 Good morning and welcome to the next lecture in our course on chemical engineering principles of CVD process. In the last lecture we looked at the mass transfer equation in a little more detail. There are essentially three contributions to mass deposition fluxes in a chemical reacting flow system such as a CVD reactor and they are basically convection forces and diffusion and we spent a little bit of time looking at how all three of these contributions can be incorporated into the mass balance equation and also how forces in particular can be expressed in terms of a characteristic phoretic velocity as well as a phoretic force and a friction coefficient and then we also define certain dimensionless numbers in particular the stanton number, the Nusselt number and the capture efficiency which can be used to effectively represent our results as well as present the data in a way which is comparable across reactors and so on. Now as we were deriving these equations there was one key assumption which we made which is that the diffusing species is present in trace amounts. In other words any of the diffusing elements a we assume that it is much smaller than 1 and while this is a good assumption in most CVD systems there are situations where this can be violated. So we need to look at how the Nusselt numbers and stanton numbers that we derived for the dilute species case will change when you have a non-dilute species diffusing in your system and also in any physical system the easiest parameter to measure is temperature. So if you look at mass momentum and energy conservation the three corresponding field density parameters are mass fraction, temperature and velocity but from an instrumentation view point temperature is much easier to measure than either mass fraction of species or prevailing velocities in the system and so ideally what you would like to do is map the temperature distribution in a CVD reactor and from that be able to estimate the prevailing distribution of velocities as well as of mass fractions of the various reacting species. Now in order to be able to do that we have to have a condition that is known as the mass transfer analogy condition or MTAC. This condition when satisfied enables us to take temperature distribution data in non-dimensional terms and from that extract corresponding distribution data for reacting species concentration. For example if we were to express the temperature distribution in a CVD reactor as some Tw minus T over Tw minus T infinity where again the designations represent how close you are to the substrate. So in a CVD reactor where you have a substrate which is heated the substrate temperature is what we call Tw and the temperature that is far away from the substrate is what we represent as T infinity. So the temperature at any location particularly within the boundary layer and by the way most of our concentration will be in this region which is the mass transfer sub layer or boundary layer around the substrate because that is really where the deposition process is happening. So if you want to calculate the temperature at any location within this boundary layer you can obtain it as in this fashion you can write this as some non-dimensional variable T star which will be a function of non-dimensional distance, non-dimensional time, Reynolds number and Prandtl number. So we know sufficient heat transfer theory and we know how to measure temperatures sufficiently well to validate our theory. So we can always derive an equation of this form. So this may be for example it could be that this may be Reynolds number to the power half, Prandtl number to the power one third right or some dependency like that. So let us assume that we know how to derive this, we know how to characterize the temperature distribution within the laminar sub layer surrounding the substrate in a CVD reactor and by the way this is for the case where force convection is dominant. You can also write this as T star of X star T star Rayleigh number and Prandtl number in the case where natural convection is dominant. Again the stars simply represent that they are non-dimensional values corresponding to the specific parameters. Of course the Rayleigh's number is defined as G beta delta T L cube by actually first you define a Grashof number or you are familiar with Grashof number and Rayleigh number. So they are used to represent just like Reynolds number is used to represent the ratio between convection and diffusion in a forced convection system. Similarly the Grashof number is used to represent the ratio of convective transfer to diffusive transfer for a natural convection system. So here G is the body acceleration, beta is a coefficient that represent change in density as a function of temperature while holding P and omega A constant. Delta T is your temperature differential so in this case it will be T infinity minus T w. L is a characteristic dimension in your system and nu of course is the kinematic viscosity mu by rho. Now if you take this Grashof number and by the way this is the Grashof number for heat transfer so you represent it with a subscript h. From this you can derive an RA h which is Grashof number multiplied by the Prandtl number and the Rayleigh's number is what you substitute into this equation. So whether it is forced convection dominated or natural convection dominated you can obtain the temperature distribution both theoretically as well as experimentally. Now the question is how do you go from there to estimating the mass fraction distributions inside the reactor. In other words what you would like to be able to do is similarly write omega A w minus omega A divided by omega A w minus omega A infinity is equal to some omega A star which is a function of T star sorry x star T star and Reynolds number and Schmidt number or in the case of forced convection this would be Rayleigh's number for mass transfer comma Schmidt number and by the way the Rayleigh's number for mass transfer would essentially have instead of delta T you would have a delta in the mass fractions instead of defining beta and again this is a beta h let us call this. You can define a beta m which would be representing the ratio of delta rho with respect to change in mass fraction actually why do not we rewrite in the case of natural convection for mass transfer the corresponding g or m value would be equal to g times beta m times delta omega A times l cube over nu square where beta m equals minus 1 by rho of del rho by del omega A holding pressure and temperature constant and Rayleigh's number for mass transfer would then be equal to Grashof number for mass transfer times the Schmidt number. So essentially substitute Schmidt number for Prandtl number. So let us say that you can now you want to write similarly the mass fraction of the diffusing species A at any location inside the boundary layer in this non-dimensional form. So you need to know what is this omega A star as a function of x star T star and the prevailing convective and diffusive dimensionless numbers. If you could say that these two are identical in other words if you can take this function which you know and simply substitute for the Rayleigh for the Prandtl number the Schmidt number in this case and in the case of natural convection substitute Rayleigh number for mass transfer in place of the Rayleigh number for heat transfer. If the same expression would then apply for example if T star goes as Reynolds number to the power half and Prandtl number to the power one-third if you can from that derive that omega A star will go as Reynolds number to the power half Schmidt number to the power one-third then you can essentially take the functional value that you have derived for how the dimensionless temperature is distributed and simply extract from that just by making the appropriate substitution the distribution of the mass fraction. If you could do that obviously it greatly simplifies our ability to simulate the system and it eliminates the need to actually measure the concentrations in the system. However it turns out that in order for this analogy this mass transfer analogy to hold there are certain requirements. One of them is that the mass fraction of the diffusing species must be in trace quantities. The second one is that the phoretic forces must be absent in other words there should be no external field which is operating on the particular species and causing its mass transfer to happen. And the third is that there should be no homogeneous reactions in the boundary layer. Now the reason for these two conditions is that in order for the analogy to hold what that implies is whatever external force or phenomenon there is it must act identically on both mass transfer and heat transfer. For example diffusion is essentially the mechanism is the same whether you are talking about heat transfer or mass transfer or even if you talk about convection the way that heat is transported convectively is very similar to the way mass is transported convectively. However if you take a phoresis force for example gravity the effect of a gravitational field on mass transfer is obviously very different from its effect on heat transfer. So when gravitational field is significant in your system you cannot use the mass transfer analogy or for example thermophoresis. Thermophoresis specifically refers to the effect of a temperature gradient on mass transfer. So clearly there is no equivalent to that for heat transfer you cannot talk about the effect of a temperature gradient on heat transfer because that is already covered under you know Fourier law and so forth. So the presence of any phoretic field violates the mass transfer analogy condition and similarly you know within this boundary layer adjacent to the substrate we like to assume from a modeling view point that the gas phase reactions are essentially frozen. In other words all the chemical reactions are occurring outside the boundary layer and then they occur at the surface in a heterogeneous fashion but in general for modeling purposes it is assumed that within the diffusion boundary layer which is very very thin we neglect homogeneous chemical reactions that enables us to apply the mass transfer analogy condition but again imagine if you allowed homogeneous reactions to occur in the boundary layer clearly that effect on mass transfer will be very different than their effect on heat transfer because the way homogeneous reactions affect the mass fraction profile within the boundary layer will be very different from their effect on the temperature profile within the boundary layer. In general the effect on the mass transfer the mass fraction gradient will be much stronger essentially species could be appearing and disappearing within the boundary layer due to homogeneous chemical reactions the corresponding effect on the heat transfer will be very muted. So there is again no clear correspondence between the two. So these are two conditions you need to bear in mind when you are trying to apply the mass transfer analogy condition. Now so we will look at the effects of all three of these conditions. What happens when omega a is not much less than 1, what happens when phoretic phenomena are not negligible and what happens when homogeneous reactions are occurring and how that affects your deposition characteristics. Before we do that just a quick comment so far we have been focusing mostly on the energy to mass transfer analogy. Now the mass to momentum transfer analogy is actually a lot more complicated it is not as easy to apply because momentum transfer particularly in the case where there is a pressure gradient along the direction of flow the mechanisms tend to be very different between momentum transfer and mass transfer. So the analogy in that particular case is not as obvious. However in the limiting case where there are no stream wise pressure gradients there is one analogy that you may be familiar with the Chilton Colburn analogy which says ST the Stanton number for mass transfer equals CF by 2 times Schmidt number to the power minus 2 third. So this is an analogy that people have used and but again recall that it has very limited validity. It requires that along the direction of flow there cannot be a pressure gradient which is a hugely limiting assumption but when the assumption is valid then if you can measure the skin friction coefficient which is not a difficult measurement from that once you know the distribution of the skin friction coefficient around an object you can from that evaluate or estimate the effect of or the associated distribution of the mass transfer Stanton number around the object. But in the case of the Nusselt number there is no well known analogy that people have been able to establish. So the only mass to momentum transfer analogy that is reported in literature and in textbooks is the Chilton Colburn analogy that says Stanton number for mass transfer equals is related to the skin friction coefficient and the Schmidt number. So let us take these 3 cases in sequence non dilute species transfer, forest is phenomena and homogenous nucleations in the boundary layer and look at their effects. So the first case when omega a is not equal to or is not much smaller than 1 physically what does that imply? CVD happens because you have a substrate with a boundary layer around it and mass transfer is occurring from the outside of the boundary layer to the substrate. Now it is a diffusive transfer but it is a net transfer of material from the fluid to the substrate. Now typically in the case of by applying mass conservation principles this assumes that there will be an equal and opposing flow which will essentially negate the diffusive flux towards the surface. Now we tend to neglect this flow in general because the diffusing species are in trace amounts. So the flow that is associated is not large enough in magnitude for us to worry about it. However as omega a the mass fraction of the departing species becomes comparable to 1 which would be the case where you do not use a carrier gas or a diluent in the CVD system. If you are using for example pure SiH4 as your depositing species and you are making Si and H2 that would be a case where SiH4 concentration is close to 1 and so you cannot say that it is much smaller than 1. Now in that case you have to take into account the flow that opposes the diffusive flow and that is called Stefan flow and it has a definite effect on the mass transfer characteristics in the system. It is typically represented by a parameter Bm which is written as Vm Vw times delta m over Da where Vw is the velocity associated with the supposing flow. So it is the velocity you are kind of you are taking the mass flux that is resulting and converting it into a velocity by dividing by an appropriate density. So essentially Vw is a velocity representation of the Stefan flux that is happening. Delta m is the thickness of the mass transfer boundary layer in the presence of Stefan flow and Da is the diffusion coefficient of species A. The effect of this flow is to alter the Nusselt number. Let us say Num A0 is the Nusselt number in the absence of Stefan flow and some Num A is the Nusselt number in the presence of Stefan flow then these two are related by Num A equals Num A0 times a correction factor due to Stefan flow and this correction factor F in the case of Stefan flow is related to this parameter Bm as ln of 1 plus Bm over Bm. So the effect of Stefan flow is to reduce the rate of deposition to the system. There are cases where this Vw can actually enhance deposition as well. For example if you have a perforated substrate and you are actually sucking air through the perforations you can actually use this flow to enhance the rate at which deposition is happening to the substrate. In that case the Vw will be a negative factor and the correction factor F will be greater than 1 but in general whenever you have diffusion happening towards the substrate inducing a convective flow which balances it the convective flow has the effect of taking some of the depositing material back to the mainstream and thereby reducing the net rate of deposition that is happening. Another way to represent this is in terms of what is called the bio number. The bio number has a representation that is very similar to Bm but with a very subtle difference it is actually written as Vm delta M0 over Da. The difference between the two is in this term. The Bm parameter which is also called the blowing parameter uses the boundary layer thickness with Stefan flow in the numerator whereas the bio number uses the boundary layer thickness without Stefan flow in the numerator. When you write the bio number in this fashion you can also write this as Vw times a characteristic dimension L over Da times Nusselt number again in the absence of Stefan flow in the denominator. So it is basically another representation of the bio number in the case where you have convective flow being the dominant effect you can also write the bio number as Vw over a characteristic velocity U times Staten number for mass transfer again under non-Stefan flow conditions. So these are various definitions of the bio number and depending on whichever is the dominant flow mechanism as well as your ability to measure the various parameters here you can choose to use any of these definitions of the bio number or you can simply use the definition of the blowing parameter. The blowing parameter is something that is unique to Stefan blowing but as we will see later the bio number is something that is common to any process that causes a difference in the transport flux. So once you have estimated this bio number let us call the Di M, A the correction factor now F Stefan becomes Bi M A divided by 1 minus exponential of minus Bi M A. So you can estimate the correction factor using either the formula for the blowing constant Bm should actually be Bm, A delta M, A or you can write the correction factor in terms of the bio numbers in either case once you have estimated the correction factor you take the Nusselt number that you have defined in the absence of Stefan flow and multiplied by the correction factor to get the actual prevailing Nusselt number from that then you can derive the actual prevailing deposition rate, film thickness and so on. Now the reason we say that Stefan flow does not violate mass transfer analogy condition is because Stefan flow actually affects both mass transfer and heat transfer identically. So just like we have defined a Bm A you can also define a BH which is the blowing parameter corresponding to heat transfer as Vw delta H over alpha A or alpha where alpha equals K by rho Cp. So you can substitute for mass diffusivity by thermal diffusivity you substitute the boundary layer thickness for mass transfer with the boundary layer thickness for heat transfer and you can estimate BH and once you have that you can again derive a correction factor for F Stefan in the case of heat transfer as ln of 1 plus BH over BH or in terms of the bio numbers again you can write in the case of heat transfer you can write a bio number for heat transfer under Stefan flow conditions as Vw delta H0 over alpha and again you can write this as Vw times L over alpha times Nusselt number for heat transfer under no Stefan flow conditions and you can write this as Vw over U times Tantan number for heat transfer under normal or nominal conditions. Yeah Vw as I was explaining is a representation of this opposing flux using a velocity so basically rho times Vw will give you the Stefan flux. So there is a flux associated with the diffusive transfer of the condensing species to the surface which is being opposed by another flux in an equal and opposite direction. So once you know what that flux is and you know the density you can estimate the velocity corresponding to that. So the point is this FH the correction factor that you obtain for Stefan flow is identical in its form for both heat transfer as well as mass transfer and therefore the correction factor will apply equally to both. Hence the mass transfer analogy is not violated okay. Now let us take another case which is phoretic transfer so in the case of phoresis our definition of its effect is very similar to what we just did for Stefan flow. So you will take the Nusselt number for mass transfer in the presence of phoresis divided by the Nusselt number for species A in the absence of phoresis and write this as a correction term Fphoresis where again this Fphoresis will be related to a bio number as a bio number for phoresis divided by 1 minus exponential of minus bio number where again this bio number will be defined similarly as now in this case instead of Vw we simply use the associated phoretic velocity. If you remember yesterday we defined a phoretic velocity in the case of sedimentation, a phoretic velocity in the case of electrophoresis, phoretic velocity in the case of thermophoresis and so on. So you simply take that corresponding velocity parameter C and supply by delta M0 over DA and everything else applies. You can again write this as C times L over DA times Num0 of A or you can write this as C over U times stanton number for mass transfer in the absence of phoresis and so on. Now phoresis as a correction factor can be positive or negative depending on which direction the phoretic force acts. If the phoresis is such that it pushes the material towards the substrate then it will enhance the deposition rate. For example in the case of thermophoresis if the substrate is hot and the reactor walls are cold which is a normal condition thermophoresis tends to happen from hot to cold. So the phoretic flux will oppose the deposition flux. So you will see a net reduction in the CVD rate when thermophoresis is happening to a significant extent. However even this is only true if the diffusing molecule is heavier than the carrier gas. When a diffusing molecule is heavier than the carrier gas it tends to go from high temperature to low temperature when you apply a temperature gradient. If the diffusing species is lighter than the carrier gas then it happens in reverse. However as you know in most CVD reactors the carrier gas tends to be a light gas like helium or hydrogen or so or species like that. So typically the diffusing species will be heavier than the carrier gas. So the effect of thermophoresis will typically be to take the material away from the surface and thereby reduce the effective diffusion rate which means that fphoresis will be typically less than 1 in the case of thermophoresis. In the case of sedimentation if the reactor configuration is such that sedimentation helps the deposition process for example in the stagnation flow configuration then this can be greater than 1. But it also requires that the diffusing species must be heavy enough that gravity does become a significant effect. So it depends once again on the nature of the deposition process. How heavy is the depositing molecule? But in general you would say that this will be greater than 1 for gravitational effects. Now for other field forces like electrophoresis that is totally up to you. For example you can actually design a CVD reactor where the electrophoretic field is oriented in such a way that it either enhances the rate of deposition or reduces the rate of deposition. So depending on the design can be greater or less than 1 for an electrophoretic field or for a magnetophoretic field. There is also a phoretic field called diffusiophoresis which by the way is simply another name for Stefan blowing. It is the phoretic flow associated with diffusive flow. So the Stefan flow example we looked at first is actually a subset of phoretic flows. However with the key difference as I mentioned Stefan flow acts on both mass transfer and heat transfer in an identical fashion. So it does not violate heat transfer analogy conditions. However the other phoretic phenomena do have a significantly different effect on mass transfer compared to heat transfer. So in all these cases the heat transfer analogy condition is violated which means you cannot take T star as being equal to omega A star. T star of R E comma P R you cannot simply take that and rewrite it with Reynolds number and Schmidt number and assume that the same non-dimensional formulation holds good. Another way of saying it is if phoretic phenomena are important in your system and you want to know the species concentration distribution in the system you do not have any other option but to actually directly instrument for it and measure it. You cannot simply measure the temperature distribution and assume that the concentration distribution will follow the same pattern okay. So is phoresis something to be desired or not in a CVD reactor? It certainly complicates your simulation, it complicates your measurements. On the other hand if phoresis can be controlled properly it can enhance your deposition rate mostly that is what you are looking for. Now if you are running a CVD reactor your intent is to get as much material to deposit and as short a time as possible. So phoresis does give you a handle to do that. However just remember that it is effects are difficult to predict and difficult to measure. So it is somewhat empirical people usually do this in a trial and error basis. So they would apply an electric field see what it does to the deposition rate. If it helps they will continue to apply it if it does not seem to help or if it is actually making it worse then they will stop using it. But if you know chemical engineering principles if you understand these phenomena these effects if you understand when you can apply the mass transfer and knowledge condition and when you cannot. If you understand how to take your baseline Nusselt numbers and Stanton numbers and correct them for the prevailing phoretic phenomena then you can use phoresis effectively and be able to model it accurately. So that your end result becomes predictable. If it is predictable it is controllable it is optimisable. So that is why it is very important to understand some of these more subtle effects of how phoretic forces play a role in affecting your deposition rates in the system. So we have essentially looked at two phenomena the Stefan blowing phenomena and the phoresis phenomena in which the former does not violate the analogy condition the latter does. In the next lecture we will take a look at a couple more of these homogeneous chemical reactions and heterogeneous chemical reactions and we will take a look at how they can be represented non-dimensionally and we can assess that effect on the prevailing mass transfer rates and processes. So let us stop at that point. Any questions? See you at the next class.