 In this lecture 7th of the module 4 on monolayers, today we continue the discussion on damping first with clean surfaces and then later when the liquid surface is contaminated with a surface active agent. Then look at some more moduli characterizing the behavior of the interfacial layer and end up this lecture with the possibility of making fibers from the surface films. So first at clean surfaces, how does the wave damping occur? We are talking here of waves which are created through a linear source which could be like a rod which is made to oscillate at the surface of a liquid creating waves which are then tracked for their wavelength and amplitude. So starting with a scrupulously clean liquid surface by collecting data you would be able to find at different distances L measured from the line source the magnitudes of the amplitude. Amplitude being maximum at the source which is at value a naught and it turns out the amplitude goes exponentially down as you move away from the line source where the waves are created. And the damping coefficient could be theoretically deduced as we will see later to be related to the viscosity, the wavelength, the frequency and for short waves to surface tension for capillary waves. These two equations as we will see later are valid only for water and for capillary waves. Let us look at the structure of the damping coefficient here. The damping coefficient is higher if the liquid viscosity is higher you expect that. The damping coefficient will be higher when surface tension is lower. Once again you would kind of expect that because the surface tension would be lower for a good reason that would be through anything which can alter our very pure liquid system with traces of contamination which is sufficient to decrease the surface tension. Then to within that approximation that a very lightly contaminated liquid may still be treated in terms of a pure liquid system we will be able to see by lower surface tension would lead to higher damping coefficient. Obviously, the best scenario for all contaminated systems all impure liquid systems would be one wherein a monolayer is visualized and the monolayer could be either insoluble or spread film or it could be soluble or adsorb monolayer. We will discuss that separately but for the time being we just make note of the structure of this damping coefficient. Damping coefficient is given by 8 pi square eta by rho L Vg lambda square which is also equal to 16 pi square eta by 3 rho L f lambda cube and that is equal to 8 pi eta f by 3 gamma. We have made use of the relation between the group velocity and the frequency over here there is a factor 3 by 2 which is involved in that relation that is why this 8 becomes 16 and a 3 appears here and we have in place of Vg now f lambda raising the lambda power to the higher 1 3. I already explained the notations here A is the amplitude of wave at a distance L from the line source and delta C is the damping coefficient for clean system C is for clean system Vg is the group velocity of the wave train and f is the frequency eta is the liquid viscosity bulk viscosity. The group velocity of a wave is the velocity with which the overall shape of the waves amplitudes known as modulation or envelope of the wave propagates through space and you have to remember that this is different from the velocity at a local point of the wave. For example, imagine what happens if a stone is thrown into the middle of a very steel pond when the stone hits a surface circular pattern of waves will appear it would soon turn into a circular ring of waves with a quiescent center and the ever expanding ring of waves is the wave group within which one can discern individual wavelengths of different wavelengths traveling at different speeds. You know the longer wavelength waves would travel quicker those are the ones which arrive at the shore quicker the smaller ones travel slower. So for the whole group as such the longer wavelengths will be the ones which will be leading to higher velocities and they will constitute the leading edge of that circular ring and the smaller waves and the capillary waves will remain lagging behind some of them may die out. The envelope as a whole the whole ring the velocity with which it travels that is a group velocity those equations were actually true only for water and for capillary waves. Now we next come to the experimental measurements on the amplitudes I may ask you a question here if you have something like a Langmuir trough with water filled in very clean water and at one end you have a line source creating the waves, waves travel downward and you got to measure what is required in this theoretical prediction the amplitude at different lengths from the source. The question is if you have to do this experimentally what would you intend to measure how would you go about these measurements. The question is clear what we are talking of is a Langmuir trough we got a source here which is made to oscillate up and down and the water here would be having these waves travelling down. Now the question is how do we measure the amplitude for this wave at any distance say l so that we can talk about how a l would vary with distance that is our expectation at any distance l from the source the amplitude would be decaying exponentially. Question is how does one measure this may be you have to fall back on the basic physics of the problem. How would you measure that that is exactly the question I want you to think like an experiment list now how would you get an idea at any location at distance l what the amplitude is. Let us get our conditions right let us understand that everything is fixed the depth of water the purity of water the line source its amplitude a naught everything is fixed and let us say you allow any transients that might be there die out which means now you have a particular pattern for the waves decaying in amplitude as you go away to greater distances from the source. You have means of calculating or predicting from theory but if you have to measure it how would you measure it and I am willing to accept any possible approach think of anything. So, you will need a pressure transducer and then from there you will extract information about the depth of liquid above that point provided you get a transducer which can differentiate between these small differences in amplitudes that will require very high sensitivity. If you look at the ordinary pressure transducers they probably will be both limited by accuracy and sensitivity in terms of their time response because the waves are travelling right. So, at some point there is a crest next instance there is a trough. So, but it is a steady state then we will expect these patterns to be stationary in time independent of time. So, one of the nice ways of measuring these amplitudes are based on the principle of stroboscopic illumination. What you do is you shine light from top or at from bottom because clean water you could even shine light from bottom at different instances with a frequency. So, light comes on goes off and so on now if you can adjust the frequency of these on off cycles for the light source you will be able to match this frequency to the frequency with which the waves are travelling which means when it is dark you cannot see obviously when it is light you are able to see and you will see the same waves. And it will appear as if these patterns are stationary right now the same light you can use together with the reflection to find out the focal lengths especially of the troughs. You might require a translucent paper sheet which can be kept at different distances and whatever is transmitted or reflected where it becomes a focus you know what distance it is from the interface from there you get the amplitudes and you could even extract out the waves from the reflections right you can do the same thing with crest. So, that is a way to go about measuring the amplitude for the successive waves along the the trough there is some difficulties with water first how to get this grouplessly clean water free of contamination even slightest of the impurities can create major differences from the theory. One might suspect looking at the values of surface tension that surface tension has dropped very little, but that is no guarantee that they would not be significant damping of the ripples. And we saw reason taking this equation to its legitimate limits provided this level of impurities is able to alter surface tension only slightly we might still be able to visualize what happens. And we see here that because of this non-linearity over here the amplitude will decay very rapidly for even small changes in surface tension on account of slightest of impurities. You could use the same old techniques purify or clean up the surface by spreading or spraying ignited talc particles on the surface and taking off the surface layer with the help of a capillary attached to a pump. When you do that when the surface is really clean then the delta C calculated and the delta C measured for this system would agree well. We can look at the values over here for clean surfaces for water. We chosen a parameter here frequency when the frequency is varied from 50 to about 920 the observed damping question goes up from 0.055 to 1.08. Calculated values follow closely these observed values. For ethanol we have only one a single data point here for a frequency of 100 there is a reasonably good agreement between the damping question measured and calculated. We began the discussion of surface waves with that question as to how the floating objects would appear for an observer who is also floating. You see that these consequences of incompressibility water would lead to circular movement of water particles because on the whole water is incompressible and the circular motion dies out as you go away from the surface at some depth it will vanish. If you had the bottom pretty close to the surface the damping of these movements would be quicker. But this is not the damping we are talking of. This was only in passing to show you the actual movement of fluid particles, water particles when the waves are passing through as a result of incompressibility. This is the same values damping questions plotted against frequency and these are the observed values. They fit very well to a straight line that would be anticipated from the theory. Many interesting departures from this clean system behavior would be expected once you have surfactants in the system and as an example we will mostly concentrate on systems which contain a single surfactant but not limited to only single component impurities surface active impurities. We will look at some data for mixtures. Now look at the range of concentrations where you will get to see the influence on damping because of these surface impurities. Concentrations as low as 10 micromolar to about 1 millimolar level of concentration of surfactant can create significant deviations in the damping behavior. To distinguish for these impure systems the damping coefficient is given subscript I. So, delta I is the damping coefficient for contaminated systems where we have an insoluble monolayer in the limit which could completely immobilize the interface. Once you have this surface layer or insoluble monolayer or spread monolayer you might have enough damping to permit no movement within the surface phase. This could happen at quite low concentrations and the calculation will be based on relevant governing equations of hydrodynamics written for the near surface behavior, near surface region and equating velocities at all points on surface is to 0 we could infer the delta I values. We will simply look at the result of this analysis delta I comes out to be equal to pi by Vg lambda which is the same group velocity as earlier eta sigma by 2 rho l to the power 1 by 2 where eta is the liquid viscosity sigma is this combination which has contributions from gravity as well as surface tension in particular it is 2 pi g by lambda plus 8 pi cube gamma by lambda cube rho l to the power 1 by 2. We could compare the damping coefficients for the two systems the one where the surface is immobilized through an insoluble monolayer and the other that we have seen formally for a clean system. So, we divide this delta I by the older delta C which was 8 pi square eta by rho l Vg lambda square which simplifies to this expression over here a factor containing a linear dependence on the wavelength and square root dependence on sigma containing gravity and surface tension contributions by the kinematic viscosity. It will take a while before you realize what range of complex behaviors could arise out of the surface active material in the surface layer. Next, the magnitude of this ratio for capillary waves with wavelength half a centimeter the ratio of delta I to delta C can be between 2 and 3. So, it is a marked dependence depending on presence of these impurities. What is the physical mechanism for immobilization? While a wave is passing through we have two factors coming into picture one the surface viscosity. I had mentioned that surface viscosity for especially the surface active material present in the surface could lead to behavior like butter or toffee. So, that in in term tells you how much will be the opposition to any movement in the surface just on account of viscosity. So, that is the first factor of course which is causing immobilization or contributing to immobilization. The second factor is the surface incompressibility. The surface incompressibility by now you would recognize would be related to the surface compressional modulus or Cs inverse which in turn is a measure of the surface incompressibility or rigidity of the surface. So, these two factors surface viscosity for the surface phase and the surface incompressibility would contribute to immobilization. And this equation over here has actually been derived as I said by equating surface velocities at all points to 0 in the surface. This is the consideration about the magnitudes. In presence of monolayers for water with spread film of a long chain acid like docosoc acid or of proteins or a shorter acid like oleic acid. The observed damping equations are in the range from 0.17 to 0.81 and they are also predictable from the theory the equations that we have just seen. The last example is again for water, but with a mixture of surfactant. Equations like Lorel sulphate and dodecanol about 100 molar Lorel sulphate and 90 micro molar dodecanol at a frequency of 150 can give you a damping cohesion about 0.67. What would be the scenario if we had to consider other kinds of monolayers? If the monolayers were made of material which would exhibit tendency to adsorb or desorb. So, we are going away from insoluble or spread films to those which are capable of adsorbing or desorbing depending on the surface pressure and availability of surfactant in the subjacent water and all this could happen during the passage of wave and why would this happen. First you have to recognize that when you have these troughs and crest then this shape has emerged from a planar interface to begin with and if you say that initially the concentration at the surface is uniform and then you have this wave passing through then we would be able to see that where the troughs are there they will be expansion with the crests are there there is a compression which will mean that these molecules which are at the surface and capable of leaving the surface for bulk would show a behavior dependent on what is the surface pressure. So, at the crest when the surface pressure gets higher by compression it is possible that some of the adsorb molecules might be ejected into the bulk. Whereas, in the troughs where the concentration drops by expansion they could be adsorption. What is happening here is a kind of short circuiting you are not permitting the surface to attain its state of compression or surface pressure history that would be present if there is no degree of freedom for either adsorption or desorption. What it means now is that while the surface wave is passing some of the molecules can actually leave the surface going to the bulk others will be leaving the bulk occupying the surface. This is not possible if you had insoluble film ok so that means you are actually short circuiting the surface stresses you do not allow the compression to raise the surface pressure to that higher value as would be possible if adsorption desorption were ruled out. In the same way at the troughs we will have corresponding deviation from the lower values of surface pressure. There is another consequence of this initially if the surface is planar surface pressure is everywhere same. As the wave passes you have the crest where the surface pressure is higher troughs where the surface pressure is lower. So, there are now these fluctuations in surface pressure created because of passage of wave. Besides the ejection from the compressed region into the bulk and adsorption in the trough region you have one more thing within a wave along the surface there is a higher pressure at the crest lower pressure at the trough that will induce movement within the surface. So you have now some energy expenditure because of the flows in the surface created by fluctuations in surface pressure ok. So apart from the compressional resistance offered by CS inverse or the surface viscosity you have these additional mechanisms for damping the waves or for dissipating the energy which is propagating in the form of waves. And therefore delta damping cohesion measured for these soluble monolayers or adsorb monolayers will be lower than the delta for insoluble monolayers otherwise the surface will may resemble the same chemistry. This brings us to the large waves on a rough sea. We had seen earlier that from Pliny's record disturb sea stormy sea was made relatively harmless by pouring oil on top of surface of such stormy sea with large waves. How is that done? We have some idea but let us look at this picture again. What would happen if we have a monolayer present in waters where we have these large waves? I will probably show you some photographs later that this is not just a theoretical argument but you can witness this in practice. The surface of a large pond of water the kind of waves you would be able to see with and without impurities will look very different when there are no impurities you see the whole surface to be relatively rough. There are lots of these short waves on top of the long waves but if you add a impurity it could be like setyl alcohol or insoluble monolayers like oils. You see that all the small scale waves get eliminated you only see a swell of large waves. The smooth surface that you have now will mean that corresponding lesser amount of energy will be transferred from action of wind to water thereby influencing both how the energy is dissipated and how the waves behave. Should we have large waves, long wavelengths for clean water you would get very rough surface because on top of these long waves will be riding the short wave. I had shown that picture in one of the diagrams earlier. If I show a large wave for a monolayer covered surface the large wave will appear quite smooth if you do not have the impurities or these monolayers but only from distance. If you were to look at it closer you would see in absence of impurities the surface to be quite rough because you have lots of these near a centimeter or sub centimeter waves on top of these. These are the ones which are causing roughness and therefore the amount of energy transferred from wind to this wave will be a lot higher. On the other hand if you have an insoluble monolayer as an extreme all these small waves are eliminated you get a smooth surface. So the amount which is actually of energy which is transferred will be lesser. If lesser amount of energy is put into these waves the tendency of these waves to break off from the crest will be lesser. This is not going to just come off from the large wave and give that energy over here. So that is the way you can actually minimize the transfer of energy from wind to the large waves and you can also minimize or reduce their tendency to break at the crest. Drag coefficient for smoother surface will be correspondingly smaller because now the ripples are either eliminated or damped out or they might even form in lesser number. Surface pressure gradient is another factor. Surface pressure gradient can be considerable and that will mean that you can have the surfaces coming to near stationary state on account of the back surface pressure stress. There could be regions where the shear stress is exactly balanced by dou pi by dou L and therefore we will not have movement within the liquid allowing transfer from the action of wind to the waves and what kind of materials could be used for this purpose. I already named sattile alcohol because that was the material we had thought of very promising compound for reducing evaporation. At a later date we will return to sattile alcohol to look at how it will reduce the transfer of oxygen from air to the water bodies. Fortunately, it turns out that the same material is able to retard evaporation very significantly but is not able to cut down the transfer of oxygen to any alarming level. So that is why this compound which spreads readily on natural bodies of water surfaces is also friendly for the aquatic fauna and flora. Other materials could be these natural materials like sail fat and blubber which is a mixture of fatty acids and triglycerides again very effective in damping the waves and compounds like kerosene, phenol and long chain acetamides are comparatively less effective. Do not be confused at what information is contained over here. In the background of how I explained the effect of back surface stress in countering the effect of wind drag, here I am stating that it is possible for slicks of hexadecannol to accelerate downstream on a lake surface. Basically if you have the action of wind then it is possible that the drag actually pushes the impurities downstream in the process it is opposed greater to a greater extent by the surface pressure gradient. But if the surface pressure gradient does not have a portion to develop to a large value its effect will not be perceived. So the drag will continue to add energy to the surface layer carrying it downstream faster and faster but it cannot be indefinite maybe this kind of behavior you would expect to see near the central part of the lake. When the region close to the shore is considered there is no opportunity for the surface active impurities to go beyond the boundary of this lake and there dou pi by dou L can increase to sufficiently large value to be equal to tau and therefore it can counter the action of wind drag completely. Once that happens then you will have immobile liquid surface unable to move. There would not be any movement of water on account of wind action underneath. Perhaps we can stop here for today.