 Today, in this third lecture in the module on monolayers, we talk about films of polymers, determination of molecular weights from the force area curves. The surface viscosity drag exerted by a monolayer or on the monolayer and one particular method, the canal method for measurement of surface viscosity for insoluble monolayers. To briefly see what we had done last time for polymer films, we first note that many of the polymers both of natural origin or those made synthetically tend to spread readily at surfaces and these polymeric films show considerable cohesion and the forces could be within molecules intramolecular or among different molecules intermolecular. Sometimes, we have both intra as well as intermolecular cohesion and the values of surface pressures at oil water interfaces would be generally greater than at air water surface. This could be seen from the data obtained on hemoglobin monolayer spread on tenth of a normal aqueous HCl at interface either with air or with cyclohexane and the surface pressure measured would depend on among other things the molecular weight and the interaction forces intramolecular as well as intermolecular forces and many at times the polymer molecules have a large bulk oriented away from the interface and that makes it difficult for such molecules to enter the water underneath. In other words, the total energy of desorption W is very high. It could be overcome by the electrical forces which could be tweaked by changing let us say the pH of the solution that way altering the electrical energy of double layer we might be able to make polyacrylic acid monolayer soluble in water or what is essentially water dilute alkaline solution. The point where I had left last time was regarding this question whether we could use the force area curves for inferring molecular weight of a polymer and it is possible how we could do it. I had asked you to think about this the hint is right here in the last paragraph. If one considers a plot of pi A versus pi then we expect information on this plot to be useful in getting an estimate of molecular weight. One could consider the relevant surface equation of state and that allowing for the limiting area could be rewritten as the first equation over here pi A equal to k t plus pi A 0. If A 0 is constant then a plot of pi A versus pi should be a straight line and if one considers the limiting value for low surface pressures then for n moles capital N moles of the film matter pi A equal to n r t should be adequate right. So, at 25 degree centigrade we would have n r t calculated to be equal to 2478 into 10 to the power 7 n Ergs. This would be the experimentally found limit to which pi A tends when pi tends to 0 or the areas are very large. It is here that the intermolecular forces will not be important the molecules are far away apart. As I suggested last time for complex molecules like proteins we may prefer to work with the areas per milligram of material rather than the area based on one particular molecule. So, in that context we could say the molecular weight will be 10 to the power minus 3 divided by this capital N which is number of moles. So, if you combine these the asymptote is 2478 into 10 to the power 7 n Ergs. We could rewrite that as equal to y into 10 to the power 4 and we could substitute for the number of moles in terms of molecular weight this 10 to the power minus 3 by capital M or that simplifies to the molecular weight equal to 2478 by the asymptote or intercept y at 25 degree centigrade. Correspondingly at 20 degree centigrade the molecular weight will be 2438 by y. So, how well this would work? We can look at some data represented by lines over here hemoglobin or a certain synthetic copolymer and a natural polymer O L albumin. They would all give you an intercept in the limiting value of pi tending to 0 which could then be used for determination of molecular weight. All these films are at air water interface and the aqueous phase here is 100 normal HCl. So, this method has been used for determination of molecular weights for many proteins and often very low pressures are not really necessary because intra and intermolecular forces are relatively small. The only assumption required or made during reduction of the molecular weight here with that seen we move on to consider surface viscosity. Now, like bulk fluids flow within a surface may also be resisted. If you make a monolayer move in the surface just like bulk liquid will oppose its movement or flow by viscous forces. Similarly, the monolayer exhibits resistance to shear stress in the plane of surface and one may use this idea to define surface viscosity and to measure it. The viscosity of a monolayer may be measured by making it flow through a canal in the surface a method which we will see later on in this lecture or the viscosity of the monolayer could be measured by its drag on a ring in the surface or from the torque which is applied on a needle rotated in the surface. These are variety of situations which will try to quantify the resistance of the monolayer to the shear stress in the plane of the surface. I would ask you here for a moment to think about a rotating needle in the liquid. Imagine that we have water with a monolayer on top and we have a needle which can be made to rotate at any location related to the interface. So, if you start in the bulk of water and try to rotate the needle to get certain number of revolutions per minute you require a value of torque that could be measured. Now, if you shift the position of the needle closer to the interface, if it is far away from interface till we may not see any change, but when it comes close to the interface and it is within the interface or the surface layer, then the torque applied would definitely be greater and that will be a measure of the resistance that is offered by the monolayer in the surface. So, the viscosity so to say has increased from the ordinary bulk value to something probably higher, majorably higher within the surface space. The question is how do we define this surface viscosity and how do we relate it to the bulk viscosity and also obtain a feel for the magnitude of the surface viscosity in terms of some physical picture. So, first we come to the definition of surface viscosity. We define surface viscosity represented as eta sub s by the following word equation. The tangential force per centimeter of surface is equal to the surface viscosity times the rate of strain. One look at this and remembering what you might have studied in transport phenomena would bring to your mind the same form of the equation. We have a certain force here it is tangential force per centimeter, rate of strain and a proportionality constant right. So, is the shear stress rate of strain and the bulk viscosity in the bulk fluids case. The form is similar where the relevant shear stress in the surface tangential force per centimeter the rate of strain and surface viscosity. Surface viscosity is that constant of proportionality. This should be reconciled with what you have been seeing again and again. When we come from bulk to the surface we now think in terms of areas or a two dimensional extent of space rather than the 3D extent of bulk phases. So, correspondingly we would be one dimension short in the corresponding quantities. So, thinking about the units of surface viscosity you could check out quite easily from here that it works out to be grams per second. We define that as surface poise to distinguish it from the poise that is used for bulk phases and you would also remember that the bulk viscosity has the dimensions of mass per unit length per unit time. Whereas, since we talk of tangential force per centimeter of surface predictably the length dimension is missing over here. So, it is mass per time grams per second right. Sir, if you can change surface quantity always be higher. We will see, we will see. If there is a monolayer then the viscosity will be not only higher, but probably to an extent that I would like you to imagine right now will come to this. But even otherwise if there is a orientation in the surface the surface viscosity would be higher. What is the looking at the super busy not going to be then we can say like viscosity is subtracting between different layer. So, suppressing viscosity will arise above that and subtracting the monolayer with the surface. So, looking at this what happens when we define. You have to be careful. Now, when we are talking of surface phase it is a very limited space. So, you may not be you may still be able to think in terms of the layers, but it is like interface and the subjection length. To think of layers within the surface phase may be difficult because you cannot have anything smaller than length of a single molecule. So, imagining that there are layers within the chain length that would be not a valid argument right. No, but between this particular layer and the water underneath you are free to think of the layers within water there are layers. This has to be regarded as a layer of very small thing. So, now the relation between the surface viscosity and the bulk viscosity eta is the following. Surface viscosity by thickness of the surface phase which is about 1 nanometer this is equal to the bulk viscosity. So, if we have to translate a major surface viscosity to equivalent bulk viscosity this is what we should be using. It is an equivalent of surface viscosity as bulk value. If you plug in the magnitude of this d about 1 nanometer, you would immediately see what kind of bulk viscosities would we expect for measured surface viscosity. That will give you a feel as to how high this surface viscosity is in equivalent terms compared to let us say the bulk viscosity of water. So, it is clear that this factor we will play a part and a significant one. The range of surface viscosity is measured is typically 1 millisurface poise to about 1 surface poise 10 raise to minus 3 to 1 surface poise. Now, if we assume the thickness of the monolayer corresponding to this uniform surface viscosity in bulk terms it will translate to 10 to the power 4 to 10 to the power 7 poises. Now, what kind of substances in common occurrence or everyday experience will have viscosities like that? The film like this is more like butter or toffee is that viscous. The values do not appear to be large 10 to the power minus 3 to 1 surface poise. The bulk values are very high. This is again reminiscent of the fact that the vapor pressures concentrations in the surfaces etcetera were very large actually than the numerical magnitudes might otherwise imply. And why does this happen? Because there is a considerable orientation of the hydrocarbon chains and there is a considerable interaction among the oriented molecules. This is more of conveying the principle of science in tune with what we are doing. Test of all knowledge is experiment. Everything begins with experiments. We then imagine deduce and make guesses and check against experiments again. So, we may need to come to measurements of surface viscosities. We will look at one particular method very easy to imagine that canal method, but after a while. Before that I want to dwell upon the concept of drag in relation to a monolayer and the liquid on top of which this monolayer is present. So, drag of a monolayer on the underlying water or drag on the monolayer or drag exerted by the monolayer on the underlying water. These are the things that we would like to now consider. So, you can think of a picture where there is water and a monolayer and in one case we look at water moving under the monolayer and tending to drag it. In the other case the monolayer is made to move and it drags water underneath it. So, the picture shown on the next slide would clarify the situation. You can think of these two situations. Let us focus on first the drag of a monolayer on water. So, to the left of this line we look at surface of water with monolayer adsorbed and made to move with some velocity. If this surface layer is made to move this monolayer moves then the water under this layer will also be dragged along. So, we see here this is the primary thing the monolayer moves and it drags water underneath. So, it moves under it or it could be a situation which is just suppose it of this, but why is water moving under the monolayer? That is because water has the normal viscosity associated with it and the monolayer is not slipping over water. It is stuck into water. So, when it moves it drags water underneath. By and large this is in tune with overwhelming amounts of data in support of the no slip condition that you study in transport phenomena in chemical engineering. However, in interfacial systems there have been exceptions and examples where the validity of no slip condition may be questionable. For the time being we will restrict ourselves to the conventional picture. So, no slippage is the reason why monolayer drags the water under it. So, we could take a typical case of a monolayer of oleic acid which is moving between velocities of about 1 to 5 centimeter per second. It is possible to measure the thickness of water layer which is entrained by this movement of monolayer of oleic acid. It is about 30 microns and it is for a situation where the film moves over a barrier that is you have a barrier underneath the monolayer and this film moves. The magnitude 30 microns is independent of velocity of the film that range 1 to 5 centimeter per second displays the same thickness of water of 30 microns dragged by the monolayer. But what would happen if we were to increase the viscosity of the solution under the monolayer? The thickness of the aqueous layer carried will be higher and that thickness is higher in direct proportion to the viscosity of the solution. So, higher the viscosity higher will be the thickness of the solution dragged by the monolayer. Yeah, we will come to this. In order to measure the surface viscosity we will have to make a monolayer flow through a certain barrier in the surface and we will have to measure the resistance offered by this monolayer to the flow. If we can measure that resistance we will be able to get the viscosity. To measure the surface viscosity however, we will have to consider this strong possibility that there will be certain water layer underneath which is also carried and that means what is the resistance attributable to the bulk of water property of bulk water is now counted in the surface viscosity that would be a mistake. Besides that I want you to understand these two sets of conditions as two distinct things the implication of these situations where the monolayer moves makes water move underneath or water moves underneath and the monolayer is made to move will have impact on many observed phenomena. We will not be able to explain many observed phenomena unless we understand this. So, it is more in terms of understanding what happens and explaining our macroscopic measurements even in process equipment that is the significance of this and the principle is very simple. The surface has a different viscosity there is no slip. So, a layer of water is associated with this moving monolayer. Similarly, the reverse is also true if water moves then again there is no slip. So, the monolayer is made to move. Now, let us focus attention on this other part if the water moves underneath then the monolayer is made to move which means this surface molecules will be getting concentrated downstream. If there is a barrier or the extremity of your apparatus or a reservoir in large scale systems thus the surface molecules will tend to concentrate on the downstream side somewhere and which means that initial uniform concentration in the surface now will lead to certain gradient of the surface concentration that will have its effect. You are already able to envisage what I am going to say next, but before I return to that let me also cover a situation where there is no barrier underneath. It is a freely flowing monolayer. The experiments are difficult to perform measurements direct measurements are difficult to get here, but the situation is more like what you study in the boundary layer theory is more like a boundary at a solid surface with the superimposed velocity is equal to the monolayer velocity. So, if the monolayer moves and there is no barrier underneath then it is similar to a solid moving on top of a liquid surface with a fixed velocity. Velocity of that solid is the velocity of the monolayer here and on that assumption the thickness of the entrained water layer will work out to this 4.8 eta L by rho L v to the power half where the thickness of the water layer entrained is delta eta and rho L are the viscosity and density of the bulk liquid. This v is the velocity of the monolayer and L is the length of the flowing sheet of film. Now, let us return to that second picture when the liquid moves underneath the monolayer. That was when we have a barrier underneath for the flowing films with a barrier underneath that range of 1 to 5 centimeter per second exhibits no variation of that 30 microns. This is the case where there is no barrier there. It is a freely flowing film. So, it is a situation like if you have a solid surface on top of which liquid is flowing. You can regard the liquid to be stationary and the solid to be moving. You can calculate the boundary layer thickness. This is the same approach. So, it is a boundary layer thickness that we are getting here and it is a situation where there is no barrier under the monolayer. That is the difference. So, I already explained to you that when we have this liquid moving at sea, it will tend to make the monolayer molecules get concentrated somewhere downstream here. And when these surface molecules are compressed here, they would tend to exert a back pressure in the surface. So, what do you expect? Surface molecules will be sucked downstream in the surface by the viscous traction underneath and they create the back surface pressure. When the two get balanced, we should have no movement within the monolayer. Is it possible and is it possible to visualize this experimentally? You can do that by sprinkling ignited tal particles, fine particles and seeing whether they move at all after certain compression has occurred. You could actually find out that the back surface pressure when it balances the viscous traction, we will get tal particles not moving at all. We can get that kind of situation. This has an importance which we will come to later when we think about how waves are created, ripples and waves are created on large water bodies by the action of wind. There is a transfer momentum from wind to the water body and the monolayers can play a significant part in modifying the formation of these waves or ripples. Now, the first phenomenon, yeah. Some part of water is starting moving and some part of motor is starting moving and life is concentrated somewhere after 10 meters or so. It will apply a back pressure and then the ripples will go down below for the monolayer. But what about this will be, this will be in the way and the same which is responsible after 10 meters or so? Until the equality of the back surface pressure stress and viscous traction is achieved, there would not be stationary monolayer. So, what you will have is the monolayer will keep moving until the concentration downstream has reached sufficiently high value such that d pi by d l is equal to tau. It is at that location with that concentration, the magnitude of the surface pressure gradient that should match the viscous force. Only then, before this, before you come to that stage, the viscous force if it exceeds the surface pressure gradient, it will, then the equality is not there. So, the viscous force will dominate, it will tend to carry the monolayer. So, monolayer will get concentrated to sufficiently high value of surface concentration, surface pressure gradient to achieve the equality with this shear stress. Now, beyond that, it will remain stationary because it has come to the extremity. If you have moved the surface layer, let us say up to here, it cannot go beyond that and in this region, the d pi by d l can balance tau. So, there would not be any movement, but here there will be. If there is a batch only of the monolayer, then of course, there is nothing to move left upstream. So, if all the surfactant gets swept to one side, then you may have a region which is stationary. But on the other hand, as will happen in natural bodies, the impurities will keep coming. So, till that balance is reached, the impurities will keep coming and will keep getting swept to the other end. So, over period of time, what you may have is the stationary region may increase in thickness, depends on what is the total extent and total supply of the impurities. Now, so these two phenomena which we have just seen, effect of monolayer, the drag of monolayer on water or drag of water on monolayer these are important. First as I said, in the study or measurement of the viscosity of the monolayer or surface viscosity, we cannot ignore the, we cannot ignore the no slippage between the monolayer and the water underneath. It will appear as a correction, as will shortly see, in the context of the canal method, how that correction comes and what is the magnitude of that correction. But practically, this phenomenon is important in foam breaking and in amplification of eddies during mass transfer. During unstable or unsteady state mass transfer, we have monolayer dragging water which influences the amplification of eddies and foam breaking. This might not be very clear, maybe in words, but I will help you understand this. The common understanding would be when you whip a soap solution, you create foam or froth. Why is foam or froth stable? Why does not it break immediately? Why does not it break immediately? If you release an air bubble under pure water, the moment it comes to the surface, it will rupture. But if it is released in a soap solution, then it forms a bubble which remains stable. If you blow large number of bubbles, you create foam or froth, it does not break that easily. However, if you want to break this foam, you may just add a drop of propenol, isopropenol and you would see that the foam breaks, foam or froth breaks immediately. What could happen? The alcohol, although soluble with water, can act for a moment like a surfactant. It will be captured by the surface and because the alcohols, you know, the polar groups will mean very high spreading cohesion. Remember the spreading cohesion s values. For alcohols, it will be very high. So, this film of alcohol will spread very rapidly on the surface, pushing out all the stabilizing soap molecules or detergent molecules, creating a situation which is like pure water surface. So, there will not be any bubbles existing there. So, that is where the monolayer has dragged water underneath. The monolayer of alcohol has swept the surface molecules at a speed which is governed by the no slippage between that monolayer and water underneath. If they were complete slippage, that would have been even faster, but that does not happen. So, it is not that any alcohol will do the trick, but many of the lower alcohols will be good enough for this purpose. Similarly, you might have seen that in context of mass transfer at times, the mass transfer equations are lot higher than what you anticipate. This is because when you have a mass transfer, let us say from gas to liquid, if because of the flow filled within the liquid, there are eddies, they will aid the diffusion. On top of molecular diffusion, we will have the eddy diffusion and whether these eddies get amplified or not, will depend on whether they can reach right up to the surface and interact with the monolayer or not. So, that unexpected influence on mass transfer, unsteady state mass transfer is also explained by the drag exerted by monolayer on the water underneath. The second phenomenon is the basis for the other methods that we will probably look at in the next lecture or so. Those are viscous traction methods, wherein we make water underneath rotate and thereby exert drag on the monolayer. And that we measure. And this is of importance in how close the eddies can approach the surface and will have a direct implication on internal circulation, reduction of internal circulation or even prevention of internal circulation within moving liquid drops. You probably are aware of these pictures, the bubble rises through a liquid or a lighter oil drop when it is moving through water, it will have a toroidal circulation inside. And if there is a mass transfer across the gas liquid or liquid-liquid interface, the circulation will affect the mass transfer. However, if you think of very fine bubbles or very fine drops, drops as fine as may be about a micron in radius. And if there are any impurities which are surface active in our system, those while getting adsorbed, those after getting adsorbed on the liquid drops, because of this movement will tend to accumulate at the lower end of a rising drop. So, if the drop is very small, the concentration of the surfactant is so high that the entire drop is covered by this monolayer of the surfactant and that prevents the circulation of liquid within. So, the drops like that, very small drops would behave like rigid drops. They would not be internal movement possible within the drops. But if the drops are larger, then you can have a sort of a stationary cap at the lower end and a distribution of concentration, which is increasing towards this cap. So, they may be partial circulation within the drop. So, the extent of mass transfer that you will have will be considerably larger in these larger drops compared to the small drops. And the difference is the extent of the surface, which is kind of stationary or unable to move because of the rigidity imparted by the surface monolayer. So, these two phenomena are both important in their own right. We then come to insoluble monolayers at air water interface and we see how we could use the canal method to measure the surface viscosity. First thing to note here is by insoluble monolayer, we mean it is a spread film. We do not have these molecules entering the bulk of water. They remain in the surface. That is why we call them insoluble. How do we measure surface viscosity? The apparatus is a standard Langmu Trap. Half of it is maybe made to occupy with a monolayer chord material and we have the other half, which is clean water. The two compartments of the Langmu Trap is connected by a narrow canal in the surface. And then we make the monolayer flow through this canal. Now, a surface film in this trap will flow through this canal at a rate depending on what is the magnitude of surface pressure gradient. It will also depend on the width of the canal and length of the canal. It will depend on the viscosity of the monolayer and lastly the drag exerted by the monolayer on water underneath. Maybe I have to repeat this. Let me show you the picture. You got Langmu Trap. That is what this is showing and that is a canal here. This is a floating canal made of plastic or mica of width W and length, which is not shown L. And then we have two barriers H and J and we can make the monolayer move from this region to this region. This is the high surface pressure region. This is the low surface pressure region. So, under the influence of surface pressure gradient, we can make the monolayer flow through this canal to this low pressure region. The width, the length, the viscosity of the monolayer, the drag underneath, drag exerted on water underneath all will come into picture in determining the surface viscosity. The question will then arise, how do we apply this surface pressure gradient and if we have to apply measurements, if it is preferable to measure the flow under constant surface pressure gradient, we have to think of making adjustments within our apparatus. We might have to make the barriers move in order to keep the surface pressure gradient constant during the experiment. So, that is what is exactly done using this floating frame of plastic or mica with W as width. The monolayer is made to flow from the higher surface pressure on the right towards the left. And this difference in surface pressure delta pi is kept constant by moving these two barriers H and J during the experiment. We take a minute here to go back and contrast this method with the equivalent method for measuring viscosity for the bulk phases. You can think of whichever methods you have used to measure viscosities. The most common one is where we make the liquid move through a capillary and we find the time for a given volume of liquid to flow through the capillary. There the viscosity is opposing the motion of the fluid liquid and the flow rate will be related to the viscosity. Here, we do not have the three dimensions, we have two dimensions, we have a canal. So, the flow is within the surface and the opposition to this flow is by the surface viscosity of the monolayer. Once again we are one dimension short because we are talking surface phase. So, the flow rate is now not centimeter cube per second, but centimeter square per second. This is the flow rate of the film through the canal in centimeter square per second area swept per unit time. And it turns out this flow rate can be related to the applied surface pressure gradient delta pi by L and the width of the canal through this equation delta pi by L w cube by 12 eta s provided that this is a theoretical case and there is slippage between the film and water. I will leave the units for you to verify common units and this is theoretical result slippage complete slippage between the monolayer and the water underneath which is not true. We know that in reality there will always be the drag on water underneath. That has to be considered. So, we are talking of the first phenomenon in the study of viscosity of monolayers. We have to consider the drag on water underneath and the correction is actually quite large. Harkins actually made the measurements on the resistance correction and he found that the resistance actually increases by this factor 1 plus w eta 0 by 3.414 eta s. So, resistance is that much higher. This is when there is no resistance, no slippage. Resistance is only the resistance offered by the monolayer. But if there is a drag Harkins found that the resistance is greater in that ratio 1 plus w eta 0 by 3.14 eta s. So, this factor should appear in the denominator. That is what is shown in this equation q now becomes equal to delta pi w cube by 12 eta sl into this 1 plus w eta 0 by 3.14 eta s. There is a restriction on the use of this equation. This formula holds if the depth and length are much larger than the width of the canal. We can play with this equation little bit. We can rearrange it in this form 1 plus w eta 0 by 3.14 eta s. q is taken on the other side. The factor is taken over here. Then we multiply by eta s and leave eta s on the left hand side to arrive at the following form delta pi w cube by 12 q L minus w eta 0 by 3.14 this one. If we did not have this correction, this would have been the original theoretical result. Now, we can plug in some values w as 10th of a centimeter, 120th of a centimeter, 100th of a centimeter and the correction factor is 3.2 10 to the power minus 4 surface points, 1.6 10 raise to minus 4 surface points. And 0.3 into 10 raise to minus 4 surface points. So, it is clear that depending on the surface viscosity that actually we are measuring, this correction can become significant. If the surface viscosity itself is of the order of 10th of a milli surface points, then this correction is huge. So, depending on the surface viscosity, one will have to make a judgment whether we can use canal method or not. And with that observation, we will stop here for today. We will take up the discussion further from this point later.