Structure of Materials II
Arrangement of atoms and probability density distribution
The solid state
The crystalline state
- Definitions and terms
- Space lattice and crystal systems
- Analytical description of the space lattice
Structures
- Single-element structures
- Alloys
- Ionic structures, Structure of inorganic glasses
- Macromolecular structures
Crystal defects
1
Alloys
Terms and definitions
Alloy
Binary alloy
Solution
Mixture
Component
System
Phase
is a compound of at least two elements from which one
must be a metal; predominant metallic bonding character.
is an alloy made of only two elements.
the compound is on atomic level.
the compound is only substantial.
a component is the pure chemical.
a system is a series of possible alloys consisting of the
same components.
a phase is the homogeneous part of a system that has
uniform physical and chemical characteristics. systems
which are of multiphase nature are heterogeneous.
phase is not equivalent to state of aggregation!
2
Alloys
Terms and definitions
A
phase boundary
Component A + Component B
B
A+B
physical mixture: two
phases, heterogeneous
solution: one phase,
homogeneous
non-miscible: [Fe/Pb, H20/Oil]
liquid: [Cu/Ni, H20/C2H5OH]
solid: [Cu/Ni]
3
Alloys
Homogeneous alloys
A. Solid solutions without a change of the crystal lattice
A1. Substitution
A2. Superstructure
A3. Interstitial
B. Intermetallic phases with a change of the crystal structure
B1. Interstitial
B2. HUME-ROTHERY phases
B3. LAVES Phases
Bx. ...
4
Alloys
Solid solution by substitution of elements
A host/solvent atom is replaced by an impurity/solute atom.
The second type of element/impurity-atom is placed on a regular lattice position.
Conditions for formation
1.
2.
3.
Atomic size factor
difference of atomic radii must be less than 15 %
Electrochemical factor there must be limited chemical affinity between
the two elements; otherwise there is formation of
an intermetallic phase Having the same crystal or chemical structure as another
Lattice factor
structures must be isotypic. elements must
crystallize in same type of lattice
5
Alloys
Solid solution by substitution of elements: Examples
System Lattice
Atomic radius
Ag
Au
fcc
fcc
1.44 Å
1.44 Å
Cu
Ni
fcc
fcc
1.28 Å
1.25 Å
Mo
W
bcc
bcc
1.36 Å
1.37 Å
6
Alloys
Solid solution
by substitution
of elements:
VEGARDs rule
10^-10 m
Characteristic for
solid solutions is
VEGARD’s rule:
lattice constants
change linear
with composition.
7
Alloys
Solid solution by substitution of elements
Practical aspect
The distribution of atoms often is not fully random. There is formation of
clusters, that is, spatially localized differences of concentration of elements.
These clusters cause a deformation of the lattice, and with that, properties
like an increased electrical or thermal resistance, or increased hardness.
8
Alloys
Solid solution by formation of superstructures
As a function of temperature & composition, formation of superstructures is possible.
The atoms arrange on preferred lattice points and cause formation of a super-lattice.
The properties of superstructures are different from that of random solutions.
Example 1: System Fe/Al
The bcc-lattice of -Fe can dissolve
more than 50% Al-atoms. Then all Alatoms may be located in the center
of the unit cell, whereas the Featoms are placed at the corners
(layer-like structure).
Fe/Al 50/50
9
Alloys
Solid solution by formation of superstructures
Example 2: System Cu/Au
Cu3Au is representing >200 alloys including Ni3Al, Ni3Fe, Ni3Cr, Ni3Pt
CuAu is representing TiAl, CoPt, FePt
Cu3 Au
Cu Au c/a = 0.932 (tetragonal)
10
Alloys
熄滅
quenched from 650°C: no superstructure
annealed at 200°C: superstructure
The thermodynamic stability
of superstructures is low
and these structures can
easily be destroyed by
thermal energy, or
mechanical deformation.
Cu Au
Superstructures preferably
form on slow cooling by
diffusion.
Cu3 Au
Specific electrical resistance (10-8  m)
解凍
Solid solution by
formation of
superstructures
Superstructures often are
brittle, and the electrical
resistance may be down.
Concentration Au
11
Alloys
Solid solution by interstitial
The second component/impurity-atom is placed at interstitial position.
Interstitial positions mainly are octahedral and tetrahedral sites.
Often occurs insertion of small non-metallic (NM) atoms in metals (M).
Condition: radius NM / radius M = 0.59 [holds often for C, H, N, B, Si and O]
The solubility is limited. Only a few percent of the non-metal can be dissolved.
The solubility, that is, how many impurity-atoms can be incorporated, is strongly
temperature-dependent since the space between host-atoms decreases on
temperature-reduction. Then the small atoms often remain dissolved, that is, there
is super-saturation, and the structure locally is strained.
Rodgers, G. E., Descriptive Inorganic,
Coordination, and Solid-State Chemistry.
3rd ed.; Brooks/Cole: Belmont, CA, p 624
12
Alloys
Solid solution
by interstitial
Example:
C in Fe (steel)
radius C
= 0.71 Å
radius Fe = 1.24 Å
superior importance for hardening of steel. mechanical/physical properties
change linear with percentage of second component.
note: in case of substitution-alloys is the change non-linear!
13
Alloys
Solid solution by intermetallic phases
Intermetallic phases are binary or polynary combinations of (metallic) elements at
fixed concentration, such that a defined formula can be assigned. It is homogeneous,
single-phase, and shows a new crystal structure, different from the crystal structures
of the pure elements.
Intermetallic phases exhibit a higher thermodynamic stability than superstructures,
are often extremely hard and brittle, and have a high melting temperature.
Intermetallic phases are labeled by a formula which shows the (average)
concentration of the different atoms.
Often: At interstitial position of the host-metal-lattice are placed small non-metals.
radius NM / radius M = 0.43 – 0.59 [ Examples: TiC, ZrN, VC, WC, Fe3C, Fe2N ]
燒合
粉末冶金
Are used as hard materials, produced by powder-metallurgy (sintering).
Application for cutting tools, end mill inserts and drills, mining tools, hot working tools
採礦工具
立鐵刀刀片和鑽頭
and rolls.
14
Alloys
Solid solution by intermetallic phases
Me
High-speed steel, coated
with TiN, Tm = 2950 °C
C,N
15
Ionic structures
Ionic structures are formed by
cation: positive charge, metal
anion: negative charge, non-metal
(often O, S, Se, F, Cl, Br, J, acidic residuals like SO42-, CO32-)
Classification of ionic structures
coordinative structures: Characterized by mutually penetrating lattices of cations
and anions. Large anions form a dense-packed lattice (fcc) with the octahedral and
tetrahedral sites filled with cations. Example: NaCl
complex structures: involving anion complexes. Example: Ca [CO3]
silicate structures: Characteristic is the sharing of anions by several cations. The
complexes [Bn Xp] are connected to larger units.
16
Ionic structures
octahedral sites: coordination number 6
an octahedron is formed
r cation / r anion = 0.732 ... 0.414
tetrahedral sites: coordination number 4
an tetrahedron is formed
r cation / r anion = 0.225 ... 0.414
Coordinative structures
17
Ionic structures
NaCl type
anion lattice is fcc
cations in all octahedral sites
r cation / r anion = 0.732 .. .0.414
coordination number 6
MgO, FeO, CuO, NiO
Coordinative structures
18
Ionic structures
Most important are silicate complexes [SiO4].
To achieve neutrality, these complexes tend forming larger
units, or even incorporating of further cations.
Si4+ O2-
The [SiO4] complex is a tetraeder and can exist in form of:
(a) islands
[SiO4]4(b) groups
[Si2O7]6(c) rings
[Si3O9]6- [Si4O12]8- [Si6O18]12(d) single/double chains
[Si2O6]4- [Si4O11]8(e) layers
[Si4O10]4(f) 3D - networks
[Si4O8] = [SiO2]
Silicate structures
19
Ionic structures
Silicate structures: Islands
Example: OLIVIN ( Mg, Fe )2 [SiO4]
The four free electrons at
the oxygen-atoms get
saturated by metal cations
most important part of
earth-crust (basalt)
20
Ionic structures
Silicate structures: Groups
21
Ionic structures
Silicate structures: Rings
Beryl Al2Be3 [Si6O18]
Massive beryl is a primary
ore of the metal beryllium.
22
Ionic structures
Silicate structures: Chains
double chains
Ca2Mg2 [Si8O22] (OH)2
single chains
Mg2 [Si2O6]
Ca Mg [Si2O6]
Example:
Wollastonite
(even used as
filler in polymers)
Example: Asbestos
By Lukasz Katlewa - Own work, CC BY 3.0,
https://commons.wikimedia.org/w/index.php
?curid=37259226
23
Ionic structures
Silicate structures: Layers
The free electrons are out of plane.
In tone minerals, the free O-electrons are
situated at the same side of the layer .
Saturation is by cations like Mg or Al, and
additional coordination of hydroxyl groups.
Examples: Kaolinite Al2 [Si2O5] (OH)4
(raw material for fabrication of porcelain)
制造
陶瓷
Mica KMg3 [AlSi3O10] (OH)2
(for capacitors, windows in high-T stoves)
MMT
24
Ionic structures
Silicate structures: 3D networks
SiO2 = Si4O8
Quartz, Tridymite, Cristobalite
Feldspar
replace Si 4+ by
Al 3+ and K+/Na+/Ca2+
K [Al Si3O8]
Na [Al Si3O8]
Ca [Al2 Si2O8]
[Orthoklas]
[Albit]
[Anorthit]
Zeolithe
molecular sieve
25
Ionic structures
Crystal
Glass
Ionic (Si-based) structures
are large molecules, which
need for formation of crystals
long time (or pressure). In
nature, these compounds are
often crystallized. The actual
use as material, however, is
in many cases as an
amorphous structure.
ZACHARIASEN and WARREN:
The structure of glassy silica (SiO2) is similar of that of crystalline quartz:
3D-asymmetric strained network of SiO44- tetrahedrons with the same
linkage between Si and O as in the crystalline state.
26
Ionic structures
Network theory by ZACHARIASEN and WARREN
Classification of elements into network formers, network
modifiers, and intermediates
Network/glass-formers: Si 4+, Ge4+, P5+, B3+ or Sb5+.
Network-modifiers are oxides of alkali and earth-alkali
elements: Na1+, Li 1+, Be1+, Ca2+
Due to the broken network, the tendency for crystallization
increases. Network-modifiers are less strong bound than
main network-forming cations, and are therefore more
mobile, e.g., in an electrical field. The viscosity, glass
transition temperature, and UV-transmission decrease.
27
Macromolecular structures
linear
branched
cross-linked
strong or weak
Lecture Prof. Binder/Prof. Kreßler: Introduction to macromolecules
Lecture Prof. Saalwächter: Mathematical and theoretical concepts of polymer science
28
Crystal defects
The perfect three-dimensional crystal does not exist.
A crystal of 1 cm3 contains 1023 atoms.
A crystalline material of 99.999% purity still contains 1018
impurity-atoms per 1 cm3.
These impurities and other types of defects largely control
the properties of materials and must therefore described
qualitatively and quantitatively.
A crystal-defect is any lattice irregularity having one
or more of its dimensions on the order of an atomic
diameter.
29
Crystal defects
Classification
Crystal defects can be classified according to their geometry:
zero-dimensional (O-D) or point-defects
phonons, vacancies, interstitial, substitution atoms
one-dimensional (1-D) or line-defects
dislocations
two-dimensional (2-D) or planar defects
stacking faults, interfaces, surfaces, grain boundaries
[three-dimensional (3-D) or volume defects]
agglomeration/segregation of vacancies
聚集
30
Crystal defects
Zero-dimensional defects
Phonons = oscillation of lattice points due to non-zero temperature.
Temperature is a measure of the average vibrational activity;
at 298 K the typical vibrational frequency and amplitude are
about 1013 vibrations/s and few thousands of a nm, respectively.
Phonons are related to the specific heat capacity.
A phonon is the quantum mechanical description of an elementary vibrational motion in which
a lattice of atoms or molecules uniformly oscillates at a single frequency.
31
Crystal defects
Zero-dimensional defects
Vacancies (SCHOTTKY defect)
An atom is missing in the lattice.
A field of local strain and stress
develops since neighbored atoms
try to fill the space. Vacancies are in
equilibrium with temperature.
Vacancies cause an increase of
internal energy U of crystal.
32
Crystal defects
Zero-dimensional defects
Vacancies (SCHOTTKY defect)
n/N = exp [ - U / (RT) ]
n/N = number-of-vacancies per
number-of-lattice-points
U =
activation energy required for
the formation of the vacancy
(in J/mol)
in metals U = 80–200 J/mol
R =
T =
gas constant (J/(mol K))
absolute Temperature (K)
n/N (Tm) = 10-4, one lattice site out of
10.000 is empty
33
Crystal defects
Zero-dimensional defects
linearization: plotting ln(n/N) vs. 1/T
The concentration on vacancies is
important for diffusion processes since
diffusion (of atoms) is faster in presence of
vacancies. Furthermore is observed an
effect on the mechanical deformation
behavior (motion of dislocations).
n/N = exp [ - U / (RT) ]
Tm
ln (n/N)
Due to restricted diffusion, there is a
deviation of experimental data observed
on cooling from the equilibrium
concentration. It is not possible to create a
crystal with low vacancy-concentration
just by cooling to low temperature. On the
other side it is possible to freeze a higher
concentration on vacancies by quenching.
experiment
theory
1/T
34
Crystal defects
Zero-dimensional defects
Self-interstitial (FRENKEL defect)
An atom is moved from a regular
position to an interstitial site, which
causes relative large distortions and
local strain/stress in the surrounding
lattice (atom is substantially larger
than interstitial position).
Thus, the energy for formation of such
a defect is higher than in case of
vacancies, and, consequently, the
number of self-interstitial defects is
much less than in case of vacancies.
35
Crystal defects
Zero-dimensional defects in ion structures
+
X
X
FRENKEL-defect
vacancy in the cation-lattice and
cation-interstitial in equal concentration
Anti-FRENKEL-defect
vacancy in the anion-lattice and
anion-interstitial in equal concentration
36
Crystal defects
Zero-dimensional defects in ion structures
+
-
X
X
SCHOTTKY-defect
Anti-SCHOTTKY-defect
vacancy in the cation-lattice and
anion-vacancy in equal concentration
anion-interstitial and
cation-interstitial in equal concentration
37
Crystal defects
1D-defects
1D, or line-defects, are dislocations.
Dislocations develop on solidification/crystallization, and during plastic
deformation. Atoms/atomic layers are dislocated from their original position
in the crystal.
There are two types of dislocations:
edge dislocation
screw dislocation
38
Crystal defects
1D-defects: Edge Dislocation
An atomic layer ends within the crystal.
It can also be considered as an additional
inserted lattice plane.
The dislocation line is perpendicular to
the plane of drawing.
The dislocation line is the center of the
lattice-distortion. It must end at a surface,
or being closed (circular).
39
Crystal defects
1D-defects: Screw Dislocation
dislocation line
40
Crystal defects
2D-defects
Two-dimensional, planar defects are
stacking faults
boundaries
crystal boundaries
grain boundaries
phase boundaries
41
Crystal defects
2D-defects: Stacking faults
hexagonal close-packed
The stacking sequence of lattice planes in fcc and hcp lattices is changed
fcc
... ABCABCABC ...
[111]
hdp
... ABABABABA...
[001]
Stacking faults develop by diffusion of vacancies, or during crystallization.
C
B
A
C
B
A
x
C
B
A
x
xxx x
x
x
B
A
C
B
A
C
B
A
42
Crystal defects
2D-defects: Grain boundaries
Grain is synonymous for crystal in (metallic) polycrystalline structures.
A grain boundary is the borderline between two or more tightly connected
crystals of different crystallographic orientation.
c
c
c
c
100-103 µm
43
Crystal defects
2D-defects: Grain boundaries
Nucleation
Growth
The grain structure (size,
shape, orientation) is the
result of the crystallization
process.
Growth
Final Grain Structure
The different orientation of the
grains causes quasi-isotropy,
that is, properties do not
depend on direction anymore.
Note: A single crystal behaves
anisotropic (crystal definition!)
Quasi-isotropic means a material having isotropic properties, but only in-plane.
The strength and stiffness are equal in all directions within the plane of the part.
Formation of grains in a polycrystalline material
44
Crystal defects
2D-defects: Grain boundaries
aluminum oxide ceramics
copper
45