 In the previous lecture we were discussing about electro osmosis and in this lecture we will make an attempt to discuss about some other electro kinetic phenomena. So we will first discuss about the concept called as streaming potential. So to understand what is streaming potential I will first explain it with a schematic in the board and then we will follow up with the slide to get comprehensive picture. So let us say that you have a channel like this the same example that we take that we have taken for the parallel plate channel for example that we have considered for describing the electro osmotic velocity profile. Let us say that the surface has negative charge both the top and the bottom just for simplicity I have drawn only one surface. Now let us say that you have a pressure – dp dx in this direction pressure gradient in this direction. So if you have a pressure gradient so in electro osmosis you are applying an axial electric field in streaming potential phenomena you are not applying any axial electric field there may be say an axial pressure gradient. So what the pressure gradient is doing this pressure gradient what it will do is it will transport the ions. So what is the excess ion in the solution excess ion in the solution is positive again there will be some positive and some negative but net will be positive. So with this fluid flow now net positive ion will be advected in the downstream direction so there will be a positive polarity generated at the downstream end of the channel relative to the upstream end like the relative to the upstream end there will be a positive polarity that is generated. So now what will happen basically you have a dynamically induced electrical potential which is called as streaming potential now what you do with this why this is so important in microfluidics. So there are many applications in streaming potential but in microfluidics you can think of some way by which now what you can do is because you have this potential now you can connect it with an external resistor and if you have an external resistor this potential will make a current flow through the external resistor that means you have converted the hydraulic energy to electrical form of energy. So you can use it as an energy conversion device if you can make it the make it a device which continuously transforms energy from hydraulic form to electrical form and that if you embed in a small chip then you can have something which is called as power plant on a chip. So instead of having a large power plant you could possibly have a small hydroelectric power plant on a chip. So now what are the fluid dynamic consequences of this now when you see that there is a pressure gradient the pressure gradient creates an electric field here induces an electric field here by virtue of the transport of ions. Now these ions by establishing a polarity relative polarity now what it will do it has created an electric field what is the direction of the electric field now the direction of the electric field is this. So in this direction what will happen there will be now an electro osmosis because of this induced electric field back electro osmosis right. So that electro osmosis will try to oppose the forward pressure driven flow right. So there is a forward pressure driven flow that induces an electric field and that induced electric field now opposes the pressure gradient that was responsible for the forward flow. So this is something like the lenses law it opposes the very cost to which it is due and this is what is very common in society in nature everywhere that is you do something good for somebody that person will oppose you that person will be the first person to oppose you in life. So this pressure gradient was a nice factor it induce the electric field now the electric field is opposing the pressure gradient driven flow. So now if you if there is an analyst in fluid mechanics who is not so much bothered about this electrical field for that person it will appear to be a mystery that the fluid flow resistance has as if increased because there is a opposing field now that opposes the driving flow due to pressure gradient. So that means for that person it will appear that as if the fluid viscosity has increased and this is known as electro viscous effect okay. So streaming potential and electro viscous effects these are very important terminologies and we will summarize this discussion through this slide consider no applied e but only applied dp dx. Advective transport of ions may occur due to the applied field. So that mobile part of the EDL moves towards the downstream end of the conduit thus an advective electrical current known as streaming current flows in the direction of the fluid motion. So there are surplus ions these surplus ions are advected with the fluid flow this current is known as advection current or streaming current these are some terminologies that we must learn. However resultant accumulation of ions in the downstream direction sets up its own induced potential field also known as streaming potential. The streaming potential field in turn generates a current to flow back against the direction of the pressure driven flow thereby opposing the primary flow field. The counteracting ion migration mechanism implies that enhanced viscous resistance is failed. In other words the reduced flow rate caused by the back EMF induced flow implies that if the reduced flow rate is compared with the flow rate predicted by conventional fluid dynamics that does not take into account the induced electric field back electric field then an adjustment needs to be made in the calculations through an enhanced effective viscosity. So that a reduced flow rate can be predicted. So this enhanced effective viscosity is due to an effect called as electro viscous effect. Now we will try to develop a mathematical framework to calculate the streaming potential because if you want to design a say small power plant on a chip the first and foremost thing that you need to do is to estimate that what is the streaming potential that will be induced. So let us try to make some calculation towards that. So let us say that you have a channel like this with a pressure gradient driven flow. Let us say that the height of the channel is 2h. So that half height is h. Let us say that u plus is the velocity of the positive ions and u minus is the velocity of the negative ions. So what is the total ionic current per unit width? So n plus is the number density. Each ion has a valency z plus. Each positive ion has a valency z plus and then e is the charge of proton. So each ion will have charge z plus into e and n plus is the number density. So basically it is the velocity of the ion times the charge that is the ionic net ionic current. Now what is the velocity of the ion? What is the velocity of the ion if there is a say na plus in water then how can you calculate that what is the velocity of ion? Velocity of ion is velocity of flow plus velocity of ion relative to flow. So let us write it in words. Velocity of ion is equal to velocity of flow plus velocity of ion relative to flow. So u plus is the velocity of positive ion. What is velocity of flow? u is the velocity of flow plus velocity of ion relative to flow. Relative to flow why ion will move? Advection is with flow because of electric field. So relative to flow ions will move because there is an electric field. This is electrophoresis okay. So what is the velocity of ion relative to flow? So if e is the electric field then what is the force on an ion? On the positive ion is z plus ee right. This force in equilibrium is balanced by the drag force. What is the drag force? The friction factor times the relative velocity. The friction factor if stroke's law is used then 6 pi mu into the radius of the ion right. See how interestingly a classical hydrodynamics is related to the motion of an ion. See this kind of multi physics based paradigm is the heart and soul of microfluidics. So f into v relative. So v relative is equal to z plus ee by f. So u plus plus z plus ee by f. What is this e in this example? In this example e is the induced streaming potential which you do not know. If it was electrophoresis you know what is e because you have applied a field. You must be knowing what field you have applied. But here you directly do not know what is this e. So what is u minus? Similarly u plus z minus ee by f. Let us assume a z is to z symmetric electrolyte. What we have considered for the example of electrophoresis? Similar thing let us consider. So if we consider that then if z plus is z what is z minus? Minus z okay. So you can write this as e z integral y equal to 0 to y is equal to 2h n plus u plus plus. So what we have is sorry minus n minus u minus. Now you write the expressions of u plus and u minus. So e is z integral y is equal to 0 to y is equal to 2h n plus minus n minus u dy. This is one term. Then plus z ee by f integral of y is equal to 0 to y is equal to 2h n plus plus n minus dy. So this current is the current due to the flow field right. This is the flow field. This is the direct movement of charges with the fluid velocity. So this is known as streaming current or advection current and this is because of what? This is because of electromigration. That means transport of ions relative to the flow field because of the electric field. So this is electromigration current and this is advection current or streaming current. Now sometimes it is difficult to calculate the expression for friction factor because electrochemists will commonly not refer to the friction factor. Of course you can calculate the friction factor by using the Stokes law but Stokes law applicability, what is the effective diameter of the ions? These are some uncertain issues. So instead of using the friction factor many people are inclined to use the ionic conductivity. So we will see that how can we express this equation in terms of the ionic conductivity. So to do that let us draw a separate sketch maybe on the left side of the board. Let us say we are writing the Holmes law. For electronic conduction as well as for ionic conduction both we can appeal to the Holmes law. Holmes law is like what? V is equal to IR. So what is V? If E is the electrical field, let us say L is the length of a conductor then E into L is the total voltage. So this is field that means voltage per unit length. So E into L is V is equal to I into R. What is R? R is the, R is equal to resistivity into length by the area of cross section of the conductor. In electrochemistry we commonly use conductivity which is 1 by resistivity. So let that be equal to lambda. Lambda is conductivity. So this is just like R equal to rho L by A. Resistivity is 1 by conductivity. So how can you get the current? Let us say, so the current, so if you have positive ion carriers in the bulk as N0 and negative ion carriers in the bulk also as N0 then you have total carrier. Number density is 2N0. Each entity as valency z and elementary charge is E. So this is the total charge per unit volume because this is number density, number per unit volume. So this is the total charge per unit volume that multiplied by A into L is the total charge in the conductor. That charge per unit time is the current. Now this L by T we can write as the drived velocity VD, the drived velocity of the ions. So you can write EL is equal to 2N0 ZEA is equal to VD. VD into L by lambda A from this expression substituting the value of I. So you can cancel L and you can cancel A. Now we have already seen that ZEE is equal to the drived velocity times the friction factor. So in place of drived velocity we can write ZEE by F. Therefore this equation will give E is equal to 2N0 ZE into ZEE by F. So Z square E square E by F lambda. So see in this expression what we need to calculate is ZEE by F. Please correct me if I make any algebraic mistake. So ZEE by F is equal to ZEE by F is equal to what? ZEE by F is equal to lambda by 2N0. Somewhat one extra ZEE has come. I do not know. Let us see. So there is another ZEE. ZEE was multiplied with this. So there is another ZEE with this. So Z square E square by F is equal to lambda by 2N0. So here it will be Z square E square not ZEE because there was an outside ZEE and with that this ZEE is multiplied. So the ionic current becomes EZ integral of Y equal to 0 to Y is equal to 2H N plus minus 2H N plus minus N minus UDY plus lambda by 2N0 E integral of N plus plus N minus DY Y is equal to 0 to Y is equal to 2H. So this electro migration of ions actually creates ionic conduction. So sometimes this component of current is also known as conduction current. So this is advection current and this is or streaming current and this is conduction current. The velocity field what is U here? U is equal to U pressure driven plus U back electro osmosis electro osmotic. So here you have a pressure driven flow in the forward direction and backward electro osmotic flow because of the streaming potential field that is developed. So let us go to the slide and we will use the expression for U. Let us go to the slide. Let us go to the, yes. So we are going to use the expression for U. So to get the expression for U what we are doing is we are solving basically the Navier-Stokes equation with the left hand side equal to 0 but now we have a DPDX. So fundamentally it is like a combined pressure driven and electro osmotic flow. So you can write the velocity because it is a linear differential equation. You can write the resultant U as a combination of U pressure driven plus U electro osmotic linear combination because it is a linear differential equation. So you can see this is the pressure driven velocity profile and this is the electro osmotic velocity profile which we derived in the previous lecture minus epsilon zeta e by mu into 1 minus psi by zeta. So this U we can write, this U is equal to 1 by 2 mu DPDX 2 h y minus y square minus epsilon zeta e by mu into 1 minus psi by zeta. So this what is this E? This E is the streaming electric field. Similarly this E is also the streaming electric field. Now we use a condition, a criterion which is physically very, very important. What is the net ionic current in the system? You have applied no external field. So when you have applied no external field that means the net ionic current must be 0. That means the advection current and the conduction current or the streaming current and the conduction current should just balance each other so that the net ionic current must be 0. So if you set this equal to 0, you will get an expression for E s, an algebraic expression which is with simple algebraic manipulation by setting this equal to 0. So this is a function of E s. E s will come out of the integration and here E s has come out of the integration. So you will get E s equal to something. I will show you the expression through the slide. It is quite a lengthy expression but I mean once you understand what is the physics, it is very, very trivial to write that equation. Just straightforward one more step you can write the expression. So that is the expression of the streaming potential as a function of the dp dx. So let us go to the slide. We will see the expression for the streaming potential. So what we have done here is we have replaced the n plus and n minus through the Boltzmann distribution. So you can see this sin h cos h these terms have appeared and you can express this in suitable non-dimensional forms also if you want by expressing y non-dimensional equal to y by h, psi non-dimensional equal to z e psi by 4 k B T like that. So this is not important. The important is the concept that we have employed that we have gone through to understand that what is streaming potential, how is it induced and what are its practical utilities. Next concept in electrokinetics that we will learn today is electrophoresis. So we have talked about electrophoresis loosely earlier but we will discuss about its mathematical treatment today. Interestingly both electro osmosis and electrophoresis refer to the relative movement of a chart surface and a liquid. The similarity between electro osmosis and electrophoresis is striking if we fix our frame of reference on the moving electrophoretic particle. So if we have a particle we fix our reference frame with that particle then from such a view point the relative motion manifest itself just like electro osmosis of liquid pass the particle. So if you attach your reference frame with the particle then sitting on the particle it will appear that as if the particle is stationary and liquid is moving past it. So it is a matter of from which reference frame you are looking into the flow. So fundamental mathematical consideration should be identical because it is the either you assume either you sit on the solid surface which is stationary and you see fluid flow past it that is electro osmosis or you sit on the moving solid body and see the relative motion of the liquid sitting on it both are the same in one way or the other. The particle which is undergoing electrophoresis can be any charged particle. It can be a colloid, a macromolecule or even a microorganism. The chart may be intrinsic like a DNA and can be an induced one like you can have an electrical double layer formed around a particular solid object so that the object gets a net charge. The spherical particle concept is representative of many physical situations for example many colloids can be approximated as collection of spherical entities and also there is ease of mathematical analysis if we consider spherical objects. So with this understanding we will first find out that so if there is a charge on a sphere which is intrinsic like the let us say the charge on an object which is intrinsic like the DNA. So that is one case but the other case which is relatively non-trivial is that there is no intrinsic charge but there is a charge formed due to electrical double layer formation. So how can you calculate the net charge for such a system? So we will consider the governing equation for electrical double layer potential around a sphere. So earlier we have considered electrical double layer potential distribution on a flat surface. Now we will consider electrical double layer potential distribution on a sphere. So for a sphere so our objective of this mathematical analysis is to obtain the electrical double layer potential distribution around the sphere. So the governing equation okay so it is just like d2 psi dy2 equal to – rho e by epsilon in the spherical coordinate system in place of d2 psi dy2 it is 1 by r square d dr of r square d psi dr. It is just a manifestation of the coordinate system that we are using. The physics remains the same. Now what is the charge density? Let us assume that you have a z is to z electrolyte, symmetric electrolyte. So what is rho e? Let us assume that the Boltzmann distribution is valid. So this equation becomes this is a minus sign right. So this equation becomes 2n0 ze sin h ze psi by kbt by epsilon. Now we will make d by Huckel linearization that is we will assume that ze psi by kbt is small. So that sin H ze psi by kbt will be approximately equal to ze psi by kbt. So this becomes 2n0 z square e square by epsilon kbt psi. What is this? 1 by lambda square where lambda is the d by length. So our governing differential equation is 1 by r square d dr of r square d psi dr is equal to psi by lambda square. To solve this equation we will apply a simple transformation z is equal to r into psi use may be a different z we have already used for valency say w is equal to r into psi. So psi is equal to w by r. So what is d psi dr? We are calculating this term. What is d psi dr? 1 by r d w dr minus w by r square. What is r square d psi dr? Multiply with r square r d w dr minus w. What is d dr of r square d psi dr? So d w dr plus d 2 w dr 2 minus d w dr. So these two get cancelled. So it becomes r d 2 w dr 2. So 1 by r square of d dr of r square d psi dr is equal to 1 by r d 2 w dr 2. So this governing equation becomes 1 by r d 2 w dr 2 is equal to 1 by lambda square in place of psi you write w by r. So this 1 by r gets cancelled. So d 2 w dr 2 minus w by lambda square equal to 0. What is the solution of this? This is very straight forward w is equal to now we can write in exponential form also. This is c 1 e to the power r by lambda plus c 2 e to the power minus r by lambda. So what are the boundary conditions? r tends to infinity what is psi? So you have a sphere. You have electrical double layer formed here. At an infinite distance the psi is 0. So at r tends to infinity psi tends to 0 that means w is what? When r tends to infinity psi tends to 0 so r w must be finite. So what is the consequence of that boundary condition? c 1 is 0. The other boundary condition at small r is equal to capital R. Capital R is what? The radius of the sphere at small r is equal to capital R psi is equal to zeta that means w is equal to r zeta. So that means you have r zeta is equal to c 2 e to the power minus r by lambda that means you have c 2 is equal to r zeta by e to the power minus r by lambda. So your potential field is w is equal to c 2 is r zeta e to the power minus r by lambda by e to the power minus capital R by lambda and w is r into psi. So psi is equal to r zeta by lambda sorry r zeta by small r into e to the power minus r by lambda by e to the power minus capital R by lambda. Now how can you calculate what is the total charge on the surface of the sphere? How can we calculate this given this information? See let us calculate what is the total charge in the electrical double layer? The charge on the surface is equal to minus of the charge in the electrical double layer because the total system is electrically neutral. So what is q ideal rho e into 4 pi r square dr because dv the elemental volume is 4 pi r square dr in the spherical space. Radius equal to capital R to radius equal to infinity. So what is rho e? 1 by r square d dr of r square d psi dr this into minus epsilon right this is the Poisson equation. So now if we write this, so this becomes minus 4 pi epsilon r square and 1 by r square gets cancelled. So integral of dr square d psi dr from r to infinity. So this is minus 4 pi epsilon. What is d psi dr at infinity? That must be 0. So this becomes 4 pi epsilon r square d psi dr at r. So let us calculate d psi dr at r. So psi is equal to r zeta by small r e to the power minus small r by lambda by e to the power minus capital r by lambda. This is that expression in the board which is already there. This expression from which we have written that. So d psi dr is equal to r zeta by e to the power minus r by lambda. So what is d psi dr at capital R r zeta by e to the power minus r by lambda minus 1. Let us take the minus common r common e to the power minus r by lambda into 1 by lambda plus 1 by r. Please let me know if there is any mistake. This is alright. So you cancel this r and cancel e to the power minus r by lambda. So minus zeta into 1 by lambda plus 1 by r. Therefore q edl is equal to 4 pi epsilon r square minus 4 pi epsilon r square minus 4 pi epsilon r square by zeta into 1 by lambda plus 1 by r. Now what is the charge on the sphere? It is minus of the q edl. So q surface this minus becomes plus. Zeta will be in the numerator right. This is the q surface. Now this we will consider 2 limiting cases. One limiting case is that the sphere is like a point mass that means r is much much less than lambda. Then you can approximate it just like a particle. This is case 1. And case 2 sphere is very large and electrical double layer is thin so that you can consider r much much greater than lambda. So we will take up this case quickly. So if you consider the sphere as a particle then you can add a small spherical particle. You can apply the Stokes law. So you can write q surface into e is equal to 6 pi mu into u sphere. So that means you can write 4 this is for case 1. 4 pi epsilon zeta r square into 1 by lambda plus 1 by r is equal to 6 pi mu u sphere into r. So from here you will get an expression for u sphere. What is that? 1 e will be there. This e has to be there. So u sphere this pi gets cancelled. One of the r gets cancelled. So 2 by 3 epsilon zeta e by mu is equal to 2 by 3 epsilon zeta e by mu is equal to into 1 plus r by lambda. And we have considered a case when r by lambda is much much less than 1. So this is approximately equal to 2 by 3 epsilon zeta e by mu. Compare with the Helmholtz-Moluchowski velocity epsilon zeta e by mu. Here a 2 third coefficient has appeared. So let us summarize this discussion so far what we had made. So we have found out what is the total surface charge on a spherical particle and then the case 1 when r is much much less than lambda and we will consider this case today before we conclude. So we have got that the u sphere which is u written here in the slide is 2 third epsilon zeta e x by mu. And this equation is very famous in the field of electrophoresis this is known as Huckel equation. But there is another case which is also a very interesting case when the electrical double layer is very thin as compared to the radius of the particle. Then in that case what happens? We will take it up in the next lecture. Thank you very much.