Structure of Metals and Alloys

 Structure of Metals and Alloys

We have learned about the crystallography, and we have seen the miller indices. [1. Structure of Materials 2. Structure of Materials : Bonding in Materials 3. Structure of Materials : Correlation between bond and physical properties 4. Crystal Structure: Lattice and Basis 5. Primitive and Non-primitive Lattices  6. Crystal Systems and Bravais Lattice 7. Bravais Lattices Symmetry in Crystals 8. Symmetry in Crystals  9. Symmetry and Correlations with the Bravais Lattices  10. Miller Indices (Planes and Directions)  11. Miller Indices (Planes and Directions) contd....  12. Miller Indices (contd). 13. Miller Indices Weiss Zone Law ] Now, we will see how atoms pack in a given structure and give rise to various structures in solids. So, in this series, the first material system that we consider is of metals and alloys, and we will see that how atoms pack in metals to give rise to various structures and what can we learn from that.

How do atoms assemble in solid structures? How does the density of a material depend upon it is the structure? And when do material properties vary with the sample orientation? I do not know whether we can be able to do this particular part or not, but we will try and do the first.

The first thing that we consider is the energetics and packing of atoms. So, if you have random packing of atoms, where atoms do not particularly follow a configuration, they are random with respect to each other. So, the neighborhood of each of the atoms is different. So, as a result, you can draw the energy versus distance configuration. These systems are typically low energy, low-density system, and they have lower bond energy. This is the equilibrium separation between the atoms, and a lower distance is not possible because there will be strong repulsion between the atoms you will immediately get into this region is very steep.

So, typically, lower separation is not possible, but higher separation is possible. So, the overall energy of the system's potential energy or the bond energy lower. So, as a result, you are not at the free energy minima, but you are at slightly higher energy, whose deprecation is lower bond energy in this system.

On the other hand, you have systems in which atoms regularly packed themselves. So, you have configurations like this in which atoms are arranged in such a fashion that they touch each other. Then they can never touch each other, but they are in close proximity to each other determined by the energy landscape, and here, the energy corresponding to this configuration is minimum.

So, as a result, they tend to have higher bond energy, their density of systems in which atoms regularly pack themselves, which have a periodicity and the same kind configuration of each atom within the lattice. So, this atom, whether you will look at these atoms, has the same neighborhood.

So, in such systems, they have higher bond energy and have better stability, and they have a high density. So, that is why metals typically tend to have higher density because atoms are regularly packed in them. On the other hand, if you look at glasses or amorphous solids, in which atoms are not regularly packed, and their density tends to be lower. So, we can summarize that structures with regular packing of atoms tend to have higher density and lower potential energy or higher bond energy.

So, we will see about the packing of atoms, crystals, the close-packed planes, and directions, packing fraction, voids in solids perhaps not possible in this lecture, but maybe in this or in the next lecture an implication of these voids, what do we mean by these voids? What is the importance of these voids in solids in metallic solids?

There are two kinds of solids, one crystalline and the second is non-crystalline. Crystalline solids have long-range periodicity, they give very sharp diffraction patterns.

So, one way to examine crystalline materials is to use the X-ray diffraction, electron diffraction and they give very sharp diffraction patterns because of periodicity you know from your twelfth standard physics, that atoms can be the regular array of atoms can be considered, if it is a regular slit pattern and you know Thomas young’s experiment when the light of a wave meets a regular array of slits it undergoes diffraction. So, and diffraction can be noted on a screen or a detector.

So, crystalline solids with a regular arrangement of atoms give very sharp diffraction patterns; they have a well defined sharp melting point. For example, aluminum has a melting point of 667oC, and copper has a melting point of 1083oC it is a well-defined melting point. So, all of them have variations within them, but since atoms are closely packed they are touching each other, and the solids are fairly well packed they have high density.

On the other hand, non-crystalline solids, they do not have long-range periodicity. So, examples are amorphous materials or glasses or polymers. Even many polymers do not have long-range periodicity, which means the periodicity of atoms does not go beyond, let us say a few tens of nanometers, it is different periodicity. So, they do not have a long-range periodicity of a particular kind, as a result, if you examine them using X-ray diffraction, they do not give very sharp diffraction patterns.

So, this is one distinction between crystalline and non-crystalline materials when you take the x-ray diffraction pattern, in this case, you will see a very sharp pattern and the non-crystalline material will get a very diffused pattern. Non-crystalline materials also do not have a very sharp melting point; we will see that later on.

So, this is another distinction between them so, when you do thermal analysis to find out the melting point, you will see that there is no very sharp peak in the thermal analysis, and as a result, they have a very low density as compared to crystalline materials. So, polymers typically are lighter, and not only because they have lighter elements, but they also have non-periodic structures that result in an even lower density.

You can do another classification based on bonding the crystals in which molecules are held together by primary covalent bonds whereas, intramolecular bonding could be of weak Van der Waals type of hydrogen type these are molecular crystals such as polymers, non-molecular crystals are held together. In crystals, atoms are held together by metallic or covalent or ionic type of bond predominantly.

So, that is why these are the three classes that you can have metallic, covalent, and ionic. Actual bonding may not be completely metallic or completely covalent or completely ionic that may be a mix of them, but it is normally dominated by one type of bonding, and there are no secondary bonds present there.

So, metallic crystals, where a free-electron cloud surrounds positive ions core, metallic bonds tend to be non-directional in nature, in case of covalency because electrons are located in certain orbitals and orbitals have a particular shape of them it is the way strong directionality with the covalent bonds, but metallic bonds do not have a directionality.

They tend to be densely packed metallic crystals, and they have several reasons for dense packing, typically only one element is present as a result, all the atoms have the same radius. If you have multiple radii is present, then there may be confused which one to choose as a neighbor. So, as a result, there may be a lack of order. But since you have only one type of element present by enlarge, they undergo dense packing.

As a result, the nearest neighbor distances tend to be smaller and smaller, nearer and nearest neighbor distances lower the bond energy, and they have simple crystal structures, and each atom tends to surround itself with as many neighbors as possible. So, these two are correlated, and then in some cases, some metals are partially covalent, and that is why some of them have BCC structure, for example, at low temperatures, and we will see that, what is the implication in terms of BCC, FCC structures in the coming slides.

We will see that the BCC structure has a lower packing density as compared to, for example, the FCC structure. So, although we are saying that they tend to be densely packed at some of the materials are not so densely packed as compared to others, and there are reasons in terms of bonding.

 So, metallic crystals typically you can divide them into 3 structures by enlarging, FCC structured metals are aluminum, iron between 910oc to 1410oC, copper, silver, gold, nickel, palladium, platinum. Body-Centered Cubic materials are lithium, potassium, sodium, titanium, zirconium, hafnium, niobium, tantalum, chromium, molybdenum, tungsten, iron at room temperature below 910oC, below room temperature it is BCC and then there are some metals which are HCP, hexagonal close-packed.

We have talked about hexagonal system hexagonal lattice, but we are not talked about hexagonal close-packed metals like beryllium, magnesium, titanium, zirconium, hafnium, zinc, cadmium they all are hexagonal close-packed structured materials. So, let us first look at the structure of metals by packing of spheres in the space.

 So, assuming all the spheres are hard, they are incompressible are of the same size. So, in 1D, you can have this kind of configuration; you can have a close-packed row of atoms. So, all the atoms are packed in such a fashion. So, they are next to each other along a row. In the 2D case, the close-packed array maybe something like that. So, you have the first, the second row goes into the position, which is the minimum energy position where it finds the highest number of neighbors with respect to each other if it goes in this position it could have gone in this position.

So, you have the first row, and then you have the second row like this, a difference is, in this case, the number of neighbors is lower, but in this case the number of neighbors is higher on this side you have only one neighbor on this site, you have one neighbor on this site, you will have one neighbor, and on the other side you will have one neighbor.

In this case, you have one neighbor, two neighbors on top, two neighbors within the row, and two neighbors at the bottom. So, you have six neighbors, and this is called a close-packed plane, the highest possible highest density plane. So, this is how if the atoms are spherical, this is the highest atomic density that you can find within a 2D plane. This will be the sort of representation of a plane, it is a hexagonal shaped plane you can also say it is an equilateral triangle if you can draw an equilateral triangle here, but you generally do not use triangle as a representation in lattice, and you have closed packed directions the directions now the depicted those depicted in blue are the directions with a highest atomic density. So, how many directions are there within planes which are close-packed?

You have three directions, and if you take the negative indices around, this is six directions six direction within a close-packed plane, but you have three rows of atoms with three distinct rows of atoms, which are close-packed and which are at equal angles to each other.

So, if you look at the close packing of equal-sized hard spheres packing in 3D, and this is the first layer which I depict by A, then I have a second layer which can go here on top of B or on top of C. So, let us say I managed to put them on top of B, third layer can go either on top of A or on top of C. So, if it is AB AB AB then what it takes is a hexagonal close-packed structure, and if it is ABC ABC ABC, it will make a cubic close-packed structure which turns out to be face-centered cubic lattice. So, this is how you will have packing.

First row, second row, the third row, then you put another layer on top, you put another layer on top, this is the AB or first layer, second layer, third layer this is ABC ABC kind of packing so, this one is as we will see it will be hexagonal close-packed.

So, this is how the packing will look like in 3D the first layer, the second layer, the third layer and then the next layer which will again be A layer third, then the next layer which will again be B, layer and C and so on.

 So, this is A and B, and C.

So, this is A, this was B, and this was C, all of these atoms are equivalent, I have chosen this to be A, this to be B and this to be C, but I could have very well chosen this as A, this as B and then A which will come on top of C will become C.

So, in that sense, all the atoms are equivalent, and their centers form a lattice. So, basically what you have is the motif, is a single atom at 0 0 0, what is the Bravais lattice? Bravais lattice, if you have ABC ABC stacking, it makes a cubic close-packed crystal. If you have a motif at a single atom, then you need to describe what the lattice type is? because without that, it is incomplete.

So, the structure which will look like is in the ABC ABC packing it is called close pack crystal or FCC lattice with a single-atom motif at 000. So, this is the answer here motif is at 0 0 0 single atom, but what is the Bravais lattice, Bravais lattice is the FCC lattice will automatically mean you have four lattice points at 0 0 0, ½ ½ 0, ½ 0 ½ and 0 ½ ½.

You can form a structure that looks like this is the A layer, this is the B layer, this is C layer, and within this, you can form a cube. So, this looks like a hexagonal pattern, but I am saying that you are forming a cubic crystal, and this is because the atoms are arranged in such a fashion if you now consider a cube. Let us say if I connect, what the arrangement of atoms along this particular plane, (111) is.

If you remember (111) type of plane had an arrangement like this, and if I make it bigger in 2D, it will be a hexagonal type of arrangement. So, basically all these atoms which are shown here ABC pattern they are nothing but (111) plane stacked on top of each other. So, this is one of the atoms of that (111) plane these are the few other atoms of that B (111) type of plane which is the B layer, this is again the C layer of (111) plane and then again you have (111) of the plane of A layer, and you can see what are these directions now which are the close-packed directions, these are the close-packed directions right what are the what is the indices of these directions, these directions are if a direction lies within a plane then the dot product should be equal to 0. So, if this plane is (111) then the direction will be (1̅10), (101̅) or (01̅1) you can write this as (1̅10) can write this as (1̅01), and you can write this as (011̅). So, these directions, which you see are the close-packed directions, and the plane that you formed within the unit cell is the close-packed plane that (111) plane.

So, this is the FCC unit cell so, this is the close-packed plane, these are the close-packed planes, this is the direction, this is not the closed pack direction, this is just a body diagonal is perpendicular to these planes is nothing but (111) type, and for a cube we know that this is perpendicular to (111) plane. So, all the close-packed planes are of (111) type. In this case, I have drawn the atoms in a different color, but they are of different, they are of the same color for different colors is drawn only for illustration. So, that you can so see two different planes. So, this is A layer, this is B layer, and this is C layer, then the again A layer this is what you form as an FCC structure.

 So, this is the stacking sequence of oranges or a set of ladoos.

 Now, let us look at the hexagonal arrangement. You have these six atoms in the A layer, then again B layer I have chosen only three atoms, but the B layer can be continued beyond the three atoms and then again A layer. So, this kind of stacking will give me the bottom layer, which is the purple color, the intermediate green layer, the top layer which I have now converted as a yellow, but it is the same layer. So, this is my A layer, B layers, and again A layer; basically on top of each other crystallographically. So, this is the hexagonal closed packed crystal unit cell, and the one unit cell that you see is the smaller one, which is a red one, which is called a rhombic prism.

In this case, there will be one atom at 0 0 0, another atom at (1/3 2/3 1/2) and this is a primitive lattice, primitive lattice with motif being a combination of these two atoms motif being a combination of atom here and combination of atom here.

So, this is basically the hexagonal unit cell they are smaller lattice ABC axis, the same orientation 1/2 0 0 between A and B and the motif is a combination of these two, and if you rotate it has 3-fold symmetry and the two atom motif, one lies at 0 0 0, another lies at (2/3 1/3 1/2) or it can write it as (1/3 2/3 1/2) depending upon the way you look at it and because of the presence of an intermediate layer, let us say you have a 6-fold you can rotate it by 6 0 0, and you still achieve the same configuration.

But because of the presence of this B layer, you have lost that 6-fold. So, what you now have is just 3-fold access. So, only a 3-fold remains in this lattice, and although I have shown the three-unit cells here, the unit cell is the smaller one. This is the primitive unit cell because both of them are needed to define the unit cell. So, these are not two different lattice points. So, you can choose the corner of the lattice between two of them it is like a dumbbell-shaped kind of thing.

So, this is that another representation of a hexagonal closed pack lattice. So, let us see the geometrical properties.

In this case, you have ABAB stacking. I have just made them a little smaller. So, here the reason why it is a primitive unit cell because A and B do not have identical orientation of neighbors, you see the definition was that each lattice point must have identical neighborhood, here they do not have an identical neighborhood because of this presence of vacant let us say C-position, or there is no C-position, as a result, you need to come as a result you need to combine both of them to have identical neighborhood. So, if you remember the discussion that we have non-primitive lattices, you need to combine both of them to call it a proper lattice. So, that is why so A and B do not have identical neighbors. So, that is why you need a unit cell contains only one lattice point, but two atoms, you do not consider both of them as different lattice points. So, this is the hexagonal close-packed structure.

So, we will close this lecture here in the next class we will look at some other variations on metallic solids like BCC, simple cubic, and we look at other properties like packing fractions, interstitial voids.

Thank you very much

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