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Physical Metallurgy of Steels - Part 7

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Physical Metallurgy of Steels - Part 7
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A series of 12 lectures on the physical metallurgy of steels by Professor H. K. D. H. Bhadeshia. Part 7 deals with the thermodynamics of irreversible processes. Growth is treated as diffusion-controlled and by a reconstructive transformation mechanism. Note: this video has the inset on thr top right missing due to a technical problem during recording
Keywords Bhadeshia, Harshad Kumar Dharamshi Hansraj

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In the case of today's lecture is going to be about the thermodynamics of the irreversible processes and everything that I'm going to teach you today may not seem related to steals but how many of you have used it that was used in this and that is because the simulation software I'm surprised about it but we had a new students but the did is so common in Chayefsky where you do calculations for Monty component diffusion and diffusion transformations in general and whenever you're dealing with her processes which involve more than 1 solid if you need to understand what I'm going to teach you today and I also give you examples in real life there things are not as simple as just looking at 1 particular process going on at 1 time there might be many different process is happening at the same time so this will lead on to the next lecture which should be about again that put in my 2 components so today's lectures about the thermodynamics all of irreversible processes so there's already a couple of them steadily technique explained to you thermodynamics and irreversible thermodynamics normally associated with the equilibrium state Sofia watching an assembly which is that equilibrium you don't see any change at all no matter how long observers because obviously you've got observe this on a large enough scale because very microscopic scale Adams that changing moving around but the 3rd anniversary dates equals macroscopic you do not see any change so thermodynamics always deal with large numbers and entities and in all cases Adams if you observe 1 of case that things are not actually still cannot steady but this dynamic equilibrium but if you observe when allowed enough scale you will not see any change in a systematic delivery and no matter how long you observe so the meaning of thermodynamics the deals with the equilibrium so quantities such as free energy and entropy do not change as a function of time if you therefore creating so and Turkey do not change with time for system which is at equilibrium the whole system at equilibrium now normally the opted thermodynamic calculations you need to do some kind of a Connecticut relations and Kennedy calculation is such a candidate phenomena such that energy is being dissipated get so I'll just write that down and explain it in a bit more detail after that so kinetics His dissipated and we need to think about things like thermal conductivity the feasibility and so on which don't appear in thermodynamic vengeance dissipated president the like conductivity if use become important but the important .period Is that energy is dissipated In the process so just to illustrate that In terms
of an analogy of a bear now this is the state of equilibrium where you have a barrel lying inside the value of some sort you know in a couple of some sort and if I observed that system I say not change OK if it is at equilibrium and this kind of an equilibrium the 1st figure here is stable equilibrium now novelist the difference between stable and unstable equilibrium if I disturb boss roots an infinitesimal perturbations that means the tiniest perturbation it will come back to its original position that unstable equilibrium is sinister figure here where this is at I haven't seen unchanged by don't deserve it but if I understand then I will lose my state of equilibrium that ball full of flat so this is unstable equilibrium and this is equilibrium and this is madness stable equilibrium because there might be a deeper drug some of them now I cannot actually distinguished matter stable equilibrium from equilibrium in real life because we do not know where the minimum and all of them the thermodynamics of exactly the same the equilibrium and met stable equilibrium so we don't need to distinguish those states except when we know that there is a further minimum and we want to work the date at which you go from Matt state to a stable state but as far as the level of the modern and acceptance and there is no difference between these 2 states so these all described it can be described by thermodynamics alone there is no time dependence in the States on the other hand this it is not at equilibrium this man is rolling down a hill right that means it's dissipating energy it's going from a higher fee energy states to allow free and fair and this is a pragmatic phenomenon which will also depend on how many other factors for example the friction between the Berlin and so on and so on you cannot describe is purely in terms of dynamics so often in the my primary process that this free energy all the time so happy with that but as unharmed to show you life which looks a bit more indeed there at the climactic state so once again we have been almost thermodynamics equilibriums date where there is no energy being dissipated these to represent cases where energy is being dissipated In this case the hearing does not have a constant slow so the rate of participation is not constant this is where we crossed steady-state reiterated his patient is constant a fireman observer located on the ball I will see no change in my environment but it's as if nothing is happening so this is quite a steady-state processor another example would be if we have a different temperature outside the room than inside the room and the constant gradient of temperature across the glass then clearly there is heat flux but if I'm inside the glass I don't see any change in the temperature gradient I don't see any change happening because the temperature inside the room and outside is constant so that's called a steady-state process and it is simply matter it then kinetic theory of every don't have said state so this is what we are going to focus on today In steady-state processes
so in between the most
general form of fanatics and there are dynamics we have a state which we could call steady state where apparently there is no change happening but energy being dissipated so we have a further stake here which is the steady state really apparently there is no change no change In surroundings but energy he has dissipated at a constant rate so don't care about the difference between a state and general panicking not only to explain to you are a focus for let me just write down that supposedly we think about a solid and liquid and equilibrium that's where easy for us to write an equation that meant a solid and liquid diet equilibrium the free energy of the liquid will be the same as that of the solid ink for abuse substance support for substance he also substance solids and liquidity but equilibrium right the free energy Of the salad is equal to the free energy of the liquid but we have an equation whenever we have free energy dissipation that becomes an inequalities funding the mounting temperature then she s is no longer equal the GAO so whenever energy is dissipated this becomes an inequality so that she does not equal g L clearly because we are dissipating energy as so mediation proceedings so it becomes much more difficult to define clearly but we are dealing with kinetic theory whether it's steady state or non-state we do not have the qualities we have inequalities to deal with but then I need to define for you what I mean by a reversible process and an irreversible process so again I'm going to use a mechanical and an analogy and I begin by defining the process and
then explaining it so a
reversible process it is 1 in which when I change the direction of the process by a very small amount and infinitesimal changing external conditions and then I that change then I will go back to the original state without this dissipating any energy so that alert through swallowed immediately but if I
show you describe this slight will become very clear so many times that we have a piston which is moving inside the cylinder and you go out there and I don't guess inside the cylinder an idea that seems to be the core of vanity yeah the relationship between pressure volume and temperature but there's no friction at all between the president and the cylinder so Scott said this point here and that change pressure so that I'm aware of this .period and then I change the pressure back then I come back exactly along that curve there is no infection here and this is an ideal gas so if I come if I follow that code exactly then I'm not dissipating any energy so this is going to enter into a civil so it's a process was that action can be changed by an infinitesimal changing in external conditions because there is no friction here and friction dissipates energy so in
reversible process that
reversible process process an infinitesimal change and conditions can be reversed without participation of the analogy used was that of a frictionless Preston in a cylinder containing an idea guests no if I could just
that With a process which is Universal there is the
person dissipates energy and he is universe about to an infinitesimal changing external conditions and an outages fiction
yes then you know what do "quotation mark pressure from here quoted the pressure I regret not changing what meant 1st because the movement of the picture of the piston is opposed by fiction back Friday's the pressure but nothing happens because of friction between the president and the Senate and then when also kind of fiction you move along this coast and not let go off the the pressure then nothing happens until I drop the passion to this point because we have fiction in the opposite direction from and you follow this and didn't come back to here now can somebody tell me what is the energy dissipated in this process is due to friction but there on the diagram what is it this is this area so clearly this is not irreversible process because if I change my pressure by an infinitesimal amount nothing happens because of the friction there so about process dissipates
energy this associated the dissipation of energy it doesn't mean that you can't reverse the process is simply means you have to spend energy in order to reverse the process OK but we are going to do processes which are dissipating energy because that's moving away now from thermodynamics 2 words kinetics but we have to stick to the steady state that means we have dissipating energy at a constant rate introduced you
to a particularly which applies to irreversible processes and then I will be right for you in the eye at approximately 40 because everything becomes approximate when the go away from thermodynamics the case
of the despair engine don't worry about this there is simple equation but extremely powerful that if I have the blocks not normally used to fluxes in terms of say the diffusion flux of the heat flux but this is a general flux which also applies to diffusion of heat but it could be anything it could be an electrical current for example that so this is a generalized locks and there's something in writing that flux and that the growth of course right so a generalized also for example in the case of diffusion it might be some sort of a gradient and in the case of God and it might be some sort of a gradient of potential so this is a generalized flux and this is the general lack of force if I multiply that you get the at which energy dissipated beaches the temperature multiplied by the rate of entropy production you know that that if temperature by entropy than I get an energy crisis so unfamiliar with that so Sigma here is not the entropy but is the remains of entropy production so that the temperature times that is the rate of energy dissipation and that is equal do but locks times the force now so we have the
temperature times the rate of entropy production of entropy production this sequel to flux the 2 college multiplied by it's fall the 2 called capital tax now if I can write any process in this regard then I'm going to say that due to the proportion of tax the deflected expressed where can express process using such an equation find flux is proportional to the courts can you give me an example let's think about diffusion what is a flux it's there is a mass flow into a unit area in a unit dynamism yeah and that is the 1st you had a very good huh OK so I thought you had it right when he said potential gradient but then you change your mind to concentration gradient so that is incorrect get concentration gradient strictly does not diffusion being on that what drives diffusion yet so you're getting near the answer is not quite activity chemical potential gradient forget the chemical potential is a free energy that if you have a gradient of free energy and you multiplied by the locks you get the rate of energy dissipation that so in the context of diffusion at the jail would represent the diffusion flux so for example J. gains and diffusion blocks the next With the chemical potential gradient so said the by the Senate revenue it is the chemical potential so I take the product of the flux and the free energy gradient then I get rate of energy dissipation any other examples so of course this is known as set out the generalized fixed law diffusion if that fluxes proportional to a chemical potential gradient fixed date according to the concentration gradient but more generally it's a chemical potential so this is if you like this series a delegation of fixes law diffusion From irreversible thermodynamics can you give me another example where you find the flux to be proportional to the fullest so is there any electrical engineering yeah you police force you are right but there is a complication the magnetic fields which have come too late again but give me another example from electrical engineering where of flux is proportional to your force I think you don't think about fluxes of flux but any think the electrical current is proportionate yet the or a lack of motive force and what is the Lord is studying the relationship between current and so we have current being proportional to the world stage so in the case of a electrical conduction current according to the flux and lacked a motive force for what this difference force which is known as the American press we call on the phone and all says that the current is proportional to the world now what happens when I'm with black and I will pitch so yet so that its power was a difference between power and energy yeah so concerned that's workers gives you energy playing unit time that the rate of energy dissipation so if you had known this agrarian here and I haven't proven to you that if you follow that equation and Javier the Russians to acts then you could on slow it is any other law many DeBonis fixes law what preceded fixes mass diffusion what came before that I about fully is no free not also hit a few yesterday said the flux of the disproportionate to the gradient of simply by multiplied that do I get the rate of energy dissipation that and unhappy soak up so I haven't explained to you the basis of what we're doing but if he is I would have said here that the cancer rate of and production equal of which is a flux standard acts if I can write equation like that then I would find that Jay is is proportional to the forces proportional to flux so I I don't have agreed that OK Click and see where we barely OK so here is
our there you have the rotating Wisconsin lonely because the times the electrical resistance if I think the product of and involved a gigantic native of energy dissipation which daytime segment temperature times the rate of and the production now I'm going to try and show you why jury is proportional to X and also when that approximation fate so we need not demonstrate that the 1st influx of proportional so let that would
prove in inverted commas that is not really approve it is proportional to facts where the signal is equal J. acts somebody write an expansion of the flux in terms of the forests using a data series and about of sequences 0 so we expansion of change 6 is equal to 0 set so when I use the races like this I mean that J is a function of people the not that Jays multiplied by fax unit was here I'm implying that functions for francs then the 1st them here the average of and the 2nd round will be 0 the 1st derivative and multiplied by fax all were men pictorial and the 3rd the double -dash the effects of 0 sorry an act Graham all this sector question but it's just a daily expansion and I can continue this fall and the number directions it has simplified Is there any term that I can beat them yeah ,comma acts so the 1st and gives the only candidate biceps yeah right but then I the GM CEO means the flux when the forces 0 so what would that be 0 did not collapse if there's no force so the candidate the 1st now supposing that we had been making only slightly from the euro yes that means the magnitude of the forces small that means that any other person that I can get rid of sir faxes small which becomes can I or that's great and I wouldn't because a small number squared with the even smaller so I can delete this 2nd and will recover I recall that the 1st Jake is proportional to ax so therefore is approximately proportional 2 tracks here Jake is proportional to ax right so what does this mean there are July 4 pixels for yesterday this isn't quite straightforward the deletion of the 1st but what what does this mean when we delete the higher-order terms what should we expect if I make the world stage difference extremely it may not be the case that the current the proportional to the world now it's very difficult or even may be impossible to do such an experiment because as soon as you apply where large world is different you get a lot of heat generated here but in the case of diffusion we can actually apply very large gradients of concentration by depositing said for example won gold on top of 1 narrow so that's the biggest gradient that you forget that and in those circumstances you diffusion will no longer have a constant mobility In a mobility is related to diffusion coefficient the diffusion coefficient itself becomes a function of the the gradient so there will be a point where is Indianapolis was the breakdown I cannot tell you at what point there will break down again but have you had just been under the composition yet as anybody who has been out of the competition so they have a solution which is such that if you give a small perturbation it will tend to break down into solitary agents are and the gradients of concentration can be very loud and you can rely on the use or in the diffusion theory for that you have to take account of effectively the dependence of the diffusion coefficient on the gradient itself so there will be a point where the thermodynamics of irreversible processes no longer applies not later than that which we can think about which is that the velocity of a boundary will depend on the driving force yet quite sensible that the velocity of the boundaries if you increase the guiding force it faster and if I might to let the velocity by but the last day of the boundary by the time for us I get the
rate at which energy is dissipated that so again we should be able to write losing hours so consider actually go to fresh page could I need to draw diagrams can so here you have a coordinated said and this is the energy parents this is the position of the this is the driving force mutual label as does the Chief and here is the activation barrier so it's huge so here is an activation barrier a the transfer of weapons from 1 side of the interface to the other activation barrier what transfer of from 1 side "quotation mark interface to other I mean I really like the last it is proportional to that achieved the meeting is the velocity of bonds and among bolivar lost about that the that I will be at the rate of energy dissipation which is the Sigma that is just like and fixes law and so on but I want to derive the equation for the velocity of a grain boundary a bit more carefully and whatever it is I look at the the rate at which jump from 1 side of the barrier to the other end in the reverse direction OK so the rate at which the Adams will jump from the left-hand side of the barrier to the right answer so the From left to right the proportional 2 exponential minus Q for please that's just Arrhenius equation and then the frequency factor in front of this so small ago which is a frequency factor the frequency of attempts the exponential per year the probability of a successful young and Omega is how many times attempting so honored at times exponential is a rate of successful films yeah not in the best direction so the lake in the reverse direction from right to left through the proportional to only times exponential minus Q just the quantities because the valuable going in that direction is larger you know that I want to go from this energy stated to the island instead of going to jump a barrier which has been achieved last year so the that it will be proportionate to the velocities so I won't be the last eh is a difference between the forward and reverse Jones so I will simplify this by taking exponential minus Q upon Haiti criminal 3 2 1 -minus exponentially felt simple the following day -minus the university the action it is not the last equal to about lasted professional due because I ignored the other frequency factors now this does not look like the last being proportional to that the G which we get from the 1st time slot the question it's quite different than but if you make that adjustment more you know expand exponentially when that no you will recover the lost being proportional to destitute so Pineda that In the 1st learned that that this becomes the state is fortunate to have achieved so you have to decide by experiment there's no way of predicting at what point that proportionality face if you do decide by experiment How are you can go forward the he said that explained so far I can tell you that you know when the velocity becomes this much you cannot use that relationship only experiment in there there were unhappy with that took look at this with
the electrical example of this is the expansion that needed with a J road we can believe because there's not locks when there is no and he can also be higher-order terms as a 1st approximation but if you're going to very large forces then you might need to take account of tradition the same thing applies to food as as he effusion where the product of the heat flux with the gradient with reiterated energy dissipation and you find that the heat flux is proportional to the gradient and in the case of diffusion the fix locked diffusion is just an empirical lower the flux is expressed as a function of the concentration gradient but that's not too rigorous we really need to express their flocks in terms of chemical potential gradient and then when you multiply that by the chemical potential gradient will get the rate of energy dissipation so this is this is to set aside side issues if you compare these 2 equations then you can define the concentration dependence of the diffusion coefficient this by comparing the terms this is the reason why the diffusion crimes the corporation depends on concentration let me make the chemical potential gradient very large it also becomes a function of the gradient itself and just to
summarize the different forces and the fluxes so we have the electorate being proportional to electrical called the temperature gradient being proportional to the heat of the heat flux being proportional to the temperature gradient diffusion blocks to the chemical potential gradient and In some cases the stress is proportional to the strain rate you might at last test by strain rate you get would you get if somewhat less stress less training rate of energy dissipation you know status as the units of energy per unit volume exactly the same as major past those of us who have energy played in volume multiplied by the strain rate which is good time gives you energy dissipation unit time OK now comes there Romanian clever step which is very simple but thing that's opposing we have a temperature gradient and the concentration gradient a temperature gradient and that diffusion just like a concentration breaking that's willing to both of them are happening at the same time how do we deal with that well the equation that we had still applies that total rate of energy dissipation it this summer Of the dissipation due to each of the processes that means jail applied at survive you know you have several different processes so if I can grants that equations then I can also right there did you actually professional on but will also depend on other forces so the diffusion plant to depend not only on the chemical potential gradient but also on the temperature so let me just write that down so for
multiple forces for multiple forces influxes it's still right there temperatures downgrade of entropy production is the sum of all the different inspections going on and then just like we did for a single was in flux this fall you go jail secretive and let let's say let me extend supposing we have 2 forces and that I can write jailed 1 is equal to an 1 groups 1 1 times and 1 plus and 1 2 plans next year that means that my that depend not only on the you do X 1 but also next to if concentration gradient and the temperature gradient then my diffusion all manganese depend on both yeah not because of a lack examples of this that you use all the time but you haven't realized it but then a couple the new measure a temperature difference between a reference and your sample actually creates a well-pitched isn't it yet so temperature gradient is creating a world stage yeah what is the opposite of that effect so we have a temperature gradient creating a world stage Is there any device which does the opposite that never worked its difference which creates a temperature that is you can indeed In another career you have every kind of gadget you can buy in the world that you can buy refrigerator according 1 the idea contender ,comma and it has no moving parts how does that work the Peltier effect but the fight but I won't difference I create a temperature difference and that's your solid-state refrigerator so the flux need not depend only on me single for us if you have several courses happening at the same time then influence flux so if you think about Monday component diffusion which is what we are after but I just right down the 2nd 1 here if you stick to the sequel to 10 to 1 In Q X 1 last 2 2 in x 2 assuming we have due process and to fluxes not if you if you close to equilibrium then the president was rates of reactions must be equal and that requires that and want to must be equal to m 2 1 I think so agenda general principle it is said in general and I want to wear the EAC would have to want except in the case of magnetic fields yeah except for magnetic fields and 1 jury pools minus and 2 1 the reason why I went through all this is you need to understand what goes into all the calculations that we do for money component diffusion and I can right now with considerable confidence that you understand if I like that the flocks of carbon sequel to and I'm going to resume fixes what now or diffusion here the diffusion coefficient of carbon times the gradient gradient of manganese so the flats of carbon not just depend on the gradient of problems but also on the gradient of manganese and this is across diffusion coefficient that means the dependence of the diffusion of carbon on the gradient of so if I have an ally which contains segregation of manganese thing so there's a different mining is concentrated in here from here and I start with the uniform concentration of carbon the carbon distributed even tho there was no originally original gradient of our the problem there might be a similarly you can write terms but the flux of manganese the miners good thank you America is a alive however smaller than those maintenance lapse is not going to be influenced greatly by gradients of problem but 1 problem there will be a big factor if there is a gradient of the fact that I had an equation like this you now completely comfortable because you know where it comes from right so this is
just the expansion of the equation the Richard radio down on the board and in general you'll find my Jake was MGI but in the case of magnetic fields we have a minus sign so I think you've learned quite a powerful principle today which is useful not just in what we are going to do but in many different aspects of these transformations and even processing of materials so that's all for today