 In this lecture, we will learn about non-wetting. What is the importance of non-wetting? So, to come to that part, we need to see how wetting can really deteriorate the properties of a material. So, in case we see like if you are drawing a vehicle, it suddenly starts pouring heavily. Then basically, it becomes very difficult to see what is happening outside or let us take an example of any material, say any metallic material. It is lying on a particular ground exposed to the environment and if it starts raining and if rain is basically wetting the material, it basically goes on chemically combining with the material and degrading the material. So, that is the reason we see that the importance of non-wetting in such engineering applications. So, non-wetting is also one of the very, becomes a very important criteria in engineering it. So, let us come back to the first example, rain, rain go away. So, if you are driving a vehicle and we see that the rain starts drizzling onto the wind shield. We see that the droplets of rain, they remain on the wind shield unless you start your wipers. But again, once you even once you are starting with the wipers and even when this droplets, they trickle down, they leave a mark, a mark of what is the edges of the water. So, this means that there is some sort of a wetting occurring with the surface of a material and that leads to the poor visibility and at some times it becomes so difficult that we cannot even see. So, overall it becomes so shady, we can also see some fogging which can occur on the wind shield and overall it is basically hindering our visibility while we are driving and because of that, we are not able to drive properly, it might lead to even accident and extended to that like we even certain sometimes we get fogging on the glasses and we sometimes get corrosion on some metallic interfaces. We might get some sticky chemicals on a particular surface and that leads to again much more wetting. So, certain certain things which can basically deteriorate the overall engineering application of a material. So, we want to go for engineering in what is nano non wetting. So, coming to the something called hydrophobic surfaces, but we here I have mentioned super hydrophobic surfaces. So, basically we have seen that in lotus leaf, if we have a lotus leaf, water will basically come trickling down to every location and it will form a droplet and this droplet will be totally spherical and it would not be wetting at all. So, we get a non wetting surface and that is that can basically utilize for say self cleaning windshields or providing lubrication of micro and of the electronic mechanical systems. And since we have water it is it is it becomes very friendly surface for contaminants or falling agents. So, once we have a super hydrophobic surface, it means it is not reacting with any water to get it that it wet it surface. We can develop non bio falling surfaces and also ultimately it can also lead to water repellent glass. So, this idea is basically coming from something called lotus leaf. So, we can say that nature is simple, but it is science, but it science is not, because there is so there is some level of complexity which is associated with the such structures in imparting hydrophobicity or non wetting. So, we do not have hydrophobicity, but we have super hydrophobicity which is associated with the lotus leaf. So, we can say that the science which is dictating this super hydrophobic nature, it is much more complex than what it appears on surface. Now, coming to what is wetting and non wetting, we see if we have water droplet say if we have a rough surface and then we have certain surface and we see that a particular droplet once it is sitting on to it, it might assume the shape while filling in those cavities. So, we have a water droplet which is basically over also taking care of filling the voids or the filling the rough grooves into the material. So, we have certain things like this and that will show certain angle. So, we have this thing this particular state we have liquid, we have a solid and this particular angle is called Wenzel state, this state is called Wenzel state WENZEL sorry this is called Wenzel state and in another case it might happen that a droplet is assuming the similar shape, but in this particular case it is not filling in those cavities. So, these cavities are not filled and in this particular case we get certain wetting angle and this wetting angle is called Cassie back star wetting angle. So, we have Cassie back star. So, we have two states Cassie back star state and Wenzel state and Wenzel state we have the droplet also filling in the grooves of the rough surfaces present Cassie back star we had this liquid and we had this solid, but the grooves of the solid they are not being filled by the liquid. So, there is some sort of a non wetting which is occurring already at this particular level. So, we have gas interrupt in this two locations. So, considering the overall wetting of this particular droplet we had fraction which stays in air and we have certain fraction which is sitting on the surface. So, while it is in air the contact angle is 180 degrees, because it is not at all wetting the surface in this particular case the liquid is not drop wetting the surface at all. So, we have contact angle of around 180 degree out here. So, once we have cos theta is equal to 180 we get minus sign. So, minus 1. So, minus f 1 plus f 2 cos theta 2 also we know that f 1 plus f 2 it is a fraction which is sitting on the air bubble and second fraction which is on the solid region this is also equal to 1. So, we get f 1 is equal to 1 minus f 2 and we can replace that part out here in this particular equation. So, we get the overall theta value or the overall theta or the wetting angle is minus 1 plus f 2 cos theta 2 plus 1. So, this is how we are defining the when the water droplet is staying in air and when the water droplet is wetting the surface the contact angle is nothing but 0. So, from that particular part we can say cos theta is equal to 1 cos theta is equal to 0 we get this value is 1 that is what we are able to see here plus f 2 cos 2 theta 2. So, once the water droplet is wetting it we are getting certain equation once it is not wetting the surface we are getting a certain equation. And obviously, in this particular case we will have theta much more as compared to the wetting state because lower theta means the surface is getting wet getting wet to a larger extent. Because once we have theta is equal to 0 it means the overall water fill or any liquid fill means wetting the surface completely and when we have theta is equal to 180 degree it means it is not wetting the surface at all. So, once we had air bubble theta was 180 and in case we have complete wetting we are seeing theta is equal to 0. So, obviously this theta which is being defined here it will lead to this one can easily define what is the overall wetting which is occurring in the occurring on the surface of a material. But there is one more aspect to it that the true contact angle the measured contact angle can be different from the true contact angle. And this measured contact angle is basically given by also a term called roughness it is the ratio between the surface and projected area. So, if we have something some surface which has a much more higher surface area and the projected area is much more like this we can see that this much area is getting reduced to this much area. So, there is some effect of roughness as well which is given by this particular term R. So, water droplet is interacting with this particular surface it is interacting with much more surface and the projected area is much lesser. So, we can see there can be difference between the true contact angle and the measured contact angle. And that part is being given by the roughness or the ratio between the surface and projected area. And this term basically remains the same which defines the overall wetting from the Young's equation. And roughness can promote either wettability or non wettability it totally depends on what the original surfaces. The original surfaces basically hydrophobic in nature roughness will induce much more hydrophobicity to it. And if the original surface is hydrophilic in nature it means it is tending to wet the surface roughness will have pronounced effect of much more wetting of the surface. So, that is what is defined by the roughness factor and that basically depends on the chemical nature of the substrate. And we already saw that Wenzel droplet will adhere strongly to the substrate, it is cassive extra it will tend to be non wetting the surface. So, we can see those two particular aspects that roughness can have a very drastic effect on what is happening at the surface level. Because roughness is the one which will keep affecting the inherent nature of the surface itself. So, non wetting will be enhanced if the surface itself was non wetting in nature or if the surface were much more hydrophilic or hydrophobic in nature then the roughness will lead to super hydrophobicity. And again this thing again comes from the lotus leaf and lotus leaf is a symbol of divine beauty. And as we see that the unfolding of the petals it basically corresponds to the expansion of the soul. So, it had much more spiritual meaning as well in the ancient history that lotus basically grows and ponds though it dwells on the dirty pond it lacks any contamination it is free from bacteria just because of its super hydrophobic nature. So, that basically is correlated with the divinity or the divine beauty or the growth spiritual growth and it also has self-cleaning properties. So, we have seen Buddha often sitting on the lotus leaf this is the lotus leaf and then it is the lotus around him and that leads to the spiritual growth as well. So, that is what out here and how does this lotus leaf will repel water? First of all the leaf itself has some chemical composition which is vaccine nature. So, it is there is something called epicuticular wax which is much more vaccine nature and additionally it has two levels of microstructures. First it has some micro scale bumps. So, it is inducing some sort of a micro roughness and as we see as we have seen that if we have a much more roughness or we have increase in the surface area the apparent contact angle will be much more higher. So, the true contact angle may be remain same the apparent contact angle will be much more high. Additionally this is the first level of roughness and additionally over those micro production itself we have something called nano hair like structure. So, we have nano scale roughness also arising which has a which is a on addition to that of a micro roughness. So, we have micro level roughness on the lotus leaf surface over that we have some sort of a nano roughness or nano hair like structure. So, we are getting two orders of magnitude two orders two lens scales of roughness being induced on the surface of a water. So, how does this cleaning effect really happen is that we have say if we have some impurities on the surface of a lotus leaf and if we have a droplet droplet will be simply roll will keep rolling on the on the on the surface of the lotus leaf. And eventually it will also capture sorry it will also capture those particles along with on surface while rolling. So, it is rolling once it is once it is encountering this particular impurity it will just tag it along with itself while rolling. So, we have this water droplet kept which which keeps rolling and it overall it keeps taking the impurities along with it. And in this particular case we have passive extra straight because the water droplet we can see how easily it can roll on the surface of lotus leaf. So, because of that we can see that the it there is much very less friction of a water droplet to roll on the lotus leaf surface even an angle of 1 2 3 degrees is enough for the water droplet to roll without wetting the surface. And in this process it also carries along the impurities which are existing on the surface of the lotus leaf. So, it is showing a very huge contact angle is the order of more than 160 degrees with the lotus leaf. So, this this is the liquid and this is the lotus leaf and we see contact angles in excess of 160 degrees with the which form which have formed with the lotus leaf. And again the raindrops also form very high contact angle which are which is much greater than 90 degrees. And this also requires that the surface of the substrate should have roughness on two length scales as we saw that lotus leaf has roughness on the order of micrometer and again some roughness on the nanometer scale as well. And we see that there are some micro protrusions which were which are dominant on the surface of the lotus leaf and they are much smaller than the raindrop droplets. So, it means that they also have to be soft enough that they do not do not puncture the raindrop and also they have to be close enough. So, that they can really let the water droplet sit on it. So, we have certain micro protrusions which are scattered throughout on the surface and once water droplet is sitting it will get support from many many of such micro protrusions. So, this is the water droplet this is there are certain supports which are which are spread 5 to 10 microns apart with a diameter of around 5 to 10 microns itself. And this one does not allow the water droplet to touch the surface. So, surface of lotus leaf remains untouched by the water droplet. And now this what this micro protrusions have to be spaced in space properly also they have to be balanced properly. So, that water will contact the surface and will just roll off without wetting the surface. So, we can see that the cassive extra state is much more predominant here. We need to induce the cassive extra state because of the roughness roughness parameter out here that we do not allow the water drop the lotus leaf surface does not allow the water droplet to touch the surface of lotus leaf. And the micro protrusions help to just balance the water droplet and it allows to it to roll it off very easily. And lotus leaf is also lotus leaf is also called lotus is also called lulumbium lutea. And lotus leaf has small small bumps which are around 5 microns in diameter around 5 to 10 microns in meters in height. And they are spread approximately 10 to 15 microns apart. So, since they are apart they can easily let the water droplet sit on to it. And then at the same time not allowing it to puncture or buckle enough that it touches the surface. So, this is a surface of lotus leaf and these are the micro bumps out here. And this is the water droplet. Since this micro protrusions of 5 to 10 micrometers will have a water droplet which is to order of 2 to 3 millimeter in diameter. So, we can see that they can many number of micro protrusions which will allow the water droplet to sit on to them. And again over the micro protrusions we also see that lotus leaf has some nano scale roughness which is basically covered with some very fine nano hairs which are around 100 nanometers in diameter. So, it is also observed that micro roughness can induce surface area increase by order of 10,000 times. And similarly going from micro to nano we can again see a factor of 10 to the power 3 increase in the surface area. So, that is being that is being generated because of the two levels of micro roughnesses micro and then nano roughnesses which are predominant in the lotus leaf structure. And seeing coming to the micro structure of the protrusions we see we have a lotus leaf surface and we see we start seeing certain micro protrusions on here. And this micro protrusions of 5 to 10 micrometers in diameter has so much their height and they are spread about 10 to 15 micrometers apart. And again if we enlarge this particular thing we will see those particular structures that it appears more like a globule which is which will have a diameter of around say 5 to 10 around 5 micrometers height of around say 5 to 10 microns. And it will be spread it approximately 10 to 15 micrometers from nearby micro protrusions. And if we again zoom this particular part this particular micro protrusions part we will see that all this micro protrusions will have very very fine nano hairs just spread it throughout the surface of the material of the protrusions. As well we see that the base part of it the base of the lotus leaf also has some nano hairs on to the base hairs as well. So, in the base as well as on the protrusions we see some nano hairs, but there is one catch to it that the overall nature of this hairs at protrusions micro protrusions and at the base they are little different in nature. So, how they are different we will see them as we go further, but there is a little bit of difference in the protrusions in the base hairs. Base hairs tend to be much more they tend to be more rigid as compared to the protrusions hairs much more they are much more flexible also because they are sitting the protrusions hairs are sitting on a particular micro protrusions. So, what is happening we are seeing dual level of flexibility because nano on the protrusions hairs they are much more finer they are much longer. So, their diameter is much finer their lengths are little longer. So, they easily undergo buckling and over that they are sitting on the micro protrusions whereas base hairs they are much more they are they are little thicker in diameter and little shorter in the length. So, they are much more rigid as compared to the protrusions hairs and the same time they are there is no second level of protrusions on which they are sitting. So, they tend to be much more rigid than the protrusions hairs. So, we can see that the base hairs they tend to be much more thicker and much more shorter whereas surface hairs they tend to be much more longer and much more thinner. So, that is what is basically providing much more flexibility to the surface hairs which are sitting on the protrusions whereas base hairs they are they tend to be much more thicker. There might be a minor difference in their diameter if this is around 100 nanometer this can go up to say around 140, 150 nanometer and so on. But again there is some difference because again there is dual level of flexibility which arises from the that we have a micro protrusions over that there is some nano hairs and in this case we have a flat leaf surface and over that we have a nano hair. So, that means out the difference in terms of the morphology of nano hairs which are sitting at the protrusions they are little different than the hairs which are present on the base. So, we have base hairs and then we have hairs which are sitting on the protrusions. So, this is how they are little different in morphology. And again to differentiate what is happening because of the two level of roughness Michigan team they had found out that the self learning property that is being performed on a lotus leaf while removing the nano hairs. So, what they have done they have taken the same lotus leaf surface and then from that they have removed the nano hairs. So, we have lotus leaf surface by some surface treatment they removed the nano hairs and they are just remaining with the micro roughness on to the particular lotus leaf surface. And in third case they have taken out the both levels of roughness. So, they are keeping the same material, but now nano hairs and micro protrusions they both are gone from the surface. And from that they have realized that if we if they just have the epicutical wax on the surface the contact angle is to the order of 74 degrees only. Whereas if they go only for nano hairs only for the micro protrusions which are sitting on the surface the contact angle is to the order of 126 degrees. And with the nano hairs the contact angle is increased to 142 degrees. So, we had one first is one surface with epicutical wax as the similar chemistry of that of lotus leaf they are seeing a certain contact angle of around 74. Once we have certain micro protrusions on the surface the contact angle basically increases to the 126 degrees. And once they also have some nano hairs on to it the contact angle is gone to 142 degrees. So, it just means that the nano hairs are responsible for the additional 16 degrees of wetting that the surface is imparting. So, we have this epicutical wax it is imparting 74 degrees. And once the nano scale here they are being melted away and while retaining the wax composition. And without any chemical change while retaining the micron roughness we are getting a contact angle of 126 degrees. And the initial contact angle of the lotus leaf was around 142 degrees. So, that has the overall role of dual level microstructure in terms of imparting hydrophobicity to the lotus leaf. And again as I stated earlier that the additional 16 degrees is coming out totally from the nano hairs which is predominant on the lotus leaf surface. And again this is itself is responsible for the rolling behavior of the drops. Because now the water droplet is undergoing a cassive extra state. And now it because very easy for the droplet to destick itself from the surface. Because there are all the cavities which are to the order of nano or nano or micro roughness. Water is not getting entrapped in those particular cavities. So, it is very easy for the water droplet to roll rather than destick itself. So, rolling becomes much more easier because the lotus leaf is predominant as a cassive extra state. So, the wetting is mostly in the cassive extra state. And rolling becomes much more easier because of the dual level microstructure. So, this particular part was more like this that in one case they had all the micro protrusions with nano hairs intact on them. So, that was the surface which was appearing in one. And in second case they had all the micro protrusions but without any nano hairs. And this one is what with nano hairs. So, in this particular case they saw a contact angle of around 142 degrees. In this case they saw a contact angle of 126 degrees. So, that is how the surface chemistry or the surface morphology of because of nano hairs can alter the overall contact angle of the lotus leaf. So, untreated lotus leaf with retain while retaining the micro nano structure it is showing 142 degrees. Whereas, the lotus if the lotus leaf is being annealed. So, that nano hairs have been melted away with significant contact angle of around 126 degrees. So, that is basically essentially telling us that the rolling tendency of the water droplet is it gets damaged in this particular case while it is being retained in this particular case. And again therefore, the secondary roughness is important in this particular case. And so is the mechanical property of the nano hairs to hold the water droplet without wetting the surface. So, apart from the roughness part we also want to see the mechanically how strong the nano hairs are. Because if the nano hairs start to buckle then again it will lead to the water droplet touching the surface and it will eventually lead to wetting. So, the nano hairs which are sitting on the surface they also have to be equally strong enough. So, that they do not buckle and they then they continue supporting the water droplet on the surface without wetting the surface. So, we had this particular state and then the passive extra state. So, we have water droplet sitting out here without wetting the surface without wetting the pores out here. In second case we have water droplet which is being again considering which is again wetting the wetting our surface. So, we had certain contact angle of around the vessel state and then again we had certain something on passive extra state. So, this is what is being attained out here. This is the liquid solid surface and again the nano hairs which are sitting on the surface they also should be able to balance the water droplet without bending. So, this is the nano hairs they do not have to buckle enough and the same time they have to support the liquid droplet which is being supported on to them. Because if you have a protrusion if you have a particular rain droplet which is falling on the lotus leaf surface it comes in contact with the nano hairs of protrusion first and if hairs which are lying on the protrusion they tend to they start buckling then there is no way that rain droplet will remain hanging on the surface without wetting the surface. So, the nano hairs they have to be strong enough to be able to support the water droplet and coming to the overall morphology the if say if we have the length part. So, we did see that the length of the protrusions it was little longer whereas the base hairs the length overall overall overall let us see again this part that if we have a protrusion and then we have base hairs the overall diameter was little longer and again the base here also have their little little longer. But the overall aspect ratio for this this is for the base and this for the protrusions this again for the protrusions and this for the base. So, we do see that the this is kind of frequency frequency we are seeing that the protrusion hairs they are they are little shorter in length as well they are shorter in diameter. But the overall aspect ratio is little different for both of them base hairs the length is around 400 to 1000 nanometer protrusions here they are around 300 to 600 nanometer whereas protrusions hairs are little slender their diameter is around 5200 nanometer whereas base hairs have a diameter of 50 to 130 nanometer. So, base hairs are longer and base hairs are thicker as well the overall thickness part basically dominates in the base hairs that makes them much more stiffer. And coming to the nano indentation of nano hairs if we indent a particular surface all this nano hairs will form will buckle. So, if we had as particular nano hair and if we tend to indentate using a nano indentor this one will basically buckle that this will basically buckle and from the low depth curve we can say which one has a higher Young's modulus. So, if we see particular low depth response for a base hair the protrusion hair will go something like this as I said earlier it is much more flexible the protrusion hair. So, we have load and then we have depth and if this is being done by an nano indentation then we will see that the protrusion hair undergo too much of deflection while the indentation is being going on. So, this happens because of the dual level of flexibility which is associated with the protrusion nano hairs. What is happening we have a micro protrusion which is much more again organic in nature over that we have a nano hair. So, once the indentation is stretching the nano hair nano hair itself will buckle at the same time the micro protrusion which is which is holding the nano hair that also will buckle. So, we will get two levels of buckling in case of a nano hair which is sitting on the protrusion micro protrusion. Whereas if we have a base hair base hair as a very solid leaf surface and we have very thicker or much more a stiffer nano hair on to it. So, if the base hair is being indented it will resist to a very large extent because we have a very stiffer leaf surface and a very stiffer nano hair with much larger diameter. So, that one can resist load deflection to a large extent. So, we see that the base hair undergoes a very low depth whereas our protrusion hair undergoes very larger depth. So, we see a pH curve or the load depth curve it is much more steeper for the base hair whereas it is very compliant for the protrusion hair and from this data we realize that the overall Young's modulus of a base hair is around 867 mega Pascal plus minus 100 and protrusion hair it is very very low it is even half as compared to that of a base hair. So, it is doubly flexible and therefore, it can get deflected by it can get deflected easily by the indented tip. So, it is highly compliant and it undergoes very high deflection once it is being indented by a nano indented and consequently fluid dynamics modeling has also been done. So, we see that a smoother surface without any roughness will tend to have the say this is the extension of a fluid one quarter of a surface and this is a water droplet this is a water droplet one quarter of the water droplet. We can see that the front of the water droplet which is flowing it keeps going without breakage without breaking in case of a smooth surface. Whereas if we start increasing the roughness we see that this particular part starts spreading out or may be starts agglomerating and starts splitting up in this particular rough surface it means that this much better wetting in smooth surface good wetting whereas as we have increased roughness this basically leads to good non wetting. So, we had this water droplet which was flowing in this direction starting from here. So, we have only one quarter of the water droplet and we see the water front starts splitting up as we have much more as the time progresses. So, with the same kind of roughness as that is available on the lotus leaf surface. So, assumptions there are certain assumptions being made for this particular type of modeling that water droplet diameter is around 2 to 3 nanometer and a contact angle of around 165 is being made with the lotus leaf surface that brings out that the contact radius which is basically supporting the water droplet is to the order of 125 to 200 micrometers. So, the contact radius at the lotus leaf surface with this type of wetting it is only 125 to 200 micrometer and ignoring that the water droplet with sag this particular radius will lead to an area of 0.049 to 0.126 micrometer square and then the simulation has been carried out using a overall wetting area of around 0.5 micrometer square. Since this one is huge then the actual what will happen. So, we can say that the assumptions are valid for this type of simulation. So, we had an area of 0.50 which is much greater than 0.126. So, this one can take care of the wetting effects which are being rendered by the water droplet. So, things are in order for this particular assumption and we see that we have maintained three kinds of roughness. In one case we have utilized 10 micrometer diameter which is being separated by a 10 micrometer of distance and all these pillars are something like 10 micrometer in height. So, we have 10 micron, 10 micron, 10 micron out here a droplet size of around 10 microns again a 10 micron of pillars 10 micrometer apart. In second case what we have tried we tried to have the distance between the two protrusions separated by 30 micrometer. So, in second case we have 10 micron size 30 microns apart again 10 micrometer of height and in third case what we tried we had utilized a 10 micrometer while remaining the 10 micrometer of distance, but increasing the height by 30 microns. So, in this case we have the protrusions are apart in this case we have the normal distance between the protrusions. In this case we have height of protrusions to the order of 30 microns. So, in this particular case once we have protrusions put far apart we can see some sort of a sagging effect. In this case we can see something called a pinching effect in this case we call something called sagging effect we can see. So, how do they really govern the overall wetting in this case we see a particular lotus leaf surface these are all the protrusions micro protrusions and once they are certain distance apart around 10 microns it shows a regular support of the water droplet. And this is the ideal distribution leading to the minimal wetting, but once we start separating the pillars by a distance we start seeing that the water droplet will tend to sag and the sagging effect will lead to the wetting of the surface. So, it might eventually happen that this sagging effect becomes so predominant that start touching the surface and eventually it is wetting the surface. And secondly if we increase the height of those pillars it might lead to some pinching effect because there is some additional term because of the potential the mass also comes into picture because the mass which is being separated by a height of h. So, overall potential energy of the water droplet is very high that can lead to a pinching effect and much more water droplet can also basically creep into out here. And this micro cavity which is around 30 microns deep can also induce some sort of a capillary action. So, we see that once the micro pillars are very they are very increased height it leads to capillary action and it can lead to pinching effect. Once the protrusions are very very further from one another it can lead to the sagging effect. And once there is some ideal distribution you can see that both these effects are basically being balanced. So, there is no pinching and no self sagging of the water droplet in terms of retaining the water at the surface. And again the energy which is being utilized by the years before buckling it is being rendered like this we have the overall overall stress which is being generated it can be related to the Young's modulus of the material. So, we require certain force to buckle the particular material. So, we have stress which is which can which particular here can take depending on this L by R ratio and the Young's modulus which is which is the property of the particular material. And again the low surface energy does not mean non wetting again the adhesive or the Vendor wall forces which are acting at such a scale they also take part in sticking of the droplet. So, apart from a low energy we also need to have minimize the adhesive or the Vendor wall forces which are dominant at nano scale. And the same time it has to be balanced so that it has a high enough and L by R low enough. So, that we have a minimum stress which is acting on the particular surface to result non buckling result non buckling of the nano here basically I will stop here. So, we also see that the effect of L by R ratio. So, if we have very high lens then overall stress will basically keep reducing. So, the flexing stress which is which can be taken by the here will reduce as we start increasing the length or the particular material will start undergoing much more buckling. So, we need to have the L by proper ratio of L by R should be low enough so that it can support the support and it should not really buckle. And the force exerted by the water droplet it comes out to be around 41.2 to 138.5 into 10 to power minus 6 Newton's which corresponds to a stress of around 2.1 to 7.06. And there are two very key features out here that protrusions are not only supporting the water droplet, but they are also resisting the puncturing of the water droplet. If water droplet gets punctured then basically there is it will basically flow out as a capillary and it will wet the surface. So, as we saw earlier the protrusion here have a Young's modulus of around 358 that corresponds to overall flexing stress of 11.5 to 47.7 with a certain L by R ratio. And as we know that the L by R ratio of the base here is very high. So, it will lead to a much more critical flexing stress and this will be very difficult to achieve. If you are able to maintain that protrusion here are not getting wet obviously the base here also will not get wet because the flexing stress is very high. At the same time they are the secondary here which on which the water droplet will eventually come to. So, we see that protrusion here have a certain flexing stress. So, this is the critical part and this should not be exceeded in order for the water droplet to remain on the surface of lotus leaf without wetting it. So, this is sort of a critical flexing stress. And as we see if we have a normal protrusion spread this 10 microns, 10 microns, 10 microns it corresponds to a number of protrusions 492, 1, 2, 5, 6 and then stress on each nano here is to the order of 4.29 to 5.65. And again coming to it the Laplace pressure will be very very high. Laplace pressure is nothing but the kind of pressure which is required to go through the go through those two protrusions here and be able to wet the surface. And since they are only 10 micrometers apart it will need a very high datum of water to be able to wet the surface. And we can see when the particular protrusions are far apart they create the number of protrusions basically reduced to 54 to 140. And therefore, because of sagging effect the Laplace pressure is very very it is little higher, but again it will lead to some sagging of the water droplet and it might even wet the surface. But when the protrusions spreads are much higher though they have similar number of protrusions stress on each protrusion on each nano here will also be remain same, but the Laplace pressure will be little lower. Why because there will be some datum effect which will also come into picture and that can also lead to the buckling of the puncture. So, puncturing effect might be much more predominant in case of a higher, but once they are sagging obviously the number of protrusions are very less and in the sagging effect will have a very low kind of Laplace pressure. And the critical flexing stress is around 11.5 to 47.7 and as we see these values are much lower than those. So, what is happening is that the micro order nano roughness it is inducing around 3 orders of magnitude reduction in the orders of reduction. And as we see the critical stress is around 10 power minus 3 times lesser and still it is being able to hold the particular droplet. So, that is the reason we are seeing the effect of roughness which is being imparted on here in terms of being able to hold the water droplet. And Laplace pressure is basically given by which is dependent on the surface tension of water minus the inclination angle and again depends on the protrusions spread and the distance which is out here the protrusions height and the protrusions distance. So, we have kept the protrusions distance of around 10 microns in one case and with the spread it becomes 30 microns. Similarly, with the height we have height of 10 microns in each and in one case we have taken height of around 30 microns to see the pinching effect. And using this particular equation we see a balance between the sagging effect and the penetration effect. So, if we see this part out here we need to keep the maximum of protrusions minimum of protrusions distance and minimum of the minimum of the high spread height. So, the overall delta p goes higher and higher it means the higher data can be supported by the particular for the particular thing for holding the water droplet. So, we need to have both high p as well as the h or the protrusions spread should be spread distance should be minimum will be minimal and protrusions height also should be minimal. So, that we have the overall delta p to be very high. So, coming on to the effect of contact and the inclination angle we can see the overall Laplace pressure how it basically delta p how it changes with the wetting angle. So, we see that for the for the for our ideal case if we have something like this like we have 10 microns of protrusions spread with 10 micrometers of the protrusions height we see that the wetting basically wetting angle remains approximately similar with once we have very much more height. So, we have 10 micrometers of spread, but 30 micrometers of height. So, height is not playing that important role in terms of being able to being able to wet it because we see there is also some dependence on the alpha where alpha is the inclination angle. So, depending on the inclination angle as well the h will keep affecting in certain manner, but protrusions spread distance that is only half of it which is basically affecting it. So, we see that once the once we have very in this case we have the spread distance as the spread distance is increasing it is affecting the delta p in a large extent. So, if you are increasing this p by say 3 times we are getting around 1.5 times in decrease in the delta p and that is what we are seeing out here. We are seeing a certain transition from cassive extra to Wenzel straight, cassive extra means non wetting straight Wenzel it means more of a wetting straight where the cavities are also being captured by the water droplet. So, we are seeing cassive extra straight which is basically incorporating the height of the particular protrusion here that we have both the cases of ideal nano hairs and something with the increased height as well. So, we have increased height and ideal protrusion and they are showing that the particular surface will remain hydrophobic in nature whereas, for the once we are increasing the protrusion spread to 30 microns. So, we have a height as 10 microns, but spread as 30 microns there are not enough protrusions to support the water droplet and that is basically making it lie below the transition layer the transition between the cassive extra and Wenzel and this particular straight remains in the Wenzel straight and this basically keeps below the non wetting line and it basically is wetting the surface. So, that is the effect of the wetting angle and the spread of the nano spread of the micro protrusions and coming to the inclination angle cassive extra straight remains obvious the cassive extra straight is again the same transition out here. We see that the ideal always remains above the cassive extra once the wetting inclination angle is couple of degrees, but as we start increasing the protrusion height we see very minor deviation from the same, but it will wet before the ideal case is achieved. So, this is for the ideal so wetting will achieve will and this case we once we have the wetting the wetting will be will initiate earlier for the higher protrusions for the higher protrusions height. Whereas, for the once the protrusions are spread far apart we see wetting at much lower inclination angles. So, this was the inclination angle this is the Laplace pressure we see the wetting will occur very easily for the when the protrusions are far apart. So, this is the overall observation of this one that we once we have wider protrusions spread there is no non wetting or there is easily wetting of the surface. And again transition is occurring in the protrusions because of the puncturing in the datum because once we start having a wetting angle of wetting angle of lower and lower this basically means that the water is being able to get trapped into the surface because because of the overall geometry and overall geometry inclination of the particular water droplet. And wetting angle is basically getting decreased and secondly the weight of water droplet itself can also play a part because while we are shifting its gravity center of gravity it means we are changing the inclination angle then water will roll off and the true contact basically will be a few degree greater than a few degrees may not even occur. So, we are seeing that true contact angle will occur in this particular region and water droplet will simply roll off. So, this inclination angle they do not make much importance in practical applications why because with certain roll of say couple of degrees will make the water droplet to go and leave the surface. So, at higher inclination angles we do not see that effect of inclination which might be dominant in inducing the wetting of the lotus leaf surface. And this is what the overall thing is all about that once we that first of all that lotus leaf surface will have effect of super hydrophobicity arising from two levels of microstructure. One is the micro protrusions which are dominated on the surface and over that they have some sort of a nano hairs and again the geometry of nano hairs is different which is lying on the protrusions surface and which are lying on the surface of the base here because base here surface leaves base here and the nano hairs which are lying on the surface of the base here they are much more rigid they are much little larger diameter and much more height as well. But, they are much more rigid because of their higher diameter and again this leads to effect of what will be the effect of protrusions spread and height on the non wetting of lotus leaf. So, a modeling has been done and which basically tells that if we have protrusions spread ideally of ideal height then it can support water droplet very easily. But, if we have protrusions which are much higher and high it can sustain it can sustain very high Laplace pressure but of course, lesser than that of ideal case and later on the pinching effect starts creeping in because of once we have some very high datum then it can lead to much more much more puncturing of the water droplet itself and it can wet the surface much earlier as compared to that of ideal case. But, the worst case if we have the supporting pillars of this micro protrusions much further apart then it can basically cannot support the whole water droplet because the number of protrusions will decrease by around 4 times and because instead we have 2 or 3 protrusions in certain distance now there will be 2 or 3 protrusions in more than 3 times of distance. So, that will basically reduce the overall holding capacity of the protrusions and then that will lead to easy wetting of the surface and sighing effect will come into play as when the protrusions are very far apart and that leads to the easy wetting of the surface both in terms of either wetting angle or the inclination angle. So, we did see that the inclination angle also plays a part in wetting the surface and it may not allow the water droplet to roll off so easily and it will the gravity effect also come into picture once we have some inclination angle and that is the basically effect of the super hydrophobicity and there have been certain ways in which a super hydrophobic surface has been engineered or has been produced synthetically. So, what they have one group has basically done that they take a lotus leaf surface and then a negative polymethyl methacrylate has been deposited on the lotus leaf surface. So, basically it is mimicking the negative replica of the lotus leaf surface. So, once it has mimicked the negative of the negative replica then this particular material has been cured and over this particular material which is a negative replica on to it we have spread polydimethyl silane and now PDMS takes the shape of the negative of negative it becomes again positive lotus leaf surface. So, this PDMS mimics what is happening at the lotus leaf surface, but again there are true transitions one PMMA flowing into the cavities of the lotus leaf and again PDMS going in going on to the cured PDM on to the cured PMMA and again pressing its topography. So, it becomes very difficult to fill in the profiles which are to the order of nanometers and in length and diameter. So, it becomes very hard to really mimic each and every contour of the surface. So, these particular coatings are super hydrophobic, but they can be made even more super hydrophobic depending on if you are able to trace those particular things, but we are able to achieve a surface which is really super hydrophobic in nature. So, this negative to negative mimicking also leads to a formation of a super hydrophobic surface again inducing some sort of a surface roughness can also make surface highly super hydrophobic. So, if we start inducing some sort of roughness inducing some sort of a micro pillars or nano pillars on a surface we can really make water droplet go from hydrophilic to super hydrophobic in nature just by altering the overall roughness of the particular surface. And in this case one group has already achieved developing negative PMMA and over that they are able to deposit PDMS and be able to mimic the what is happening at the lotus leaf surface. So, this is what with I will end my lecture with thank you.