okay so we'll make a start today I'm going to finish off all the phase transformations and we are going to do perlite which is the last of these transformations and perlite as you can see is all about diffusion but it's

unique amongst all the phases that we have treated so far that you have two phases growing at the same time from the parent phase so you have perite and cementite growing together cooperatively from the parent phase so

on the time-temperature transformation diagram obviously because this is a transformation which requires diffusion it will happen at higher temperatures above around 600 degrees centigrade at a reasonable rate of course you can get it forming at a lower temperature but the rate of transformation will be very very slow so for example in 0.4 carbon and

three manganese and two silicon steel it took me 45 days to begin to produce perlite at 450 degrees centigrade yeah so when diffusion is very slow low temperatures its rate of reaction will be incredibly slow especially if you are below about 600 degrees centigrade so there are two conditions

you require to form perlite first that the carbon concentration of the austenite must be sufficient to allow both ferrite and cementite to grow simultaneously and secondly that the temperature must be high enough to

allow diffusion to happen so we'll just write that down so there are two conditions needed for perlite to form number one is that everything diffuses

during the growth of perlite iron atoms carbon atoms and any alloying elements that you can care to think about there is never a case where there is

paraequilibrium in other words you simply do not get the case where substitutional alloying elements do not partition during the growth of perlite so the first is that there must be sufficient diffusion there must

be sufficient diffusion generally at a reasonable rate that means you'll be at temperatures above 600 degrees centigrade so approximately greater than

600 degrees centigrade so iron atoms carbon atoms and manganese or whatever you have in your material will need to partition during the

growth of perlite and the second is that the conditions must be such that both cementite and ferrite can precipitate from the austenite so it must be possible for alpha and theta to precipitate simultaneously from

austenite now I explained to you that perlite happens in the vast majority of steels that are produced every year but it's usually there in

combination with ferrite because the vast majority of steels that we make are fairly low strength so most of the steels that go into high-rise buildings and so on they are of the order of 400 to 500 megapascals in strength and similarly when you make bridges the decks and so on a fairly low strength steels however there are certain steels which are fully politic

and are incredibly strong so this is what a partially transformed specimen of

perlite looks like you see these nodules or what we call colonies of perlite which on a microscopic scale consists of cementite and ferrite growing together from austenite and this rope here is one of the

strongest steels available you know the strength typically is between two and a half to three and a half gigapascals so all the suspension bridges etc contain lots and lots of pearlitic wire which you transform at a low temperature and then you draw it out to increase the strength further

drawing means cold deformation to stretch it out and work hard in it and make the structure even finer to get very high strength and in principle you should be able to get to five gigapascals okay but you require the ability to draw the wire a great deal and that means you have to have very

pure steel so generally speaking the strongest of pearlitic steel wires is of the order of three gigapascals the stuff that you see on suspension bridges on cables and cranes and so forth so that can be incredibly strong because you can make the spacing between the cementite and ferrite very

fine indeed I'll give you some examples of that later on but first let's go into the theory of the pearlite transformation and let me begin by a question what is the condition which ensures that you can precipitate both cementite and ferrite from austenite so you've got able to precipitate

both cementite and ferrite yeah don't worry just take a guess you know

don't worry about getting things wrong I often do get things wrong yeah remember you corrected my mathematics so this is what the iron-carbon phase diagram looks like roughly the carbon temperature so this is austenite ferrite

plus austenite austenite plus cementite and the concentration here is

0.8 weight percent so I repeat my question what is the condition that allows both ferrite and cementite to precipitate simultaneously from austenite right so the concentration has to be at this eutectoid point

where all three phases can coexist in equilibrium okay so that temperature is about 723 degrees centigrade but let me ask you then is it the case that I can

only produce a ferrite fully pearlitic steel when my carbon concentration is the eutectoid composition so what do I need to do to represent that on this diagram see the condition is as follows if I want

to precipitate cementite from austenite then my alloy must lie in the gamma plus theta phase field right if I want to precipitate ferrite from austenite then my alloy must lie in the alpha plus gamma phase field yeah

but if I want to precipitate all three then on this equilibrium phase diagram there's only one point where all three phases are in equilibrium and that's this eutectoid however if I extrapolate these phase boundaries to

lower temperatures so that's an extrapolation then any alloy lying in this shaded region can form a fully pearlitic structure okay because any alloy in that region is super saturated with respect to ferrite and

cementite right it's both in the alpha plus gamma and gamma plus theta phase fields so this region is known as the Hultgren extrapolation the Hultgren so even if I have an alloy which has only 0.4 weight percent carbon if I

super cool it the austenite sufficiently it'll come within that extrapolated region and I can make a fully pearlitic steel right so what

will be the difference between a fully pearlitic 0.4 weight percent carbon steel and a fully pearlitic 0.8 weight percent carbon steel what do

you expect to see as the difference between 0.4 and 0.8 weight percent carbon fully pearlitic steel exactly right the amount of cementite will be much smaller in the 0.4 weight percent carbon steel because carbon cementite comes from carbon so all you do is you increase the spacing

between the lamellae so that the volume fraction of cementite is consistent with the concentration and similarly if I raise my carbon concentration to one weight percent then I will get much more cementite than I would with a 0.8 carbon steel in a fully pearlitic case okay so that's

very good what this diagram also tells us is that if I have a composition let's say 0.8 weight percent that is my C bar then the equilibrium concentration in the ferrite which grows from cementite will be given by

the extrapolated alpha plus gamma phase boundary and the equilibrium composition of the cementite will be the extrapolated gamma plus theta phase boundary okay so we use the same terminology as before the C gamma theta and C gamma alpha for the equilibrium compositions so this is a nicely drawn

diagram where you can see that if I'm forming pearlite at this low

temperature well below the eutectoid temperature then the composition of the austenite which is in contact with cementite will be given by an extrapolation of the gamma plus theta phase boundary and the composition of the austenite which is in contact with ferrite will be higher given by the extrapolation of the gamma plus alpha phase field. So if you've got two phases

growing from austenite then carbon will tend to diffuse in the austenite which is in contact with ferrite towards the austenite which is in contact with cementite because cementite is absorbing that carbon right so this is a

schematic diagram of pearlite growth which is I think it's given in your notes somewhere where we have these lamellae of cementite and lamellae of ferrite when we look on a fine enough scale and this is the parent

austenite and the two phases are growing together in such a way in an iron carbon steel in such a way that the average composition of the pearlite is the same as the average composition of the austenite. So what does that tell you about the growth rate? If the average composition of the pearlite is the

same as that of the austenite then what does that tell you about the growth rate? We haven't got any alloying additions other than carbon but how do you expect the growth rate to vary with time? Yeah it will be constant

because there's no change in the composition of the austenite as the pearlite grows. The average composition of the pearlite is the same as that of the austenite so we expect to derive an equation where the growth rate is

constant. Okay so bear that diagram in mind the term S here is what we call the inter lamellar spacing and S alpha and S theta clearly depend on the amount of carbon you have in your steel because that is related to the

volume fraction of cementite right. So we are mostly interested in this inter lamellar spacing and the way in which pearlite grows is that the carbon that is rejected by the ferrite diffuses towards the cementite which is absorbing the carbon and you can see that the diffusion path unlike all the

transformations that we've discussed so far the diffusion part is parallel to the transformation front. The carbon is going from the ferrite towards the cementite okay. Okay so let's derive the growth rate of pearlite so I can

do this in two ways I can work out the growth rate of the ferrite I can work out the growth rate of cementite it doesn't matter does it because they're both growing at the same rate right. So supposing we focus on the cementite

then V which is the velocity of growth so the velocity times the rate at which the cementite is absorbing carbon because there will be a

concentration profile this is C bar this is the composition in the austenite which is in contact with the cementite and this is the composition of the cementite which is in contact with austenite so here we have

cementite and here we have gamma and as this interface advances the

cementite will be absorbing that much carbon okay so I can write that the velocity times C theta gamma minus C gamma theta okay that's the rate at which cementite is absorbing carbon so this is the rate at which cementite

absorbs carbon that must be equal to the flux that is coming from the

ferrite to the cementite yeah so that must be equal to the diffusion coefficient so we have the diffusion coefficient times the flux from the

ferrite to the cementite in other words the austenite that is in contact with the ferrite so this is C gamma alpha minus C gamma theta divided by

the diffusion distance which I will call Phi into s okay so Phi s is the diffusion distance and s is the interlamellar spacing so we have a

ferrite here cementite and ferrite of theta and alpha and the interlamellar spacing here is s and the reason why I'm using this parameter Phi is because

diffusion distance is not exactly s you know it could be from the middle of the ferrite to the cementite in which case it would be half but you need to think a bit more carefully because diffusion is happening from every point on the ferrite towards the cementite okay so it will be some

fraction of s everyone happy with that so we have a very simple equation that the rate at which cementite is absorbing carbon from the austenite here is equal to the diffusion coefficient of carbon in the austenite times the

flux from the ferrite towards the cementite okay so the concentration in the austenite next to the ferrite is greater than the concentration in the austenite next to the cementite and therefore you get a flux parallel to the interface okay so we can simply write the velocity is equal to the diffusion

coefficient divided by Phi times s into C gamma alpha minus C gamma theta divided by C theta gamma minus C gamma theta so once again we have

terms from the phase diagram okay so all the thermodynamics is taken care of the C the solubilities of carbon in ferrite which is in contact with

austenite the solubility of carbon in cementite which is cost in contact with the austenite and we have the interlamellar spacing and we have a diffusion coefficient

why isn't the volume fraction the average carbon concentration in there it determines the

interlamellar spacing doesn't it yeah so it's implicitly in in this you know it determines the thickness of the cementite relative to the thickness of the ferrite okay so there's no time dependent term here so the growth rate will be constant so it's a constant

growth rate okay let me just plot what that graph looks like if I plot the velocity

versus the interlamellar spacing then I get a curve which looks like this as the interlamellar spacing decreases the velocity increases indefinitely because the diffusion

distance decreases the diffusion is parallel to the transformation front so if you decrease the interlamellar spacing then the diffusion distance decreases so can you see that there is a problem here which exactly the same problem that we had with Wiedemann-Sadden ferrite

I don't actually have a unique velocity here right why is that yeah yeah you know when we

grow perlite we are creating ferrite cementite interface and we and that is a cost which we haven't accounted for okay so let's just quickly work out how much interface we are creating so if I take a cube of perlite and let's say the side is a and I have a lamellar of cementite and

the spacing here is the interlamellar spacing then the volume of that cube is simply a cubed right so the volume is equal to a cubed and the amount of area between the cementite and ferrite

is a cubed to a cubed upon s now how do I get that well the area of the lamellar here

the face is a squared right and the number of these lamellar I have per unit length is a

divided by s right s is the interlamellar spacing if I divide a by s I get the number of lamellar I have in the cube but we have two interfaces per lamellar this side and that side and

therefore we have a factor of two here so if I cancel out the terms which are common then that comes to two upon s which is equal to the amount of surface per unit volume so this is the amount of theta alpha interface per unit volume actually I've derived this in a little

bit of a simple way but this relationship is generally true that if you measure a mean

linear intercept for a grain size then the amount of grain boundary area per unit volume is two divided by the mean linear intercept so we know how much surface we have created as we've grown the pearlite so the amount of free energy that's consumed in creating the cementite

interface is delta G and we call it I is the energy consumed in creating theta alpha interfaces

which is equal to two upon s which is the amount of surface per unit volume multiplied by the interfacial energy okay so sigma is interfacial energy per unit area so the units

of delta G I which is the cost of creating those interfaces are joules per meter cubed because

sigma is joules per meter squared and s is a meter therefore when I do two sigma upon s that gives me the amount of energy locked up in interfaces inside the pearlite and that's going to take that's going to reduce the driving force for transmission so if delta G

if that is the total chemical free energy change then the actual driving force is equal

to delta G T minus two sigma upon s so this is the actual the free energy change now when we get

to a particular interlamellar spacing which we'll call s with a subscript C a critical spacing all of the free energy is used up in creating interfaces so delta G will be zero okay so at a

critical spacing SC at critical spacing SC delta G equals zero and we get delta G T will be equal

to two sigma upon SC so I can rewrite delta G is equal to two sigma into one upon SC minus

one upon s so SC is simply the interlamellar spacing where all of the driving force is used

up in creating interfaces and delta G over delta G total is equal to so one minus SC upon s is

the fraction of the driving force available to drive the interface everyone happy with that

okay so if I just take the equation we derived earlier and multiply it by the term in square brackets then we are we have taken into account the cost of creating interfaces because you know

that velocity is proportional to driving force in the first approximation right so for now rewrite the equation that we had on the previous page this equation here then I will get the

velocity is equal to the diffusion coefficient divided by pi into s times C gamma alpha minus

C gamma theta divided by C theta gamma minus C gamma theta okay and then multiply that by

one minus SC upon s okay so all we've done is we've taken that previous equation multiplied it by the fraction of free energy that's available to drive the perlite and if I plug

this equation out velocity versus s then instead of getting a curve which looks oops a day let

me just get rid of that instead of this which is not correct we will get a curve which goes to a maximum and the growth rate is zero when the spacing is a critical spacing it goes through

a maximum so the perlite will try to find an optimum spacing now in the case of Wiedemann Stein Ferrite we picked the maximum as the growth rate right because the experimentally

measured values of the growth rate of Wiedemann Stein Ferrite was slightly higher than the maximum this is not actually the case for perlite but we have two choices for picking a velocity okay so we could simply assume s corresponds to maximum velocity and in this case you know s

will be equal to twice SC you can prove that for yourself and the second is the growth occurs at

a rate which dissipates the maximum amount of free energy so the maximum free energy dissipation rate is assume s which leads to maximum free energy dissipation rate so the most rapid

decrease in free energy and you remember this equation where we had t into I will use of entropy production equals a flux times the force you remember that right now if you are

doing this transformation isothermally then the rate of entropy production times temperature is also the free energy dissipation rate so there's some logic in choosing this my my own favorite

is the second condition where we get the maximum free energy dissipation rate and experiments seem to confirm that but as I explained to you last time there is no fundamental theory which will tell you that this should be picked or that should be picked the theory that is there

is simply too complicated to represent real situations okay so if you want to do a calculation your safe bet is to pick a spacing s which leads to the maximum rate of free energy dissipation

okay even if you try to do calculations of solidification you know the rate of dendrite growth is not what you get but you get the rate of dendrite growth as a function of the dendrite tip radius so you have to have some condition to pick the actual velocity at which the material at which the transmission happens okay so that's enough of theory and this is

an actual growth rate calculation would look like for an iron-carbon steel where the growth

rate of the pearlite is of the order of you know 20 micrometers per second and interlamellar spacing is below a micrometer okay that's quite typical now we have dealt with pearlite growth

with diffusion happening in the austenite parallel to the transformation front but there's also a grain by a bound interfaces here right and interfaces are easy diffusion parts so we can

help this parallel flux by also diffusing through the boundary at a much greater rate yeah so all we have to do is also add another term which represents the flux through the grain

through the boundary between the pearlite and the austenite so there's also an additional diffusion part which we haven't taken into account and just like we had the flux going

through the volume so this is the volume diffusion coefficient and this term and then we had the diffusion distance we have a grain boundary diffusion coefficient and a grain boundary thickness and this term here is exactly the same as that because we are assuming that the equilibrium there is also the same but this diffusion coefficient times the boundary thickness

which is what you can measure experimentally is actually a much faster diffusion path so when you go down in temperature boundary diffusion will dominate over volume diffusion okay because

volume diffusion becomes much slower the activation energy for volume diffusion is much greater than through the boundary boundaries are loosely packed structures a flux not only has an diffusion coefficient in it but also an area through which the flux happens right and the area through

which the boundary flux happens is much smaller than the volume of the material ahead of it so as you raise the temperature even though the volume diffusion coefficient is smaller than the boundary diffusion coefficient the area through which volume diffusion can happen is greater and therefore at high temperatures volume diffusion will dominate at low temperatures

boundary diffusion will dominate so let me just write that down so the transformation front itself is a diffusion path so when we had the equation that the velocity times the rate at

which the cement diet is absorbing carbon sorry that's a minus it's equal to the diffusion coefficient in the austenite times C alpha gamma minus C theta gamma no so it's C gamma

alpha minus C gamma theta divided by s and there was a phi here so this is through the volume of austenite we simply add another term here which is the green boundary diffusion coefficient

times the thickness of the boundary times the same gamma alpha minus C gamma theta divided by pi into s so dB has units of meters per second and dB times Delta which is the thickness of

the boundary has units of meters squared per second so Delta is the thickness of the boundary and the diffusion coefficient through the volume of the austenite has units of

meters squared per second so you can see that dB times Delta has the same units as DV the volume diffusion coefficient so I'm not going to go through you know including the interface

energy term and so on it's fairly easy to add on a second diffusion path and by doing this you don't need to say okay my reaction is interface controlled or its volume diffusion controlled this equation automatically tells you how much flux goes through the boundary and how much flux goes through the volume as you alter the transformation temperature okay right so this

is the same thing expressed again that we have our flux through the volume and the flux through

the boundary and the red points here represent experimental measurements on iron-carbon alloys this is the calculated growth rate as a function of temperature assuming that diffusion only happens

through the volume of the material and this curve here represents the growth rate calculated assuming that diffusion only happens through the boundary and the curve in the middle allows both of them to happen automatically so you can clearly see at low temperatures the proper model is consistent

with all the points and it's closer to the interface controlled diffusion okay because most of the flux goes through the interface at low temperatures whereas at high temperatures you have much more

austenite ahead of the transformation front so you have a greater area through which diffusion can happen and therefore volume diffusion dominates the growth rate but if you use the proper equation you don't need to worry about whether it's volume diffusion controlled or interface diffusion control you just add the two fluxes okay so it's possible to calculate the growth

rate of perlite using this simple theory the only problem that remains which I'm not comfortable with and which many people are not comfortable with is how to select the interlamellar spacing from the function of velocity versus the spacing if you have to use

the maximum free energy dissipation rate as the choice for picking s so this is just telling you the ratio of the volume flux through the boundary flux is greater at higher temperatures because

you have a greater amount of austenite through which diffusion can happen than through the boundary now there are papers in the literature which propose also a third diffusion part because diffusion in ferrite is much faster than in austenite diffusion of carbon so it's possible

that carbon from the austenite gets into the ferrite and then gets to the cementite what that would predict is that the cementite thickness increases behind the transformation front and there is no evidence that that is happening but supposing you needed to do this calculation

you just add a third term yeah so it's not a problem but it's not believable that you are getting thickening of cementite behind the transformation front when we quench the specimen and we look at the shape of the cementite is more or less parallel okay right we can also

take into account ternary systems just like we did for a lot of morphic ferrite in this case in the case of where you have added manganese or substitutional solids you can

forget more or less about volume diffusion the difference between volume diffusion and boundary diffusion is so large that for all of these cases it's really boundary diffusion control growth okay substitutional solids are really happy to be and move about in the boundary compared with the volume of the austenite so the fact that they have to diffuse dramatically

slows down the perlite reaction okay and here's case for chromium okay let me just show you I explained to you that strength depends on inter lamellar spacing right and

we can reduce that spacing by drawing the steel but obviously if you start with a finer spacing then you end up an even finer spacing when you draw the wire out right how can I reduce

the inter lamellar spacing okay so I'll just move back there forget this slide what in your equations allows you to reduce the inter lamellar spacing by phase transformation you know what

limits the inter lamellar spacing right what limits the finer spacing that I can get what

determines the value of the critical inter lamellar spacing at which growth rate becomes zero why does the growth rate become zero at a certain inter lamellar spacing

yep go ahead no no it can't have a time defense here why does the growth rate become

zero when the spacing becomes a critical value SC yeah so when the spacing becomes so fine that all the driving force is used up in creating interfaces you get a zero growth rate so how can I

make it even finer I provide extra driving force right now one way of providing extra driving force is to use a magnetic field so this is now coming back to your area of expertise okay why would a magnetic field increase the driving force for the transformation of austenite so if

I take austenite I impose a magnetic field on it why would that increase the driving force for transformation so generally speaking austenite is not ferromagnetic right yeah ferrite and

cementite actually both can be ferromagnetic so I impose a magnetic field it favors the formation of ferromagnetic phases so that increase the driving force yeah so by applying

a very large magnetic field this slide shows you some work we did a few years ago this is the steel transformed without a magnetic field as soon as you apply a magnetic field you form pearlite instead okay because it's favoring the formation of pearlite and if I show you

a transmission electron micrograph you get incredibly fine pearlite so this is a spacing of about 50 nanometers this had not been drawn it's just by phase transformation right now I'm asking you again 30 Tesla magnetic field is that easy to produce very very

difficult is hugely expensive right is there any other way in which I can increase the driving force you would require the power of a small city to consistently produce a field of 30

Tesla's right how can you alter the free energy difference between austenite and ferrite normally without magnetic fields hmm temperature but we have a limit because diffusion must happen yeah

so you already actually the wires are already transformed at about 450 degrees C by patenting they call it a patenting process where you go from a hot to a lead bath or some other bath

but what about alloying yeah what elements can I add to the austenite which will make the ferrite more stable aluminium or cobalt those are basically the only two alloying elements which will increase the free energy change so this is completely without a magnetic field

right and here we also have a spacing of the order of 50 nanometers by altering the chemical composition of the steel just by natural cooling you get to 50 nanometer spacing

so this paper will come out soon in scripter material yeah I'm not saying this is practical because cult is expensive but it's interesting you know to start with a spacing which is 50 nanometers you you know we don't even know how much we could get to if we now wire draw it

so well it hasn't had much attention in recent years but it could have some major technological advances if we can create without wire drawing a fine structure because then

afterwards if you draw it you'll get it even finer okay that's all for today