00:05

it's so just in case any of the mystery that is a homework assignment which you should hand in tomorrow it should take no more than half an hour we can't just just look at how we derived a growth rate of fare from Austin and follow the procedure drawl composition profile at the interface and then balance 2 terms which is the rate at which carbon petitioned are absorbed against the rate at which it is taken away by diffusion OK so today I am going to cover 2 things venture transmission diagrams and I'm going to make a start on the transformation in his past history but 1st time temperature transmission diagrams where you soul the DTD diagram of difficulty keeping diagram will have to seek goes 1 for for the reconstructive transformations and the other 1 for the displays of transformation and bad times like these are used widely in industry but what actually the Germans those codes for on the campus of assistance plan that's what we're after is how do we actually calculate that we've got a growth I'm going to briefly introduce you to nuclear fission because it's a combination of new creation and growth which gives us but temperature transmission diagram now most of you will have done UPS in theory but I'm going to go through it again at the feet you mediation is as follows supposing I'm observing my boss tonight and I got as much resolution as I need and I'm observing at that temperature then you suddenly see that in some regions of new structure and chemical composition of recreated by random fluctuations but the system is a homogeneous but suddenly you will see a fluctuation of structure and chemical composition which is consistent with fairer because composition of that region and the crystal structure is consistent policy it's a random fluctuations Adams of moving about and you just find a region which has the structure and composition of Ferrari but of course if you created by Friday announced 90 will also have an interface between the 2 phases that and that interfaces is badly fitting so there is a certain energy to defect and that fluctuation will not be able to grow and dinner guest to a size where the cost of creating interface is less than the cost of creating and then begin from the creation of the right that's basically new creation theory that by random fluctuations we get regions which have the right structure and right composition for the product base and if that fluctuation is large enough then it will grow into a phase so we can

03:14

expressed that very simply let's imagine that you got a spherical UPS forming an hour Austinite so this is the free energy CO Austinite and prearranged code of practices the equilibrium temperature and below that you can see that the free energy of priority smaller than that of course tonight and that's where we might be able to see the fluctuations so in the creation of this sphere I will get a reduction in free energy so this term is negative that chemical is negative because you can see that locally energy then Austinite below the equilibrium temperature and this is simply before partly by Q is simply the volume all the nuclear but when we create a spike nuclear is also creates a surface and the area of the surface is fooled by R Squared and the cost of creating that interface is signal which is induced by me great this in the face of energy per unit area so that is positive this is negative and you might also have some strain because the wording changes shape change over whatever the strain energy scales with the 1 in all your material so this chemical-free free energy change once the transformation to happen strain opposes it and the creation of interface supposedly so that's the balance the free energy change strategy when we create a nuclear self size are radius if

04:50

I dropped that expression the a plant on the

04:58

vertical axis the free energy so Taliban chief and it's 0 yeah negatives and positive and along here is the radius r then the chemical free energy change will tend to promote the transmission and it would be something like this so this is 4 . 3 Our queued into the that tends to favor and it raises without you and art usually smaller than r-squared for a small values of our so the surface energy terms really didn't that small radio the 4 firepower square In signal individual energy problems and the state-managed Dembele follow curve which made it argued I'm not going to let that on you this fall just to for the sake of simplicity not that effect of these 2 terms used to produce the kind of mission excited so it isn't until the radius becomes greater than hostile that the free energy changed that the actually decreases and this height here is called the activation energy the GE stock is the activation energy of nuclear fission so simply because you are below the equilibrium transmission temperature doesn't mean that the transformation happens instantaneously you've actually got to overcome the cost of the interface so you need fluctuations which are greater than asked for them to develop into a successful nuclear threat now in order to find G-Star and Austin also that Austin is incorrect London why but let me just tell austerity is actually here this point again now define the

07:42

value of the in Austin this is the 1st

07:48

equation is simply that energy change many former particle operator stock if I differentiate badminton Specter and said this 2 0 then I get hostile varies with the individual energy so the facial energy 0 then any fluctuation will grow into Our product phase of it but it never really zeroing practice and of course if I have a larger diving for stand the critical size of fluctuation which is successful will be smaller and if I'm not substitute are starting to the 1st equations and I can get the value of GE which varies

08:29

with the queue of official energy so it's very sensitive to the individual energy the larger into facial energy will make it much more difficult for new creation to happen and it raised with the square off the chemical driving force so if I increase my driving force than the activation barrier becomes smaller now this is

08:56

just a summary of the variation of individual energy and a chemical-free energy change the strain energy as a function of the body courageous and this is the critical size are stopped and here we have the activation energy we can now say OK the probability of a successful event is given by exponential minus the star upon taking the usual thermally activated by unions that equation yeah and new over there is the jump frequency is the attempt frequencies so it's attempting to get over the barrier not all of those attempts are successful the probability of a successful attempt is that exponential minus you stop on Katie and it is simply the number density of new sites for unit volumes and we have a 2nd exponential them here which is exponential minus Q upon Katie and that is the activation energy for the transfer of atoms across interface because you know you've got a goal from this crisis structure of Austinite do that effect when you transfer inactive that has a certain barrier associated with it and that's a constant value cute GE stock will vary video transmission conditions if you transform larger and cooling induce other this morning because the driving force 3 so this is the equation for new creation rate by unit volume so now we have the growth rate that we've done many growth rate calculations and we have the nuclear fission rates so we should be able to calculate the volume infractions as a function of time temperature composition accepted but we can calculate the volume fraction that gives us the PTT diagram so just the problems that arise when 1 to calculate what infractions so imagine that

10:54

I have I because here there so here we have 2 particles which have grown inside the Austin and that is at a certain time Deke and I look at the body close a smaller time interval later then they will have become bigger and also I might have new created new particles it's these 2 because here I knew those so is there anything wrong with that diagram on your right hand side what do you think is wrong that diagram yeah despite that

11:44

and despite the new by because these 2 have grown because what is wrong with the the go-ahead .period I think you know the answer I think yeah it doesn't make sense to have about new creating inside the typewritten already grown right and similarly we've got this problem of overlap so once we start to look at volume fraction you've got to think about particles touching each other and how can it take account of fight because touching each other and this is a good

12:33

illustration of this is glass crystallizing like window glass if you hit it long enough then it crystallized and these are like those of Chris lines last growing and you can see that the problem here that Dutch didn't start growing to each other but your stopping them from growing and eventually ran all of this is filled it looked like any creates great structure know the sort of thing that you see when you look in your microscopes so the problem is that we need to take account of what's known as hard impingement that means growing by both physically touching each other and we cannot allow new to happen in regions which have already transformed OK so you

13:22

just write that down so hard impingement means hitting each other market mutated from different positions positions Dutch so let me know that diagram again have good reason here Of the let's say at this time the the close and this is at the time he was tour and I couldn't do particles here a short time later they have grown and I might have you created 2 nearby not let's ignore at 1st the fact that some by because a growing another body goes over that but because of Dutch if I calculate the change in the fraction all the right in going from door to door opener it would be wrong but let's do that anyway let's call that the change in extended volume that means we allow quite to grow to each other so the changing extended volume change extended volume please and you call transmission products began and he for extended this is that the cafe's growing so this will be the wrong while you may have calculated because obviously you can't transform regions which have already transformed and a real change in the volume of the German right as we so this is the true change and obviously they're not going to be equal but if I multiplied the right-hand side here why the probability of finding on transformed material then that correct fall the extended volume because what we want is the change in volume which falls into 1 transform material what is the probability of finding on transformed material so this will not be equal to this effect but if I multiplied the extended volume by the probability of finding on transformed material the probability on transformed materials is equal to what what's the probability of finding on material and there is no transformation 1 was the probability of finding undone for material when half of it is transformed half yet so simply given by 1 minus the 1 infraction of the so that's 1 minus 1 move the dead divided by the total it because we beat that all of the it is a volume fraction of the Beacon there were unhappy with that so we can write that's the real change in volume His equality the change in extended volumes multiplied by by 1 minus the the upon so this is a really important equation it allows you To look at many particles growing together and calculating volume fraction and this is quality of equation and we do like that again on the next stage but rearranging get the extended volume on the left-hand side so you have the lead Peter extended the the widened by 1 we need to find me that this rearranged the previous the BBB down the left-hand side and now I can integrate that so that get the extended volume z quotas now I was saying to grow up the XOR acts yeah but it yeah I love it yet so we will have logarithms needs some space after equals the number them all off 1 minus the veto on and we have to With a minus me

21:20

there because 1 1 mind speed so this is the relationship between the true volume of media and the extended volume of the 2 it's a very very simple equation but extremely powerful this is possibly the most useful kinetic equation that I know far better than face field theory of any fancy methods for calculating infractions so lessons assume that we now have completed the theory for impingement of bicycles now let's do a calculation if we are observing a system transforming and if I plot the size of a particle sighs also by for us is the time to USA a particle nucleated analysts assumed that the growth rate is constant then this particle which negotiated at this point "quotation mark court told 1 and then there might be another product of its nuclear it's at a different time going to and then another 1 4 or 3 and so on this is what you would actually observed particles started at different times before all the time .period Tower 1 there was no particle 1 so the site 0 at that point and then it starts to grow and there is even a constant growth this is constant growth interior quality then the volume of a particle new created at a particular point of view the this is a small beach With the substrate .period and nearly you that the isotropic that means the same in all directions and that will be for upon 3 GQ and the growth rate Q team -minus tell because before the . towers there is no but of it only increases in size beyond the point Dow here before that it doesn't exist and you that's an individual particle but of course we will have many because new creating at different times right so in that time in the world of small time interval number of particles that reform is the new creation rate times the volume OK because it's a new creation rate per unit volume than the 1 in times a time interval the number of particles articles you created In he called a sequel to the mutation rate times the volume because it's a new creation rate for unit .period impossible a multiplied by the number density of nutrition sites In did 2 and it's the number density of nuclear fission site the 100 that 1 to the next so the changing extended volume that it's quality the mutation rate per unit volume times the volume dancing number density of nutrition sites multiplied by the volume of each fighter which is given by 4 . 3 why she here T minus Q times the time detour In this a number of articles that have been created in that time into 0 times the William what each of those and all have to do now is to integrate this From time equals 0 to any particular value of time set by integrate this From he was 0 2 he was a particular kind but I get the extended volume of the sequel to and In 1 3rd chief the team to the power of 4 show that for yourself later on but the fact disappeared because we're integrated together that time exponents it should be acknowledged to be the towel because the integrating pal From the equal 0 a value so Dow will disappear and these days a dowries at the point at which a particular product comes into existence OK so that's so the extended volume and we already have a relationship between the extended volume and available in if I go back

29:20

yes it is the extended volume as a function of the real 1 where we can simply substitute for the extended volume instead of extended Volume I write mines of one-liners the upon receipt the was In fact unlocked this then I would get 1 minus the volume fraction of sequence exponential -minus by the hand on the street so there is no need to pick up into groups and that defines the volume fraction as a function of time the driving force the driving force comes into the growth equation into the new creation equation and the number density of nutrition site safe nutrition happened that grain boundaries than the grain size determines the number density of nutrition site so everything is in that region and taken account of hard impingement but plot that aggression out and this will be that difficult shape Of course time here and mediator upon reasonable infractions and it's what's known as a signal idle CO going from 0 2 1 so that's how the the volume fraction involving time and the incubation period store for a product to come into existence is not fixed it depends on at what point that part of the family existence so when people talk about time temperature transformation diagrams they made a mistake all saying that the incubation period is between the Seiko and the word Lexus that is not an incubation period that is the time required to achieve a detectable amount of transformation depends on your technique if you have a digital the 1 per cent 1 infraction measurement accuracy and I take 1 % here then you start to plot Yoko at the time corresponding to 1 % I can't got another code which corresponds to 5 % 10 per cent and on and that regenerate the whole of the diagram by calculations now he had in this example the growth rate to be constant so many Downie rich phase transformation in steel gives your constant growth rate yet so 1 which doesn't involve diffusion we can alternatively where the composition of the product faces the same as that of the parent so any NEW very constructive transmission with the composition of the parent is the same as the product well I it you know it's a mixture of seamen died and identified with the same average composition as the Oscar nite said it was like got colony will grow at a constant rate yeah so this would be a model for poor life and if I if I obtain an experimental curve of the 1 infection was his time and I get an exponent of time of 4 then that indicates that the growth rate is constant this is probably because there might be another mechanism which also leads to an exponent of fall and the exponent of 3 comes from the girl because we had a growth rate you a and 1 from the constant nuclear generator that have you so supposing now that we have a parabolic growth rate that means that besides various repeated the power of a half then what do I expect the exponent of being the parabolic growth rate and a constant mutation rate what would the time exponent almost there so we've got size rating with key to the heart so what want 1 reiterated that you of the site 3 of 1 to record the half c upon you and then what With come from the fact that we have a new creation times time so the explaining that he will get his 5 yeah there unhappy about that so parabolic growth will lead to a different dynamic women so that if you measure experimentally and the sea at the time exponent is 5 . 2 that indicates that the growth rate he is bearable at the time and the nuclear nuclear-generated sponsors so it's an indication about there may be ambiguous it may be ambiguous because another mechanism might also lead to keep .period the European Christians theory for transformations in methadone allies to find a table of different combinations of new creation and growth functions which give you different values of the time exports if you analyze it didn't me grows more carefully you'll be able to study also the mechanism of the transformation products using this analysis I'm just going to repeat all that using my slides so this

36:17

is this is the what happens after a small-time these particles have grown so the dark-blue regions that represents the increment of consummation and we've also nucleated another 2 particles but the calculation of the change in volume fraction is going to be wrong because we've got all of facing this because here should not play in an area which has already transformed so the correct for that by multiplying the incorrect change in volume fraction by the probability of finding on transform material and that really is the essence of programming period that tells you how to convert extended volume into real volume so you don't need to worry about impingement between part of the extended volume is very much easier to calculate because you just calculate the growth rate in the nuclear fission great and it all up and then make a correction and of course

37:23

that this is just an integration of that equation

37:27

this is showing you observing the system you would see a variety of particle size is growing from the Matrix because they started different values of the incubation period and the volume of a single particle assuming a constant growth rate given by at GQ into the minor stroke you stands for bonds 3 by because we are assuming isotropic growth you don't need to assume isotropic grows you can energy 1 G-2 G-3 and growth rate being different in different directions extremely flexible theory and then

38:05

final already equation and it contains everything contains the growth rate nuclear fission rates dying temperature everything now 1 bad thing but you will find in the literature is that they expressed the immigration as a 2nd equation riches I just saying Look there's a constant care and there's a time exponent and so the use it completely and basically and they don't realize that they are using it and particularly that think that is the Army's theory so I don't remember but there was a talk given a seminar given where the person was comparing you know the model of the overall transformation kinetics and then said local this equation is completely empirical work that person didn't realize that he is using it empirically the question itself is perfectly OK if you put in the right terms it's very powerful equation much more powerful than face filled air right so these because

39:17

of codes and to find the

39:20

time exponent you take a double logarithm of the volume fraction is fairly straightforward

39:25

mathematics and then you plug your sentence and should

39:28

transportation director what I'd like you to tell me now at this this was

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1 phase growing from the Austinite just 1 thing what happens if I had more

39:49

than 1 place so supposing that I have felt like growing from Austin and in another region I also have simmered and growing because of government concentration is very high so independently growing How can I modified the army creation to take account of more than 1 faced growing very important yet because when do pampering all create powerplants Steelers began many different kinds of provides forming not just 1 at a time and for more than 60 years after the army equation there was nothing in it for so I wanted to create that teary right now How can I modified equations to deal with more than 1 face at the time so distinctly in that diagram on the right In addition to the we'll buy because I also have some red particles would have have to do well I think about how this probability dome changes no what does that represent that just represents the amount of on transformative so if I have another place forming do modify that prevented them yes but every year so if I faced the death will mean then I have been out of last week at the top In the probability but of course we only have enough on both sides here so I need a 2nd equation says the figure equals 1 minus we are published weekly delving into DVD the debt and then I saw all those equations simultaneously if I have 6 phases out 6 equation aid makes a remarkable difference To the accuracy of prediction if you do it properly because said this was sort just a few years ago friendly was modeled precipitation kinetics and power plants the Israelis get many different beams of particles forming like seamen died and when didn't carbide and printed 386 and so on so modify that equation for multiple phases is very

42:27

simple so they have DVD

42:35

offer I this equality one-liners the health last week the death toll with a total volume In the the Alpha extended and we need a 2nd encouraging which is theater the quota 1 minus 3 of the last 3 upon the total volume times the Standard and need to solve these equations simultaneously which is not a problem what is more you know you can do this numerically do it step by step it or you can do it analytically if you if you know the dependence of the going out there but numerically is the best procedure because supposing that the composition of the major changes while the faces a growing then you can alter the composition of the matrix needs that and that will influence the growth in new creation so that is "quotation mark soft impingement where particles all the composition of the matrix therefore they will grow on you created a different rates so it's the only lap of diffusion fields they have attached but the overlap of diffusion fields which is called soft right as opposed to actually physically touching so this is the most popular theory to deal with any number of reaction happening at the same time and last year in GFT we applied this would song applied this you oxidation but we have a steel containing silicone and manganese and so on and even the silicon is an extremely strong oxidizing agent the combination of silicon oxygen produces a much bigger free energy changed and I ended up the iron oxide forms for us and the reason is that there is a complication is much more iron than silicon and also the silicon as diffuse longer distances and equations like these automatically predict that the can if you you can download it from our website and you can see exactly this theory being applied for oxidation doesn't necessarily have to be presentation so forth again so and then Japan's mentioned diagrams are extremely and if you use of running theory properly that means not empirically then it is the perfect for modeling face transmission connected and of course once you have a time temperature transmission diagram you can easily converted into a continuous schooling transmission diagram because computers could be simply a set of eyes steps you can add them all up there now I want to start make a start today on and

45:50

I'm not going to teach you things which you probably already know that it's a wonderful steel is using cars to save the world yet to reduce the weight and so on but actually you know you're you're wrong the rate of cars has actually increased over the last 10 years it hasn't decrees in spite of all your research the why is why is the rate of a car in the average rate of a car in Europe has increased not decreased Weiss said city that's the thing so we see this this model here yes this is picture which I have taken this body there so they you get a side impact than the deflection is limited so that fewer safe your car will be a total right because the spreads the deformation across the but of the car in you will be safe but your carbon along the usable or repairable so the addition of said and the additional safety features have meant that the rate of a car has actually increased not decreased so all the pieces that you see in GFT which start off by saying Oh we've got to save fuel and save the world and so on this isn't going to save the world is simply improving your safety at this stage but of course you can also reduce the rates

47:19

but this is this is another piece of stealing a car where you're joining up 2 pieces of steel using a laser In of and then you make the component from a complicit steel so this is called a tailored blank this is the black that means it hasn't been formed but you don't require the same properties in every region of fuel component so you choose this deal which has the right strength right there ability cost accepted and formed a component part of a composition so this is now actually very common in the manufacture of costs and it doesn't really do a action in rates because you don't need the same thickness even in all regions of your car so here for example this

48:08

here was made from that tailored blanks which assure during the previous life this is a amended but actually it's a BMW death made by BMW so we are not succeeding actually in reducing the rate of a cop in terms of what the customer buys and in terms of safety regulations but if he didn't have trips steal the problem would be much worse yeah that's the main point the writing of these introductions we justify work now I'm going to

48:45

start uh by distinguishing between a trip steel and effective assistance to you you know the difference what's the difference between a trips the lender to persistent Steve exactly right there so the original work on trips deals was by sacking and his group and it involved 100 percent Austinite the students is 100 per cent of knitting so obviously to get a steel which is 100 per cent of Snedeker of a large concentration of following elements there so for example this is an example of a very very expensive material in the context of steels because we have about the debate Percent of nickel I don't think possible will be very happy if I suggest that you make such as steel right but if you think about carbon nanotubes and so on they are far far more expensive and they are not worried about the costs of you have to actually charge much more for a they have much better then carbon nanotubes and yet we charge a lot less OK so let's take this deal 1st I'm only going to talk about the kids give passes that means you have a small 1 infraction of Austinite and the other phases present to preventing in any assistance from the trip but it's not a trip the and this is a modern science .period temperature of minus 80 degrees centigrade endit you can see that the transformation produces a shape change yet so this is the southwesterly produced by the forming of modern science in this material and that is the deformations is the transformation plasticity transformation plasticity state means plastic strain caused by a phase change that there is a crisis structure change but it's still a physical deformations now what I want

50:55

to do In just to introduce you you how to calculate the strain along any particular direction His of we've got a transformation happening we know the nature of the changing it's an amendment stating that the share and share to the habit planes and William changed normally the have and the value of of the order of 0 . 0 0 I'm sorry value of this year's trainees of the order of 0 . 2 6 is the William changed Delta is about 0 . 0 3 3 % so I imagine that that this blue region here is a square of Austin 9 of them mentioned 1 and I define my goodness here as Z 1 along the horizontal axis said 3 is normal To the horizontal plane and said to is poking out of the plane of the board and this set of facts it is called off the normal because I've said that the last nite has dimensions 1 and the sexes are all at 90 degrees to each other for the coalition worked on almost set of taxis so what I want to do is I want to define a defamation matrix so that if I multiplied making by any direction I'll get the change in that sector due to this information

52:31

the case of got an what set of coordinates said long I said to you and to eat said 1 isn't along here and said 3 along with his unit vectors and this is my last nite and when it transforms change shape there this is the sheer strength and here In the delegation of strained the 1 in change so Z 1 has has been lent off 1 and said do and at 3 also have lent someone and they're all mutually perpendicular so I because that 1 the 1 0 0 direction similarly said a 0 1 0 direction and said Teresa 0 0 1 direction so the vectors at we can write 1 0 0 I said to the question is 0 1 0 instead 3 0 0 1 straightforward OK so when this deformation happens what happens to the records at 1 what it's coordinates when the defamation happens going the yeah so there change for the electors that 1 lies here again and I'm sharing this change it doesn't change so it remains as 1 0 0 by this defamation so I will define my defamation Matrix said said the 1st Rector said 1 remains as 1 0 0 other ads at it is coming out of the plane of the board again it lies in the invariant playing so it's not going to change so it's 0 1 here and Howard said what happens to the component of that treaty in this direction "quotation mark initially 3 has no component elements at 1 but by this year work what happened so this has changed you do this so what's the component elements at 1 yes so I returned tests and along that there's no deformation along said and along that high we want less than the correct so that is the matrix which defines a defamation by multiply that matrix now by any other then not be able to you predict what happens to that effect as a function of the deformation self supposing that I want to calculate what happens when I multiplies said BZ by electing 1 0 1 I think just arbitrarily take any direction 1 0 1 1 0 1 is a column back to case a column the I multiplied the matrix by the column would do I get as the 1st 1st roll by :colon 2 1 bent 1 plus 0 times 0 last estimate and 1 1 plus asterisk so become 1 less and then 0 1 0 times that it is 0 on this times this will be 1 less so as a consequence of the defamation 1 0 1 becomes new 1 says 0 and 1 them it will not change direction it would change my look out its magnitude I just take the sum of squares and square root because we've got autonomic coordinate system so I wanted to prove to yourself for the next lecture yet given that ass is equal to 1 . 0 2 so the quota 0 . 2 6 and Delta is equal to 0 . 0 3 what is the change in plans remind myself clear that you get 14 per cent illumination again to prove to yourself the younger patient is 14 % when 1 0 1 it is deformed by this defamation again so I'm going to use this information in the next lecture I want to discover you know supposing that all of my Austinite transformed into modern site and distressed what is the maximum elongation I can get from In order to do that I need to work out the long mission under stress exits which would have arbitrary coordinate so I take my defamation matrix multiplied by elected gives me the new act which would have a different lands and 5 take its land divided by the original land then I get the nomination but if that's all for today