 Today, we will start discussing about very important topic in microfluidics which is surface tension driven flows. As we have discussed earlier that as we go down to smaller and smaller scales, surface effects become more and more important as compared to volumetric effects. So there are possibilities by virtue of which one can exploit surface effects to a benefit or surface effects may be used as a control mechanism for controlling or driving or actuating fluid flow. For example, when we think of a pressure gradient driving a flow, it is really a challenging proposition to create a good enough pressure gradient to drive a good amount of flow through a microfluidic system because of huge frictional losses. So what one can do instead of using a pressure gradient directly, one can exploit a pressure gradient that is implicitly induced because of a surface tension gradient. So it may be possible that naturally occurring surface tension gradients are there or naturally occurring surface tension forces are there or surfaces can be engineered to create surface tension gradient and because surface tension occurs so nicely in small scale systems, it is very important that we study the behavior of surface tension or the characteristics of surface tension driven flows in general and surface tension driven micro flows in particular. So a brief motivation what I have discussed that in small dimensions surface tension dominates over other forces. So it is not that it is a very new kind of force, it is something which nature has been knowing for a very long time. Like if you think of the transmission of water from the lower level to the topmost level of a tall tree, how that is possible? There is no pump, there is no pump made by engineers by which water is pumped from the root to the topmost level of the tree but how it is possible? It is possible by surface tension, it is possible by capillary action. So because of the smallness of the capillaries that we will see that because of the smallness of the capillaries, surface tension force can lift the water to a significant height and that significant height is good enough for supplying the tallest, the topmost branches of the tree with the necessary water that is sucked from the ground. So there are many examples where surface tension forces are important like one can use surface tension force to manipulate droplets, to manipulate bubbles and these droplets and bubbles are also very important in microfluidics based applications. So we will briefly revisit the fundamentals of surface tension that is what is the surface tension and why is it important. So I will come to the board, of course it is represented in the view graph also but we will utilize this view graph for mostly summarizing our observations but just for simplicity we will not consider a complicated scenario but consider a very simple scenario when you have a container there is an interface, the interface separates the vapour from the liquid. Now I mean whether the interface is flat or curved forget about that at this moment, we will discuss about all these issues in much more details later on but for the time being let us assume that the interface is of some form, some shape. Let us consider some liquid molecule here, when the liquid molecule is here this molecule is acted upon by forces equally from all possible sides because all sides are surrounded equivalently by other liquid molecules. So there is equal attraction from all the sides, however consider a molecule on the interface when you are considering a molecule on the interface what is happening on one side you have liquid and on another side you have vapour which is more dense liquid or vapour, liquid is more dense. So liquid being more densely packed it is more likely that this molecule is attracted more strongly from the liquid side as compared to the vapour side, if that be so then what is the possibility then the possibility is that this liquid this molecule at the interface has a very special character we do not call it liquid or vapour but it is an interface but it is also having molecules and this molecule will have a tendency to dissolve into the liquid phase but that does not happen because if that happens no interface can be formed. So interface is formed that means that the molecules at the interface are having grossly speaking from the system level they are having some energy so that they can overcome that attraction this net attraction and remain at the interface. So this is called as interfacial energy or surface energy. Now not only that how this energy is expressed typically it is expressed like that so when you have an interface because of this reason the interface is considered to be under some tension and since the interface is considered to be under some tension this state of the interface is described by a force which is also called as surface tension and it is expressed in terms of force per unit length. So the surface tension or the surface tension coefficient sigma between 2 phases is expressed as a force per unit length of the interface so its unit is Newton per meter okay. So this is what is summarized in this view graph so you can see that like there is a volume element which is identified and we are that volume element crosses the interface or cuts across the interface and we are seeing that there is a differential force or extra force towards the liquid as compared to the force towards the vapor because of the differential density okay. So this is what we have discussed I am not going to reiterate I mean this view graph discusses about what is surface energy and what is surface tension. So of course there are more formal definitions associated with this but this is a qualitative way of understanding the concept which will be useful for us for our analysis. Now we will of course start with the basic understanding of surface tension but the question is how can surface tension actuator control fluid flow that is what is of very important concern for us. So of course surface tension itself is like a force and as a force it can actuate the flow by itself. On the other hand there are other subtle means by which you can control flow by surface tension. How other than surface tension being utilized as it is you can use the gradients in surface tension not just the surface tension itself but gradients in surface tension to manipulate the flow. So that is possible by using a gradient in temperature. So gradient in temperature can create a gradient in surface tension this is known as thermo capillary flow we will discuss about this in details and the corresponding convection is known as marangoni convection is one of the like classical mechanisms by which you can control surface tension driven flow by virtue of temperature gradient. So not only that you can create a gradient in concentration because gradient in concentration can create a surface tension gradient. You can use electric field to modulate surface tension we will see how that is possible. You can modulate contact angle through design hydrophobization or hydrophilization of solid substrates. So you can control the surface tension force by altering the contact angle and the contact angle can be altered by making the surface or designing the surface either hydrophilic or hydrophobic with a particular contact angle or a combination of above effects. So these are some of the possible ways by which you can create gradients in surface tension. So these are some of the ways by which you can modulate the flow other than of course using surface tension itself to manipulate the flow. So using surface tension itself is something obvious that is what I have not kept that in this slide but I mean the creating gradients of surface tension to manipulate the flow is something which is not always that intuitive. Now we will start with a very basic law or very basic equation governing surface tension and that equation is known as Young Laplace equation. So Young Laplace equation what it does it relates the pressure difference across an interface with the surface tension. So to understand this I will also draw some schematic in the board to supplement what is there in this view graph but first I am explaining you what is there in the view graph because the schematic which is there in the view graph I will not reproduce that in the board I will reproduce only a part of that in the board. So let us say that there is a membrane with dimensions x and y. So this dimension is x this dimension is y. So this is a membrane and this is a curved membrane. So the side with x has a radius of curvature r1 and the side with dimension y has a radius of curvature r2 okay. So these are sometimes called as the principle radii of curvature and like these similar considerations are used in many other aspects of engineering like for example if you are talking about pressure vessels and all these things different radius of curvature they come into the picture. Now forget about that let us come to this particular example. Now because of a pressure difference between the inside and the outside there is a change in the area of the membrane and there is a displacement also of the membrane. So how that is possible? So because of the pressure difference there is a displacement delta z and because of the work done by virtue of this pressure difference there is an additional energy imparted to the interface and that makes the interface assume a greater area because the surface energy has increased. So what has helped in increasing the surface energy that is the work that is input to the system by virtue of the pressure difference between the inside and the outside. I will explain you that how to qualitatively assess that like which side is of higher pressure which side is of lower pressure and all those things I will discuss about that in details in the board. But now so the membrane is stretched now once the membrane is stretched because it has now more surface energy as work is input to it. So you have now the dimensions of the membrane as x becomes x plus delta x and y becomes y plus delta y okay. So with this little bit of understanding let us come to the board and we will figure out the eraser just bring the eraser. So let us say that this is one of the curves of dimension x. I am not drawing the other one of dimension y. So this becomes x plus delta x this is r1. I have just magnified the figure to make it clear to you of course like these are all differentially small changes so this is delta z. So from similar triangles you can write r1 by r1 plus delta z is equal to x by x plus delta x. So this becomes r1 by delta z is equal to x by delta x. If that be the case we can similarly write that r2 by delta z is equal to y by delta y right because the radius of curvature r2 is associated with that side with dimension y and radius of curvature r1 is associated with that side with dimension x. Now let us apply the work energy consideration. What is the work done to push this interface from the initial location to the dotted location delta p into A into delta z right. Delta p is the difference in pressure across the 2. So that time say is the net force and force time displacement is the work right. And this is equal to the increase in surface energy because of the stretching of the membrane. So surface energy is surface tension coefficient or surface tension. Surface tension and surface tension coefficient are the same. They are just same things expressed in 2 different terminologies. So surface tension coefficient times the change in area delta A. So what is the change in area x plus delta x into y plus delta y minus xy. So this is sigma x into y gets cancelled with x into y then x delta y plus y delta x. Now the neglect delta x delta y this is a small lower order term as compared to the other terms. And what is this A here A is x into y. So we can write from here what is delta p sigma x into delta y divided by xyz xy delta z. So sigma delta y by y delta z plus delta x by x delta z right. Now look at this what is delta x by x delta z this is 1 by r1 and this is 1 by r2. So we can write finally that delta p is equal to sigma into 1 by r1 plus 1 by r2. This equation is known as Young Laplace equation and a very very important equation because it relates the surface tension with a pressure differential. And you know that by a pressure differential you can even create a flow. Now so surface tension as a driving force what it is fundamentally doing? Surface tension as a driving force is actually creating a pressure differential. So although it is a force acting on the interface but it is eventually giving rise to a pressure differential across an interface and that pressure differential is actually manipulating the flow okay. So now if you consider the surface tension as a force therefore this will appear to be a force in the Navier-Stokes equation only at the interfacial control volumes where there is a sudden jump in pressure. So that is the way in which you can treat it numerically but in this chapter we will mainly concentrate on how to handle these issues analytically. Now let us ask ourselves certain very simple questions. These questions might appear to be too simple but let us try to answer that even if the question is simple I mean it is okay if we attempt to make an answer. So the first question is that we are writing delta p. So when there is an interface there is an inside of the interface and there is an outside of the interface right. So delta p we are talking about what p in-p out or p out-p in so and how do we know that? So whenever we are asking any question this is what I mean for any examination or any tutorial or any discussion I would always encourage that whenever I am asking a question I am not minding a wrong answer because many times I can also give wrong answer or wrong interpretation but it is more important that what is the logic that you are applying to give the answer and that is where actually the concept lies. Sometimes we may use the logic little bit incorrectly so that final answer becomes wrong but sometimes the final answer is right but we just know it as an information rather than something which is of scientific origin then that kind of knowledge actually has no value. So let us try to see what is the scenario. Let us say that this is an interface okay. Now can you tell that if this be the interface then what is the direction of the surface tension force? So the surface tension force just like if you have cut the interface here it will be the interface will be in some form of tension. So now if you resolve this force in various components you can see that this component and this component they may cancel and along this direction let us say this is x direction along this x direction you have a resultant force due to surface tension right. So if you have a resultant force due to surface tension here then you also have some force due to pressure. So you have a resultant force due to pressure inside from this always remember that pressure is by definition positive pressure means acting inward normal to a surface. So this is a force due to inside pressure this is a force due to outside pressure if this is inside and this is outside. So for equilibrium we can say that force for pressure inside is equal to force for pressure outside plus force surface tension right this being a positive quantity that means force for pressure inside is greater than force for pressure outside that means pressure inside is greater than pressure outside because they are acting on the same area okay. So do not just make it like an information that you want to remember if you use this logic you can come to this conclusion by very simple mechanics that where should the pressure be more inside or outside. So this is nothing but Pi-Po. Now let us see that how do we apply this equation. Let us assume that there is a channel in this channel there is an interface this is liquid and this is vapor okay. So now you can clearly see that this interface can be of any shape right. You cannot presume that it is spherical or whatever but if this thickness is small if this height is small then it can be thought of as a part of a large sphere without making very significant amount of error if this gap is small then like this can be thought of as a part of a sphere. So this can be thought of if you imagine a large sphere this can be a part of the surface of this of the large sphere. So if you have that kind of a situation then for a sphere R1 is equal to R2 is equal to R sphere okay. So I am so much emphatic with this subscript R sphere here because R sphere is not same as the radius of this capillary that you must understand right. So the radius of this capillary let us say is this one and the radius of curvature of the sphere is something different from that okay. So now if R sphere is something which is known and you can clearly find it out provided you know what is the contact angle and I will discuss about what is the contact angle later on. I have not yet introduced the definition of contact angle. So we will stop there for the moment but for a sphere we know that because R1 is equal to R2 is equal to R so delta P is equal to sigma into 1 by R plus 1 by R, R sphere plus 1 by R sphere. So 2 sigma by R sphere this is the formula that you commonly use in undergraduate texts for finding out the pressure difference between inside and outside of a droplet spherical droplet. So this is how the formula comes. So one important thing we can get from here is that in many cases assuming the interface to be of spherical shape is not something which is pragmatically bad. So we can assume that and let us go to the next view graph where we do the same thing the previous derivation that we made that delta P equal to sigma into 1 by R1 plus 1 by R2 is general. It does not assume any shape of the interface but if you assume the shape of the interface you can do the same thing for a droplet or for an interface which is a part of a sphere and for that we can use the considerations of energy minimization for making the derivation and I will show you that how the same see the first derivation that we did purely from work energy principle right. The work done due to the pressure differential is adding to the surface energy that is the work energy principle that we used. Now we will use the energy minimization principle what is that that a droplet in its equilibrium shape will try to adhere to a configuration with minimal interfacial energy because that is its stable configuration. So what is displayed in this view graph is a droplet which is a part of a sphere then there are certain parameters which are defined we will first define the angle theta which is called as the contact angle. So again I will come to the board to discuss a little bit more about this figure and more about this derivation. So let us say this is a substrate and a droplet is sitting on this substrate which is a part of a sphere. Let us say that this is like a hemisphere up to this is like a hemisphere. This black is actually a part of the hemisphere this is liquid this is vapour and this is solid. So you have a triple phase contact actually. Now what you do you draw a tangent to the liquid vapour interface and the angle that is made from the liquid side with that that angle is called as the contact angle with respect to the substrate. So this theta is called as the contact angle. Now different surface tension forces act along the interfaces. So for that see when we say sigma it actually has no meaning we have to say sigma with respect to which interfaces solid liquid vapour solid vapour whatever. So when we say the surface tension force here so what should be the subscript? So sigma between liquid and vapour so you can write sigma vl or lv that makes no difference but commonly we write sigma lv. So in the formula where we write 2 sigma by r that sigma what we write is commonly that is actually sigma lv. Now what is this? This is sigma sl or ls whatever you say and this is sigma sv. So what is the surface energy of the droplet? Sigma sl-sigma sv into Asl-sigma lv Alv. What is Asl? Asl is the solid liquid interface area. So if this is say some radius then pi into that radius square right this is remember this is a part of a sphere. So its footprint on the solid surface will be a circle right you cut the sphere from anywhere you will get a circle. So then what is the liquid vapour interfacial area? The liquid vapour interfacial area is the area of this curved surface the black curved surface. Now this energy is minimized subject to some constraint. What is that constraint? Volume of the droplet is conserved right so the volume of the droplet must be conserved. So this has to be minimized subject to constant volume. So it is as good as minimizing this minus lambda v equal to 0 where this lambda is called as Lagrange multiplier. This is one of the ways in which we solve optimization problems in mathematics that minimizing this with a constraint that v is a constant is as good as minimizing this minus lambda into v. Where lambda is a parameter which is known as Lagrange multiplier. We will see that what is the physical significance of this Lagrange multiplier in this case. Now to proceed further we have to figure out what is ASL and what is ALV. So let us construct some more lines in this figure. So we join this line like this. So this blue dotted line is perpendicular to the green dotted line because the green solid line because the green solid line is tangent to the surface of the sphere and this blue dotted line is the normal that means this angle is 90 degree minus theta. So this angle is also 90 degree minus theta okay. So ASL so what is this particular radius if r is the radius of the sphere what is this radius r sin theta right. So ASL is pi r square sin square theta. This black curved surface will be a full hemisphere when this theta is equal to 90 degree okay. So when this theta is equal to 90 degree then this will be just pi r square where r is the radius of the sphere. Now let us find out what is ALV. To find out ALV what we do let us consider that at an angle phi this angle is phi arbitrary angle we consider a small strip of d phi. So what is the surface area of the shaded region? So it is a small part or small slice from the sphere is actually like a small cylinder with some radius and some lateral dimension right. So 2 pi what is this radius r cos phi. So 2 r cos phi 2 pi r cos phi into r d phi ALV becomes this integral with phi from what to what phi is equal to 90 degree minus theta to 90 degree. So this is what 2 pi r square sin phi from 90 degree minus theta to 90 degree. So 2 pi r square into 1 minus cos theta check for theta equal to pi by 2 that is the surface area of a hemisphere that is 2 pi r square double of that is the area of a sphere 4 pi r square. So this is ALV what is the volume of the sphere I mean part of the sphere v is equal to so it is like pi r square h right. So pi r square means r square cos square phi what is the height of this one r d phi is the lateral dimension cos of that. So r d phi cos phi again from 90 degree minus theta to 90 degree. So pi r cube integral cos cube phi d phi from 90 degree minus theta to 90 degree. Now we can express cos cube theta in terms of cos 3 theta. So we can write cos 3 theta is equal to 4 cos cube theta minus 3 cos theta not visible in the board write it little bit below cos 3 theta is equal to 4 cos cube theta minus 3 cos theta. So instead of cos cube phi we can write cos 3 phi plus 3 cos phi by 4. So this will become pi r cube by 4 cos 3 phi will become sin 3 phi by 3 plus 3 sin phi 90 degree minus theta to 90 degree. So pi r cube by 4 minus 1 third plus 3 then minus with another minus we will make it plus so cos 3 theta by 3 minus 3 cos theta right if I make any algebraic mistake please correct. So pi r cube by 4 8 by 3 plus 1 by 3 cos 3 theta minus 9 by 3 cos theta. So pi r cube by 12 8 plus cos 3 theta minus 9 cos theta substitute theta equal to 0 sorry theta equal to 90 degree that is the hemisphere you substitute theta equal to 90 degree this becomes 8 pi r cube by 12 that is 2 by 3 pi r cube double of that is 4 by 3 pi r cube which is the volume of a sphere okay. So let us write the simplified expression for E sigma SL minus sigma SL minus sigma SV into pi r square sin square theta plus sigma LB into ALB 2 pi r square into 1 minus cos theta minus V into lambda. So minus lambda into pi r cube by 12 8 plus cos 3 theta minus 9 cos theta this is what is the expression for E. For minimum E what you have one is del E del r equal to 0 so that means sigma SL minus sigma SV into 2 pi r sin square theta plus sigma LB into 4 pi r into 1 minus cos theta minus 3 lambda pi r square by 12 into 8 plus cos 3 theta minus 9 cos theta equal to 0 and you also have del E del theta equal to 0 that means sigma SL minus sigma SV into 2 pi r square sin theta cos theta plus sigma LB into 2 pi r square sin theta minus lambda pi r cube by 12 is equal to minus cos 3 theta is minus 3 sin 3 theta and cos theta this will become plus 9 sin theta that is equal to 0 right. Please check carefully whether there is any algebraic mistake because we will continue with these expressions for finding out an expression for like the equilibrium condition for the droplet. So to summarize what we have discussed so far in this particular lecture we have discussed about what is the work energy principle that governs the relationship between the pressure difference across an interface and the surface tension coefficient and then for a droplet which is a part of a sphere we have tried to derive the same consideration from energy minimization principle. So we have written an expression E which is the energy minus lambda into volume so for a constant volume that needs to be minimized so for minimization of that we have set the partial derivative of that with respect to r and theta equal to 0 and we will derive our subsequent relationships or subsequent expressions by using these two final equations. We will take it up in the next lecture. Thank you very much.