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In our previous lecture we started discussing about nano fluidics 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 when those issues may appear to be important typically some interaction forces and so on then we started discussing about a problem of filling of a nano scale capillary Or a nano Boltzmann which is very important from the nano technological point of view and we looked into we rather revisited the Lucas Washburn was born equation with a modification to take into account the nano scale effects, So, in the Lucas Washburn was the only 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 to slip length we can use the Lucas Washburn equation to predict that 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 dynamics simulations in the latter 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 these particular lecture is to give you some glimpse of the physical phenomena at the nano scale So, if you find if you look into this graph you will see that L square versus T See the Lucas Washburn 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 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 of it the filling rates are high slope 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 a more or less all these characteristics show L square proportional to T is similar to Lucas Washburn model the classical Lucas Washburn model but whatever is the classical Lucas Washburn model prediction the filling rate is slower than that predicted by the classical Lucas Washburn model And there is a gradual decrease in the slope of the meniscus as seen in these 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 sorts of data possible So, we can ask our self 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 simulations So, how do you predict slip playing 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 the 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 weightability of the substrate and the weightability of the substrate has a strong role to play in deciding the slip So, slip length should be depend on the contact angle B 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 in which determines roughness of the substrate So, it is roughness weightability combination along with the driving acceleration that derives 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/ acceleration acceleration is surface tension force/ mass So, it is a sort of driving acceleration because the driving force is the surface tension force that divided by mass is a driving acceleration Now you can see that 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 generally characteristic if you non-dimensionalise the plot that means the y axis is already non dimensionalised if you make the x axis or 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 in the denominator you have the viscous resistance scale 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 now we have got 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 our slip length non-dimensional slip length as a universal function of non-dimensional acceleration and this functional form we will use in the Lucas Washburn equation So, just to recollect this is the Lucas Washburn equation and here in the slip length we have plotted it as a function of acceer non-dimension acceleration in which there is theta So, it is 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 we integrate can be there to see equation now you will get Lt 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 dynamic scattered data points or molecular dynamic simulations you can see that there is such a nice match between the Lucas Washburn model predictions And the molecular dynamics simulation so I just want to give you a philosophical outlook that this is many times the hallmark of Nano fluidic 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 acceleration driving acceleration non-dimension 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 the simple 1 dimensional model in which you plug in that like a like a constitutive behavior you plug in that information and then get the result which can actually reproduce the molecular dynamic simulation results the advantage in this process is that You do not have to do molecular dynamic simulations all the time at i mean 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 Washburn model in which you feed this data and you see that it remarkably agrees with the molecular dynamic simulation So, this is a kind of paradigm which we very commonly used for research in molecular simulations our research in nano fluidics So, it is not just brute force molecular dynamic simulations but some information from the dynamic molecular 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 nano fluidics 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, if the computational time is very significant and you cannot stimulate a large system the number of atoms have to 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 signs 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 we can use even in the nano fluidic domain So, this first example talked about the weightability issue now the roughness so you can use the roughness parameter in 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=A star smooth+a correction parameter That depends on the roughness which varies with varies exponentially with the roughness parameter So, we can incorporate roughness weighability 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 Washburn model fits the final results very nicely “Professor – Student conversation starts” yes that functional form we we we take the functional the universal slip length versus a star that I have shown you the graph this graph okay so you can see even it is fitted form is written in the legend I mean it may not be twenty easily visible this one okay so you have a fitted form of this so you can

use that fitted form directly so that it becomes analytically tractable or if you may have to do a simple numerical integration at the most so that reduces the cost significantly “Professor –Student conversations ends” Now the next example of nano fluidics modification in the nano fluidics domain that I will discuss about is electrical double layer phenomenon how is it modified in the nano scale preview So, just a quick revisit these things we have discussed in length in these course but like if we have our charge surface then there may be a charge interfacial layer in vicinity of the charge surface so that the entire system is electrically neutral These charged interfacial layer is also known as the electrical double layer discharge layer means the charge surface+ the fluid together is known as the electrical double layer So, if you have the electrical double layer have shown that under certain assumptions and we will discuss about the sanctity of these assumptions and what are the issues when they have modified so under those assumptions you can write the Boltzmann distribution for the number density distribution Within the electrical double layer and the poison equation see one pretty important thing that you should keep in mind that the Boltzmann equation has a lot of assumptions we had discussed about that the poison equation does not have any assumption because it follows directly from the Gauss law basically if you start with a integral form it is their differential form of the gauss law which can we 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 maybe questionable So, 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 the sanctity of this equation that many come into question Why that may come into equation is because we had made some assumptions while deriving the Boltzmann distribution these assumptions that I had discussed about discussed in the class it is exactly the same slide what is presented while discussing but I just want to recap which you led that because we will see that what happens if some of these assumptions are not valid So, the assumptions is Ions are point charges And the system is in equilibrium with no macroscopic advection diffusion The solid surface is microscopically homogeneous The charged 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 far 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 are 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 know 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 device length you have seen are of the order of few nano meters Now if the channel height itself is of the order of few nanometers then the characteristic length scale of the channel an 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 garnish and that the psi=0 at the center line

The EDL potential is 0 at the center 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 or hydrated by water water molecules in a solution and then solvation interactions can play a big role in altering the net electro chemical 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 with the Poisson equation that is the whole philosophy Based on the subsequent discussion is evolved So, some additional consideration so what I have done is I have not purposefully gone into the mathematical 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 color So, that if you are interested you should read these papers for getting state of the art understanding So, I will go through these considerations one by one I am seeing 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 polarized These reflected field this is a concept called an 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 Ions solid attractive and repulsive interaction that is in combination you can talk about that as a Lennard Jones interaction We will discuss about linear jones potential when we discuss about molecular dynamics that it is basically a combination of attractive repulsive and potential very commonly used for molecular dynamics 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 these we have discussed earlier see now how we can incorporate that in electro kinetics see this is this is what is or what is so nice about this physical chemistry over small scales That you can incorporate or you should attempt to incorporate some of these issues in a electro kinetic model which is based on a different formalism but you I will show you 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 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 forces that usually decays exponentially with surface separation There is an additional free energy component to create an iron sized cavity in the fluid that is to solvate a salute with the 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 theory

Mobile counterions in the electrical double layer diffuse part of the electrical 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 even 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 this ions correlation forces can be important with <4 nanometer Effects of finite sizes of ions this is important we considered ions as point to point masses point charges while deriving the Boltzmann distribution because now the diameter of ion explicitly came inti the picture while making the derivations But effect of finite sizes of ions, what these effects do they tend to enhance the repulsion between 2 surfaces this is analogous to the increase osmotic pressure of van der Waals gas due to the finite sizes of the gas molecules In a very similar manner finite sizes of coions and counterions contribute to enhanced repulsion In cases of coions absorbed 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 absorbing ions The ionic concentration close to the surface can easily exceed the maximum allowed coverage because typically if a point charge that can be infinite charge density that can be allowed But in reality you cannot allow that so it 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 these 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 have modified the EDL potential with some different potential with some augmented potential that incorporates these interactions through a modified free energy description We can start with this with the fundamental free energy itself and then we can find we can find a derivative of the free energy and set it to 0 to set the condition for equilibrium and then we will get someone 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 okay So, we start with the free energy in the free energy we first write 1 component I will explain all these terms self-energy of the electrical field-epsilon/2 grad psi square then electrostatic energy of the ions if you have z is to z symmetric electrolyte zen+psi-zen-psi So, fundamentally it is summation of zi e ni psi Okay, so this is about the electrical component now what I want a 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-TS right So, the –Ts component is this so to write this component in fact one can correct these and there can be several directions of research towards that For a simple analytical deviation 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 as 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+Cl- their sizes are grossly different They are not of the same size So, this is just for analytical description without sacrificing the essential physics So, this is the entropy of the positive ions this is c so this is n+ in the number density So, this is entropy of the positive ions so you can recall that in the free energy expression there was term log off 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 these are the 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 these formula comes pretty simple and straightforward only assumption that laying scale of the positive and negative ion are the same and the volume scales with a3 a is the length scale So a is not normally ionic radius It is an equivalent length scale again to capture it may also include the hydrated radius instead of the normal radius So, sometimes it is the effective radius so not 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 No how do you define the chemical potential we basically differentiate the free energy with respect to the number density or their 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 then tropic term So, these two terms for this and this is constant for equilibrium “Professor – Student conversation starts” No no this is Helmholtz U-TS I mean see let me let me answer this question more carefully I mean this deals with I mean one has to have very vigorous thermos dynamic background to appreciate that when do you use Gibbs free energy and when you use Helmholtz free energy right it depends on what it depends on the context like if you are in 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 flowing system but when you were discussing about a system which just give the thermodynamic picture but not the transport picture then we are essentially bothered about a system about a system with which is not a flowing system So, if we do the non-flow of a system we do not use the enthalpy we use the internal energy even if you look at the fossil of thermodynamics You see that when it is a non-flow process we use our 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 is 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 these because this is not a thermo dynamics class Because you raised this question I made this important remark so it does not matter whether you are using Helmholtz or 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 considerations So, Helmholtz and Gibbs give the same thing here there is no question of any PB term here So, you can write this as Helmholtz or Gibbs whatever for this case ”Professor – Student Conversion ends” Modified Poisson Boltz equation this is= to derive that so this is =constant the potential is constant now we would differentiate that and state 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 the 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=1-2n0a cube So, 2 n0 a 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 1 n0 a cube for the positive ions another n0*a cube for the negative ions So, total 2 n0 a cube this is called as steric factor So, this steric factor if you include you can write equations for +and n- so you can see these are modified versions of the Boltzmann equation If you set a=0 then you will find that this points down to the Boltzmann distribution So, your modified Boltzmann equation is n+=this and n-=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 valley with electrical double layer overlap one slight overlap is fine slight overlap will still work but with strong electrical double layer overlap if you can modify these 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 field overlapping phenomena in the same framework as these with analytical attract ability without analytical attract ability people have done but with this kind of analytical attract ability 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=0 there were 2 parameters parameters were n+ n- and psi, so we said already that derivatives with respect to n+ n-0 now we said derivative with respect to psi=0 SO that gives 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 straight forward there was an entropic correction but in the entropic correction nowhere you had a dependence on psi right because there was nowhere a dependence on psi the entropic correction will not correct the Poisson equation so that is why it is 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 okay So, you can see that this is the modified Poisson Boltzmann equation that takes the steric effect or the finite size effect of the ionic species into our account So, how can you get a generalized EDL model so we have discussed that how to incorporate one of the effect but we could see there could be many other effects like the solvation interaction and 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 electro chemical potential by differentiating the above with respect to NI constancy of chemical potential gives the eauilibrium NI distribution Set derivative of F with respect to psi to 0 to get the governing differential equal equilibrium for the 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 two 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 the 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 potentials 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 a non-zero potential at the centerline and non-bulk value of the density charge density at the centerline 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 our big open area of research of electro kinetics in a nano fluidic domain So, for some applications when a general chemical 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 very important considerations if you come down to the nano domain So, when our general chemical sample is introduced in a channel it is a mixture of a number of analytes Within the channel sudden effects are imposed on these analytes And they respond differently to these effects For example electrical effect so based on size the electro phoretic velocity size will be different So, as a result they acquire different velocities so they reach the channel exit at different times and allowing them to be collected separately Thus we obtain each individual component of the mixture and the components are said to be separated okay the principles by which they are 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 of dispersion because we have already discussed about this but what I want to say is that now in nano fluidic domain additional factors will affect this One important factor is hindered diffusion we have discussed about hinder diffusion in the nano channel Now you do not get the diffusion co efficient as the bulk diffusion co efficient because of the extreme hindrance created by the narrow confinement So, you have an altered diffusion co efficient So, that will result in a different dispersion characteristic Now you in addition to that in a electric kinetic separation you will have the hydrodynamic and the electro phoretic effects I have discussed about hydrodynamic interaction 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 Poisuille flow if it is a pressure driven flow or it will disturb the Helmholtz Smoluchowski type of velocity if it is a electro osmotic flow In addition you have electro phoretic effects, an electro phoretic effects acts 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 null plug like velocity profile So, the uniformity of the velocity profile in the electro osmotic 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 level 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 Wall interaction and the electrical double

layer interaction between the EDL of the particle and EDL 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 hinder 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 phenomena 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 microfluidics 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 dynamic simulations which we will cover partly in this lecture and partly in the next lecture Scales of analysis concept of mean or average flow modeling beyond continuum, basics of molecular dynamic 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 fluidic domain and in the microfluidics 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 viewpoint 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 that actually 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 average behavior then in the microscopic in the macroscopic 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 in properties and you 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 microfluidics and the nano fluidics domain like the Navier Stokes systems of equations In the microscope a description we can use as some sort of statistical description or deterministic description of molecules But in between you can have a intermediate picture where you neither talk about the full microscopic description Or nor you talk about the macroscopic description you talk about an intermediate description which is called as mesoscopic description and one very important modelling paradigm which falls into mesoscopic description is the Lattice Boltzmann model So, I mean there are various modeling considerations and if someone is working with simulations in microfluidics and nano fluidics I mean there are specialists who work maybe in the continuum domain with modifications or molecular dynamics or Monte Carlo simulation 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 these 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, as like 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 yes 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 1 student and the 1 student there is a high probability that the 1 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 that not below average or what it all the super super but the genius nothing in between and we will see that how these did this kind of data distribution may be problematic So, you have average 1 very clear everybody has the 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+1/2 is the average is 2 So, now if you consider that total leverage that 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 genial student has appeared so 2*3 +14*1 .So, the total value is twenty 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 taught by the teacher? How do we estimate the intellect of the average student? So, average value of 3 semester this is one possibility Right so, this will give you the concept of a Aesop in symbol in a different pronunciation So, average values of the 3 semesters 3+1++2/3 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 have appeared+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 two 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 total number of particles in the system and make the averaging because these two 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 one genius you may have? So, that means that with a 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 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 So you can have a cellular approach and you can have average particle velocity So center of mass velocity in the cell is this is based on compartmentalized concepts like the semester wise breakup 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 it 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+1+2/3 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 transporter equation which is known as Kinetic Boltzmann money 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 converged to the idea of Lattice Boltzmann equation In 1960 on the other hand I mean these years may not be exact but like just to give you the rough idea of the era Molecular dynamics was 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 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 1990 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 Knudsen number with different orders of the Knudsen number you can do this 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 Okay 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

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