Uncategorized 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         