noc18-mm20 Lecture 10-Thermodynamics of Real Solution

[Music] [Music] okay so now we begin with the new lecture in phase equilibria module and that is lecture number 10 so I just briefly recap what we did in the last lecture so in the last lecture we basically discussed about effect of impurity addition on stability of solutions so basically when you add a small amount of impurity to a to a pure phase it is common and equally stable and mainly driven by even though Delta H of mixing can be may be positive the impure phases slightly in pure phase is more stable because of increase in the configuration entropy which drives the free energy of mixing to become negative in the impure phase and then secondly we introduce the concept of activity a I so you have activity is basically related to mu I so you have element mu I which is equal to G I plus RT Ln a I so this is the relation of chemical potential with respect to activity so higher the activity of a system higher the chemical potential is so and it basically depicts the tendency of an element to leave or remain within the system so if the activity of a component is high which means it doesn't like to stay in the solution it wants to get out of the solution and what that depicts is basically that Delta H mixing is higher is positive and an activity is lower what it means is that the chemical potential is also lower and the element likes to remain within the solution with other elements lowering the free energy of that state and which is manifested in lower enthalpy of mixing as well so this is what we discussed in the last lecture now we will move on to the koalas real solutions now real solutions basically are different from ideal and regular solutions because whatever we have discussed previously they do not really describe the reality so previous models do not describe reality and this is because we have not considered we have assumed that a and B do not have size differences but if you have two elements they are going to have size differences so the factors which we didn't consider first was the size we didn't consider any strain fields when you have size differences you will have strain and this strain leads to enthalpy of mixing so these size differences and corresponding strain fields lead to changes in the enthalpy of mixing we didn't consider differences and structures differences in crystal structure of the elements we assume that both of them are similar structure but they may not necessarily be the same that structures may be different similarly you can have differences in structure not only structure but also electronic configurations electronegativity and so on and so forth so all these differences lead so when we even we so as we saw in that structure of materials module the formation of solid solution is governed by certain rules called as um ro 3 rules which say that you know solid solution will form when there is a certain size difference is smaller than a certain number and the valence is of those two elements ideally similar or the electronegativity is as similar and their structures are similar so these factors which were not considered in the first two models of idea and regular solutions make the situation far more complicated and as a result you don't necessarily form a substitution solution your solute solute solution could be interesting for its solution and this interstitial solid solution will have completely different free energy considerations enthalpy of mixing for instance as compared to regular or ideal substitutional solid solution so these factors were not taken into account as a result the solid solution and thermodynamics associated also is different and then we also didn't consider whether two elements we considered that they both the elements will extend the solution solid solution all throughout whereas in reality they could be formation of other phases so at certain compositions these two may react with each other and they may lead to formation of phases whose a structure and properties and bonding etcetera are very different from the solid solution itself so for instance they may form so these phases could be for instance intermetallics which are compounds basically intermetallics or line compounds they may form ordered phases they may form various other kinds of phases okay so these kind of differences may lead to a completely different situation as far as for energy considerations are concerned so so the assumption that you know random arrangement of atoms will always provide a most stable structure is not necessarily true because of these differences and actual arrangement of the atoms and such situations will be the one which yields minimum free energy with sufficient amount of entropy so as we know that entropy cannot be zero so entropy makes a way entropy so it's a balance between Delta H mixing and entropy changes considering the differences that I just narrated that will determine the minimization of free energy so the possible you may have various possible solid solutions for instance the ordered solid solution as we know is something like this my apologies for the differences in sizes but assume that sizes are same so hang on let me just change the first row and make it slightly bigger okay now here let's say the first arrangement could be like this so these are basically ordered substitutional Solutions you can have another scenario in which you have situations like these and you can have some sort of clustering let's say few items here are clustered in this fashion so they don't have as such absolute randomness they have some sort of clustering so this is you can have a clustered solid solution so it's not exactly periodic because there is no ordering as such here but atoms tend to cluster at various places so you can see that B atoms have clustered so this is if this is B this is a you can see that B have clustered in different pockets whereas a is present as well in certain different pockets so this is for example pocket of a this is pocket of B so this is pocket of B okay this is again a pocket of B it's again a pocket of a on the other hand there's the pocket of a this is the pocket of a so they have clustered each other in different regimes and this may happen in cases where a and B do not like each other which means a B bonds is not preferable if you have excess of size differences you may have completely different scenario for instance you may have these atoms present somewhere here and this will lead to either a contraction so this will this is basically interstitial and this will lead to contraction or dilation of the lattice depending upon the size difference and the size of the void and these are basically again interstitial solid solution could well again be random or ordered depending upon the enthalpy of mixing and also this is further complicated by differences in electronegativities as well as differences in the bonding of between different atoms so there are various factors which are which which go behind defining the phase formation what kind of solid solution will will form eventually so let's first define the kind of phases that can form if they do not form completely at random solid solution so in this series the first type of solid solution first type of mixture that we consider so it's not solid solution as such it's ordered phases now in this class of ordered phases there are two categories now ordered phases are are those phases in which atomic arrangement is is periodically ordered so that we saw in crystal structures module 1 that you can have arrangement like for instance you can have a phase like this so basically the manifestation of ordered solute solution is this you have an ordered solid solution so you have this is a atom this is B atom you can see that a and B are arranged in such a fashion so that you can define a lattice either like this or like this and in this you can see that if you if you choose this as a blue one as a unit cell then a is at the corner and B is at the face center and this is always the case so you can have either be at the corner or a a at the corner or be at a at the center of B at the centre depending upon which one is that but the pattern remains same all throughout so this is basically ordered solar solution in three dimensional for example materials like so this is Cu a u so this atom is Cu and this is a L okay see you L so this makes a structure which is periodic but it is not a BCC lattice it is a simple cubic and this is what repeat itself so you can see that a all throughout the structure the bond that is preferred is see you al bond instead of see you see you or al al bond and this is because of stability of these bonds this is because of stability of thermodynamic stability of see you al bond because the delta h mixing in such case is low as a result copper likes to be in close to in in close proximity with the aluminum however you can have a disordered form disordered form would be in which in the same lattice so each atom is now 50% Cu and 50 percent L so in such a scenario each site is occupied the probability the occupation and the probability of occupation of the site is 50 percent copper 50 percent aluminum as a result this makes a disordered structure which means a a a b and b b bonds are equally probable in such a scenario this makes a BCC structured lattice but in a disordered form so each of each item is mathematically considered as 50 percent copper and 50 percent aluminum so in this case and the perfectly ordered phase there is no Cu Cu bond there is no aluminum aluminum born nearest neighbor as far as considered all the bonds that are considered are CU l bond okay in this case there is a possibility that you can have see you AC you see you born there is a possibility you can have a Lal bond there is a possibility you can have Cu L bond so this is so depending upon so now you go to generalized a be solid solution go back to generalized a B solution and let's say in a number of number of a B bonds in a completely random solution that would be PAB and this is nothing but in a Z X a xB na is the Avogadro number Z is the number of bonds per atom XA and xB are atomic fraction so for instance if you have for example if you have this as 100 this has 4 this has 0.5 this has 0.5 then PAB would be 100 okay but if the propensity of if there is a propensity to but if let's say propensity of a B bond formation then Delta H mix is less than zero so this says that a B bonds form more than what they wouldn't in a random solution so basically more a B bonds will be formed than what you will form in a random solution in such a case the alloy may form either short-range order or long-range order and this is depicted by a parameter called as degree of ordering and this is determined as s is equal to PAB observed minus PA be random divided by PA be maximum in our Loy / - PA be random okay so basically this would be state of complete ordering this would be random solid solution and this is the observed so now let's say we have a situation case one in which let's say we have PA B as 100 which is the observed total number of atoms let's say is is equal to 100 so in that case PAB random was equal to n a into Z into n into Z into X a xB so consider n a as 100 z as 4 bonds per atom and if it is a Qui atomic alloy then 0.5 in 2.5 so this will become 100 and PA B max so you have you know let's say 100 atoms so 100 atoms means 10 row so 10 this way ten this way so you will have 100 in place so total number of 100 200 bonds that you will have so total number of 10 horizontal bonds and vertical bonds sorry 100 horizontal hundred vertical bonds as a result PA B max is 200 so this small s in this case will be equal to 100 minus 100 divided by 200 - 100 this will be equal to 0 so which means it is completely random solid solution disordered solid solution so this means this ordered where if you have s to be equal to let's say P a be observed which is equal to however so let's so case two let's say so if P a be observed as let us say 130 PA be random that we have worked out the 100 and PA B max so this is random this is max it's 200 in such a case s would be equal to 130 minus 100 divided by 200 - 100 this would be zero point three zero so this will have certain degree of ordering it's complete so it's partial ordering which is called a short-range ordering and this would look something like you know if you have so in certain regions you will have ordering here also let's say you have ordering but on other regions you have dis watering so they do cluster tend to form so you can see that you can see that certain regions are ordered certain other regions are disordered or you can say sorry certain regions are perfectly ordered so there is short-range order but other regions are random so clustered there so this is sort of example of partial or short-range ordering and of course perfect ordering will have case three perfect ordering or long-range you can say long-range in that case PAB observed would be same as Bab max so s would be equal to PAB max minus PA be random divided by PA be max minus PA be random and this will be equal to one so for long-range ordered solid solution s will be equal to one for a completely random solid solution si be equal to 0 and for short-range ordering it would be intermediate values so these are the mathematical tools to define the ordering disordering however in the context of in the context of thermodynamics basically what it means is that ordered structure defines low entropy of mixing and this ordered structure has high entropy of mixing so if you just consider the entropy because high high disordered phase will have higher confusion entropy as a result higher configure entropy of mixing so what is that now but in reality what is observed is that generally at high temperature your disordered phase is stable whereas low-temperature ordered phase is stable and we know that Delta G Delta H mix - date T Delta s mix so as far as the comparison of two phases is concerned real data as mixed term is surely higher for the disordered phase however this enthalpy term is also is an important factor so it is a relative competition between Delta H mix and T Delta s mix which governs at which temperature which phase is stable so as far as ordered phase is concerned for them they are stable generally at low temperature and for them - T Delta s mix is not so I can say that plus delta T not large but Delta H mix is less than zero so it typically happens at lower temperatures when Delta H mixed is lower T Delta s mix is not large and in comparison to disordered phase Delta H mix term of this phase dominates as a result they are more stable whereas disordered phase high temperature t delta s mix larger as a result entropic term is bigger but Delta H mix is relatively it's higher than Delta H mix of ordered so it generally happens at high temperature so it is the competition between Delta H mix and T Delta s mix of two phases which dictate which term will be stable so in general for example CUA you see you al at high temperature it is the BCC phase which is stable disordered and at low temperature it is the simple cubic phase which is stable which is ordered and there are various other systems in which these phases exist for example copper gold system and various other systems in which ordering this ordering happens and which is determined by competition of enthalpy of mixing and entropy of mixing so we will finish clear this lecture in which we have discovered we have talked about the that the real solutions in which enthalpy and entropy term can be very different from the or substitution from the ideal and regular solutions which can see which did not be did not consider the factors such as size different valence difference electronically different strain difference and so on and so forth so we will take up in the next class some other issues related to the real solutions thank you [Music] [Music]