 In this lecture number 11, we will continue discussion on spreading contact angles, work up addition, not only in context of smooth surfaces, but also on rough and composite surfaces. And as mentioned last time, we will take recourse to free energies considerations rather than surface tension, because first free energies are difficult to measure for solids and they are not a precise concept to define in terms of solid surface tensions. To keep this lecture complete in itself, we go a few slides back and resume from the discussion on magnitudes of contact angles of liquids on solids. We had seen that Young's equation allowed the contact angle theta to be related to the work up addition between the liquid and solid and the work up cohesion for the liquid which is trying to spread on to the solid. And if it fails to do so, it would be restrained in the form of a finite lens in its equilibrium shape. So, Young's equation would relate the contact angle to WSL and WL which is the work up cohesion of liquid and you know that is twice the surface tension of the liquid. And we had seen that for a given surface tension the contact angle would increase as the addition between the liquid and the solid decreased and that was indicated by three pictures in which the contact angle made by the tangent to the surface of liquid at the three phase contact line made an acute angle for the greatest work up addition. That was 90 degrees angle when the work up addition was relatively less and the angle became obtuse when the work up addition for the liquid on solid decreased further. And one could take that argument further to determine what would be the maximum contact angle possible, how obtuse can it get theoretically one may expect a contact angle of 180 degrees. However, in all real liquid solid systems this would not be achieved. Having said that we will actually use a contact angle of 180 degree in one of the examples to follow in the later parts of this lecture. So, in practice you have mercury resting on steel giving you a contact angle of about 154 degrees. If you place a water drop on paraffin wax the theta could be measured up to 110 degrees that would indicate the work up addition from Young's equation as WSL equal to 48 Ergs per centimeter square. That is pretty close to 43 Ergs per centimeter square measured for liquid paraffin wax liquid paraffin in contact with water that is pretty close to the work up addition for the solid wax that we have cited here. On the other hand water resting on polyethylene will give you a theta of 94 degrees. High theta or contact angle is also attainable on surfaces which are normally hydrophilic. I hope this statement is clear to you. Once we say that we are looking at water and a solid, if the solid surface is hydrophilic then the necessary implication is that contact angle would be very small. If water has a positive spreading tendency on this hydrophilic surface the contact angle should be 0. However, what we are trying to imply here is that it is possible to attain a high contact angle by modifying this otherwise hydrophilic solid surface. So, we take the example of glass clean carefully with hot chromic acid that would give theta equal to 0 degrees per water. However, a very thin layer of silicone or a derivative of dimethylsilane would completely alter the weighting properties. What now happens could be shown in the form of diagram. Here we can see what would happen if you have a very thin layer of dimethyl dichlorosilane laid on top of the hydrophilic glass surface. So, if this is a clean glass surface this should be hydrophilic. If we had to spread water or place a drop of water on this surface it would just flatten out into a thin film of water right across this hydrophilic surface. However, if you look at the structure of this surface you would have a silicate lattice which gets chemically modified if we were to place on the clean glass surface this dimethyl dichlorosilane. What this does is that the OH groups on the outside of the clean glass surface would get reacted with dimethyl dichlorosilane releasing HCl and then bonding this group over here. So, this is a case of chemical bonding this unit is repeated all over the surface what will the drop of water placed on top of this chemically modified surface C it would see the methyl groups that is about it. What has happened the clean hydrophilic glass surface has been rendered completely hydrophobic now because of these hydrocarbon groups methyl groups even a monolayer of dimethyl silane will render glass or steel completely hydrophobic or completely lipophilic. That means now the glass will be weighted by oil this knowledge is used often in preparing suitable rings or dropping tips for surface tension measurements. When we go into the details of surface tension measurements will be spending quite some time on the procedures preparing our equipment and surfaces for actual measurements. So, we continue in this lecture the concentrations of contact angles and spreading also into rough surfaces. This is what I have just explained dimethyl dichlorosilane attaches so strongly to the surface of glass that now you would not have contamination of liquids which are brought in contact with such a modified surface. They were certainly other options possible for example, heavy metal soaps and long chain fatty acids and amines could also produce similar effect rendering and otherwise hydrophilic surface hydrophobic. But there is a risk involved here because these compounds are not chemically bonded to the surface. So, they may actually contaminate the liquids which are being studied for their spreading tendencies contact angles etcetera. They have been commercial products available under various trade names Tadol trifilm M441 of ICI these were the materials available which could be applied from solution in carbon tetrachloride making the silicate lattice now hydrophobic. The conclusion here is that silicone and silane treatments are preferable because they would necessarily avoid contamination of adjacent liquids. We talked about Teflon a few lectures ago this polytetra fluoroethylene PTFE also known as Fluon or Teflon you would know is strongly water repellent. I had mentioned that nothing sticks to Teflon. With respect to water the contact angle is 108 degrees which is about 14 degrees higher than for polyethylene. The higher contact angle is again a reflection of less work of addition or less tendency of water to adhere to Teflon compared to polyethylene. But it does not mean that Teflon surface cannot be made wettable with water. But you need an intervention like a strong wetting agent like sodium, dioptylsulfo succinate to be used along with water. If you make a solution of STOS then the aqueous solution can actually wet Teflon. At this point we logically move into a somewhat more difficult question that is with the observation that so far we have restricted ourselves to only smooth surfaces. We only talked of smooth solid surfaces. What would happen if we were to have a rough surface? Same liquids that we have been talking about in terms of contact angles or works of addition or spreading cohesions how would these magnitudes get changed? What principles would govern this and maybe some of the common observations that we make in everyday life perhaps the answer could lie in the reality of rough surfaces. I intend to take you through the very fundamentals of these considerations for rough surfaces. So we may go slow here but hopefully we will get a hang of what are the chief considerations which make rough surfaces different from smooth surfaces. I already primed this thought that contact angles would depend on roughness of surface. Rough surfaces will be perhaps also have accompanied adsorption of impurities of water maybe even air and if we have these adsorbed impurities especially compounds like water we might even have the possibility of solid surface molecules reorienting themselves. These aspects would have to be addressed in at least some sense. Let us see how far we go how far we can go into these details. First to note is this very broad observation made by Ray and Bartel in 1953. Surface of a surface makes contact angle further from 90 degrees. What it means is if on a smooth solid you get a contact angle greater than 90 degrees if you take a rough equivalent of that solid same composition except that surface is roughened then the contact angle will become even more greater than 90 degrees. However if the smooth solid has a contact angle for a given liquid less than 90 degrees then the roughness will decrease that contact angle further. So it is like a situation the contact angle relative to 90 degrees if greater than 90 degrees it swings further away is less than 90 degrees again swings further away from 90 degrees that is the way roughness seem to seems to affect the contact angle could we quantify this. It is fine as a matter of experimental observation but we would need theoretical substantiation too. So if this is clear to you clear enough also to remember we can move further. Let us look at some magnitudes. If we take smooth paraffin wax for water the contact angle is above 110 degrees roughen the wax surface and the contact angle will become about 132 degrees it has taken theta further away from 90 degrees is an application to this while measuring contact angles and surface tensions using a method called Willem's plate method we actually use roughening. This will ensure that the contact angle of thin mica plate in contact with the surface active solution is always close to 0. We need it fine there are applications in measurements of contact angles measurements of surface tensions. So we can keep this in mind as a way of ensuring whenever we deal with solid liquid systems and we need the solid to be weighted by the liquid as one possible avenue in order to get the weighting of the solid by the liquid. Next we come to the question that before we can exemplify the effect of roughness on contact angle we got to quantify roughening. We must be able to define the roughness or roughening in some quantitative manner. So the way to do that is we define a factor called rugosity or roughness simply as roughness as the ratio of the real surface to apparent surface. Let me explain this supposing we have a smooth solid surface alright and we are looking at an equivalent which is rough. So let me draw some kind of roughness equivalent here the same solid but rendered rougher. Now how do we define the roughness here quantitatively? So we define this rugosity r as the actual surface which will be corresponding to this crevices and valleys here as it is. How we measure is a different matter we might use a BET apparatus to know whatever is the actual surface area which is again based on adsorption. How much gas is adsorbed and knowing the molecular cross section using the BET theory we can estimate what is this area. Apparent area that we have in this context is whatever is the area which is the area of projection. Looking from top if we were to look at just unit area which is projected then between these boundaries look from top we will have 1 centimeter square apparent area 1 centimeter by 1 centimeter square is what we are looking at. But the same apparent area corresponds to much larger area because of this roughness. So the rugosity is the real to that projected or apparent area. This is what we could use as a means of quantifying the roughness of a solid surface. So the question next is what would be the contact angle for the same liquid on the same solid but with rougher surface. Historically this relation was given by Wenzel in 1936 and the relationship is as simple as this cosine of contact angle for a rough surface is equal to the rugosity times cosine of angle for a smooth solid surface. So we have cos theta bar corresponding to the rough surface theta bar is the contact angle for rough surface, theta is the contact angle for the same liquid for smooth solid r is this rugosity real to apparent surface area ratio or real to projected area ratio. We should be able to derive this though. So that is our next task to show that Wenzel's relation is actually logically possible to derive. So let us prove Wenzel's relation for rough surface the contact angle is theta bar and for smooth surface the contact angle is theta. What we use a starting point here is the equation for infinite simul change in an equilibrium system on similar lines as we had addressed last time too. But now remember that we are dealing with rough surfaces we are actually referring to the same expression or same kind of equation which for infinite simul change in equilibrium system adds up the free energy change and work done. Infinite is the same diagram we are referring to we have a solid plate immerse in a liquid and inclined to the horizontal at angle theta such that the liquids meniscus is planar right up to the three phase contact line ok. So maybe I can show you that again. This is the same diagram we have the liquid surface and we have the solid plate deliberately inclined to make angle theta here with respect to the horizontal. This angle is theta and the liquid surface is planar right up to the three phase contact line. We do not worry about this part and in our thought experiment we say that this plate will be immersed through a differential length delta L and in the process some work done by the contact angle by the surface tension component acting at an angle corresponding to contact angle will tend to pull the plate inside ok. So that is in this case surface tension is trying to assist the immersion. So gamma L a cos theta will tend to help immersion associated with this work also is the change in energy because initially one into delta L unit width in the direction perpendicular times delta L that area which was in contact with air is now finally in contact with liquid. So there is a surface energy change for smooth solid we had made the balance which now has to be modified because now this surface is no longer smooth that is the only difference otherwise this same picture is valid for our concentration here. So how do the concentrations change? We can look at the next slide where we find that the first term FSLS the final interfacial energy between liquid and solid minus FSAS the surface free energy of solid in contact with air is now multiplied by 1 into delta L and that area apparent area 1 unit width times delta L now needs to be replaced by the real change in area and that will be real by apparent into the apparent that is R into 1 into delta L this real change in area should replace 1 into delta N in original equation. When we do this since we are now dealing with the solids which are rough we would necessarily replace theta by theta bar. So those are the two changes we have made we replaced 1 into delta L by R into 1 into delta L and we replace theta by theta bar. So new equation that we have here would appear this way we are careful here in noting that this would be justifiable only if the roughening is small scale which means we might still be able to think of some contact angle for the rough surface. This will not be true at all scales of roughening for the time being let us say that the roughening is able to permit application of this method of virtual displacement of three phase contact line. We will add a few explanatory sentences here a short while later, but at this stage if we simplify this cancel of delta L from these two terms and we get R equal to FSLS minus FSAS plus gamma LA cos theta bar right. Sir due to the roughening when the FSLS and FSAS change I mean not in terms of magnitude but in terms of direction. Please remember that now we are talking of energies. So it is a scalar ok. These are only magnitudes works per centimeter square ok. The only difference we are conceiving here is of solid either being smooth or rough that is it ok. This is another merit of method of virtual displacement or the alternative of speaking in terms of the surface energies rather than tensions because with tensions you would have immediate difficulties. So just note that the dimensional equivalence of tensions and pre energies should guide you as a reliable shortcut a mental shortcut, but do not take that analogy very far. Once we decide to drop the concept of tension we not only get rid of the imprecision about the idea of tension for solids but we also now switch over to more general free energies which can be measured and they are scalar so we do not have to worry about those. So I am not saying that the tensions should not be thought of but they should be at a subconscious level guiding a mental check on what we are doing. The precision like will lie in working with the free energies alright. So your question I will take further and say that will roughening change the free energies and my response to that is that since roughening is not altering the chemical composition the free energies should remain same ok or if the changes are there they are minuscule we are not giving much weightage to those changes right ok. So we have now a new relation which brings in the contact angle effective or apparent contact angle for rough surfaces theta bar in relationship with rugosity or roughness R this is it. Now is it possible to establish a relation between theta bar R and theta that is what Wenzel's relation states we need to relate theta bar not only to R but also to theta. So that is here that we can take help of this equation which was written for smooth surfaces. So that is exactly what we do we substitute for F s l s minus F s a s this quantity minus gamma l a cos theta F s l s minus F s a s is equal to minus gamma l a cos theta from here. Once we substitute for the difference required here the energy difference is same. So that gives you R times minus gamma l a cos theta plus gamma l a cos theta bar that term remains same and we cancel off gamma l a here and we get the Wenzel relation cos theta bar is equal to R cos theta. Now this will bear out all your expectations as to what roughing would do experimentally in relation to the magnitude of contact angle against 90 degrees. Now I add that explanation for the caveat that the surface presumed here is necessarily submicroscopic in the scale of roughing. Why is it necessary? This is not difficult to see. If we were to think of a solid which is roughened to a very coarse level what difficulties will we have? Close to the solid the liquid boundary will necessarily become ragged with one more consequence that the contact angle theta would not be constant independent of delta l and that was the assumption in the derivation of Wenzel relation. So we require the roughing to be of micro scale. This leads to a further thought. If there is a pattern roughing as often happens in practice I will just ask you to think of rough surfaces in everyday experience that you come across and when you think of the different rough surfaces you can think of the patterns of roughing. In most of the rough solids manmade solids we see a regular pattern. If there is a pattern of roughing and if there is any grooving then one may require to know how that pattern of roughing is or how that grooving has been done. What is the periodic variation of the surface with respect to the planar and grooved parts? If there are only a few grooves on the surface of Teflon the mean contact angle is still given by Wenzel's relation. But we may not always be so lucky we might have patterns which are experimentally possible to measure or quantify but they may not be as periodic or as less frequent as we may want in the normal Wenzel relation. So it is here we think about one more complexity. The surface might be composite. First we talk about a smooth composite surface and we consider a finely constituted smooth composite surface which means we have two or more kinds of components making up the surface. Let us limit ourselves to only two components and the composite surface is finely constituted in the sense the length scales over which these different solids occur at the surface that scale is small it is a fine scale. In addition when we look at the contact angle of a given liquid on this composite surface which is smooth let us say the contact angle is defined as theta prime what is indicated here on the left hand side and if we were to take these solids separately the contact angles that the liquid will exhibit on these solids will be theta 1 and theta 2 ok. If fraction f 1 of this composite surface is made up of constituent 1 and rest 1 minus f 1 of the constituent 2 the apparent contact angle for such a smooth finely constituted composite surface will be given by this relation f 1 cos theta 1 plus 1 minus f 1 cos theta 2 that is equal to cos theta prime. The reality forces us to look at more complex situations. Once again the idealization of a smooth composite surface might no longer be true what if we have finely constituted rough surface for a composite. So, for a finely constituted rough composite surface a fraction f 1 has roughness r 1 and the remaining fraction 1 minus f 1 has roughness r 2 and the corresponding contact angles r theta 1 and theta 2 as earlier. Then the apparent contact angle now becomes modified as for this right hand side cos theta prime is now r 1 f 1 cos theta 1 plus r 2 1 minus f 1 cos theta 2 that is for rough composite surface sub microscopic roughening. If we say f 1 is 1 the second term drops out and we have Wenzel's relation back again theta prime will be cos theta, theta prime will become theta bar cos theta bar is equal to r 1 cos theta 1. Let us see whether this simple relation can help us understand some nice observations that you get to make in practice even in nature. Let us take the case of air solid composite. What I mean here is that the solid surface has a porosity and the pores within the solid are occupied by air. So, if you have a part of surface composed of air spaces like on a porous solid what does our relation for composite rough surface yield to us. There is a mention here of theta to equal to 180 degrees. I had made a tangential reference to that theoretical expectation of contact angle equal to 180 degrees and that will not be possible for any liquid and real solid surface, but since a water drop in air is spherical one may say with relation to air the contact angle of water is 180 degrees. So, cos theta 2 if 2 is air in our rough composite surface with the roughness is actually air spaces cos of 180 degrees minus 1. So, our relation gets modified to this cos theta prime which is f 1 cos theta 1 plus 1 minus f 1 cos theta 2 if we could regard this as a smooth solid that is solid in air providing a smooth surface then f 1 cos theta 1 plus 1 minus f 1 cos theta 2 will simplify to this relation. Cos 180 degrees minus 1. So, you see that will make this combine into f 1 cos theta 1 plus 1 that minus 1 here. Let us put some numbers here. If most of the surface consists of air spaces and you see that I am sure you have thought of certain textiles you can easily get most of the surface occupied by air and it actually is an interesting fact made use of by certain plants and insects to remain unwetted. Theta prime can easily become of the order of 150 degrees even when the basic contact angle for the solid theta 1 is only 90 degrees. All it will require is f 1 is equal to 0.134. So, you can design surfaces or textiles by using the patterns occupied by air spaces to make a fabric unwetable by common liquids. You just take away the contact angle to the region where we achieve unwetable surface. It is here that we are forcing ourselves to consider a certain interesting experimental fact that is related to the hysteresis of contact angle. In an experiment similar to what we imagined in the method of virtual displacement, one may be able to measure the contact angle between the liquid and solid while the plate is getting immersed into the liquid. So, there is a part of cycle in which we are moving forward with immersion. We would measure a certain contact angle and in the other part we might have withdrawal of the plate. So, if we measure the contact angle while advancing the plate into the liquid and we measure the contact angle again when we withdraw the plate from the liquid, the contact angles seem to be different. The difference is illustration of hysteresis of contact angle. So, the fact is that cleaner the surface smaller is the hysteresis of contact angle. We might want to understand this theoretically. Advancing contact angle is generally larger, receding contact angle is smaller and the hysteresis is lesser if we have a cleaner surface. So, we can think of possible explanations for this. In the advancing cycle when we measure the advancing contact angle, we might have to begin with the plate with some material adsorbed on it. So, there may be a film of some material which prevents the liquid from adhering to the solid. We will see how this complexity can develop further even when accidentally you are extremely careful. So, if the liquid cannot adhere to the solid, we will measure a larger contact angle. Then what is actually likely to be the real contact angle if the material will were clean. But after contact with the liquid in the forward or immersion cycle, this film may be either completely or partially get removed. If that happens, then the receding cycle will see a cleaner solid surface in contact with the same liquid. But now that the film is gone on the surface, the liquid will be able to adhere to the solid better and therefore, the contact angle will get smaller. This is the general explanation for hysteresis contact angle. And if you use cleaner material, then of course, this effect is lesser. So, that will be in tune with the experimental observations. Can we rationalize this? That is the next question. We may be able to do this in terms of the concentration of the Helmholtz free energy per area FSLS between the liquid and solid. This may become reduced. We can offer the following argument to explain the hysteresis contact angle. We may say that in the advancing cycle, part of cycle, the adsorb film is spontaneously getting removed. If this happens, FSLS will decrease. Therefore, cos theta must become greater. And you know, if you look at the cos theta versus theta, if cos theta becomes greater, theta must become less. So, in the forward cycle, the film spontaneously leaves making from free energy concentrations the effective FSLS for the receding cycle smaller, which will mean that the receding cycle will see a smaller FSLS. And therefore, FSAS minus FSLS will be higher and therefore, cos theta will be higher and theta smaller. So, in the receding cycle, you have a reduction in the contact angle. So, that is how the hysteresis of contact angle could be explained. Let us see whether we can stretch this understanding to a finer level. The adsorb solid, adsorb material on the solid is in the form of film. When you bring this film covered solid in contact with the liquid, this film leaves the solid liquid interface and occupies the air liquid interface. It leaves spontaneously. When it leaves this solid, it means that this is a spontaneous process and therefore, the solid surface energy must diminish. Everything happens in direction of decreasing free energy. So, that means now FSLS is lower, FSAS minus FSLS is higher, therefore, cos theta is higher. If gamma L A has not changed much and therefore, theta must have become reduced. So, in the receding part of the cycle, we will see a more complete contact or lesser value for the contact angle. Clear? If the film consists of a greasy material adsorbed in small traces from air, we could confirm whether this happens. We are little bit ahead of these lectures, but we will see later that it is possible to measure the potential surface and interfacial potentials to detect these impurities at clean water surface. So, by measuring potentials after a few minutes of exposure of a clean air water surface, we may be able to confirm that there is a adsorbed film on solid which is leaving it and occupying the air water surface. And because it is a spontaneous thing, it is indicating a preferential adsorption of this material, greasy material at the air liquid surface. So, what effect can this have? This could cause the surface tension of water to diminish. After all, this film has come spontaneously to the surface, so it must have also reduce the surface energy of the liquid. Therefore, next part of argument is gamma LA will be reduced. And then again, by the same equation we can say if gamma LA is reduced, FSL is reduced. Actually, this quantity has become higher, we will require cos theta higher. So, that means theta must be reduced. So, this is a consistent rationalization which can explain the hysteresis of contact angle. Are there practical examples which can substantiate these kind of phenomena? In the literature on floatation, you see that in practice if solid surface is covered, even up to the extent of about 10 percent using a monolayer of long chain compounds, it can drastically increase theta. So, one commonly makes use of this understanding in floatation practice. Taken together, we may say that this explanation of hysteresis is reasonable. The liquid surface in such extents must be very carefully cleaned. And this older practice of taking the mean of advancing and residing contact angle has practically nothing to recommend it. They are actually two different contact angles. The solid surface with adsorbed impurities is actually a different solid surface. We cannot take an average. This is the last slide for today. Even if we are extremely careful in removing the contamination, we may still observe the advancing contact angle to be too large. How could that happen? That may be result of adsorption of a film of air on rough solid. Especially rough solids will have larger area, so greater probability of adsorption of air. That would decrease the work of addition between the liquid and this air covered rough surface solid. And if WSL decreases, then that may make the contact angle too large. Remember Wenzel's relation. This is the last slide for today. WSL is gamma L a cos theta plus 1. By adsorption of air, WSL between the given liquid and solid is reduced. That is what this arrow is supposed to tell you. And therefore, we require cos theta to decrease. If this relation is to be obeyed, cos theta has to be decreased, which corresponds to theta increase. So, in practice we might have much larger observed contact angle even when you carefully treated the liquid surface and the solid surface to get rid of impurities, but the ambient air still can play a part. So, very clean and very smooth solid surfaces are shown to display very little or no hysteresis of contact angle. We will stop here for today and resume our discussion further considering the nature of solids and the way you cut out a solid surface before you make measurements on contact angles. So, we will stop here for today.