 In our previous lecture, we started discussing about nanofluidics and we were essentially discussing about some of the important physical issues which may not appear to be important over the micro scale, but in the nano domain those issues may appear to be important typically some interaction forces and so on. Then we started discussing about problem of filling of a nanoscale capillary or a nanopore which is very important from the nanotechnological point of view and we looked into the or we rather revisited the Lucas-Warsburn equation with a modification to take into account the nanoscale effects. So, in the Lucas-Warsburn equation we got something we got an expression which is a modified version as compared to its original form to include the effect of slip. Now the question was that how can we describe the slip that is what should be that appropriate slip length, how can we get the slip length. So, if we can get the slip length we can use the Lucas-Warsburn equation to predict the capillary filling characteristic the displacement velocity acceleration as a function of time, but the question is how do we get the slip length. So for that we need to do molecular dynamic simulations and we will discuss about the molecular dynamic simulations in the later part of this lecture we will start with that, but now I will try to interpret some of the important results because the objective of this particular lecture is to give you some glimpse of the physical phenomena at the nanoscale. So, if you find if you look into this graph you will see that L square versus T see the Lucas-Warsburn model gives L scales with square root of T right, so L square versus T is a sort of linear is a linear in fact it is linear because L scales with square root of T. So, the filling rates what we have found that like if you see the filling rates the filling rates are higher for greater weightability, so for different contact angles the graphs are plotted and this is quite obvious that like the because the capillary action is stronger. However, one has to keep in mind that although the capillary action is stronger, but you also have a resistance force and the resistance force will depend on the slip and slip depends on weightability. So, it is a complex coupling actually you cannot just give reasoning from a straight forward intuitive argument that if this is the weightability this should be the capillary filling because with weightability issues slip issues also come into the picture and based on the slip you have the resisting viscous force. So, all these things have to be taken into account, but you can see that more or less all these characteristics, so L square proportional to T is similar to Lucas-Warsburn model the classical Lucas-Warsburn model, but whatever is the classical Lucas-Warsburn model prediction the filling rate is slower than the that predicted by the classical Lucas-Warsburn model and there is a gradual decrease in slope of the meniscus as seen in this plots. So what we try to do is that like we try to look into various issues like whether there is any role played by the dynamic contact angle with molecular dynamics simulations which are essentially like simulations of the direct dynamical features or explicit dynamical features of the molecules you can extract all sort of data possible. So we can ask ourselves a question that is is dynamic contact angle important is the variation of viscosity close to the wall important is the slip important what is the slip and so on. So, to get the slip it is very difficult to directly get the slip length from capillary filling problem, so what we try to do is that we try to apply an equivalent driving force and try to predict the slip length from the corresponding pressure driven flow simulation. So how do you predict slip length from a pressure driven flow simulation, so basically you have a pressure driven flow you extrapolate the velocity profile and see where it matches the 0 condition from wall to that length is the physically the slip length. So the slip length will vary with the contact angle and it is quite intuitive that slip length will vary with the contact angle. Now that is true and it has been known for quite a long time and the reasoning is quite obvious because the contact angle will determine the wettability of the substrate and the wettability of the substrate has a strong role to play in deciding the slip. So slip length should be depend on the contact angle but what we have found out is that slip length is not just a function of the contact angle but also a function of the driving acceleration. So in this graph you see we plot the slip length as a function of contact angle for different non-dimensional driving acceleration a star which we will define in the next slide what is a star. So and there is a parameter n which determines the roughness of the substrate. So it is a roughness wettability combination along with the driving acceleration that decides the slip length. So to understand that we have first plotted the slip length so all these are normalized with respect to some parameter. So this sigma is a molecular length scale so the slip length is normalized with respect to the molecular length scale it is plotted as a function of 1 by acceleration. Acceleration is surface tension force by mass. So it is a sort of driving acceleration right because the driving force is the surface tension force that divided by mass is a driving acceleration. Now you can see that the data is scattered depending on smooth surface like for different values of the surface roughness parameter and so on. Now the scattering of the data shows almost a universal genetic characteristic if you non-dimensionalize the plot that means the y axis is already non-dimensionalized if you make the x axis a non-dimensional acceleration a star okay. So if you so you can see here that it is basically you normalize the driving force with the viscous resistance. See in the numerator you have the driving force and the denominator you have the viscous resistance scale wise. So it is a relative driving force. So if you normalize the a in terms of the relative driving force that is the driving force as compared to the resistive force then the entire set of scattered data is fitted by a common function. So this LS versus A star is a universal variation and this variation you can see does not depend on the depend on anything else. But it depends on the contact angle because in the A star there is contact angle okay. So we have got now the slip length. So we have now got the answer to the question where from we will get the slip length. From synthesizing the molecular dynamic simulations and by casting these simulations in a non-dimensional form we have been able to get a slip length non-dimensional slip length as a universal function of non-dimensional acceleration. And this functional form we will use in the Lucas-Warshman equation. So just to recollect this is the Lucas-Warshman equation and here in the slip length we have plotted it as a function of non-dimensional acceleration in which there is theta. So it is not a very simple function of theta because here also there is contact angle in the right hand side also there is contact angle. So dependence on contact angle is something which needs to be worked out depending on how Ls varies with A star. Now with that variation if you plot this theoretical estimation that is the prediction. So if you integrate this equation now you will get L as a function of t and then L square as a function of t. So if you plot L square as a function of t that is the form lines and the scattered lines are molecular the scattered data points are molecular dynamics simulations. You can see that there is such a nice match between the Lucas-Warshman model prediction and the molecular dynamics simulation. So I just want to give you a philosophical outlook that this is many times the hallmark of nanofluidics research in using molecular dynamics. So using molecular dynamics we try to get some information which is not possible to be obtained by continuum calculations. There is no way by which by continuum calculations you could get slip length as a function of driving acceleration non-dimensional acceleration. Once that information is obtained then the history of that information is no more important and then you can forget about the molecular dynamics. You can use a simple one-dimensional model in which you plug in that like a constitutive behavior. You plug in that information and then get the result which can actually reproduce the molecular dynamics simulation results. The advantage in this process is that you do not have to do molecular dynamics simulations all the time. Once you get the slip length versus non-dimensional acceleration data ready with you it is like a database then you do not have to do molecular dynamics simulations again and again. You can just use the Lucas-Warshman model in which you feed this data and you see that it remarkably agrees with the molecular dynamics simulation. So this is a kind of paradigm which we very commonly use for research in molecular simulations or research in nanofluidics. So it is not just brute force molecular dynamics simulations but some information from the molecular dynamics simulations to be plugged in with a continuum model. So it is a modification of the continuum model so that if the continuum model is able to get a better predictive capability that is in turn important and interesting for using in nanofluidics applications without requiring molecular dynamics all the time because molecular dynamics you know is a very good tool but it is computationally very expensive. So the computational time is very significant and you cannot simulate a large system the number of atoms that is restricted. So there are several restrictions although information wise you can gather molecular level information and that has tremendous like fundamental or basic principle level information within the science that is addressed by the problem but we have to understand that it is computationally quite involved. So if we can somehow make a sort of like an arrangement where we use molecular dynamics for a specific purpose and then use the molecular dynamics information to modify the continuum model that sometimes serves as a modified continuum model which you can use even in the nanofluidic domain. So this first example talked about the weightability issue now the roughness so you can use a roughness parameter n there is actually an interaction function in the molecular dynamics which in which there is a parameter n. So the L square versus T for different values of n so we can see that the roughness we can incorporate in the acceleration parameter by defining a non-dimensional acceleration a star rough equal to a star smooth plus a correction parameter that depends on the roughness which varies with varies exponentially with the roughness parameter. So we can incorporate roughness weightability coupling which is very important in the small scale domain and the effect of the driving force. So all these parameters come into the picture to decide the slip length and that slip length when incorporated in the Lucas-Walshburn model fits the final results very nicely. Yes that functional form we take the functional the universal slip length versus a star that I have shown you the graph this graph. So you can see even its fitted form is written in the legend I mean it may not be very easily visible this one. So you have a fitted form of this. So you can use that fitted form directly so that it becomes analytically tractable or you may have to do a simple numerical integration at the most. So that reduces the cost significantly. Now the next example of nanofluidics or modification in the nanofluidic domain that I will discuss about is electrical double layer phenomenon how is it modified in the nanoscale per view. So just a quick revisit these things we have discussed in length in this course but like if we have a charge surface then there may be a charge interfacial layer in vicinity of the charge surface so that the entire system is electrically neutral. This charge interfacial layer is also known as the electrical double layer this charge layer means the charge surface plus the fluid together is known as the electrical double layer. So if you have the electrical double layer we have shown that under certain assumptions and we will discuss about the sanctity of these assumptions and what are the issues when they are modified. So under those assumptions you can write the Boltzmann distribution for the number density distribution within the electrical double layer and the Poisson equation see one very important thing that you should keep in mind that the Boltzmann equation has a lot of assumptions we have discussed about that. The Poisson equation does not have any assumption because it follows directly from the Gauss law right it is basically if you start with the integral form it is a differential form of the Gauss law which can be derived by using the divergence theorem in association with the integral form of the Gauss law. So the Poisson equation is a bit is much more universal than the Boltzmann distribution right. So there is absolutely no question on applicability of the Poisson equation in the nano domain but substituting the Boltzmann distribution in the Poisson equation that may be questionable. So it is not the sanctity of the Poisson equation but sanctity of the Poisson Boltzmann model that is the Boltzmann equation substituted in the Poisson equation. So it is a sanctity of this equation that may come into question. Why that may come into question is because we had made some assumptions while deriving the Boltzmann distribution. These assumptions I had discussed about discussed in the class it is exactly the same slide what we presented while discussing but I just want to recapitulate that because we will see that what happens if some of these assumptions are not valid. So the first assumption is ion surpoint charges and the system is in equilibrium with no macroscopic advection diffusion. The solid surface is microscopically homogeneous. The charge surface is in contact with an infinitely large liquid medium. The strength of the electrical double layer field significantly overweighs the strength of the applied electric field and we have seen that that is quite justified and the first stream boundary condition is applicable. Now can you say that out of these assumptions which are the assumptions which are likely to be strongly violated as you go down to the nano domain. See the first assumption is one of the big sources of the discrepancy. Ions or point charges these assumption is valid provided the system length scale is significantly greater than the ionic length scale. But if the system length scale itself is few nanometers then you no more can consider ions as point charges and then finite size effects of the ions needs to be appropriately considered. Then some of the other assumptions are still okay but the first stream boundary condition that may be questionable if there is electrical double layer overlap and electrical double layer overlap is important in the nano scale domain because you say typical divine lengths you have seen are of the order of few nanometers. Now if the channel height itself is of the order of few nanometers then the characteristic length scale of the channel and electrical double layer length scale are comparable. Then it is possible that electrical double layers formed at the opposing walls they tend to interfere therefore that will not rise to a condition that the psi equal to 0 at the center line. The ideal potential is 0 at the center line. So that is one restriction. Not only that the Boltzmann distribution does not consider any other form of interaction which may be important in the nano scale domain. Some of the interactions that we have discussed like the solvation interaction, structural interaction these things the Boltzmann distribution does not understand all this. But this may be important because see why solvation interactions may be important because ions may form hydration shells and they may be solvated by or hydrated by water molecules in a solution. And then solvation interactions can play a big role in altering the net electrochemical potential or the interaction potential that is not considered explicitly by the Boltzmann distribution. So you have to make certain modifications to the Boltzmann distribution to make it compatible with the nano domain before integrating it with the Poisson equation that is the whole philosophy based on which the subsequent discussion is evolved. So some additional considerations. So what I have done is I have not purposefully gone into the mathematics behind some of these additional considerations because I mean those are not for elementary level understanding. But I have jotted down the physics which is responsible for the modified understandings. I have marked the corresponding references the papers with red colour so that if you are interested you should read these papers for getting the state of that understanding. So I will go through these considerations one by one. Ions in a polar fluid experience a reduced dielectric response near the solvent substrate interface. Ionic charges interact with the surface because of the field reflected by the surface on being polarised. This reflected field this is the concept called as image charge. What is the concept? This reflected field is the same as if it is like a reflection there was an image charge on the other side of the surface at the same distance. It is a hypothetical concept. If epsilon be the permittivity of the aqueous medium and epsilon prime be the permittivity of the surfaces then an additional repulsive force due to this image charge interaction will occur if epsilon prime is less than epsilon. So this is an additional interaction beyond the Poisson Boltzmann picture. Ions in iron solid attractive and repulsive interaction that is the in combination you can talk about that as a Lennard Jones interaction. We will discuss about the Lennard Jones potential when we discuss about molecular dynamics that it is basically a combination of an attractive and repulsive potential very commonly used for molecular dynamic simulations is an additional interaction that can be considered beyond the Poisson Boltzmann description. Then the third point certain short range forces come into play when two surfaces approach closer than a few nanometers. Short range oscillatory solvation forces of geometric origin arise this we have discussed earlier. See now how we can incorporate that in electrokinetics. See this is what is so nice about this physical chemistry over small scales that you can incorporate or you you should attempt to incorporate some of these issues in a electrokinetic model which is based on a different formalism. But I will show you that how can you incorporate that. So short range oscillatory solvation forces of geometrical origin arise whenever liquid molecules are induced to order into a quasi discrete layers into quasi discrete layers between two surfaces or within highly restricted spaces. This is ultra narrow confinement. Additionally surface solvent interactions can induce orientational reordering of the adjacent liquid all we have discussed all these and this can give rise to a monotonic solvation force that usually decays exponentially with surface separation. There is an additional free energy component to create an ion size cavity in the fluid that is to solvate a solute with no attraction to the solvent. This is known as hydrophobic solvation energy. So you can get an expression of the hydrophobic solvation energy if you go through this interesting reference published in PRL. Mobile counter ions in the electrical double layer diffuse part of the electrical double double layer constitute a highly polarizable layer at each interface. These two opposing conducting layers experience an attractive van der Waals force known as the ion correlation force which becomes significant only for distances less than typically less than 4 nanometer. So these kinds of forces we normally do not bother at all about in the macro in the micro scale domain. Even in the nano scale domain greater than 4 nanometer separation we do not care about this. So but physics tells that these ion correlation forces can be important with less than 4 nanometer. Effects of sizes of the ions this is important we considered ions as point so point masses point charges while deriving the Boltzmann distribution because nowhere the diameter of the ion explicitly came into the picture while making the derivations. But effect of finite sizes of the ions what they what these effects do they tend to enhance the repulsion between 2 surfaces. This is analogous to the increased osmotic pressure of a van der Waals gas due to finite sizes of the gas molecules. In a very similar manner finite sizes of co ions and counter ions contribute to enhance repulsion. In cases of co ions adsorbed on the surface the repulsion is nothing but steric repulsion between the overlapping stern layers. As such since Poisson Boltzmann equation does not take into account the finite size of the adsorbing ions the ionic concentration close to the surface can easily exceed the maximum allowed coverage. Because technically if it is a point charge there can be infinite charge density that can be allowed but in reality you cannot allow that so that is a big limitation of the Poisson Boltzmann model. This anomaly may be resolved by considering an entropic entropic means basically size based contribution to the free energy that is repulsive in nature. These interactions mimic the fact that ions of finite sizes undergo hindered transport in the concentrated solution without having specific interactions with the substrate and again there are several references which talk about this and there are models. How to overcome these limitations? So we understand that the Poisson Boltzmann model needs to be corrected. Question is how it can be corrected? Through a modified potential in the Poisson Boltzmann equation that is one possibility that we modify the ideal potential with some different potential with some augmented potential that incorporates these interactions. Through a modified free energy description we can start with the fundamental free energy itself and then we can find the derivative of the free energy and set it to 0 to set the condition for equilibrium and then we will get some modified version of the Poisson Boltzmann equation. So as an example take the second approach that we modify the free energy. So I will give you one particular example of how to take into account the finite size of the ions. So we start with the free energy. In the free energy we first write one component I will explain all these terms. Self energy of the electrical field minus epsilon by 2 grad psi square then electrostatic energy of the ions if you have z is to z symmetric electrolyte it is z en plus psi minus z en minus psi. So fundamentally it is summation of z i en i psi. So this is about the electrical component. Now what about the finite size that will give rise to an entropic component. So you know the free energy like if you discuss about the Helmholtz free energy it is the internal energy minus Ts right. So the minus Ts component is this. So to write this component in fact one can correct this and there can be several directions of research towards that. For a simple analytical derivation what people have done and we have shown that in the slide that you have taken the size of the positive ions and the negative ions are the same. But in reality there is a big difference in paradigm the anions and cations in a system they may be grossly varying in size. So even in a simple system like NaCl like Na plus Cl minus their sizes are grossly different they are not of the same size. So this is just for analytical description without sacrificing that essential physics. So this is the entropy of the positive ions this is see so this is n plus in the number density. So this is entropy of the positive ions. So you can recall that in the free energy expression there was a term log of concentration when we derived it in the Boltzmann distribution that essentially is modified with the size parameter effect okay. So this is entropy of the positive ions this is entropy of the negative ions and there is also a solvent. So 1- so if 1 is the total fraction then basically 1- positive ions- negative ions is the contribution of the solvent. So that is how this formula comes very simple and straightforward. Only assumption that length scale of the positive and negative ions are the same and the volume scales with A cube if A is the length scale. So the A is not normally ionic radius it is an equivalent length scale again to capture the it may also include the hydrated radius instead of the normal radius. So sometimes it is the effective radius. So one should not confuse it with the actual ionic radius better to say an effective length scale representing the ionic size that is the fundamental way of looking into it. Now how do we define the chemical potential? We basically differentiate the free energy with respect to the number density or the concentration. So that will give rise to these kinds of terms. So you can see that with number density when you differentiate this term does not come into the picture but this term comes and the entropic term comes. So these 2 terms are there and this is constant for equilibrium no, no this is Helmholtz u-ts. I mean see let me answer this question more carefully. I mean this deals with I mean one has to have very rigorous thermodynamic background to appreciate that. When do you use Gibbs free energy and when do you use Helmholtz free energy? It depends on what it depends on the context like if you are interested to couple it with the transport phenomena we commonly use the Gibbs free energy. The reason is that there you are talking about enthalpy which basically deals with the thermal energy of a flowing system or a flowing system. But when you are discussing about a system which just gives the thermodynamic picture but not the transport picture then we are essentially bothered about a system which is not a flowing system. So if it is a non-flow type of a system we do not use the enthalpy but we use the internal energy. Even if you look into the first law of thermodynamics you see that when it is a non-flow process we use a internal energy based consideration for the first law. When it is a flow process we use the enthalpy based consideration because in the flow process you require an additional form of energy which means the energy that the fluid must have to maintain the flow in presence of pressure that is called as flow energy or flow work. So all these things I did not want to bring all this because this is not a thermodynamic class but because you raise this question I made this important remark. So does not matter whether you use the Helmholtz or the Gibbs free energy provided you know what is the context in which you apply. So in this particular case there will be no difference because you are not having any flow in your consideration. So Helmholtz and Gibbs will give the same thing here. There is no question of any PV term here. So you can write this as Helmholtz or Gibbs whatever for this case. So the modified Poisson Boltzmann equation this is equal to derive that what is so this is equal to constant the potential is constant now you differentiate that and set it to 0. So to do that you will get this I am not going into the algebra I am just giving the concept. So then you can make this substitution this is just for the algebraic understanding then you can get this nice differential form. So a logarithmic form with a correction factor now p is equal to 1-2n0a cube. So 2n0a cube n0 is what n0 is the bulk concentration. 2n0a cube is a factor that comes into the picture because of the size effect of the ions. So like 1n0 into a cube for the positive ions another n0 into a cube for the negative ions. So total 2n0a cube this is called as steric factor. So this steric factor if you include you can write equations for n plus and n minus. So you can see these are modified versions of the Boltzmann distribution. If you set a equal to 0 then you will find that this boils down to the Boltzmann distribution. So your modified Boltzmann distribution is n plus equal to this and n minus is equal to this. Obviously there are certain issues that come into the picture. One important consideration that leads to this derivation is that we have used the far stream boundary condition at the center line. So this model is not essentially valid with electrical double layer overlap. Slight overlap is fine. Slight overlap it will still work. But with strong electrical double layer overlap if you can modify this and bring an analytical formalism to this that actually is a new research topic and that has not yet been done in the literature. To bring the electrical double layer overlapping phenomena in the same framework as this with analytical tractability. Without analytical tractability people have done. But with this kind of analytical tractability but with electrical double layer overlap something which is not that straightforward to do. So this does not consider electrical double layer overlap. Now for equilibrium you should also have the derivative with respect to psi equal to 0. The two parameters were n plus n minus and psi. So we said already that derivatives with respect to n plus n minus 0. Now we set the derivative with respect to psi equal to 0. So that gives the nothing but the Poisson equation. So see the sanctity of the Poisson equation is not disturbed and you get the same thing. Why? Can you tell why? The reason is very straightforward. There was an entropic correction but in the entropic correction nowhere you had a dependence on psi. Because there was nowhere a dependence on psi the entropic correction will not correct the Poisson equation. So that is why it is a equation of such a fundamental importance that is the Gauss law. So Poisson equation is actually the mathematical form of the equation. Better to say it is a Gauss law. So you can see that this is the modified Poisson Boltzmann equation that take the steric effect or the finite size effect of the ionic species into account. So how can you get a generalized DDL model? So we have discussed that how to incorporate one of the effects. But we could see there could be many other effects like the solvation interaction, image charge effects, so many other effects are there. So describe the free energy functional concerning containing the pertinent interactions. So in the free energy there may be additional expressions because of additional interactions. Obtain the chemical potential which is essentially electrochemical potential by differentiating the above with respect to ni. Constancy of chemical potential gives the equilibrium ni distribution. Say derivative of f with respect to psi to 0 to get the governing differential equilibrium for potential distribution. This is nothing but the Poisson equation. Modify the potential to accommodate additional effects if you have not already accommodated through the free energy. So there are 2 ways in which you can do. One is you directly write the corresponding contribution in the free energy. Or if it is difficult for you to express that in form of the free energy, you write the free energy of the base case, find the chemical potential by differentiating this and add additional interaction potential with that chemical potential. So that is another way of looking into this. So then for modeling electrical double layer overlap, you have to write additional equations. Write an equation representing the global number conservation of particles. Write an equation representing the global charge neutrality constraints. Because like if you have non-zero potential at the center line and non-bulk value of the density, charge density at the center line, somehow you should constrain those values. These value constraints should come from the overall mass balance and overall charge balance. So that needs to be worked out carefully. So then you can get a closer model and describe the most general form. I can tell you that it is easier said than done. There are many effects which people know but they have not been yet formulated in the framework of a modified Poisson Boltzmann model and that gives a big open area of research of electrokinetics in the nanofluidic domain. So some applications, when a general chemicals, so separation. So let us say when, so this is something to do with the concept of dispersion that we have discussed. Now we will see that what are the important considerations if you come down to the nanodomance. So when a general chemical sample is introduced in a channel, it is a mixture of a number of analytes. Within the channel, certain effects are imposed on these analytes and they respond differently to these effects. For example, electrical effect. So based on size, the electrophoretic velocity will be different. So as a result, they acquire different velocities so that they reach the channel exit at different times allowing them to be collected separately. Thus we obtain each individual component of the mixture and the components are said to be separated. The principles by which the imposed or induced effects operate on the analytes depend on a complicated interplay between the analyte the channel and the flow field characteristics and decide the efficiency of the separation process. So the analytes form distinct bands. Actually we discussed this in the context of dispersion and these bands reach the channel exit at different times and we say that the analytes have got separated. There are 2 factors which are of importance. Average velocity of the band which is called as band velocity and spread of the band which is called as dispersion. So I will not go through the details of dispersion because we have already discussed about this but what I want to say is that now in nanofluidic domain additional factors will affect this. One important factor is hindered diffusion. We have discussed about hindered diffusion in the nano channel. Now you do not get the diffusion coefficient as the bulk diffusion coefficient because of the extreme hindrance created by the narrow confinement. So you have an altered diffusion coefficient. So that will result in a different dispersion characteristic. Now you in addition to that in a electrokinetic separation you will have the hydrodynamic and the electrophoretic effects. I have discussed about hydrodynamic interactions and hydrodynamic interactions in a confinement are different from hydrodynamic interactions in a bulk scale. So hydrodynamic interaction will influence the flow field. So there you can remember that because of the presence of a nearby particle there is a perturbation in the flow field and that perturbation in the flow field will disturb the Poiseuille flow. If it is a pressure driven flow or it will disturb the Helmholtz-Moluchowski type of velocity if it is an electroosmotic flow or in addition you have electrophoretic effects and electrophoretic effects will act on the particle. Then so molecular size in nano channels molecular size may become comparable to the channel height ensuring a greater hydrodynamic influence. Significant protrusion of electrical double layer in the bulk ensures a non plug like velocity profile. So the uniformity of the velocity profile in the electroosmotic flow that is based on thin electrical double layer assumption because the gradient will be there within the electrical double layer. If the electrical double layer is thick then the gradient will be very strong and then the uniform velocity profile will be lost that will significantly affect the dispersion process and the separation process okay. So not only that interaction of molecules with the channel wall will become important in the nano channel base separation. So the interaction is basically the van der Waals interaction and the electrical double layer interaction between the ideal of the particle and ideal of the wall. Now all these interaction forces are important in the nano domain. So while working out the dispersion characteristics you have to include these effects in addition to the hindered diffusion. So I mean you can look into a significant amount of significant volume of research article in the literature on nano channel base separation under electrical double layer phenomenon. So there are significant issues of that. So there are several applications of nano fluidics like desalination of water, DNA transport in nano channel, energy applications and so on. We will discuss about some such applications in the subsequent lectures but before that since we introduced at least the idea of molecular dynamics I will try to share with you some important aspects of molecular dynamics before we get into the applications of micro fluidics and nano fluidics and we will mainly discuss about bio and energy related applications in this particular course. So we will now move on to the molecular simulations. So please do keep in mind that this is not a course on molecular dynamics. So I do not have the opportunity of giving you all the details which should go with understanding molecular dynamics but my objective is I want to give you some fundamental ideas and concepts based on which if you are interested you can easily get started with molecular dynamics. So the outline of the molecular dynamics simulations which we will cover partly in this lecture and partly in the next lecture, scales of analysis, concept of a mean or average, flow modeling beyond continuum, basics of molecular dynamics simulations, interaction potentials and running a simulation, some practical issues and post processing of the molecular dynamics. So now scale issues like I want to discuss about this because although we are discussing about molecular dynamics there are several other modeling strategies which could be of importance and those are also commonly used in the nano fluidics domain and in the micro fluidics domain. So if you have a description of a system you can have a microscopic description. So microscopic description will essentially mean you have discrete particles which you can use in a Lagrangian view point which you can analyze. So just like that is what is commonly used in molecular dynamics. So discrete molecules or atoms, so these are like particles and you directly capture their influence. Sometimes you directly capture their influence, sometimes you do a statistical representation. That means instead of modeling individual molecules you statistically model a group of molecules having same statistically averaged behavior. Then in the microscopic picture you use the laws of continuum mechanics. So there you use a field basically. So you have a force field. So you use the well known rules of differential calculus to describe the variations or the gradients in properties and you consider the system as a continuous medium. So there are models that describe the macroscopic picture and some of these models we use even in the micro fluidics and the nano fluidics domain like the Navier-Stokes systems of equations. In the microscopic description we can use some sort of statistical description of molecules or deterministic description of molecules. But in between you can have an intermediate picture where you neither talk about the full microscopic description nor you talk about the macroscopic description. You talk about an intermediate description which is called as mesoscopic description. And one very important modeling paradigm which falls into mesoscopic description is the lattice Boltzmann model. So I mean there are various modeling considerations and if somebody is working with simulations in micro fluidics and nano fluidics I mean there are specialists who work maybe in the continuum domain with modifications or molecular dynamics or Monte Carlo simulation which is a statistical simulation or lattice Boltzmann or its variants. So there are whole ranges of simulations possible that we can discuss. Now to make the things a little bit light I will start with the concept of averaging and this few slides that I will be presenting I have borrowed from the internet from like from a particular presentation. So but I want to show these to you not for academic purpose. So do not take it in a very academic or a very serious note but it will lead to a concept of averaging which we can commonly use for post processing the data. So like as teachers you know we I mean our regular work is to teach students. So now there is a university in which you have a student who is ranked or who is rated by a number from 1 to 3. So there is a genius you can see the like example Einstein type of personality who is having intellect of 3 out of 3. And there is a student not so genius maybe the facial appearance reflects it. So this student has an intellect of 1. Now there is a teacher who has 3 semesters of teaching experience and this is the summary of his or her experience. In the first semester he gets a class with one student and the one student there is a high probability that the one student can be the genius student and the genius student is actually there. So the class average is 3. In the second semester there are so many students but I mean everybody is not so genius. So I mean we are talking about extreme cases when there is no average data either the not below average or the super about the genius nothing in between. And we will see that how this kind of data distribution may be problematic. So you have average 1 very clear everybody has intellect 1 so average 1. In the third semester it is a little bit balanced there are 2 students in the class one is the genius another is the not so genius so 3 plus 1 by 2 is the average is 2. So now if you consider the total average there are 16 students in 3 semesters. So if you count you will find there are 16 students total value is 2 times the genius student has appeared so 2 into 3 plus 14 into 1. So the total value is 20. Now how do we get the intellect of the average student taught by the teacher? How do you estimate the intellect of the average student? So average value of the 3 semesters this is one possibility right. So this will give you the concept of an ensemble in different pronunciation. So average values of the 3 semesters 3 plus 1 plus 2 divided by 3 which is 2. If you take semester wise average on the other hand if you take average over students this is also averaging. So you have 2 times the genius student has appeared plus 14 times the ordinary student has appeared that divided by 16 the total number of students that is 1.25. So you can see that none neither of these 2 averaging techniques are wrong fundamentally but these have given rise to drastic averaging results. And this thing is important because from molecular information if you want to get a continuum average velocity then which technique we should use? Do you take small groups and do the averaging based on the individual group behavior or you take the total number of particles in the system and make the averaging because these 2 things are grossly differing. And this difference is there because of a fundamental statistical reason that there is a correlation between the class size and quality of students in the class. So if you have a very small class size you have a high probability of having the genius. If you have a large class size you have very low probability of having the genius. So what happens having many geniuses actually one genius you may have. So that means that there is a correlation between the class size and the intellect of the students. So this leads to the conceptual paradigm how should one measure local fluid velocities from particle velocities because in molecular dynamics you will get molecular information. Now how you synthesize the local fluid velocity from the particle based information? So you can calculate the center of mass velocity in a cell. So you have a cellular approach and you can have an average particle velocity. So center of mass velocity in the cell is this is based on compartmentalized concept like the semester wise break up type of thing. So then if you have S number of samples then this is a sample average okay. So if you have a sample average sample average means basically the semester wise type of average. Then you can have alternative estimate of average from cumulative measurement that is total number of student based average. So the sample average is 3 plus 1 plus 2 divided by 3 we are coming back to the same example and the cumulative measurement based average is 1.25. So you can see that with the same concept if you apply for measuring fluid velocities based on statistical processing of particle velocities there can be discrepancies. And we will see that this kind of discrepancy is because of wrong choice of the sample data where there is extreme bias between the size of the sample and the number of entities in a sample. So with this little bit of background we will enter into a more serious note that is development of discrete models of medium. So in 1872 Boltzmann first described the transport equation which is known as kinetic Boltzmann equation. But this equation is based on a probability distribution and it is a complex integral differential equation. So this equation actually could not be solved. Although Boltzmann proposed this equation this equation could not be solved until in 1964 Boltzmann equations with discrete set of velocities instead of a continuous velocity space was introduced. And this paradigm eventually converts to the idea of lattice Boltzmann equation. In 1960 on the other hand I mean this years may not be exact but like just to give you the rough idea of the era. Molecular dynamics was first coming into the picture because you know molecular dynamics the idea has been there for a long time. But what do you have the computational resource to solve the equations of motions of molecules or atoms. So then this has evolved to a paradigm which is called as lattice gas automata and that has converged again to the lattice Boltzmann model. So in early 1990s one started with the people researchers have started extensively in working with the lattice Boltzmann model and one can show that this lattice Boltzmann models are so good because with a expansion of the variables in the equation in terms of the Knudsen number with different orders of the Knudsen number you this is called as Chapman-Enskog expansion it is possible to recover various macroscopic equations. For example the Euler equation the Navier-Stokes equation or some higher order equations. So it is sort of a good bridge between the microscopic consideration and the macroscopic consideration ok. I think we stop here for the time being and we will continue with the molecular simulations in the next lecture. Thank you very much.