 Alright, we come to this second lecture in the second module on measurements and since we are meeting after a few days of interval, we will have a quick recap of the first method that we had seen for measurement of surface tensions the ring method. Basically, here you measure the force which is required to detach a metal ring from the surface of a liquid and is one of the oldest methods used for measurement of surface tension and as we will see later of interfacial tensions as well. And what may be done is a wire ring connected to either an arm of a balance or to a torsion wire through a light weight beam could facilitate the measurement of this vertical pull required to snap the ring of the liquid. And at the point of the detachment of the ring from the liquid the vertical pull required would be equal to the total perimeter operative of the ring times the surface tension. And it becomes clear from this diagram that the total perimeter will be the total length of the wire times 2 because the meniscus in action is one on inside of this wire and the other on the outside. So, the perimeter is actually to be counted twice and we had seen that once we connect the ring to a balance it becomes this name tensiometer do not tensiometer. Equating that vertical pull F to twice 2 pi r times gamma we get the surface tension as F over 4 pi r and in view of the fact that the surface tension may not act vertically exactly vertically and the contact angle may not necessarily be 0 it would mean that we may have to use some corrections. One would have obviously ensure the 0 contact angle by gently flaming the platinum ring which is used for measurement of surface tensions, but the correction factor beta is still necessary and that would depend on the 2 radii the radius of the wire radius of the ring and the density of the liquid. The correction factors for standard rings used and one requires the apparatus to be calibrated with the help of standard tensions for pure liquids. It was here that we stop last night and we continue from alright we go into the measurement of interfacial tensions with the ring method. Basically same methodology could be used for measurement of interfacial tension with the ring method, but the only necessary requirement now would be that in this 2 liquid system the lower liquid should preferentially weight the platinum ring. If we take for instance the benzene water and carbon tetrachloride water systems as shown in this diagram you would have on the left benzene lighter than water floating on it and carbon tetrachloride which is much heavier than water sinking below it. So if this platinum ring which is prepared to be clean by gentle flaming if it is weighted by water and if you start with this ring within the lower phase water the pull required to snap off this ring from the interface will be giving you indication of the interfacial tension. So that is quite clear and very similar to the measurement of surface tension in a similar situation in an equivalent surface tension measurement system instead of benzene we had air. So the water would be trying to pull the ring inside against the vertical pull trying to snap it away from it. So at the moment of the snapping of the ring the vertical pull just would be able to overcome the surface tension force. Here the equivalent is interfacial tension force otherwise we have similar approach to writing the force balance and even the notations could be retained to be similar with different meaning. However if we take the carbon tetrachloride water system clean platinum ring will not be weighted by CCl4. So if we were to start with platinum ring submerged in carbon tetrachloride and if we were to look at the situation of the force balance at the time this ring comes out of the interface because now the platinum ring is initially in carbon in contact with carbon tetrachloride it would come out clean in water the interface will not be exert will not be able to exert its action towards retaining this ring in a position to the vertical pull required. So clearly if we have a situation like this then it is an indication that we have to do something about the ring right. So I suppose you have started thinking about what modifications we may have to make perhaps we should choose a different ring a ring which is made of stainless steel and weighted with silicone would make it oleophilic oil weighted and then we have the situation that we desire we start with the silicone coated steel ring submerged in carbon tetrachloride weighted by it and when it is attempted to snap it off from the interface will have to apply a pull equal to the existing interfacial tension force. Best results are obtained using large rings of fine wire, representative examples are about 3 centimeters diameter for the ring and 200 super centimeter of the wire radius it is here that we move on to a different method again I think a method known to you and in principle not very different when it comes to writing the force balance of course the physical picture is somewhat different. I may quickly add here that almost all methods of measurement of surface and interfacial tensions will require corrections. So in the context of ring method even while measuring the interfacial tension we will have to allow for a correction similar to what we had earlier and then the same steps like measuring the interfacial tensions for standard systems to calibrate the method that would be applicable here too ok. Now moving on to the drop weight method I believe this will be the force method to come to anybody's mind given the task of measuring the surface tension. About interfacial tensions we will think a little later however as I made a previous remark these methods being simple they are also deceptive you do not realize what extent of precautions one may need to take. So as I mentioned in context of ring method we would have other considerations here. So let us see what are the details of this method and what precautions we need to take. Here we measure the weight or volume of each drop of liquid when it detaches itself from the tip of a vertical tube we have to understand that this weight or volume of the drop will be determined largely by the surface tension of the liquid. So again we visualize the surface tension coming to picture and in contrast to the applied vertical pull in the ring method the gravity is doing the job for you. If these drops are formed extremely slowly please note the choice of word extremely here because I want you to think how slow the drop formation could be. At this point if we allow for this situation that the drops are forming extremely slowly then they would detach themselves completely from the tip when this gravitational pull balances the restraining surface tension force. What complexities could there be there which we do not envisage? We could look into this sketch to get an idea. We have here a capillary with a radius 2a and here the drop is about to fall as a rough sketch this should be ok and a moment a drop is detached the situation might look somewhat like this. Do not worry about exact precision in the drops shape etcetera but we could roughly say this is what it is. The more important point is that we see what is largely spherical portion which is the drop which is getting detached and a neck here there is a certain amount of liquid which is still sticking to the tip of the tube even while the drop is beginning to fall over here we see that some of the liquid in this neck becomes a part of the drop that is what I try to indicate by this conical portion. The remaining part is still hung up with the tip of the capillary ok. So, this gives you an idea as to what might be happening over there. If mass of the drop which has detached itself is capital M and G is the acceleration due to gravity, M G is the weight of the drop that is the volume of the drop times density of liquid rho L times G and so, this weight of the drop is being opposed by the surface tension force that may be given as the right hand side of this equation 3 2 pi A gamma where A is the radius of the tip of the tube. Those of you who are beginning to discern about this might see another potential problem a departure from reality in mathematical effort to mimic it in the force balance. I will just point that out and catch up with it later for the time being if we rearrange our equation the surface tension force gamma is M G by 2 pi A or V rho L G by 2 pi A. So, it would appear that if we can measure the volume of the drop or weight of the drop and if we know the radius of the capillary then it should be possible to measure the surface tension force that is the essential part, but there are details. So, what are those details? In this case we require corrections to estimate the surface tension because the liquid forming the drop does not leave the tip in totality and like earlier the surface tension force does not act exactly vertically. The correction factors are rather important in this method because the drops can be smaller than what is predicted by this equation by as much as 40 percent. This liquid and what is going with the drop in addition to what happens when the drop is detached from the tip in relation to liquid within the capillary those are the sources of complications. The drop is significantly smaller than what is predicted in this relation. So, what do we expect the correction factor to depend on first we name it phi we expect it to depend on drop volume, we expect it will depend on the tip of the capillary. Very careful measurements have revealed that correction factors greater than 1 on this plot the magnitudes are spanning range from 1.5 to 1.70 for the correction factor of phi. Correction factor greater than 1 is to be incorporated you would see logically in the numerator because the volume of the drop or mass of the drop is actually smaller than what is predicted in the equation. So, you need to correct for that smaller volume. So, phi comes here so, phi times mg by 2 pi a or phi times v rho lg by 2 pi a should give you the surface tension. In this plot you see the correction factor phi represented as a function of the tip radius divided by volume of drop to the power 1 by 3. So, you note that this is dimensionless and the value for a by v to the power 1 by 3 spans 0.5 to about 1.25 0.5 to about 1.25 and these have been obtained by Harkins and Brown way back in 1919-1928 have been checked several times by others and one could take these as authentic correction factors one could depend on. Here we try to ask ourselves the question as to how should we go about doing our experiment including the choice of the tip that is used for forming the drops. Obviously, we would have a choice for what tip radius to start start with. So depending on what age we choose we will get the corresponding volume we choose a different age we will get different volumes. So our choice of the tip will determine this factor depending on this factor we will need a different magnitude of correction factor phi. Now one glance at this figure coupled with your understanding of school level calculus will automatically suggest that we should be aiming at probably the flatter part of this curve somewhere may be between 0.75 to 0.95 as our operating range. We should choose such a tip radius such that A by V to the power 1 by 3 is between 0.75 and 0.95 so that the correction factor is least sensitive to any variations in this parameter A by V to the power 1 by 3 there you are likely to be having most reliable measurements alright. The correction factor is incorporated here right now you have other ways of measuring surface tension. So, for pure liquids we know surface tensions and then you actually experimentally find out what M and V are once you know that and you do it for several liquids and find out what the correction factor is for several tips of the capillary or different radii and then you get the data on phi versus A by V to the power 1 by 3. Obviously yeah so the much older method ring method is one which is common standard and standard surface and interfacial tensions have been known for much longer time. It is the convenience or speed with which we can make measurements those are the factors which come and there are variety of methods depending on the circumstances one could choose one of one of those methods which is more suitable. The ring method or do now tensiometer based method measurements are the ones which are fundamental. So, this gives you an idea that when you aim at this A by V to the power 1 by 3 to range between 0.7, 0.75 and 0.95 we are likely to have least dependence of phi on variations in A by V to the power 1 by 3. You see the gradient is 0 at the extremum maximum so the values of D phi by this dimensionless tip radius will change magnitudes from minus negative to positive through 0 and these values are likely to be quite small the gradient is close to 0. In view of what comments I made perhaps you might think about this and it was intentional to say that different liquids have been used and different red eye for tips. Those measurements reveal that for all liquids tested we get the same value for phi it means differences in viscosities of different liquids, differences in other properties for different liquids need not be allowed for in this method that is a fortunate set of circumstances a method is not critically affected by differences in properties. So, this correction factor of phi could be looked at as generic values for all liquids. The only condition is and I repeat the drops must be formed very slowly and then we are guided by A by V to the power 1 by 3 in the range 0.75 to 0.95. Now, at this point it is always interesting to ask this question how slow should the drop formation be could you make a guess maybe you can think about what you might want to do. If I ask you to measure the surface tension and all I want to tell you is that you got only very few simple expedience at hand in lab capillaries liquids weighing bottles weighing balances maybe rings how do you go about measuring the surface tension. I think 9 out of 10 times you will think of just having a capillary letting it form a drop of liquid measure the volume or weight of the drop weight of the drop will be easier to measure. And if you do it in an uninformed way then you might find that even for pure water surface tension values are very different from what you expect and it is very common to make this mistake. I do not want you to remember this equation this is only an equation of fit which gives you a good correlation between external data and the prediction, but it could be alternatively used for correct finding the correction factor. The critique was if it was not formed slowly then the balance of that particular thing is not valid. Is somehow is somehow not correct. Is the balance not valid or is it the other reason of the slower? The volume that you measure of the drop is a result of experiment right. So, whether the weight balance is the surface tension force or not is the only issue ok. So, you have to think of whether that assumption that the weight of the drop is balanced by the surface tension force or not is the only issue. So, if those are balanced you are ok and that happens when the drop is formed very slowly if not and with the corrections though and if not then of course, something is going wrong and that is what I am asking you to think before I flash it in the next slide yeah. First force is also generated if it is not very slowly that is do balance by the surface tension. Right and there is nothing in our equation to take care of anything related to that momentum of the liquid right. So, I will just come back to this slide again the how slow those drop formation sequences should be. We can collect a number of drops in a weighing bottle each drop should form slowly so slowly that not more than 1 every 3 minutes would fall and that I think in most people mind most people's minds would be not coming easily. If we can ensure that more not more than 1 drop forms every 3 minutes then accuracy to a few tenths of a percent is attainable. The drop is formed too quickly it will be actually too heavy because of the taint tail being blown up with more liquid coming down the tube ok. So, that is the reason why we need to have the drops forming very slowly now the same approach will suggest that we could use drop formation drop weight method for measuring the interfacial tension also. One will have to choose the heavier liquid to form drops within the lighter liquid. So, the heavier liquid will come down the capillary and form a drop at the tip and in our equation this would not be any longer the true mass of drop, but apparent mass and rho L will be replaced by the density difference. So, M is the apparent mass of the drop rho L is the density difference between the two liquids and phi is the correction factor found in the same manner as we did for surface tension estimation. Now, there is another observation we should be making here experimentally we must ensure that the outside of the tip from where the drops are forming is either completely wet or is completely non wetted by this liquid if it is completely wetted by the liquid it is fine if it is not wetted by completely non not wetted by the liquid then there is a problem because now the drops will form with an uncertain A. We do not know what effective radius is operative for the surface tension force. So, we have to ensure that the tip is either completely wetted or not wetted. Second we must not have too hydrophobic a surface of the tube or the capillary if it is too hydrophobic and you have a liquid like water coming down it then you might have seen this you would expect the drops to actually form inside the capillary itself because the contact angle does not permit continuous wetting of the inside surface. And therefore, one can never think of making a surface or interfacial tension using the drop wet method when the tube or capillary is made of Teflon. Teflon will be not wetted by practically any liquid. So, we cannot use that generally very lightly siliconized stainless steel is reliable. And these tips have to be made very carefully using watchmaker's lathe such that the ends are sharp and regular and the tips may be made hydrophobic with silicone and in handling these tubes you have to exercise a great deal of care because there should be no contamination. There is any contamination of the tips the surface or interfacial tensions you measure will come out to be spurious results. I already explained the frequency the small frequency of formation of the drops at the capillary tip is here that I may add a detail which is not indicated here. If you have to weigh something like 50 drops it would take a long time to conduct one measurement. So, in practice what can be done is that the initial part of drop formation that could be gone through without much difficulty relatively quickly. I might make a few remarks here while we are discussing the drop wet method about the drop formation. Now that is a generic feature of how drops or bubbles form here this rough diagram sketch has been made deliberately so that you can see what might be happening. I did not indicate a drop forming only at the tip and expanding I could have shown a sequence of drop shapes leading up to this I have chosen to show that there is a neck here. Now that is generic about drop and bubble formation there is a stage 1 in which the bubbles or drop form at a tip of an orifice remaining attached to the perimeter and then it is followed by a stage 2 in which somehow neck forms and keeps elongating. So this dropper bubble at the tip is growing but at the same time moving away from the tip of the orifice and at some point of time it snaps off leaving a daughter droplet or a daughter bubble in relevant situation. If you want to know more about the details of drop or bubble formation you should be seeing the mathematical models proposed by Professor Kumar from Indian State of Sands Bangalore and Professor Davidson from University of Cambridge. Their research groups have published a series of papers in 70s and especially a readable account may be found in the advances in chemical engineering series of 1970 in a chapter written by Professor Kumar and Kulur on drop and bubble formation. Well the same method can be used for measuring the standard interfacial tensions. I would like to explain a little bit more on how to save on this time may be through an experimental intervention. What could be done is a tip like this can be attached preferably through a ground glass joint to a micro pipette which allows you to feed liquid into this tube at a control rate and one could do that relatively quickly in the initial part when the bubble when this drop is expanding. So there is no error involved if we do not push the liquid with too much force at the moment of detachment. During the formation you can form the drop at a relatively rapid rate without having any problem but then you slow down and make the drop detachment very slow. So while you are forming one drop in every 3 minutes in normal circumstances you might be able to save on time you might form much of that drop much quicker and then slow down using the micro pipette and control you can slow down the drop formation at the later stages such that you do not blow up the tail into the bulk of the drop at the moment of detachment. So that way you may be able to do a better job of achieving the same precision of measuring the surface or interfacial tension but then also achieving a saving in time. Once again for interfacial tensions we have to calibrate the method and apparatus with the help of standard values. Now it is here that we move into the discussion of willy me plate method. Now this is kind of universal method to check surface tension over long time intervals. What is done is a very thin plate is attached to an arm of a balance or like in the tensiometry suspended from a light beam attached to a torsion wire and we see what is the additional pull required when this plate gets immersed in the liquid. If you start with the plate descending through air and entering the liquid at the point when it is partially immersed in the liquid whatever is the additional pull which is exerted by the liquid on the plate if that is measured that could be equated to the perimeter times the surface tension force. Now once again there are certain guidelines for achieving more accurate results it is preferable not to detach the plate from the surface but to keep it immersed partially in the liquid and make corrections for buoyancy. You could view this statement in the sense of what is the difference in the vertical pull when this plate is coming out of the liquid. So you measure what pull is required to take this plate make this plate coming out of the liquid but actually not snapping up from the surface rather allow for corrections related to buoyancy and this latter effect also can be made negligible if you suitably choose the mica plate. A very thin mica plate with the dimensions such as 10 centimeter width 5 centimeter height and 200 super centimeter in thickness may be adequate to minimize the correction by buoyancy. Let me show the diagram such that this discussion makes a clearer sense. We have here two sketches for a plate of solid being withdrawn from the liquid. In the first part A this solid is completely weighted so theta equal to 0 and therefore you see the vertical force applied is F the surface tension is almost acting vertically here if theta is 0 the surface tension will act vertically downward this arrow is only to be shifted to the solid surface for contact angle theta equal to 0. However, if the contact angle is not 0 then we have certain theta here the surface tension would not act vertically. We require for this method for F to give you the surface tension we require contact angle to be 0 or this liquid to weight the solid so that angle is actually 0. What if we suspect that complete weighting will not occur for some reason? We need to ensure now that that is achieved by rubbing this mica plate with some fine grade emery paper so that it is roughened and we have contact angle becoming 0. Mica would normally be weighted by clean water pure water but if it does not then roughening of mica could facilitate it and the way to do that is we should take the mica plate and rub it with a fine grade emery paper and there is also particular direction moving it in small circles to make roughening as uniform as possible in all direction. Those of you who have prepared samples for let us say making a scanning electron micrograph you would be aware of the kind of elaborate preparations that go in preparation of the samples you have to actually follow a similar method to prepare the substrate for making the SA measurements. If the perimeter is unity then f should give you gamma in general however the meniscus takes the shape which is as shown in this part b there is a angle theta the liquid makes with the solid and therefore, what one measures here is the weight of liquid pulled up above the main level by the liquid surface of the liquid surface and that would be gamma cos theta per unit perimeter or if theta is ensured to be 0 degrees then it will be equal to gamma. So, f is equal to gamma if theta is 0 degrees otherwise will be gamma cos theta. Given suitable weighting characteristics such that the plate is completely weighted by one liquid we should be able to use this method also for measuring the interfacial tensions right. So, it is a similar situation as we have consider in the context of the ring method. To obtain a plate strongly water weighted one uses glass carefully cleaned with chromic acid a cover slide may be a good example or it could be rough and mica as I just explained. However, if it is to be oleophilic oil weighted then mica coated with lamp black from a smoky Bunsen burner flame is adequate we will get theta equal to 0 degrees in oil and close to 180 degrees in water. I think we can conclude our lecture logically at this point and start with the Pendant drop method next time.