Peter's Physics Pages

Physics for Civil Engineering

This is an introduction to Electricity, Strength of Materials and Waves.

Lecture 11 (Microscopic Solid Structures)

In this lecture the following are introduced:
•Crystal Structures
•Body Centred Cubic Unit Cell
•Face Centred Cubic Unit Cell
•Hexagonal Close Packed Unit Cell
•Diamond structure Unit Cell
•Simple Cubic structure
•Packing Density in the Unit Cell (FCC and BCC)
•Ionic, Covalent and Metallic Crystals
•Intermediate Bond types
•Amorphous Solids
•Polymers

Crystals, Amorphous solids and Polymers

At the microscopic level, three main types of solid structure emerge:
crystals, with long range order,
amorphous solids, with no long range order, and
polymers, which have long carbon chains.

Crystal Structures

All ionic solids and most covalent solids are crystalline.
All solid metals, under normal circumstances, are crystalline.
The ideal crystal has an infinite 3D repetition of identical units, which may be atoms or molecules.

Crystals have a lattice and a basis.
 Lattices are different types of regular 3D arrays of points in space. They may be cubic, rhombic, hexagonal etc. A basis, or unit cell, is either an atom, or 3D geometrical arrangement of atoms, which is repeated identically at every lattice point.

The way atoms are stacked in the unit cell is set by the type(s) of bonding, and tends to determine the properties of the solid as a whole. The unit cell will often tell in what directions the solid as a whole may be easily stretched or broken.

Body Centred Cubic Unit Cell

 This is a cube with an atom at each of the 8 corners, and another atom at the centre of the body diagonals. The co-ordination number is the number of nearest neighbours. Here the atom at the centre has 8 nearest neighbours so bcc structures have a co-ordination number of 8.

Face Centred Cubic Unit Cell

 This is a cube with an atom at each of the 8 corners, and 6 other atoms at the centre of the face diagonals. The co-ordination number here is 12.

Hexagonal Close Packed Unit Cell

 This is a hexagonal prism with: • one atom at each of the 12 corners, • one atom at the centre of each of the 2 hexagonal faces, and • a triangle of atoms in between the hexagons, which rest in the shaded valleys. The co-ordination number is 12. Note: This is closely related to fcc, because if the bottom hexagon is displaced so that the apex atom of the triangle rests in the non-shaded valley, then the structure is fcc!

Diamond Structure Unit Cell

 This has tetrahedral units linked to each other. It can also be described as two interpenetrating fcc structures, one at co-ordinates (0,0,0) and the other at (1/4,1/4,1/4). The co-ordination number is 4.

Simple Cubic structure

 The simple cubic structure has alternating atoms at each cubic lattice point. It can also be described as two interpenetrating fcc structures, one at co-ordinates (0,0,0) and the other at (1/2,1/2,1/2). The co-ordination number is 6.

Packing Density in the Unit Cell

When thinking of the way atoms stack together in the unit cell, it is usual to think of the atoms as solid spheres. The packing density refers to the fraction of space occupied by these solid spheres in their unit cell. The packing density controls the volume density of the solid and influences the way a solid either deforms plastically or fractures.

Packing density for fcc
 In the face centred cubic unit cell, an atom has to be at the centre of a cubic face. If the atoms are taken to be all the same size, they fit as shown below. To fit the same size spheres along the face diagonal, the face diagonal must be four times the radius of the spheres, i.e. d=4r From Pythagoras the face diagonal is : hence

Now the cube will contain a half atom from each of the six faces and 1/8 of an atom from each of the 8 corner atoms, i.e. four whole atoms. The volume of four of these atoms is given by:

The atoms occupy 74% of the cube.

Packing density for bcc
 In the body centred cubic unit cell, an atom has to be at the centre of a body diagonal. If the atoms are taken to be all the same size, they fit as shown below. To fit the same size spheres along the body diagonal, the body diagonal must be four times the radius of the spheres, i.e. d=4r From Pythagoras the face diagonal is : hence

Now the cube will contain the centre atom and 1/8 of an atom from each of the 8 corner atoms, i.e. two whole atoms. The volume of two of these atoms is given by:

The atoms occupy 68% of the cube.

Packing density summary

Extending these calculations, the following table is given.

 Unit cell fcc hcp bcc sc d Co-ord. No. 12 12 8 6 4 Packing density 74% 74% 68% 52% 34%

Ionic Crystals

Sodium chloride has ionic bonds and forms an ionic crystal with simple cubic structure. The Na+ and Cl ions by themselves are both fcc structures and they interpenetrate to form simple cubic.

Caesium Chloride also has ionic bonds but the relative ion sizes are more similar. CsCl crystal structure is bcc with alternating sites filled by Cs+ and Cl ions.

Zinc Sulphide (Zinc-blend) has ions that are very different in size. It forms the Diamond structure with alternating lattice sites occupied by positive and negative ions.

NaCl, CsCl and ZnS have atoms that form singly charged ions, so their crystal structures have equal numbers of positive and negative ions. Ionic crystals are also be formed with ions of different valence e.g, CaF2, TiO2 and Na20.

Covalent Crystals

The covalent bond is directional which means that covalent crystals will not pack tightly together, i.e. they will have a low co-ordination number (i.e. 4). Diamond, Silicon and Germanium form covalent networks with a single atomic species. Quartz (SiO2) forms a covalent network with different species of atoms.

Metallic Crystals

Most metals form close packed crystals because the metallic bond has few restrictions. This is reflected in their large mass densities (eg Lead and Gold).

Metals with fcc structures:
Copper, Gold, Aluminium, Lead, Silver, Nickel, Platinum, and Iron between 9100 and 14000 C.

Metals with hcp structures:
Magnesium, Zinc, Titanium and Beryllium, etc

Metals with bcc structures:
A number of metals are not close packed and form bcc structures:
Iron below 9100 C, Tungsten, Molybdenum, Chromium and the Alkali metals, Lithium, Sodium, Potassium.

Intermediate Bond types

Crystals often have bond characteristics which merge between ionic,covalent and metallic. This can be due to the time an electron spends with one or other of the two bonded atoms, or the electron may be shared between three or more atoms. Intermediate bond types are the rule rather than the exception.

Amorphous Solids

Solids that do not have long range atomic order are called amorphous solids. They often have sub-units that have consistent form, but their long-range order is disturbed because the sub-units pack randomly. Amorphous solids are formed when liquids are cooled too quickly from the molten state to allow the sub-units to arrange themselves in the low energy, crystalline state.

Solids with pure ionic bonds do not form amorphous solids but all the other bond types can produce amorphous solids.

Silica (SiO2) can form either covalent amorphous solids, usually called glasses or regular crystal structrures (Quartz).

 In glasses, the tetrahedral SiO2 structure forms the sub-unit, and it is the flexibility of corner to corner links that accounts for the ability of SiO2 to form the random structures shown below. If the molten Silica is cooled very slowly, then the sub-units fall into the regular crystal structure of Quartz, shown below. Impurities in SiO2 hinder crystallisation. Common window glass (soda lime glass) has Na20 and CaO added. Ovenware glass (borosilicate glass) has B203 added.

Polymers

Polymers are useful because they are non-reactive, non-conducting, easily shaped, and some can even emit light! Polymers have long chains of carbon atoms with side groups. The bonds between the backbone carbon atoms and the side groups are covalent.

The chains in a polymer are held together by:

• chain entanglement, and/or
• Van der Waals bonds, and/or
• Hydrogen bonds, and/or

Chain entanglement and Van der Waal's bonds gives loose packing.

The most common polymer is polyethylene, which has only hydrogen atoms are attached to its carbon chain. Others are the structural polymers polyvinylchloride (PVC), polystyrene and the glue polyvinyl acetate (PVA).

The more branches a polymer has, the greater the tendency for the polymer to be in the amorphous state.

Important branched polymers are polyethylene and polypropylene.

Two common examples of cross-linked amorphous polymers are rubber and fibre-glass resin (phenylformaldehyde).

Rubbers are generally cross-linked by a process known as vulcanisation in which a sulphur atom forms the bridge between chains.

Resins are heavily cross-linked chemically forming mammoth 3D amorphous structures.

Summarising:

There are three main types of microscopic structure: crystals, amorphous solids, and polymers.
Crystals have a geometrical arrangement of atoms, which is repeated indefinitely.
The bond type and atom size determine the crystal structure.
Some crystal structures are fcc, bcc, hcp and diamond.
Amorphous solids do not have long range atomic order because the sub-units are disordered.
Polymers have long Carbon chains which are held together by a combination of weak and strong bonds.

 Copyright Peter & BJ Eyland. 2007 - 2015 All Rights Reserved. Website designed and maintained by Eyland.com.au ABN79179540930. Last updated 20 January 2015