 Good morning and welcome to the next lecture in our course on chemical engineering of CVD processes. I had circulated some links for lectures that are in the NPTEL course on advanced transport phenomena. So I hope that you have had a chance to look at these lectures. There was one lecture on the CVD process involved in deposition of tungsten on bulb walls and the other one was on the heart corrosion process. And then there were additional lectures on some of the fundamentals of transport phenomena which we need to understand for the remaining portion of this course. We are going to concentrate quite heavily on the transport aspects for the remainder of the course predominantly. Towards the end we will go back and discuss a few more applications of CVD technologies but before we do that it is important to understand the transport phenomena that are involved. Now in order to fully understand the transport processes in a CVD reactor you have to first understand how these transport phenomena are interlinked in a chemically reactive flow system. Because although the CVD process itself is a mass transfer process, you are taking some material from the gas phase and depositing it on a substrate in solid phase. So it is clearly fundamentally it is a mass transfer process but you cannot study the mass transfer process in isolation. You have to really study it in terms of its relationship to the other prevailing transport phenomena such as momentum transfer and heat transfer and entropy conservation and so on. So it is important to get a full systematic understanding of the conservation laws. So we have mass conservation, momentum conservation, energy conservation, entropy conservation and so on which you have studied in your earlier courses but it is good to spend a few minutes kind of recapping and realizing how they all kind of hang together. In any conservation equation there are four terms as we have discussed earlier. There is a net rate of change term, there is a convective term, there is a diffusive term and there is a source term. So regardless of whether you are looking at mass conservation or momentum or energy or entropy all these terms appear in the equation. Conservation equations are those that apply to all materials. They are not specific to the fluid or system under consideration. Later on we will deal with constitutive laws which are specific for the material that is being studied or the fluid that is being studied or the system that is under analysis. So typically you start by writing the conservation equations and you use your constitutive relationships to obtain a closed system of solutions for the set of equations. So the two kind of go together. So when you talk about conservation laws you have to first understand the assumptions that are involved in framing these laws. So one of the modules that I circulated has a discussion of various assumptions that are typically made when we try to describe flow systems. For example the simplest one is continuum versus free molecular which depends on the Knudsen number. If the Knudsen number is in excess of 1 you have to consider it to be a free molecular flow. If the Knudsen number is much lower than 1 then you consider it a continuum flow where the Knudsen number of course is the ratio between essentially the distance between adjacent molecules to a characteristic dimension of the system. The other assumption that is typically made is incompressible versus compressible. In most flow situations the incompressible flow assumption certainly simplifies the analysis but the requirement for incompressible flow is that density must be constant across stream lines not with time. In fact that is important to understand that incompressible flow density at any location can change with time. However across the flow lines the stream lines of flow density must remain constant at any instant in time. Another assumption is viscous versus inviscid flow which is really described by the Reynolds number. The lower the Reynolds number the more viscous the flow and the higher the Reynolds number the more inviscid the flow and of course as the Reynolds number becomes very high you also reach a transition between laminar and turbulent. So another way to characterize flow is laminar versus turbulent. The other ways of describing flow are based on for example dimensionality. Is it one dimensional or multi dimensional or can we characterize it as quasi one dimensional. So what we mean by these cases is in a one dimensional flow the variations in properties primarily occurs along one direction. So you can assume that the variables are constant with respect to the other spatial dimensions whereas in a multi dimensional flow you have to take into account variables across all dimensions. Multi one dimensional essentially suggests that even though the variability along let us say 2 out of 3 dimensions may not be 0 as long as the variability is negligible compared to the variability in one dimension you can treat it as a quasi one dimensional flow by assuming that the property variations in the other two directions are negligible compared to the variation in the one direction and similarly you have steady state versus unsteady flow. If flow properties are constant with respect to time you call it steady state. If they are changing or varying with time you call it unsteady and again we can we can define a quasi steady state flow where properties are changing but very very slowly. So these are all ways of classifying flow systems and making certain assumptions about how flow occurs. Another example would be Newtonian versus non Newtonian right. In a Newtonian fluid there is a linear relationship between stress and strain whereas in non Newtonian fluid it can be highly non linear and another way to discuss this is in terms of fluid mechanics versus rheology. Fluid mechanics is usually used to study Newtonian fluids and rheology is the name we give for studying the fluid mechanics of non Newtonian fluids normally. Another classification that is sometimes relevant is memory effects. Does the system remember what happened to it sometime ago and also is there an action at a distance effect? Does the fluid at one location react to a stress put up on it at a different remote location? So these are action at a distance effects and memory effects can also be important in certain classes of fluids. So the first thing you need to do whether you are trying to analyze a CVD system or any system is go through this list of how flows can be classified and develop a set of assumptions that you can make regarding the flow. When you start your analysis you can start with setup with this set of assumptions however you do need to go back at the end and validate the assumptions. For example if you have assumed that something is a one dimensional flow but then when you actually do your calculations or your experimental measurements if you find that changes are happening in more than one dimension then you have to go back and relax your assumption and start treating the problem as a two dimensional or higher order problem okay. So that is kind of step number one. Step number two is to define control volumes and a control volume is very critical in analysis because you essentially treat it as a system and everything outside of the control volume you treat as surroundings and you try to solve your equations for the system. So how you define that control volume can have a large bearing on how you actually approach the treatment of the problem and there are essentially four types of control volumes there is the Eulerian or fixed control volume where the control volume is fixed in space and the fluid flows through it. The second type of control volume is called the material control volume or Lagrangian control volume in which the control volume itself moves along with the fluid flow at the same velocity as the fluid. So essentially the same parcel of fluid is contained within this control volume throughout its transit through the system. The third type of control volume is what we call a hybrid control volume in which certain parts of the control volume are fixed and certain parts of the control volume move with the fluid. So it is a combination of the Eulerian and Lagrangian control volumes and finally there is the completely arbitrary control volume which is not fixed in space but which is not moving with the fluid either. It has a completely different configuration and it moves but at a different velocity vector compared to the fluid that you are treating. So these are four types of control volumes and again you have to make the decision of which control volume is appropriate for the system you are studying. For instance in a CVT reactor I would assume that you would want to use an Eulerian control volume because a CVT reactor for the most part is a batch system. Material comes in, reactions happen, deposit forms and the byproducts leave. So this type of system is reasonably convenient to describe with a fixed or Eulerian control volume. However in the case where you have for example an atmospheric pressure CVT reactor where it is essentially an open flow kind of a situation you might want to consider using a material control volume. The material control volume is also more appropriate when there are fairly rapid changes in the geometry of the system which makes it difficult to construct a fixed control volume which can capture all these local changes. So there are some pluses and minuses to both approaches and you need to be aware of them as you formulate your control volumes. So you have looked at your system, made certain assumptions, you have defined your control volume, the third step is to start writing your conservation equations, right. And when before you write the conservation equations the first thing you have to do is define something called the field density. Associated with each of the conserved quantities we have something called a field density parameter which is a representation of its quantity per unit volume of the fluid. So for example when we talk about total mass conservation the field density parameter corresponding to that is the density of the fluid itself rho. If we talk about species mass the corresponding field density parameter is rho i which you can also write as omega i times rho where omega i is the mass fraction of species i in the mixture. And similarly element mass is rho of k which is equal to omega k rho where this is the mass fraction of the kth element which is estimated as omega k equals omega k i times omega i where omega k i is the mass fraction of the kth element in the ith species multiplied by omega i. Then you have linear momentum which is another conserved quantity. So what would be the corresponding field density parameter here can somebody tell me. What is momentum per unit volume? How would you write it? What is momentum? m v right. So m v divided by capital V so it is rho v right. What else? Let us say total energy will be rho times E plus v square by 2 where E is the specific internal energy and rho E is the specific internal energy and rho times v square by 2 is the specific mechanical energy. So similarly for kinetic energy you can write this as rho times v square by 2. Another quantity that we will be interested in is entropy and again this you can write as some rho times a specific entropy value S. So once you have defined these appropriate field density parameters it is actually quite easy to write the corresponding conservation equations. So in general as I said the format of this equation is an accumulation term plus a convective term equals a diffusive term plus source term and by the way when we talk about control volumes you know it is important to realize that let us say that you know this is a control surface. Control surface is essentially the outline or contour of your control volume in 2 dimensions. So what is contained within this is what we call the control volume right. So by convention when we talk about the convective term here we take it as an outflow. So convection in its with a positive sign is denoted as material leaving the system and diffusion by convention again is always has a negative sign associated with it because diffusion is encountered as an inflow term. So convection is regarded as an outflow diffusion is regarded as an inflow diffusion and of course when you have net consumption of a species the source term then becomes a symptom. So the source term can also have a plus sign or a minus sign. So now that is the general equation. Now you can write conservation equations in 2 ways you can write them for integral volumes I mean you can write the conservation equation for the entire control volume in which case it will be an integral expression or you can write it for a small differential volume within the control volume. So you can either have an integral equation or a differential equation and typically conservation equations are written as differential equations because our methodologies for solving differential equations is fairly well developed and also writing it as a differential equation enables us to track changes that occur within the control volume instead of treating it as a black box. So there are certainly several advantages to treating to solving this set of equations in a differential control volume rather than a macroscopic or integral control volume. So in a differential control volume this would typically have the format of del by del T of some quantity plus divergence of another quantity equals minus divergence of a quantity plus a source term which is typically written as you know let us call it some S and it will have units of per unit time per unit volume okay that is the general formulation so you can take any conservation law and write it in this form. Now what goes in here are these field density parameters okay so for total mass this would simply be del rho by del T plus divergence of rho B so the field density parameter will go here the field density dot B will go here and at the diffusion term will have different forms for the different types of conservation equations and the source term again will have different forms for the different conservation equations. But in general the left hand side of the equation once you have defined your field densities then the first term is simply del by del T of the field density parameter the second term is the divergence of the field density parameter times V which is a vector by the way you know what is divergence right have you now in the case of total mass what will be on the left on the right hand side in this conservation equation when you are doing mass balance total mass balance it is 0 plus 0 total mass cannot diffuse and total mass cannot be created or destroyed unless you have achieved light speeds or something and you are transferring mass into energy and vice versa but under normal conditions total mass cannot be created or destroyed and total mass cannot diffuse only the components of the total mass can diffuse okay let us say let us take this equation now the species mass conservation equation so this will be del rho i by del T plus divergence of rho i times V equals now what do you have on the left hand side on the right hand side would you have your terms yeah because species can diffuse and species can be created and consumed so you will have a minus j i dot double prime which is a diffusive term plus r i dot triple prime which is the source term now this is what we call the diffusive flux of species i it is a flux so this dot represents per unit time the double prime represents per unit area so that the diffusional the species mass conservation equation can simply be written in this fashion now where do the constitutive laws apply where do we apply the fluid specific relationships this and this the diffusive flux term for species mass as well as the chemical reaction term for the species involve constitutional relationships for example the source term requires knowledge of chemical kinetics so you obtain that reaction rate term by applying chemical kinetic laws and similarly the flux term the diffusive flux term involves knowledge of flux laws for example in the case of species mass flux you need to know thick diffusion flux laws right so the constitutive laws are typically applied on the right hand side of the conservation equations in order to obtain a closed setup solutions okay how about element mass how would this equation look so again the left hand side is fairly simple del rho k by del t plus divergence of rho k times v now which terms will be on the right hand side for chemical total mass we said nothing on the right hand side for species we said both the diffusive and the source terms will be there for element mass which will be the terms that appear can element mass be created or destroyed unless it is a nuclear reaction right but can can elements diffuse yeah so only the j k dot double prime term will be there but again remember that in a CVD system an element cannot really diffuse by itself it is somewhat of a fictitious representation when you talk about element fluxes it is basically a weighted sum of the fluxes of all species containing that element right so j k dot double prime that we have written there will be the summation over i of omega k i times j i dot double prime so you take all the species diffusion fluxes weight them by the mass fractions of the element in the corresponding species add it all up that is what gives you the total species or total element mass flux and as we saw in the in the last few lectures when you are depositing a certain film like in a CVD reactor this is the most important quantity because this has to satisfy the stoichiometric requirements for example if your the deposit has let us say silicon and let us say there is a composite deposit of silicon tungsten carbide right what this would mean is first you calculate the diffusive flux for Si as an element calculate the diffusive flux of tungsten as an element the ratio of those diffusive fluxes of the molar fluxes has to be the same as their ratio in the stoichiometric compound otherwise you will not get a stoichiometric deposit. So the thing that we are really going to focus on mostly is how to calculate this but to get to that point you need to be able to do everything else you need to be able to calculate all the other quantities that we are interested in and then home in on this particular quantity and how you calculate it and so on okay. So let us see let us get back to completing our discussion of conservation equations how do you write the let us say the linear momentum conservation equation so following the same format del by del p of rho v plus divergence of in this case rho v v equals now you have a diffusion term for momentum I mean how what is how does momentum diffuse it is clear how momentum gets convected right it is basically as something flows there is a flow velocity and there is a mass so you multiply by 2 you get momentum convection but how what is momentum diffusion how does momentum diffuse any idea in other words supposing we write minus divergence of something what should it be and what is the source for momentum you know is there a source term when we talk about diffusion remember that we are talking about something that occurs around the contours or boundaries of the control volume you know around control surface so when you have momentum that is acting on this control surface we know that it can be transported in 2 ways one is by convection the other is by diffusion so diffusion necessarily must be a surface process so it is essentially related to surface stresses so the term that you use here is called pi, pi is a combination of viscous stresses and atmospheric pressure so you write pi as minus p times the identity vector or matrix I plus what is called an extra stress operator tau which is actually viscous stress operator so that is where the viscosity term comes in so essentially momentum is dissipated or diffused by building up a stress along the surfaces that the momentum comes in contact with okay. So what is the source term in this case the source term arises because of the body forces that are acting on the species that are present in the system so this term is given by summation over i of rho i g i it is a summation of all the gravitational body forces that are acting on the species that are present in the system okay. Again I am kind of going at this at a very high level just catching the highlights if you want to understand a lot of this in more detail I mean you are certainly welcome to take my course on NPTEL advanced transport phenomena which I think devotes 10 lectures to each of each of these you know mass conservation momentum conservation energy conservation and so on or you can also welcome to read the textbook that I had given a reference for very early on transport phenomena in chemically reacting flow systems by Daniel Rosner but I think for the purposes of this class the CVD class I think as long as you have a good overall understanding of how various quantities are conserved and particularly we will focus more on their interaction effects later on in this course okay. Let us talk about energy then how would you write the conservation equation for energy. Again the left hand side is you know reasonably obvious del by del t of rho e plus v squared by 2 plus divergence of rho e plus v squared by 2 times v equals now you have to have again a diffusional term and a source term. Now one of the diffusional terms is can be written simply as minus q dot double prime which is a diffusive flux of for heat and there is also a volumetric heat source term which you can write as q dot triple prime. So these are essentially heat sources and heat fluxes which are present in the system for example this heat flux is what you would write as you know the Fourier law of conduction right and there is another component to it as species diffuse there is an accompanying heat diffusion that happens because of the enthalpy of the species. So the heat flux term we will see later actually has 2 components to it one is the simple molecular thermal flux and the other is the heat flux that is associated with mass flux right and the q dot triple prime term is you know it is any volumetric source of heat for example it could be a radiative heat that is present in the system or it could be because of interaction between cosmic rays and chemical species that are present. So we try to consolidate within that term all the possible volumetric sources of heat. Now but what are the other terms here that we need to take into account also. For example these stresses that are developed are doing some work right so that has to be accounted for in your energy conservation equation and similarly these body forces that are giving rise to force terms are also doing work. So these 2 terms also have to be accounted for in your energy conservation equation. So you would then write this as plus divergence of pi dot v which is the work done by the stress operators plus summation over i of rho i v times g i by the way g i is not necessarily gravity say any volumetric body force most likely it is a gravitational force but there can be situations for example in a centrifugal reactor the body force term can be much higher than gravitational force. So these are the 2 extra terms that are added for the energy conservation term and what else okay entropy we will write as del by del t of rho s plus divergence of rho s dot v equals minus j s dot double prime plus j s dot triple prime where j s dot double prime is again the diffusive flux term for entropy which is again associated with both mass flux as well as energy flux. They both give rise to an entropy flux as well and similarly the entropy source term includes all volumetric contributions to increase in entropy of the system. Now these are conservation laws that are used in transport mechanism studies how do they relate to thermodynamics I mean can you restate some of these equations as thermodynamic equations obviously there is a relationship you know energy to work conversion how about entropy what is the thermodynamic requirement for system entropy should be the source term must be positive right. So we can rewrite this equation as an inequality to say that for a system to be thermodynamically feasible the source term in entropy must be greater than 0 which means that if you go back to this equation these 2 terms plus the diffusive flux terms if you add them all up it must be greater than 0. So it is a self-consistency check you know when you are doing analysis of a C V D system you have to verify not only the thermodynamic validity and the transport validity but also the entropic validity to make sure that the C V D film that you are trying to achieve is feasible and will happen if one of these 3 is violated it would not happen. So it is important to consider all modes of conservation and do a comprehensive calculation for the system as a whole okay. So these are all the various conservation laws again kind of in a nutshell and you know you can certainly write them in different coordinate systems we have used Cartesian you can use cylindrical coordinates spherical coordinates so you know depending on the geometry of your C V D reactor you will have to restate them different ways and wherever possible actually if you can apply an integral formulation of these equations and get away with it fine because it is simpler to solve the integral equations and the differential equations in terms of numerical complexity and so on. So the next thing that we want to focus on is on the right hand side of this equation you know assume that the left hand side is something we know either how to measure or how to calculate I mean you can certainly look at the time rate of change of mass by various analytical techniques you can look at time rate of change of species mass element mass momentum energy and so on. So the left hand side of these conservation equations is not too difficult to either calculate or measure but the right hand side of these equations is where the challenges are because they require specific information about the fluid at hand or the material at hand or the system at hand for example if you want to calculate in energy conservation you want to calculate this heat flux term you have to know the thermal conductivity of the system right and similarly if you want to calculate momentum diffusion you have to know the viscosity that is prevailing if you want to calculate the diffusional flux you have to know the diffusion coefficients that are prevailing. So quantities such as mu viscosity k thermal conductivity and d diffusivity are the very key properties that either you have to be able to estimate from the molecular properties of the system or measure based on well designed experiments and for these constitutive coefficients either approach is considered valid. So the three primary types of constitutive relationships that we deal with are equations of state for any fluid you have to be able to write its equation of state so that you know what its prevailing state is for a given temperature and pressure and essentially what the phase diagram looks like what is going to be the composition of the system what is going to be the solids fraction liquid fraction and so on. So the equation of state is the most fundamental constitutive relationship the second most fundamental at least for chemical systems is your kinetic rate loss whether you assume that thermo chemical equilibrium applies or you assume that it is kinetically controlled you still need to know the corresponding rate coefficients. So the rate coefficients themselves as well as the actual formulation of the kinetic model are also very important aspects of the constitutive relationships and again in a CVD reactor you will realize the complexity of this because even though the overall reaction may be quite simple you know SiH4 going to Si plus H2 the actual steps involved the actual reactions involved can be many many and to obtain a kinetic description may be the most difficult part in closing a CVD problem and then the third important constitutive relationship is the diffusion loss how does mass diffuse how does energy diffuse how does momentum diffuse and so on. So without a good understanding of how to write the constitutive loss you cannot really proceed very far I mean it is great to be able to formulate the conservation laws I mean that is clearly the it is an essential step but not sufficient. Once you have written down the conservation laws you know you are kind of halfway across the well but you still have to take the other step to the other side of the well and that is to be able to write the constitutive relationships to be able to estimate these parameters you know how do you estimate viscosity of a fluid or thermal conductivity of a fluid and especially diffusivity is particularly complicated because every species can have a different diffusivity right so you have to be able to estimate diffusivities of individual chemical species that are present in the system some of them are dilute species and some are present in higher concentrations and the equations for deriving the diffusion coefficients are very different as the concentration of the species gets higher the diffusion of one species can affect the diffusion of the other species. So these types of coupling effects have to be taken into account so in the next lecture we will spend a little bit of time particularly talking about the constitutive relationships focusing on the diffusive loss that apply for mass flux okay any questions on what we have covered today okay see you.