 start with a new topic which is electrokinetics. So in some of our previous lectures we made a passing remark that in microfluidics flow can be controlled by electrical fields and the science behind that mainly lies on electrokinetics which is basically a combination of electrostatics and hydrodynamics. So with the reasonable introduction on hydrodynamics that you have we will start with some basic issues in electrochemistry, thermodynamics and electrostatics that will lay the foundation of electrokinetics. So what is electrokinetics? It is not a very stringent definition but in one of the very famous books authored by Hunter who is well known scientist in this area in the book foundations of colloid science, he has referred electrokinetics to all the processes in which the boundary layer between one charged phase and another is forced to undergo some sort of shearing process. The charge attached to one phase will move in one direction and that associated with the adjoining phase will move more or less tangentially in the opposite direction. This is just a qualitative remark but frankly speaking this does not give too much of information of what is electrokinetics all about. So we will start with some of the basic issues. Now one of the important considerations of electrokinetics is the formation of a charge layer close to the fluid solid interface. So the question is first that what is this charge layer? So think of a micro channel. So if you have a micro channel made of material like glass say for example. So we expect that there will be some surface charge and there will be a layer of fluid which is close to the solid in which there will be some kind of net charge. So the question is that how will this charge layer be formed and what are the typical dimensions of this charge layer and like what are the details of the charge layer that are formed. So this charge layer which is called as electrical double layer some of the important terminologies associated with that are as follows. Most solid surfaces tend to acquire a net surface charge may be positive or negative when brought in contact with an aqueous solvent. Various surface charging mechanisms are there like ionization of covalently bonded surface groups, ion adsorption this is totally chemistry basically. So we will not get into the details of that but we will use that in the context of fluid mechanics for the purpose of modulating flow through micro channels and nano channels. Now aqueous solutions generally have dissolved ions. There are some terminologies. So if the surface has a negative charge then in the fluid if you have positive charge ions those are called as counter ions. Counter ions means ions charge opposite to the surface charge. So if the surface charge is negative then positive ions in the fluid is counter ion. If surface charge is positive then negative charge in the fluid is counter ion. So counter means opposite sign and co-ion means ions having the same polarity as that of the surface charge very straight forward terminologies. Charge surface attracts counter ions it is obvious because of columbic attraction charge surface will attract counter ions and they will repel co-ions. Now as I told you that it is the chemistry that decides that there will be a net surface charge and there are several mechanisms. I will not get into the details of the mechanism but some of the mechanisms are highlighted here like ionization of surface groups. So magnitude of the surface charge depends on the acidic or basic strengths of the surface groups and on the pH of the solution. Charge crystal surfaces may be there or isomorphic substitution may be possible specific ion adsorption may be possible. So these are some of the basic chemical features or chemistry based phenomena that in turn dictate that it is possible that there will be a net surface charge. Question is in microfluidics we do not so much bother about what is the mechanism by which the surface charge is formed but we are more bothered in how we can exploit the surface charge to control or manipulate the fluid flow. Now there are some more issues about the electrical double layer. So I will try to give you a qualitative picture that what we are bothered about. Let us say we have a surface like this. Let us say this is a glass surface. I am talking about a single surface if there are multiple walls same phenomenon will happen at all the walls. So this surface has say because of some ion adsorption this surface has acquired some negative charge. Now the remaining the entire system being electrically neutral one would expect that all the positive charge whatever it is will fall on the surface so that it will neutralize. But neutralization will not happen just like that because all the ions would have fallen on this surface provided the ions had no thermal energy but ions because of their temperature have their own thermal energy. So that will resist all the ions from falling on the top of this. So there will be some fixed layer of ions some fixed positive charges which will be bound to the surface. But there so this will not totally neutralize this and there will be some in the outer layer which is not I will tell you the names of this layers the technical names but I am just trying to give you a qualitative picture. So there will be some positive ions as well as some negative ions. There will be some draw the negative ions by green color. So there will be some negative ions in the bulk but there will be some positive ions in the bulk and those will dominate I mean I have not drawn it in the proper way actually because there will be a distribution it is not that it will be stacked like this. But what I want to convey is the important thing that this bulk plus subsurface in totality will neutralize the system. So in the surface there will be some charge on the surface there will be some charge some immobile ions these ions are sort of immobilized because they are strongly bound to the surface. But in the bulk there are some additional positive ions there are some negative ions but positive ions surplus the negative ions because you know eventually because the surface had acquired a net negative charge the fluid should have a net positive charge so that the system is electrically neutral. So the question is that like how far this distribution of ions is there that means how far from the wall you see the effect of this charge layer? We will try to answer this question first qualitatively and then more and more quantitatively. Then what are the names of these different layers? So we will discuss about that through the slide which is there displayed. So we will refer to the slides. So now if ions had no thermal motion all the counter ions would have stacked against the surface right. But that is a hypothetical picture in reality ions have non-zero absolute temperature so they have thermal motion. So there is a balance between Coulombic attraction and thermal interaction. So by Coulombic attraction what will happen? All the positive ions will tend to fall on the negative surface but because of thermal energy what will happen is that ions will try to escape from that attraction and will try to remain in the bulk. So there will be a balance at equilibrium there will be a equilibrium distribution of ions in the fluid. So there is a charge distribution that prevails adjacent to the surface. So you have a charge surface and an ion distribution. So basically you are considering 2 layers. One is a charge surface layer and another is the ion distributed fluid layer and these 2 together we call it electrical double layer. So there are various theories of electrical double layer and these theories can be from as simple to more and more complicated. Complex form. So we will discuss about a very simple model which is called as Gauss Chapman model with stern modification. So to give you a qualitative picture and this is a very important qualitative picture you see that like if you assume the surface to be of negative charge like very commonly we give an example of negative charge because with neutral water pH glass surface will develop negative charge. So that is a very common example practical example that is why but do not keep any prejudice in mind that it has to be negative charge. It can be negative positive anything depends on the pH of the solution. So even it can be 0 charge if the pH of the solution corresponds to point of 0 charge. So it can be anything. Now if you see that there are if the surface has a negative charge then positive charge one layer of positive charge ions gets stacked against the surface. So this layer does not move and this layer is called as stern layer. The name of this layer is called as stern layer or Helmholtz layer stern layer or Helmholtz layer. This layer is few angles and stroms in thickness and the reason is quite obvious that this the few angstroms is like basically represents one entity of ions they are the ionic diameter. Beyond the stern layer you will see that ions are mobile. In the stern layer the ions are not mobile but I mean advanced research shows that in the stern layer ions are not mobile provided there is no hydrophobic interaction. If there is hydrophobic interaction then you may even immobilize the stern layer but I will not try to confuse you with those research issues. So under circumstances when this the wall is hydrophilic then you have the stern layer that is immobile. So you can see there is a layer which is called as diffuse layer shown in the diagram where you have both positive and negative ions as I mentioned in the schematic also that I drew in the board. There are both positive and negative ions and those ions are mobile that means if you can apply a field then you can make the field may be whatever electric field whatever field you can make ions in this layer move okay. So one strategy of actuating fluid flow could be that in the diffuse layer whatever are the ions if you now apply electric field if this ions move then because of viscous interaction between the ions and the polar water molecules the water molecules will also start moving and this is the process called as electro osmosis. So we will discuss about this in details but this is just to give you a picture that why are we studying this. There must be a motivation for which we are going through the description of the electrical double layer. So this diffuse layer is also known as the gauch-chapman layer. So there are various names. So you can see that there is an interface between the diffuse layer and the stern layer. There is an interface between the diffuse layer and the stern layer and this that interface is given by this dotted line whatever dotted line is there okay this dotted line whatever this cursor is indicating this dotted line is the interface between the 2 layers and this interface is called as shear plane and the potential. So if you now see the potential distribution across if the transverse direction of the solid boundary is y then you can see that how the potential varies with y. So you have a potential drop across the stern layer and then you will have the there is a potential drop and you come to a location where you find that the potential has almost asymptotically reach to 0. So that potential which has asymptotically reach to 0 is called as I mean the distance up to which you go to reach the potential approximately equal to 0 is the span of the electrical double layer. So the electrical double layer is a sort of analogous to boundary layer. Just like outside boundary layer you do not feel the effect of the viscous interaction between the wall and the fluid. Similarly outside the electrical double layer you do not feel the effect of surface charging. Now the electrical double layer has a characteristic length scale which is called as lambda. Now this is not the thickness of the electrical double layer. You can see that actually the electrical double layer spans somewhat beyond that but this is a characteristic length scale that denotes the length scale of the electrical double layer and that is called as Debye length. So these are very important terminologies. Sometimes Debye length is confused as thickness of the electrical double layer that is not correct. It is just a length scale characterizing the electrical double layer. It is not exactly the thickness of the electrical double layer. These are some of the misunderstandings or mis-concepts that must be clarified. The other concept is that in electrochemistry there is a very important terminology called as zeta potential. So the zeta potential, what is zeta potential? Zeta potential is the electrical potential at the shear plane, this dotted line. At this dotted line whatever is the potential that is called as zeta potential. So the shear plane is a very important location because in case of a hydrophilic surface you can see that anything to the left to the left of the shear plane in this diagram, anything to the left is immobile. So the shear plane effectively acts like a plane on which you apply the no-slip boundary condition not on the wall because anything left of the shear plane is actually immobile in this case but when it is mobile, when you make the stern layer mobile then that consideration is not true. So these are certain things that you have to keep in mind. Now with the electrical double layer formation associated may be several electrokinetic effects there are 4 primary electrokinetic effects which are considered in the literature. What are these? Electroosmosis, it refers to the relative movement of liquid over a stationary charged surface with an external electric field acting as an actuator. Streaming potential, it refers to the electrical potential that is induced when a liquid containing ions is driven to flow along a stationary charged surface. So it is like sort of invariant reverse of electroosmosis. Electrophoresis, it refers to the movement of a charged surface relative to a stationary liquid due to the application of an external electric field. And sedimentation potential, it refers to the potential that is induced when a charged particle moves relative to a stationary liquid. So interestingly electroosmosis or electrophoresis it depends on the reference frame from which you are looking into it. So electroosmosis will refer to movement of liquid relative to a solid boundary. So the solid boundary is stationary but the liquid is moving and electrophoresis refers to movement of a charged particle relative to a liquid that means as if the liquid is stationary and the charged particle is moving both are subject to electric field. So in that sense electroosmosis and electrophoresis are related and streaming potential and sedimentation potential are related. Out of all these phenomena we are going to study electroosmosis in some details, streaming potential in some details and some aspects of electrophoresis in this particular course. Of course we will also study some other electrokinetic phenomena but in significantly less details as compared to this. Now every study in science requires a background. So we will study the electrical double layer phenomena with a suitable background that requires some considerations in thermodynamics. Why we require the considerations in thermodynamics? The reason is straight forward that like if you have a system of different chemical entities the system is in equilibrium chemical equilibrium when there is no gradient in chemical potential because chemical potential is the driving force for a chemical change to take place that is called as chemical potential. So if there is chemical potential gradient then that will create a driving force and that will create a change. So equilibrium picture with chemical entities should have like no gradient of chemical potential. However in our system it is not merely chemical entities. In our system you have ions which are chemical entities but with charge. So we have to augment the chemical potential with electrical effects and that is called as electrochemical potential. So we have to first establish an expression for the electrochemical potential within the electrical double layer and for that we have to first refer to the chemical potential and there we have to refer to the thermodynamic issues. So to understand the thermodynamic issues we will go to the board and try to discuss about this. We will start with because I mean different students in the class may have different backgrounds in thermodynamics. I am not quite familiar with what kinds of backgrounds all of you have. So I will start with something which is very basic. So we will start with the first law of thermodynamics. So let us say that we are having a system with some heat transfer delta Q and with some work delta W. So the first law for the system tells that delta Q is equal to De plus delta W where this E includes internal energy plus kinetic energy plus potential energy okay. Very often in thermodynamic processes the change in internal energy is much more important as compared to changes in kinetic and potential energy and then that will approximate to du okay. So importantly it is not the kinetic energy and potential energy that is small but changes in kinetic energy and potential energy are small. So many times I mean this is from my experience. I have asked students to write what are the assumptions behind this equation and they will straight away write neglecting kinetic energy and potential energy. I mean that is not a correct thing because kinetic energy and potential energy may be very important in many processes. Their changes may not be important because it is the D of that what is of our concern. Now let us write let us assume that the system is a simple compressible substance undergoing a quasi equilibrium process or a quasi static process. So if the system is a simple compressible substance means the changes in pressure volume temperature are much more significant as compared to other effects like electrical effect, magnetic effect and so on. So for such a case if it is a quasi equilibrium process we can write this as PDV. So again the restriction is important. Simple compressible substance undergoing a quasi equilibrium process. So you can write this as PDV okay. Now this quasi equilibrium process is also one type of internally reversible process. So a process is reversible when once having taken place it can be reversed but in doing so it leaves no change in the system and in the surroundings. So when the system comes back to its original state there is no net change in the surroundings also. So if you do that then this kind of a process so you can realize this process by a very slow expansion of gas in a piston cylinder arrangement and that makes sure that internally it is a reversible process but external reversibility, rather external irreversibility cannot be precluded. The reason is there could be a finite temperature difference between the system and the surrounding across which the heat transfer can take place that will make it externally irreversible. Now a system is, a process is reversible if it is both externally and internally reversible and for a reversible process you can make an additional modification in this that you can write in place of del Q Tds. This is for a reversible process. Now you can get away with the internal, external irreversibility if you choose a proper temperature so that your system considers the only the system it neglects the surrounding and you are talking about the system boundary temperature but we will not get into that complication. There are several processes like which are only internally reversible but externally reversible I mean there are some terminologies associated called as endo reversible processes but I will not get into such a complication. So I will simply say that if it is a reversible process del Q is Tds. Now what is this T? So typically temperature of the system boundary across which the heat transfer is taking place. Now if you write now so these are like the extensive properties we can divide by the mass and write Tds is equal to du plus pdv. Now although for deriving this equation we have used various assumptions once this equation is derived it can be used to calculate the change in entropy for any process. The question is why because for deriving it we have used several assumptions now we are claiming that it can be used for any process so why it should be like that. See the reason is as let us say there is a system in any plane PV plane Ts plane whatever we draw the system changing state between 1 to 2 okay. So there is an irreversible path which is this typically irreversible processes in thermodynamic planes we draw by dotted lines because we do not know exactly what are the intermediate thermodynamic states. Now we construct a hypothetical reversible process between 1 to 2 say this is a reversible path. So over that hypothetical reversible path we can integrate this equation but once it is integrated it will give you s2-s1 because s is a point function that will not depend on whether we have gone by this dotted path or this continuous path that is the spirit by which we say that it is valid for any process. So the integration has to be carried over a reversible path but once the integration has been evaluated that change in entropy can be used to evaluate the change in entropy for any process that connects the two end state points okay. So with this little bit of a background now we can write the alternative forms of this and because of its universal nature this kind of thermodynamic relationship is very important and in place of u you can write H-PV right by using the definition of enthalpy which is H. So this is DH-PDV-VDP-PDV so TDS is equal to DH-VDP that is another way of writing the same thing in terms of enthalpy rather than internal energy. Now when we are thinking of chemical potential we are interested with two other important functions the Gibbs function or the Gibbs free energy or the Helmholtz function or the Helmholtz free energy. So as an example we will talk about the Gibbs function or the Gibbs free energy and the chemical potential associated with that. So what is G that is H-TS so G is defined as H-TS so this is called as free energy that is so called an energy that is freely available for a chemical change to take place. So you can write DG is equal to DH-TDS-SDT DH-TDS is what VDP right from this equation DH-TDS is equal to VDP so this is VDP-SDT the chemical potential of a pure component chemical potential of a pure component pure substance this is defined as mu is equal to G bar bar at the top when I use it means molar quantity per mole that is and capital is the extensive property. So G by the number of moles so this equation is written on a mass basis but you can easily write it on a molar basis. So for a molar basis just write in terms of per unit mole in terms instead of per unit mass okay. So just convert from mass basis to mole basis all quantity all parameters will be change equivalently. So you can write DG at constant T which is equal to what this is equal to VDP because at constant T DT is equal to 0 okay. Now let us take an example of an ideal gas if you take the example of an ideal gas then you can write V is equal to RT by P this R bar is the universal gas constant that is that 8.314 okay. So R bar T by P so this DG bar this is D mu because G bar is mu D mu at constant temperature is equal to RT because T is constant it is like D P by P is D of LNP. So for an ideal gas we can express the chemical potential of the pure substance exclusively in terms of pressure and typically the ideal gas quantities we denote by star. So just I am writing D mu star, star will mean because every time I will not write in words ideal gas whenever I write star you must understand it is ideal gas whenever I give bar at the top you must understand that it is molar quantity okay. So now the question is that this is true for an ideal gas but what is the same thing if it is not an ideal gas or what is the similar type of expression when it is not an ideal gas. So just for mathematical convenience for a substance which is not ideal gas we cannot write D mu is equal to RT D LNP right. So what do we write? Instead of using the pressure as a quantity we use a different quantity which is a pseudo pressure that means it becomes pressure only when the state of the substance corresponds to an ideal gas state otherwise it is not pressure and that pressure like quantity or pseudo pressure is known as fugacity. So it is called so you for any general substance you can write D mu is equal to RT D LNF this F is called as fugacity. So it is like pseudo pressure but when does fugacity become pressure when it becomes an ideal gas at what pressure it becomes an ideal gas technically ideal gas becomes ideal gas at what pressure low or high very low pressure. So when you consider the limit as P tends to 0 F by P that tends to 1 because fugacity tends to pressure as P tends to 0. So the complete definition with fugacity is that in the asymptotic limit when you tend to ideal gas behavior the fugacity should tend to pressure okay. Now so this is about a single component but single component is just to give you an idea of the basic definitions we are interested about not a single component system but a multi component system. So how do we characterize a multi component system? Now for a multi component system let us say that we have an extensive property always extensive properties we write by upper case alphabets. So let us say that capital X is an extensive property capital X is a function of let us say there are 2 components A and B just for simplicity that in the mixture there are 2 components A and B. So capital X is a function of see this is the difference between the pure substance thermodynamics that you have learnt in undergraduate curriculum and you have the now the concept of a multi component system. The property of a pure substance will not depend on the composition of the individual constituents but when you have the multi component system it will in addition to the thermodynamic states it will also depend on the composition NA and NB are the number of moles of A and B okay. So you can write DX at T and P at fixed T and P identically there would have been no change in property if it would have been a pure substance with T and P as independent thermodynamic properties. But now there will be a change because of the change related to the composition first let us write DX not at fixed T P because it will give you a clear idea where from we are getting the expression. So you can write so this is just mathematics rule of partial derivatives the total derivative is sum total of all the total differential is sum total of all the partial differentials okay. Now for the pure substance these were the terms right fixed NA and NB okay. So just like DG is equal to minus SDT plus VDP you can cast that in this form right. But you are now having some extra terms here. So what are the extra terms? The extra terms so you can if you write DX at T and P constant T and P the extra terms will be important because at constant T and P these terms will be 0. So this term then use a color to represent it this term del X del NA at T P and NB this is called as partial molal property A partial molal property and remember partial molal property only refers to extensive property it does not refer to any intensive property. So what this is the change in extensive property X per unit change in the number of moles of the component A that is the partial molal property for A keeping what fixed temperature pressure and number of moles of B fixed. So if in place of B there are in addition to B there are CDEF many other components that means you are keeping T P and all other NJs other than the component I that is only the component I which is A for this case gets change in number of moles and all other number of moles are fixed okay. Now this being a property this is a property so it should depend on composition any property will depend on composition. So when you are allowing the change in number of moles of A that will change the composition. So technically how can it be a property when you are allowing it change in composition to take place. So to resolve this basically we are assuming that an infinitesimal number of moles of A is either added or taken away. So the change in number of moles is infinitesimal practically it is not infinitesimal but practically considering the number of total number of moles which are present in the system the actual change in number of moles may be very insignificant as compared to that. So that practically it does not change the composition of the system. So you can write this as okay. So now if you integrate this so you can write X at T P is equal to X bar A NA plus X bar B NB. With the integration the assumption is that this partial molar properties are constant these do not do not change with the change in the number of moles okay and the assumption that I have told that it is actually the number of moles the change is insignificant as compared to the total number of moles that you are handling in the system okay. So the partial molar property is very important. So you can write in place of capital X if you write the G at T P so you can write G bar A NA plus G bar B NB. What is this G? This is the Gibbs function or the Gibbs free energy. This is the partial molar Gibbs function for the component A in the mixture okay partial molar Gibbs function for the component A in the mixture and this is alternatively given by the symbol mu A. So you can see the analogy between the pure component and the mixture. So with this definition now just like that for the pure component we define something which is a pseudo pressure or a pressure like quantity which is fugacity of a pure component. Now for a mixture we should define something which is fugacity of the component in the mixture. So that we will do in the next slide. So fugacity of a component in a mixture. So consider a component I in the mixture. So for that mu I so I is like the component A mu I is equal to GI bar is equal to del G del Ni at T P NJ, J not equal to I. So you can write this as D mu I is equal to RTD ln f bar I okay that f bar I which is this one f bar I this is fugacity of component I in a mixture that is how you write it. Now look at the limit definition with the limit as what when you are considering an ideal gas state what will be the fugacity? Fugacity of a component in the mixture will become now the partial pressure of the ideal gas that is there. So when you have a mixture of ideal gases you use the concept of partial pressure that is the pressure that it would have exerted had it occupied the entire volume of the mixture. So now what is the partial pressure of an ideal gas that is the total pressure times the mole fraction. So you can write that limit as P tends to 0 fi bar by Yi is the mole fraction of I Yi is the mole fraction of I so the mole fraction of I into pressure okay. So the difference between the pure component and then and the mixture for an ideal gas is for a pure component you have the pressure as it is for an ideal gas the pressure is for a mixture the for an ideal gas the pressure is replaced by partial pressure which is the total pressure times the mole fraction that is the change in basic paradigm. Now we will define or we will try to find out what is the change in volume when you mix that is a very important concept that we will discuss. So when you mix say 2 chemical entities these entities may be ions or whatever when you mix from their individual constituent state to a mixture state there are changes in various properties and one of the very important properties volume. So we will try to figure out that what is the change in volume as you mix. So to do that so let us write this G as a function of T P N A N B in the slides I have generalized by N number of components and used in index I for the ith component and J for any other component but I think that it will you will find the derivation easy to follow if I just use 2 components and then you can generalize it for N number of components that is why in the board I am working with only 2 components. So D G is equal to now let us consider D G at fixed T and N B okay D G at fixed T and N B at fixed T and N B D T will be 0 and D N B will be 0. So this will become del G del P can you tell what is this very easily you can tell I will tell you that how you can quickly tell this. These are the multi component terms forget about this. So the pure substance terms are these 2. So try to see an analogy between D G is equal to minus S D T plus V D P right. So this is basically minus S and this is V. So this will be equal to V and what is this? This is partial molar Gibbs function of the component A or equivalently the chemical potential of A. Now in thermodynamics if you have or just in differential calculus if you have d z is equal to m dx plus n dy then this m is basically del z del x and this is del z del y. So for continuity you must have del m del y is equal to del n del x for the continuity in the second order partial derivative. So take m is equal to this and n is equal to this. So you can write del m del y del v del n A at T P n B is equal to del n del x del mu A or del G A bar whatever you write which one T n A n B yes not P right. So what is this? By the definition of partial molar property this is partial molar volume. So V A is equal to del G A del P at T n A n B. We can write the same thing for the pure component. For the pure component in place of partial molar property it will be molar property. So V A with a bar at the top is equal to del G A del P at constant T constant T n A n B of course because for the pure component n A n B are anyway fixed okay. Now what is the change in volume due to mixing? What is volume before mixing? What is V before? What is volume before mixing? N A V A plus N B V B individual sum of the volumes. What is the volume after mixing? N A partial molar volume plus N B partial molar volume right. This is the mixture property. So the change in volume due to mixing is the difference between these 2. So you essentially require to calculate the difference between capital V A bar and small V A bar and that we will take up in the next lecture. Thank you.