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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