Materials Science

Structure of crystalline solids

The structure of solids

We understand how individual atoms may bond What about real materials made of multiple

atoms? How do atoms arrange to form solid structures? How do these structures influence a material’s

density? How do material properties vary with these different

structures?

The structure of solids Crystalline materials have atoms situated

in a periodic array over large atomic distances Metals and most ceramics

Some properties depend on the crystal structure of the material (e.g. density and ductility)

Amorphous materials have no long range order (non-crystalline). Glasses – Silica Plastics Rapidly cooled metals (1x105 °C/s)

The structure of solids

Steel, PEO and Silica glass structures (left-right)

Polycrystalline materials ‘Nuclei’ form during

solidification, each of which grows into crystals

Crystals grow and meet These grains are separated

by an amorphous grain boundary

Energy and packing of atoms in solids

Dense, regular-packed structures tend to have lower energy

• Dense, regular packing

Energy

r

typical neighbor bond length

typical neighbor bond energy

• Non dense, random packing

Energy

r

typical neighbor bond length

typical neighbor bond energy

Crystalline solids Lattice: 3D array of regularly spaced

points Crystalline material: atoms situated in a

repeating 3D periodic array over large atomic distances

Hard sphere representation: atoms denoted by hard, touching spheres (a)

Reduced sphere representation (b) Unit cell: basic building block unit (such

as a flooring tile) that repeats in space to create the crystal structure; it is usually a parallelepiped or prizm

The unit cell

Smallest repeating arrangement

Metallic crystal structures

Repetitive patterns – unit cells (represents the symmetry of the crystal structure) FCC – Face Centred Cubic BCC – Body Centred Cubic HCP – Hexagonal Close Packed

Tend to be densely packed due to non-directional bonding in metals

Simple crystal structures, for important engineering materials

NOTE: HARD SPHERE MODEL

Simple cubic structure (SC)

Cubic unit cell is 3D repeat unit Rare (only Po has this structure) Coordination number (CN)

Number of the nearest neighbors or touching atoms

Easy for civil engineers

SC has CN=6

Simple cubic structure (SC)

Simple Cubic has CN=6 Close-packed directions (directions along which

atoms touch each other) are cube edges How can we describe packing of these atoms?

Coordination # = 6 (# nearest neighbors)

Atomic packing factor Fill a box with hard spheres

Packing factor = total volume of spheres in box / volume of box Question: what is the maximum packing factor you can expect?

In crystalline materials: Atomic packing factor = total volume of atoms in unit cell /

volume of unit cell A value <1 (APF is the fraction of solid sphere volume in a unit

cell)

APF = Volume of atoms in unit cell*

Volume of unit cell

*assume hard spheres

Atomic packing factor

APF for a simple cubic structure = 0.52

APF = a3

4

3(0.5a)31

atoms

unit cellatom

volume

unit cellvolume

contains 8 x 1/8 = 1 atom/unit cell

Lattice constant

close-packed directions

a

R=0.5a

Face centred cubic (fcc) Unit cell of cubic geometry with atoms located at each

corner and the centre of the cube faces E.g. copper, aluminium, silver, gold, (ductile metals) Coordination number ?... APF ?...

a

Exercise: Atomic packing for fcc

APF for fcc structure ~ 0.74, CN=12

APF = a3

4

3( 2a/4)34

atoms

unit cell atomvolume

unit cell

volume

Unit cell contains: 6 x 1/2 + 8 x 1/8 = 4 atoms/unit cell

Close-packed directions: length = 4R

= 2 a

Slip systems Why are fcc metals ductile? Ductility (ease of plastic deformation) is linked to crystal

structure and close packed planes Slip occurs on specific atomic planes and in specific

crystallographic slip directions, i.e. slip systems Slip planes – most densely packed planes, and in that

plane the closely packed direction

Face centred cubic (fcc)

For fcc there are 4 close packed planes (face diagonals) and 3 close packed directions, making 12 slip systems

Ductile metals as there are many opportunities for planes to slide over each other

Defining crystal directions

What do we mean by a [111] plane ?

We assign x,y,z a magnitude (0 to 1) for the unit cell

This is sufficient to define a plane or a direction

Body centred cubic (bcc) Unit cell of cubic geometry with atoms located at

each of the corners and the cube centre E.g. chromium, -iron, tungsten Experience:

Ductile-Brittle transition Feature a fatigue limit

CN = 8 APF ?...

Body centred cubic (bcc)

The Atomium, André Waterkeyn, 1958

Brussels The bcc unit cell of an

iron crystal magnified 165 billion times… shiny!

0.1nm

x165 billion

aR

Atomic packing factor – bcc

APF for a body-centered cubic structure ~ 0.68

Close-packed directions: length = 4R

= 3 a

Unit cell contains: 1 + 8 x 1/8 = 2 atoms/unit cell

APF = a3

4

3( 3a/4)32

atoms

unit cell atomvolume

unit cell

volume

Slip systems in bcc crystals

6 slip planes x 2 slip directions = 12 slip systems

{110} planes in the direction of

1 11

Hexagonal close packed (hcp) Hexagonal unit cell Top and bottom faces of hexagonal cell

consist of 6 atoms (hexagon) with an atom in the centre

Middle plane consists of 3 atoms E.g. cadmium, magnesium, titanium

and zinc. Least ductile metals (only 1 close

packed plane) CN=12, APF~0.74 (like fcc)

Slip systems in hcp crystals

1 slip plane x 3 slip directions = 3 slip systems Most brittle of the metals, but subtle difference

between fcc structure

{0001} plane

Comparison of crystal structures

Crystal structure CN APF CP directions

Simple Cubic (SC) 6 0.52 cube edges

Body Centered Cubic (BCC) 8 0.68 body diagonal

Face Centered Cubic (FCC) 12 0.74 face diagonal

Hexagonal Close Pack (HCP) 12 0.74 hex side

Other crystal structures There are many structures for

Compounds Polymers (molecular crystals) More complex configurations Polymorphism, etc.

• Structure of NaCl

• Structure of Carbon

Graphite

Diamond

Other crystal structures

Single crystal materials When the periodic and repeated

arrangement of atoms is perfect and extends throughout the entirety of the specimen

Benefits extreme technology Electronic and optical material (Si wafers) High performance turbine blades Abrasive materials (synthetic diamond)

Crystal properties reveal features of the atomic arrangement Anisotropic (directional) properties

Polycrystalline materials ‘Nuclei’ form during

solidification, each of which grows into crystals

Crystals grow and meet These grains are separated

by an amorphous grain boundary

Polycrystalline materials Most engineering materials are polycrystalline Collections of very small crystals (grains) Most metals, alloys and ceramics Isotropic properties (same properties in all directions)

Single- vs poly-crystalline• Single Crystals

-Properties vary with direction: anisotropic.

-Example: the modulus of elasticity (E) in BCC iron:

• Polycrystals

-Properties may/may not vary with direction.-If grains are randomly oriented: isotropic. (Epoly iron = 210 GPa)-If grains are textured, anisotropic.

E (diagonal) = 273 GPa

E (edge) = 125 GPa

200 mm

Amorphous materials

• atoms pack in periodic, 3D arrays• typical of:

Crystalline materials...

-metals-many ceramics-some polymers

• atoms have no periodic packing• occurs for:

Noncrystalline materials...

-complex structures-rapid cooling

crystalline SiO2

noncrystalline SiO2"Amorphous" = Noncrystalline

Summary Atoms may assemble into crystalline,

polycrystalline or amorphous structures Material properties generally vary with single

crystal orientation (i.e., they are anisotropic), but properties are generally non-directional (i.e., they are isotropic) in polycrystals with randomly oriented grains.

We will next look at how defects in these structures governs the most important mechanical properties of materials: strength and toughness