Part III:- Solid Solution and Alloys

 Part III:- Solid Solution and Alloys

We are again going to talk about solid solutions Alloys. In the last lecture, [you can see here Part I:- Solid Solution and Alloys, and Part II:- Solid Solution and Alloys] we said that when you put an impurity in a material, the impurity may go to either Substitutional site or Interstitial site, if it goes to substitutional site makes a substitutional solid solution, and if it go to goes to interstitial site, it makes an Interstitial solid solution and that that where it goes is to determine by the size. So, there are some rules for substitutional solid solutions, which were called as Hume Rothery rules.

As per Hume Rothery's rules, extent of solubility is based on the size difference, depending upon the valence difference. So, if you want to make a substitutional solid solution, extensive soluble solid solubility is permitted only when the size difference is less than 15%, and the valances are nearly similar, or similar crystal structures and electronegativity difference is very small. There are certain violations, but by and large, the rules are followed.

So, the examples which we took were Au-Ag, Si-Ge, and Cu-Ni; these are solid solutions. So, if you look at the phase diagram of Si-Ger, this is composition excess % Gefrom 100% Si to 100% Ge. This is the temperature axis. So, Si has a melting point that is nearly 14000C. This is a Ge melting point and what they form is a structure that is throughout the same structure. So, this is let us say it is a Si-Gealloy, which is solid solution.

So, it is a disordered solid solution, and it is not an ordered solid solution. So, at 0% Germanium, it has a structure that is FCC structure. At 100% Germanium, it has a structure that is 100%, which is FCC, and intermediate phases also have intermediate compositions also have FCC structure throughout. It is happening within the same structure, same lattice it is just that some of the sites are being replaced by Ge. If you move from Ge side, some of this Ge atom is going to be replaced by Si. So, even at 50-50, it is 50 Ge, 50 Si with each atom having a probability of being 50% Ge or 50% Si each site occupancy. So, this is a disordered solid solution that shows extended solid solubility. So, this is true for Au-Ag; this is true for Si-Ge; this is true for Cu-Ni.

What happens in partial solid solubility? It is slightly different. So, in partial solid solubility, let us say you take an example of A and B, this is 100% A, this is 100% B, and on the left side on the vertical axis, you have temperature. Now, what happens in partial solid solubility? Solid solubility typically increases as the temperature increases. So, this line determines up to which point in the composition, the solid will dissolve.

Similarly, on this side, you will have these boundaries. So, between this and this it will be let say α-phase. Α-phase is a solid solution of B in A. So, it is A with some of the A atoms replaced by B. This is β-phase, this is a solid solution of A in B. Essentially, it means nearly B with some atoms replaced by A and this is a substitutional solid solution which is a disordered substitutional solid solution. Between these two, it gives a mixture of α and β. This is one simple example, there are more complicated examples, where some other phases may form, but this is one simple example of where solid solubility is limited.

Let us say, at this point, it is 10%, it is 5%. It increases as you increase the temperature, typically solid solubility increases as the temperature is increased. So, because the lattice dilates and there are other thermodynamic factors, the competition between entropy and enthalpy eventually determines what the solid solubility will be? But this boundary is the boundary beyond which you cannot add any more B to A. If you add any more B to A, so, up to this point at a fixed temperature, let us say this is room temperature. So, if you keep adding B to A, it will remain in the structure of A up to this point, beyond this point, it will form β-phase.

So, as soon as you cross this point, it will start converting into α+β, and this α+β will continue to remain in existence until you reach this point. The proportion of α and β will change, which will be determined by Lever rule. The proportion of α and β will change in this regime, but you will have two-phase coexistence because the solid solubility of α and β are limited. So, this is the limit of solid solubility of β, this is the limit of solid solubility of α and this happens when you have deviations from Hume Rothery rules, when the size differences are larger, when crystal structures are not similar, when valances are not similar and when electronegativity difference typically gives the information of compound, but these are the factors when you deviate from extensive solid solubility. So, this is what happens in most cases. So, as I said in case of Cu-Zn you can have 35% Zn in Cu. And in the case of Zn, you have 1% Cu in Zn.

So, you can see if, now you can correlate to the previous diagram that on the Cu side, the boundary can extend up to 35%, but on the Zn side, it can extend only up to 1%, and this is because Cu and Zn have different structures, they have different valences. So, that is why when you mix two elements sometimes as I said in Cu-Zn example 50%, you have ordering.

So, in the case of Cu-Zn, what happens is that, so, when you add 35% Zn, it remains as FCC. When you are up to 1%, it remains as HCP. But in between, as I said, it starts forming other phases which are different. So, for example, you had BCC phase in between, or you had a simple, cubic phase in between depending upon the temperature. So, you no longer have FCC, HCP structured phases, you have some other phases. These phases are formed because Cu and Zn tend to form secondary phases. So, especially at lower temperatures, when you mix two elements they tend to arrange in themselves in an ordered fashion. So, this is called as ordering in alloys.

So, as I said at lower temperatures, it had an ordered structure due to free energy changes, and this is mainly determined by ΔH, which is dependent upon the nearest neighbor. The nearest neighbor interactions determine enthalpy because it is enthalpy is a manifestation of bond strength and bond strength is a manifestation of the nearest neighbor. What is the configuration? Cu-Zn bond will have different bond energy, Zn-Zn bond will have different bond energy, Cu-Cu bond will have different bond energy. So, ΔH and then you will have ΔS, which is predominantly configurational entropy, 𝑠=𝑘 𝑙𝑛𝑤. So, when you mix A and B, you can arrange A and B number of different ways. So, you calculate various permutation combinations, and then you calculate the configuration entropy. Then we know that ΔG = ΔH - T ΔS. So, at a fixed temperature, whether you will have a disordered structure or ordered structure will be determined by minimization of free energy at a given temperature.

Usually at high temperatures, random arrangements are preferred typically. And lowering of temperature leads to leads to order which is mainly determined by lowering enthalpy. Enthalpy is lowered, as a result, your ordering takes place. It can also function of concentration. But typically, we talk in the context of fixed compositions. So, we take a composition, and then we see whether it is ordered or not at a given temperature. So, thermodynamic variables are temperature and concentration and pressure, but in most metallurgical processes, the pressure is kept as a constant. So, temperature and concentration both are variable.

In the case of metallic alloys, they are often called Intermetallics. These Intermetallics are strange compounds with a fixed composition. If you go slightly away from the composition, they will decompose into something else, or they will be present in a two-phase mixture where this particular compound will be present or something else. So, let me draw a phase diagram, for example, AB let us say, So, this is α-phase, this is β-phase, this is liquid, So, this is liquid+α. This is a compound or AxBy, which is an intermetallic compound. So, this is liquid+AxBy, this is again liquid+AxBy, and this is liquid+β, on this side, you have α + AxBy, here you have β+AxBy. So, you can see that at this particular composition, which is let us say AxBy, you have a fixed composition. If you are standing at this composition, you are in two-phase regime where you have α-+AxBy. You can see that the solid solubility of α is changing as a function of composition and temperature. The solid solubility is moving towards right as you increase the temperature. So, this will be 100% B, and this is the this is let us say Cα, this is let us say Cβ.

The α-solid solubility is increasing with increase in the. So, at this temperature, if you compare these two temperatures at T1 and T2, at T2, α has more amount of B than at T1. Similarly, at T2 β-has, more amount of A, but you can see AxBy, remains vertical. The composition of AxBy does not change, it is a line compound. When you are some other composition, you have α+AxBy, but composition of AxBy will remain same. For example, at this temperature, if you keep the composition fixed, again you have α+AxBy, but composition of α has changed according to this composition. So, at this temperature, α had a composition of C1, at this composition, α-has a composition of C2, but AxBy has same composition throughout. This is a line compound, which means it has a fixed composition, and it forms at only at that particular composition.

So, in two particulars, you will see a composition. You will have two phases, one of may one of them may be a line compound, for example, Cementite, in Steel. Cementite is a fixed composition. So, in Steel, that is a very good example.

So, if I draw the phase diagram of steel, this is Iron, this is carbon. So, steel has a eutectoid reaction. I will only draw the low-temperature reaction. I will draw them. So, this is α, and this is α+γ, and, this is γ +Fe3C and, this is α-+Fe3C. Fe3C, cementite, whose composition is given by this line, it is a fixed composition.

So, Fe3C as you change the temperature. So, if I am at let us say 0.5% carbon- steel at temperature T1 and a temperature T2, I have α and Fe3C in both cases, but here α has a different composition, this is number 1, this is number 2, but Fe3C has the same composition, which is determined by this vertical line, which is Fe3C. It is a line compound. So, this is the line compound or an intermetallic compound.

So, Intermetallics are compounds of fixed composition. Typically they do not have any variation in the composition, and they are called as line compounds. The examples are first one is Fe3C or Cementite in Steel. In Ni-Al system, Ni3Al is intermetallic compound. The line compounds are very strong. They have much higher strength, higher hardness, and strength over host lattices. So, for example, in case of Ni3Al, has a yield strength which is of the order of 850 MPa. This is nearly five times more than stainless steel. If you look at the composition Al, the soft alloy. Ni has a strength, which is of the order of Iron-Steel, but Ni3Al, the line compound is very strong.

Similarly, cementite in steel is extremely hard. That is why we prefer high carbon steels for application requires high hardness, and this high hardness comes from cementite. And similarly, Cu3Au. These precipitates form in Cu-Au systems, and they again give rise to high hardness. Similarly there are a lot of other systems, like Cu-Al, Cu-Zn. So, all of these Intermetallics of fixed composition have high hardness.

So, Ni3Al, when it is disordered, where Ni is FCC, Al is FCC. So, when it is disordered, then each atom can be considered as 75% Ni and 25% Al in disordered form. This is Ni3Al. In case of order, let me draw all the six atoms; this is order structure. So, this is ordered Ni3Al where Ni goes to ½ ½ 0, ½ 0 ½ and Al goes to 000. So, this is FCC lattice. FCC lattice with motif at 000 and this motif is 70% Ni and 20% Al. In this case, it is simple cubic with four atom motif. And this Ni3Al is very useful because it is used in superalloys to strengthen the superalloys. Ni-based alloys are superalloys, and if you add some Al to Ni, they form Ni3Al precipitates, and these precipitates provide resistance to dislocation motion, and they make the Ni-Al alloys and Ni alloys stronger for high-temperature applications.

Then, there is another category, the last category I will say is called Electron Compounds. These are the compounds that do not obey the rule of valency. They occur at certain electron to atom ratio, and these ratios are typically 3:2, 21:13, and 7:4. And they are formed in same system, for example, Cu-Zn. Cu-Zn could be β-Cu-Zn, Cu3Zn2, γ-CuZn, Cu5Zn8, Cu Zn because Cu has one valence, and Zn has two valencies. And then third one is ɛ-CuZn3, which has 7:4. So, these are called as Electron Compounds. So, what we have looked at is various type of compounds Line Compounds, Electron Compounds, Solid Solutions, how they follow Hume Rothery Rules? And why Solid Solutions form? And what are the guidelines?

Finally, to calculate the theoretical density of a material, it is given as,

 

Where, n = number of atoms per unit cell, A is atomic weight, Vc is the volume of unit cell, NA is the Avogadro number.

For the Previous Lecture Click below

Part II:- Solid Solution and Alloys

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