 Good morning and welcome to the next lecture in our course on chemical engineering principles of C V D process. In the last lecture we started covering the basics of transport processes that occur in C V D systems and we started out by classifying various types of chemical reacting flow systems in terms of their basic characteristics. We discussed how to define control volumes for various applications and then we discussed the conservation laws governing mass conservation, momentum conservation, energy conservation and entropy conservation. Towards the end of the lecture I mentioned that the right hand side of the conservation balances reflect the constitutive equations that are needed to close the system of equations. Constitutional laws actually the name itself constitutional means that it depends on the constitution of the system or the fluid. So constitutive law by definition unlike conservation laws is specific to the system or fluid under consideration. It is necessary to provide closure to the conservation equations. The 3 most predominant types of constitutive laws are the equations of state, chemical kinetic equations and rate expressions and diffusive fluxes. And when we talk about diffusive fluxes in particular parameters such as viscosity, thermal conductivity and diffusivity become important. Now in general a diffusive flux law can be stated in a linear fashion. You can say that the flux is proportional to the prevailing gradient of the field, the density parameter that is relevant. So for example the mass transfer diffusive flux can be taken to be linearly proportional to the prevailing concentration gradient. The heat flux can be taken to be proportional, linearly proportional to the prevailing temperature gradient and so on. Now this linearity is a critical assumption because it implies several things. Most importantly it says that the flux that is happening, diffusive flux that is happening at any location at any instant in time is independent of events that are occurring far away from that location and events that occurred far back in time. So you know as we were discussing yesterday the distance, action at a distance effect and also the memory effect are presumed to be absent when you define diffusive flux laws and that greatly simplifies our definition of the diffusive flux law. Now if you take momentum diffusion as we mentioned yesterday momentum diffusion is reflected as a stress term and in particular the pi operator which has the two components a thermodynamic pressure component and the extra stress or viscous stress component is a reflection of how momentum diffuses. And if you look at the second component tau which is the stress term as you know there is two types of stresses there is a normal stress and the shear stress. Normal stress is typically expressed as some tau xx and the shear stress would be expressed as tau yx. Shear stress by definition is the action of a force in a direction x which is acting or a y directional force acting on a surface in the x direction right as you will see when you have flow past a flat plate and things like that and you know that this tau yx term is related to mu times del vx by del y where this mu is called the viscosity coefficient and of course mu is different for different fluids every gas has a different viscosity every liquid has a different viscosity and so on. Now in order to then close the momentum conservation equation you need to know what is the tau term in order to know that you need to know what is mu and how do you estimate mu there is basically two ways mu can be derived from the kinetic theory of gases you know from first principles or it can be experimentally determined essentially you run shear stress for example as a function of various velocity gradients and take the slope and that will give you the viscosity coefficient. So in constitutive laws the constitutive coefficients can be derived either theoretically from first principles or phenomenologically from observations of experiments. I am sure you are familiar with the Chapman-Enskog theory have you read that in your fluid mechanics Chapman-Enskog which says that for viscosity particularly of low density gas mu can be related to the other properties by the following equation pi m k B T to the power half over pi sigma squared omega mu which is a function of k T B over epsilon mu. Have you seen this expression does it look familiar? So it is basically a way of relating viscosity to the molecular weight of the species or the liquid or the fluid the Boltzmann constant the prevailing temperature. Sigma is the molecular size or it is the spacing you could say between adjacent molecules and this parameter is actually a correction factor for the fact that not all molecules behave like hot spheres. So it is called the hot sphere correction and this depends on this parameter k T sorry it is k B T not k T B Boltzmann constant times temperature. So the omega mu can be written as some 1.16 times k B T by epsilon mu tie to the power minus 0.17 I think where this epsilon mu is basically an activation energy barrier for fluidity of the fluid. So this is a general expression governing the dynamic viscosity of a fluid. If you have a mixture of fluids you can estimate the mixture viscosity as summation of Mi to the power half Yi mu i divided by summation Mi to the power half Yi where again m's are the molecular weights and y's are the mole fractions. If you look at the temperature dependence of mu you can see that for a gas the temperature dependence is fairly weak it is about 1.5 to 1.8 because according to this you have a T to the power half dependence but there is also a temperature dependence that is built in here and so the net effect is actually a fairly weak dependence of mu on temperature. Of course for a liquid you know that the viscosity has a significant dependence on temperature as the temperature increases viscosity will decrease and again that depends on the activation energy for fluidity of the fluid. When you have turbulence in the system there is a turbulent viscosity mu T that is introduced and mu T plus mu is the mu effector which is the prevailing viscosity of a turbulent fluid is essentially the sum of its viscosity under laminar conditions plus a turbulent induced viscosity of the fluid. So the net viscosity can be written as mu effective equals mu plus mu T. So that is just a very quick recap of how the dynamic viscosity of a fluid can be calculated essentially from the kinetic theory of gases. Now when we look at heat flux you know that there is the Fourier heat flux law which says that q dot double prime equals minus k gradient in temperature where k of course is the thermal conductivity. However this is for essentially a situation not involving mass transfer as I mentioned yesterday when you have heat fluxes in systems that involve multi component mass transport you need to include another term summation over i j i dot double prime h h i which accounts for the fact that as species diffuse enthalpy also diffuses and that sets up a heat flux as well. So in general the expression for heat flux in a multi component chemically reacting flow system such as a CVD reactor has to have a mass diffusion flux related term also built into it. So here obviously the key parameter is your thermal conductivity k and again the Chapman-Enskog theory says once you know what viscosity is you can calculate k based on the viscosity of the fluid and the formula for that is 15 by 4 times r by m times mu times 1 plus 4 by 15 times Cp by r minus 5 by 2 where Cp of course is the heat capacity under at constant pressure and k mix is summation over i in this case it is m i to the power one-third y i k i divided by summation over i m i to the power one-third y i. So thermal conductivity as well it can be measured from experimental observations because thermal conductivity is fairly easy to measure simply by measuring temperature gradients which is much easier to measure compared to velocity gradients obviously. So thermal conductivity is much easier to measure experimentally. So typically the route that one would take is if you do not want to calculate mu and k from first principles then you would run experiments to measure k and then use this formula now that you know k you can go back and estimate mu right because as I said mu viscosity is very very difficult to determine through experimental observations whereas thermal conductivity can be obtained much more easily and just like with viscosity there is also kt parameter which is the thermal conductivity associated with turbulence plus k will give you k effective. And by the way the turbulence terms mu t and kt and we will see later on that similarly for diffusivity there is a di t plus di which is equal to di effective where di is the fict diffusivity under laminar conditions and di t is the turbulence enhanced fict diffusivity. The interesting thing to note is that di t has absolutely no relation to di. Similarly kt has no relation to k, mu t has no relation to mu they are completely independent of each other. Mu t kt and di t only depend on the prevailing turbulent conditions and also interestingly mu t kt and di t are very close to each other in terms of the relevant ratios. So if you take mu t by rho t which is of course the kinematic viscosity mu t you know if you take the dynamic viscosity and divide by density you get the kinematic viscosity. If you take the ratio of nu t to di t what do we call that nu by d any idea? It has got a number named to it kinematic viscosity divided by diffusivity it is called the Schmidt number s c i t and similarly if you take kt by rho c p it is called alpha t thermal diffusivity and there is a name for alpha t by di t any idea what that is? Lewis number. So the point is all these are actually close to unity. In other words once turbulence sets in the associated kinematic viscosity thermal diffusivity and fict diffusivity are all almost equal. In other words once turbulence sets in the effect of turbulence is much more predominant compared to the effect of the fluid itself. So the nature of the fluid does not matter as much as the nature of the turbulence. However under laminar conditions it is the nature of the fluid that really determines these transport coefficients okay. So let us talk about diffusivity which is obviously the parameter that is most relevant to this particular course. So we should talk about it in a little more detail. Now according to Fick's diffusion law which he introduced back in 1885 the diffusive mass diffusive flux of a species in the case of an isotropic isothermal fluid is proportional to the prevailing concentration gradient and it occurs in the direction of decreasing concentration right. But again these are very important definitions. When we say isotropic what it means is that the diffusivity is equal in every direction and when we say isothermal what we mean is there is no temperature gradient which can also drive mass flux according to a phenomenon known as thermal diffusion. So under these restrictive conditions you can write the diffusive flux minus J i dot double prime as equal to D i rho gradient of omega i. Again this is for isotermal and isotropic conditions and also there is an assumption here that it is a pure material that it is not a mix of species. But in a real system particularly in a CVD reactor you know that there are going to be hundreds of species. So the effective diffusion coefficient of a single species is going to be dependent not only on the prevailing temperature and pressure conditions but also the mole fractions of all the species comprising the system. Because when you have a case where you have a dilute species which is diffusing in a low density gas which is typically the case in a normal CVD reactor. So for that particular combination of a dilute diffusing species in a low density gas you can write the effective or prevailing diffusion coefficient as the aggregate of all the binary diffusion coefficients. In other words as these dilute species move around they are going to be encountering other molecules that are present in the system and each binary encounter imposes a certain diffusion coefficient because diffusion coefficient is basically something that reflects random walk. A random walk is basically the distance that is covered between two, between one collision to the next. And so you can essentially estimate a binary diffusion coefficient Dij which simply reflects the diffusion path associated with binary collisions and from this you can estimate Dimix that is the diffusion coefficient of species i in a mixture of gases rather than in a binary system. So there is an expression here for Dij which goes as 3 times kBt over 8p times pi kBt Mi plus Mj over Mi Mj the whole thing to the power half 1 over pi sigma ij squared omega D which is a function of kBt over epsilon D. So if you compare this expression to what we wrote down for viscosity and thermal conductivity the key differences are pressure is now a key term and it is the denominator. As pressure increases diffusivity decreases which is why if you want to enhance diffusion rates you use low pressure system. So that is an important thing to keep in mind. The second thing to keep in mind is here the temperature dependence is basically close to T times T to the power half or roughly 1.5 but again there is some temperature dependence here. So the actual temperature dependence of the diffusion coefficient is roughly T to the power 1.5 to 1.8. The other thing you need to keep in mind is because we are talking about a binary diffusion process there is this term called sigma ij. What that reflects is the equilibrium distance of separation between two molecules. So if you look at the force between any two adjacent molecules Van der Waals force for example it will follow this typical pattern where 0 is here and there is a particular distance of separation between the two molecules. So what we are plotting here is essentially the epsilon ij or the intermolecular potential as a function of the distance of separation. So as the molecules come very close together there is a repulsive force as they move farther away there is an attractive force so there is a particular distance of separation between two molecules which results in a net force of 0 which is the equilibrium position of rest between these two molecules. So that is essentially obtained by plotting the intermolecular potential between molecules i and j which is epsilon ij as a function of distance of separation and the distance at which it reaches 0 is what we call sigma ij. So it is the intermolecular distance of separation corresponding to the point where epsilon ij equal to 0. And again this omega D parameter is written as 1.16 times k B T by epsilon ij to the power minus 1.7 and again it corrects for the fact that the diffusing molecules are not hot spheres. The closer they are to a hot sphere condition the closer omega D will be to 1 but in general you have to include this correction factor. So once you have estimated this parameter the binary diffusion coefficient the diffusivity of species i in the mixture of species can be derived as 1 minus yi times summation of j equal to 1 to n where j is not equal to i of yj over Dij the whole thing to the power minus 1. So it is essentially estimated as a weighted sum of the binary diffusion coefficients which enables us to calculate the prevailing diffusion coefficient in the case where the collisions are not only binary but multi species are involved. So once you have estimated the mixture or the diffusivity of species i in the mixture this again we are assuming that it is a single parameter for the system which means we are still assuming isotropic. The only assumption we have relaxed here is the you know the binary assumption we are allowing multiple species to be present and we have estimated the diffusion coefficient in the presence of multiple species but the fact that we only have a single scalar value for Di mix essentially implies that we are still assuming it is an isotropic fluid and that the diffusivity in every direction is the same but in reality for many fluids particularly fluids that are under high shear stresses extreme turbulence may not be a good assumption. So we will have to in such cases realize that what we have defined here is like an effective or average diffusion coefficient which is a scalar representation of the diffusion vector that is present in the fluid. Now in the case where you have a heavy species let us say that your molecular weight of the diffusing species exceeds the molecular weight of the carrier gas which can very easily happen in CVD systems because we use H2 as a carrier gas but the depositing species may be SiH4 which is much heavier and larger than the H2 molecule. In such cases how do you estimate the diffusion coefficient? So here you do not really use a random walk principle to estimate diffusion coefficients. Instead you go to the Stokes Einstein theory if you remember which is based upon inertial effects. So you essentially try to track each molecule like a particle and you estimate its diffusivity just like you would estimate the diffusivity of a particle in a fluid. So you may recall some of the experiments you might have done in mechanical operations and so on to estimate the diffusivity of a particle and if you remember the expression that comes out so this is for heavy vapors in a gas or it could even be for solids in a gas or in a liquid the expression for the diffusion coefficient Di equals KBT over 3 pi mu sorry 3 pi mu sigma i effective right. So in the case of a particle you would simply substitute sigma with the particle diameter dp but in the case of a heavy molecule you use the molecular size. The effective molecular size of the species appears in the denominator and so you are modeling diffusion process in this particular case as akin to an inertial process. You are assuming that particles are diffusing or heavy molecules are diffusing in a mechanism that is very similar to how inertial settling happens. So you are kind of modeling it as a sedimentation process is another way to put it and this turns out to be a good way again to estimate the diffusivity of heavy vapors in a lighter gas or as I said particles in a gas or a liquid. So the other type of diffusion process that can happen is after the molecule has adsorbed on a surface it has diffusion along the surface and in fact that is a very critical process in achieving the final state of the CVD film on the substrate. So when you talk about surface diffusion it has very different characteristics when you write the equation because in this case what you are really trying to model is the migration of an adsorbed species from one interstitial site to another and you are trying to estimate the flux of that and from the flux you try to estimate the diffusion coefficient. The diffusive flux associated with the species essentially jumping from one site to the next available site depends on you know several parameters including the activation energy that is required for the molecule to jump from one interstitial site to another and also the number of attempted jumps you know how frequently does the molecule even attempt to jump. So in this case the diffusion coefficient for adsorbate can be written as a combination of a geometric factor times a lattice dimension squared times an attempted jump frequency parameter times an activation energy parameter. So the activation energy parameter can be written as exponential of epsilon d over k b t. So it is the energy that is required that is that you have to get over if you want to move one molecule from one interstitial site to the next one. Now if you look at this expression the interesting thing of course is that the geometry plays a huge role. For example if you have features on the substrate that make it difficult for diffusion to happen that can have an inhibiting effect on the diffusion process. On the other hand if you have certain well defined diffusion pathways to begin with it will actually speed up the diffusion process. So this all this is clubbed into this geometric factor term. The lattice dimension reflects the crystallinity of the substrate. A crystalline structure is one where the energy required to go from one interstitial site to the next can be quite high the energy barrier. So the degree of crystallinity which is reflected in a lattice dimension is another key parameter in determining the rate of diffusion. Of course the attempted jump frequency now that is a temperature dependent parameter. The higher the temperature the more will be the frequency of attempted jumps from one place to the other which is why if you have a CVD film on a surface and you want to make it more uniform if you want the surface spread to be equal and if you want the film to be more adherent you typically raise the temperature because it facilitates the movement of atoms across the surface after they have been absorbed. It just energizes the atoms and makes them more willing to jump from one place to the next. Of course finally again as the temperature increases it becomes easier to overcome the activation energy barrier and move atoms from one place to the other. So the thing to remember with diffusivity is that there is no single expression for it depending on the particular physical situation at hand you know whether you are talking about the diffusion of the precursor vapors to the substrate that is governed by a different diffusivity coefficient and then the actual migration of agglomerated or heavier atoms is governed by a different diffusion equation and finally once the diffusing molecule has been absorbed on the surface its diffusivity on the solid surface is governed by a different equation and the effect of temperature and pressure are very different in each of these cases. For example surface diffusion is dominated by temperature effects but pressure has very little influence whereas diffusion in the gas phase as we saw there is a very strong effect of pressure inverse effect whereas the temperature effect is more moderate between T to the power 1.5 to 1.8 and in the case of heavy vapors again the temperature effect is quite significant but pressure is not. By the way I do not think you need to remember all these actual equations but what you should recollect are some of the critical dependencies you know how do how does viscosity thermal conductivity and particularly diffusivity depend in various situations on system parameters like temperature and pressure and concentrations and so on. So what we have quickly gone through in this lecture is ways to estimate the constitutive coefficients particularly the viscosity thermal conductivity and diffusivity which need to be plugged into the conservation laws in order for us to be able to solve the equations and obtain the velocity distribution, the temperature distribution as well as the concentration distribution of the reacting and diffusing species. Once you have obtained these coefficients you can solve the conservation equations but in the case of a CVD process obviously you need to focus on the transport mass transfer mechanisms involved. So if I look at the overall rate of deposition of a CVD film the deposition flux it has a convective component to it plus a diffusive component to it, right. Mass can be transported by both convection and diffusion but there is also a third effect which is called fluorescence. So there are actually three phenomena that govern the overall mass transport process. Now this is fairly obvious when we talk about convection if rho V is the convective flux of the fluid system as a whole if you take this and multiply this by omega i which is the mass fraction of the ith species that gives you the convective flux of species i can also write this as rho i times V, V is a vector. So the convective flux is associated with molecules that are essentially following the motion of the fluid itself. You know they are just moving along with the carrier fluid but because the carrier fluid has a velocity and a mass associated with it there is a convective flux of the depositing species also. And we just saw that the diffusive flux can be expressed as minus omega i rho di sorry just minus rho di gradient in omega i. So once you know the diffusivity of the diffusing species and you know its gradient you can calculate the diffusive flux. The new term is the Forrest system. So what do we mean by that? What is Forrest's why does it happen and why is it important in CVD systems? Forrest's refers to the movement of species under the effect of an applied force. So it is distinct and different from the convective motion associated with the fluid flow itself. For example if you have a flow of a fluid with a certain velocity V and let us say that its density is rho, rho times V gives you the convective flux and rho i times V gives you the convective flux of species i simply associated with the enforced flow of the fluid. But supposing now I impose a temperature gradient across it. So this is T2 T1 where T2 is let us say much greater than T1 then what happens? Or let us say that I impose an electric field across it right or a magnetic field or a pressure field. So I can on top of the basic flow situation I can impose various external fields. When I do that that field induces a motion as well and it is more akin to a convective motion rather than a diffusive motion. The field that you can think of is just gravity. You know in all our discussions of CVD we have kind of neglected gravitational force but it is present and it can have an effect. So the gravitational field itself induces a phoretic effect. For example in a CVD reactor what would be some of the dominant phoretic fields? Certainly thermal because particularly in a cold wall reactor you know that the temperature gradient between the substrate and the walls of the reactor can be in the range of hundreds of degrees. So the thermal gradient we do need to account for. Pressure is typically not you know hugely varying I mean the entire reactor will be set at a constant pressure. So there is no pressure involved convection within the reactor. Magnetic fields very rare I am not aware of too many CVD reactors which use magnetic fields to drive the process. However electric fields are used because it has been found that by applying an electric field you can actually and setting up an electric field gradient you can speed up a CVD process and so electrophoresis can be an important effect. How about gravity? Is gravitational field an important parameter in CVD? Typically not because it is a diffusion based process and gravity does not play a significant role in enhancing diffusion but when you start talking about very heavy molecules that are diffusing in a very light gas gravity can also become a parameter to consider and then there is something called the diffuse euphoresis which is a forces associated with diffusion because as diffusion occurs there is a concentration gradient being set up that actually diffusion is conveying mass in one direction and according to the principle of mass conservation there must be an equal and opposite flow in the opposite direction and that is what we call diffuse euphoresis it is also known as Staphon flow. So that is another parameter that can influence your net deposition flux. So what we will do in the next few classes is really concentrate on this phoretic phenomena. What are the different types of fields that can be applied and what are the velocities and fluxes associated with these phoretic fields and you have to essentially learn to superimpose. So for example associated with each phoretic field there will be a velocity C. So just like you have a convective velocity V which induces a convective flux the phoretic velocity C will induce a phoretic flux which will be you know just like we have written the convective flux like this we would essentially write this as rho C times omega i or rho i times C as so this is the convective flux and this is the phoretic flux and of course this is the diffusive flux. So the challenge really becomes how do you estimate this phoretic velocity C for various fields. In some cases it is obvious some cases it is not. So we will deal with that in a couple of lectures because it is these correction factors to the deposition flux which if you are not careful and you neglect you will not be able to estimate the deposition flux in a CVD reactor very accurately and that can lead to very misleading conclusions about the design and operation of your CVD reactor. So we will stop at this stage any questions okay so I will see you at the next class.