Physical Metallurgy of Steels - Part 6
Bhadeshia, Harry
A series of 12 lectures on the physical metallurgy of steels by Professor H. K. D. H. Bhadeshia. Part 6 deals with the formation of allotriomorphic and idiomorphic ferrite in steels. Growth is treated as diffusion-controlled and by a reconstructive transformation mechanism.
 Okay, we'll make a start. So we have actually finished all of the displacive transformations which lead to a shape change with a large shear component martensite, bainite and weavement center ferrite. And we're going to raise the temperature again and talk about ferrite formation by a diffusional mechanism. So it involves a lot of diffusion. And the vast majority of the 1.3 billion tons of steels that we make every year contain as a majority phase what we are going to talk about today. Not bainite, not martensite, not weavement center ferrite. The vast majority of, I would say 90% of all the steels that we make, the dominant phase is the ferrite that we are going to talk about. So it's a very important phase in steels. And the second most important phase is probably pearlite and I will deal with that after another two lectures. So today's talk is about ferrite. And I'm going to use some specific terminology. A lotomorphic ferrite. It's a long word but it's important to understand what that means. A lotomorphic ferrite is ferrite which forms at the austenite grain boundaries. And because austenite grain boundaries are easy diffusion parts, it tends to grow more rapidly along the boundaries than into the grain. So the shape of a lotomorphic ferrite does not reflect its crystalline symmetry. You know, you don't get nice beautiful shapes with straight facets because it's dominated by the grain boundary. And again, we are looking at a partially transformed sample here. On the other hand, idiomorphic ferrite nucleates probably on dirt particles inside the grains of austenite. And therefore it has a shape which is a reflection of the symmetry of the ferrite and of the austenite in which it grows. So we see some nice faceted shapes. Everything means you develop some plain faces with particular crystallography. The mechanism of transformation is otherwise the same. The shape is different because these grains of ferrite grow inside the austenite, intragranulally nucleated and these are nucleated at the austenite grain boundaries. The austenite grain boundaries are the easiest nucleation sites in steels. Inclusions are the next easiest nucleation sites. So just to summarize that, a legeomorphic ferrite has a shape which does not reflect its crystalline symmetry. It's basically dominated by easy growth along the austenite grain boundaries. So it's dominated by easy growth along the austenite grain boundaries. And idiomorphic ferrite in contrast forms intragranulally. And therefore its shape reflects the symmetry of the ferrite and of the austenite in which it grows. It's the superimposed symmetry of alpha and gamma. And shape reflects the superimposed of alpha and gamma. If it was growing in a liquid then obviously its shape would be determined just by the symmetry of the ferrite but it's not. It's growing in another solid and therefore it's a superimposed symmetry of the two phases. And both of these transformations can only happen at temperatures where iron itself can diffuse over the length scale of your transformation product. So both transformations can only occur at temperatures typically above 600 degrees centigrade where the iron atoms are mobile. So the growth doesn't just involve the diffusion of carbon. You must also be able to move the iron otherwise you get a shape deformation and strain energy and all the rest of it. So there is no shape deformation here with a shear component. There's only volume change. So if you polish a crystal of austenite completely flat and you transform it to one of these ferrites it will not lead to a shape change other than volume change. So there is no shape deformation other than volume change. So this is an illustration of idiomorphic ferrite where you can see there are some nice flat interfaces with particular crystallographic indices. This is martensite. Of course you can only see these shapes when you've partially transformed the material. If you've got 100% transformation and the ferrite grains have touched each other then the shape is not determined by symmetry but by space filling. And this is an allotium morph of ferrite which is growing along an austenite grain boundary. And the thing that I wanted to notice is that it's like a thin layer which forms at the boundary and it grows rapidly along the boundary but the growth rate normal to the boundary is slow because this is a diffusional transformation and you're building up carbon in front of the interface. So unlike what we did in the last lecture where you had a thin plate growing and the partitioned carbon is being left behind with the tip advancing into fresh austenite. Here you have a plane which is moving. So the carbon piles up in front of the ferrite and therefore this growth rate will not be constant. In yesterday's lecture the lengthening rate of the plate of ferrite was a constant rate. In the equation for velocity we did not have a time dependence of velocity. The velocity only depended on the concentration terms, the diffusion coefficient and the tip radius. The diffusion distance was constant equal to the radius. Here the situation is completely different because we are moving effectively a plane and there's no leaving behind of carbon so you accumulate carbon in front of the interface. This is another example of allotrimophic ferrite where you can see that the thickening rate normal to the boundary is much slower than the lengthening rate. And notice also that this layer is not a single crystal of ferrite. You've nucleated many grains of ferrite along the boundary but they all are limited by thickening. So I'm going to treat this problem as one-dimensional growth of ferrite because it's effectively a plane moving normal and diffusion is only happening in one direction. Right so the difference between the calculations we did yesterday and what we are going to do today is that the velocity of growth will not be constant. It will change as the thickness of the ferrite increases. Now do you think that the velocity should decrease as the particle becomes thicker or increase? Yeah, correct. It will decrease and why is that? Yeah, you're absolutely right. The velocity will decrease as the particle becomes thicker. Any ideas why? So it's because you are accumulating more and more carbon in front of the interface so you have to diffuse over longer and longer distances. So imagine that you're looking at a pond. Yeah, a pond is a pool of water and it's winter and you form a layer of ice on the surface. The ice then becomes thicker so it takes longer for heat from the water to diffuse to the surface and more solidification to happen. So the freezing rate of the pond slows down as the ice becomes thicker. That's very lucky because otherwise the whole of the water would freeze and any fish inside would suffer. If you solder something and you form an inter-metallic compound between the solder and the wire that inter-metallic compound will slow down as it becomes thicker because you need to diffuse material between the wire and the solder. If you look at oxidation and you form an oxide layer on the surface, if oxygen has to diffuse through the oxide to the metal, as the oxide becomes thicker, oxidation rate will slow down. So we need to discover that relationship in any theory that we form. We are going to assume again as we did yesterday that local equilibrium exists at the interface. In other words, the interface compositions are given by a tie line of the phase diagram. So we will consider diffusion control growth, consider diffusion control growth with local equilibrium at the interface. And by local equilibrium, we mean that the composition of the ferrite at the interface will be given by this phase boundary and the composition of the austenite which is in contact with the ferrite will be given by this concentration here from the phase diagram. So these points are determined by the common tangent touching to the two free energy surfaces as we did in the last lecture. So when these two phases are in contact, the composition of the austenite cannot exceed C gamma alpha and that of the ferrite will be C alpha gamma and this is our C bar is the average carbon concentration in the steel. Okay so let's proceed as we did in the last lecture and define the concentration profile at the interface. So if I draw a graph of distance z and concentration plotted along here and as a result of the partitioning of carbon, I will get a profile which looks like this. This is C bar, here we have ferrite and here we have austenite. This point here is z star as we did in the last lecture. The concentration here is C gamma alpha and here it's C alpha gamma. And this distance here is the diffusion distance. So far there is nothing here which is different from what we had for the lengthening rate of Weidmüsten ferrite. And I can immediately write down an equation for the rate at which carbon is being partitioned into the austenite, being equal to the rate at which it's being taken away by diffusion from the interface. So as this interface moves, we will have to put that much carbon into the austenite but to maintain C gamma alpha constant, it has to be carried away by diffusion away from the interface. So the velocity, so this is again C gamma alpha and this is C alpha gamma and this is C bar. So the rate at which the interface moves, which is d z star by dt multiplied by the amount of carbon that we are partitioning, that is C gamma alpha minus C alpha gamma must equal to the rate at which it's being carried away from the interface by diffusion, which is the diffusion coefficient times C gamma alpha minus C bar divided by the diffusion distance. So these two terms must be equal, this is how much carbon is being dumped into the ferrite in a unit of time and this is how much carbon is being carried away from the interface in that time period. If those two terms are not equal, gamma alpha will not be constant and that's not diffusion control growth. And notice that this gradient, again we are assuming to be constant but in practice it will be like an error function. So this gradient is really evaluated at the position of the interface. Everyone happy with that? Now the problem is that this time the diffusion distance is not constant because we've got a plane moving and there is nowhere for the carbon to go except in front of the interface. So we are accumulating carbon as the interface moves. So we've got to have another condition to solve for the diffusion distance. So that other condition comes from mass balance. So if I draw this concentration growth again, this is the distance and this is concentration. Here I have ferrite. This is the amount of carbon, total amount of carbon that has been rejected as ferrite growth and this is the amount of carbon that has accumulated inside the austenite. So if I label this as A and this as B, then the areas A and B must be equal to conserve mass. So the areas A and B must be equal to conserve mass. So I can write another equation that C bar minus C alpha gamma into the thickness of the ferrite Z star must equal the area B which is C gamma alpha minus C bar into ZD upon 2 because it's a triangle so it's half. So the area A here is simply given by C bar minus C alpha gamma, C bar minus C alpha gamma times Z star which is the thickness of our ferrite. And the area of this triangle is C gamma alpha minus C bar times this diffusion distance here divided by 2 because it's a triangle. So now I've got an equation for the diffusion distance. I can write the diffusion distance as ZD is equal to C bar minus C alpha gamma into Z divided by C gamma alpha minus C bar and multiplied by 2. So just rearrangement of this equation here. Is that okay? Yeah? Okay let me just summarize how far we've got. So this is the local equilibrium condition. Here we have the diffusion distance. The concentration in the austenite cannot exceed C gamma alpha and this is the concentration in the ferrite. This is the relationship of the concentration terms with respect to the phase diagram and this is the rate at which solute is partitioned which is this concentration minus this concentration times the velocity of the interface, the rate at which solute is pushed into the austenite. And that has to be carried away by this flux which is because we are assuming a constant diffusion coefficient that's given by Fick's first law as the diffusion coefficient times the gradient. I've lost this minus sign because this gradient is actually negative. So these two terms must be equal. This must equal this and therefore we get the equation which has an unknown which is the diffusion distance. And to get the diffusion distance we have mass balance that this area must be equal to this area and this is the mass balance equation which allows us to substitute for delta Z or ZD as we are using into the first equation. Everyone care about that? Okay so all we have to do is to substitute this equation which we'll call equation 2 into equation 1 here. So ZD here is underneath, let me see if I can put up two pages at a time. Okay it doesn't help because I have to turn it. Right, let me just write equation 1 out again. So equation 1 was that the velocity which is dz star by dt into c gamma alpha minus c alpha gamma will equal the diffusion coefficient times the gradient which is c gamma alpha minus c bar divided by ZD. So this is equation 1. And now I'm going to substitute for ZD into this equation. So I want you to check in case I make a mistake. From now on it's your duty because you identified the mistake the last time. So we have dz star by dt. I want to isolate the velocity on one side of the equation. So I'm going to take c gamma alpha minus c alpha gamma onto the other side. So that's just taking this term here onto the other side. Diffusion coefficient comes on top and we have a term c gamma alpha minus c bar. And since ZD is at the bottom I will have another term here which is c gamma alpha minus c bar divided by c bar minus c alpha gamma and a Z star at the bottom and a 2 at the top. So this is a 2. Is that correct? 2 at the bottom, yes you're right. Come on rub out. Okay, right so I can simplify this by taking Z star onto this side. So I have Z star into dz star is equal to and a dt over here and then a term which I'm not going to rewrite but it's basically this part and the 2. So you've got Z star dz star on one side and dt on the other side. So if I integrate that I get Z star squared being equal to something times d, right? So if I integrate this therefore Z star squared is proportional to time or in other words Z star is proportional to t to the power of a half. So the thickening thickness of the ferrite will vary with time to the power of a half. So this is called parabolic thickening. So in oxidation the oxide is known as parabolic thickening in some cases, right? Where the material has to diffuse through the oxide to reach the metal or the metal has to diffuse through the oxide to reach the surface. Exactly the same. Similarly the thickening of inter metallic layer when you solder something will vary parabolically with time. So this is called parabolic thickening. And if I plot the thickness versus time then it will follow something like this, okay? Now clearly the slope of that line is decreasing as time increases that means the growth rate is slowing down, yeah? So growth rate slows down as the ferrite, sorry, alpha, okay, as ferrite decreases. So this is an important result. If you have a very large austenite grain size then it will be very difficult to complete the transformation because you will have thin layers growing from the surface. It takes a very long time for them to reach the center of the grain. So in large grain size specimens in the same amount of time as a small grain size specimen you will get less ferrite. So this is the same calculation where we now have the two equations. One that the rate that is partitioned must equal to the flux from the interface and this is our mass balance equation and you go through the maths and you find that the velocity will vary with the thickness of the sample and the velocity of growth decreases as the sample thickness increases and that the thickness is proportional to the square root of time. Once again this is a very powerful equation because we have the diffusion coefficient, we have terms from the phase diagram and the average composition of the steel here and we can predict the thickness as a function of time. Everyone happy with that? We are only considering a binary alloy at the moment which is iron carbon. So this is for a binary alloy, there is only one solute which is controlling the thickening rate of the ferrite and this is actually how the thickness would vary with time and the slope of this curve is the velocity of the growth and clearly the slope is decreasing as it thickens and thickens and thickens. Now I said to you it is a very important equation because it takes account of the alloying elements, you know the phase diagram and this is quite an interesting calculation and it is actually reflected in practice as well. Look if I plot the ferrite thickness here versus the time I am plotting actually the half thickness because diffusion transformations like ferrite are not limited by the austenite grain boundary. If you nuclearize at the boundary you can grow this way and this way but martensite plates and bainite plates and Wiedemann's Sun ferrite will not cross an austenite grain boundary because they are displacive and involve a systematic movement of atoms which cannot be sustained across a change in crystallography. But diffusion transformations can go right across austenite grain boundaries. So here is the half thickness being plotted versus time and what I want you to note is that here I am changing the carbon concentration by equal amounts. So here the difference is 0.02, 0.02, 0.02 and 0.02 but there is a very rapid change in kinetics here compared with here. So at low carbon concentrations the growth rate is far more sensitive to the concentration of carbon than at high carbon concentrations. This is exactly what is found in practice. And the reason for this goes back to our equation here that when your average carbon concentration approaches the solubility of carbon in ferrite there is no need to partition carbon. So if you are getting very close to the solubility of carbon in ferrite then very little carbon is partitioned into the austenite so the growth rate becomes very large. If C bar is equal to C alpha gamma then the growth rate is infinite. Now it is not possible to get a growth rate of which is infinite right? So what happens? What becomes the limiting factor instead of the diffusion of carbon in the austenite ahead of the interface? Well, you know you still have to change the crystal structure as the iron atoms cross the boundary. So there is a certain rate at which that can happen and that becomes the limiting factor and that is called interface control growth instead of diffusion control growth where some other process other than diffusion of carbon becomes the rate limiting factor. So let us just write that down. So the growth rate is very sensitive to the carbon concentration at low carbon concentrations. Growth rate is very sensitive to carbon concentration as C bar approaches the solubility of carbon in the ferrite because then there is very little carbon partitioned into the austenite because then very little carbon is partitioned into gamma. When C bar becomes close to C alpha gamma some other process will become rate limiting instead of the diffusion of carbon. So when C bar becomes equal to C alpha gamma growth rate tends to infinity if it is diffusion controlled. But in practice some other factor will become rate limiting. For example the transfer of iron atoms across the interface. So obviously if you have completely pure iron the ferrite is not going to grow at an infinite rate from austenite. It will be the transfer of iron atoms across the boundary which is a thermally activated process which will limit the growth rate of ferrite. So the model we have derived is a diffusion control growth model which assumes that the growth process is controlled by the diffusion of carbon in the austenite ahead of the interface. So I am sorry my handwriting is not very good but I am trying to record everything. So many of the steels that we produce today for structural applications have low carbon concentrations you know not very far from 0.05 or 0.03 weight percent carbon. So it is extremely important to control that concentration accurately even though you are producing hundreds of millions of tons of this kind of material every year. If you don't control it you will get different properties and variations in properties during your steel production process. Now the reason why it is parabolic is illustrated here that as my ferrite becomes thicker you see this area here must be equal to the area under the triangle. So as the ferrite becomes thicker the area under this triangle must also increase and therefore the diffusion distance is becoming longer and longer as the ferrite thickens and therefore the growth rate slows down as the ferrite becomes thicker. Now today's lecture we have dealt with the binary system and it is very easy to fix the concentrations at the interface just by drawing a tie line on the phase diagram for the transformation temperature here and the average carbon concentration here. So everybody familiar with the term tie line? So it is basically saying look I have got a phase diagram here if I have a steel of this composition then the equilibrium composition of ferrite is given by this point and the equilibrium composition of austenite by this point and this line is what connects the compositions of the two phases which are in contact and in equilibrium. So that is called a tie line because it ties them together they depend on each other. So it is very easy for a binary to pick this. I am just going to introduce you to what happens when we add other alloying elements but before we can treat that problem I need to teach you something called irreversible thermodynamics which I will cover in the next lecture but I just want to introduce you to the problem. Okay so that tie line or the phase boundaries are obtained by drawing a tangent which is common to the free energy surfaces of alpha and gamma and that gives us the equilibrium compositions. Yeah, straight forward. What happens when I add another alloying element along here? So here we are plotting carbon and let us say I add manganese along this axis. And what should we get? Well these free energy curves will become surfaces in three dimensions so something like this here in both in the third dimension and the tangent becomes a contact plane which is touching those two free energy surfaces. But when you have a contact plane there is not going to be a unique answer for this and this because I can rock that plane and still get it to touch both the surfaces at a variety of points. So we don't have a unique answer. You actually at a constant temperature will have an infinite number of tie lines. So this is what it looks like for a ternary system where we have say iron and manganese and carbon. These are actually surfaces and this is my tangent plane. If I touch it to these surfaces, yes I can get a tie line which I plot on this horizontal plane but I can actually rock that plane like this. Can you imagine that? And it will touch other places so there is a whole set of tie lines which defines an alpha plus gamma phase field at a constant temperature. So in the binary system there was a unique temperature and a unique set of equilibrium compositions but here we have a lot more choice. So it is not how we define the interface compositions when it comes to more than one solute diffusing. And in order to treat that problem I need to introduce you to multi-component diffusion and in order to do that I need to treat irreversible thermodynamics which is a very easy subject in the next lecture. That is all for today.
 Okay, we'll make a start.  So we have actually finished all of the displacive transformations which lead to a shape change  with a large shear component martensite, bainite and weavement center ferrite.  And we're going to raise the temperature again and talk about ferrite formation by a  diffusional mechanism.  So it involves a lot of diffusion.  And the vast majority of the 1.3 billion tons of steels that we make every year contain  as a majority phase what we are going to talk about today.  Not bainite, not martensite, not weavement center ferrite.  The vast majority of, I would say 90% of all the steels that we make, the dominant phase  is the ferrite that we are going to talk about.  So it's a very important phase in steels.  And the second most important phase is probably pearlite and I will deal with that after another  two lectures.  So today's talk is about ferrite.  And I'm going to use some specific terminology.  A lotomorphic ferrite.  It's a long word but it's important to understand what that means.  A lotomorphic ferrite is ferrite which forms at the austenite grain boundaries.  And because austenite grain boundaries are easy diffusion parts, it tends to grow more  rapidly along the boundaries than into the grain.  So the shape of a lotomorphic ferrite does not reflect its crystalline symmetry.  You know, you don't get nice beautiful shapes with straight facets because it's dominated  by the grain boundary.  And again, we are looking at a partially transformed sample here.  On the other hand, idiomorphic ferrite nucleates probably on dirt particles inside the grains  of austenite.  And therefore it has a shape which is a reflection of the symmetry of the ferrite and of the  austenite in which it grows.  So we see some nice faceted shapes.  Everything means you develop some plain faces with particular crystallography.  The mechanism of transformation is otherwise the same.  The shape is different because these grains of ferrite grow inside the austenite, intragranulally  nucleated and these are nucleated at the austenite grain boundaries.  The austenite grain boundaries are the easiest nucleation sites in steels.  Inclusions are the next easiest nucleation sites.  So just to summarize that, a legeomorphic ferrite  has a shape which does not reflect its crystalline symmetry.  It's basically dominated by easy growth along the austenite grain boundaries.  So it's dominated by easy growth along the austenite grain boundaries.  And idiomorphic ferrite in contrast forms intragranulally.  And therefore its shape reflects the symmetry of the ferrite and of the austenite in which  it grows.  It's the superimposed symmetry of alpha and gamma.  And shape reflects the superimposed of alpha and gamma.  If it was growing in a liquid then obviously its shape would be determined just by the  symmetry of the ferrite but it's not.  It's growing in another solid and therefore it's a superimposed symmetry of the two phases.  And both of these transformations can only happen at temperatures where iron itself can  diffuse over the length scale of your transformation product.  So both transformations can only occur at temperatures  typically above 600 degrees centigrade where the iron atoms are mobile.  So the growth doesn't just involve the diffusion of carbon.  You must also be able to move the iron otherwise you get a shape deformation and strain energy  and all the rest of it.  So there is no shape deformation here with a shear component.  There's only volume change.  So if you polish a crystal of austenite completely flat and you transform it to one of these  ferrites it will not lead to a shape change other than volume change.  So there is no shape deformation  other than volume change.  So this is an illustration of idiomorphic ferrite where you can see there are some nice  flat interfaces with particular crystallographic indices.  This is martensite.  Of course you can only see these shapes when you've partially transformed the material.  If you've got 100% transformation and the ferrite grains have touched each other then  the shape is not determined by symmetry but by space filling.  And this is an allotium morph of ferrite which is growing along an austenite grain boundary.  And the thing that I wanted to notice is that it's like a thin layer which forms at the  boundary and it grows rapidly along the boundary but the growth rate normal to the boundary  is slow because this is a diffusional transformation and you're building up carbon in front of  the interface.  So unlike what we did in the last lecture where you had a thin plate growing and the  partitioned carbon is being left behind with the tip advancing into fresh austenite.  Here you have a plane which is moving.  So the carbon piles up in front of the ferrite and therefore this growth rate will not be  constant.  In yesterday's lecture the lengthening rate of the plate of ferrite was a constant rate.  In the equation for velocity we did not have a time dependence of velocity.  The velocity only depended on the concentration terms, the diffusion coefficient and the tip  radius.  The diffusion distance was constant equal to the radius.  Here the situation is completely different because we are moving effectively a plane  and there's no leaving behind of carbon so you accumulate carbon in front of the interface.  This is another example of allotrimophic ferrite where you can see that the thickening rate  normal to the boundary is much slower than the lengthening rate.  And notice also that this layer is not a single crystal of ferrite.  You've nucleated many grains of ferrite along the boundary but they all are limited by thickening.  So I'm going to treat this problem as one-dimensional growth of ferrite because it's effectively  a plane moving normal and diffusion is only happening in one direction.  Right so the difference between the calculations we did yesterday and what we are going to  do today is that the velocity of growth will not be constant.  It will change as the thickness of the ferrite increases.  Now do you think that the velocity should decrease as the particle becomes thicker or increase?  Yeah, correct.  It will decrease and why is that?  Yeah, you're absolutely right.  The velocity will decrease as the particle becomes thicker.  Any ideas why?  So it's because you are accumulating more and more carbon in front of the interface so  you have to diffuse over longer and longer distances.  So imagine that you're looking at a pond.  Yeah, a pond is a pool of water and it's winter and you form a layer of ice on the surface.  The ice then becomes thicker so it takes longer for heat from the water to diffuse to the  surface and more solidification to happen.  So the freezing rate of the pond slows down as the ice becomes thicker.  That's very lucky because otherwise the whole of the water would freeze and any fish inside  would suffer.  If you solder something and you form an inter-metallic compound between the solder and the wire  that inter-metallic compound will slow down as it becomes thicker because you need to  diffuse material between the wire and the solder.  If you look at oxidation and you form an oxide layer on the surface, if oxygen has to diffuse  through the oxide to the metal, as the oxide becomes thicker, oxidation rate will slow  down.  So we need to discover that relationship in any theory that we form.  We are going to assume again as we did yesterday that local equilibrium exists at the interface.  In other words, the interface compositions are given by a tie line of the phase diagram.  So we will consider diffusion control growth, consider diffusion control growth with local  equilibrium at the interface.  And by local equilibrium, we mean that the composition of the ferrite at the interface  will be given by this phase boundary and the composition of the austenite which is in contact  with the ferrite will be given by this concentration here from the phase diagram.  So these points are determined by the common tangent touching to the two free energy surfaces  as we did in the last lecture.  So when these two phases are in contact, the composition of the austenite cannot exceed  C gamma alpha and that of the ferrite will be C alpha gamma and this is our C bar is  the average carbon concentration in the steel.  Okay so let's proceed as we did in the last lecture and define the concentration profile  at the interface.  So if I draw a graph of distance z and concentration plotted along here and as a result of the partitioning  of carbon, I will get a profile which looks like this.  This is C bar, here we have ferrite and here we have austenite.  This point here is z star as we did in the last lecture.  The concentration here is C gamma alpha and here it's C alpha gamma.  And this distance here is the diffusion distance.  So far there is nothing here which is different from what we had for the lengthening rate  of Weidmüsten ferrite.  And I can immediately write down an equation for the rate at which carbon is being partitioned  into the austenite, being equal to the rate at which it's being taken away by diffusion  from the interface.  So as this interface moves, we will have to put that much carbon into the austenite but  to maintain C gamma alpha constant, it has to be carried away by diffusion away from  the interface.  So the velocity, so this is again C gamma alpha and this is C alpha gamma and this is  C bar.  So the rate at which the interface moves, which is d z star by dt multiplied by the  amount of carbon that we are partitioning, that is C gamma alpha minus C alpha gamma  must equal to the rate at which it's being carried away from the interface by diffusion,  which is the diffusion coefficient times C gamma alpha minus C bar divided by the diffusion  distance.  So these two terms must be equal, this is how much carbon is being dumped into the ferrite  in a unit of time and this is how much carbon is being carried away from the interface in  that time period.  If those two terms are not equal, gamma alpha will not be constant and that's not diffusion  control growth.  And notice that this gradient, again we are assuming to be constant but in practice it  will be like an error function.  So this gradient is really evaluated at the position of the interface.  Everyone happy with that?  Now the problem is that this time the diffusion distance is not constant because we've got  a plane moving and there is nowhere for the carbon to go except in front of the interface.  So we are accumulating carbon as the interface moves.  So we've got to have another condition to solve for the diffusion distance.  So that other condition comes from mass balance.  So if I draw this concentration growth again, this is the distance and this is concentration.  Here I have ferrite.  This is the amount of carbon, total amount of carbon that has been rejected as ferrite  growth and this is the amount of carbon that has accumulated inside the austenite.  So if I label this as A and this as B, then the areas A and B must be equal to conserve  mass.  So the areas A and B must be equal to conserve mass.  So I can write another equation that C bar minus C alpha gamma into the thickness of  the ferrite Z star must equal the area B which is C gamma alpha minus C bar into ZD upon  2 because it's a triangle so it's half.  So the area A here is simply given by C bar minus C alpha gamma, C bar minus C alpha gamma  times Z star which is the thickness of our ferrite.  And the area of this triangle is C gamma alpha minus C bar times this diffusion distance  here divided by 2 because it's a triangle.  So now I've got an equation for the diffusion distance.  I can write the diffusion distance as ZD is equal to C bar minus C alpha gamma into Z  divided by C gamma alpha minus C bar and multiplied by 2.  So just rearrangement of this equation here.  Is that okay?  Yeah?  Okay let me just summarize how far we've got.  So this is the local equilibrium condition.  Here we have the diffusion distance.  The concentration in the austenite cannot exceed C gamma alpha and this is the concentration  in the ferrite.  This is the relationship of the concentration terms with respect to the phase diagram and  this is the rate at which solute is partitioned which is this concentration minus this concentration  times the velocity of the interface, the rate at which solute is pushed into the austenite.  And that has to be carried away by this flux which is because we are assuming a constant  diffusion coefficient that's given by Fick's first law as the diffusion coefficient times  the gradient.  I've lost this minus sign because this gradient is actually negative.  So these two terms must be equal.  This must equal this and therefore we get the equation which has an unknown which is  the diffusion distance.  And to get the diffusion distance we have mass balance that this area must be equal  to this area and this is the mass balance equation which allows us to substitute for  delta Z or ZD as we are using into the first equation.  Everyone care about that?  Okay so all we have to do is to substitute this equation which we'll call equation 2  into equation 1 here.  So ZD here is underneath, let me see if I can put up two pages at a time.  Okay it doesn't help because I have to turn it.  Right, let me just write equation 1 out again.  So equation 1 was that the velocity which is dz star by dt into c gamma alpha minus  c alpha gamma will equal the diffusion coefficient times the gradient which is c gamma alpha  minus c bar divided by ZD.  So this is equation 1.  And now I'm going to substitute for ZD into this equation.  So I want you to check in case I make a mistake.  From now on it's your duty because you identified the mistake the last time.  So we have dz star by dt.  I want to isolate the velocity on one side of the equation.  So I'm going to take c gamma alpha minus c alpha gamma onto the other side.  So that's just taking this term here onto the other side.  Diffusion coefficient comes on top and we have a term c gamma alpha minus c bar.  And since ZD is at the bottom I will have another term here which is c gamma alpha minus  c bar divided by c bar minus c alpha gamma and a Z star at the bottom and a 2 at the  top.  So this is a 2.  Is that correct?  2 at the bottom, yes you're right.  Come on rub out.  Okay, right so I can simplify this by taking Z star onto this side.  So I have Z star into dz star is equal to and a dt over here and then a term which I'm  not going to rewrite but it's basically this part and the 2.  So you've got Z star dz star on one side and dt on the other side.  So if I integrate that I get Z star squared being equal to something times d, right?  So if I integrate this therefore Z star squared is proportional to time or in other words  Z star is proportional to t to the power of a half.  So the thickening thickness of the ferrite will vary with time to the power of a half.  So this is called parabolic thickening.  So in oxidation the oxide is known as parabolic thickening in some cases, right?  Where the material has to diffuse through the oxide to reach the metal or the metal  has to diffuse through the oxide to reach the surface.  Exactly the same.  Similarly the thickening of inter metallic layer when you solder something will vary  parabolically with time.  So this is called parabolic thickening.  And if I plot the thickness versus time then it will follow something like this, okay?  Now clearly the slope of that line is decreasing as time increases that means the growth rate  is slowing down, yeah?  So growth rate slows down as the ferrite, sorry, alpha, okay, as ferrite  decreases.  So this is an important result.  If you have a very large austenite grain size then it will be very difficult to complete  the transformation because you will have thin layers growing from the surface.  It takes a very long time for them to reach the center of the grain.  So in large grain size specimens in the same amount of time as a small grain size specimen  you will get less ferrite.  So this is the same calculation where we now have the two equations.  One that the rate that is partitioned must equal to the flux from the interface and this  is our mass balance equation and you go through the maths and you find that the velocity will  vary with the thickness of the sample and the velocity of growth decreases as the sample  thickness increases and that the thickness is proportional to the square root of time.  Once again this is a very powerful equation because we have the diffusion coefficient,  we have terms from the phase diagram and the average composition of the steel here and  we can predict the thickness as a function of time.  Everyone happy with that?  We are only considering a binary alloy at the moment which is iron carbon.  So this is for a binary alloy, there is only one solute which is controlling the thickening  rate of the ferrite and this is actually how the thickness would vary with time and the  slope of this curve is the velocity of the growth and clearly the slope is decreasing  as it thickens and thickens and thickens.  Now I said to you it is a very important equation because it takes account of the alloying  elements, you know the phase diagram and this is quite an interesting calculation and it  is actually reflected in practice as well.  Look if I plot the ferrite thickness here versus the time I am plotting actually the  half thickness because diffusion transformations like ferrite are not limited by the austenite  grain boundary.  If you nuclearize at the boundary you can grow this way and this way but martensite  plates and bainite plates and Wiedemann's Sun ferrite will not cross an austenite grain  boundary because they are displacive and involve a systematic movement of atoms which cannot  be sustained across a change in crystallography.  But diffusion transformations can go right across austenite grain boundaries.  So here is the half thickness being plotted versus time and what I want you to note is  that here I am changing the carbon concentration by equal amounts.  So here the difference is 0.02, 0.02, 0.02 and 0.02 but there is a very rapid change  in kinetics here compared with here.  So at low carbon concentrations the growth rate is far more sensitive to the concentration  of carbon than at high carbon concentrations.  This is exactly what is found in practice.  And the reason for this goes back to our equation here that when your average carbon  concentration approaches the solubility of carbon in ferrite there is no need to partition  carbon.  So if you are getting very close to the solubility of carbon in ferrite then very little carbon  is partitioned into the austenite so the growth rate becomes very large.  If C bar is equal to C alpha gamma then the growth rate is infinite.  Now it is not possible to get a growth rate of which is infinite right?  So what happens?  What becomes the limiting factor instead of the diffusion of carbon in the austenite ahead  of the interface?  Well, you know you still have to change the crystal structure as the iron atoms cross  the boundary.  So there is a certain rate at which that can happen and that becomes the limiting factor  and that is called interface control growth instead of diffusion control growth where  some other process other than diffusion of carbon becomes the rate limiting factor.  So let us just write that down.  So the growth rate is very sensitive to the carbon concentration at low carbon concentrations.  Growth rate is very sensitive to carbon concentration  as C bar approaches the solubility of carbon in the ferrite  because then there is very little carbon partitioned into the austenite because then very little  carbon is partitioned into gamma.  When C bar becomes close to C alpha gamma some other process will become rate limiting  instead of the diffusion of carbon.  So when C bar becomes equal to C alpha gamma growth rate tends to infinity if it is diffusion  controlled.  But in practice some other factor will become rate limiting.  For example the transfer of iron atoms across the interface.  So obviously if you have completely pure iron the ferrite is not going to grow at an infinite  rate from austenite.  It will be the transfer of iron atoms across the boundary which is a thermally activated  process which will limit the growth rate of ferrite.  So the model we have derived is a diffusion control growth model which assumes that the  growth process is controlled by the diffusion of carbon in the austenite ahead of the interface.  So I am sorry my handwriting is not very good but I am trying to record everything.  So many of the steels that we produce today for structural applications have low carbon  concentrations you know not very far from 0.05 or 0.03 weight percent carbon.  So it is extremely important to control that concentration accurately even though you are  producing hundreds of millions of tons of this kind of material every year.  If you don't control it you will get different properties and variations in properties during  your steel production process.  Now the reason why it is parabolic is illustrated here that as my ferrite becomes thicker you  see this area here must be equal to the area under the triangle.  So as the ferrite becomes thicker the area under this triangle must also increase and  therefore the diffusion distance is becoming longer and longer as the ferrite thickens  and therefore the growth rate slows down as the ferrite becomes thicker.  Now today's lecture we have dealt with the binary system and it is very easy to fix the  concentrations at the interface just by drawing a tie line on the phase diagram for the transformation  temperature here and the average carbon concentration here.  So everybody familiar with the term tie line?  So it is basically saying look I have got a phase diagram here if I have a steel of this  composition then the equilibrium composition of ferrite is given by this point and the  equilibrium composition of austenite by this point and this line is what connects the compositions  of the two phases which are in contact and in equilibrium.  So that is called a tie line because it ties them together they depend on each other.  So it is very easy for a binary to pick this.  I am just going to introduce you to what happens when we add other alloying elements  but before we can treat that problem I need to teach you something called irreversible  thermodynamics which I will cover in the next lecture but I just want to introduce you to  the problem.  Okay so that tie line or the phase boundaries are obtained by drawing a tangent which is  common to the free energy surfaces of alpha and gamma and that gives us the equilibrium  compositions.  Yeah, straight forward.  What happens when I add another alloying element along here?  So here we are plotting carbon and let us say I add manganese along this axis.  And what should we get?  Well these free energy curves will become surfaces in three dimensions so something like this  here in both in the third dimension and the tangent becomes a contact plane which is touching  those two free energy surfaces.  But when you have a contact plane there is not going to be a unique answer for this and  this because I can rock that plane and still get it to touch both the surfaces at a variety  of points.  So we don't have a unique answer.  You actually at a constant temperature will have an infinite number of tie lines.  So this is what it looks like for a ternary system where we have say iron and manganese  and carbon.  These are actually surfaces and this is my tangent plane.  If I touch it to these surfaces, yes I can get a tie line which I plot on this horizontal  plane but I can actually rock that plane like this.  Can you imagine that?  And it will touch other places so there is a whole set of tie lines which defines an  alpha plus gamma phase field at a constant temperature.  So in the binary system there was a unique temperature and a unique set of equilibrium  compositions but here we have a lot more choice.  So it is not how we define the interface compositions when it comes to more than one  solute diffusing.  And in order to treat that problem I need to introduce you to multi-component diffusion  and in order to do that I need to treat irreversible thermodynamics which is a very easy subject  in the next lecture.  That is all for today.
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Okay, we'll make a start. So we have actually finished all of the displacive transformations which lead to a shape change with a large shear component martensite, bainite and weavement center ferrite. And we're going to raise the temperature again and talk about ferrite formation by a diffusional mechanism. So it involves a lot of diffusion. And the vast majority of the 1.3 billion tons of steels that we make every year contain as a majority phase what we are going to talk about today. Not bainite, not martensite, not weavement center ferrite. The vast majority of, I would say 90% of all the steels that we make, the dominant phase is the ferrite that we are going to talk about. So it's a very important phase in steels. And the second most important phase is probably pearlite and I will deal with that after another two lectures. So today's talk is about ferrite. And I'm going to use some specific terminology. A lotomorphic ferrite. It's a long word but it's important to understand what that means. A lotomorphic ferrite is ferrite which forms at the austenite grain boundaries. And because austenite grain boundaries are easy diffusion parts, it tends to grow more rapidly along the boundaries than into the grain. So the shape of a lotomorphic ferrite does not reflect its crystalline symmetry. You know, you don't get nice beautiful shapes with straight facets because it's dominated by the grain boundary. And again, we are looking at a partially transformed sample here. On the other hand, idiomorphic ferrite nucleates probably on dirt particles inside the grains of austenite. And therefore it has a shape which is a reflection of the symmetry of the ferrite and of the austenite in which it grows. So we see some nice faceted shapes. Everything means you develop some plain faces with particular crystallography. The mechanism of transformation is otherwise the same. The shape is different because these grains of ferrite grow inside the austenite, intragranulally nucleated and these are nucleated at the austenite grain boundaries. The austenite grain boundaries are the easiest nucleation sites in steels. Inclusions are the next easiest nucleation sites. So just to summarize that, a legeomorphic ferrite has a shape which does not reflect its crystalline symmetry. It's basically dominated by easy growth along the austenite grain boundaries. So it's dominated by easy growth along the austenite grain boundaries. And idiomorphic ferrite in contrast forms intragranulally. And therefore its shape reflects the symmetry of the ferrite and of the austenite in which it grows. It's the superimposed symmetry of alpha and gamma. And shape reflects the superimposed of alpha and gamma. If it was growing in a liquid then obviously its shape would be determined just by the symmetry of the ferrite but it's not. It's growing in another solid and therefore it's a superimposed symmetry of the two phases. And both of these transformations can only happen at temperatures where iron itself can diffuse over the length scale of your transformation product. So both transformations can only occur at temperatures typically above 600 degrees centigrade where the iron atoms are mobile. So the growth doesn't just involve the diffusion of carbon. You must also be able to move the iron otherwise you get a shape deformation and strain energy and all the rest of it. So there is no shape deformation here with a shear component. There's only volume change. So if you polish a crystal of austenite completely flat and you transform it to one of these ferrites it will not lead to a shape change other than volume change. So there is no shape deformation other than volume change. So this is an illustration of idiomorphic ferrite where you can see there are some nice flat interfaces with particular crystallographic indices. This is martensite. Of course you can only see these shapes when you've partially transformed the material. If you've got 100% transformation and the ferrite grains have touched each other then the shape is not determined by symmetry but by space filling. And this is an allotium morph of ferrite which is growing along an austenite grain boundary. And the thing that I wanted to notice is that it's like a thin layer which forms at the boundary and it grows rapidly along the boundary but the growth rate normal to the boundary is slow because this is a diffusional transformation and you're building up carbon in front of the interface. So unlike what we did in the last lecture where you had a thin plate growing and the partitioned carbon is being left behind with the tip advancing into fresh austenite. Here you have a plane which is moving. So the carbon piles up in front of the ferrite and therefore this growth rate will not be constant. In yesterday's lecture the lengthening rate of the plate of ferrite was a constant rate. In the equation for velocity we did not have a time dependence of velocity. The velocity only depended on the concentration terms, the diffusion coefficient and the tip radius. The diffusion distance was constant equal to the radius. Here the situation is completely different because we are moving effectively a plane and there's no leaving behind of carbon so you accumulate carbon in front of the interface. This is another example of allotrimophic ferrite where you can see that the thickening rate normal to the boundary is much slower than the lengthening rate. And notice also that this layer is not a single crystal of ferrite. You've nucleated many grains of ferrite along the boundary but they all are limited by thickening. So I'm going to treat this problem as one-dimensional growth of ferrite because it's effectively a plane moving normal and diffusion is only happening in one direction. Right so the difference between the calculations we did yesterday and what we are going to do today is that the velocity of growth will not be constant. It will change as the thickness of the ferrite increases. Now do you think that the velocity should decrease as the particle becomes thicker or increase? Yeah, correct. It will decrease and why is that? Yeah, you're absolutely right. The velocity will decrease as the particle becomes thicker. Any ideas why? So it's because you are accumulating more and more carbon in front of the interface so you have to diffuse over longer and longer distances. So imagine that you're looking at a pond. Yeah, a pond is a pool of water and it's winter and you form a layer of ice on the surface. The ice then becomes thicker so it takes longer for heat from the water to diffuse to the surface and more solidification to happen. So the freezing rate of the pond slows down as the ice becomes thicker. That's very lucky because otherwise the whole of the water would freeze and any fish inside would suffer. If you solder something and you form an inter-metallic compound between the solder and the wire that inter-metallic compound will slow down as it becomes thicker because you need to diffuse material between the wire and the solder. If you look at oxidation and you form an oxide layer on the surface, if oxygen has to diffuse through the oxide to the metal, as the oxide becomes thicker, oxidation rate will slow down. So we need to discover that relationship in any theory that we form. We are going to assume again as we did yesterday that local equilibrium exists at the interface. In other words, the interface compositions are given by a tie line of the phase diagram. So we will consider diffusion control growth, consider diffusion control growth with local equilibrium at the interface. And by local equilibrium, we mean that the composition of the ferrite at the interface will be given by this phase boundary and the composition of the austenite which is in contact with the ferrite will be given by this concentration here from the phase diagram. So these points are determined by the common tangent touching to the two free energy surfaces as we did in the last lecture. So when these two phases are in contact, the composition of the austenite cannot exceed C gamma alpha and that of the ferrite will be C alpha gamma and this is our C bar is the average carbon concentration in the steel. Okay so let's proceed as we did in the last lecture and define the concentration profile at the interface. So if I draw a graph of distance z and concentration plotted along here and as a result of the partitioning of carbon, I will get a profile which looks like this. This is C bar, here we have ferrite and here we have austenite. This point here is z star as we did in the last lecture. The concentration here is C gamma alpha and here it's C alpha gamma. And this distance here is the diffusion distance. So far there is nothing here which is different from what we had for the lengthening rate of Weidmüsten ferrite. And I can immediately write down an equation for the rate at which carbon is being partitioned into the austenite, being equal to the rate at which it's being taken away by diffusion from the interface. So as this interface moves, we will have to put that much carbon into the austenite but to maintain C gamma alpha constant, it has to be carried away by diffusion away from the interface. So the velocity, so this is again C gamma alpha and this is C alpha gamma and this is C bar. So the rate at which the interface moves, which is d z star by dt multiplied by the amount of carbon that we are partitioning, that is C gamma alpha minus C alpha gamma must equal to the rate at which it's being carried away from the interface by diffusion, which is the diffusion coefficient times C gamma alpha minus C bar divided by the diffusion distance. So these two terms must be equal, this is how much carbon is being dumped into the ferrite in a unit of time and this is how much carbon is being carried away from the interface in that time period. If those two terms are not equal, gamma alpha will not be constant and that's not diffusion control growth. And notice that this gradient, again we are assuming to be constant but in practice it will be like an error function. So this gradient is really evaluated at the position of the interface. Everyone happy with that? Now the problem is that this time the diffusion distance is not constant because we've got a plane moving and there is nowhere for the carbon to go except in front of the interface. So we are accumulating carbon as the interface moves. So we've got to have another condition to solve for the diffusion distance. So that other condition comes from mass balance. So if I draw this concentration growth again, this is the distance and this is concentration. Here I have ferrite. This is the amount of carbon, total amount of carbon that has been rejected as ferrite growth and this is the amount of carbon that has accumulated inside the austenite. So if I label this as A and this as B, then the areas A and B must be equal to conserve mass. So the areas A and B must be equal to conserve mass. So I can write another equation that C bar minus C alpha gamma into the thickness of the ferrite Z star must equal the area B which is C gamma alpha minus C bar into ZD upon 2 because it's a triangle so it's half. So the area A here is simply given by C bar minus C alpha gamma, C bar minus C alpha gamma times Z star which is the thickness of our ferrite. And the area of this triangle is C gamma alpha minus C bar times this diffusion distance here divided by 2 because it's a triangle. So now I've got an equation for the diffusion distance. I can write the diffusion distance as ZD is equal to C bar minus C alpha gamma into Z divided by C gamma alpha minus C bar and multiplied by 2. So just rearrangement of this equation here. Is that okay? Yeah? Okay let me just summarize how far we've got. So this is the local equilibrium condition. Here we have the diffusion distance. The concentration in the austenite cannot exceed C gamma alpha and this is the concentration in the ferrite. This is the relationship of the concentration terms with respect to the phase diagram and this is the rate at which solute is partitioned which is this concentration minus this concentration times the velocity of the interface, the rate at which solute is pushed into the austenite. And that has to be carried away by this flux which is because we are assuming a constant diffusion coefficient that's given by Fick's first law as the diffusion coefficient times the gradient. I've lost this minus sign because this gradient is actually negative. So these two terms must be equal. This must equal this and therefore we get the equation which has an unknown which is the diffusion distance. And to get the diffusion distance we have mass balance that this area must be equal to this area and this is the mass balance equation which allows us to substitute for delta Z or ZD as we are using into the first equation. Everyone care about that? Okay so all we have to do is to substitute this equation which we'll call equation 2 into equation 1 here. So ZD here is underneath, let me see if I can put up two pages at a time. Okay it doesn't help because I have to turn it. Right, let me just write equation 1 out again. So equation 1 was that the velocity which is dz star by dt into c gamma alpha minus c alpha gamma will equal the diffusion coefficient times the gradient which is c gamma alpha minus c bar divided by ZD. So this is equation 1. And now I'm going to substitute for ZD into this equation. So I want you to check in case I make a mistake. From now on it's your duty because you identified the mistake the last time. So we have dz star by dt. I want to isolate the velocity on one side of the equation. So I'm going to take c gamma alpha minus c alpha gamma onto the other side. So that's just taking this term here onto the other side. Diffusion coefficient comes on top and we have a term c gamma alpha minus c bar. And since ZD is at the bottom I will have another term here which is c gamma alpha minus c bar divided by c bar minus c alpha gamma and a Z star at the bottom and a 2 at the top. So this is a 2. Is that correct? 2 at the bottom, yes you're right. Come on rub out. Okay, right so I can simplify this by taking Z star onto this side. So I have Z star into dz star is equal to and a dt over here and then a term which I'm not going to rewrite but it's basically this part and the 2. So you've got Z star dz star on one side and dt on the other side. So if I integrate that I get Z star squared being equal to something times d, right? So if I integrate this therefore Z star squared is proportional to time or in other words Z star is proportional to t to the power of a half. So the thickening thickness of the ferrite will vary with time to the power of a half. So this is called parabolic thickening. So in oxidation the oxide is known as parabolic thickening in some cases, right? Where the material has to diffuse through the oxide to reach the metal or the metal has to diffuse through the oxide to reach the surface. Exactly the same. Similarly the thickening of inter metallic layer when you solder something will vary parabolically with time. So this is called parabolic thickening. And if I plot the thickness versus time then it will follow something like this, okay? Now clearly the slope of that line is decreasing as time increases that means the growth rate is slowing down, yeah? So growth rate slows down as the ferrite, sorry, alpha, okay, as ferrite decreases. So this is an important result. If you have a very large austenite grain size then it will be very difficult to complete the transformation because you will have thin layers growing from the surface. It takes a very long time for them to reach the center of the grain. So in large grain size specimens in the same amount of time as a small grain size specimen you will get less ferrite. So this is the same calculation where we now have the two equations. One that the rate that is partitioned must equal to the flux from the interface and this is our mass balance equation and you go through the maths and you find that the velocity will vary with the thickness of the sample and the velocity of growth decreases as the sample thickness increases and that the thickness is proportional to the square root of time. Once again this is a very powerful equation because we have the diffusion coefficient, we have terms from the phase diagram and the average composition of the steel here and we can predict the thickness as a function of time. Everyone happy with that? We are only considering a binary alloy at the moment which is iron carbon. So this is for a binary alloy, there is only one solute which is controlling the thickening rate of the ferrite and this is actually how the thickness would vary with time and the slope of this curve is the velocity of the growth and clearly the slope is decreasing as it thickens and thickens and thickens. Now I said to you it is a very important equation because it takes account of the alloying elements, you know the phase diagram and this is quite an interesting calculation and it is actually reflected in practice as well. Look if I plot the ferrite thickness here versus the time I am plotting actually the half thickness because diffusion transformations like ferrite are not limited by the austenite grain boundary. If you nuclearize at the boundary you can grow this way and this way but martensite plates and bainite plates and Wiedemann's Sun ferrite will not cross an austenite grain boundary because they are displacive and involve a systematic movement of atoms which cannot be sustained across a change in crystallography. But diffusion transformations can go right across austenite grain boundaries. So here is the half thickness being plotted versus time and what I want you to note is that here I am changing the carbon concentration by equal amounts. So here the difference is 0.02, 0.02, 0.02 and 0.02 but there is a very rapid change in kinetics here compared with here. So at low carbon concentrations the growth rate is far more sensitive to the concentration of carbon than at high carbon concentrations. This is exactly what is found in practice. And the reason for this goes back to our equation here that when your average carbon concentration approaches the solubility of carbon in ferrite there is no need to partition carbon. So if you are getting very close to the solubility of carbon in ferrite then very little carbon is partitioned into the austenite so the growth rate becomes very large. If C bar is equal to C alpha gamma then the growth rate is infinite. Now it is not possible to get a growth rate of which is infinite right? So what happens? What becomes the limiting factor instead of the diffusion of carbon in the austenite ahead of the interface? Well, you know you still have to change the crystal structure as the iron atoms cross the boundary. So there is a certain rate at which that can happen and that becomes the limiting factor and that is called interface control growth instead of diffusion control growth where some other process other than diffusion of carbon becomes the rate limiting factor. So let us just write that down. So the growth rate is very sensitive to the carbon concentration at low carbon concentrations. Growth rate is very sensitive to carbon concentration as C bar approaches the solubility of carbon in the ferrite because then there is very little carbon partitioned into the austenite because then very little carbon is partitioned into gamma. When C bar becomes close to C alpha gamma some other process will become rate limiting instead of the diffusion of carbon. So when C bar becomes equal to C alpha gamma growth rate tends to infinity if it is diffusion controlled. But in practice some other factor will become rate limiting. For example the transfer of iron atoms across the interface. So obviously if you have completely pure iron the ferrite is not going to grow at an infinite rate from austenite. It will be the transfer of iron atoms across the boundary which is a thermally activated process which will limit the growth rate of ferrite. So the model we have derived is a diffusion control growth model which assumes that the growth process is controlled by the diffusion of carbon in the austenite ahead of the interface. So I am sorry my handwriting is not very good but I am trying to record everything. So many of the steels that we produce today for structural applications have low carbon concentrations you know not very far from 0.05 or 0.03 weight percent carbon. So it is extremely important to control that concentration accurately even though you are producing hundreds of millions of tons of this kind of material every year. If you don't control it you will get different properties and variations in properties during your steel production process. Now the reason why it is parabolic is illustrated here that as my ferrite becomes thicker you see this area here must be equal to the area under the triangle. So as the ferrite becomes thicker the area under this triangle must also increase and therefore the diffusion distance is becoming longer and longer as the ferrite thickens and therefore the growth rate slows down as the ferrite becomes thicker. Now today's lecture we have dealt with the binary system and it is very easy to fix the concentrations at the interface just by drawing a tie line on the phase diagram for the transformation temperature here and the average carbon concentration here. So everybody familiar with the term tie line? So it is basically saying look I have got a phase diagram here if I have a steel of this composition then the equilibrium composition of ferrite is given by this point and the equilibrium composition of austenite by this point and this line is what connects the compositions of the two phases which are in contact and in equilibrium. So that is called a tie line because it ties them together they depend on each other. So it is very easy for a binary to pick this. I am just going to introduce you to what happens when we add other alloying elements but before we can treat that problem I need to teach you something called irreversible thermodynamics which I will cover in the next lecture but I just want to introduce you to the problem. Okay so that tie line or the phase boundaries are obtained by drawing a tangent which is common to the free energy surfaces of alpha and gamma and that gives us the equilibrium compositions. Yeah, straight forward. What happens when I add another alloying element along here? So here we are plotting carbon and let us say I add manganese along this axis. And what should we get? Well these free energy curves will become surfaces in three dimensions so something like this here in both in the third dimension and the tangent becomes a contact plane which is touching those two free energy surfaces. But when you have a contact plane there is not going to be a unique answer for this and this because I can rock that plane and still get it to touch both the surfaces at a variety of points. So we don't have a unique answer. You actually at a constant temperature will have an infinite number of tie lines. So this is what it looks like for a ternary system where we have say iron and manganese and carbon. These are actually surfaces and this is my tangent plane. If I touch it to these surfaces, yes I can get a tie line which I plot on this horizontal plane but I can actually rock that plane like this. Can you imagine that? And it will touch other places so there is a whole set of tie lines which defines an alpha plus gamma phase field at a constant temperature. So in the binary system there was a unique temperature and a unique set of equilibrium compositions but here we have a lot more choice. So it is not how we define the interface compositions when it comes to more than one solute diffusing. And in order to treat that problem I need to introduce you to multi-component diffusion and in order to do that I need to treat irreversible thermodynamics which is a very easy subject in the next lecture. That is all for today.
formation by a diffusional mechanism. So it the easiest nucleation sites in steels. because this is a diffusional transformation and velocity. The velocity only depended on the been rejected as ferrite growth and this is the with the thickness of the sample and the velocity