma L A plus F S L S. F S A is replaced by F S L S plus gamma L A cos theta. So, F S L S gets cancelled, gamma L A can be taken common and we get S S A as now gamma L A into cos theta minus 1. Please remember that this is for non-spreading system in which the liquid is able to maintain a drop whose form and shape will attain the equilibrium contour. So, here is a way we can estimate the negative spreading cohesion provided we know the surface tension of the liquid and the contact angle. We do not need the properties otherwise directly dependent on the solid energies. Restriction is this is valid only for theta greater than 0 and it cannot predict positive spreading cohesions. At this point I would like you to think of the following question. Is it possible to see this relation in light of what you know as deducible from some other considerations? We have seen many equations by now. Here we have followed the usual approach to get S S A is equal to gamma L A cos theta minus 1, but is it possible to regard this as directly deducible from other considerations we have including the physical insight that we have for the spreading cohesion. It is interesting that most of these equations are quite simple and you understand them individually. You also understand their physical significance individually, but it is a good idea to connect different pieces of information contained in the various equations. I may prompt your thinking a bit by saying that think of the equation which otherwise has a similar right hand side except the minus sign is replaced by plus and the left hand side of course changes. What is that? Right. So, the work of addition is gamma L A cos theta plus 1. Now, the next suggestion is consider that equation together with the physical significance of spreading cohesion. Could we have obtained this S S A equal to gamma L A cos theta minus 1 from Young's relation without actually making the surface energy balance or the parallel of the balancing of surface and interfacial tensions. It is probably not obvious, but if you have initiated some ideas let me complete those. Remember the way we rationalize spreading cohesion at one point of time in terms of the works of addition and cohesion. The spreading cohesion was shown to be the difference between work of addition and work of cohesion of the liquid addition between liquid and solid work of cohesion of the liquid. So, if you plug in for the work of addition from Young's equation gamma L A cos theta plus 1 and for work of cohesion twice gamma L A you get this. The two expressions work for work of addition on a solid for a given liquid and the spreading cohesion have very similar equations except that in the expression for the spreading cohesion we have a minus sign and let me complete this mnemonic further so that you never forget this maybe after 30 years. Because this equation is applicable for negative spreading cohesions we have a minus sign here. You will know that in the Young's equation it has got to be the other sign. Many of these mnemonics are not necessary if you can just think in terms of the basic equations and are ready to work out their logical implications correctly and quickly. All right. So, we have now a little more insight into this expression for the spreading cohesion negative spreading cohesion of a liquid on a solid that is what I had here in the second point. But what if the spreading cohesion is positive? In that case we go back to this equation which is the basic fsis minus the sum of gamma L A and fsls. We directly make use of this equation because now we cannot be limited by that equilibrium kind of situation. When the positive spreading cohesion is positive the liquid spreads indefinitely on the solid until a thin or thick layer is obtained. As you will see later the last expression for spreading cohesion can also be used to calculate the spreading cohesion of liquid on solid from measurements on surface tension and contact angle separately. Or as we will see later in some methods we can measure gamma L A and gamma L A cos theta directly especially in a Wilhame plate method. So, whether we choose to measure the surface tension gamma L A and contact angle separately or surface tension separately and the product of surface tension and cos of contact angle. Whatever way we choose we should be able to use those extended measurements to evaluate the negative spreading cohesion. There is yet another alternative for determination of this negative spreading cohesion. The theory of which we will look into later and that method is based on a large drop of liquid called Caesile drop method. We could look at some numbers here. If we consider oil spreading on chromium then the spreading cohesion is positive. If we take the oil as 7-butyl tri-decane the spreading cohesion works out to be 25 dimes per centimeter. If we were to choose effect of chain branching instead of taking the straight alkane we had branching. The spreading is somewhat reduced spreading cohesion is now lower to about 17 dimes per centimeter. Still positive, but reduced compared to the straight chain compound. Then again the history of metal surface is important. Think of this situation. Chromium surface could be cleaned by ion bombardment and then immediately wetted by water. The metal surface now will be seen to be oleophobic. It will not be wetted by oil, oil heating and therefore the spreading cohesion S is less than 0 for oils for such a metal surface. We surely understand here that the impurities can play a large role in governing the weighting of high energy surfaces, especially metals. Since in practice we do deal with mechanical parts lubricated by oils, the design of oil, the design of parts both will have to have consideration for spreading of oils. Lubricating oils must spread readily over steel and they do so especially if they are somewhat oxidized to polar compounds. And the spreading cohesions or pressures you get on steel will then be pretty large perhaps only slightly lower than on water. How does one explain the experience that some oils don't spread on metals at all? It may mean that the metal surface is not clean. For clean surfaces the surface energies FSAS are very large. Several hundreds to several thousands of oaks per centimeter square and therefore the spreading cohesion is always positive. If a given oil doesn't spread on a metal surface it is a clear indication that metal has some adsorbed impurities. For example, adsorption of chlorinated hydrocarbons and phosphate esters can give you S less than 0 for oils on metals. Sometimes holotel liquids also cover the surface and if it happens to be even a monolayer adsorption in the region of measurement we might encounter negative spreading cohesions. For otherwise expectedly positive spreading systems. If one measures contact angle here then it will be for the liquid against the monolayer covered solid. If that's a situation one would have to correct for the spreading cohesion for clean surfaces. And there we will need the work of adhesion. The adhesion between the impurity adsorbed and the metal. Let's see how we can do this. That brings us to the consideration of addition of liquids to solids analogous to liquid-liquid interfaces. For liquid adsorbed on solid or adhering to solid the equivalent of Dupre's equation would be WSL equal to FSAS plus gamma LA minus FSLS where WSL is the work of addition of liquid to solid. FSAS the surface is the surface energy free energy per solid area. Gamma LA surface tension and FSLS is the surface energy per unit solid liquid interfacial area the usual terms. But the spreading cohesion is given by FSAS minus sum of gamma LA and FSLS or alternatively we can write that as FSAS minus FSLS minus gamma LA. And substituting for FSAS minus FSLS in the work of addition equation here FSAS minus FSLS we find SSA is equal to FSAS minus gamma LA plus FSLS giving you the same equation I refer to a few minutes back. WSL gives you SSA plus 2 gamma LA. We are substituting for FSAS minus FSLS as SSA plus gamma LA. So, that gamma LA adds up with this and we get WSL equal to SSAS plus 2 gamma LA that is what I had referred to while asking you to relate Young's equation to the spreading cohesion for an equilibrium situation non-spreading liquid on solid. Let us see how this equation compares in the face of external data. We might estimate the work of addition for a liquid like water on paraffin wax and then compare the work of addition with the measurement on liquid paraffin against water. Since the chemistries are similar we should get a near proximity of these numbers as is seen here. For water on paraffin wax solid paraffin wax there is a non-spreading tendency minus 98 irks per centimeter square comes out for spreading cohesion. If you plug it in here twice 72 for water gives you 46 irks per centimeter square as the work of addition for water placed on paraffin wax solid and you compare it with water on liquid paraffin that value is 43 irks per centimeter square. So, that kind of confirms that these independent measurements point out to a general acceptability of our analysis so far. Do not worry about the arrowheads here WSL equal to gamma LA cos theta plus 1 may also be used for obtaining the work of addition and the quantities all refer to the measured S and theta. If they were adsorption of a monolayer the work of addition will refer to the work of addition of liquid to the monolayer covered solid surface. One may have to do the correction the correction is done in this fashion whatever is the measured work of addition for the monolayer covered surface that is this WSL measured to that you add the change in free energy of the surface due to adsorption of vapor. So, this is measured we need to find out the change in energy due to adsorption of vapor that can be done using Gibbs's equation which reads as follows p the vapor pressure multiplied by the partial derivative of surface tension with respect to the vapor pressure at constant temperature this will give you this will be given to you by minus k T n where n is the number of adsorbed molecules per centimeter square k is the Boltzmann constant 1.38 10 to power minus 16 works per Kelvin. So, from here we can estimate the correction this is Gibbs's equation previous one. Think of a solid surface if it is having a high surface free energy then they may be adsorption here this adsorption could be just the monolayer. So, let us say we indicate the monolayer may be this way. So, if we are looking at the measured work of addition if we place a drop of liquid on top of this let us say of water this could have been metal this liquid is water and this is adsorb impurities with hydrocarbon chains facing air. So, if that is the case water will see only a monolayer which is hydrophobic over here. So, water will not be able to spread on this high energy surface although the expected high energy the surface would have called for spreading of water. This would be the situation like one I described you take chromium metal subject it to ion beam bombardment and then immediate contact with water that would have made the chromium surface water wettable and oleophobic ok. But if you have not done this special treatment screw plus cleaning by followed by immediate contact with water the high enough surface energy of metal would have demanded some molecules floating in air. If there is nothing else as hydrocarbon even components of air would get adsorbed. So, surface energies are too high to remain in that higher energy state without adsorption. So, given just enough time this monolayer would appear once this appears then water drop will not be able to spread on it ok. However, if you are given such a surface which is pre coated with the monolayer measured work will be this WSL which is for monolayer covered surface. This is what is measured and this will be lower because this is now not for the solid the virgin solid, but for a reduced energy state when the monolayer has managed to diminish the surface energy of solid to an extent. So, this will be necessarily smaller than the actual work of addition between the same liquid and solid it could be water and fresh chromium surface. If it is monolayer covered then we will get a smaller work. It is easier to separate this water when it has to be separated from the adsorbed impurity compared to its separation from the clean metal surface. And it would be then addition of the change in free energy that correspond to the adsorption of monolayer. If we add this free energy change which the surface has managed to achieve by adsorption of impurities even in the form monolayer. This is the deficit between expected work and the measured work. So, the actual work WSL should have been what is measured plus the change in energy that has come up. And you know that the free energy is related to surface tension. So, the gives us gives us equation which gives you P dou gamma by dou P at constant temperature equal to minus k T n can be used to estimate this delta F s. Surely it will call for measurement of a new entity that is the number of adsorb molecules on surface n on the right hand side. So, this measured quantity will be an additional measurement that will be required in making this correction corresponding to the decrease in free energy of the native solid surface. This is the way we go about in estimating the work of addition between a given liquid and a clean solid surface. One thought which may cross your mind at this stage is there is a considerable role for extreme precision and necessity of taking precautions for rigorous measurements in interfacial phenomena. If you are not careful, it is very easy to have spurious conclusions even from experiments. Normally, it is the experiments we trust more as a rule, but we got to be careful in getting reliable external data here. This brings us to another interesting phenomenon called de-weighting. By now you all know that clean glass surface will be weightable by pure water, but consider this experiment. You take pure water, clean glass surface and now you add a small quantity of a surfactant like a cationic surfactant. Long chain, quaternary, ammonium ions will be an example. For very low concentrations of surfactant, these quats will find now the same glass surface is no longer weighted by the aqueous solution, a very dilute aqueous solution of the cationic surfactant. What would happen if we were to attempt to weight this glass surface after adding some more surfactant to the water? As we increase the concentration of surfactant in water, we find that surprisingly the solid surface becomes weighted again by this solution. Originally for pure water, clean glass was weightable. In between for certain small range of concentrations of the cationic surfactant, the surface became unweightable. When we increase the concentration of surfactant further, it becomes weighted again. So, we have certain zone here, a region of concentration in which we may say that the glass surface exhibits a de-weighting phenomenon. What is expectedly weightable is no longer weighted by the dilute solution. It is interesting enough conceptually that this thing happens, but it will be even more interesting if we can rationalize this phenomenon in terms of our equations. So, let us attempt to do this. We would also eventually try to glean out of it some physical insight. Let us see how we could explain this. We may start with the work of addition between the solid and liquid. WSL is gamma LA cos theta plus 1 Young's equation. This should be valid in the region where the glass is de-weighted by the solution. So, the small concentrations region is where this equation will apply. So, what we could do is rearrange Young's equation to have cos theta as WSL by gamma LA minus 1. And now, argue in view of what would happen to the quantities on the right hand side with pure water. The work of addition WSL is high for clean glass. Contact angle is therefore, 0. Note that it is not going to be the equilibrium contact angle. It is a spreading system. Water would wet the glass, positive spreading. So, this equation actually does not apply. The contact angle is not an equilibrium contact angle. But what happens if you add this tiny quantity of cationic surfactant? It reduces the work of addition nearly to the new value of surface tension. You have to be careful here. When you add a small quantity of surfactant to water, the high surface tension of water will get reduced from 72 dimes per centimeter to mid-20s about 25 dimes per centimeter less than that. So, first thing is gamma LA is lower for the surfactant solution considerably lower compared to pure water. But what we are saying is, as a result of this small quantity of surfactant, even WSL is lowered quite a lot. If it comes down to the level of gamma LA, then we have this ratio comparable to 1, which means cos theta will be a quantity close to 1 minus 1 very close to 0. And that is the result of contact angle nearly 90 degrees. Cos of 90 degrees is close to 0. So, we understand here that the addition of surfactant can actually make the anticipated hydrophilic looking glass surface modified in such a way that it is no longer weighted by the aqueous solution. We should be able to explain what happens if we continue to add surfactant to the aqueous solution. If we add further amount of surfactant, it would decrease the surface tension somewhat further, but not alter the already lowered work of addition, which means this has been lowered to some value comparable to surfactant solution surface tension at low values. However, if we continue adding the surfactant, this is now not going to change much. This will continue to diminish. And if it does so, then we have cos theta going up again or theta decreasing. The rise in value of cos theta corresponds to corresponding fall in the contact angle. And when the contact angle again comes close to 0, our solid this glass plate with modified surface is becoming weightable again by the higher surfactant concentration solution. So, that explains why d weighting can happen followed by return to weightability. We may look into this phenomenon a little further. When a surface active agent is present in small amount, the work of addition, which is the sum of the surface free energy of solid per area and the surface tension of liquid less the interfacial energy for the liquid solid interface, that becomes nearly equal to the surface tension of the liquid. Because of a much greater reduction in FSS compared to the reduction in FSLS. And therefore, making these quantities comparable at their new values and therefore, canceling them out. Once the surface and interfacial energies of solid per area are cancelled out of this equation, the chains of the surfactant would not be able to distinguish much between air and liquid. Or we may state it in the form of our equation WSL as FSAS plus gamma LA minus FSLS. The bigger arrow next to FSAS indicates that the free energy of the solid is diminished considerably coming down from the large value that it has. Whereas FSLS, which is already modded, which is already modest between the solid and liquid would also be lowered, but not that much. So, that can allow for WSL becoming comparable to gamma LA. That was the first step in the explanation on why the contact angle can become nearly 90 degrees in view of Young's equation. But we may want to go further and derive a physical insight in terms of what may happen. The thought experiment could be as follows. We have a monolayer covered interface shown here, the solid with the adsorb monolayer. At low concentrations, the surfactant is an effective way of lowering the high energy of solid by virtue of its amphiphilic nature. Even when the concentration is low in the aqua solution, these are not acceptable in the bulk of water. So, either the chains have to hang out in air or they have to occupy some other surface where the total system free energy will diminish by a greater amount. Considering that energy for the surface of liquid is high, but the surface energy of solid is comparatively much larger. This intermediate position is still a situation which is preferred by the surfactant. Even though these tails have to disrupt water structure, the energy decrease achieved for the solid surface is far greater than whatever is the unfavorable nature of conditions pertaining to retaining these tails in water. So, this is what is preferable. Monolayer does get formed on the high energy solid surface. What does the water see or the surfactant solution see? It sees a monolayer of hydrophobic compound, the tails. Clearly, while the system responded this way in the greater objective of reducing the free energy, it finds itself unfavorable for spreading of the solution on this hydrophobic surface, which means now this has become de-weighted. What is supposed to improve the ability of the aqua solution in terms of weighting of solids has worked against that objective, which means in practice when you are attempting to weight a solid, there is a limit to how low the surfactant concentration you can get down to. This might be a practical consideration in formulation of detergent powders or liquids. You cannot diminish the more expensive surfactant by the fillers to an indefinite level. You cannot even use water as a means of dilution leveraging profits. Having said that, it is a sad reflection that one comes across profit margins as high as 2100 percent in the surfactant industry, but perhaps you will trade off with the quality of the product. How does the reweighting occur? This is the explanation. If the concentration of surfactant is high, then system can respond by having additional surfactant molecules reorienting themselves to form a hydrophilic effective surface again. The second layer of surfactant now is oriented such that the tails are hidden, hydrophobic tails are hidden from water and locked up with the tails which were otherwise facing water. This is a very good situation. Once this happens, the surface has become weighted by water again. That is how beyond that narrow zone of low concentrations, the hydrophobic hydrophilic surface is retrieved and the glass with its invisible bilayer of surfactants is weightable again by the same concentration, higher concentration of surfactant solution. This is how we could visualize the deweighting followed by reweighting phenomenon. At higher concentrations, polymolecular adsorption may occur with the tails of new surfactant molecules, preferring the tails of the previously adsorbed surfactant molecules rather than water, thereby presenting the hydrophilic head groups at the outer surface of the double layer of surfactant. This will give you FSLS with a very low value. Water facing the newly rendered hydrophilic bilayer covered glass surface will have very low free energy. There is one more factor which comes into play here. With increase in surfactant concentration, the surface tension would decrease further and that means any tendency of a drop-up aqueous solution of surfactant to cohere to itself will get reduced. 2 gamma LA is now much lesser. So, we have again conditions favorable to spreading and S becomes positive again. We may sum up this as follows. The work of addition of solid and liquid is nearly equal to FSAS plus gamma LA and decreases lower than gamma LA and therefore, according to our equation cos theta increases again and theta decreases leading to weighting again. So, the external phenomena can be neatly understood in terms of our equations. Perhaps, we can stop here for today.