 We begin here the next lecture, which somehow still runs into spreading contact angles free energies, considerations which are quite common in interfacial systems, but I would like to begin here with some more general introduction and not necessarily my observations. You might be interested in knowing that this seminal journal primarily catering to science and one of the highest ranked journals with greatest impact factor over 30. Nature brought out a certain collection of papers which pertain to the latest advances in the interfacial science and engineering and clubbed under nature insights. What you see here is a graphic wherein cobalt nanocrystals have been organized in the picture is one taken by transmission electron microscopy. It is a self-assembled super lattice of mono disperse colloidal cobalt nanocrystals and some water droplets have been added. Those are visible as these blue spots over here. It is clear that water droplets are retaining their identity. That is about the thing that will pertain to a lecture, but I do not leave it here. I take you through these observations made by the senior editor for this Nature Insights collection Magdalena Hema on surfaces and interfaces. She observes that the importance of interface cannot be overestimated. I hope we all agree with this. They play a vital role in technological applications as diverse as catalysis, microelectronics, lubrication, corrosion, photography and in many environmental processes involving infamously the stratospheric ozone destruction. Moreover, many of the biochemical reactions that occur or that sustained life occur at surfaces and interfaces. Although their significance has been realized for centuries, surfaces and interfaces long evaded detailed scrutiny at the atomic scale. After all, they are not simply the final layer of say a piece of metal or liquid in contact with air, but an exceedingly thin region with properties distinct from those of the bulk material on either side. That is very important for us to understand that interfacial properties or surface phase properties are very different from the bulk phase properties. But the past 30 years have seen the development of increasingly sophisticated techniques that have delivered incisive insight into the composition and structure of a wide variety of surfaces and interfaces. As our understanding of these peculiar regions of matter grows along with the range of characterization and manipulation tools at our disposal, we are in a position to use the unique environment of surfaces and interfaces to explore the fascinating science and new applications. She goes on, this is with discontinuity. Flavor of the challenges and opportunities in our quest to fully understand surfaces and interfaces and to precisely control the properties for applications that range from materials processing and information technology on one hand to biology and medicine. It goes almost without saying that richness of the field in terms of the systems and processes studied, the tools being used and the diversity of applications that are being targeted is superb. The choice of themes being reviewed here is thus inevitably somewhat eclectic. There is a spelling error eclectic, but we hope you enjoy the offering and it is also my hope that you will enjoy this lecture, Surya. We begin from where we left last night. We have been talking about the work done in separation of an oil against water or between any two liquids, one of them having a very high surface tension. And we ended that last lecture observing that the liquid of higher surface tension retains a thin layer of liquid of lower surface tension. And that is the reason when we separate the oil-water interface, it is akin to separating oil from itself. Therefore, the work of addition comes out to be pretty close to work of cohesion of the liquid of lower surface tension. Continuing on that example, we note here in passing that the liquid of lower surface tension sometimes would have its surface tension change, but only little and that can be predicted from Gibbs's adsorption relation. This indeed is found in practice when you dissolve water to saturation in benzene, the surface tension of benzene is only raised by about 1.1 dynes per centimeter from 28.2, 29.3 and for chloroform and ether, the increase may be as little as 0.8 and 1 dynes per centimeter. We may relook into Antonov's relation. It could be stated as an observation of facts that when a liquid of high surface tension is contacted with a liquid of lower surface tension, this liquid of high surface tension tends to radially adsorb a monolayer of the lace polar substance. The lace polar substance has its surface tension raised only slightly by presence of the more polar molecules. This is an alternative way of looking at the facts which lead to Antonov's relation being borne out from measurements. This relation however breaks down when at equilibrium neither liquid spreads on the other and the examples could be higher paraffins and water or methylene iodide and water. We had seen in the context of methylene iodide and water, the final spreading coefficient was large negative value of the order of minus 24 dynes per centimeter. So, let us quantitatively deal with these aspects and derive some more insight. We now have non-spreading conditions. We are looking into when Antonov's rule is not obeyed. So, we have non-spreading conditions which will be the first inequality for water drop place on oil. Gamma W B A plus gamma B W is greater than gamma B W A. You may mentally rearrange this inequality slightly transposing the terms on the left hand side to the right hand side. That would be gamma B W A minus gamma W B A plus gamma B slash W. We are looking at water being placed as a drop on oil. So, the first term on the right hand side of this inequality gamma B W A reflects the action of the surface tension of the oil saturated with water tending to spread the drop. Spread the drop of water and the terms which are transposed to the right hand side that is minus sum of this gamma W saturated with B slash A plus gamma B slash W. That is the tendency of water through its action of surface tension and interfacial tension with oil which is trying to cohere the drop. So, gamma B W A is trying to spread it. These two terms are trying to help the drop cohere to itself. That is what your spreading coefficient is and expectedly the spreading coefficient is less than 0 or 0 is greater than this spreading coefficient. Likewise, if we consider considered a drop of oil placed on water then we will have gamma B W A plus gamma B W that is same as gamma W B greater than gamma W B slash A. Explanation is identical. We have only interchanged the notations B and W over here and understood that gamma B slash W is same as gamma W slash B. The interfacial tension between the two liquids is irrespective of how you look at it. Where do we go from here? We recall Dupres equation and that is W B slash W is equal to gamma B W slash A plus gamma W B A minus gamma B W or from here we get gamma B W as gamma B W A plus gamma W B A minus gamma B W as gamma B W A plus gamma W B A minus W B W the work of addition between oil and water. If we now substitute for this interfacial tension in terms of the work of addition in the first inequality over here for gamma B W we have gamma B W plus gamma W B A plus gamma B W A plus gamma W B A minus W B W greater than gamma B W A. Then we realize here that gamma B W A is common and then can be cancelled. So, we get 2 gamma W B A minus W B W greater than 0 or in other words the work of addition W B W is less than 2 gamma W B A. But the right hand side over here of the inequality 2 gamma W B A is nothing, but the work of cohesion of water when it is saturated with oil. Similarly, from the second inequality we get W B W work of addition less than work of cohesion of the oil twice gamma B W slash A. We interpret this as follows the addition between 2 mutually saturated liquids is less than cohesion of either liquid. This is the statement of the non-spreading of the oils. And if Antonoff's relation where to hold quite generally we would expect to find the spreading to at least just occur on one of the 2 liquids after mutual saturation is established because the S final should be 0. This is where that last column in the table was showing S final large negative and then there was no match with expectation from Antonoff's relation. And those are the cases where this may not happen. For slightly puller oils like chloroform or benzene this is approximately true and we find Antonoff's relation is ok, but not for more puller oils on one hand like normal heftyl alcohol nor for initially non-spreading oils like carbon disulfide or methylene iodide. This is the new insight into the interfacial tensions as they are related to the surface tensions of mutually saturated liquids. From here we move on to another interesting consideration of drops of oils on water. Consider for example, a drop of oleic acid floating on water and presume that it is an equilibrium with monolayer covered surface. This system has been studied extensively. Now, if you consider this equilibrium drop shape the surface between the floating oil drop and air makes an angle theta 1 with the horizontal that is a tangent drawn from the 3 phase contact line along the surface of the oil drop makes an angle theta 1. The interface between an between oil and water makes an angle theta 2. Let me draw it here. Let us say we have water here and an oil drop over here then you have gamma o a and gamma o w over here. The angle here is let us say theta 1 the angle here is theta 2. If it is equilibrium situation we have to understand that gamma o a and gamma o w besides gamma w a would all correspond to mutual saturation. Oil saturated with water water saturated with oil. So, this is the kind of system we are looking at we return to the slides. So, if you have such a drop floating on water then both these angles theta 1 and theta 2 must be adjusted such that the forces balance. And this oil drop must be limited in size if it were to be in equilibrium. We could consider demonstration of drop of spreading oil being placed on a clean water surface. Spreading is generally rapid enough to carry out enough liquid from the drop to form a multi layer multi molecular layer on water surface. We can even see interference colors color patterns interference colors on the liquid film spread film at this stage after a few seconds. The excess of oil retracts it retracts into drops leaving the rest of the surface covered with a monolayer. The energy of the system will be minimum for a certain drop size. Think about this we might have something familiar from earlier lectures at work over here. If we have the spectrum of droplets of oil floating on water the smaller oil drops may exhibit enhanced spreading pressures reminding you that this is actually similar to the higher solubility of particles of smaller radii or of higher curvature. The theory behind equilibrium existence of droplets of different sizes is not adequate yet, but the film pressures of monolayers in equilibrium with floating drops of oil have received much attention of researchers. I had broach the issue of retarding evaporation of water water losses from large water bodies. When we think of large water bodies and we will talk about this in more detail later we have to understand that these large water bodies would have their own share of contamination. If one way to require to measure surface pressure for these impurities adsorbing onto the water surface in large ponds or lakes we will need special tools for measurement of these surface pressures. There are two ways of going about it one related to the interference other related to a concept of bracketing a given quantity between two known quantities. So, let us take indicator oils to begin with. You could take this simple example from crank case oil of a car you could take a filtered sample dilute it with new gel which is non-spreading viscous white paraffin oil. The spreading tendency and thickness of the film will depend on surface pressures on the surface where they are spread we might have no visible lenses here. However, the interference colors of multi molecular patches of indicator oil from previous calibration would be able to give you the spreading pressure of ponds or lakes accurately. The idea is you just have this indicator oil prepared such that it can form a thin layer still multi molecular thick and then make use of the interference color patterns to relate those patterns to the thicknesses. So, if you do the previous calibration from the observed interference color patterns we should be able to estimate the thickness. So, surface pressure is used in different contexts differently you can think of this. Suppose, you have clean water the surface tension at 25 degrees will be 72 dimes per centimeter. However, if you have impurities adsorbed on this water surface the surface tension will drop down and it will drop down by a value equal to the surface pressure of the impurities which are spread as a adsorbed film which is trying to repel its own molecules. So, it is kind of offsetting the contractile tendency of water by exerting this surface pressure which is oppositely directed. Now, take the other context you already have water which is contaminated. So, there is certain surface pressure which is operative. Now, if I place an oil drop on such water then in order to spread on the surface of water it must push behind this surface film. Now, if this oil can exert a pressure a film pressure greater than the surface pressure of impurities only then the spreading can occur. It is like now surface pressure is now opposing the spread of the new impurity which may be oil. Unless that is done we cannot have spreading against the contamination. So, concepts like surface pressure film pressure they have to be understood in different context accordingly. So, the first notion was the impurities have caused the surface pressure surface tension to reduce by certain magnitude that will be equal to surface pressure of the impurities. If we have a contaminated surface of lowered surface tension then if you place an oil drop which has a tendency to spread this oil drop will be able to spread provided it can exert a film pressure which is greater than the surface pressure. It is only when the spreading pressure or film pressure exceeds the surface pressure of impurities the spreading can occur. If not the drop will be forced to be constrained in size as a lens the equilibrium shape that way. So, it will occur the surface tension rather than the surface pressure because surface pressure is not going to depend right. No, no, no I think you are you are basically not got the concepts right this may be because of first few lectures miss, but I will do a quick recap for you. Surface tension is trying to contract a surface surface tension is trying to contract a surface. The impurities lower the surface tension they have a positive repulsive pressure which offsets the action of surface tension. Now, when you have a surface which is covered with a monolayer or a multi layer whatever in order to occupy a position on water surface and keep spreading it must be able to oppose this barrier. This is the diagram ahead of time, but let me draw it. I will need the visualizer here supposing I have this surface of water. Now, I have these adsorbed impurities to begin with which have their mutual repulsion. Therefore, the surface tension of this system will be lower than the surface tension of pure water. If now I place a drop of oil here it will tend to spread oppose against whatever is the position of the monolayer molecules. So, if the film pressure or surface pressure exerted by the drop spreading pressure exerted exerted by the drop is greater than the surface pressure of these impurities which is pi only then I will have this going into a thick or thin film all over the surface of water. So, that is the action. The spreading liquid has to overcome the resistance of the adsorbed impurities or their surface pressure. Now, depending on the kinds of impurities we have we will get different thicknesses. Different thicknesses will correspond to different interference color patterns. So, once you do the calibration of the thickness versus the interference from measured interference pattern you can estimate what the thickness is. This is what indicator oils will suppose will supposedly do for you. So, that crank is oil diluted with new gel may just spread into a thick film thickness of which is characteristic of the position posed by the contaminants. So, that will be a measure of the level of impurities or the surface pressures of impurities on contaminated water surface. In a sense you can look at this as a usual way of measurement that is your method directly gives you what is the existing surface pressure of impurities ok. So, indicator oils do this for you. Indicator oils give you directly from prior calibration the level of impurities or their surface pressure. As opposed to this we have the next category or next method for measuring the surface pressures on ponds or lakes and that is by use of what are called piston oils. What pistons piston oils do is that they spread only to a monolayer on any surface which may be contaminated and the excess oil remains as a lens. This lens acts as a reservoir. It supplies whatever is required to maintain a given constant surface pressure as long as drops are present. So, the oil lens acts as a reservoir supplying whatever is required to maintain a given constant surface pressure and the spreading is only to a monolayer. It is important to note that it is only a monolayer which is spread. This will happen provided the surface pressure existing around does not exceed their own spreading pressure ok. So, now you have a different method. Here one oil will not do the trick for you. You will need probably a range of different oils which can exert different constant spreading pressures. So, will require a number of formulations, number of piston oils capable of exerting different magnitudes of spreading pressures. So, when you take a contaminated surface you have to now choose at least two such piston oils which may exhibit a way to bracket the existing pressure. What it means is that you take one piston oil which cannot spread even to the extent of a monolayer. Another which pushes out the impurities. That means, the impurities must have that unknown surface pressure in between what is supplied or provided by these two oils. So, if pi 1 and pi 2 are two spreading pressures of oil 1 and oil 2 and the oil 1 is able to spread on the contaminated surface, oil 2 is not able to spread then the actual pi must be in between these pi 1 and pi 2. Now, if you want a greater accuracy we will need closer range of spreading pressures for the piston oils used. This is what I mean by bracketing the unknown surface pressure between two arbitrarily close spreading pressures, but which are known to you. So, you can determine the unknown pressure to within a certain accuracy provided by the proximity of the spreading pressures of these piston oils. If this is all clear then we go to another basic concept contact angles. If a small drop of liquid is placed on a uniform perfectly flat solid surface, it may not spread completely on this surface. If that is the case the edge of the drop may make an angle theta with the solid as I will show you in this figure. Let us say we have three solids on solid 1 we place a given liquid let us say oil which shows this shape. The liquid is L the solid B represent as capital S. This is the first scenario and the surface tension of liquid in contact with air for all these cases will be this gamma L A making a liquid making an angle theta through the liquid. We will choose to be somewhat crude and approximate here and refine our understanding as we go along. So, far there is no approximation involved no inaccuracy whatsoever. For liquid we have legitimate surface tension gamma L A with contact angle theta. However, the next two quantities I write should be taken with a pinch of salt. First along the surface I am going to write gamma S A along the surface we have surface tension of solid in contact with air acting parallel to the surface and then inside the liquid along the interface that is over here we have gamma S slash liquid L gamma S A between the solid and air gamma S L between solid and liquid. The direction is gamma S A is acting this way gamma S L is acting this way gamma L A is acting along the tangent. This may be one kind of situation where the liquid actually does not spread on the solid. So, there is an equilibrium shape exhibiting a contact angle theta. Let us see what would happen if we choose a solid which has a slightly different level of molecular attraction. Supposing the solid is chosen now such for the same liquid that the molecular attraction is somewhat lesser then perhaps we may get a drop of this shape if you draw the tangent along the liquid air surface we have gamma L A making 90 degrees angle with the solid. Once again we have gamma S A over here and gamma S L over here that leaves the third category. The molecular attraction the solid has for liquid is now much lesser if that is the case the same liquid probably will exhibit a different shape of the drop something like this may be gamma L A acts along the tangent gamma S A along the solid gamma S L along the interface inward towards the liquid. In each of these cases we have a drop of liquid resting on the solid and each solid is having a lesser and lesser molecular attraction for the liquid than the preceding one. If this picture is clear we should be able to go forward it is quite clear now what the contact angle is and we drop roughly understand the effect of the molecular attraction that solid has for liquid as the attraction diminishes the liquid tends to retain its own identity more and more. And then you can how do the forces balance in the same way as we did earlier but we will go into that analysis shortly we need that here and we need to refine that also right and it is here that I will have a chance to explain to you something more than what I did last time because there are many misconceptions about surface tension interfacial tension contact angles I want to dispel all of those hopefully here. One of the questions was which direction we take balance in that clarity was not there for some of you but that should emerge now soon. So, the drop of liquid resting on solids will have a smaller molecular attraction than the previous case and it is exactly analogous to the behavior of one liquid on another it is important to note whether we are spreading one liquid on another or one liquid on solid the behavior is analogous here the theta will be equal to theta 1 plus theta 2 that we considered earlier in the liquid drop on another liquid case. In its simplest statement the theory of contact angles allows us to resolve the equilibrium tensions horizontally which means gamma s a let us return to the visualizer gamma s a gamma s a acting horizontally is balanced by gamma s l in the opposite direction and gamma l a cos theta. So, component of this in this direction added to interfacial tension will be equal to surface tension of solid likewise. So, we return to the slides we understand that gamma s a is equal to gamma s l plus gamma l a cos theta. Theta may be less than 90 degrees as in the first case equal to 90 degrees as in the second or greater than 90 degrees as in the third case right and appropriate form of Dupras equation is W s l equal to gamma s a plus gamma l a minus gamma s l. The interfacial energy of the separated system solid and liquid will be the surface tension of solid plus surface tension of liquid minus the interfacial tension or interfacial energy in the beginning per area. What we do next is we substitute for gamma s a from the previous equation substitute for gamma s a as gamma s l plus gamma l a cos theta over here. Once we do that we get W s l equal to gamma l a cos theta plus 1, gamma s l plus gamma l a cos theta will have gamma s l cancelled and gamma l a cos theta plus 1 clear. This is claimed to be more useful than the previous equation why because now it does not contain the surface and interfacial tensions which anyway cannot be easily or accurately measured for solids. In fact, the surface tension of solid or interfacial tension concept for solid liquid interface itself is not precise measurements apart. So, in a sense this is a preferable equation which would apply only if the liquid can rest on the solid and we can have an equilibrium that balance of forces this must be possible. This is possible only if the liquid does not actually spread on the solid. If it were to spread then this force balance would never be met. Besides the concept of tension not being possible to measure we have this certain amount of inaccuracy of imprecision related to tensions for solids. So, one may say that it is preferable to deduce the expressions for gamma s a and W s l in terms of surface energies of the solid. There is no issue with surface energies of solid those are perfect entities to understand define or measure. So, the next effort will be to get expressions equivalent to gamma s a and W s l in terms of surface energies of the solid. It is here that the work of Daryagin in 1957 and Fawkes and Harkins in 1940 is of utility to us. Their approach is known as the method of virtual displacement of equilibrium three phase contact line or boundary. Here we imagine that the solid is available in the form of a flat plate and we immerse this solid into the liquid in a special manner ensuring that the liquid surface is horizontal on the left hand side of this surface. Let me draw a diagram here so that this point is quite clear to you. Suppose we take our liquid surface here that is our liquid l. Now, the solid in the form of plate is immerse in this liquid this is the solid plate for you. What we done is we have inclined this plate such that the plate makes an angle theta with the liquid theta is not arbitrary. This plate is actually oriented such that the liquid surface is planar right up to this three phase contact line. We do not worry about this part that is not our concentration that will be necessarily curved meniscus, but we have to ensure that this surface on this side is planar right up to the contact line. So, all of this is planar ok. Now, we have air over here in contact with the solid. We have the liquid in which the solid is immerse and we have the solid in the form of this flat plate which we orient this way and are able to let us say advance or withdraw. We will see later how that could be done, but right now it is only a thought experiment. So, the plate can be immerse or withdraw. Now, let us see what this leads to. So, the thought extent is clear to you the immersed inclined plate is being pushed into the liquid. Now, we imagine that a length delta l of the upper solid gas interface solid air interface is in head invaded by liquid by immersion of the solid sheet for which we presume the width is 1 centimeter unit width. So, we have this unit width of the solid plate we immerse it into the liquid through a length which is this differential length delta l in the downward direction theta which is ensuring the liquid surface is planar right up to the 3 phase contact line on the left hand side upper surface. Now, during immersion what role does the tension surface tension plate if it is a usual liquid the surface tension will tend to pull the plate inside. It will assist this immersion of the plate and the work done will be delta l into gamma l a cos theta gamma l a is along the surface over here gamma l a cos theta will be the component tending to assist the immersion. So, that component will be gamma l a cos theta this one and there is a unit width of the plate perpendicular to the plane of paper there is unit width and the immersion is through the length delta l delta l is the length of immersion in this differential step. So, the work done on the plate by the surface tension we return to the slides is delta l times this force gamma l a cos theta per unit width right ok. So, this is one thing now we recognize that this is an infinite symbol change in an equilibrium system which means from thermodynamics we should say the total work done is equal to 0. Remember the motivation is at any cost to avoid the surface tension for solid or the interfacial tension between solid and liquid. So, while we now want to complete this total work equal to 0 this equation we realize that the upper surface is initially in contact with air delta l into 1 was in contact with air now it is invaded by the liquid. So, it has come in contact with the liquid. So, the free energy change is there and there is a work done these are the two quantities we need to plug in for completing this equation in our symbolic form. So, if f s is the surface energy per unit area of solid then delta l into 1 is the area which is invaded initially it is in contact with air. So, delta l into 1 this area multiplied by this f s surface energy per unit area is the initial surface energy of the solid later the same area is in contact with the liquid. So, the energy is now f s l s. So, we realize here that the first contribution to our equation will come from over here 1 into delta l is the area final energy is f s l s initial energy is f s s. So, the change in the energy is f s l s minus f s s that is the change in energy plus the work done is delta l into gamma l a cos theta that should be equal to 0. We have been able to obviate the difficulty of defining surface tension and interfacial tension for the solid air solid liquid system. Now, we simply cancel delta l and write this as f s s minus f s l is equal to gamma l a cos theta or we can write f s s is equal to f s l s plus gamma l a cos theta. Now, we do not have surface and interfacial tension. We have surface energy for solid in contact with air interfacial energy for solid in contact with liquid. As an after thought we could say I could have done a smarter thing. I could have in my mind thought of surface tensions need not have spelled them. I could have written this from the force balance and then realizing the equality of tensions with free energies I could have written this. But this method is a formal method for allowing you to do that without at any stage invoking the concept of tension or interfacial tension for systems involving solids. So, that is the method of virtual displacement for you. I believe we have about a minute so let me see. Sir, suppose we consider that that was an equilibrium. Right. Suppose that was not an equilibrium what all terms are we selecting as in. If it is not in equilibrium then of course, there are forces which will make the system not exhibit time independence right. Here there is there are only two forces the force which is two quantities. The force which is trying to immerse it and the work which is done by surface tension and the change in energies. If it is not in equilibrium this balance would not be there. So, we will have extra terms ok. And we can actually write this total work equal to 0 only for equilibrium system. We could have obtained this equation by understanding the equivalence of F's and gammas, but it would not have in any easier visualizing visualizing a solid in a state of tension. So, by this more rigorous method of virtual displacement of three phase contact line we can arrive at the Young's equation which comes out to be same as earlier WSL is equal to gamma L a cos theta plus 1. So, that equation remains unchanged. We can alternatively obtain it through the thought experiment or the theory of method of virtual displacement of three phase contact line. The last thing I would like to cover here is the ignored component. It is amazing how things which are not taught to us remain in our minds. So, when one of you had asked me this question that I believed that surface tension had to do with the vertical components. Practically nowhere would that be mentioned, but one has to watch out for your imperfect recollection from memory based on maybe understanding. That is the only way to dispel the misconceptions. For the time being here since we have not discussed the vertical component at all what we should be talking about is the gamma L a sin theta. The same plate inclined at theta has one component acting along the surface of the solid. So, that is gamma L a cos theta, but there is another component which is acting perpendicular to this that will be gamma L a sin theta. What do we do with the vertical component and why is it that nobody talks about it? Gamma L a sin theta is actually of interest in context of solids, but to study that you require a special experiment. You could study what a drop would do to a thin mica sheet and to get the best measurable conditions. We take liquid like mercury with very high surface tension and place a drop on a very thin mica sheet about 1 micron thick. This was exactly what was done by Bailey in 1957. What you now get is what could be shown in another diagram. You have a thin mica sheet and not even showing the thickness and you place a mercury drop on this. The vertical component of surface tension actually makes the solid bulge like this. The mica sheet is now deformed this way and now what you have is gamma L a acting along the tangent, gamma S a acting along tangent to the solid and gamma S l acting along tangent to the interface between mercury and solid. So, that is gamma S l for you. One may say this angle is phi 1 and this angle is phi 2. So, the vertical component gamma L a sin theta is causing this deformation. Unfortunately, for most solids the deformation is very difficult and therefore, you would not observe this. That is the only reason why people do not probably talk about it, but it does not mean that it does not exist. It does exist, it deforms the solid whether the deformation is measurable or not may be an issue, but if you make a special extent like this, Bailey's extent we can detect what the action of the vertical component is. So, in general the angles which I have shown phi 1 and phi 2 for most solids will be close to 0 not detectable. Perhaps now it is time to take a logical break. We will talk about magnitudes of contact angles of liquids and solids and a variety of interesting things which arise with respect to considerations like waiting or non waiting of solids by liquids. So, we will stop here for today.