Materials Science
Joachim Rösler, Sebastian Piegert,
Britta Laux, Michaela Necker
July 18, 2024
This script written by me, Joachim Rösler, assisted by Sebastian Piegert, Britta Laux
und Michaela Necker, is subject to German copyright law. It may only be used for
personal use for the purpose of studying. In particular, it is not allowed to edit, translate,
or copy the script or parts of it and to pass it on to other persons, neither as a copy
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Contents
1 Introduction
4
2 Elastic stiness, atomic bonding and material structure
10
1.1
1.2
2.1
2.2
2.3
Importance of materials science in mechanical engineering . . . . . . . . .
Material classes, prices and availability . . . . . . . . . . . . . . . . . . . .
Phenomenological description of elastic deformation . . . . . . . . . . . .
2.1.1 Nominal stress, nominal strain, elastic constants . . . . . . . . . .
2.1.2 True stress and strain . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.3 The tensile test . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The atomic bond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 The ionic bond . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.3 The covalent bond . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.4 The metallic bonding . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.5 Secondary Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The structure of the materials . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.2 Ceramics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.3 Polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Structure of the polymer groups . . . . . . . . . . . . . . . . . . .
Examples of important polymers . . . . . . . . . . . . . . . . . . .
3 Strength and phase diagrams
3.1
3.2
3.3
3.4
The ideal strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Dislocations in crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Forces between dislocations . . . . . . . . . . . . . . . . . . . . . .
Measures to increase strength . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 Work hardening . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.2 Strengthening by grain size reduction . . . . . . . . . . . . . . . .
3.3.3 Solid solution strengthening . . . . . . . . . . . . . . . . . . . . . .
3.3.4 Particle strengthening . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.5 Quenched and tempered steels . . . . . . . . . . . . . . . . . . . .
Phase diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.1 Solubility, states of aggregation, Gibbs phase rule . . . . . . . . . .
3.4.2 Binary systems and microstructure formation . . . . . . . . . . . .
3.4.3 The Iron-Carbon Diagram . . . . . . . . . . . . . . . . . . . . . . .
2
4
4
10
10
14
15
18
18
20
22
27
29
31
31
36
39
39
54
62
62
64
68
69
69
70
72
74
83
87
87
89
97
Contents
4 Processes at High Temperatures
104
5 Oxidation and corrosion
122
Bibliography
137
4.1
4.2
4.3
4.4
5.1
5.2
Vacancies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
Diusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
Recrystallization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
Recovery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
Oxidation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
Corrosion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
5.2.1 Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
5.2.2 Kinetics of corrosion . . . . . . . . . . . . . . . . . . . . . . . . . . 127
5.2.3 Corrosion protection measures . . . . . . . . . . . . . . . . . . . . 129
5.2.4 Degradation of polymers . . . . . . . . . . . . . . . . . . . . . . . . 133
3
1 Introduction
1.1 Importance of materials science in mechanical engineering
During the design and construction of components, an ever increasing number of possible
materials are available to the mechanical engineer. While at the end of the nineteenth
century there were still several hundred, today there are more than one hundred thousand. For example, synthetically produced polymers, technical ceramics, titanium alloys,
high-temperature resistant nickel alloys or modern composite materials have essentially
been developed over the past fty years and new material variants are constantly being
added. Accordingly, there is great potential for improving products by selecting the right
material, reducing costs or enabling previously unimaginable properties. For example,
eyeglass frames can now be made from high-strength titanium alloys or so-called shape
memory alloys, which do not break or bend permanently even in the worst cases of misuse. On the other hand, wrong decisions can also have catastrophic consequences. One
example is the so-called Liberty ships built during the Second World War. Due to the
use of an unsuitable welding material with too low fracture toughness, the ships broke
apart at sea along the welded joints.
The aim of teaching materials in mechanical engineering is to impart the necessary
knowledge for proper material selection in the design and construction of systems and
assemblies. The lecture Materials Science gives a rst overview of the dierent material
classes and their most important properties.
1.2 Material classes, prices and availability
Materials can be divided into three large groups, each with typical properties: metals,
ceramics and polymers.
Signicant examples of metals are steels (i.e. ferrous materials containing carbon
as an additional element within certain limits), the especially light metals (aluminium,
magnesium and titanium), nickel and copper alloys. As shown in Tab. 1.1, they are
characterized by high strength and good formability. This good formability - often
called ductility - leads to good processability of metals. For example large steel bars
may be rolled into thin sheets and then deep drawn to car body parts with complex
nal shapes. In addition, metals have good electrical and thermal conductivity. This
is especially the case for copper, aluminium, silver and gold. Therefore, these materials
can be used, for example, as conductor paths in the semiconductor industry. However,
metals are usually chemically unstable and have a relatively high density.
With a global annual production of about 1.6 billion tons, steels are by far the most
important material group in mechanical engineering. They can be found in all areas of
mechanical engineering. In comparison, several tens of millions of tons of aluminum and
4
1 Introduction
group elements
main
1
IA
H
VIIIA
2
hcp
3
IIA
4
bcc
11
hcp
12
bcc
19
hcp
20
IIIB IVB VB VIB VIIB
21
22
23
24
25
26
VIIIB
27
28
bcc
37
fcc
38
hcp
39
hcp
40
bcc
41
fcc
42
cub
43
bcc
44
hcp
45
bcc
55
fcc
56
hcp
57
hcp
72
bcc
73
bcc
74
hcp
75
hcp
76
bcc
87
bcc
88
hcp
89
hcp
104
bcc
bcc
hcp
hcp
(bcc) krz
fcc
Li
Be
Na Mg
K
Rb
Cs
Fr
Ca
Sr
Ba
Ra
VIA VIIA hcp
8
9
10
tet
13
dia
14
hcp
15
cub
16
mon
17
fcc
18
IB
29
IIB fcc
30
31
dia
32
cub
33
ort
34
ort
35
fcc
36
fcc
46
fcc
47
hcp
48
ort
49
dia
50
rho
51
hcp
52
ort
53
fcc
54
fcc
77
fcc
78
fcc
79
hcp
80
tet
81
dia
82
rho
83
hcp
84
ort
85
fcc
86
fcc
fcc
fcc
rho
hcp
fcc
rho
cub
transition metals
Sc
Y
La
Ac
Ti
Zr
Hf
V
Cr
Nb Mo
Ta
W
Mn
Tc
Re
Fe
Ru
Os
B
Co
Ni
Rh
Pd
Ir
Pt
Cu
Ag
Au
Al
Zn
Cd
Hg
Ga
In
Tl
C
Si
Ge
Sn
Pb
N
P
As
Sb
Bi
O
S
Se
Te
Po
F
Cl
Br
I
At
Ne
Ar
Kr
Xe
Rn
(fcc)
Rf
metal
semi-metal
hcp hexagonal close-packed
fcc
He
IIIA IVA VA
5
6
7
face-centered cubic
nonmetal
bcc body-centered cubic
kub cubic
ort orthorhombic
tet
rho rhombohedral
dia diamond lattice
tetragonal
Figure 1.1: Periodic table of the elements excluding lanthanides atomic numbers 58 to 71) and
actinides (atomic numbers 90 to 103). The crystal structures will be explained in
section 2.3. Semi-metals have bonds of a mixed covalent-metallic type. Some materials exhibit dierent crystal structures depending on the temperature [26].
a little over hundred thousand tons of titanium are produced each year.
Taking the high conductivity as an example of a typical characteristic of metals, the
elements of the periodic table can be easily divided into metals and nonmetals (Fig. 1.1).
Apparently most elements belong to the metals, being located on the left side of the
periodic table. However, many metals have not achieved any signicance in mechanical
engineering, e.g. because, like mercury, they are already liquid at ambient temperature,
toxic (e.g. beryllium) or extremely expensive (e.g. gold or platinum).
Important representatives of ceramics include aluminium oxide (Al2 O3 ), silicon nitride
(Si3 N4 ), silicon carbide (SiC) and carbon, which is used in particular in the form of
carbon bres and hard coatings (diamond coatings). There are also a number of mixed
oxides which are of great technical importance. These include glass, which consists
mainly of silicon oxide (SiO2 ) but also contains sodium oxide (Na2 O) and calcium oxide
(CaO), as well as cement, which consists mainly of calcium oxide, silicon oxide and
aluminium oxide.
Ceramics are extremely hard, wear- and heat-resistant. For this reason they are used,
for instance, to make cutting tools for metalworking. However, they are also brittle
(the opposite of ductile). They are electrically isolating or semiconducting and, with a
few exceptions, have low thermal conductivity. In addition, they have good chemical
resistance. As can be seen from the examples, ceramics are mostly inorganic compounds
between a metal (e.g. Al) and a non-metal (e.g. O). As already mentioned the carbon
materials are also counted as ceramics.
5
1 Introduction
Polymers are synthetically produced organic materials (i.e. materials composed of
compounds of carbon with other elements). They consist of long chain molecules that
are woven together like a ball of wool threads. With a few exceptions, there are only
carbon atoms along the threads. In addition, one or two side groups hang on each
carbon atom. In the simplest case, polyethylene, the side groups consist only of hydrogen
(H). However, the side groups can also contain other atoms, such as chlorine (Cl), or
consist of molecules, such as CH3 . Other examples of technically important polymers
are polycarbonate, from which compact discs or the glazing of automobile headlights
are made, and polyamides, from which radiator fans for automobiles are made. Because
polytetrauoroethylene (PTFE) has a very low coecient of friction, it is used, for
example, to make plain bearings. The Teon coating of the frying pan also consists of
this material, whereby the term Teon is only a trade name for PTFE.
Polymers are characterized by a low density (polyethylene is even slightly lighter than
water). They are generally insulating both electrically and thermally, have good chemical
resistance and are partly optically transparent. In addition, they are very easy to shape
and to process, which is why it is particularly easy to produce components with complex
geometries from them. However, their strength is limited and they are very exible.
This is a disadvantage when a particularly rigid structure, such as an automobile body,
is required. However, for an snap closure this exibility is advantageous. In addition,
their thermal durability is low because they melt early or decompose through reaction
with atmospheric oxygen.
Figure 1.2: Classication of materials adapted from [1].
A material does not have to consist of only one representative of the material groups
mentioned above. Often particularly advantageous properties are achieved in so-called
composite materials, which are composed of more than one representative (Fig. 1.2 ).
6
1 Introduction
The following are examples:
ˆ glass/carbon ber-reinforced polymers (e.g. tennis rackets), which are characterized by high stiness and low weight,
ˆ WC-Co "cemented carbides" which are characterised by high hardness and sucient toughness, polymer-steel composites (e.g. car tires) and
ˆ reinforced concrete.
Table 1.1: Material classes, typical representatives and their most important characteristics.
material class
typical representative
metals
iron, steel
high thermal and
titanium alloys
electrical conductivity,
ceramics
polymers
characteristics
magnesium alloys
high tensile strength,
nickel alloys
good formability,
copper alloys
mostly chemically unstable
aluminium oxide (Al2 O3 )
isolating,
silicon carbide (SiC)
high compression strength,
silicon nitride (Si3 N4 )
brittle,
cement (CaO-Sio2 )
generally hard to process,
glass
chemically resistant,
diamond
high melting point
polyethylene (PE)
mostly isolating,
polymethylmetacrylat
low specic weight,
(Plexiglas or PMMA)
chemically resistant,
polypropylen (PP)
gut verarbeitbar,
polyamide (PA)
low thermal stability,
rubber
(melting, decomposition)
An important criterion for the selection of a material is usually the price. For example,
assuming a price of 10,000 e and a weight of 1,000 kg for a small car, a material with
a price of 10 e/kg would already represent 100 % of the estimated costs. In reality, a
material for the automotive industry may cost a maximum of several e/kg in order to
be able to manufacture a reasonably competitive product.
The costs of important structural materials are shown in Fig. 1.3. If we compare the
prices between steel and aluminium, it is clear, why aluminium bodies are currently
oered in the top market segment exclusively. As expected, the most common materials
(concrete, steel, wood, polyethylene and polypropylene) belong to the lowest price segment. Assuming the costs of iron and nickel would be reversed our environment would
be predominately determined by nickel constructions instead of steel constructions.
The costs of a material are partly determined by its availability (other factors: mining
costs, energy costs, processing costs). Tab. 1.2 summarizes the materials most commonly
present in the terrestrial crust.
7
1 Introduction
Figure 1.3: Approximate cost of important materials.
It is obvious that the components for ceramic materials (silicon, oxygen, nitrogen,
carbon) and polymers (carbon, hydrogen) are present on a large scale. The same refers
to the metals aluminium, iron, magnesium and titanium. However, other metals such as
cobalt, copper or rhenium occur in far lower amounts and may be considered strategically
important.
Table 1.2: Occurrence of the elements in the terrestrial crust in weight percentages.
element
occurrence / wt.-%
oxygen
47
silicon
27
aluminium
8
iron
5
calcium
4
sodium
3
potassium
3
magnesium
2
titanium
0,4
hydrogen
0,1
carbon
0,02
Especially helpful for material selection are material diagrams in which the target
values for dierent materials are plotted. As an example, if good electrical conductivity
combined with low material costs are required for a material of a power line, copper and
aluminium are possible choices according to 1.4.
8
1 Introduction
Figure 1.4: Ashby diagram: price vs. specic electrical resistivity. Created with help of the
Cambridge Engineering Selector Edu Pack 2008.
9
2 Elastic stiness, atomic bonding and material structure
If you bend a wire or a paper clip and the stress is low, any deformation caused by
the load is completely reversed when the load is released. The material reacts like a
spring which returns to its original shape when the load is released. The deformation is
therefore reversible and is referred to as elastic deformation. On the other hand, if the
load is increased above a critical value, the wire (or paper clip) will not fully return to
its original shape. A part of the imposed deformation remains. Those deformations are
called irreversible or plastic deformation. The example shows that plastic deformation
is always combined with elastic deformation (the wire always springs back a little when
released), whereas elastic elongation can also occur alone. This must be considered, for
example, if one wants to produce a body sheet of a car by deep drawing, because the
nal contour of the sheet does not correspond exactly to the contour of the forming tool
due to elastic spring back when released. This is taken into consideration in practice by
designing the forming tool suitably.
In Chapter 2 we initially focus on elastic deformation. Chapter 2.1 is about describing
and calculating elastic deformation. This will lead to an important material parameter,
the Young's modulus. It indicates whether a material is particularly sti or particularly
exible. This material parameter, which is particularly important for designing structural elements, is determined experimentally by the so-called tensile test, as discussed in
Chapter 2.1.3. This is followed by the discussion of the reasons for elastic deformation
and the question why polymers can be elastically deformed much easier than metals and
ceramics. To understand this, we have to study atomic bonding (Chapter 2.2) and the
structure of materials (Chapter 2.3).
2.1 Phenomenological description of elastic deformation
2.1.1 Nominal stress, nominal strain, elastic constants
The elastic stiness of a spring is determined by the spring constant, which means a
linear relationship exists between load F and elongation ∆l:
D=
D
F
∆l
F
.
∆l
(2.1)
spring constant
load
elongation
As explained above, the term elastic means that the elongation is reversible when the
load is released. Likewise, the Young's modulus E is dened as a measure of the elastic
10
2 Elastic stiness, atomic bonding and material structure
Figure 2.1: Illustration of the elastic stiness using a spring model.
stiness of materials. If imagining the material as a three-dimensional arrangement
of springs (Fig. 2.1), it is obvious that the elongation at a certain force is inversely
proportional to the cross-sectional area S0 of the material. In order to eliminate this
geometric dependence, the term normal stress is introduced:
σ=
F
,
S0
σ
S0
(2.2)
normal stress
initial cross section
where F is the applied load oriented perpendicular (i.e. normal) to the surface (Fig. 2.2(a)).
The unit of the stress σ is 1 N/m2 = 1 PASCAL (Pa). It is often given in Megapascal
(MPa) or Gigapascal (GPa), where 1 MPa = 106 Pa and 1 GPa = 109 Pa. Obviously, the
elongation ∆l of the material is proportional to the length l0 in the direction of the load.
Therefore a further normalized variable, strain, is introduced:
ε=
l − l0
∆l
=
.
l0
l0
ε
l
l0
(2.3)
strain
length in loaded condition
initial length (unloaded)
Under a normal load, which can be both a tension and a compression load, the socalled Hooke's law results in analogy to Equation 2.1:
(2.4)
σ =E·ε ,
E
Young's modulus
in which the Young's modulus E is a material-dependent parameter. Since the strain ε
is a dimensionless value, the Young's modulus, like the stress, has the unit N/m2 or Pa.
11
2 Elastic stiness, atomic bonding and material structure
∆x
S0
l0
F
S
F
γ
y
∆l
(a) normal load
(b) shear load.
Figure 2.2: Basic load situations, adepted from [26].
If a material is extended as shown in Fig. 2.2 (a), the atomic distances are elongated
and the atoms are moved out of their equilibrium distance r0 . As a result, the equilibrium distance decreases slightly in the transverse direction. This means that the atoms
feel more comfortable to move closer in the transverse direction. The elongation in
the longitudinal direction imposed by the stress σ thus results in a contraction in the
transverse direction. The so-called Poisson's ratio indicates the ratio between transverse
strain and longitudinal strain:
ν=−
εtrans
.
εlong
ν
εtrans
εlong
(2.5)
Poisson's ratio
transverse strain
longitudinal strain (strain)
Since εtrans und εlong always have dierent algebraic signs (+-), the minus sign is
inserted so that ν is always positive. The Poisson ratio is another material constant
characterizing the binding characteristics. Tab. 2.1 shows some typical values for the
Poisson's ratio.
Table 2.1: Poisson ratio for selected metals, ceramics and polymers.
Material
Poisson's ratio ν
Al
0,34
α-Fe
0,29
Ni
0,31
Al2 O3
SiO2
0,23
0,20
Polystyrene
0,33
Polyethylene
0,40
As a rule of thumb you can remember:
ν Metal ≈ 0.3,
ν Ceramics ≈ 0.2,
ν Polymers ≈ 0.4.
12
2 Elastic stiness, atomic bonding and material structure
For ν < 0.5 tensile loading leads to an increase in volume and compressive loading to a
decrease in volume.
In a similar way to the normal load, where the stress vector is oriented perpendicular
to the loaded surface, a relationship for a shear load can be established. In this case, the
applied force F is parallel to the surface under consideration S (see Fig. 2.2(b)). The
resulting shear stress is therefore dened as follows:
τ=
F
.
S
(2.6)
In contrast to a normal load, a shear load causes no deformation of the surface under
consideration S . The surface S remains unchanged and no transverse contraction occurs.
The shear stress τ is related to the shear strain γ via the shear modulus G:
(2.7)
τ =G·γ .
shear stress
shear modulus
shear strain
τ
G
γ
γ corresponds to the angle change of an initial right angle (Fig. 2.2 (b)). For small angle
changes tan γ is approximately γ and therefore:
γ=
∆x
.
y
(2.8)
The shear modulus and the Young's modulus are related by the Poisson's ratio ν :
G=
E
.
2(1 + ν)
(2.9)
This means that the elastic behaviour of an isotropic material is determined by two independent variables. A material is dened as isotropic if its properties are not dependent
on the direction of loading, i.e. the same Young's modulus is measured in all directions
of loading. This is usually the case for metals, ceramics and polymers with sucient accuracy. Many composite materials, such as ber-reinforced polymers and woods, on the
other hand, exhibit a pronounced directional dependence of elastic stiness. They are
anisotropic and the description of the elastic behavior becomes much more complicated.
By plotting the Young's modulus E for typical structural materials (Fig. 2.3), a remarkable range can be observed. Obviously there is a dierence of up to ve orders of
magnitude between the most elastically sti material (diamond) and the most exible
materials (elastomers). If a polypropylene rod (E = 1000 MPa) with a length of 100 mm
and a cross-sectional area of 10 mm2 were extended by 1 mm under a load of 100 N, this
would only be 1 µm for a diamond rod.
Furthermore, a clear demarcation between the material classes can be seen. Ceramics
and metals are obviously sti, whereas polymers are elastically compliant. These very
13
2 Elastic stiness, atomic bonding and material structure
Figure 2.3: Young's modulus for selected structural materials, adapted from [1].
large dierences are related to the dierent atomic structures of ceramics, metals and
polymers. In order to gain a deeper understanding of the nature of the elastic behaviour
of materials, the structure of matter is discussed in Chapter 2.2.
2.1.2 True stress and strain
During normal loading, the transverse contraction causes the cross sectional area of the
rod shown in Fig. 2.2 (a) to change with the load. Under tensile loading, the actual
cross-sectional area S becomes smaller than the cross-sectional area in the initial state
S0 . In case of compressive stress it is exactly the other way around. Strictly taken
this should be considered when calculating the existing normal stress. However, if the
elongation is small, as is usually the case with elastic loading, the change in cross-section
is also small, so that the failure is negligible. Therefore, it is quite possible to work with
eq. 2.2 and the calculated stress is called nominal stress.
However, if the normal stress is calculated by using the actual cross-sectional area S
it is called the true stress σw and the following applies:
σw =
F
.
S
(2.10)
The situation is similar for the strain. In eq. 2.3, the elongation is related to the initial
length l0 and the strain ε thus dened is called nominal strain. It would be more accurate
to imagine the elongation ∆l divided into many small increments dl and to relate this
14
2 Elastic stiness, atomic bonding and material structure
incremental change in length to the current length l. The true strain change dφ occurring
with each incremental length change is therefore given by:
dφ =
dl
.
l
(2.11)
In order to calculate the total strain φ from the initial length l0 to the nal length l1 ,
the strain increment dφ must be summed up by integration:
Zl1
φ=
dl
l1
= ln = ln
l
l0
l1 − l0 + l0
l0
= ln (1 + ε) .
(2.12)
l0
Fig. 2.4 shows the nominal and true strain as a function of the normalized elongation
∆l/l0 . It can be seen that the dierence for small elongations is minimal, so that the
nominal strain ε is a good approximation for the true strain φ. In general, the true
strain for positive strains (tensile range) is less than the nominal strain, for negative
strains (compression range) its value is greater than the nominal strain.
ε, ϕ
1,0
0,5
−1,0
−0,5
0,0
0,0
0,5
−0,5
1,0
∆l/l0
ε
ϕ
−1,0
Figure 2.4: Comparison of nominal and true strain [26].
Under shear stress, the surface on which the shear stress τ occurs does not change and
no deformations transverse to the direction of stress occur. Therefore the distinction
between nominal and true values is not necessary.
2.1.3 The tensile test
Material characteristics, such as the Young's modulus E , must be determined experimentally. By far the most important test to determine such characteristic values and to
describe the deformation behaviour of a material is the so-called tensile test according
to DIN EN ISO 6892-1.
In tensile tests, specimens are clamped in a testing machine and pulled apart at a
constant speed (Fig. 2.5). The prevailing force is measured using a load cell mounted
15
2 Elastic stiness, atomic bonding and material structure
Figure 2.5: Strain measurement in tensile test [26].
with the load rod. The specimen strain is determined with the use of a displacement
transducer. The sensor consists of two measuring blades whose distance determines the
initial length l0 . These measuring blades are pulled apart with the specimen and their
extension ∆l is determined with the use of a measuring bridge. For a known crosssectional area S0 of the specimen, all information is available to display the result of
the tensile test in the so-called stress-strain diagram. The value pairs for stress σ and
strain ε measured at each point are entered into the diagram and the stress-strain curve
which is characteristic for the respective material is obtained. In order to understand
the stress-strain curves, it is important that the nominal stresses and strains are plotted.
Fig. 2.6 shows typical stress-strain curves for two ceramics (Al2 O3 and Si3 N4 ), two
metals (the structural steel S 355 and a so-called stainless steel with the designation
X 5 CrNi 18 9) and two polymers (Polymethylene methacrylate and Polyethylene). It
can clearly be seen that there is a linear relationship between stress and strain at low
loads. In this area the material behaves elastically and obeys Hooke's law according to
Equation 2.4. In the case of ceramics, this relationship persists until fracture. Thus,
they only deform elastically.
The other materials have a curve above a critical stress that deviates from the elastic
straight line due to plastic deformation in addition to elastic deformation (Fig. 2.7).
This critical stress is known as the Yield Stress Rp . It is important that the elastic
strain εel can also be determined for a stress σ > Rp using Hooke's law. If the total
strain εges is also determined, the plastic strain εpl can be calculated from the dierence
between εges and εel . If the specimen is unloaded before it ruptures, the total strain does
not remain constant. Instead, it decreases with the elastic strain. During unloading,
the stress-strain curve thus forms a straight line with the slope dened by the Young's
modulus (Fig. 2.7). The maximum stress the material can withstand is known as the
tensile strength Rm . As the examples show, the material does not necessarily have
to fracture when reaching Rm . In fact, ductile metals generally deform further before
fracture occurs. From this point on, however, the deformation is no longer uniformly
16
2 Elastic stiness, atomic bonding and material structure
600
σ / MPa
Si3N4
Si3N4
500
400
X5 CrNi18-9
S 355
300
200
Al2O3
Al2O3
100
Poly(methyl methacrylate)
0
0,0 0,002 0,0 0,1 0,2 0,3 0,4 0,5 0,6
Polyethylene
3,0
3,5
ε/ −
Figure 2.6: Stress-strain curves of dierent materials, in the left part a detail view. Al2 O3 and
Si3 N4 are ceramics. S 355 is a structural steel, X 5 CrNi 18 9 is an austenitic steel.
Poly(methyl methacrylate) (PMMA) and Polyethylene (PE) are polymers. [26].
distributed over the specimen volume, but the material begins to contract at a certain
point. This results in a strong local contraction of the cross-section and is the reason
why the nominal stress decreases. If, on the other hand, the true stress were plotted, it
would rise until fracture.
σ
Rm
R p0,2
0,2%
E
εel
εpl
εges
Ag
A
ε
Figure 2.7: Schematic stress-strain diagram of a ductile material, important characteristic parameters: Rm : tensile strength, Rp0,2 : Yield stress, (Ag : uniform strain), A: fracture strain, E : Young's modulus, adapted from [26].
In practice, there is a problem in determining the yield strength Rp : because plastic
deformation slowly sets in with increasing stress, it is not possible to determine the exact
17
2 Elastic stiness, atomic bonding and material structure
beginning of plastic deformation. This is solved by a pragmatic approach by drawing a
parallel to the elastic line which is shifted by 0.002 (i.e. 0.2 %) to larger strain values
(Fig. 2.7). At the point of intersection between this parallel shifted linear line and the
stress-strain curve, the corresponding stress value is read o and referred to as the yield
strength Rp0,2 . The numerical value i.o. thus indicates how the yield strength was
determined.
σ / MPa
Rm
fracture
Rp0,2
σ / MPa
necking
Rm
R eH
R eL
fracture
necking
E
E
A
A
ε / %
ε / %
(a) Ductile material without apparent yield
poin
(b) Ductile material with apparent yield point
Figure 2.8: Schematic stress-strain curves, adapted from [26] (see also DIN EN ISO 6892-1).
Many steels, such as the structural steel S 355 shown in Fig. 2.6, show a specic
property when the critical stress is reached, from which on plastic deformation occurs:
the stress suddenly decreases and stabilises at a lower level before increasing again. In
this case we do not speak of the yield strength Rp , but - as shown in Fig. 2.8 (b) - of
the upper yield strength ReH and the lower yield strength ReL .
The tensile test is generally terminated by the fracture of the specimen. The remaining strain after fracture is referred
√ to as elongation after fracture A or Ak , where k is
calculated from the ratio k = l0 / S0 . If no value is given for k , a ratio of 5.65 is used
according to DIN EN ISO 6892-1.
2.2 The atomic bond
We return to the question of why the Young's Modulus of polymers is by far lower than
that of ceramics and metals. To address this, we have to consider the atomic bonds
between the atoms, which can be quite dierent in the various materials. The occuring
bonding types include the ionic bond, the covalent bond, the metallic bond and the
secondary bond. As we will see, the type of bonding determines quite substantially the
elastic stiness of the material and also other properties - e.g. electrical conductivity .
Therefore, the individual types of bonds will be discussed in the following.
2.2.1 Basics
Atoms consist of a positively charged atomic nucleus surrounded by negatively charged
electrons. The electrons of an atom cannot arrange themselves in an arbitrary way
18
2 Elastic stiness, atomic bonding and material structure
around the atomic nucleus. Rather, they are located on so-called electron shells, which
are arranged at an increasing distance from the atomic nucleus and can each only hold
a limited number of electrons. The further away an electron shell is from the atomic
nucleus, the greater the energy of the electrons in this shell, so that electrons on the
outer shells are bound to the nucleus weaker than those on the inner shells. Only these
outer, energy-rich electrons participate in the chemical bond, because only they feel
their environment and can possibly form bonds with other atoms.
Electrons cannot be localized at a certain position. One can only state a certain
probability that an electron is at a certain position. So there are places in the vicinity of
the atomic nucleus where an electron prefers to be, while it avoids other places. The area
where the electron can be found is called the electron cloud or orbital. Fig. 2.9 shows
some examples of electron orbitals. It can be seen that there are spherical orbitals as
well as directional orbitals. An electron shell is generally composed of dierent orbitals.
Each of these orbitals can take a maximum of two electrons.
z
z
y
z
y
z
y
x
x
(a) s - orbital
y
x
x
(b) p - orbitals.
z
z
y
z
y
x
x
z
y
y
x
x
(c) d - orbitals.
Figure 2.9: Schematic representation of some electron orbitals, adapted from [26].
The basic structure of the electron shell is the same for all atoms. The innermost
electron shell of all atoms, called the K-shell, can hold a maximum of two electrons,
since it has only one spherically symmetric orbital (the s-orbital). The next shell, the
L-shell, holds up to eight electrons, two of which occupy a spherically symmetric sorbital, while the other six take up the three p-orbitals. Since nature strives for low
energy states, the electron shells of an atom are always lled from inside to outside
according to the energy scheme shown in Fig. 2.10. In the periodic table, the elements
where the rst electron shell is lled (H and He) are found in the rst period. The eight
elements (Li, Be, B, C, N, O, F, Ne) where the second shell is lled, in the second period
and so on. Energetically particularly favourable are states in which the electron shells
are fully occupied. This is the case with noble gases, such as helium, neon or argon.
Therefore, they are essentially chemically inert. Fluorine, on the other hand, is very
19
2 Elastic stiness, atomic bonding and material structure
reactive, because it is anxious to capture an additional electron and, by that, also to
form a completely lled electron shell.
6p
6s
5p
5s
4p
energy
4s
5d
4f
4d
3d
3p
3s
2p
2s
1s
Figure 2.10: Energy levels of electron orbitals in the periodic table.
In principle, a bond between atoms takes place when the electron arrangement is
energetically more favourable in the bound state than in the unbound state. There are
two important principles leading to a low energy electron arrangement:
1. The above mentioned attempt to achieve completely lled electron shells.
2. The attempt of the electrons to occupy a space as large as possible. Ultimately
this is responsible for the fact that the electrons do not fall into the nucleus despite
attraction by its positive charge.
2.2.2 The ionic bond
The ionic bond is formed between two dierent types of atoms. It is always an atom
with an almost lled and one with an almost empty outer electron shell such as the alkali
metal sodium and the halogen chlorine1 . If the sodium atom gives an electron to the
chlorine atom, then the chlorine atom can completely ll the outer electron shell with
electrons and also the sodium atom reaches this state because the formerly outermost
electron shell is now completely empty and the underlying shell is completely lled. As
explained above, this is particularly favourable in terms of energy and therefore an ionic
bond is formed between the atoms involved. In the case of sodium and chlorine, common
salt (NaCl) results.
If one imagines one sodium and one chlorine atom each, which are far away from each
other, the formation of the ionic bond between the two elements can still be inspected
more closely: First we imagine to remove from the sodium atom the electron in the
1
In this case, only one electron binds to the 3s orbital of sodium. Chlorine requires eight electrons to
completely ll the 3s- and 3p-orbitals with electrons, but for this purpose it has only seven electrons
available.
20
2 Elastic stiness, atomic bonding and material structure
outermost electron shell. This leads to a positively charged Na+ -ion and the ionization
energy required for this is 5.1 eV (1 eV = 1.6 · 10−19 J). If the electron is transferred to
the chlorine atom, this is energetically favourable because the outermost electron shell
of the chlorine atom is now completely lled with electrons. The energy gain is -3.6 eV
and a Cl− -ion is created. Altogether, this results in an energy expenditure of 5.1 eV
- 3.6 eV = 1.5 eV. However, there exists an attractive electrostatic force between the
positively charged Na+ -ion and the negatively charged Cl− -ion. It is given by
Fatt =
q1 q2
.
4πε0 r2
Fatt
q1 , q2
ε0
r
(2.13)
attractive electrostatic force
atomic charges
dielectric constant
ion spacing
Thereby, q1 and q2 are the atomic charges of the two ions, ε0 is the dielectric constant
and r is the ion spacing. If both ions are now approached to a distance of 0.4 nm, an
energy of -3.6 eV is released. Therefore, the energy gain is 1.5 eV - 3.6 eV. = -2.1 eV
and an ionic bond is formed. In common salt, the actual energy gain is even a little bit
higher, because not only two but many ions are connected to each other and a Na+ -ion
surrounds itself with several Cl− -ions (Fig. 2.11). The same applies to a Cl− -ion.
Figure 2.11: Schematic representation of the ionic bond using sodium chloride (NaCl) as an
example.
The electrostatic attraction is countered by a repulsive force Frep ∼ r−n (n ≫ 2).
Therefore, the atoms or ions do not come arbitrarily close, but, as sketched in Fig.
2.12, reach an equilibrium distance r0 , at which the force of attraction and the force of
repulsion are balanced. If one looks at the cumulative curve, which results from Coulomb
attraction and the repulsive force, one nds out that its shape around r0 resembles a
parabola. The parabola can be described by the relationship U = U0 + 1/2 k(r − r0 )2 ,
where k is a constant. By dierentiating once according to the local coordinate, one
obtains the force Fi = −dU/dr, which results when the atoms are deected from their
equilibrium position r0 . In the case of the parabola, Fi = k(r0 −r) is obtained. As shown
in Fig. 2.12, this equation describes the behavior of a spring with the spring constant k .
The behaviour of atomic bonds under mechanical stress is therefore similar to that of a
21
2 Elastic stiness, atomic bonding and material structure
spring. Accordingly, we can symbolically represent atomic bonds by elastic springs, as
in Fig. 2.1. This simple model is suitable for all bonds.
If one dierentiates again according to the local coordinate, one can directly make a
statement about the stiness S of the atomic bond and, thus, of the materials:
S=−
S
Fi
dFi
.
dr
(2.14)
stiness of the atomic bond
interaction force between the atoms
Now the elastic behaviour of materials becomes understandable. Under mechanical
stress, the atomic bonds are obviously stressed and the atomic distances change. If the
load is removed from the material, the original bond spacing r0 is established, i.e. the
deformation is reversible (= elastic), just like with a spring. Since the elastic strain
of technical materials is also very low (it is no more than a few tenths of a percent for
metals and ceramics), the rigidity S practically does not change under stress (Fig. 2.12).
Therefore, in the tensile test we nd an elastic straight line and with the modulus of
elasticity E a constant material property.
The dierences in the modulus of elasticity of dierent materials that have already
been noticed are now also becoming clear. If the bond between the atoms is particularly
strong, the curvature of the potential curve around the equilibrium position r0 also
increases. This results in a greater stiness S of the atomic bond and a greater spring
constant k of the associated parabola (Fig. 2.13), which is macroscopically reected in
a larger Young's modulus.
As we have seen, common salt is an inorganic compound between a metal (Na) and
a non-metal (Cl) and thus a ceramic, although without technical relevance. Ionic bonds
therefore occur in ceramics and belong to the strong bond types, as shown in Table 2.3.
This is in accordance with our observation that strong bonds lead to high moduli of
elasticity and that ceramics have a high have elastic rigidity.
2.2.3 The covalent bond
Also in covalent bonding, the bonding principle is to achieve electron shells that are
completely lled. However, in this case atoms are involved with little dierence in electronegativity. The electronegativity is a measure of the force with which an additional
free electron is attracted to an atom. In this case, the transfer of an electron from one
atom to another does, thus, not succeed. Nevertheless, the partners involved nd a way
to ll their electron orbitals completely by sharing electrons. For example, hydrogen with
a single electron on the K -shell needs another electron to ll this shell. Two hydrogen
atoms can therefore combine and share their electrons. This creates a hydrogen molecule
H2 and the bond type is called covalent. Carbon has four electrons on the L-shell, so
it must form four covalent bonds to ll it completely with eight electrons. This is the
case with the CH4 molecule, for example. As shown in Figure 2.14,this creates four
club-shaped orbitals, each of which has a hydrogen atom at its ends, whereas the carbon
22
2 Elastic stiness, atomic bonding and material structure
energy
U
repulsion
spring
U = U0 + 1/2 k (r-r0)2
∆E
distance r
Coulomb attraction
r0
rD ~ 1,25 r0
force
Fi
distance r
stiffness
S
r0
rD ~ 1,25 r0
Technically
relevant
range
distance r
Spring analogue:
F = k (r0 - r)
U = 1/2 k (r0 - r)2
r0
r
Figure 2.12: Interaction forces between two atoms with equilibrium distance r0 (Bonding
energy U , Interaction force between the atoms Fi
d2 U/dr2 ), adapted from [26].
23
= −dU/dr, Stiness S =
2 Elastic stiness, atomic bonding and material structure
energy
U
Frep1 << Fatt2
k1 << k2
Frep
spring
U = U01 + 1/2 k1 (r-r01)2
r01
distance r
U = U02 + 1/2 k2(r-r02)2
Fatt1
r02
Fatt2
Figure 2.13: Eect of the atomic bond strength on the modulus of elasticity E : A larger
Coulomb attraction Fatt leads to a more strongly curved potential curve around
r0 and thus to a larger spring constant k .
atom is located at the center. The covalent bond is thus directed and the arrangement
of the atoms involved is precisely determined by it. Since each orbital is occupied by two
electrons, two electrons are arranged around each hydrogen atom and eight electrons
around the carbon atom. The principle of completely lled electron shells is fullled.
Figure 2.14: Electron orbitals of the methane molecule [8].
24
2 Elastic stiness, atomic bonding and material structure
Another example is diamond, which consists only of carbon atoms. Here, too, each
carbon atom forms four club-shaped orbitals, but now there is a further carbon atom
at each end, which itself is surrounded by four carbon atoms. The resulting spatial
arrangement, in which each carbon atom has four covalent bonds to adjacent carbon
atoms, is shown in Figure 2.15.
Figure 2.15: Diamond structure with electron orbitals [26].
The example of the diamond shows that the covalent bond also occurs in ceramics
and Table 2.3 demonstrates that this type of bond is also one of the strongest bonds. In
fact, the carbon-carbon bond is the strongest known bond. This explains why diamond
has the highest modulus of elasticity (approx. 1000 GPa) and also the highest hardness
and melting temperature (3727 C) of all substances known to us. Technically this is
used, for example, by depositing diamond layers on tools (e.g. tools for metal cutting),
which enormously improves their wear resistance. Diamond coatings are also used in
piezo injection systems for diesel engines in order to minimize friction and wear between
components moving against each other.
Carbon bers have a dierent arrangement of carbon atoms than diamond. Nevertheless, the strong carbon-carbon bond leads again to high E-moduli, which are between
140 GPa and 820 GPa depending on the ber type. Considering that steel has a modulus of elasticity of about 200 GPa at four times the density, one recognizes the great
potential of using carbon bers to create particularly light and rigid structures (such as
an aircraft fuselage).
Figure 2.15 shows that all orbitals of the carbon atoms in diamond have exactly the
same shape. Nevertheless the electrons on these orbitals feel slight dierences. For
example, an electron that is bound to the diamond surface notices that it is only halfsided surrounded by neighboring electrons, whereas an electron in the interior nds
adjacent electrons on all sides. Therefore, the energy of these electrons is no longer
identical. This fact is illustrated in Figure 2.16: When electrons approach from a great
distance and combine to a solid, the individual energy levels split into energy bands.
The last energy band lled with electrons is called the valence band. In diamond it
°
25
2 Elastic stiness, atomic bonding and material structure
is completely lled with electrons, since all orbitals are completely lled with electrons.
Generally speaking, it can be said that both the covalent as well as the ionic bond lead to
a completely lled valence band, since they are based on the principle of completely lled
electron shells. As Figure 2.16 shows, above the valence band (i.e. at higher energies)
further permissible energy levels occur, but they are empty.
energy
U
vacant
vacant
filled
partially filled
r
0
distance r
energy levels
insulator
metal
Figure 2.16: Energy levels of the metallic and covalent bond.
Using the example of common salt and diamond, we have seen that ceramics can have
covalent or ionic bonds. On closer inspection, however, we nd that the chlorine atom
is not able to completely bind the outermost electron of sodium to itself. This means
that this electron also orbits the sodium atom, although it is located much more often
around the chlorine atom. Thus, there is a smooth transition from the ionic bond to the
covalent bond and Table 2.2 shows that ceramic materials generally have partly ionic
and partly covalent bonds. However, for diamond the ionic share is zero, because it only
consists of one type of atoms.
Table 2.2: Percentage bond fraction of the ionic bond in various ceramic materials.
material
ionic bond fraction /%
CaF2
89
NaCl
67
Al2 O3
SiO2
Si3 N4
63
SiC
12
diamond
0
51
30
26
2 Elastic stiness, atomic bonding and material structure
2.2.4 The metallic bonding
Already in chapter. 1.2 we have seen that most elements are metals and that they are
found on the left side of the periodic table. As the example of sodium shows, they have
far too few electrons on the outermost electron shell for it to be possible to completely
ll the outermost shell by covalent bonds. This principle of reaching an energetically
favourable state obviously does not work with metals. Therefore, metals have no other
choice than to fall back on the second principle, namely to distribute the electrons of
the outermost shell over as large an area of space as possible. This is easy for them,
because these electrons are only weakly bound to the atoms anyway. Therefore, one
has to imagine that these outermost electrons in the metal are practically able to move
freely. Since they move inside the metal like a gas in a container, one often speaks of an
electron gas
Figure 2.17: Schematic representation of the metallic bond using sodium as an example.
Figure 2.17 illustrates the bonding state schematically, whereby the uniform grey
value between the atomic nuclei is intended to symbolize that the probability of the
electrons of the outermost shell being present there is the same. This makes clear that
the metallic bond is not directed. It is not quite as strong as the ionic and covalent
bond. However, the metals compensate for this by a trick, as we will be discussed in
chapter 2.3 more closely: While a carbon atom in diamond surrounds itself only with
four nearest neighbours and forms four covalent bonds, an aluminium atom, for example,
has 12 nearest neighbours. So more bonds are formed and in the nal result the binding
energies of metals are, thus, comparable to those of ceramics, as shown in Table 2.3.
This makes it understandable, why the elastic rigidity of both groups of materials is at
a comparable level (Figure. 2.3).
In the case of metals, too, discrete energy levels give rise to energy bands as shown in
Figure 2.16. However, since metals are not able to completely ll the outermost electron
shell with electrons, the valence band (i.e. the last band still containing electrons) is not
completely lled with electrons (Figure 2.16). This has an important consequence if an
electric voltage U is applied against a metal. As a result, a force E · e is exerted on the
27
2 Elastic stiness, atomic bonding and material structure
electrons, where E is the electric eld strength and e is the charge of an electron, and
they are accelerated in the direction of the electric eld. So there is a current, because the
electrons of the valence band now move in a certain direction. Through this movement
the electrons absorb kinetic energy, i.e. their energy increases. This is not a problem,
because in the valence band, unoccupied states are available that allow a slightly higher
electron energy. Accordingly, metals have good electrical conductivity. The situation
is completely dierent with ceramics. Here the valence band is completely lled and,
therefore, the electrons cannot absorb any additional energy. The energy jump to the
next unlled band is too great for this. The electrons have no other possibility than to
stay close to the atomic core. Therefore, ceramics are electrical insulators.
The energy gap between the valence band and the next higher band (the so-called
conduction band) is not the same for all ceramics. With the so-called semiconductors,
such as silicon, it is relatively small. Therefore, a few electrons are still able to jump on
this conduction band and conduct some electricity. The resulting current conductivity
is nevertheless much lower than that of metals.
Figure 2.18: Ashby-diagram: Thermal conductivity vs. specic electrical resistance. Created
with the Cambridge Engineering Selector Edu Pack 2008.
28
2 Elastic stiness, atomic bonding and material structure
Electrons can transport not only electricity but also heat. For this reason, all metals
are also good heat conductors as Figure 2.18 shows. The dierences, however, are
considerable. For example, the thermal conductivity of many steels is almost an order
of magnitude lower than that of aluminium or copper. That is why you dont burn your
ngers when you touch the steel handle of a cooking pot. As Figure 2.18 shows, the
reverse conclusion that all electrically insulating materials should also be good thermal
insulators is not necessarily correct. This is because the atoms are excited by heat
to oscillate. If one heats a material one-sided, atoms there start swinging strongly.
Because of the spring-type bonding between the atoms they also excite neighbouring
atoms to oscillations and thus heat is transported from one end to the other. This works
particularly well with diamond, aluminium nitride and silicon carbide (SiC), so that
their thermal conductivity is at the level of aluminium alloys, although their electrical
conductivity is lower by many orders of magnitude. With ceramic brakes in expensive
cars, consisting of a carbon bre-reinforced SiC matrix, this fact is exploited. The
advantages over brake discs made of grey cast iron are the lower weight (the density of
SiC is less than half of grey cast iron) and the particularly high temperature resistance of
SiC. However, because the manufacturing is complex, such a brake is also quite expensive.
2.2.5 Secondary Bonds
Water (H2 O) is a covalently bonded molecule, with both oxygen and hydrogen completely
lling their outer electron shells with electrons. Accordingly, one wonders how bonding
between water molecules is possible in ice. Obviously, there is no longer a possibility
to form covalent bonds between the molecules. In this context, it is important that
the oxygen atom attracts electrons more strongly than hydrogen. So oxygen is more
electronegative. This results in a charge shift as shown in Figure 2.19. Although the
water molecule is electrically neutral (it is not positively or negatively charged), it is
an electric dipole. Therefore, dierent H2 O molecules try to arrange themselves in such
a way that sides with positive and negative excess charge are directly adjacent (Figure
2.19). This results in an attraction between the molecules, which is called dipole bonding.
Table 2.3 shows that the dipole bonds are much weaker than the other three types
of bonds. It is also evident that the binding energy between water molecules is comparatively high. This is due to the fact that the electronegative oxygen atom deprives the
hydrogen atom of its electron shell (it only has one electron). Therefore, the hydrogen
atom can come very close to the oxygen atom of a neighbouring molecule, resulting in a
comparatively large binding force. This type of bond is called hydrogen bond.
At the moment it is not yet understandable why there is a weak attraction even
between molecules that do not have a permanent dipole moment. For example, this weak
attraction is the reason why even noble gases are liquid at suciently low temperatures
and eventually become solid. So let us imagine, for example, two neighboring argon
atoms (Figure 2.20). The electrons around the atoms are on average equally distributed.
But at a certain point in time, the left atom just happens to have an excess of electrons
on the left side. So for that moment there is an electric dipole and the adjacent atom to
the right reacts to this with a charge shift, so that an attractive force is created between
29
2 Elastic stiness, atomic bonding and material structure
Figure 2.19: Schematic representation of the dipole bond.
r
-
-
+
statistically derived dipole
+
induced dipole
Figure 2.20: Schematic representation of the van der Waals bond: The atoms are kept together
by time-varying dipole moments, adapted from [4].
the two atoms. This type of dipole bond, which is based on temporal uctuations in
the electron distribution, is also called van-der-Waals bond. It is even weaker than the
binding based on permanent dipole moments.
The dipole bond is mainly responsible for the binding in polymers, as explained in
chapter 2.3.3. Since the bond energy is 10 to 100 times lower than that of ionic,
covalent and metallic bonds, the low modulus of elasticity of polymers can be understood.
A direct correlation between the melting temperature of the materials and the bond
strength can also be seen in Table 2.3. This explains why polymers have a low thermal
stability and why application temperatures above 100 C are the exception.
°
30
2 Elastic stiness, atomic bonding and material structure
Table 2.3: Bonding energy and melting point of dierent materials.
type of bond
material
bonding energy
/(eV/atom)
ionic
covalent
metallic
dipole bond
melting point
‰
/
NaCl
3.30
801
MgO
5.20
2800
Si
4.70
1410
C (diamond)
7.40
Hg
0.70
> 3550
−39
Al
3.40
660
Fe
4.20
1538
W
8.80
3410
Ar
0.08
Cl2
NH3
H2 O
0.32
0.36
−189
−101
−78
0.52
0
2.3 The structure of the materials
By discussing the bond types in chapter 2.2 , the origin of elastic strain and the reason
why dierent materials have dierent elastic stinesses could be explained. It also became understandable why metals are good electrical and thermal conductors, whereas
ceramics insulate electrically. However, many things are still unclear; for example, why
metals but not ceramics can be plastically deformed, why high-strength aluminium alloys can be ten times stronger than pure aluminium, or why elastomers can often be
elastically stretched to several times their initial length, whereas the elastic elongation
of metals and ceramics is no more than a few tenths of a percent. To understand this, it
is now necessary to deal with the structure of materials from a microscopic to a macroscopic scale. It will be shown that the type of bond and the structure of the materials
are closely related.
2.3.1 Metals
As discussed in chapter 2.2.4, metal atoms have the tendency to surround themselves
with as many neighbouring atoms as possible in order to be bound as rmly as possible
in the material. This is easy for them because the metallic bond is not directed, in
contrast to covalent and ionic bonds. If one imagines the atoms as spheres and reects
how to pack them as tightly together as possible, it is easy to see the resulting atomic
arrangement. If one places spheres with the same diameter in a plane, the tightest
possible packing is achieved by arranging the spheres in rows whereby all spheres touch
each other and each row is shifted by half the diameter of a sphere relative to the adjacent
one (Figure 2.21). This is exactly how billard balls are arranged at the beginning of the
game. Each sphere is surrounded by six nearest neighbours in the plane.
To obtain a close-packed arrangement of spheres in three dimensions, such close-packed
planes are stacked on top of each other in such a way that the spheres of the following
31
2 Elastic stiness, atomic bonding and material structure
Figure 2.21: Close-packed sphere packing of stacking sequence ABC.
layer come to rest in the depressions of the preceding layer (Fig. 2.21 ). If the lowest level
is called the A level and the second level is called the B level, the atoms of the third level
can be placed in the depressions of the B level so that they lie exactly above the A atoms.
If one continues this construction principle in thought and stacks the fourth plane on the
positions corresponding to the second plane etc., the stacking sequence ABABAB... of
the planes follows. However, as shown in Fig. 2.21, the third plane can also be placed in
the troughs of the second plane in such a way that it takes up a new position C, which
does not coincide with either the rst or the second plane. Continuing this construction
principle, the stacking sequence is ABCABCABC... . Both arrangements are closepacked arrangements with a space lling of 74 % of the total volume. A larger lling
of the space is not possible with spheres of the same size. Since each atom has not
only 6 closest neighbours in the plane but also three in the plane above and below, the
total number of closest neighbours is 12. This multitude of nearest neighbours results,
as mentioned above, in the fact that the metal atoms are strongly bonded although the
metallic bond is not quite as strong as the covalent or ionic bond.
Aluminium, copper and nickel are examples of metals that have an atomic arrangement
according to the stacking sequence ABCABC. Titanium and magnesium are examples for
metals with the stacking sequence ABAB. The metal atoms are thus arranged according
to a regular recurring pattern. Therefore, there is not only a short range order, by which
the arrangement of atoms in the immediate vicinity is dened, but also a long range
order, because also the atomic arrangement in a far distance is exactly dened by the
periodic arrangement of the atoms. Whenever such a long range order exists, one speaks
of a crystal or a crystalline material. Therefore, metals such as aluminium, nickel or
titanium are crystalline materials.
In order to be able to describe the respective atomic arrangement in a simple way, socalled unit cells are used. Fig. 2.22(c) shows the one for the tightest sphere packing with
stacking sequence ABAB. Here the closest packed A and B planes are easily recognizable.
Because the unit cell has the shape of a straight prism with a hexagon as basis, the atomic
arrangement is called hexagonal close packed crystal structure. For the edge lengths of
the unit cell, a ̸= c applies. Since the atoms are very small, the same applies to the edge
32
2 Elastic stiness, atomic bonding and material structure
lengths. They are typically 0.2 nm to 0.6 nm.
The unit cell for the stacking sequence ABCABC is shown in Fig. 2.22 (a). Because
it is a cube with edge length a, the atomic arrangement is called the cubic close packed
crystal structure. Since each cube face has an atom arranged in its center in addition to
the corner atoms, one also speaks of a face-centered cubic crystal structure. The closest
packed planes are not so easy to see here. Their normal is oriented along the space
diagonal of the unit cell. Thus, the closest packed planes lie diagonally in the unit cell
as shown in Fig. 2.22(b).
Another important structure of metals is the body centered cubic crystal structure
(Fig. 2.22 (d)). Its name is derived from the fact that the unit cell is again a cube,
where besides the corner atoms an atom is also located in the center of the cube. With
a space lling of 68 %, this crystal structure is no longer packed most closely. But the
packing density is still very high (compare NaCl structure: 52 % space lling). Apparently, additional covalent bonds come into play, which stabilize this structure in certain
metals. Iron is the most important metallic element, which forms this crystal structure
at ambient temperature. However, it transforms to the face-centered cubic crystal structure at 911 C. This transformation is of tremendous technological importance. Thanks
to it, steels are so important and high-strength steels can be made.
If one takes unit cells and links them together in all three directions of space, a crystal,
e.g. from aluminium, is created. Hereby, one has to imagine that contact areas between
adjacent unit cells merge to one area. Once again, the long range order can be seen
in this: If the rst unit cell is placed in space with a certain orientation, the position
of all other unit cells is xed. The formation of such crystals occurs in reality during
solidication from the liquid or, in special cases, from the gaseous state. Everyone knows
about the formation of snow crystals, where each of the snowakes shown in Fig. 2.23
is a single crystal. In Fig. 2.23, the long range order and the resulting unique building
principle can be clearly recognized. Also precious stones consist of a single crystal.
In technical applications, however, single crystalline materials are the exception. The
most important examples are silicon wafers for the semiconductor industry and single
crystalline turbine blades in aircraft engines or gas turbines for power generation. In
contrast, polycrystalline materials, i.e. materials that consist of many crystals, are the
rule. This is so because the solidication of a melt normally produces many small crystal
nuclei (Fig. 2.24 (a)), which grow during solidication (Fig. 2.24 (b-e)) until they touch
each other and the material completely solidies (Fig. 2.24 (f)). In Fig. 2.24 each
square shall represent a unit cell. The dierent crystals, also called crystallites or grains,
dier in the position of their unit cells in space. Within a grain the long range order is
given. If, however, one crosses the boundary of a grain, the orientation of the unit cell
changes in an unpredictable way. Thus, one gets from one grain to another. The border
between the grains is called the grain boundary. As Fig. 2.25 shows, the periodicity
is disturbed here. Grain boundaries have a thickness of about two elementary cells, i.e.
about 1 nm (1 nm = 10−9 m). In contrast, typical grain diameters are 0.01 mm - 0.1 mm.
If the surface of a polycrystalline metal is etched with an acid, a preferred attack occurs
along the grain boundaries, because the atomic arrangement is disturbed there and the
atoms can be attacked more easily by the acid. As Fig. 2.26 (d) illustrates, light rays
°
33
2 Elastic stiness, atomic bonding and material structure
A
a
ne
pla
C
a
A
eB
n
pla
a
a
a
a
(a) Unit cell of the face-centered
cubic crystal structure.
(b) Closest packed planes in the
face-centered cubic crystal structure.
plane A
a
plane B
c
plane A
a
a
a
(c) Hexagonal closest packed crystal structure.
(d) Body-centered cubic crystal
structure.
Figure 2.22: Structure of the metals, adapted from [26].
34
2 Elastic stiness, atomic bonding and material structure
Figure 2.23: Snowakes [21].
(a)
(b)
(c)
(d)
(e)
(f)
Figure 2.24: Schematic representation of nucleation and nucleation growth of crystals during
solidication from the molten state, adapted from [22].
35
2 Elastic stiness, atomic bonding and material structure
Figure 2.25: Structure of grain boundaries in metals and ceramics [18].
incident there are therefore scattered and the grain boundaries appear dark when viewed
under a light microscope. The light microscopic image of polycrystalline, essentially pure
iron is shown in Fig. 2.26 (a) and the grain boundaries are visible as dark lines. Because
one is on the microscopic length scale, one also speaks of the microstructure of the
material. The microstructure of most technically important materials is not as simple
as that of pure iron. For example, the steel C45, which contains 0.45 percent (wt.%)
carbon in addition to iron, also contains lamellar iron carbide Fe3 C, which appears black
in Fig. 2.26 (b). In so-called nickel-based superalloys, which are used to make turbine
blades for aircraft engines, one nds extremely small cube-shaped precipitates in the
nickel-rich matrix (Fig. Abb. 2.27).
Imagine a large block of a polycrystalline material, e.g. an aluminum block. If tensile
specimens are cut from this block in completely dierent directions, in most cases identical values for the E-modulus, the yield strength Rp or the tensile strength Rm result. In
this case the properties of the examined material are the same in all directions in space
and one refers to this, as already mentioned, as isotropic behaviour. This was not to be
expected a priori because if we look at the unit cells in Fig. 2.22 or the spring model
in Fig. 2.1, one would rather expect that, for example, the stiness of a crystal should
depend on the loading direction, i.e. that the behaviour is anisotropic. However, the
crystallites in polycrystals are usually arranged randomly, so that these dierences are
averaged out. In contrast, a single crystalline turbine blade actually behaves anisotropically. Its modulus of elasticity in the longitudinal direction of the blade is signicantly
smaller than perpendicular to it.
2.3.2 Ceramics
The structure of crystalline ceramics is similar to the structure of metals. However, the
atoms in ceramics are not arranged in a close-packed structure, and the covalent or ionic
bond between the atoms lead to dierent crystal structures than in metals.
The unit cell of diamond has already been shown in gure 2.15. Figure 2.28 (a)
36
2 Elastic stiness, atomic bonding and material structure
(a) Iron with a carbon content of 0.03 wt.-%.
(b) Iron with a carbon content of 0.45 wt.-%.
microscope
polished and
etched surface
surface notch
grain boundary
(c) Grain structure of Al2 O3 . Scanning elec- (d) Image generation at the optical microscope,
tron microscope image. The horizontal white adapted from [10].
bar has a length of 1 µm [26].
Figure 2.26: Microstructure of metals and ceramics.
shows the unit cell of NaCl. Since the ionic bond predominates here, the atoms are
arranged in a way that each atom is neighboured by oppositely charged atoms, whereas
equally charged atoms are arranged in a longer distance to each other. In crystalline
SiO2 (Quartz), the covalent bond predominates, where every Si atom is neighboured by
four oxygen atoms, and every oxygen atom has two bonds with every adjacent Si atom
(g. 2.28 (c)).
Fig. 2.29 shows the mineral calcite as an example of crystalline ceramics. Each calcite
platelet is a single crystal. Fig. 2.26 (c) shows the polycrystalline microstructure of the
important technical ceramic Al2 O3 . Due to the etching contrast, the grain boundaries are
clearly visible as groves. As with metals, technical ceramics are normally polycrystalline
From the above examples, it is apparent that the unit cells of ceramics are much more
complicated than those of metals. If a ceramic solidies from the liquid state, it is thus
37
2 Elastic stiness, atomic bonding and material structure
Figure 2.27: Microstructure of a nickel base superalloy.
much more dicult for the atoms to be arranged in space according to the requirements
of the unit cell. This needs much more extensive atomic rearrangements. If there is
not enough time for this, the long range order cannot be established and only a short
range order occurs. In the example of SiO2 , every Si atom is surrounded by the required
number of oxygen atoms. However, this short range order does not repeat periodically,
so that there is no long range order (Fig. 2.28 (b)). In contrast to crystalline materials,
such materials are called amorphous or glasses. Window glass is an amorphous material
that consists mainly of SiO2 .
If window glass were to be kept for a long time at several hundred degrees Celsius in
a furnace, it would begin to form crystallites, which would then grow until the material
is completely polycrystalline. The glass would thereby lose its optical transparency and
become opaque, because the grain boundaries would scatter the incident light. This
example shows that the crystalline state is the stable condition. The only reason it did
not form during solidication is because the time was not enough for that.
If we consider the specic volume (i.e. the reciprocal of density) as a function of
temperature, important dierences between solidication to a crystalline or amorphous
material can be illustrated once again. For this purpose, we assume that the melt
solidies into a crystalline solid at the temperature Tm , the melting temperature, if
there is sucient time for this to happen. Hereby, the atoms are regularly arranged
and, thus, the specic volume decreases sharply during crystalline solidication (Fig.
2.30). If, on the other hand, cooling is rapid so that crystallization does not succeed,
the liquid state is initially retained. The atoms coordinate themselves as shown in Fig.
2.28 (b). However, the thermal movement of the atoms is so strong that atomic bonds
are constantly broken and reattached. Thus, the atoms are constantly in motion. As the
temperature decreases, this movement decreases until it suddenly becomes impossible
when a critical temperature, the glass transition temperature Tg , is reached. The melt
freezes, so that we can imagine the amorphous state as that of a frozen melt. From what
has been said it follows immediately that the glass transition temperature Tg (often also
called glass temperature) must be lower than the melting temperature.
38
2 Elastic stiness, atomic bonding and material structure
Na+
Si
O
Cl–
3r
2r
r
r
(a) Crystalline ceramic with
predominantly ionic bonding
components (NaCl), adapted
from [26].
(b) Amorphous structure of
a ceramic.
Due to the
two-dimensional representation only three bonds per silicon atom are shown [3].
Si4+
O2–
(c) Crystalline ceramic with covalent bond character (SiO2 ), adapted
from [6].
Figure 2.28: Examples of the structure of ceramic materials.
2.3.3 Polymers
Structure of the polymer groups
The backbones of almost all polymers are carbon chains (Fig. 2.31 (a)), i.e. chains
of covalently bonded carbon atoms. Hereby, the tetravalent carbon atoms form two
covalent bonds to adjacent carbon atoms along the chain and two covalent bonds to side
groups. In the simplest case, the side groups are hydrogen atoms, so that polyethylene
results (Table 2.4). However, they can also consist of other atoms (e.g. Cl, F, O) or
molecules (e.g. CH3 , OH). The molecular chains have a three-dimensional structure with
binding angles of 109 . Of decisive importance for the mechanical behaviour is the fact
that the chains are able to rotate along the carbon-carbon bonds (Fig. 2.31 (b)). Due
to this ability, they can be either stretched or entangled (Fig. 2.31 (c)).
°
39
2 Elastic stiness, atomic bonding and material structure
Table 2.4: Structure and properties of polymers (Tg : glass transition temperature, Tm : melting
temperature), adapted from [26].
Name
Structural formula
Applications
Tg /
‰
Tm /
‰
−25 −15
150 175
−110 −90
125 132
Thermoplastic polymers
H CH3
Polypropylene, PP
Polyethylene, PE
Polystyrene, PS
Tubing, bottles,
insulation
C
C
H
H n
H
H
Tubing, bottles,
electrical insulation
C
C
H
H n
H
H
C
C
H
Thermal insulation
(foam)
74 100
usually
amorphous
n
usually
H
Polyvinyl chloride,
PVC
H
C
Flooring, window
frames
C
75 105
methacrylate), PMMA
(acrylic glass)
Transparent
H CH3
C
phous
(210)
Cl H n
Poly (methyl
amor-
window-glass (e.g.
C
windows in planes)
85 165
usually
amorphous
H C
CH3 n
O
O
usually
Polycarbonate
CH3
O
C
O
Lenses, helmets, CDs,
C
laminated glass
O
CH3
142 205
amorphous
(220)
n
Elastomeres
Polybutadiene
H
H
H
H
C
C
C
C
H
H
Tires
−108 −75
147
207
n
Thermosetting polymers
Polyester
O
I
R
C O R
Fiberglass
II
n
40
2 Elastic stiness, atomic bonding and material structure
Specific volume (1 / density)
Figure 2.29: Naturally grown calcite crystals [25].
ous
h
amorp
crysta
lline
Tg
Tm
Temperature T
Figure 2.30: Volume change due to fast and slow cooling, adapted from [2].
The polymers are formed by polymerization of so-called monomers. This is illustrated
in Fig. 2.32 where formation of polyethylen from the monomer ethylene (C2 H4 ) is
shown. It is important that ethylene has a double bond. Through addition of a radical
former, which contains an unbound, i.e. particularly reactive electron, this double bond
is broken as shown in Figure 2.32. This produces again a radical, which in turn reacts
with an ethylene molecule. By continuation of this reaction scheme, long polyethylene
chains are formed. However, they are not arbitrarily long, since there are also termination
reactions. For example, two chains with one unbound electron at each end of the chain
can form a covalent bond, in that the two unbound electrons react with each other.
The resulting new chain can no longer take part in the polymerization reaction, since
41
2 Elastic stiness, atomic bonding and material structure
(a) Structure of a polymer chain. The large spheres correspond to the C-atoms, the small ones can, for
example, be H- or Cl-atoms.
(b) Rotation possibilities along the carbon chain [10].
(c) Schematic representation of a
warped chain molecule [11].
Figure 2.31: General structure of the polymer chains.
it does not have any other unbound electron. Typically, the chains are made of 103 to
105 monomers, wherein the C-C bond spacing is 0.15 nm. The chain length is therefore
between 0.3µ and 30µ. The fact that the polymers are made up of monomer units that
are repeated many times, is symbolized in the structural formulas (Table 2.4) by the
addition n.
The polymers are divided into three large material groups: The thermoplastic polymers (also called thermoplastics), elastomers and thermosetting polymers (also called
thermosets). As they are very dierent in structure and mechanical properties, they are
discussed separately in the following sections.
42
2 Elastic stiness, atomic bonding and material structure
R
H
H
H
H
R + C
C
C
C
H
H
H
H
H
H H
H
C
C
C
C
H
H H
H
H
H
H
H
C
C + C
C
H
H
H
H
R
R
Figure 2.32: Catalytic polymerization of ethylene to polyethylene. The symbol R denotes the
free radical former containing a free electron.
Thermoplastic polymers
As we have seen above, polymer chains are built by covalent bonds. In order for the polymer chains to become a solid material, there must also be bonds between the polymer
chains. In the case of polyethylene (PE), only dipole bonds are available for this purpose, because all possibilities to form covalent bonds have already been used up within
the chain. This is exactly the characteristic of thermoplastic polymers: They have exclusively dipole bonds between the chains. Polyethylene is therefore a thermoplastic
polymer. Because in this case only hydrogen atoms are present as side groups, there
is no electronegativity dierence between dierent side groups. So only very weak van
der Waals bonds develop. In the case of polyvinyl chloride (PVC), on the other hand,
a hydrogen atom is replaced by a chlorine atom (Table 2.4). The latter is much more
electronegative so that dipole bonds between the polymer chains form, which are based
on permanent dipole moments (Fig. 2.33). Therefore, PVC chains are more strongly
bound to each other than polyethylene chains.
H
C
H
H
C
C
-
H
C
-
Cl H Cl H
Dipole
interaction
+
H H H+ H
C
C
C
C
Cl H Cl H
Figure 2.33: Illustration of the dipole interaction on the example of polyvinyl chloride (PVC).
The family of thermoplastic polymers is again divided into amorphous and semicrystalline thermoplastics. To understand this dierence, we consider the solidication
of a thermoplastic polymer from the molten state. As Table 2.4 shows, polyethylene is
in the molten state above 130 C. The temperature is then so high that the weak dipole
bonds between the chains are no longer able to keep the chains together. Therefore, the
polymer chains can slide freely against each other and a melt is present. However, the
viscosity is much higher than that of water or a metal melt, because it is much more
°
43
2 Elastic stiness, atomic bonding and material structure
(a) Arrangement of the polymer chains in the crystalline areas. Gray
spheres represent C-atoms and red spheres H-atoms [11].
(b) Spherulite structure in polyethylene [26].
(c) Schematic representation of a single
spherulite [10].
Figure 2.34: Structure of thermoplastic polymers, adapted from.
44
2 Elastic stiness, atomic bonding and material structure
dicult to move long polymer chains past each other than small molecules or single
atoms. It is about the same level as honey. If one cools the polymer slowly from the
molten state, as shown in Fig. 2.24(a) for the metals, crystallization nuclei are initially
formed which grow during solidication. Hereby, the polymer chains arrange themselves
in a regular, periodically repetitive form (Fig. 2.34 (a)). Thus, a crystalline structure
is created, which can be imagined as built by periodically shifting a unit cell.
Starting from the crystallization points, solidication proceeds radially symmetrically
outwards. However, it can be seen that the periodic folding of the polymer chains to
crystalline areas is quite dicult. Therefore, it does not succeed everywhere and, as
shown in Fig. 2.34 (c), sandwich-like packages consisting of crystalline and amorphous
regions are formed. In the amorphous regions, the polymer chains are not regularly
folded, but are disordered. According to the denition of the amorphous state in chapter
2.3.2, there is no long-range order. If one starts at one end of a polymer chain and runs
through the chain to the other end, one notices that the chain folds many times in a
crystalline package, then passes through an amorphous region to merge into the next
crystalline area again. A chain thus bridges many areas, so that the amorphous and
crystalline areas are closely interlinked. This is important to understand the mechanical
behavior discussed later on.
The radially symmetric solidication does not continue arbitrarily, since the growing
crystallization nuclei eventually touch each other. Looking at the solidied polymer at
a magnication of about 500 times in the light microscope (Fig. 2.34 (b)), one can see
this and the microstructure resembles that of polycrystalline solidied metals. As we
have seen, the exact structure is nevertheless very dierent from that of polycrystalline
metals. Therefore, one does not speak of individual grains but of spherulites.
The slow solidication process discussed above results in a semi-crystalline thermoplastic polymer consisting of crystalline and amorphous areas. If, in contrast, one imagines
fast solidication by rapid cooling, the time for a regular folding of the polymer chains
is not sucient. Thus, the polymer chains do not crystallize below the melting temperature Tm , but they remain randomly entangled and become more and more immobile as
the temperature decreases, until they nally freeze at the glass transition temperature
Tg . The viscosity increases to such an extent in a narrow temperature interval around
Tg that we perceive this as a transition from a liquid to a solid state. This process is
completely analogous to that discussed for ceramics in section 2.3.2. Accordingly, we
now have an amorphous thermoplastic polymer, whereby Tg < Tm also applies here.
Whether a thermoplastic polymer solidies amorphously or semi-crystalline does not
only depend on the solidication rate but also very much on the structure of the polymer
chain. Polyethylene is a very simple polymer which can easily fold into crystalline areas.
Thus, it is always semi-crystalline at technically usual solidication rates. Hereby, the
crystalline portion is between 45% and 80%, depending on the process parameters. On
the other hand, polymethyl methacrylate (PMMA), which is also shown in Table 2.4, has
much larger side groups, making regular folding very dicult. Under normal technical
solidication conditions PMMA is, thus, always amorphous.
The mechanical behaviour of thermoplastic polymers is strongly dependent on temperature. This can be illustrated particularly well if the modulus of elasticity is plotted
45
2 Elastic stiness, atomic bonding and material structure
E
/ GPa 1
E
/ GPa 1
10-3
Glass transition
Glass transition
energy elastic entropy
elastic
10-60
1
amorphous areas:
entropy elastic
crystalline areas:
energy elastic
10-3
viscous
2
10-6
0
3
energy elastic
viscous
1
2 Tm
(a) Amorphous thermoplastic.
E
/ GPa 1
(b) Semi-crystalline thermoplastic.
E
/ GPa 1
Glass transition
Glass transition
10-3
10-3
10-60
3
T/Tg
T/Tg
energy elastic
entropy elastic
1
2
10-6
0
3
energy elastic
1
2
T/Tg
3
T/Tg
(c) Elastomer.
(d) Thermosets.
Figure 2.35: Plot of the modulus of elasticity as a function of temperature, adapted from [26,
9].
as a function of the temperature. For amorphous thermoplastics, the curve shown in
Fig. 2.35 (a) is obtained. Below Tg , the modulus of elasticity is several gigapascals.
Thus, it is about two orders of magnitude smaller than that of metals and ceramics.
This is due to the weak dipole bonds between the polymer chains. They are stretched
strongly even at low stresses, so that the polymer stretches strongly elastically even at
low stresses.
Below the glass transition temperature, many amorphous thermoplastics, such as
PMMA or polystyrene, are hard and brittle (i.e. their elongation after fracture determined in a tensile test is low). This is understandable from the idea that the polymer
chains freeze below Tg and remain immobile in their positions. Since the glass transition
temperature of amorphous thermoplastics is generally signicantly higher than ambient
temperature (Table 2.4), this is unfavorable for the majority of technical applications.
However, there are also some amorphous thermoplastics which are ductile below Tg .
These include, for example, polycarbonate (PC) already mentioned in chapter 1.2. As
Table 2.4 shows, polycarbonate has a comparatively complicated structure. As a result,
certain areas of the chain are rigid and give strength up to the glass transition temperature, while other areas are already exible well below ambient temperature, thus giving
the material ductility and toughness. However, this more favorable material behavior
resulting from the more complicated structure leads also to signicantly higher material
costs compared to amorphous thermoplastics with a simple structure (Fig. 1.3, 1.4).
If the glass transition temperature is reached from low temperatures, then the Young's
46
2 Elastic stiness, atomic bonding and material structure
modulus decreases rapidly by several orders of magnitude in this temperature regime
(Fig. 2.35). The same applies to the strength. This is understandable, since the dipole
bonds break up and the transition to the molten state takes place. Thus, the scope
of application of components made of amorphous thermoplastics is below Tg while the
softening above Tg is used to produce components. The property to soften at elevated
temperatures has given thermoplastics their name.
Fig. 2.35 (a) shows another detail: Above Tg the course of the modulus of elasticity
has an S-shape, i.e. the decrease of the elastic modulus with temperature slows down at
rst and then accelerates again. This is somewhat surprising, because with the transition
to the liquid state the elastic stiness should become essentially zero. However, it should
be remembered that the polymer chains are still entangled. Like in a ball of wool,
the resulting anchoring points between the individual chains lead to a certain residual
elastic stiness, although there are no longer any binding forces between the polymer
chains. For the purpose of dierentiation, this type of elasticity is referred to as entropy
elasticity, which we will look at in more detail in connection with elastomers. In contrast,
the elasticity associated with the stretching of atomic bonds is refered to as energy
elasticity. Correspondingly, the three areas marked in Fig. 2.35 (a) as energy-elastic,
entropy-elastic and viscous result.
If one imagines that the same thermoplastic polymer is transformed once into the
amorphous and once into the semi-crystalline state by dierent cooling rates, the following dierences result for the semi-crystalline material:
ˆ Well below T , the mechanical behaviour corresponds largely to that of the amorg
ˆ
ˆ
phous thermoplastic polymer, i.e. it is generally characterised by brittleness. However, the modulus of elasticity is somewhat higher. This is because the crystalline
regions are packed tighter than the amorphous ones. Accordingly, more dipole
bonds can be formed between the chains and also the binding forces are stronger
due to smaller bond spacing, although the bond type is the same. Ultimately, this
is the reason why the crystalline regions only melt at Tm > Tg .
When the glass transition temperature is reached, the Young's modulus also decreases (Fig. 2.35 (b)). However, the decrease is much smaller. This is because the
amorphous areas soften, but not the crystalline areas. Since the crystalline regions
are interlinked by the fact that individual polymer chains link several crystalline
regions (see above), they provide for the rigidity and strength of the material,
whereas the softened, amorphous areas for ductility. Due to this favourable combination of strength and ductility, partially crystalline thermoplastics are often
applied between Tg and Tm .
Only when the melting temperature Tm is reached do the crystalline regions also
melt. Accordingly, the modulus of elasticity drops rapidly and a viscous melt is
formed.
Previous considerations have shown that the mechanical behaviour of thermoplastics
is strongly dependent on temperature. This is further illustrated in Fig. 2.36, showing
the results of tensile tests on the amorphous thermoplastic polyvinyl chloride (PVC).
47
2 Elastic stiness, atomic bonding and material structure
Figure 2.36: Temperature and time dependence of mechanical properties on the example of
polyvinyl chloride (PVC) [13].
°
°
While the material behaves brittle at -30 C and 25 C, a distinctly ductile behavior can
be observed at 40 C and 70 C. At the same time, the strength decreases considerably. If
this is compared with the glass transition temperature of PVC, which is approximately
90 C, it becomes apparent that the polymer chains do not freeze completely below
Tg . If a suciently high tensile stress is applied, the polymer chains can obviously
move past each other at 40 C and thus cause plastic deformation. Ultimately, however,
this is understandable: Freezing increases the viscosity by several orders of magnitude.
However, if the material is stressed long enough and the temperature is not too low,
the viscosity is not high enough to completely prevent the polymer chains from slipping
and plastic deformation occurs. This consideration suggests that for thermoplastics, the
duration of stress is also a very important factor. If the stress duration is very short,
viscous slipping of the polymer chains past each other is not possible. The result should
be a brittle material behaviour. If, on the other hand, the stress is applied slowly,
sucient time should be allowed for viscous slipping and the polymer should behave
ductile. This is conrmed by Fig. 2.36, where tensile test data for PVC at constant
temperature (25 C) but dierent speeds of loading are shown. This time dependence
of the mechanical behaviour, which already exists at room temperature, is a special
feature of polymers (i.e. also of elastomers and thermosets). For metals and ceramics
comparable phenomena only occur at signicantly higher temperatures, so that they
usually do not have to be taken into account.
In order to design a component made of a thermoplastic polymer, a stress-strain curve
determined in a tensile test is obviously not sucient due to this time and temperature
dependence. That is why a group of so-called isochronous stress-strain curves is often
°
°
°
°
°
48
2 Elastic stiness, atomic bonding and material structure
Figure 2.37: Isochronous stress-strain curves of PMMA [26].
used(Fig. 2.37). Each of these curves indicates the stress as a function of strain for
a given loading duration. The temperature eect can also be taken into account by
dierent scales of the ordinate. The component design is therefore much more complex
than when using a metallic or ceramic material, as a much more extensive data set
must be used and the component service life must be included as a further parameter.
For example, if the component is designed for 100 operating hours, signicantly higher
stresses are permissible than is the case with a required service life of 10000 hours.
Elastomers
Elastomers, i.e. rubber-like polymers, show some remarkable dierences compared to
thermoplastics. Firstly, their modulus of elasticity measured at ambient temperature is
lower by a factor of 100 (Fig. 2.3). Secondly, elastomers can achieve elastic elongations of
several hundred percent. No other material group is capable of this. To understand this,
it is helpful to look at the production of elastomers using an example. The starting point
is again polymerization from a monomer2 , which in this case has two double bonds (Fig.
2.38). In this case, polymerization produces a polymer chain whose chemical structure
corresponds to natural rubber obtained from the rubber tree. To make an elastomer
from natural rubber it must be vulcanised. In this process the material is heated by
adding sulphur molecules. This occurs, for example, in the form of sulphur dichloride
(S2 Cl2 ), which decomposes to Cl2 and S2 molecules in the heat. As shown in Fig. 2.38,
the sulphur molecules break the double bond of the polymer and thus cause covalent
cross-linking between the individual polymer chains. However, the cross-linking density,
i.e. the number of covalent bonds in relation to the number of carbon atoms, is very
2
The exact chemical description is 2-methyl-1,3-butadien.
49
2 Elastic stiness, atomic bonding and material structure
low at 10−3 to 10−4 . Only every ten thousandth to one thousandth carbon atom along
a polymer chain has a covalent bond to an adjacent chain.
Isoprene
Polyisoprene
CH3 H
H CH3 H H
H2C C
C CH2
Polymerisation
C
Polyisoprene
H CH3 H H
C
H
C
C C
H
C C
H
H
H CH3
H
C
C C
H S
H H
C
Vulcanisation
C
H CH3 S
H
C
C
C
H
H
C
H
n
Figure 2.38: Polymerization of 2-methyl-1,3-butadien followed by vulcanization.
This special structure of the elastomers is also noticeable when the elastic modulus is
plotted over temperature (Fig. 2.35 (c)). Below Tg , the behavior corresponds to that
of an amorphous thermoplastic polymer. The material is hard and brittle. There is no
rubber elasticity. When the glass transition temperature is reached and exceeded, the
Young's modulus also decreases by several orders of magnitude. However, it stabilizes
on this low level and even increases slightly with temperature. In this temperature range
above Tg , elastomers are rubber elastic, i.e. that's where their application area is.
Above Tg , as with amorphous thermoplastics, all dipole bonds are dissolved. However,
in contrast to the thermoplastics, the polymer chains cannot slide freely against each
other because they are still connected by the covalent bonds. Therefore, an elastomer
is solid above Tg , whereas an amorphous thermoplastic polymer is a viscous liquid.
However, the covalent bonds are far apart. Thus, there are relatively long chain segments
between these bonds, which are tangled together in the absence of an external load. If,
however, an external load is applied, these chain segments stretch (Fig. 2.39) and
so does the elastomer. If the load is released again, one might think that the chains
simply remain in this stretched state. But this is not the case, because all atoms and
molecules oscillate above 0K around their rest position due to the thermal energy. As a
result, the molecule chains tangle up again by themselves and the elastomer is returning
to its original form. The reason for the reentanglement is that one can imagine much
more entangled than stretched states. If one imagines a thousand dice for comparison,
the stretched state would correspond to a six on all dice, whereas the tangled state
corresponds to any combination of numbers. So because entangled is much more probable
than stretched, the molecules entangle themselves again. This eect of nature choosing
the more probable, disordered state by itself is described in thermodynamics by entropy.
The cause of rubber elasticity is thus nature's endeavour to adopt a state of greatest
50
2 Elastic stiness, atomic bonding and material structure
possible entropy (i.e. greatest possible disorder). Therefore, the rubber elasticity of
elastomers is also called entropy elasticity. Since the atoms vibrate more and more with
increasing temperature, the tendency to assume the disordered state becomes stronger
and stronger with increasing temperature. In this respect it is understandable why
Youngs modulus rises slightly with temperature above Tg . In the regime of energy
elasticity, i.e. below Tg , Youngs modulus, on the other hand, decreases with increasing
temperature, because the dipole bonds between polymer chains weaken with increasing
temperature. When Tg is reached, the thermal oscillations become so strong that the
dipole bonds are no longer able to hold the polymer chains together. We have refered
to this as breaking of the dipole bonds.
Figure 2.39: Stretching of the chain segments by applying an external load, adapted from [6].
It follows from the above that the glass transition temperature of elastomers must be
below room temperature, otherwise one would not be able to make use of the rubberelastic behaviour in most applications. This is achieved by choosing polymer chains
with a very simple structure and non-polar side groups (especially H, CH3 ). This results
in very weak van der Waals bonds with glass transition temperatures far below room
temperature. The Young's modulus in the entropieelastic regime is 0.0001 GPa - 0.1 GPa,
depending on the degree of cross-linking. Weak cross-linked elastomers show very high
elastic elongations at particularly low elastic rigidity. With increasing cross-linking, the
Youngs modulus increases, while the elastic stretchability decreases.
The dierence in the elastic behaviour of amorphous thermoplastics, semi-crystalline
thermoplastics and elastomers is summarized in Fig. 2.40, using polystyrene as an
51
2 Elastic stiness, atomic bonding and material structure
Figure 2.40: Inuence of the degree of crystallization and crosslinking on the shear modulus G
using the example of polystyrene, adapted from [19].
example. Polystyrene is normally amorphous, since the phenyl side group shown in
Fig. 2.41 leads to limited chain mobility. However, it can also be converted into
the semi-crystalline state by cooling particularly slowly from the molten state. Since
the aromatic ring has unsaturated double bonds, it can also be crosslinked to form an
elastomer (this state is referred to as crosslinked in Fig. 2.40)3 . In case of the amorphous
material, the S-shape of the curve above Tg can be seen very clearly, which is the more
pronounced the longer the polymer molecules are. In the crosslinked state, however, the
Youngs modulus remains almost constant at a low level. In the semi-crystalline state,
the Youngs modulus remains at a comparatively high level until melting occurs above
200 C.
°
H
H
C
C
H
H
H
C
C
H H C
H
C
C
n
H
C
C
H
C
H
n
Figure 2.41: Simplied (right) and explicit (left) representations of the structural formula of
polystyrene with phenyl side group.
3
Technically this is not important, because Tg is well above room temperature Therefore, the rubber
elastic state can generally not be used.
52
2 Elastic stiness, atomic bonding and material structure
Thermosetting polymers
Like elastomers, thermosets have covalent bonds between the polymer chains. However,
the density of crosslinks is much higher at 10−2 to 10−1 . This leads to a rigid network of
covalent bonds between which the polymer segments can no longer move. For this reason,
thermosets behave energy-elastically both above and below Tg and are comparatively
brittle. If we examine the Youngss modulus as a function of temperature, the glass
transition temperature can be seen from a kink in the curve. The associated decrease
in elastic stiness is due to the fact that the dipole bonds can no longer transmit loads.
However, as the covalent bonds are still present, the Young's modulus remains at a high
level. Since the covalent bonds are very strong, with increasing temperature there is also
no melting, but the thermoset burns at several hundred degrees in air.
O
O
O
C CH CH C O CH2 CH2
+
O
n
O
CH2
O
O
+
O
C CH CH C O CH2 CH2
CH2
CH
O
O
CH O
C CH CH C O CH2 CH2
O
C CH CH C O CH2 CH2
n
Figure 2.42: Crosslinking of polyester and phenylethene (solvent).
The production of thermosets is shown in Fig. 2.42 on an example. The starting point is a polyester resin. It is highly uid because with 10-20 repeat units the
polyester chains are much shorter than those of thermoplastics and they are also dissolved in phenylethene. As already discussed above, the crosslinking reaction is started
by adding a radical former (here called hardener). It ensures that the double bonds of
the phenylethene and polyester molecules break up, so that free electrons result at these
molecules which react with each other. This leads to crosslinks as shown in Fig. 2.42.
Hereby, a tightly connected network is formed that does not allow the polymer chains
to unraveling, as mentioned above.
In mechanical engineering, thermosets are used to produce bre-reinforced polymers.
Since the resins have a low viscosity, bre meshes can be well inltrated before the resin
is hardened. Although the mechanical properties of such polymer matrix composites
are essentially determined by the bres, the matrix also fulls important functions: it
xes the bres spatially, enables load transfer onto the bres, protects the bres against
buckling under a compressive load and prevents damage of the bres (e.g. by aggressive
substances).
53
2 Elastic stiness, atomic bonding and material structure
The most important reinforcing bres are listed in Table 2.5. Glass bres are most
common because they are the most cost-eective. Particularly high strengths and stinesses can be achieved with carbon bres, while polymer bres (PE-, aramid bres) give
the composite particularly high impact strength. It is interesting to note that polymer
bres have much greater elastic stiness and strength than normal polymers. This is
due to the extreme stretching of the polymer chains along the ber direction, which can
be imagined schematically as shown in Fig. 2.43. Accordingly, especially the covalent
bonds along the polymer chains are stressed when pulling at a ber. Because these
are much stronger than the dipole bonds between the chains, the elastic stiness and
strength is much greater. This eect is also exploited, for example, with nylon thread,
which is nothing more than a stretched polyamide material.
Table 2.5: Properties of reinforcing bres.
Material
ϱ /(g/cm3 )
E /GPa
Rm /MPa
A /%
1.8...5.3
Glass bres
2.5...2.6
69...85
1500...4800
Aramid bres
1.4...1.5
65...147
2400...3600
1.5...4.0
0.97
62...175
2200...3500
2.7...4.4
1.75...2.2
140...820
1400...7000
0.2...2.4
Polyethylen bres
Carbon bres
Figure 2.43: Structure of a polymer bre, consisting of bre bundles [26].
Examples of important polymers
Because of the great importance of thermoplastics, some important polymers of this
material group are discussed in more detail in this chapter. Important parameters
are summarized in Tab. 2.3.3. Fundamentally, one distinguishes between so-called
commodity plastics and engineering plastics. The former are inexpensive to produce
but have limited mechanical properties, i.e. they are either not very strong or brittle.
The four most important representatives of this group are polyethylene, polypropylene,
polyvinyl chloride and polystyrene. Together they cover about 85 % of the total market
54
2 Elastic stiness, atomic bonding and material structure
for thermoplastics. Their eld of application is mainly outside mechanical engineering (construction, electrical and packaging industry, household goods). But one nds
also applications in mechanical engineering. By contrast, engineering plastics such as
polyamide, polyoxymethylene or polycarbonate have improved mechanical properties.
In the following, important semi-crystalline thermoplastics will be discussed rst, before examples of amorphous thermoplastics are given.
Semi-crystalline thermoplastics
Polyethylene (PE)
As already mentioned, PE is a semi-crystalline thermoplastic which has a high crystalline
content due to its simple chain structure. Since the dipole bonds between chains are very
weak, the Youngs modulus, the yield strength Rp , the glass transition temperature Tg ,
and the melting temperature Tm are comparatively low (Table 2.3.3). It is therefore a low
strength material, but inexpensive to produce and very easy to form. For this reason, PE
products are often used in the household sector (bottles, pipes, containers, etc.) and the
construction industry (e.g. drinking water and sewage pipes). An application example
in mechanical engineering is the lower engine compartment cover, which is used in newer
cars to reduce ow resistance. Likewise, the cups of hip joint implants are partly made
of PE, because this material has a very low coecient of friction and is, thus, very
suitable as a counterpart to the metallic or ceramic femoral head. In this case, so-called
UHMW PE is used (UHMW: ultra high molecular weight), which has an improved wear
resistance compared to regular PE due to its extremely long molecular chains4 . This is
easy to understand, since with increasing chain length (i.e. increasing entanglement) it
becomes more and more dicult to rip a chain out of its surrounding. However, the melt
viscosity also increases with increasing chain length, so that processing in the molten
state becomes increasingly dicult.
A distinction is often made between so-called LD-PE (LD: low density) with a crystalline content of 45 % - 55 % and HD-PE (HD: high density) with a crystalline content
of 70 % - 80 %. Since the crystalline areas are more densely packed than the amorphous
areas, the density increases with the crystalline portion and due to the stronger and
more numerous dipole bonds in the crystalline than the amorphous areas, the strength
and elastic stiness are also greater in case of HD-PE. In terms of chain structure, the
two variants dier in that the chains in LD-PE are not linear, as we have imagined so
far, but individual hydrogen atoms are replaced by short polyethylene side chains (Fig.
2.44). This branched structure of the chain makes regular folding more dicult and
consequently the crystalline content decreases. Since the branching also increases the
bond spacing, it is understandable why LD-PE also has slightly lower glass transition
and melting temperatures.
4
UHMW PE has 1 − 2 · 105 monomer units along a molecular chain, whereas HD-PE and LD-PE have
103 to 104 .
55
2 Elastic stiness, atomic bonding and material structure
(a) unbranched
(b) branched
Figure 2.44: Unbranched and branched polymer chains. Due to the presence of side chains the
formation of a crystalline structure is hindered [26].
Polypropylene (PP)
This material, which also belongs to the semi-crystalline thermoplastics, is very similar
to PE. However, the CH3 side group is much larger than a single hydrogen atom. If the
chain wants to rotate along the C-C bonds, this is more dicult, because this large side
group has to be rotated as well, which can get caught on neighboring chains more easily.
This spatial obstruction leads to less exible chains and thus to higher strength and
elastic stiness. For the same reason, the material only softens at higher temperatures,
i.e. Tg and Tm increase.
(a) isotactic
(b) syndiotactic
(c) atactic
Figure 2.45: Geometric arrangement of side groups using the example of polypropylene (PP).
Figure (a) shows the isotactic arrangement in which the CH3 side groups are regularly arranged on the same side, Figure (b) the syndiotactic arrangement with
alternating CH3 side groups and Figure (c) the disordered, atactic arrangement
[26].
Polypropylene has one CH3 side group per monomer unit. There are basically four
dierent positions available for this and a distinction is therefore made between dierent
56
2 Elastic stiness, atomic bonding and material structure
arrangement types. If the CH3 group ends at the same position for each monomer unit
(Fig. 2.45 (a)), this is referred to as an isotactic arrangement. If, on the other hand, the
CH3 group alternates regularly on dierent sides of the carbon chain, the arrangement is
called syndiotactic (Fig. 2.45 (b)). If it is completely irregular (Fig. 2.45 (c)), it is called
atactic. It is easy to see that the folding to crystalline regions is the easiest in case of the
isotactic arrangement and the most dicult for the atactic one. Accordingly, isotactic
polypropylene, which is technically the most important, has a crystalline portion of 70 %
- 80 %, whereas in an atactic arrangement it is only 50 % - 60 %.
The elds of application of PP are similar to those of PE. In mechanical engineering,
for example, car bumpers can be made of PP.
Polyamide (PA)
Like PE and PP, polyamides also have a very simple chain structure and are therefore semi-crystalline. However, the NH-CO group produces hydrogen bonds between
the polymer chains, as shown exemplarily in Fig. 2.46. Hereby, the electronegative
oxygen produces a negative charge surplus while the likewise electronegative nitrogen
atom causes the corresponding hydrogen atom to be positively polarized. Therefore,
the binding forces between the polymer chains are much greater than is the case with
PE or PP. Accordingly, Tg , Tm and the mechanical properties increase strongly (Table
2.3.3). Since the polyamides are also ductile, they play an important role in mechanical
engineering. For example gear wheels, housing parts such as motor covers and cooling
fans for automobiles are made of PA. It is often also reinforced with glass bres, which
signicantly increases the elastic stiness and strength.
Figure 2.46: Molecular structure of PA 6 and PA 66 [12].
CH2 chain segments of dierent lengths can be between the NH-CO groups. This
can inuence the density of the hydrogen bonds and, thus, the mechanical and thermal
properties. Accordingly, polyamides are designated by the number of carbon atoms
between the nitrogen atoms. PA6, for example, has the structural formula [NH-CO(CH2 )5 -]n , PA12 the formula [NH-CO-(CH2 )11 -]n and PA66 has the formula [NH-(CH2 )6 NH-CO-(CH2 )4 -CO-]n . In the latter case, the CO group is located once to the left and
57
2 Elastic stiness, atomic bonding and material structure
once to the right of the NH group, which is why the structure may have two CH2 chain
segments of dierent lengths.
A disadvantage of polyamides is that they tend to absorb water. This is easy to
understand, since in both cases the hydrogen bond is the decisive intermolecular bond
type and water molecules feel particularly comfortable in polyamides for this reason.
The more hydrogen bonds are formed between dissolved water molecules and polymer
chains, the less is available for bonding between the polymer chains. In addition, water
absorption increases the distance between the polymer chains, which also weakens the
bonds. As a result, strength and elastic stiness decrease. Therefore, the mechanical
properties of polyamides are dependent on the ambient humidity, which must be taken
into account in the component design.
Polyoxymethyle (POM)
POM also has a very simple monomer unit and, consequently, a high crystalline content
of 70 % - 80%. The special feature is that the chain is built by a sequence of carbon
and oxygen atoms. In this particular case, relatively strong dipole bonds are formed
between the oxygen atoms along the chain and the hydrogen atoms bound to the carbon
atom. Therefore, the strength and stiness values are much higher than those of PE or
PP (Table 2.3.3). They almost reach the level of polyamides. This is why POM and
PA compete with each other and their elds of application are similar. POM has the
advantage that it does not tend to absorb water like PA. However, it is less temperature
resistant.
Amorphous thermoplastics
Polymethyl methacrylate (PMMA)
This plastic, often referred to as plexiglass, is characterised by a very large side group.
Due to the spatial obstruction already described above this impedes chain rotation. Consequently, the polymer chains are relatively sti, so that regular folding to crystalline
regions during solidication is not possible. Thus, PMMA is an amorphous thermoplastic polymer. However, due to the relatively high stiness of the polymer chains, the
glass transition temperature of about 100 C is also signicantly higher than is the case
with the semi-crystalline thermoplastics discussed above. As to be expected, PMMA
behaves strong but also comparatively brittle at ambient temperature. Since PMMA is
optically transparent and has a very good resistance to ultraviolet radiation, it is used,
for example, as glazing in aircrafts.
°
Polycarbonate (PC)
Polycarbonate has a complex polymer chain structure (Table 2.3.3). In addition to
stiening aromatic rings, there are also exible elements (especially the oxygen atoms
along the chain), so that not only a major softening occurs when the glass transition
temperature of about 140 C is reached, but also a minor softening at about -60 C. If the
°
°
58
2 Elastic stiness, atomic bonding and material structure
Youngs modulus is plotted over the temperature, this is noticeable by a small S-shape
in the curve. As already discussed, certain chain segments become already exible above
-60 C, giving the polymer a high impact strength, whereas the still immobile segments
provide strength. This results in superior mechanical properties compared to PMMA,
while still maintaining good optical transparency. Consequently, PC is used, for example,
to produce headlight glazings for automobiles, CDs and protective helmets.
°
Polyethylene terephtalate (PET)
PET has a similar chain structure to PC. It therefore belongs those amorphous thermoplastics, which combine good toughness and strength at ambient temperature. Since
there are more exible elements along the chain than with PC, the glass transition
temperature is somewhat lower than with PC, but processability is easier for the same
reason. Beverage bottles are, for example, made from PET.
Polyimide (PI)
These amorphous thermoplastics are an example of how the stiening of the polymer
chain by aromatic rings can be taken to the extreme. Accordingly, polyimides have very
high glass transition temperatures and the highest possible operating temperatures of up
to 250 C among all thermoplastics. However, polyimides are also particularly dicult
to process. Components made from these materials are therefore very expensive.
°
59
H n
H
C
methylene
(POM)
C
(PS)
H
H
Polystyrene
H
H
Polyoxy-
H
C
(PA)
C
H
O
n
n
(CH2)x
H n
H
O
C
C
(PP)
Polyamide
H CH3
Polypropylene
N
C
C
(PE)
H
H
formula
Structural-
Polyethylene
Name
n
‰
74 110
−18 −8
44 56
−25 −15
−110 −90 ∗
/
Tg
‰
160 184
210 220
150 175
125 132
/
Tm
140
115
110
103
97
‰
λ
/ W/mK
20
3 · 10
21
3.3 · 1020
1.4 · 10
1.5 · 1019
3 · 1023
3.3 · 1022
3 · 1024
3.3 · 1022
76 202
144 149
122 180
126 198
0.12 0.13
1025 1027
90 153
Amorphous thermoplastic polymers
0.22 0.35
0.23 0.25
0.11 0.17
0.40 0.44
1.2 2.6
2.5 5
2.6 3.2
0.9 1.6
0.6 0.9
E
/ GPa
−6
/K
α
/ 10
ρe
/ µΩ cm
Semi-crystalline thermoplastic polymers
/
Tmax
29 56
49 72
50 95
21 37
18 29
/ MPa
Rp
bridge Engineering Selector Edu Pack 2008, Engineered Materials Handbook, Vol. 2, ASM International.
36 57
60 90
90 165
28 41
21 45
/ MPa
Rm
1.2 3.6
10 75
30 100
100 600
200 800
/ %
A
sion, E : Young's modulus, Rp : Yield strength, Rm : Tensile strength, A: Strtain to rupture, ρ: Density), References: Cam-
maximum service temperature, λ: Thermal conductivity, ρe : Specic electrical resistivity, α: Coecient of thermal expan-
Table 2.6: Characteristic properties of some important polymers. (Tg : Glass transition temperature, Tm : Melting temperature, Tmax :
ρ
3
1040 1050
1390 1430
1120 1140
890 910
939 960
/ kg/m
Polyester
Polybutadiene
(PET)
thalate
-tereph-
Polyethylee
(PC)
carbonate
Poly-
(PMMA)
methacrylate
Polymethyl-
(PVC)
chloride
Polyvinyl-
Name
C
R
O
C
C
H H
O
C
n
II
H
C
H
n
n
H H n
O C C
H
O
O
C O R
O
C
C
H
H
H
I
CH3
O C
C
H C
CH3 n
O
O
C
H CH3
CH3
O
C
C
Cl H n
H
H
formula
Structural-
147 207
−108 −75
68 80
142 205
85 165
75 105
‰
Tg
/
‰
Tm
/
150
120
87
144
57
70
/
‰
Tmax
λ
3 · 1021
3.3 · 1020
1020 1021
3 · 1024
3.3 · 1023
1020 1022
1020 1022
0.29 0.3
3 · 10
19
3.3 · 1018
Thermosetting polymers
0.2 0.22
99 180
130 150
115 119
120 137
72 162
100 150
2.1 4.4
0.001 0.0021
2.8 4.1
2 2, 4
2.2 3.8
2.1 4.1
E
/ GPa
−6
/K
α
/ 10
ρe
/ µΩ cm
Elastomers
0.14 0.15
0.19 0.22
0.08 0.25
0.15 0.29
/ W/mK
Rp
33 40
1.5 7
57 62
59 70
54 72
35 52
/ MPa
Rm
41 90
1.5 7
57 72
60 72
54 80
41 65
/ MPa
A
2 2.6
500 550
30 300
70 150
2 10
12 80
/ %
ρ
1040 1400
910 940
1290 1400
1140 1210
1160 1220
1300 1580
/ kg/m
3
3 Strength and phase diagrams
Please note: To understand chapters 3.3.3 and 3.3.4, it is helpful to be familiar with phase
diagrams (Chapter 3.4). This is absolutely necessary for Chapter 3.3.5. It is therefore
recommended that after Chapter 3.3.2, you rst familiarize yourself with Chapter 3.4
before working on Chapters 3.3.3 - 3.3.5.
3.1 The ideal strength
In chapter 2 we have seen that the elastic stiness of materials, which is an important
property in the construction and design of components, is decisively dependent on the
structure of the materials and the atomic bonds. Because the atoms in ceramics and
metals are more strongly bound than in polymers, the former have a higher Youngs
modulus than polymers.
Designs must also meet strength criteria. They are generally required to only deform
elastically, but not plastically. With the yield strength Rp and the tensile strength
Rm determined in the tensile test, important characteristic values have already been
introduced and discussed in connection with the polymers. Here, too, it turned out that
strength and structure of the materials are closely related. For example, polyamides are
much stronger than polyethylene due to stronger dipole bonds. HD-PE is stronger than
LD-PE due to a larger crystalline fraction. It also became clear that plastic deformation
of polymers is caused by the mutual sliding of the polymer chains, i.e. a mechanism
that is fundamentally dierent from the mechanism of elastic deformation. Analogous
dependencies can also be found in metals and ceramics. However, aspects come into play
here that were not yet accessible. For this reason, the questions of what determines the
strength of metals, what causes plastic deformation when the yield strength is exceeded
and how the strength of metals can be raised are now discussed in chapters 3.2 and 3.3.
A thought experiment is helpful as an introduction to this topic. Here the question
is asked of how strong a material should be, i.e. what tensile stress it should withstand
before it breaks. If we look at Fig. 2.12, we could arrive at a simple answer: As the
stress increases, the atomic bonds are increasingly loaded and the atoms are deected
from their equilibrium distance. If this deection is about 1.25 times the equilibrium
distance, the restoring force that tries to move the atoms back to their original position,
reaches a maximum. If this force is exceeded, the bonds break apart. Therefore, the
material cannot withstand a higher load without rupturing. From σ = E ·ε and ε ≈ 0.25,
the ideal strength ideal σideal ≈ E/4 follows. More precise calculations with real atomic
potentials result in approximately E/10.
Accordingly, a steel or zirconia ceramic with a Youngs modulus of ≈ 200,000 MPa
each should have a tensile strength of about 20,000 MPa. However, simple deep-drawing
steels only reach values of about 300 MPa and also the strength of high-strength steels is
62
3 Strength and phase diagrams
no more than 2,000 MPa. The compressive strength of zirconia is 4,000 MPa (Abb. 3.2).
Yet, under a tensile load, the strength is considerably lower and the ceramic breaks at an
elongation of about 0.5 %, instead of the expected 25 %. Metals, on the other hand, do
achieve rupture elongations of 25 %, but by plastic and not elastic deformation. Polymers
which usually have Youngs moduli of a few gigapascals, should have tensile strength
values of a few hundred megapascals, which are, however, not achieved in reality.
These examples show that there may be something wrong with the above consideration. A central assumption was that all atoms are loaded equally and failure occurs
when all atomic bonds are loaded at the same time with the maximum possible value.
However, this idea is obviously not right. For example, if we consider plastic deformation of polymers by slipping of polymer chains past each other it is not necessary for
the slipping chain to overcome all dipole bonds to adjacent chains at the same time to
move a little bit. Rather, initially only a short segment of a polymer chain moves and
this displacement then propagates through the chain, so that nally the whole chain has
moved. So only few bonds are overcome at the same time. Thus, it is understandable
that the plastic deformation of polymers begins far below the theoretical strength. This
plastic deformation is ultimately responsible for failure, so that the tensile strength Rm
is also far below the theoretically expected value.
This fact can be illustrated once again with an analogue, the carpet trick. For this,
imagine a very large carpet that is to be moved a little. If one wanted to move it in
one piece, the eort would be very high, because the frictional forces would have to be
overcome over the entire surface. If, on the other hand, one lifts the carpet only on one
location (i.e. the bonds are only released locally there), this fold can be pushed through
much easier, so that in the nal eect the whole carpet is moved a little (Fig. 3.1).
Accordingly, the polymer chain is not moved as a whole, but shifted segment by segment.
Hereby, the shifting propagates through the entire chain. Just as the displacement of
the carpet is permanent, so is the displacement of the polymer chain: When the load is
relieved, it does not move back. Thus, a plastic deformation results.
Figure 3.1: Analogy to the sliding of polymer chains: Movement of a carpet, adapted from [4].
63
3 Strength and phase diagrams
Also with metals, the carpet trick is the reason why the ideal strength is far from
being achieved. Here, certain crystal defects, so-called dislocations, are responsible.
They are discussed in the next chapter.
3.2 Dislocations in crystals
In chapter 2.3 crystals were dened as periodically built structures, which result by
periodic arrangement of the unit cell in all three dimensions. However, this perfect
order is not maintained during solidication from the melt. Often half-planes are formed
(Fig. 3.2), which end in the crystal. They are called edge dislocations. The perfect crystal
order is disturbed where the inserted half-plane ends. Therefore, a dislocation is a onedimensional, i.e. line-type, defect while a grain boundary is a two-dimensional defect.
Consequently, a dislocation is represented by a so-called dislocation line. In case of an
edge dislocation, it runs along the end of the inserted half-plane as illustrated in Fig. 3.5
(a).
(a) Crystal containing a half-plane.
(b) Perfect crystal.
Figure 3.2: Edge dislocation and denition of the Burgers vector, adapted from [14].
If a shear stress is applied to the crystal (Fig. 3.3) and if the edge dislocation is moved
from left to right through the crystal, the upper half of the crystal is shifted by the
vector b relative to the lower half of the crystal.
One calls b the Burgers vector of the dislocation. As shown in Fig. 3.2 (a), the size
and direction of the Burgers vector can be recognized by moving along a line around the
dislocation line and doing the same number of steps down/up and right/left. One will
notice that the line is not closed. The vector needed to close the line is the burger vector.
Characteristic of edge dislocations is that the Burgers vector is oriented perpendicular
to the dislocation line.
Fig. 3.3 illustrates that even in the case of plastic deformation of metals the carpet
64
3 Strength and phase diagrams
(a)
(b)
Figure 3.3: Movement of an edge dislocation through a crystal, adapted from [4, 26].
trick is used. To displace two crystal halves permanently past each other, not all bonds
have to be released at the same time (and reestablished after the displacement). Instead,
this only occurs at the position of the dislocation. Hereby, the dislocation moves by the
Burgers vector b. By repeating this movement many times, the crystal halves nally
slide past each other. The resulting deformation is plastic, because dislocations do not
move back on unloading. Because of this carpet trick, plastic deformation occurs at
stresses ≪ E/10.
As Fig. 3.2 shows, the atoms near the dislocation line are displaced from their equilibrium position. Where a half-plane is inserted, the atoms are pushed together, i.e. the
atomic bonds are compressed. Accordingly, there is a compressive stress. But where the
half-plane is missing, the atomic spacing is too big. The atomic bonds are stretched and
there is a tensile stress. Therefore, a dislocation is surrounded by a distortion eld which
increases the energy of the crystal. Since the displacement of atoms in the vicinity of
the dislocation is proportional to the Burgers vector b and the force for the displacement
of an atom is proportional to G · b (Fig. 2.12), the stored energy per unit length of a
dislocation must obviously be proportional to G · b2 . In fact, one nds for this so-called
line tension T of the dislocation.
65
3 Strength and phase diagrams
T ≈
T
b
Gb2
.
2
(3.1)
line tension
length of the Burgers vector
Eq. 3.1 shows that energy is required to produce a dislocation. Because nature tries
to minimize this energy expense, which increases with b2 , the crystal lattice only contains dislocations with the smallest possible burger vector. A further condition for b is
apparent from Fig. 3.3. The Burgers vector must be a translation vector that translates
the crystal lattice into itself. Therefore, it must start at one atomic position and end
at another. Otherwise, dislocation motion would displace atoms to positions that are
not allowed in the crystal lattice. In the face-centered cubic crystal lattice the smallest
possible distance between atoms is that between a corner atom of the unit cell and an
atom arranged in the middle of a side-face. Correspondingly, only burgers vectors exist
which connect a corner atom with an atom arranged in the middle of a side-face. Such
a burger vector is shown in Fig. 3.4 (a). For the same reason, burger vectors in the
body centered cubic crystal lattice always connect a corner atom of the unit cell with
the atom arranged in the centre of the unit cell, as shown in Fig. 3.4 (b). Plastic deformation always occurs by motion of dislocations with these burger vectors. Also, the
dislocations only move along certain planes. These so-called slip planes are always the
closest packed planes in the respective crystal lattice (i.e. the planes that have the most
atoms per area) because dislocation motion is easiest along these planes. Fig. 3.4 shows
one slip plane for the face-centered cubic (a) and body centered cubic (b) crystal lattice.
Lattice constants, i.e. the edge lengths of the unit cells, and burgers vectors are listed
in Table 3.1 for some metals.
Table 3.1: Lattice constants and Burgers vectors of some metals.
Metal
Structure
Lattice constant
Fe
bcc
2.87 × 10
2.48 × 10
W
bcc
3.17 × 10
2.74 × 10
−10
−10
−10
Burgers vector
−10
−10
−10
Al
fcc
4.05 × 10
2.86 × 10
Ni
fcc
3.52 × 10
2.48 × 10
−10
−10
Dislocations in ceramics have larger burger vectors, because the crystals are more
complicated. This is associated with a higher line tension. Accordingly, greater forces
are required to move dislocations through the crystal lattice. In addition, the dislocation
movement is made more dicult by covalent bonds. Therefore, dislocation motion in
ceramics is virtually immobile at ambient temperature. This explains why ceramics fail
in a brittle manner and are not plastically deformable like metals.
In addition to the edge dislocation shown in Fig. 3.5 (a), where b is perpendicular to
the dislocation line, so-called screw dislocations also occur (Fig. 3.5 right). As with a
66
3 Strength and phase diagrams
(a) Face-centered cubic crystal
structure with slip plane and
Burgers vector.
(b) Body centered cubic crystal
structure with slip plane and
Burgers vector.
Figure 3.4: Representation of a slip plane and a Burgers vector, adapted from [26].
spiral staircase, the planes are displace parallel to the dislocation line. Consequently, the
Burgers vector b is oriented parallel to the dislocation line. Figure 3.6 shows schematically the slip process when a screw dislocation moves through the crystal.
(a) Edge dislocation
(b) Screw dislocation
Figure 3.5: Comparison of the two basic types of dislocations. The drawn straight lines indicate the course of the dislocation lines.
67
3 Strength and phase diagrams
Figure 3.6: Slip process when moving a screw dislocation through the crystal.
3.2.1 Forces between dislocations
Dislocations distort the crystal lattice and are therefore surrounded by a stress eld.
At the location of the inserted half-plane of an eddge dislocation, the crystal lattice is
compressed resulting in compressive stresses. Where the inserted half plane is missing,
the crystal is widened. So there are tensile stresses. Due to these stress elds, dislocations
exert forces on each other. If two dislocations with the same sign, i.e. the inserted half
plane points in the same direction, are positioned on the same slip plane, they repel each
other (Fig. 3.7). In contrast, dislocations with inverse sign attract each other. If they
are positioned on the same slip plane, they annihilate each other.
Figure 3.7: Interaction between dislocations.
A frequently required quantity is the force F , which is exerted on a dislocation by an
external shear stress. Let the crystal shown in Fig. 3.8 be represented by the dimensions
l1 and l2 . Then, external work τ · l1 · l2 · b is expended when the upper crystal half slips
by the Burgers vector b. Herby, τ · l1 · l2 is the external force acting on the slip plane
and b the resulting displacement. The displacement is caused by the movement of an
edge dislocation of length l1 over the distance l2 . If F is the force on the dislocation,
68
3 Strength and phase diagrams
Figure 3.8: Illustration of a crystal loaded by a shear stress τ which slides by Burger's vector b
[15].
then the expended work is F · l2 . Equating both terms leads to:
(3.2)
F =τ ·l·b .
l
length of the edge dislocation
3.3 Measures to increase strength
In Section 3.2 it became clear that dislocations are the carriers of plastic deformation in
metals. The movement of dislocations is associated with the breaking and reestablishment of bonds at the location of the dislocation core. Therefore, a critical shear stress τ0
must be exceeded to enable dislocation motion through the crystal. The strength of pure
metals, which is approx. 1 MPa to 10 MPa, results from that. If one wants to further
increase the strength of metals, that is, their resistance to plastic deformation, measures
must be taken, to further hinder the movement of dislocations. These are discussed in
the following.
3.3.1 Work hardening
The simplest strengthening method is work hardening (also refered to as strain hardening). This is based on the fact that metallic materials become stronger when they are
plastically deformed. This increase in strength with increasing plastic deformation can
also be seen from the stress-strain curve: If one unloads a plastically deformed material
and loads it again, the yield strength (ReL or Rp0,2 ) rises to the previously achieved stress
level. The reason for the increasing strength is that the dislocation density has greatly
increased due to plastic deformation. Because of that, the dislocations get more and
more entangled, thus making dislocation motion more and more dicult (see Fig. 3.9).
The strength increases with the square root of the dislocation density ρ. Accordingly, the increase in strength due to work hardening ∆σwh compared to the imaginary
dislocation-free state σ0 (which does not occur in reality) is:
69
3 Strength and phase diagrams
∆σwh = const ·
∆σwh
ρ
√
(3.3)
ρ.
Strength increase due to work hardening
∆σ = σ1 - σ0
dislocation density
Figure 3.9: A dislocation (gray) moves through a eld of obstacles caused by other dislocations, adapted from [26].
In Fig. 3.10, the yield strength Rp0,2 , the tensile strength Rm and the elongation at
fracture A is plotted for a copper alloy containing 30 % nickel as a function of the degree
of rolling. The degree of rolling indicates how much the thickness of a sheet is reduced by
plastic deformation during rolling. A considerable increase in strength with increasing
degree of rolling is clearly visible. Since such forming processes are easy to realize
and often part of the manufacturing process, work hardening is a particularly simple
measure to increase strength. This is a particular advantage. However, an important
disadvantage is also evident in Fig. 3.10: The elongation at fracture, i.e. ductility, of the
material decreases drastically due to work hardening.
This shows that strain hardening is a cost-eective way to increase the strength of
relatively soft materials with high ductility The resulting loss of ductility can often
be accepted. High-strength materials, on the other hand, already exhibit a relatively
low elongation at fracture. In this case, there is no chance of achieving an additional
strengthening contribution through work hardening.
3.3.2 Strengthening by grain size reduction
In reality, crystalline materials do not consist of one crystal with one orientation, but of
many crystals with dierent orientations. Since dislocations are bound to their slip planes
(this corresponds to the closest packed plane of the unit cell, see section 2.3.1), they
cannot easily pass from one grain to the neighbouring grain. Therefore, grain boundaries
act as obstacles where dislocations accumulate (Fig. 3.11(a)) and this strengthening
mechanism is, thus, also referred to as grain boundary strengthening. The ner the
grain, the more grain boundaries, i.e. obstacles, stand in the way of the dislocations
and the stronger the material. Correspondingly, strengthening by grain size reduction
results.
70
3 Strength and phase diagrams
Figure 3.10: Strength and strain to rupture of a copper alloy containing 30% nickel as a function of the degree of rolling, adapted from [6].
As more and more dislocations accumulate at a grain boundary, they can exert such
a high stress on the neighbouring grain that it in turn emits dislocations from the grain
boundary and the deformation is propagated through the metal (Fig. 3.11(b)).
Empirically, there is an increase in strength ∆σgsr compared to an imaginary reference
material with innite grain size according to the so-called Hall-Petch relation:
k
∆σgsr = √ .
d
∆σgsr
k
d
(3.4)
Strengthening by grain size reduction
Hall-Petch-constant (also called kHP )
grain size (diameter)
Typical values for the Hall-Petch constant are:
Steel
N
19 mm
3/2
Aluminium
N
3.5 mm
3/2
Copper
N
5.5 mm
3/2
As can be seen from the Hall-Petch constants mentioned above, the increase in strength
by grain size reduction, especially in the case of metals with face-centered cubic crystal
structure, is rather small. However, strengthening by grain size reduction has a signicant advantage: while the elongation at fracture decreases with all other strengthening
71
3 Strength and phase diagrams
Slip plane
Grain 2
Slip plane
Grain 1
(a) Dislocations pile up in grain 1 and
are blocked at the grain boundary.
Slip plane
Grain 1
(b) Increasing dislocation pile up leads
eventually to dislocation motion in grain
2 along a dierent slip plane.
Figure 3.11: : Dislocations in grain 1 accumulate at a grain boundary due to the applied shear
stress and eventually cause dislocation movement in grain 2 on a dierent slip
plane, adapted from [26].
measures, it increases with this measure. For this reason, the aim is to use materials with
the smallest possible grain size. Grain sizes between 0.01 mm and 0.1 mm are typical.
3.3.3 Solid solution strengthening
A good way to increase the strength of metals is to contaminate them with other
elements. The alloyed elements go into solution in the solid, similar to how sugar dissolves
in water. Now that the crystal is made of a mixture of several elements, one speaks of a
solid solution and solid solution strengthening. Aluminium strengthened by magnesium
is a technically relevant example. As the Al-Mg phase diagram (Fig. 3.12) shows, approx.
1 % magnesium is soluble in aluminium at room temperature. In practice, however, up to
5 % magnesium will remain in solid solution if cooled quickly from elevated temperature
(e.g. 400 C), as the driving force for the formation of the equilibrium phase Al3 Mg2 is
not sucient.
The magnesium atom with an atomic radius r of 0.16 nm is approx. 10% larger than
the aluminium atom (r = 0.143 nm) and, therefore, distorts the lattice locally. Similar to
the interaction between two dislocations, a dislocation feels this distortion eld. Thus,
an additional force is required to move the dislocation past the dissolved magnesium
atoms. This increases the yield strength Rp0,2 with the magnesium content by up to
100 MPa as shown in Fig. 3.13.
Fig. 3.13 shows that Rp0,2 increases approximately with the square root of the concentration c of the dissolved element. Correspondingly, the general rule for the increase
in strength due to solid solution strenghtening ∆σsss compared to the non solid solution
strengthened material (c=0) is:
°
∆σsss = const ·
∆σsss
c
√
(3.5)
c,
Strength increase by solid solution strengthening
Concentration of the dissolved element
72
3 Strength and phase diagrams
Figure 3.12: Binary Al-Mg phase diagram, adapted from [24].
Figure 3.13: Yield strength Rp0,2 of aluminium as a function of the dissolved magnesium content, adapted from [1].
73
3 Strength and phase diagrams
Hereby, c is the concentration of the alloyed element. In the case under consideration, magnesium atoms replace the aluminum atoms in the host lattice. This is called
substitutional solid solution strengthening (Fig. 3.14(a)).
The situation is quite dierent when carbon is added to iron (steel = iron alloyed with
carbon). Since the carbon atom (r = 0.077 nm) is much smaller than the iron atom
(r = 0.124 nm), it is placed in the interstices, i.e. the gaps between the host atoms. This
is known as interstitial solid solution hardening (Fig. 3.14(b)). Since the carbon atom
does not t exactly into these interstices, this again leads to lattice distortion and, thus,
to an increase in strength. As an interstitially dissolved atom, carbon is already relatively
easily mobile in the host lattice at ambient temperature. Since it is somewhat too large,
it tends to move to the area of the dislocation core where the bonds are stretched, thus,
taking up an energetically favourable position. At ambient temperature, a few days are
required for the carbon atoms to travel the distance to the dislocations. The dislocations
are now locked by the carbon atoms and a relatively high level of stress is required before
dislocations can be torn o from the carbon atoms. This causes the upper yield strength
ReH shown in Fig. 3.15. Once dislocations have been torn o, they are already mobile
at a lower stress, the lower yield strength ReL . This special feature of steels leads to
inhomogeneous deformation. It produces so-called Lüders bands, also refered to as slip
bands or stretcher strain marks. When deep-drawing sheet metal these stretcher strain
marks disturb the surface nish. To avoid this, sheets can rst be rolled to mobilize all
dislocations and then deep-drawn.
(a) Substitutional solid solution.
(b) Interstitial solid solution.
Figure 3.14: Dierent types of solid solutions in a crystal.
3.3.4 Particle strengthening
If a solid solution, consisting of the host atoms A and the dissolved atoms B, is cooled
down, it may happen that the B atoms can no longer be completely dissolved. Precipitation of B-rich particles then occurs. This is illustrated in Fig. 3.16. While Fig. 3.16 a
shows a solid solution in which the solute atoms (bright) are randomly distributed on
the sites of the host lattice, Fig. 3.16 b, c show two examples of precipitates in a crystal
74
3 Strength and phase diagrams
Figure 3.15: Technical stress-strain-diagram of steel.
lattice. In Fig. 3.16 b, all lattice planes of the matrix continue in the particle and vice
versa. Then one speaks of coherent particles. This situation can occur when the crystal
structures are very similar. In the case shown here, they are even identical, so that matrix and precipitate particles dier only because of their dierent compositions. In Fig.
3.16 c, on the other hand, the two crystal structures are so dierent that no lattice plane
of the matrix is continued in the particle and vice versa. Then one speaks of incoherent
particles. Something analogous to the case described here happens when you dissolve a
lot of sugar in hot water and then let it cool down. At some point, the sugar molecules
are no longer completely soluble and sugar crystals precipitate out. The temperature at
which the solute is just soluble is called the solvus temperature or solubility limit.
(a) Solid solution.
(b) Coherent precipitate
(c) Incoherent precipitate.
Figure 3.16: Comparison between solid solution and dierent precipitates in a crystal lattice.
As the example of Al-Mg in section 3.3.3 has already shown, it is necessary to fall below
the solubility limit in order to form precipitates. Under certain circumstances, however,
this still does not happen. To understand this, it is necessary to look more closely at
the energetic conditions involved in the formation of precipitates. Two terms need to be
considered. First, we imagine that a solid solution is cooled below the solubility limit.
75
3 Strength and phase diagrams
Then, the transformation of the volume V from the state of disequilibrium (supersaturated solid solution) to the equilibrium state (solid solution with reduced concentration
of dissolved atoms plus precipitates) is associated with a decrease in free enthalpy. This
is what nature strives for. If ∆Gv is the free enthalpy dierence between the nal state
and the initial state, then ∆Gv < 0 applies. The absolute value of ∆Gv is greater, the
greater the supercooling below the solubility limit is. For spherical particle with radius
r, the following applies:
4
∆Gv = πr3 ∆gv ,
3
∆Gv
r
∆gv
(3.6)
Dierence in free enthalpy
Particle radius
specic change in free enthalpy
∆gv represents the change in free enthalpy related to the precipitated volume. At the
same time, an interface is created due to the formation of a particle (Figure 3.16). This
is associated with an increase in free enthalpy. For a spherical particle, the following
holds:
(3.7)
∆Gs = 4πr2 γs .
∆Gs
γs
Dierence in free enthalpy
Specic interphase energy
The specic interfacial energy γs is typically between 0.2 J/m2 and 2 J/m2 . It is
particularly low for coherent particles, but particularly high for incoherent particles.
This can be easily understood from Fig. 3.16 b, c. All atoms of a precipitate particle
would like to nd conditions as it is the case for the atoms inside the particle. In the case
of an incoherent particle, on the one hand, an atom directly at the interface with the
matrix "notices" that both the chemical composition and the coordination with nearest
neighbors are not as it would be desirable. Accordingly, the formation of interface
is very unfavorable. A large interfacial energy is generated. In the case of coherent
precipitation, on the other hand, the coordination with nearest neighbors ts at the
interface. Only the chemical composition does not match. Consequently, the interfacial
energy is signicantly lower.
Comparing Eq. 3.6 with Eq. 3.7, it is obvious that ∆Gv is proportional to r3 , whereas
∆Gs increases with r2 . Since the quadratic term predominates for small particle radii,
an initial expenditure of free enthalpy is necessary to form a particle. This is shown in
Fig. 3.17, where besides ∆Gv and ∆Gs also the total change of the free enthalpy ∆G
with
4
∆G = ∆Gv + ∆Gs = πr3 ∆gv + 4πr2 γs
3
76
(3.8)
3 Strength and phase diagrams
Free enthalpy
Effort for
interface
Total energy
(Critical
radius)
Particle radius r
Gain from
chem. bonds
Figure 3.17: Energy balance during precipitation of a particle, after [20, 26].
is shown. Thus, a critical radius r⋆ results, which must be reached before the precipitate continues to grow with a decrease in free enthalpy. Since r* occurs at the maximum
of the curve, the following applies:
d∆G
= 0 = 4πr⋆2 ∆gv + 8πr⋆ γs
dr
(3.9)
i. e.
r∗ = −
r∗
2γs
.
∆gv
(3.10)
Critical radius
According to eq. 3.10, r⋆ becomes smaller, the larger the absolute value of ∆gv is and
the smaller γs is. The negative sign results because ∆gv is negative.
An important result of this observation is that precipitates do not form immediately
despite the existence of a thermodynamic driving force (∆Gv < 0), because the free
enthalpy rst increases until r⋆ is reached. At this point, one may ask how precipitates
can be formed at all, if ∆G increases rst. This is due to the fact that the atoms
can move in the crystal lattice. Responsible for this is another type of defect in the
crystal lattice, namely unoccupied lattice sites, which are called vacancies (Fig. 3.18).
In this context, it is important to realize that the atoms in the crystal lattice are not
at rest but vibrate, and the higher the temperature, the stronger the vibration. The
amplitude of oscillation varies from oscillation to oscillation. In Fig. 3.18, top left, the
77
3 Strength and phase diagrams
atom marked with an arrow has such a large amplitude at the time under consideration
that the atom overcomes the bonding forces to the neighboring atoms and jumps into
the neighboring vacancy. The vacancy and the atom have changed places and moved. In
the next moment (top right image), the same happens for an atom dissolved in the host
lattice (blue) and then again for host atoms (bottom row of images). These processes
will be discussed in more detail in Chapter 4.1. At this point, it is sucient to realize
that atoms can change places in the crystal lattice and that this occurs more frequently
the higher the temperature is. This change of places causes a random movement of the
atoms in the crystal lattice, which is called diusion. As a result, it happens by chance
that a precipitate particle with the critical radius r⋆ forms and then continues to grow
because this is now energetically favorable.
Figure 3.18: Crystal lattice consisting of host atoms (black), dissolved atom (blue) and a vacany. Due to lattice vibrations, atoms can move into a void. As the example
shows, this can also lead to the movement of the dissolved atoms in the crystal
lattice.
We now imagine an aluminum alloy consisting of aluminum and 4 % Cu. As illustrated
by the phase diagram shown in Fig. 3.19, the solubility limit for this alloy is 500 C. At
a temperature T ≥ 500 C, the copper content is completely soluble and a solid solution
is present. Below 500 C, ∆gv is negative. Thus, there is a thermodynamic driving force
for the formation of precipitates, in this case CuAl2 . As mentioned, the precipitates
nevertheless do not form immediately, because the energy hill must rst be climbed.
The question now is how long this takes. First, it is clear that no particles can form at
500 C and above because ∆gv is zero at 500 C and positive above. Below 500 C, on the
other hand, ∆gv is negative and becomes more negative as the temperature decreases.
It is then only a matter of time before precipitates form. Let us rst imagine a relatively
high temperature a little below 500 C. Because the solubility limit is only slightly undershot, the absolute value of ∆gv is small and thus r⋆ is very large. Therefore, it takes
a very long time for a particle with critical radius r⋆ to be formed by random change of
the atom positions, even though the velocity at which the atoms change their places, in
°
°
°
°
°
°
78
°
3 Strength and phase diagrams
the following referred to as place change velocity, is large to the high temperature. This
is shown in Fig. 3.20. The curves labeled 1 and 99 respectively indicate the time
after which 1 % and 99 % of the precipitable volume of CuAl2 has formed at a given
temperature. The two curves thus represent approximately the beginning and the end
of the precipitation process. Also plotted is the solubility limit above which CuAl2 can
never form.
T,°C
single phase
(Al-solid solution)
700
600
Liquid +
Al-solid solution
500
400
300
two phase
(Al- solid solution +
CuAl2-solid solution)
200
100
0
0
4
8 12 16 20 24 28 32 36 40
weight-% Cu
Figure 3.19: Binary phase diagram for the system aluminium-copper (adapted from [26]). The
alloy Al-4Cu is marked.
If we next imagine a low temperature, e.g. room temperature, then the absolute
value of ∆gv is large and r⋆ is very small. However, it takes a very long time for the
precipitation process even in this case, because now the place change velocity of the atoms
is extremely low. On the other hand, as Fig. 3.20 shows, the precipitation process is
fastest at intermediate temperatures because then the place change velocity is relatively
large and r⋆ is relatively small. Thus, the precipitation process is not extremely retarded
because neither r⋆ is huge nor is the place change velocity extremely small. Accordingly,
Fig. 3.20 shows a curve which has the shape of a nose. Such diagrams are called
time-temperature-transformation diagrams (TTT diagrams for short), because they give
information about phase transformations (here the formation of CuAl2 precipitates) as a
function of temperature and time. TTT diagrams thus indicate the rate at which phase
transformations take place.
The outstanding feature of particle strengthening is that particularly large increases in
strength can be achieved by this method. However, this requires very small precipitates
with diameters that are typically a few to a few tens of nanometers. Accordingly, the next
question is how such small diameters can be realized. In this context, two parameters
are important: First, the number of particles with radius r⋆ that form per unit time in
a volume under consideration, hereafter referred to as the nucleation rate. Second, the
growth rate of these particles. Initially, we imagine that the nucleation rate is small and
79
3 Strength and phase diagrams
Figure 3.20: Time-temperature transformation diagram for the formation of CuAl2 in Al-4Cu,
after [2]. The two red curves indicate the beginning and the end of the precipitation process. Also plotted (see blue curve) are the heat treatment steps in particle
strenghtening, consisting of solution heat treatment, quenching and precipitation
heat treatment.
the growth rate is large. So it takes a long time to form a particle that can continue
to grow on its own. It then grows rapidly because the growth rate is large. In the case
of CuAl2 precipitates in the aluminum solid solution, the area surrounding a particle
is depleted of Cu, because this element accumulates in the particle. As a result, it is
no longer possible to form further CuAl2 particles in the neighborhood of this particle,
because there is no longer sucient copper for this. After some time, another particle
may form at a great distance, which then also grows rapidly and prevents further particle
formation in its vicinity. So it is clear that in this case few, large particles are formed,
which is undesirable. Now let us imagine the opposite. Let the nucleation rate be large
and the growth rate be small. Then very many particles are formed in a short time, but
they grow only slowly. Before they can grow to any appreciable extent, other particles
have already formed in their neighborhood, so that, as desired, the precipitated particle
volume consists of many small particles. Thus, a large nucleation rate coupled with a
low growth rate is desirable. Of course, one can wish for many things. The question is
what can be achieved in reality. For this purpose, one must be clear about what both
variables depend on. The smaller r⋆ is, the higher is the nucleation rate, because the
easier it is to form a growing particle. Because random place changes of the atoms are
required for this, the nucleation rate also increases with the place change velocity of
80
3 Strength and phase diagrams
the atoms. Likewise, particle growth increases with the place change velocity, because
it is these place changes that, in the example of the CuAl2 particles, allow copper to
be transported to the particle. Thus, if the place change velocity is increased, this
has a desirable and an undesirable eect. These two eects cancel each other out, so
that the place change velocity has no inuence and r⋆ remains as the only relevant
variable. The smaller r* is, the ner the precipitates are. Thus, to ensure ecient
particle strengthening, we need a small r⋆ .
Based on this observation, the question arises how to ensure a small critical radius.
This is shown by Eq. 3.10. Obviously, the specic interfacial energy γs must be small
for this purpose. Therefore, not all particles of a second phase, which precipitate in
a matrix, are suitable for particle strengthening, but only coherent ones, because for
them γs is small. Particle strengthening is therefore possible, for example, only for very
specic aluminum alloys and not for others. As the above example shows, Cu-containing
aluminum alloys obviously belong to the former group. Additionally, the critical particle
radius becomes the smaller the more negative ∆gv is, i.e. the lower the temperature
is. Therefore, the lower the temperature at which the precipitation reaction takes place,
the smaller the precipitated particles become. Thus, one could conclude that it is best
to simply wait at room temperature until the CuAl2 particles have precipitated out in
our example. However, Fig. 3.20 shows that this is not too good an idea, because the
precipitation process would start only after a waiting time of about 8000 hours (i.e.,
almost a year) and would be completed only after many years. The premise is therefore
to choose a temperature that is as low as possible, but also as high as necessary, so
that the precipitation process takes place in a technically reasonable period of time. As
Fig. 3.20 shows, temperatures around 150 C are a reasonable choice for the Al-Cu alloy
under consideration.
From what has been learned, the following sequence of temperature control must be
followed during particle strengthening. First, the material must be heat treated in a
single-phase eld to dissolve particles formed in an uncontrolled manner and to achieve
an initial single-phase state. In Fig. 3.20, approx. 530 C was chosen for this purpose
(see starting point of the blue curve). After this, the material must be quenched. The
quenching rate must be high enough to pass the "nose" in the TTT diagram, as shown
in Fig. 3.20 by the blue curve, so that the single-phase state is maintained. If one were
to cool more slowly than shown in Fig. 3.20, the precipitation process would begin at
high temperatures, resulting in large particles. Then, the material is heat treated at a
temperature that, as mentioned, is chosen as low as possible but as high as necessary.
This heat treatment step is called precipitation heat treatment. In summary, there are
three steps:
°
°
ˆ Solution heat treatment
ˆ Quenching
ˆ Precipitation heat treatment
Finally, a subtlety in the TTT diagrams should be pointed out. In Fig. 3.20, on
the one hand, a cooling curve is drawn in which the temperature changes continuously
81
3 Strength and phase diagrams
with time. On the other hand, the temperature does not change with time during the
precipitation heat treatment. Of course, the temperature-time history has an eect on
the time at which phase transformations occur. Thus, it makes a dierence whether one
cools continuously from a certain temperature (about 530 C in Fig. 3.20) and observes
when a phase transformation occurs as a function of the cooling rate, or whether one sets
(very quickly) a certain temperature, keeps it constant, and then determines the time at
which the phase transformation occurs. In practice, both temperature control methods
are commonly used. Therefore, there are so-called continuous TTT diagrams, which are
based on experiments with continuous cooling, and isothermal TTT diagrams, which
are based on isothermal heat treatments. However, the dierences are mostly small
and irrelevant for qualitative considerations, as they are in the foreground here. For this
reason, no distinction is generally made here between these two types of diagrams. If you
want to have exact information, however, you have to use the TTT diagram applicable
to the respective case.
Due to the logarithmic time scale, it is always the case that continuous cooling curves
in the TTT diagram are convexly curved and, hence, always reach the nose above
the nose tip. Because the phase transformation in the temperature range above the
nose tip is slower the higher the temperature, the continuous TTT diagram above the
nose tip shifts to somewhat larger times compared to the isothermal one. For example,
isothermal holding at 400 C would result in faster precipitation of CuAl2 particles than
cooling from 530 C to 400 C. This situation is illustrated in Fig. 3.21. There, the red
lines indicate the beginning and end of the precipitation process of CuAl2 particles for
isothermal heat treatment, and the green lines for continuous cooling. It can be seen
that the green curves above the nose tip are shifted to the right to longer times. Below
the nose tip the eects reverse, because below the nose tip phase transformations are
slower the lower the temperature. Accordingly, the two green curves below the nose
tip shift forward relative to the red ones. The green, dotted line indicates that here
the precipitation process comes to a standstill during cooling because diusion is now
too slow. Unlike the green solid lines, the precipitated volume of CuAl2 particles is not
constant along the dotted line. Rather, it changes from one percent of the precipitable
volume on the far left to 99 % on the far right. In the middle, at the position of the
number 50, it is 50 %.
In Fig. 3.21, two continuous cooling curves are also shown in green with dashed
lines. If one follows the continuous cooling curve 2 , the precipitation process starts at
approx. 450 C and ends at approx. 400 C. The precipitable content of CuAl2 has now
been completely formed. If one follows the continuous cooling curve 1 , the precipitation
process starts at approx. 400 C. However, the second curve, which shows that the entire
precipitable CuAl2 volume has now been formed, is never reached. Rather, the green,
dotted line is intersected at about 260 C. As already mentioned, the precipitation process
comes to a halt here and the number 50 indicates that half of the precipitable volume
of CuAl2 particles has formed at this point.
Finally, a possible misinterpretation when interpreting cooling curves on the basis of
isothermal TTT diagrams must be pointed out. For this purpose, we once again consider
the cooling curve 2 . The beginning and end of the precipitation process are described
°
°
°
°
°
°
O
°
°
O
82
O
3 Strength and phase diagrams
qualitatively correctly using the red curves for the isothermal diagram. Only slightly
too high temperatures result, for example approx. 430 C instead of approx. 400 C for
the end of the precipitation process. The red curves are intersected by the cooling curve
again at approx. 250 C and approx. 150 C. However, this is meaningless. It would be
a misinterpretation to conclude that CuAl2 dissolves again between 250 C and about
150 C. As the phase diagram shows (Fig. 3.19), CuAl2 is thermodynamically stable
below 500 C for the alloy Al-4Cu considered here. Once formed, it therefore does not
dissolve on cooling.
°
°
°
°
°
°
°
Figure 3.21: Isothermal (red lines) and continuous (green lines) TTT diagram of the alloy Al4Cu, after [2]. Green, dashed lines show two cooling curves with dierent cooling
rates.
3.3.5 Combined measures to increase strength: the quenching and
tempering of steels
In the case of steels, there is a technologically very important process for achieving a
favorable combination of strength and ductility, known as quenching and tempering.
Due to the outstanding importance of quenching and tempering, this chapter will take
a closer look at the processes involved. We will see that several hardening mechanisms
interact in this process, i.e. the resulting increase in strength is made up of various
contributions from the individual mechanisms discussed in sections 3.3.1 to 3.3.4.
To get started, we look at the iron-carbon phase diagram (Fig. 3.36) and set ourselves
the goal of realizing particle strengthening by precipitating Fe3 C particles in the αphase. If we proceed according to the procedure worked out in chapter 3.3.4, we rst
83
3 Strength and phase diagrams
have to transform the steel into the single-phase state by heat treatment in the α single
phase eld. According to Fig. 3.36, the maximum carbon solubility in the α-phase is
0.02 %. It is reached at 723 C. Thus, in order to heat treat in the α single phase eld,
we must choose a steel with a carbon content not exceeding this value. After solution
heat treatment, the steel is quenched and precipitation heat treated, e.g. at 300 C. If
we were now to measure the strength, the disappointment would be great, because the
increase in strength would be very small. This is due to the low solubility of carbon
in the α-phase. As can be seen immediately from the lever rule, this leads to a very
low volume fraction of precipitated Fe3 C particles and therefore to a small increase in
strength.
Looking at the Fe-C phase diagram again, one can come up with another idea. Since
the γ -phase can dissolve much more carbon than the α-phase, one could choose a steel
with a higher carbon content (e.g. 0.8 %) and set a single-phase condition by solution
heat treatment in the γ single phase eld, e.g. at 1000 C. The material is then quenched.
Fig. 3.22 shows the corresponding TTT diagram on the left. It can be seen that the
nose in the TTT diagram can be avoided when the steel is quenched out of the γ
phase eld in about one second (see step 1 to 2 in Fig. 3.22). In this case, too few
atomic place changes take place, i.e., diusion is too slow. Therefore, α-phase and Fe3 C
cannot form by a transformation that requires diusion. Thus, one would expect single
phase γ to continue to exist after quenching. However, this is not the case, because the
γ -phase nds another way to transform into the α-phase without requiring atomic place
changes (for which there is not enough time). Such phase transformations (here from
the γ -phase into the α-phase), which take place without the atoms changing places, are
called martensitic transformations. In contrast, phase transformations that require place
change processes (i.e. diusion) are called diusion-controlled.
Because martensitic transformations do not require diusion, they cannot be suppressed by rapid quenching. They always take place when a thermodynamic driving
force is present. This is also illustrated in Fig. 3.22. On the basis of the horizontal
line marked Ms at approx. 230 C, it can be seen that the martensitic transformation
of the steel shown with 0.8 % carbon starts at approx. 230 C, even at extremely rapid
cooling rates. The only prerequisite is that one passes the nose in the TTT diagram,
as shown by the cooling curve marked with 1 in Fig. 3.22, so that a diusion-controlled
transformation has not already taken place beforehand. Ms stands for martensite start
temperature. Thus, the martensitic transformation starts at this temperature. M50 and
M90 , on the other hand, denote the temperatures where 50 % or 90 %, respectively, of
the γ -phase is martensitically transformed. Mf indicates the martensite nish temperature, where 100 % is transformed into martensite. With increasing carbon content,
the martensitic transformation temperatures are reduced because carbon stabilizes the
austenite with respect to the ferrite (see g. 3.36). At 0.8 % carbon content, Mf is at
room temperature. Thus, the austenite is barely transformed completely into martensite.
At higher carbon content, this would no longer be possible.
During the martensitic transformation, the atoms within the austenitic crystal reorient themselves slightly so that the face-centered cubic structure becomes body-centered
cubic. This so-called displacive transformation can best be illustrated by comparing the
°
°
°
O O
°
O
84
°
3 Strength and phase diagrams
Temp., °C
723°
3
1
1s
2
1 min
1h
1a
0,8
Figure 3.22: Schematic TTT diagram of an eutectoid steel with a carbon content of 0.8 wt.-%
(left) with corresponding phase diagram (right), after [27].
elementary cells of the face-centered cubic γ -phase and the body-centered cubic α-phase
sketched in Fig. 3.23. If we look at two adjacent unit cells of the face-centered cubic
structure, it is clear that there is a distorted body-centered cubic unit cell between two
unit cells, which is compressed along two axes and stretched along the third axis. A
slight displacement of the atoms in such a distorted cell can transform the crystal into
a cubic body-centered one without the need for atomic place changes. The transformation is therefore diusionless. It begins at numerous sites within a crystal and spreads
from there almost at the speed of sound. The movement of the atoms is coordinated,
whereby the atoms always rearrange themselves around the area that has already been
transformed. Also, the orientation of the new crystal lattice is not arbitrary, but depends on that of the old crystal lattice. The new α-crystals grow in a roughly lenticular
shape. In a metallographic cross sectional cut through such a structure, typical so-called
martensite needles can be observed (Fig. 3.24). Because very many martensite needles
form in an austenite grain, a very ne structure is created.
Not least because the body-centered cubic lattice is somewhat less densely packed than
the face-centered cubic lattice, considerable internal stresses and local plastic deformation occur during martensitic transformation. The latter causes a high dislocation density in the martensitic microstructure. In addition, a high carbon content is now forcibly
dissolved in the α-phase. This leads to an extreme distortion of the crystal lattice and
thus to an unusually strong solid solution strengthening, which increases strongly with
85
0, 355nm
3 Strength and phase diagrams
1 nm
0, 25
0, 355nm
(a) Two face-centered cubic
cells from the γ -phase (austenite).
(b) Tetragonal
cell from (a).
(c) Martensite
cell
Figure 3.23: Illustration of the martensitic transformation from the γ -phase to the α-phase.
Hereby, the face-centered cubic unit cell is compressed in one direction and elongated in the others [26].
Figure 3.24: Micrograph of a quenched steel containing 0.6 % carbon. Martensite needles can
be seen. [26].
the carbon content. Finally, the martensite needles are very ne, so that a ne-grained
structure with corresponding strengthening by grain size reduction is present here. Thus,
solid solution strengthening, work hardening and strengthening by grain size reduction
interact here, with the former mechanism having by far the greatest strengthening eect.
Above a carbon content of about 0.2 %, this results in an extremely strong and hard,
but also very brittle material state. In this state, the material has no useful properties
due to its brittleness. Therefore, a heat treatment is carried out, which is called tempering (see Fig. 3.22 Step 3 ). Depending on the application, temperatures between
150 C and 600 C are typical. In this process, part of the lattice distortion is removed
by precipitation of nely distributed carbide particles (Fe3 C). This results in a very ne
microstructure overall (nely distributed martensite needles, nely distributed precipitates), which leads to a very favourable combination of strength and ductility. Naturally,
the strength decreases with increasing tempering temperature, whereas the ductility increases. Steels with this favourable combination of properties are called quenched and
tempered steels. It is important to note that not every steel is suitable for such heat
treatment. In addition to a sucient carbon content (see above), a homogeneous dis-
°
°
O
86
3 Strength and phase diagrams
tribution of carbon in the structure is particularly necessary. If this were not the case,
strong uctuations in the properties of the material would result. At places of locally
increased carbon concentration, the material would be disproportionately strong and not
very ductile, while at places with too low carbon content, the result would be too low
strength with high ductility. This would not be acceptable for practical use.
The possibility of hardening and tempering steels is decisive for the special importance
of steels in mechanical engineering. This is only possible because iron has two dierent
phases (α and γ ) and the phase existing at higher temperatures has a much greater
carbon solubility. If this were not the case, steels would not be as important as they are
today.
Finally, the TTT diagram of Fig. 3.22 should be considered again. There, on the
right, the designations coarse pearlite, ne pearlite and bainite are given for the microstructure transformed by diusion, whereby it is sucient here to imagine bainite
as particularly ne perlite. In analogy to particle strengthening, where the precipitated
particles become ner and ner with decreasing precipitation temperature, here the
pearlitic microstructure becomes ner and ner (i.e. the α- and Fe3 C-lamellae become
narrower and narrower), the lower the temperature for the diusion-controlled transformation is. The beginning and end of the diusion-controlled transformation is depicted
by two solid lines. The dashed one indicates an intermediate state.
3.4 Phase diagrams
For the description of solid solutions and precipitates (sections 3.3.3 and 3.3.4) the Al-Mg
phase diagram (Fig. 3.12) has already been mentioned. In the following, phase diagrams
are explained in more general terms.
3.4.1 Solubility, states of aggregation, Gibbs phase rule
The term phase can be illustrated by the dierent behaviour solutions in the liquid state
(Fig. 3.25). If alcohol is added to water, the alcohol dissolves. The composition is the
same everywhere in the liquid, so there is only one liquid phase. If, on the other hand,
oil and water are mixed, the oil does not dissolve, and there are two phases. Common
salt dissolves in water. At high concentrations, the solubility limit is exceeded, and salt
is precipitated as a solid phase from the supersaturated solution.
In general, a phase is matter of spatially constant physical state (e.g. state of aggregation, crystal structure). The phase diagram shows which phases are stable at a
given composition, temperature T and pressure p. Gibbs' phase rule gives a correlation
between the number of the equilibrium phases P , the number of degrees of freedom f
and the number of components n of a system:
(3.11)
f =n−P +2 .
Hereby are:
87
3 Strength and phase diagrams
(a) States of aggregation of water.
(b) Water and alcohol.
(c) Water and salt.
(d) Water and oil.
Figure 3.25: Phases and solubility, adapted from [6].
f
n
P
number of the degrees of freedom
number of components
number of equilibrium phases
Degrees of freedom are freely selectable system variables (e.g. composition of phases,
temperature, pressure). Examples of components are elements or stable chemical compounds (molecules).
For pure elements or water as a molecule n = 1 holds and the composition does not
change. The three states of aggregation (solid, liquid and gaseous) represent possible
phases. In the corresponding single-phase regions (P = 1) in the phase diagram of water
(p-T diagram, Fig. 3.26) there are two degrees of freedom (f = 2) according to the phase
rule, i.e. pressure and temperature can be changed without leaving the region. Along
the lines that dene the areas of existence of the phases, two phases are in equilibrium
(P = 2). This results in a degree of freedom (f = 1), i.e. one system quantity (e.g.
pressure) can be specied and, thus, the other system quantity (temperature) is xed.
88
3 Strength and phase diagrams
Pressure p
At the triple point all three phases are in equilibrium (P = 3) and all system variables
are xed (f = 0).
Liquid
Solid
(Ice)
1 bar
Vapour
TSm T3
TS
Temperature T
Figure 3.26: Schematic phase diagram of water. The meaning of the symbols is as follows: TSm
- melting temperature, T3 - triple point, TS - boiling temperature
If one considers materials in mechanical engineering, such as steels, only the solid and
liquid states of aggregation are of interest. Since the phase boundaries between solid and
liquid substances depend only to a small extent on pressure and applications usually take
place at 1 bar, the pressure is generally xed so that one degree of freedom is already
consumed. Thus, the phase rule can be written as follows:
(3.12)
f = n − P + 1 (p = const.).
For a pure metal (n = 1) at the melting point the following applies: (i) two phases are
stable (solid and liquid, P = 2); (ii) there is no longer a freely selectable system variable,
i.e. the number of degrees of freedom is f = 0.
3.4.2 Binary systems and microstructure formation
Systems of two substances consist of two components (n=2; e.g. Cu-Ni, Fig. 3.27). If
one xes the pressure, which is always assumed in the following, the temperature and
the compositions of the phases are the remaining degrees of freedom.
In the phase diagrams, the temperature T is plotted on the ordinate and the concentration c of the added element (in weight-% or atom-%) on the abscissa.
Binary phase diagrams with complete miscibility
Fig. 3.27 shows the binary phase diagram for the copper-nickel system. On the left and
right ends of the diagram there are 100 % copper and 100 % nickel, respectively. From
89
3 Strength and phase diagrams
Figure 3.27: Binary phase diagram for the copper-nickel system. L: melt; α: solid, adapted
from [7].
left to right, the nickel content increases, which lowers the copper content. The sum of
both concentrations is always 100 %.
The phase diagram for copper and nickel is particularly simple because in the solid
state there is only one phase (α). Therefore, there is a complete miscibility between
the two metals. If one starts on the left end of the phase diagram, pure copper with a
face-centered cubic crystal structure exists. If one moves then to the right, more and
more nickel is dissolved in the face-centered cubic crystal until one nally arrives at
pure nickel, which is also face-centered cubic. Requirements for complete miscibility are
the same crystal structure of the components and an identical or only slightly dierent
atomic radius r.
Basically, such a phase diagram can be divided into three regions: a single-phase region
of the melt (L = liquid), which is delimited by the so-called liquidus line, a single-phase
region of the solid solution (α), which is delimited by the so called solidus line, and the
lenticular, two-phase region enclosed by the solidus and liquidus line. This region is
two-phase, since both liquid melt (L) and solid solution (α) are present. For a certain
temperature, the concentrations of both phases are given by so-called tie-lines. These
are horizontal lines (i.e. lines at a certain temperature) extending over the entire twophase eld, with the end points of the tie-lines giving the respective concentrations. In
Fig. 3.27 this is exemplary shown for T=1300 C. The end point of the tie-line on the
liquidus line denotes the concentration of the melt (cliq ) and the end point at the solidus
line the concentration of the solid solution (csol ), which is in equilibrium with the melt.
°
90
3 Strength and phase diagrams
If the Gibbs phase rule is applied to the two-phase region, the degree of freedom is
f = 2 − 2 + 1 = 1. For example, one can freely choose the temperature. Then, the
compositions of the two phases are determined by the corresponding tie-line. It is also
possible to freely select the concentration of a phase, e.g. the concentration cliq shown in
Fig. 3.27. Then, the position of the tie-line is dened, i.e. temperature and concentration
of the second phase are xed. Further examples for complete miscibility are silverpalladium (face-centered cubic; rAg = 0.145 nm; rPd = 0.138 nm)) or molybdenumtungsten (body centered cubic; rMo = 0.136 nm; rW = 0.137 nm).
Figure 3.28: Equilibrium solidication by slow cooling of a copper-nickel alloy. a and b
stand for the length of the lever arms, explanation see text, adapted from [7].
Let us now consider the cooling of a Cu-Ni melt containing 40 wt.% nickel (Fig. 3.28).
At a temperature above 1280 C, only the melt is present. When cooling down to 1280 C
the two-phase eld is reached. The rst crystals precipitate and oat in the melt.
According to the associated tie-line, they have a nickel concentration of 52 %, i.e. they
are enriched with the higher melting element nickel. With further cooling, the proportion
of the solid solution increases, whereby the composition of the melt and solid solution
changes. Both phases become poorer in nickel. When the solidus temperature is reached,
the last melt solidies and the copper alloy containing 40 wt.% nickel is completely
solidied. It is now a single-phase solid solution.
°
°
Lever rule
The previous example has shown that the proportions of melt and solid solution change
while the two-phase eld (α+L) is passed through during solidication. How large these
proportions are at a certain temperature can be calculated using the so-called lever rule,
which is explained using the Cu-Ni system presented here. According to Fig. 3.28 the
following is dened:
c(Ni)
concentration of nickel in the alloy (gross concentration)
cliq(Ni) concentration of nickel in the melt
csol(Ni) concentration of nickel in the solid
Furthermore, the masses are dened as:
91
3 Strength and phase diagrams
mliq
msol
m = msol +mliq
mass of the liquid phase
mass of the solid phase
entire mass
Since the mass of nickel must be preserved, the following applies:
csol(Ni) · msol + cliq(Ni) · mliq = c(Ni) · m = c(Ni) · (msol + mliq ).
(3.13)
Rearrangement results in:
(csol(Ni) − c(Ni) ) · msol = (c(Ni) − cliq(Ni) ) · mliq .
°
(3.14)
If we inspect the example in Fig. 3.28 (tie-line at approx. 1260 C), the tie-line can be
interpreted as a seesaw, which is supported at the gross concentration (c(Ni) =0.4). The
length of the lever arms a and b of this seesaw result as:
Lever arm a
Lever arm b
= c(Ni) − cliq(Ni)
= csol(Ni) − c(Ni)
And the length of tie-line is:
Tie-line length a + b
= csol(Ni) -cliq(Ni)
Similar to the equilibrium of moments in mechanics, the lever rule says (eq. 3.14),
that the product of the right lever arm length b with the mass of the phase at the right
hand side (msol ) must be as large as the product of the left lever arm length a with the
mass of the left phase (mliq ). When solidication begins, the lever arm (c(Ni) − cliq(Ni) )
is very short. Correspondingly, the mass of the melt is large. The opposite is true at the
end of the solidication.
Adding the term (c(Ni) − cliq(Ni) ) · msol on both sides of eq. 3.14, results in:
(csol(Ni) − c(Ni) ) · msol + (c(Ni) − cliq(Ni) ) · msol
= (c(Ni) − cliq(Ni) ) · mliq + (c(Ni) − cliq(Ni) ) · msol
(3.15)
msol · [(csol(Ni) − c(Ni) ) + (c(Ni) − cliq(Ni) )] = (c(Ni) − cliq(Ni) ) · m
(3.16)
(c(Ni) − cliq(Ni) )
msol
=
m
(csol(Ni) − cliq(Ni) )
(3.17)
And in analogy:
92
3 Strength and phase diagrams
(csol(Ni) − c(Ni) )
mliq
=
m
(csol(Ni) − cliq(Ni) )
(3.18)
This relation says that the mass fraction of one phase (msol /m respectively mliq /m) is
given by the ratio between the length of the opposite lever arm (a respectively b) and
the total length a+b. This calculation of the phase fractions can be used in any twophase area. For example, it can also be used to determine the proportion of precipitates
during precipitation hardening.
Non-equilibrium solidication with complete miscibility
In the case of very slow cooling, concentration dierences in the solid state are compensated by diusion, i.e. the movement of atoms in the solid. In this way, the Ni
concentration of the solid solution formed in Fig. 3.28 changes from 52 % to 40 % and at
the end of solidication a homogeneous solid solution is obtained (Fig. 3.28) which has
the same concentration everywhere.
As shown in Fig. 3.29, the conditions for rapid cooling (non-equilibrium solidication)
are dierent. At rst, crystals precipitate again with a nickel concentration of 52 %.
After further cooling the equilibrium concentration of nickel in the solid is 46 % and the
already solidied material should actually release nickel by diusion in order to reach
this equilibrium concentration. However, as the solidication is fast, this does not occur.
Only the newly solidied material is deposited with this concentration at the previously
solidied one. Say, the resulting average nickel concentration in the solid is 51 %. After
further cooling to the solidus temperature the equilibrium concentration of nickel is
40 % and say the average is 48 %. Under equilibrium conditions, solidication would
be now fully completed. Here, however, this is not yet the case, because the average
concentration of the solid does not yet correspond to the gross concentration of 40 %.
Accordingly, the solidication process continues until the average nickel concentration in
the solid material has reached 40 %. Thus, in the case of rapid cooling the solidication
interval is given by the thin dash-dotted line in Fig. 3.29. Due to this non-equilibrium
solidication, concentration dierences are formed within the solidied solid solution.
They are referred to as segregation.
Figure 3.29: Non-equilibrium solidication through rapid cooling of a copper-nickel alloy,
adapted from [7].
93
3 Strength and phase diagrams
Binary phase diagrams with limited miscibility
If the solubility of the components in the solid state is limited, miscibility gaps occur, i.e.
there are T -c areas in which two solid phases coexist. An example of this is the eutectic
system, which is described below on the example of the Pb-Sn phase diagram (Figs.
3.30 to 3.33). In the medium concentration range lies the two-phase eld ((α + β)),
which reaches up to the solidus line. A small quantity of tin is soluble in the α-Pbphase and up to a tin content of 2 wt.% a single-phase solid solution is formed from the
melt during solidication (Fig. 3.30). Alloys with a tin content between 2 and 19 wt.%
solidify rst to a single-phase solid solution, but if the solution falls below the solvus line,
the second phase is precipitated (Fig. 3.31). This behaviour is utilized for precipitation
strengthening (chapter 3.3.4).
Figure 3.30: Solidication of a Pb-2 wt.% Sn alloy, adapted from [7].
For Sn contents between 19 wt.% and 61.9 wt.%, solidication begins, as before, with
the precipitation of the α-phase (Fig. 3.32). At 183 C, however, the eutectic reaction
now takes place, in which the residual melt, having a Sn content of 61.9%, solidies into
two solid phases.
A Pb-Sn melt with eutectic composition (61.9 wt.% Sn) solidies at a certain temperature (the so-called eutectic temperature) completely (without solidication interval),
and there are two solid phases after solidication. Therefore, three phases are in equilibrium at the eutectic temperature (melt + 2 solid phases, P = 3). According to the phase
rule (eq. 3.12), the number of degrees of freedom is f = 0. The two-phase eutectic
microstructure (Fig. 3.33) is characterized by a lamellar arrangement of the two phases.
Elements which are hardly soluble in the solid state generally stabilize the more tol-
°
94
3 Strength and phase diagrams
Figure 3.31: Solidication of a Pb-10 wt.% Sn alloy, adapted from[7].
Figure 3.32: Solidication of a Pb-30 wt.% Sn alloy, adapted from [7].
erant melt and lower the liquidus temperature. This is exploited for solder materials,
which often have a composition close to eutectic.
In addition to the eutectic solidication, which follows the reaction scheme L ↔ α + β ,
there is another reaction, the so-called peritectic solidication, in which three phases are
in equilibrium, i.e. the number of degrees of freedom is zero. This is shown in Fig. 3.34
on example of the Ni-Re phase diagram. If an alloy with a rhenium content of 60% is
imagined, it begins to solidify at 2481 C, forming the β -phase (= Re-rich solid solution).
At the peritectic temperature of 1620 C, the residual melt solidies by reacting with
the β -phase to form the α-phase (= Ni rich solid solution). The reaction scheme is now
L + β ↔ α.
Phase diagrams often consist of a sequence of peritectic and eutectic reactions. An
example is the phase diagram for nickel and aluminium shown in Fig. 3.35. On the Al
rich side, one can see a eutectic reactionL ↔ α + δ at 639.9 C. The phase α stands for
the Al-rich solid solution. Because the solubility for nickel is very low, the expansion
°
°
°
95
3 Strength and phase diagrams
Figure 3.33: Solidication of a Pb-61,9 wt.% Sn alloy, adapted from [7].
Figure 3.34: Binary phase diagram of the Ni-Re system, adapted from [23].
96
3 Strength and phase diagrams
of this single-phase eld is also very small and can hardly be seen in the diagram. At
854 C, a peritectic reaction is present. Here, the melt reacts with the ε-phase to the
δ -phase (reaction scheme: L + ε ↔ δ ). While the ε-phase exists from about 57 % to
61 % nickel, the range of existence of the δ -phase is so narrow that it only appears as
a line in the diagram. A further peritectic reaction occurs at 1133 C (reaction scheme:
L + β ↔ ε). At high nickel contents (to the right of the β -phase) a peritectic reaction
(reaction scheme: L + β ↔ γ ') and a eutectic reaction (reaction scheme: L ↔ γ ' +γ )
occur. In addition, at 700 C the β -phase reacts with the γ '-phase under formation of
the η -phase (reaction scheme: β + γ ′ ↔ η ). Since only solid phases are involved, one is
not talking of a peritectic reaction but a peritectoid reaction. The so-called nickel-based
superalloys are based on this phase diagram. With the help of the alloying element
aluminium, the Ni-rich solid solution (γ -phase) is precipitation strengthened by the γ 'phase, resulting in outstanding strength at operating temperatures up to about 1000 C.
°
°
°
°
Figure 3.35: Binary phase diagram of the Al-Ni system, adapted from [24] .
3.4.3 The Iron-Carbon Diagram
The iron-carbon diagram is the most important phase diagram in engineering because
steel, as an alloy of iron and carbon, is based on this diagram and, with an annual world
production of about 1.6 billion tons, is the most frequently used material in mechanical
engineering. Accordingly, the iron-carbon diagram (Fe-C diagram for short) will be
discussed in more detail here. In Fig. 3.36 it is shown up to a carbon content of about
6.6 %. According to the diagram, the melting temperature of pure iron is 1536 C.
At this temperature, the melt (denoted by the symbol L for liquid) solidies into
solid iron with a cubic body-centered crystal structure. This rst solid phase is called
°
97
3 Strength and phase diagrams
°
the δ phase. At a temperature of 1392 C, a transformation of the crystal structure
from the body-centered cubic to the face-centered cubic takes place. This face-centered
cubic modication of the iron is called the γ phase or austenite. If one cools further,
the face-centered cubic lattice transforms again into the body-centered cubic lattice at
911 C. The resulting phase is called α-phase or ferrite. The respective lattice structures
have already been presented in Chap. 2.3 (Fig. 2.22 on page 34).
°
Figure 3.36: Metastable iron-carbon diagram.
If carbon is now added to the iron, it is dissolved within certain limits in the α, γ or
δ phase. Solid solutions are formed. It is noticeable that the γ phase can dissolve much
more carbon than the α phase. In the γ phase, up to 2.06 % carbon is soluble at 1147 C,
whereas in the α phase a maximum of 0.02 % at 723 C is soluble. As already mentioned
in section 3.3.5, the existence of these two phases with their enormous dierence in
solubility for carbon is the decisive reason why the mechanical properties of the steels
can be adjusted within wide limits depending on the application (see Chap. 3.3.5).
In Fig. 3.36, three reactions can be seen in which three phases are in equilibrium, i.e.
the number of degrees of freedom is zero. First, a so-called peritectic reaction occurs
at 1493 C (top left in the diagram). Here, melt L and δ phase transform according
to the reaction L + δ ↔ γ into the γ phase. However, this reaction is not important
for understanding the steels. Secondly, a so-called eutectoid transformation takes place
at 723 C. In this process, the γ phase with a carbon concentration of about 0.8 %
decomposes into the α phase and the iron carbide Fe3 C, which is also called cementite,
°
°
°
98
°
3 Strength and phase diagrams
according to the reaction γ ↔ α + Fe3 C. A eutectoid reaction diers from a eutectic
reaction by the fact that here, a solid phase (instead of a liquid phase) decomposes into
two solid phases. Third, a eutectic reaction occurs at 1147 C where melt with a carbon
concentration of 4.3 % transforms according to the reaction L ↔ γ + Fe3 C into γ phase
and Fe3 C. According to the phase diagram, the solidication of the melt ends with this
reaction when the carbon content is more than 2.06 %. At lower contents, solidication
occurs either via the peritectic reaction described above or by the melt transforming
directly into the γ phase when passing through the two-phase eld L + γ .
Steels are all Fe-C alloys with a carbon content of not more than 2.06 %, whereas Fe-C
alloys above 2.06 % C are called cast iron. In steels, the carbon always precipitates as iron
carbide Fe3 C, whereas in cast irons both Fe3 C and graphite can form. In the rst case,
one speaks of white cast iron. In the second, technically more important case, which
generally requires the addition of graphite-stabilising elements (especially silicon), one
speaks of grey cast iron. Surprisingly, graphite is slightly more thermodynamically stable
than Fe3 C, although precipitation of graphite occurs only under special circumstances
due to kinetic inhibition. As Fig. 3.37 shows, the phase boundary lines shift slightly
when graphite forms instead of Fe3 C. These changes are shown by the blue dashed lines.
They occur only where carbon / Fe3 C are involved in the phase formation. Where this
is not the case, the phase diagram remains unchanged and the black lines continue to
apply. The relevant diagram for steels with formation of Fe3 C is therefore the metastable
diagram (solid lines in Fig. 3.37), whereas the diagram with the changes according to
the blue dashed lines is the stable one.
In the following we restrict ourselves to the steels, i.e. carbon contents not exceeding
2.06 %. In accordance with our considerations above, we assume that the γ phase has
formed after solidication of the melt, and we now follow the microstructural transformations during slow cooling from the γ single-phase eld to room temperature. We
progressively move from low carbon contents to higher ones. Fig. 3.38 represents a section of the Fe-C diagram in which a composition containing 0.3 % carbon is plotted.
When the line GS is intersected (here: 820 C), the transformation from austenite to ferrite begins. The transformation temperature has lowered due to the addition of carbon
compared to pure iron. Point 2 shows the microstructure just below the GS line. The
rst α crystallites have formed along the γ grain boundaries. In principle, the α phase
can also precipitate in the grain interior. However, the grain boundaries are preferred
here because they represent favourable nucleation sites for the α-phase. Since ferrite
cannot dissolve as much carbon as austenite, the remaining austenite is enriched with
carbon by diusion processes. With further cooling, the proportion of ferrite increases
and the γ phase accumulates even more carbon (point 3 in Fig. 3.38). At 723 C the
γ -phase has nally reached eutectoid composition with 0.8 % C. It now decomposes eutectoid as described above with the formation of a lamellar structure consisting of ferrite
and Fe3 C. This phase mixture has been given the name pearlite due to its pearlescent
sheen under the light microscope. The nal microstructure (Fig. 3.38, right) consists of
ferrite (light) and pearlite (dark), the lamellae can be seen under higher magnication.
The microstructure is therefore also called a ferritic-pearlitic microstructure. It is also
important to note that in the eutectoid reaction, the carbon must split between the α
°
°
°
99
(Temperature in °C)
3 Strength and phase diagrams
(Austenite)
(Ferrite)
Carbon content in wt.-%
Metastable system
Stable system
Figure 3.37: Comparison between metastable (solid lines) und stable (blue dashed lines) ironcarbon diagram.
phase and the cementite (Fe3 C). This happens by diusion and takes time. If it is not
available, other transformation processes take place, which are described in chapter 3.3.5.
If we increase the carbon content to 0.8 %, as shown in Fig. 3.39, it is noticeable that
no transformation interval is passed through during cooling where α-phase and γ -phase
coexist. Above 723 C the single phase region of austenite is present. When the PSK
line is reached at 723 C, the austenite disintegrates lamellar into ferrite and cementite
(Fe3 C) and a pearlitic microstructure is formed (see Fig. 3.39, right). The process of
pearlite formation is like that discussed above for the steel containing 0.3 % carbon.
If the carbon concentration is greater than 0.8 %, the phase transformation occurs as
shown in Fig. 3.40 for a concentration of 1.1 %. Again, we start by looking at the single
phase region of austenite (γ ) at about 900 C. Cooling slowly from this temperature,
the SE line is intersected at 825 C. The γ solid solution is now no longer able to absorb the entire carbon content and cementite (Fe3 C) precipitates on the austenite grain
boundaries. This is similar to the discussed α precipitation for the steel containing 0.3 %
carbon, but there the C content increased in the γ phase, while here it decreases. Finally,
the γ phase reaches the point S at 723 C and the remaining austenite decomposes to
pearlite (ferrite + cementite). A typical microstructure consisting of cementite along
the former austenite grain boundaries (light) and pearlite is shown in Fig. 3.40 on the
right.
°
°
°
°
°
100
3 Strength and phase diagrams
1100
E
γ
γ
(Austenite)
1000
γ
γ
γ
γ+Fe 3C
γ
γ
G
900
γ
γ
1
Temperature, °C
α
800
γ
2
700
γ
S
P 3
γ
γ
K
4
(a) Ferritic-pearlitic microstructure
α
(Ferrite)
600
Perlit
Fe 3C
α
α
500
20µm
α+Fe 3C
400
(b) Magnification of ferritic-pearlitic
microstructure from (a)
0
1,0
1,0
2,0
Carbon content, wt.-%
Figure 3.38: Solidication of a steel with 0.3 wt.-% C. Right: Optical micrographs of a ferriticpearlitic microstructure, after [10].
101
3 Strength and phase diagrams
1100
E
γ
(Austenite)
1000
γ+Fe 3C
G
900
Temperature, °C
γ
800
γ
1
γ
γ
S
P
700
K
2
α
α
(Ferr ite)
600
Fe 3C
500
α+Fe 3C
400
0
1,0
1,0
2,0
Pearlitic microstructure
Carbon content, wt.-%
Figure 3.39: Solidication of a eutektoid steel, 0.8 wt.-% C. Right: Optical micrograph of a
pearlitic microstructure, after [10].
102
3 Strength and phase diagrams
1100
γ+Fe 3C
γ
(Austenite)
1000
γ
γ
γ
900
γ
1
γ
γ
800
2
γ
Fe 3C
α+γ
K
3
700
Perlit
Temperature, °C
γ
α
(Ferr ite)
600
Fe 3C
α
Fe 3C
500
α+Fe 3C
400
Pearlitic microstructure with cementite
(light) on the grain boundaries
0
1,0
1,0
2,0
Carbon content, wt.-%
Figure 3.40: Solidication of a hypereutectoid steel, 1.1 wt.-% C. Right: Optical micrograph of
a hypereutectoid microstructure, after [10].
103
4 Processes at High Temperatures
In the previous chapters, we have already encountered processes at high temperatures.
We have seen that atoms can migrate in the crystal lattice because there are vacancies
in the crystal lattice, but this only happens to a signicant extent at high temperatures.
This allows, for example, to smooth out segregations occuring during solidication (see
chapter 3.4.2) or to form precipitates (see chapter 3.3.4). These important processes at
high temperatures will be considered in more detail in this chapter. First, chapter 4.1
explains why vacancies form in the crystal lattice and why their concentration increases
drastically with temperature. Then, in chapter 4.2, we will discuss the laws governing
diusion in the crystal lattice, which is based on vacancy migration. Finally, chapters
4.3 and 4.4 discuss the recrystallization and recovery processes that take place at high
temperatures.
4.1 Vacancies
As already mentioned, vacancies are places in the crystal lattice which are not occupied
by atoms. Fig. 3.18 shows such a vacancy. It is created when an atom is removed from
its place in the lattice. Since the atom has bonding forces to its neighbouring atoms,
an energy must be expended to remove the atom. This energy expenditure is called the
vacancy formation energy Qb . Despite this energy expenditure, vacancies form because
this increases entropy. For example, if one imagines a crystal lattice with N sites,
where all sites are occupied by the same atoms (e.g., Fe), there are no distinguishable
arrangements, i.e., the number of distinguishable arrangements is one. If, on the other
hand, one adds a vacancy, there are N distinguishable arrangements, because the vacancy
can reside in N dierent places. Accordingly, the entropy increases.
The entropy increase due to the insertion of a vacancy in the above example can be
easily calculated. In general, the entropy S is
(4.1)
S = kB · ln(W ) + b
where kB is the Boltzmann constant and b is an arbitrary constant. W is the probability of the considered state, dened by the number of distinguishable realization possibilities for this state (in this context one speaks of the number of so-called microstates). The
entropy of a state is therefore larger, the more distinguishable realization possibilities
there are for a state.
If the situation in which all N lattice sites are occupied by atoms is state 1, then
W1 = 1, because there is only one distinguishable realization possibility. For the state 2
with one vacancy, on the other hand, there are N distinguishable realization possibilities,
i.e. W2 = N . Thus, the entropy change ∆S when inserting a vacancy results in
104
4 Processes at High Temperatures
W 2
∆S = kB · ln
= kB · ln(N )
W1
(4.2)
Following this example, the question is now how many vacancies form in the crystal lattice in thermodynamic equilibrium. Here we can assume constant volume in a
good approximation and therefore use the free energy F as the relevant thermodynamic
quantity with the help of which we can answer this question. It holds:
(4.3)
F = U − TS
U:
T:
S:
internal energy
temperature
entropy
Let the energy of the crystal with N lattice sites, all occupied by the atomic species
A (no vacancies), be U0 . For the case that n lattice sites are occupied by vacancies, the
probability W is given by
W = ln
N!
n! · (N − n)!
(4.4)
In accordance with the above example, W = 1 if there is no vacancy, i.e. for n = 0
(note: 0! = 1). If there is a vacancy (n = 1), W = N . One way to understand the formula
in general is as follows: For each distinguishable arrangement of the n vacancies and
(N −n) atoms on the N places, there would be further n! arrangements, if the n vacancies
would be distinguishable among themselves, respectively (N − n)! further arrangements,
if the (N − n) atoms would be distinguishable. Accordingly, for each distinguishable
arrangement there would be n!(N − n)! additional arrangements, if vacancies and atoms
were distinguishable. In this case, all objects would be distinguishable and the number
of possible arrangements on N places is then N ! The number of possible arrangements
for the case of indistinguishability at hand here multiplied by n!(N − n)! must therefore
be N ! Converted, this results in the quotient in equation 4.4.
With the help of this probability W , the entropy results to
S = kB · ln
N!
n! · (N − n)!
(4.5)
where the arbitrarily constant b was set to zero for simplicity.
Since n vacancies increase the internal energy by nQb , the free energy is given by
F (n) = U0 + n · Qb − kB T · ln
N!
n! · (N − n)!
105
(4.6)
4 Processes at High Temperatures
In thermodynamic equilibrium, F (n) is minimal, i.e.
following holds
dF (n)
dn
= 0. For dFdn(n) , the
dF (n)
d N!
d = Qb −kB T ·
ln
= Qb −kB T ·
ln(N !)−ln(n!)−ln(N −n)!
dn
dn
n! · (N − n)!
dn
(4.7)
With the approximation ln(x!) ≈ xln(x) for not too small numerical values x, the
derivative results to
d d ln(N !) − ln(n!) − ln(N − n)! =
N ln(N ) − nln(n) − (N − n)ln(N − n)
dn
dn
(4.8)
N −n
N −n
= 0 − ln(n) − 1 + ln(N − n) −
· (−1) = −ln(n) + ln(N − n) = ln
N −n
n
and thus
N −n
dF (n)
= Qb − kB T · ln
dn
n
(4.9)
n
Because of n ≪ N , N n−n ≈ N
= cV results, where cV is the vacancy concentration.
dS
Thus, the entropy change dn is given by the term kB ln c1V . Because of cV < 1, dS
dn is always positive. Thus, under the condition n ≪ N used here, the entropy increases steadily
with increasing vacancy concentration. However, dS
dn becomes smaller and smaller with
increasing vacancy concentration. Thus, the increase in entropy with each additional vacancy becomes smaller and smaller. In contrast, the increase in the energy of the crystal
lattice with each new vacancy is constant, namely Qb . The equilibrium concentration
of vacancies is reached when the increase in the energy of the crystal lattice with each
new vacancy equals the increase in entropy with each additional vacancy multiplied by
the temperature. If the temperature increases, this equilibrium is reached at a smaller
value of dS
dn , i.e., a larger vacancy concentration. This qualitative observation can be
conrmed on the basis of the equilibrium condition dFdn(n) = 0 as follows.
0=
dF
= Qb + kB T · ln(cV )
dn
(4.10)
Rearranging equation 4.10 leads to
ln(cV ) = −
Qb
kB T
(4.11)
respectively
106
4 Processes at High Temperatures
Q b
cV = exp −
kB T
(4.12)
°
For example, Qb for copper is 1.6 · 10−19 J. Copper melts at 1083 C, i.e. 1356 K. Accordingly, using the above equation and kB = 1.38·10−23 J/K, the vacancy concentration
is 6.5 · 10−18 at room temperature and 1.9 · 10−4 at the melting temperature. This illustrates the drastic increase in vacancy concentration with temperature. For all metals,
the vacancy concentration is about 0.0001 when the melting temperature is reached, i.e.
about every ten thousandth lattice site is occupied by a vacancy.
If the crystal lattice is heated, the atoms start to vibrate. Hereby, it can happen
that an atom absorbs so much energy that it jumps from its place in the lattice to a
neighboring vacancy (see Fig. 3.18). If we look at the vacancy, it jumps in the opposite
direction. In the following, we consider the processes from the point of view of the
vacancy. Since the vacancy occupies a lattice site before and after the jump, the energy
of the vacancy before and after the jump is identical. In between, however, the vacancy
is in an energetically unfavorable state. Thus, as Fig. 4.1 shows, the vacancy must
overcome an energy mountain before it moves from its original equilibrium position
at location x1 to the next equilibrium position at location x2 . The magnitude of the
energy required for this is called the migration energy of the vacancy Qw (Fig. 4.1). As
mentioned, this migration succeeds from time to time only because the atoms oscillate
and thus occasionally have a sucient amount of energy in the right direction so that
the atom or vacancy can overcome the energy mountain. Related to one oscillation, the
probability that the
will migrate by overcoming the energy mountain is given
vacancy
Qw
by the term exp − kB T . On the one hand, the probability is the larger the higher
the temperature is. Thus, as the temperature increases, not only does the number of
vacancies increase dramatically. The vacancies also move much more frequently. On the
other hand, the probability of vacancy migration increases with decreasing migration
energy Qw .
Figure 4.1: Energy of a vacancy as a function of the spatial coordinate x. If the vacancy is
located at a lattice position (coordinates x1 and x2 ), its energy is minimal. If the
vacancy moves from lattice site to lattice site, it is in an energetically less favorable
state in between. It has to expend the migration energy Qw in order to overcome
the energy mountain located between the equilibrium positions.
107
4 Processes at High Temperatures
The chapter started with the statement that a vacancy is formed when one removes
an atom from its lattice site. One may wonder at this point what has become of the
atom that was previously present at the site of the newly formed vacancy. The answer
is that dislocations and grain boundaries are sources and sinks for atoms and vacancies,
respectively. Thus, they can absorb atoms while emitting vacancies or absorb vacancies
while emitting atoms. In this way, the crystal lattice is able to adjust the vacancy
concentration as the temperature changes to match the thermodynamic requirement
given in Equation 4.12. Fig. 4.2 shows the process of vacancy absorption or emission
using the example of an edge dislocation. In the left image, three vacancies can be seen
next to the edge dislocation. One of the vacancies moves towards the edge dislocation
and replaces an atom there. This changes the position of the edge dislocation (see middle
image). It no longer ends in its original slip plane, which is horizontal in Fig. 4.2, but one
level higher. This movement of the dislocation from one slip plane to another is called
climbing of the dislocation. In the case considered here, one vacancy has disappeared
because there are only two vacancies left in the middle image. This process is repeated
from the middle image to the right one, so that there is only one vacancy left on the right
and the dislocation has climbed up one level again. Because vacancies have disappeared
in this example, the dislocation acted as a vacancy sink. But one can also read the
image sequence from right to left. Then, the edge dislocation climbs down and creates
vacancies. So in this case, the dislocation acts as a vacancy source.
Figure 4.2: Absorption or emission of vacancies at an edge dislocation (the symbol ⊥ indicates
the location of the dislocation line, see Fig. 3.6) [26].
4.2 Diusion
Diusion is the movement of particles occurring without external inuences, in this case
the movement of atoms or vacancies in the crystal lattice. As already explained, diusion
is necessary, for example, in order to compensate for concentration dierences or to be
able to form precipitates. We now imagine that we observe a certain volume over a
certain time and note how many vacancies change places. The larger the number Z
of these events, the faster the vacancies move in the crystal lattice, so the greater the
diusion rate. On the one hand, Z is the greater the more vacancies there are in the
lattice, i.e., the greater their concentration cv . On the other hand, the more frequently
a vacancy moves from one lattice site to another, the greater
Z is. Thus, the diusion
Qb
rate increases with the vacancy concentration cv = exp − kB T and the rate a vacancy
108
4 Processes at High Temperatures
changes places, which is proportional to exp − kQBwT . If, for example, the concentration
of vacancies and the rate a vacancy changes places were both increased by a factor of
10, the number of place changes in a given time would increase by a factor of 100. Thus,
there is a multiplicative relationship and we can state:
Q Q Q +Q w
w
b
b
Diffusion rate ∼ exp −
· exp −
= exp −
.
kB T
kB T
kB T
(4.13)
We now imagine a solid solution consisting of the host atoms and the atoms dissolved
in the host lattice. Let the concentration of the dissolved atoms be the same everywhere.
In this case, diusion can only be recognized by a random motion of the atoms in the
crystal lattice. This changes if there is a concentration gradient in the solid solution,
e.g. due to a non-equilibrium solidication. Such a situation is shown in Fig. 4.3. In
this case, a solid solution with complete miscibility of the two elements, namely the
gray atoms and the red atoms, is assumed, as is the case, for example, with copper and
nickel (compare with Fig. 3.27). On the far left only the gray atoms are present, on
the far right only the red atoms. As Fig. 4.3 shows, the concentration c of the gray
dc
atoms decreases from left to right. The concentration gradient dx
is negative because
c decreases in the positive x-direction. If we now look at the area drawn in blue, there
are more gray atoms to the left of it than to the right. The motion of the atoms is
also in this case random. But because there are more gray atoms to the left of the blue
area than to the right of it, on average more gray atoms move from left to right through
this area than vice versa. As the arrows show, in the case of Fig. 4.3, four gray atoms
move from left to right through the blue area but only three from right to left when all
atoms change places once across the area. In this way, there is a total ow J of one gray
atom from left to right (for the red atoms it is the other way around). J is dened as
the number of gray atoms moving from left to right across the area under consideration
per unit time. Naturally, the larger the area A under consideration is, the larger is J .
J
To eliminate this relationship, we dene the current density j = A
. Thus, the current
density j indicates the number of gray atoms moving from left to right per area and
time unit. Accordingly, its unit is m12 s . The greater the concentration gradient and the
diusion velocity, the greater the current density. Accordingly, with the proportionality
factor D0 we obtain
Q + Q dc
Q dc
dc
w
b
j = −D0 · exp −
= −D0 · exp −
= −D
kB T
dx
kB T dx
dx
(4.14)
This relationship is called Fick's rst law after its discoverer. The negative sign results
dc
from the above example: While dx
is negative, the current density is positive because
it points in the positive x-direction. The parameter D is called diusion coecient. It
contains the so called diusion constant D0 and in the exponential term the so called
2
activation energy for diusion Q = Qb + Qw . D and thus also D0 have the unit ms .
The concentration is the number of atoms per volume. Accordingly, the concentration
gradient has the unit m14 . The e-function is dimensionless (numerator and denominator
109
4 Processes at High Temperatures
have the unit of one energy). As required, this results in the unit of the current density
on the right side, i.e. m12 s .
Figure 4.3: Emergence of a diusion current in a concentration gradient, after [5].
Table 4.1 gives some values for D0 and the activation energy for diusion Q′. Here, Q′
does not refer to an atom as before, but to a mole of atoms. The unit of Q′ is therefore
J
′
23 gives the number
mol . Thus, Q = Q · L, where Avogadro's number L = 6.022 · 10
of atoms per mole. Considering that for the ideal gas constant R = kB L holds, the
diusion coecient can be written as follows
Q Q′ D = D0 · exp −
= D0 · exp −
kB T
RT
(4.15)
The temperature-dependent diusion coecients resulting from Table 4.1 are shown
in Fig. 4.4. The following becomes clear:
ˆ Dierent elements diuse at dierent rates in a given host lattice. In iron, for
example, Cr diuses particularly slowly and carbon particularly quickly. The latter is due to the fact that carbon is interstitially dissolved. While a substitution
atom, such as chromium, can only change its place if one of the rarely occurring
vacancies is present at a neighboring lattice site, an interstitially dissolved atom
can do so if a neighboring interstitial site is unoccupied, which is almost always
the case. For this reason, interstitially dissolved atoms diuse as a basic principle
faster than substitutional atoms. This has an important technical consequence in
connection with the hardening of steels. As can be seen from Fig. 3.22, a simple
steel containing essentially only carbon as an alloying element must be cooled very
rapidly to get past the "nose" in the TTT diagram in order to harden the steel
110
4 Processes at High Temperatures
by the martensitic transformation. These high cooling rates are necessary because
carbon diuses very quickly. Thus, it is easy to partition carbon between the αphase and Fe3 C by diusion, and the diusion controlled transformation is fast.
While suciently fast cooling rates to prevent diusion controlled transformation
can be realized in thin-walled components, this is not possible inside thick-walled
components. This phenomenon is referred to as insucient hardenability, because
hardening by the martensitic transformation does not reach the interior of the
component. One way out is to use slower-diusing alloying elements, such as Cr.
These must also be partitioned between the α-phase and the carbide by diusion.
Because they diuse more slowly, considerably more time is required for this. Accordingly, the nose in the TTT diagram shifts to the right to longer times, so that
lower cooling rates, which can also be realized for thick-walled components, are
sucient for the martensitic transformation to take place. Hardenability is thus
improved.
ˆ In dierent host lattices, the atoms diuse at dierent rates. For example, it is
noticeable that diusion is much slower in iron than in aluminum. This is because
the iron atoms are more strongly bonded in the crystal lattice than the aluminum
atoms (see Table 2.3). This makes the formation and migration of vacancies, and
thus diusion, more dicult. The stronger bonding also causes iron to melt later
than aluminum (see Table 2.3). Thus, there is a tendency for the rate of diusion
at a given temperature to decrease as the melting temperature (i.e. the bond
strength) increases. Consequently, iron must be annealed at a higher temperature
than aluminum, for example, to compensate for concentration dierences.
ˆ The diusion rate depends on the crystal lattice. This is particularly well seen
in the fact that the diusion rate in iron changes abruptly at the transformation
temperature from α-Fe to γ -Fe. In the fcc γ -Fe it is obviously much lower than
in the bcc α-Fe. This is because the face-centered cubic crystal lattice is more
densely packed than the body-centered cubic one. This makes vacancy migration
more dicult because there is less space available for atomic motion. In addition,
the denser packing leads to more nearest neighbors. As a result, an iron atom
is more tightly bound in the fcc lattice than in the bcc lattice. Accordingly, the
formation energy for vacancies Qb is larger in the fcc lattice, which also slows down
diusion.
The atoms dissolved in the host lattice, such as C or Cr in iron, are also called impurity
atoms. If we consider their diusion, we therefore speak of impurity diusion. If, on the
other hand, we consider the diusion of the host atoms themselves, i.e. in this example
the diusion of the iron atoms, we speak of self-diusion. At this point, one may ask
how one can observe self-diusion. To do this, one uses an isotope of the host element
and observes its motion.
The example in Fig. 4.3 shows a concentration gradient pointing only in the direction
dc dc dc
of the x-axis. In general, the concentration gradient has components dx
, dy , dz in the
direction of all three coordinate axes. This results in components jx , jy , jz of the current
111
4 Processes at High Temperatures
′
Table 4.1: Diusion constants D0 and activation energies Q for diusion of dierent elements
in dierent host lattices. Sources: [16] for Cr in Fe; [10] for all other data.
2
kJ
Q′ [ mol
]
C
D0 [ ms ]
2.8 · 10−4
6.2 · 10−7
Cr
3.0
343
Fe
5.0 · 10−5
2.3 · 10−5
2.3 · 10−4
284
Matrix
Diusing element
α-Fe
α-Fe
α-Fe
γ -Fe
γ -Fe
Fe
Al
Al
C
251
80
148
144
Figure 4.4: Diusion coecient D of dierent elements in dierent host lattices as a function
of 1/T (T : temperature).
density. Thus, generalized, the current density is a vector with these three components.
Accordingly we can write
 


jx
dc/dx
jy  = −D · dc/dy 
jz
dc/dz
(4.16)


d/dx
j = −D · d/dy  c = −D · grad c
d/dz
(4.17)
or
112
4 Processes at High Temperatures
where grad c is the concentration gradient and the underscore symbolizes that the
current density (like the velocity) is a vector.
For example, if we consider segregation due to non-equilibrium solidication, we are
interested in how long it takes for concentration dierences to be equalized by diusion.
In other words, we are interested in how the concentration at a location changes over
time. This information can be derived from Fick's rst law. For now, we again restrict
ourselves to the one-dimensional case and consider the volume shown in Fig. 4.5 with
area A and thickness dx. We imagine that we observe the ow of foreign atoms B in a
concentration gradient. We note that on the left side B atoms ow into the volume with
current density j(x) and on the right side they ow out with current density j(x + dx) ̸=
j(x). Accordingly, j(x) · A gives the number of atoms owing into the volume per unit
time, whereas j(x+dx)·A shows how many atoms ow out per unit time. Consequently,
the change in the number of B atoms per unit time in the volume under consideration is
(4.18)
− A · j(x + dx) − A · j(x)
Figure 4.5: Diusion current density j through a thin-walled cuboid of cross-sectional area A
and thickness dx changing with spatial coordinate x.
In the case shown, j(x + dx) and j(x) are positive because they point in the positive
direction of the x-coordinate. For example, if j(x + dx) is larger than j(x), as shown
in Fig. 4.5, the number of B atoms in the volume under consideration decreases. This
proves that the minus sign in Equation 4.18 is necessary. Dividing equation 4.18 by the
volume A · dx gives the change in the number of B atoms per unit time and volume, i.e.
the change in concentration with time ∂c∂tB :
∂cB
A · j(x + dx) − A · j(x)
j(x + dx) − j(x)
∂j
=−
=−
=−
∂t
A · dx
dx
∂x
113
(4.19)
4 Processes at High Temperatures
Whereas previously the concentration gradient was assumed to be invariant with time,
c and j are now quantities dependent on location and time. Therefore, Eq. 4.19 contains
partial derivatives, represented by the symbol ∂ . If we now use the right term of Fick's
rst law (Eq. 4.14) for j and generalize ∂c∂tB to ∂c
∂t , we obtain Fick's second law
∂ ∂c ∂2c
∂c
=−
−D
=D 2
∂t
∂x
∂x
∂x
(4.20)
It can be seen that the concentration at a given location always changes over time when
the second derivative of the concentration with respect to the location is not zero, i.e.
when the concentration gradient changes with the location. A constant concentration
gradient, as in Fig. 4.3, does not lead to this. This fact is also easy to understand with
the help of Fig. 4.5. If the concentration gradient were constant there, j(x + dx) = j(x)
would follow from Eq. 4.14 and the concentration would not change in the considered
volume. Obviously, however, there is a situation in Fig. 4.5 where the concentration
gradient on the left side of the cuboid is smaller in magnitude than on the right and
therefore fewer B atoms ow into the volume than out of it.
Extending the relationship shown in Eq. 4.20 to the general three-dimensional case
is straightforward. In general, not only along the x-coordinate dierent atomic currents
will ow into and out of a volume element. This will also happen along the y and z
coordinates, and this will change in an analogous way the concentration of the atomic
species under consideration. If we imagine, for example, that along the x-coordinate
5 atoms less ow into the considered volume element than ow out, but along the yand z-coordinate 4 and 2, respectively, more ow in than ow out, the total change is
−5 + 4 + 2 = 1. Thus, all contributions have to be added, resulting in the following
relation for the three-dimensional case:
∂2c
∂c
∂2c
∂2c =D
+
+
∂t
∂x2 ∂y 2 ∂z 2
(4.21)
An important example that can be considered with the help of Fick's second law is the
carburization of so-called case-hardened steels. Case-hardened steels are characterized
by a low carbon concentration. Typical values are between 0.1 % and 0.2 %. In case hardening, the steel is heated in a furnace to such a high temperature that it is in the γ single
phase eld and the carbon solubility is correspondingly high, e.g. to 950 C. In addition,
the steel is exposed to a carburizing atmosphere. This can be achieved, for example,
by introducing carbon monoxide and hydrogen. The reaction CO + H2 −→ H2 O + C
creates the desired carburizing atmosphere with a certain constant carbon concentration
in the gas. This also establishes a certain constant carbon concentration cs on the steel
surface. Starting from the surface, the carbon now diuses into the steel and a surface
layer with increased carbon content is formed. The steel is then quenched and subsequently tempered at a relatively low temperature, e.g. 180 . The high carbon content
in the surface layer coupled with the low tempering temperature results in a hard and
wear-resistant surface layer. Because the steel continues to have a low carbon content
°
°
114
4 Processes at High Temperatures
in its interior, a ductile but also strong material state is produced there despite the low
tempering temperature. This combination of properties is ideal for components, such as
gears, whose surfaces are subject to wear.
We now take a closer look at the carburization process with the aid of Fick's second
law. As already mentioned, a certain constant carbon concentration cs is established
on the steel surface. Starting from the steel surface, which is at the spatial coordinate
x = 0, the carbon now diuses into the steel. For x > 0, we are in the steel, which we
imagine to be innitely extended. This assumption is generally well fullled, because
the penetration depth of the carbon is generally small compared to the steel dimensions.
Thus, c(x = 0) = cs for all times t ≥ 0 and c(x) = c0 for t = 0 and x > 0 holds. Here,
c0 is the carbon concentration of the not yet case-hardened steel. What is sought is the
carbon concentration c(x, t) in the steel for arbitrary time and location. As solution,
which satises Fick's second law and the given boundary conditions, one obtains:
x c(x, t) − c0
= 1 − erf √
cs − c0
2 Dt
(4.22)
with Gaussian error function
2
erf(z) = √
π
Z z
(4.23)
exp −y 2 dy
0
The quotient on the left side of Eq. 4.22 gives the dierence between searched concentration c(x, t) and initial concentration c0 , related to the concentration dierence
cs − c0 initially prevailing everywhere. For example, if the value of the quotient is 0.1,
the concentration at the considered location and time increases by 10 % of the initial
dierence. It is noteworthy that c(x, t) depends only on the quantity √xt . For example,
c(x1 , t1 ) = c(2x1 , 4t1 ) is therefore valid. Thus, at twice the distance, four times the time
is required until a certain concentration of the atomic species under consideration (here:
carbon) is reached. The resulting temporal and spatial variation of c(x, t) is shown in
Fig. 4.6. The course of the Gaussian error function is shown in Fig. 4.7.
In our example, let cs = 0.01 (i.e. 1 %) and c0 = 0.001. To ensure sucient wear
resistance over the lifetime of the component, we require a carburized surface zone of at
least 0.2 mm thickness, which should have a carbon content of at least 0.5 %. Insertion
into Equation
4.22 gives the value 0.50 for the quotient on the left-hand side. Accordingly,
√
√
erf(x/(2 √Dt) = 0.5 must hold. This is satised to a good approximation for x/(2 Dt) =
0.5 or x/ Dt = 1. The thickness of the carburized boundary layer is thus
x≈
√
(4.24)
Dt
This formula is often used to estimate the distance atoms travel by diusion.
For economic reasons, we limit the carburization process in our example to a duration
of one hour, i.e. t = 3600 s. Logarithmizing equation 4.24 we obtain
115
4 Processes at High Temperatures
Figure 4.6: Concentration c(x, t) calculated with equation 4.22 depending on the local coordi-
°
nate x and time t. Values used: T
= 950 C, diusion coecient D for carbon in
γ -Fe according to Tab. 4.1, cs = 0.01, c0 = 0.001.
Figure 4.7: Gaussian error function erf(z) in dependence of the parameter z .
Q′
1
ln(D0 ) −
+ ln(t)
2
RT
(4.25)
Q′
R · ln(D0 ) + ln(t) − 2ln(x)
(4.26)
ln(x) =
i.e.
T =
°
Using the values from Table 4.1 for D0 and Q′ , 951 C results as annealing temperature
in order to obtain the required surface layer thickness of 0.2 mm after one hour. This
116
4 Processes at High Temperatures
is also shown by the blue curve in Fig. 4.6, which was calculated for almost identical
parameters (T = 950 C instead of 951 C). Now, for example, one could still aim to
realize a thicker surface layer by annealing longer. Therefore, instead of one hour, four
hours shall be annealed in the furnace. Inserting this new annealing time in Eq. 4.24
gives a resulting surface layer thickness of 0.4 mm (compare also with the orange curve in
Fig. 4.6). This result could have been obtained without recalculation,
because we have
√
already seen that c(x, t) depends only on the parameter x/ t. Thus, for a given value
of c(x, t) (here: 0.5 %), the surface layer thickness x doubles for a quadrupled annealing
time t. Thus, if one wanted to increase the surface layer thickness tenfold to 2 mm,
one would need 100 hours instead of one hour. This shows why such surface layers are
limited in practice to a few tenths of a millimetre.
°
°
4.3 Recrystallization
In chapter 3.3.2, the special importance of strengthening by grain size reduction became
apparent, which leads to an increase in strength without loss of ductility. Therefore,
this strengthening method, i. e. a particularly small grain size, is highly desirable.
First of all, the solidication conditions have an inuence on the grain size. However,
solidication does not lead to grain sizes of about 10 micrometers and smaller that are
desired for pronounced strengthening by grain size reduction. If one wants to achieve
such a ne grain size, the method of recrystallization, leading to the formation of new
grains, must be used. This is the subject of this chapter.
First, let us imagine a metal that has been strongly deformed. As explained in section
3.3.1 and shown schematically in Fig. 4.8 (a), this leads to a high dislocation density
of up to 1016 m−2 . According to section 3.2, dislocations distort the crystal lattice and
thus increase the energy stored in the crystal. The high dislocation density has thus
created an energetically unfavourable state. If it were possible to reduce the dislocation
density, the material could be converted to a more energetically favourable state. One
way to accomplish this is by recrystallization. As Fig. 4.8 (b) illustrates, individual
grains reform in this process. This happens analogous to the formation of precipitates
by a nucleation and growth process. However, here we do not have particles of a second
phase. Instead, grains are formed which exactly correspond to the already existing grains
of the host lattice with respect to composition and crystal structure. These newly formed
grains grow as atoms from the neighboring, highly deformed grains migrate across the
grain boundaries and are incorporated into the new grains. The growth process of the
new grains is not defect-free and therefore dislocations also occur in the newly formed
grains. However, their dislocation density of about 1012 m−2 is much lower than that in
the highly deformed grains. Consequently, recrystallization leads to a more energetically
favourable state.
In particle hardening, the goal was to produce small precipitates. To understand how
this can be achieved, it was necessary to look at the energy contributions in this process.
Here, the goal is to form grains that are as small as possible. Accordingly, it is also
necessary here to be clear about the energy contributions. There are many analogies to
particle strengthening. Here, too, for example, it is necessary to form as many nuclei as
117
4 Processes at High Temperatures
(a) Initial situation with a high dislocation density due to plastic deformation
(the symbol ⊥ indicates the location of
the dislocation line, see g. 3.6).
(b) Formation of a new grain (blue) with
reduced dislocation density by recrystallization.
Figure 4.8: Schematic representation of the recrystallization of a polycrystalline metal.
possible, because the nuclei formed grow until all the grains of high dislocation density
have been replaced. Accordingly, the more nuclei formed, the smaller is the grain size
after recrystallization is complete. Let ρ1 be the dislocation density of the deformed
grains and ρ2 that of the newly formed ones with ρ2 < ρ1 . According to Section 3.2,
the energy stored per unit length of a dislocation line is given by the line tension T .
Because the dislocation density gives the dislocation line length per volume, the product
T · ρ gives the energy stored per volume (T · ρ = (energy/length) (length/volume)). If
we think of the newly formed grain simplied as a sphere with radius r, the dislocation
density in this spherical volume has changed from ρ1 to ρ2 . The associated change in
free enthalpy is therefore1
4
∆Gv = πr3 · T · (ρ2 − ρ1 ) < 0
3
(4.27)
Again, the analogy to equation 3.6 is obvious. However, the causes of the driving
forces are dierent. Here, the reduction of dislocation density leads to an energetically
1
For those with a good knowledge of thermodynamics: Strictly speaking, the term 43 πr3 T (ρ2 − ρ1 )
only indicates the energy change U of the crystal lattice. As said, however, no other phase is formed
here but grains of the same crystal structure and composition are formed. Therefore, in contrast to
particle strengthening, the entropy change ∆S is negligible. Because of ∆G = ∆H −T ∆S, ∆G ≈ ∆H
(H : enthalpy) follows here rst of all. Furthermore, because recrystallization does not lead to any
appreciable volume change (because of identical crystal structure, the packing density of the atoms is
unchanged before and after recrystallization) and thus the work of volume change is negligible, it also
follows ∆H ≈ ∆U , thus also ∆G ≈ ∆U . Consequently, it is irrelevant here whether one considers
the change in energy U or in free enthalpy G. To make the analogy to particle strengthening clear,
we stay here with ∆G.
118
4 Processes at High Temperatures
more favorable state, whereas in particle strengthening the chemical disequilibrium is decisive. Since both cases involve a volume term (i.e., a term proportional to the converted
volume), the same symbol ∆Gv is used here.
As in the case of particle strengthening, the formation of the new grain creates an
interface. In the case of a particle of a dierent phase, we call it a phase boundary; here
we refer to it as a grain boundary. The associated increase in free enthalpy is given by
Eq. 3.7 as follows
(4.28)
∆Gs = 4πr2 γs
(γs : specic interfacial energy of the grain boundary). Accordingly, an energy balance
completely analogous to that shown in Fig. 3.17 results2 . Thus, a nucleation barrier
must be overcome here as well. From
4
∆G = ∆Gv + ∆Gs = πr3 · T · (ρ2 − ρ1 ) + 4πr2 γs
3
(4.29)
follows the condition for the maximum of ∆G, at which r corresponds to the critical
radius r⋆ :
d∆G
= 0 = 4π(r⋆ )2 · T · (ρ2 − ρ1 ) + 8πr⋆ γs
dr
(4.30)
This leads to the critical radius
r⋆ =
2γs
2γs
≈
T (ρ1 − ρ2 )
T ρ1
(4.31)
Because ρ2 ≪ ρ1 applies in general, the expression on the right-hand side can be
further simplied as indicated. Thus, the larger the dislocation density generated by
deformation, the smaller is r⋆ .
The result obtained here for r⋆ is analogous to that of Section 3.3.4. However, the
denominator in the above equation depends on the dislocation density and not on the
driving force due to a phase transformation. As in particle strengthening, the smaller
r⋆ is, the greater is the nucleation rate of new grains. Thus, the greater the dislocation
density before recrystallization, the ner the grains afterwards.
The existence of a nucleation barrier has another important consequence: as in particle strengthening, recrystallization does not take place immediately but with a time
delay. This is illustrated in Fig. 4.9 by the example of a nickel alloy deformed at room
2
As Fig. 4.8 (b) shows, the new grains are generally formed at the grain boundaries of existing grains.
Strictly speaking, one would have to take into account that not only new grain boundaries are formed,
but also that previously existing ones are removed. For reasons of simplication, this is neglected
here, especially since more new interface is formed than old interface is removed, so that the statement
that an energy expense occurs here is correct.
119
4 Processes at High Temperatures
temperature to a nominal strain of -0.7 (in a rolling process, this corresponds to a thickness reduction of 70 %). The higher the temperature, the earlier recrystallization begins,
because the atomic motion becomes faster and faster with increasing temperature and
thus less and less time is required until a critical nucleus is formed. In contrast to
the formation of precipitates, this trend toward higher temperatures continues on and
on because the denominator in the above equation does not decrease with increasing
temperature, whereas this is the case for the denominator of Eq. 3.10. It is true that
the onset of recrystallization depends not only on temperature but also on dislocation
density, because r⋆ changes with dislocation density. However, referring to the absolute
temperature scale in Kelvin, it is a rule of thumb that one should reach at least 40 % of
the melting temperature of the metal in question in order to start the recrystallization
process in a reasonable period of time. For nickel with a melting temperature of 1726 K,
this is approx. 700 K.
Figure 4.9: Annealing times tStart , tF inish to the start and end of recrystallization, respectively, as a function of temperature for a cold-worked nickel alloy (after [17, p. 85]).
In summary, the steps to achieve the smallest possible grain size by recrystallization
can be summarized as follows:
ˆ Introduction of a high dislocation density by as much plastic deformation as possible.
ˆ Annealing at elevated temperature to recrystallize the material completely within
a technically reasonable period of time.
Since much greater plastic deformation can be achieved under compressive stress than
120
4 Processes at High Temperatures
under tensile stress before the material fails, forming under pressure such as rolling or
forging is normally used for the rst step.
4.4 Recovery
As was pointed out in chapter 4.1, moving vacancies can be absorbed by dislocations.
The dislocations thus leave their original slip plane and end up on another one. We have
called this process climbing. It can only occur at elevated temperatures, because only
then do vacancies migrate to a signicant extent. In Fig. 3.7, it became clear that two
edge dislocations with their half-planes can unite to form a perfect crystal plane. The
two dislocations originally present have then disappeared. They have annihilated each
other. The prerequisite is not only that the two half-planes point in opposite directions.
The two dislocation lines must also lie on exactly the same slip plane. If this is not the
case, the two half-planes cannot unite to form a complete plane and the dislocations
with their respective dislocation lines remain.
This dislocation annihilation is extremely rare at room temperature because it is very
unlikely that two oppositely oriented dislocations share the same slip plane. At elevated
temperatures, however, the dislocations can help themselves because they are able to
climb. This allows them to leave their original slip planes and reach the same plane. Once
this is achieved, annihilation takes place. In this process, the dislocations move towards
each other by climbing and gliding because attractive forces prevail, i.e. the dislocations
attract each other. This process of dislocation annihilation, which is called recovery,
reduces the dislocation density. Thus, there are two processes by which dislocation
density can be reduced at elevated temperature: recovery and recrystallization. In
contrast to recrystallization, recovery begins at somewhat lower temperatures (rule of
thumb: from about 30 % of the melting temperature, based on the absolute temperature
scale).
If one wants to produce a grain as ne as possible by recrystallization, recovery is
undesirable, because part of the originally existing dislocation density is thereby reduced
before recrystallization begins. Because no nucleation barrier has to be overcome for
recovery to take place, recovery begins immediately at elevated temperature. Thus, it
is inevitable that the dislocation density is somewhat reduced before recrystallization
begins. On the other hand, recovery can also be useful. For example, if a metal was
strongly plastically deformed and, thus, its deformation capability largely exhausted, the
deformation capability can at least be partially regained by recovery. The metal thus
partially recovers from the previous exertion of cold forming. The strength decreases
in the process because work hardening is reduced.
121
5 Oxidation and corrosion
Materials are surrounded by media that can lead to material attack on the surface. If,
for example, an iron rod is held in a ame, it tends to scale, i.e. a layer of iron oxide is
formed, which often akes o during cooling and leads to consumption of the material.
This type of attack is called oxidation because it is caused by oxygen in the air. Our
experience shows that this type of attack requires high temperatures.
In contrast, the rusting of a steel car body sheet, for example, already takes place
at ambient temperature. The main dierence is that the attack is not caused by atmospheric oxygen alone but also by contact with an electrically conductive liquid, the
electrolyte. This process is called corrosion. Both processes will be discussed below. It
will be explained why oxidation processes only occur at much higher temperatures than
corrosion processes.
The economic loss through corrosion/oxidation damage is very signicant. About
2% of the annual production of steel is lost through rust every year. For the Federal
Republic of Germany this means a steel loss of about 0.9Mt/a or 30 kg/s. Therefore, it is
not surprising that about 2500 to 5000e per inhabitant are spent annually on corrosion
protection measures.
5.1 Oxidation
All materials that can form a covalent or ionic bond with oxygen under an energy gain
are at risk of oxidation in air. Considering the energy conversion during the reaction
(5.1)
Material + Oxygen −→ Oxide
it can be seen that all metals except gold tend to oxidize (Fig. 5.1), negative values
correspond to energy gain). This is why gold is the only metal that occurs in the earth's
crust in elemental form.
With the exception of PTFE, polymers also tend to oxidize. They burn under formation of CO2 and H2 O. The reason for the stability of PTFE is that the electrons of
carbon are already bound by the extremely electronegative uorine. Nitride and carbide
ceramics also tend to oxidize, since the oxidation products (oxide plus CO2 or NOx ))
are more energetically stable. Only oxide ceramics remain as engineering materials that
are resistant to oxidation.
This raises the question why, in our experience, oxidation at ambient temperature in
dry air does not play a role, although the oxidation reaction should take place from an
energy point of view. To answer this question, it is necessary to consider the reaction
kinetics. It has already become clear from the example of the scaling of iron that an
oxide layer forms during the reaction with oxygen. This separates the reaction partners.
122
5 Oxidation and corrosion
Resistant against
oxidation
Vulnerable for
oxidation
Energy [kJ/mol]
SiC
Diamond
Al2O3
SiO2
(no reaction)
Ceramics
Magnesium
Aluminium
Titanium
Chromium
Zinc
Tungsten
Iron
Nickel
Copper
Platinum
Gold
Metals
Wood
CFRP
GFRP
other
Polymers
PTFE
Polymers
Composites
Figure 5.1: Formation energy for oxides.
Further oxidation can only occur to the extent that oxygen ions or metal ions can
diuse through the layer (Fig. 5.2). Since diusion processes are very slow at ambient
temperature, even a few nanometers of oxide layer thickness are sucient to separate
the reactants and to prevent oxidation.
(a) Metal ions diuse faster than oxygen ions.
(b) Oxygen ions diuse faster than metal
ions.
Figure 5.2: Kinetics of oxidation using the example of a metallic material.
If a material is to be resistant to oxidation at high temperatures the following is
required from the oxide layer:
ˆ Low diusion coecient (i.e. generally high melting point);
ˆ Low electrical conductivity;
ˆ Firm adhesion to the base material.
123
5 Oxidation and corrosion
When comparing the melting points of important oxides (Tab. 5.1), it can be seen
why Al2 O3 , Cr2 O3 and SiO2 protect very eciently, whereas FeO and Cu2 O are less
eective. Al2 O3 has the additional advantage of a very low electrical conductivity (109
times lower than FeO). Therefore, it is not surprising that virtually all metals used
at high temperatures are alloyed with Cr, Al and, in rare cases, silicon. Ceramics for
high temperature applications are either oxide-based (e.g. Al2 O3 ) or silicon-based (SiC,
Si3 N4 ). In the latter case, it is the formation of a SiO2 top layer that protects against
oxidation.
Table 5.1: Melting temperatures of important oxides.
Material
Melting temperatures /
FeO
1369
Cu2 O
Cr2 O3
Al2 O3
SiO2
1235
‰
2266
2072
1702
Tab. 5.2 summarizes the temperature limits for important engineering materials. It is
clear that the oxidation resistance increases steadily with the Cr+Al+Si content. This is
understandable, since alloys generally do not form pure surface layers, but rather mixed
oxides, e.g. (FeO + Cr2 O3 ), and as the content of Cr, Al, Si increases, the proportion of
the less protective oxide is reduced. Unfortunately, refractory metals such as tungsten
(Tm = 3422 C) form extremely unstable oxides. For this reason, they have not yet gained
any practical signicance in high-temperature applications, although their mechanical
strength is exceptionally high. Polymers, whose main components are C and H, do not
form a protective oxide layer. As a result, they usually decompose above 250 C.
°
°
Table 5.2: Temperature resistance of some engineering materials.
Material
Oxidation resistant up to /
Alloy steels
500
(1 5 % Cr)
High temperature steels
650
Heat resistant steels
1100
Ni-Cr-alloys
950
(ca. 15 20 % Cr)
Ni-Cr-Al-alloys
1100
(ca. 15 20 % Cr + 5 % Al)
SiC, Si3 N4
1300
124
‰
5 Oxidation and corrosion
5.2 Corrosion
5.2.1 Fundamentals
In the previous chapter it became clear that high temperatures are necessary for the
oxidation of materials, because diusion of ions and electrons through oxide layers is
required. Consequently, alloyed steels are suciently resistant to dry air up to 500 C
(Table 5.2). If, on the other hand, they are simultaneously exposed to moisture or
wetness, the attack can take on dramatic proportions even at room temperature. As Fig.
5.3 shows, the attack of iron by oxygen containing water is very similar to oxidation.
°
Figure 5.3: Reaction scheme for the corrosion of metal in water.
Also here an oxidation of the metal takes place by giving o electrons
Fe −→ Fe2+ + 2e− ,
(5.2)
as well as a reduction reaction under absorption of electrons
O2 + 2H2 O + 4e− −→ 4OH−
(5.3)
However, if the liquid is electrically conductive, i.e. an electrolyte, it can dissolve
the charged metal ions and move them away from the metal/electrolyte interface. This
means that no protective surface layer is formed on the metal so that the attack can
take place even at low temperatures. This corresponds to our experience of rusting
steel. First the iron ions go into solution. If the water evaporates, their solubility limit
is reached and loose rust is deposited on the metal, which, however, does not form a
protective surface layer. Simplied, it can be understood as 4Fe(OH)2 (compare with
Fig. 5.3). Fig. 5.3 also shows that the two partial reactions according to eq. 5.2,
5.3 can take place at dierent locations. This is because the electrons in the conductive
metal can move easily. Lateron, it becomes clear that this has important consequences
for corrosion protection.
4Fe(OH)2 + O2 + 2H2 O −→ 4Fe(OH)3 −→ 2Fe2 O3 + 6H2 O ,
125
(5.4)
5 Oxidation and corrosion
In summary, the following requirements are necessary for corrosion attack:
ˆ The presence of a conductive medium;
ˆ The presence of an oxidizing agent (in water this is the dissolved oxygen);
ˆ A good solubility of the oxidation product in the corrosion medium.
In addition to the reduction reaction in water containing oxygen, as shown in eq. 5.3,
corrosion in acids according to
2H+ + 2e− −→ H2
(5.5)
often plays an important role.
As with oxidation, the driving forces of the reactions (eq. 5.2- 5.5) shall be analysed
in the following. The following thought experiment is used for this purpose: A metal
plate Me is immersed in an electrolyte containing the same metal as cation. Depending
on the oxidation tendency of the metal, a certain amount of positively charged metal ions
is dissolved in the electrolyte during immersion. As the electrolyte becomes increasingly
positively charged and the metal sheet increasingly negatively charged, the metal ions
are increasingly attracted to the metal surface and the reverse reduction reaction leading
to metal deposition is accelerated (Fig. 5.4). After a short time, an electrical potential
is established so that the two partial reactions are in equilibrium. If this potential is
measured against a reference, the tendency of the metal to release electrons, i.e. to
oxidize, can be quantied. It is the greater the more negative the potential of the metal
plate is.
Figure 5.4: On the driving force of corrosion.
126
5 Oxidation and corrosion
As a reference, a second half cell, the standard hydrogen electrode, is used. Its reaction
is according to eq. 5.5 (Fig. 5.5). Thereby, the H+ -concentration is arbitrarily set to
1 mol/l (pH=0), the H2 -pressure is set to 1 bar and the potential to 0V. The voltage
dierence between both half cells is measured by a voltmeter. Negative voltage means
that an electron pressure towards the reference electrode exists. In analogy to the
energy scale in Fig. 5.1, Fig. 5.6 shows voltages. An important result is, that only the
noble metals Au, Pt, Ag are resistant against corrosion in aqueous media and acids.
Figure 5.5: Measurement of the voltage dierence between two half cells. 1: Measuring electrode, 2,3: counter platinum electrode, 4 electrolyte (pH=0), 5 electrolyte bridge,
6 voltmeter, 7 ne-pored glass frits
5.2.2 Kinetics of corrosion
According to Fig. 5.6, it would be expected that the corrosion in water containing
oxygen would proceed faster than in acids. However, this contradicts the experience
that a millimetre thick iron sheet dissolves in an acid (e.g. HCl) within a few hours.
If, on the other hand, the same sheet is held in water containing oxygen, the corrosion
loss is less than one millimetre per year. The cause is immediately apparent from Fig.
5.7. Whereas in acid corrosion the positive H+ ions are drawn to the negatively charged
metal sheet by the electrostatic forces, the O2 molecule uncharged. It diuses in the
aqueous solution and meets the metal sheet only randomly.
Another important aspect is shown in Table 5.3. Here the corrosion rates of dierent
materials in agricultural and industrial air are compared. First of all, it is noticeable
that the less noble materials zinc and aluminium have lower removal rates than steel.
As with oxidation, this is due to the formation of relatively dense oxide/hydroxide layers
which inhibit corrosion. These layers can be unstable against further substances in the
corrosive medium. For example, zinc hydroxide layers are attacked by SO2 (component
of industrial air). A steel plate, containing a zinc coating of 40 µm (hot dip galvanizing),
is protected outside of industrial areas for about 25 years. However, in industrial areas
127
5 Oxidation and corrosion
Figure 5.6: Standard electrode potentials.
(a) Metal in Acid.
(b) Metal in oxygenated water.
Figure 5.7: Representation of the corrosion of metals in acid and oxygenated water.
128
5 Oxidation and corrosion
the protective eect can be reduced to a few years.
Table 5.3: Material loss due to atmospheric corrosion.
Material
In industrial air / µm/a
In country air / µm/a
Aluminium
0,7
0,05
Steel
40 170
10 65
Copper
1,3
0,5
Zinc
3 22
1 2
A very important and aggressive component of aqueous solutions is the Cl− -ion (in
seawater, through road salt on roads). It is able to attack many layers, such as Al2 O3 .
Therefore, aluminium is at risk of corrosion in seawater or in contact with road salt,
whereas it is very stable in the absence of Cl− -ions (Table 5.3).
5.2.3 Corrosion protection measures
It is often unavoidable to use metallic materials in media that present a risk of corrosion.
It is therefore extremely important to take measures which reduce the corrosion attack to
the extent permissible for the planned component life. There are three main possibilities:
ˆ Selection of suitable materials;
ˆ Application of a protective coating;
ˆ Corrosion-resistant design.
These possibilities will be discussed in the following.
Application of a protective coating
A very common measure is the galvanizing of steel. Hereby, the kinetic inhibition of
the atmospheric corrosion of zinc (Table 5.3) is utilized. Typical layer thicknesses are
5 to 50 µm. This results in a corrosion protection that can last for several decades
outside industrial areas. However, in industrial areas the protective eect may be only
a few years. Comparable processes are also phosphating of steel and the anodizing of
aluminium. In anodizing, aluminium is immersed in an electrolyte and additionally
positively polarised. This leads to the formation of a relatively thick Al2 O3 top layer,
protecting the base material.
Coating processes have the advantage that they are cost-eective. Their disadvantage
is that their protective eect is limited in case of local damage (no self-healing eect).
Coatings which are less noble than the base material (e.g. Zn on Fe) are preferable. In
case of small damages, the oxidation reaction still occurs by dissolving zinc. As electrons
are pressed from the less noble zinc into the more noble iron, a negative polarization of
the iron occurs. This suppresses the oxidation of the iron and at the iron surface almost
exclusively the reduction reaction takes place (Fig. 5.8 (b)).
129
5 Oxidation and corrosion
(a) Noble coating.
(b) Non-noble coating.
Figure 5.8: Corrosion attack in case of noble and non-noble coatings.
In the case of damage to more noble coatings (e.g. Cu, Fig. 5.8 (a)), the entire
surface is available for the reduction reaction so that a large amount of oxygen can be
converted. The electrons required for this must be made available by oxidation of the
exposed metal. The metal dissolution is therefore highly localized and accelerated, so
that a deep corrosion hole can form.
Selection of suitable materials
The most important element for increasing the corrosion resistance of steels is Cr. As
already discussed in chapter 5.1, it forms a dense Cr2O3 layer which is relatively resistant
to aggressive media. Steels with a chromium content of at least 10.5% are therefore
designated as stainless steels in accordance with DIN EN 10020 (further requirement: Ccontent < 1.2%). Strictly speaking, however, this only applies to atmospheric corrosion.
In aggressive media, such as sea water, corrosion phenomena can nevertheless occur. A
very common steel is the so-called 18-8 or V2A steel (18 Cr, 8 Ni). Cutlery, for example,
is made of this material. Since nickel has a much lower tendency to form chlorides than
iron, Ni-Cr alloys provide even better corrosion resistance. However, this is achieved at
considerably higher costs (cf. Fig. 1.3).
A very important aspect in the selection of corrosion resistant materials is to avoid
microstructural inhomogeneities. They always reduce the corrosion resistance locally.
This is caused by the formation of so-called local cells, whereby the reduction reaction
takes place spatially separated on the surface of the noble microstructural components.
If the area ratio between noble and less noble microstructural components is large, very
high local corrosion rates result (analogy: failure of noble coatings). For this reason,
precipitation strengthened alloys generally have worse corrosion resistance than solid
solution strengthened alloys. Solid solution strengthened Al-Mg alloys, for example, are
signicantly more corrosion resistant than precipitation strengthened Al-Cu-Mg alloys.
Another important example of the occurrence of local cells is the intercrystalline corrosion of stainless 18-8 Cr-Ni steels under certain conditions (Fig. 5.9). If the carbon
content is well above 0.03% and the material is heated to temperatures between 600 C
and 800 C, as is the case for example during welding, chromium-containing carbides
(Cr23 C6 ) are precipitated under these conditions (Fig. 5.10). As a result, the regions
around grain boundaries become depleted in chromium. If the chromium content falls
°
°
130
5 Oxidation and corrosion
below approx. 12%, the passivity is lost locally and the steel is attacked locally. The conditions under which this phenomenon occurs can be read from the so-called sensitization
diagram (Fig. 5.11).
(a) Initial stage.
(b) Final stage.
Figure 5.9: Grain decay in a 18-8 Cr Ni-Steel.
Figure 5.10: Schematic representation of the microstructural causes of grain decay.
Shown in Fig. 5.11 is the range, dependent on time, temperature and carbon content,
in which the material is sensitized, i.e. susceptible to intergranular corrosion. Fig. 5.11
131
5 Oxidation and corrosion
is similar to the time-temperature diagram discussed in section 3.3.5. This is not
surprising, as it is the formation of (Cr23 C6 ) particles at the grain boundaries that is
responsible for the sensitization. However, the upper branch of the curves does not run
horizontally, as is the case with the TTT-diagrams. This is due to the fact that Cr
diuses from the grain interior to the depleted grain boundary areas at suciently high
temperatures so that the Cr content increases again over time.
Figure 5.11: Sensitization diagram (18-8 Cr-Ni steel).
Accordingly, the sensitization is lost again if the annealing time is sucient, as can
be seen from the falling upper curve branch of the curves. The section of the associated
phase diagram inserted in Fig. 5.11 shows the relevant phase elds (γ and γ + Cr23 C6 ).
Measures to counteract the sensitization are:
ˆ Reduction of the carbon content below 0.03%
ˆ Adding elements having a higher anity for carbon than chromium. As a result,
special carbides of these elements are formed and the chromium remains in solution.
Suitable carbide formers are titanium and niobium.
Corrosion protection by design
It has already become clear that pairs of dierent metals under corrosion attack are
always unfavourable, since the more noble metal provides additional surface for the
reduction reaction. Particularly unfavourable are situations in which the area of the
more noble material is large compared to the area of the less noble material. So it would
be a bad idea to x a copper rain gutter with a steel nail. Also the connection of steel
pipes with copper pipes can lead to contact corrosion, also called galvanic corrosion.
However, this is not a problem with closed water circuits (central heating), as no oxygen
is supplied.
If pairings of dierent metals are unavoidable, the following aspects should be considered under corrosive conditions:
132
5 Oxidation and corrosion
ˆ Combine metals which are as close as possible with respect to their standard electrode potentials;
ˆ The area of the more noble metal should be small compared to the area of the less
noble metal (area rule);
ˆ If the above points are not fullled, the contact points should be isolated.
In the case of corrosion in oxygen-containing water, areas with dicult access to
oxygen (e.g. at crevices caused by the design) are also particularly at risk. This is
particularly true for corrosion-resistant materials (e.g. stainless steels) which would be
protected by a stable surface layer under normal conditions. Because of the impeded
oxygen access, the crevice initially depletes oxygen. As a result, only metal dissolution
takes place in the crevice, but no oxygen is available to maintain the protective oxide
layer (Fig. 5.12).
Figure 5.12: Corrosion and degradation of the protective layer in the crevice due to lack of
oxygen.
Therefore, the protective layer increasingly dissolves, resulting in the formation of an
active local element (metal oxidation in the crevice; oxygen reduction at contact surfaces
outside the crevice). This results in another important rule:
ˆ Crevices due to the design should be avoided in contact with corrosive media.
Examples of good and bad design solutions are given in the following gures. Figures
5.13 to 5.15 give examples of how to avoid crevices and contact corrosion in welded,
riveted and bolted connections.
5.2.4 Degradation of polymers
Strictly speaking, corrosion is the electrochemical attack of a material. Since electron
transport in the solid and electrolyte is a necessary part of the corrosion reaction, polymers are immune against electrochemical attack as they are electrical insulators. However, polymers are not resistant to purely chemical attack by aggressive substances. This
will be discussed in the following. An essential feature of polymers, making these materials sensitive to chemical attack, is their relatively open structure (not most densely
133
5 Oxidation and corrosion
Figure 5.13: Comparison of welded designs regarding crevice corrosion.
Figure 5.14: Comparison of riveted joints regarding crevice corrosion (equal materials).
Figure 5.15: Comparison of bolted connections with regard to contact and crevice corrosion.
The situation shown on the left is unfavourable even though insulating material is
used, because its application leads to crevices. This is avoided on the right hand
side.
134
5 Oxidation and corrosion
packed). This allows molecules (water, alcohol etc.) to penetrate into polymers and lead
to chemical change. Due to dierent packing densities, amorphous thermoplastics are
generally more susceptible than partially crystalline thermoplastics.
Damage is caused by the reaction of the penetrated substance with the bonds of the
polymer. Since the carbon-carbon bond is very stable, it is not attacked. In thermoplastics, the points of attack are rather the dipole bonds. The rule of thumb is that
substances which are polar themselves (alcohols, acetone etc.) interact with these dipole
bonds and, thus, lead to chemical attack. Therefore, polymers are generally sensitive to
polar substances (Fig. 5.16), but more resistant to non-polar substances (e.g. fats).
Figure 5.16: Breaking of dipol bonds in PMMA by methanol.
Chemical decomposition can also be caused by UV radiation or prolonged exposure
to heat. One example is the decomposition of PVC. Thereby, C-H and C-Cl bonds are
broken. The released radicals react to form HCl, which is released.
135
5 Oxidation and corrosion
Acknowledgement
This script was translated from its German version with the help of DeepL Translate,
the translator in Microsoft Word, and the dictionary LEO.
136
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