Materials and biomaterials engineering
Albert Marin – Universitat de Barcelona
2014 / 2015
CONTENTS
I
Materials engineering
1
1 Introduction
1.1 Types of materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Discoveries and evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Crystals and amorphous structures
2.1 Quantum physics review . . . . . . . .
2.1.1 Main points . . . . . . . . . . .
2.1.2 Bonding . . . . . . . . . . . . .
2.2 Lattices . . . . . . . . . . . . . . . . .
2.2.1 Main crystallographic concepts
2.3 Crystalline structures . . . . . . . . .
2.4 Structure geometry . . . . . . . . . . .
2.4.1 Directions . . . . . . . . . . . .
2.4.2 Planes . . . . . . . . . . . . . .
2.4.3 X-Ray diffraction . . . . . . . .
2.5 Polimorphism . . . . . . . . . . . . . .
2.6 Amorphous materials . . . . . . . . . .
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5
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11
11
3 Crystalline solidification
3.1 Introduction . . . . . . . . . . . . . . . . . . . . .
3.2 Solidifation of metals . . . . . . . . . . . . . . . .
3.2.1 Homogeneous nucleation . . . . . . . . . .
3.2.2 Heterogeneous nucleation . . . . . . . . .
3.2.3 Grain formation . . . . . . . . . . . . . .
3.3 Solid crystals . . . . . . . . . . . . . . . . . . . .
3.4 Crystalline imperfections . . . . . . . . . . . . . .
3.4.1 Identification of crystalline imperfections
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13
13
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14
14
14
14
15
4 Diffusion in Solids
4.1 Introduction . . . . . . . . . . .
4.2 Atomic diffusion in solids . . .
4.2.1 Self-diffusion . . . . . .
4.2.2 Interdiffusion . . . . . .
4.3 Quantification of rate diffusion
4.4 Temperature influence . . . . .
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16
16
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17
17
5 Mechanics
5.1 Introduction . . . . . . . . . . . . .
5.2 Stress and strain in metals . . . . .
5.3 Deformation . . . . . . . . . . . . .
5.3.1 Dislocations . . . . . . . . .
5.3.2 Strategies for Strengthening
5.3.3 Heat treatment . . . . . . .
5.4 Hardness . . . . . . . . . . . . . .
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19
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23
24
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Biomedical Engineering
5.5
5.6
5.7
CONTENTS
Fracture of metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Fatigue of metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Rupture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
25
26
6 Phase Diagrams
6.1 Phase diagrams . . . . . . . . . . . . . . . .
6.2 Gribbs phase rule . . . . . . . . . . . . . . .
6.3 Cooling curves . . . . . . . . . . . . . . . .
6.4 Alloy systems . . . . . . . . . . . . . . . . .
6.4.1 Determination of phases(s) present .
6.4.2 Determination of phase compositions
6.5 Ternary phase diagrams . . . . . . . . . . .
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27
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7 Engineering alloys
7.1 Introduction . . . . . . . . . . . .
7.1.1 Steels . . . . . . . . . . .
7.1.2 Cast irons . . . . . . . . .
7.2 Fe production . . . . . . . . . . .
7.2.1 Fe-C phase diagram . . .
7.2.2 Stainless steel . . . . . . .
7.2.3 Martensitic stainless steel
7.2.4 Austentitic region . . . .
7.3 Biomedical alloys . . . . . . . . .
7.3.1 Mg alloys . . . . . . . . .
7.3.2 Ti alloys . . . . . . . . . .
7.3.3 Ni alloys . . . . . . . . . .
7.3.4 Intermetallic alloys . . . .
7.3.5 Biomedical applications .
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31
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34
8 Polymeric materials
8.1 Introduction . . . . . . .
8.2 Polymerization reactions
8.3 Processing . . . . . . . .
8.4 Mechanical properties .
8.5 Deformation . . . . . . .
8.6 Polymer types . . . . . .
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35
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. . . . . .
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of carbon
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9 Ceramics and composites
9.1 Introduction . . . . . . . .
9.2 Structures . . . . . . . . .
9.2.1 Oxide structures .
9.2.2 Silicate ceramics .
9.2.3 Polymorphic forms
9.2.4 Point defects . . .
9.3 Properties . . . . . . . . .
9.4 Intrachapter summary . .
9.5 Applications . . . . . . . .
9.5.1 Applications . . .
9.5.2 Fabrication . . . .
9.6 Composite materials . . .
10 Yuru
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45
Part I
Materials engineering
1
CHAPTER
ONE
INTRODUCTION TO MATERIALS SCIENCE AND
ENGINEERING
As all the disciplines in the world, we must the piece, and last but not least, the processfirst define the contents of the course:
ing we make to the device will give a particular shape or a specific finishing coat, which will
• Materials science: is primarily concerned be clearly seen if we are to modify the surface
with the search for basic knowledge about of the object. So the properties depend on the
the internal structure, properties and pro- structure (ex. hardness vs structure of steel) and
cessing of materials.
processing can change structure (ex. structure
• Materials engineering: is mainly con- vs cooling rate of steel)
cerned with the use of fundamental and applied knowledge of materials so that the
materials can be converted into products
needed or desired by society.
• Materials Science and Engineering is,
then, a discipline that combine both materials science and materials engineering in
order to develop new applications and/or
new materials.
Figure 1.1: Different processed steels
So as seen above, one can clearly see that the
conjunction of both disciplines is what creates
this subject. A clear example for this application is a hip replacement: firstly, one needs 1.1
Types of materials
to know the structure of the broken material
in order to try to find another one with simi- There are 4 main types of materials with their
lar characteristics. Once it is found, the inven- main characteristics below:
tion of a device with which we can assume the
• Metals:
same function is required – with specific addons such as lubricants, etc. In this example, the
– Strong, ductile.
requirements would be a material with a high
– High thermal & electrical conductivmechanical strength (as it is assuming many cyity.
cles), good lubricity and, of course, biocompati– Opaque, reflective.
ble. The key problems to overcome would be a
need of a fixation agent do hold the acetabular
• Polymers/plastics: with covalent bonding
cup, a lubrication material, the fixing agent to
→ sharing of e−
the femoral stem and it must avoid any debris
– Soft, ductile, low strenght and density.
in the cup.
The main procedure to choose a material goes
– Thermal & electrical insulators.
in 4 steps: firstly, one has to think about the
– Optically translucent or transparent.
performance that the material has to do. Sec• Ceramics: with ionic bonding (refractory).
ondly, as we have chosen the function of the implant, these abilities will be given by the propThey are compounds of metallic + nonerties of the chosen material. On the same way,
metallic elements (oxides, carbides, nitrides, sulphides...)
these properties are given by the structure of
3
Biomedical Engineering
CHAPTER 1. INTRODUCTION
– Brittle, glassy, elastic.
– Non-conducting (insulators).
• Composites: materials which are not made
by single and simple atom unions but for
many other things. The easiest example to
picture it: wood.
The main properties of these materials can be
plotted so as to be easier to remember them:
Figure 1.5: Resistances to fracture
Figure 1.2: Densities
Figure 1.6: Conductivity
would cover up to more than a chapter. Finally,
the optical features will be studied slightly in
further chapters as well as deteriorative effects
of the use of the material.
1.2
Figure 1.3: Stiffness modulus
Recent discoveries and
material evolution
A number of exciting initiatives in materials science have been undertaken that will potentially
revolutionize the future of the field. Smart
materials and devices at micrometric size
scale and nanomaterials are two classes of materials that will critically affect all major industries. The first ones are materials that have the
ability to sense external environmental stimuli
(temperature, stress, light, humidity, and electric and magnetic fields) and respond to them by
changing their properties (mechanical, electrical or appearance), structure or functions. The
nanomaterials are generally defined as those materials that have a characteristic length scale
smaller than 100 nm. At the nanoscale, the
properties of the material are neither that of
the molecular or atomic level nor that of the
bulk material: new properties can be achieved
by tuning the size of a structure.
Figure 1.4: Strengths
As an example of some other properties: the
electrical resistivity of copper increases if we add
impurity to a piece of pure copper or if we deform it. On the other hand, thermal conductivity decreases when zinc is added. Some materials
can have magnetic storage and magnetic permeability but it is a very complex process which
4
CHAPTER
TWO
CRYSTALLINE AND AMORPHOUS STRUCTURES IN
MATERIALS
2.1
Quantum physics review
to have discrete energy states and to occupy the
lowest available energy state – they prefer to
have the last shell full rather than to be ran2.1.1 Main points
domly distributed. Most of the elements are in
As a quite fast review of basic physics, the pe- an unstable electron configuration because the
riodic table is a chart of atoms sorted by elec- outer shell (valence) is not usually completely
tronegativity, χ, which ranges from 0.7 (Fr) to filled.
4.0 (F). Depending on their value, an atom will
tend to lose an electron or to gain it. There are
inert gases which don’t gain or give any elec- 2.1.2 Bonding
tron. Atoms are made of electrons, protons and
There are a few types of bonding:
• Ionic bond: usually occurs between a metal
atom (which donates the electrons) and
a non-metal atom (which accepts the electrons). This kind of binding only happens
between positive and negative ions, requires
an electron transfer and a large difference in
electronegativity. There is a balance of at-
Figure 2.2: MgO bonding
Figure 2.1: Periodic table
tractive and repulsive terms of the relation
which makes it to be stable when it reaches
the minimum energy state. It is the predominant bonding in ceramics.
neutrons with difference both in charges and in
mass1 . The characterization of the number of
protons in the nucleus (or electrons of neutral
species) is determined by the atomic number, Z.
The atomic mass unit, A, is the twelfth of the
mass of 12 C and the relation with the grams is
made through the Avogadro’s Number NA .
Depending on the valence of the electrons, the
properties of the atoms relative to chemistry,
electricity, thermal and optical processes vary.
It could be useful to study this fact taking into
account the quantic numbers (n, l, m and spin)
which define electron energy states. Atoms tend
• Covalent bond: they must have similar electronegativity so they share electrons. The
bonds are determined by the valence (s & p
orbitals).
• Metallic bond: this two types of primary
bonding are not always 100% of one type
in the case of metallic bonding. One has
to think the electrons as a cloud of probability in which the bond is de-localized. So
if we want to determine the % of the ionic
1 e−
−31 kg p+
mass = 9.11 × 10
mass = nmass = 1.67 ×
10−27 kg
5
Biomedical Engineering
CHAPTER 2. CRYSTALS AND AMORPHOUS STRUCTURES
character of the bond we must apply the
formula:
−(χA −χB )2
4
%= 1−e
× (100%) (2.1)
where χi is the Pauling electronegativity.
The secondary bondings arise from interaction between dipoles, which may be fluctuating (i.e. when there are two asymmetric electron
clouds in molecules) or permanent (such as HCl,
polymers . . . ).
The difference of this bondings create different
properties. For example, the length (r) and the
bond energy (E) can be modified if we increase
the melting temperature (Tm ), for instance, or
depending on the coefficient of thermal expansion (α) it will be more or less modified by it.
To sum it all up and to introduce the next
section, there is a scheme about bondings and
materials just below the text.
Figure 2.4: Cubic symmetries
Figure 2.5: Linear symmetry
of atoms. The points could go on and on until
the infinity (minus or plus) when we consider it
Figure 2.3: Summary of bonding/materials
as a never-ending distribution. Assuming that
the distance between points is a and the atomic
radius, Ra ranges between 1 ∼ 5 Å2 we could
that the magnitude order of these rela2.2 Thspe ace lattice, unit estimate
tions (interatomic interactions) goes to a mean
cell and Bravais lattices of 1 ∼ 5µm.
If we move on to the 2D dimension, we would
When studying materials, it is very useful how get the same result but with three different conelectrons are distributed but if there is any sym- figurations as the angles between adjacent sides
metry pattern in it. For instance, in a cubic can be equal or different to 90◦ so we could get
structure we can find many symmetries depend- three types of motifs.
ing on if we search for plane or point symmetry. This leads us to define punctual groups
and spacial groups. Its symmetry and electronic location influences the external appearance, which can be classified into six different
classes (cubic, tetragonal, orthorhombic, hexagonal/trigonal, monoclinic and triclinic).
The space lattice, or crystalline lattice, is
a periodic repetition of a motif in a threedimensional space. The simplest motive is a linFigure 2.6: 2D symmetries
ear net: a one-dimensional succession of equidis2 1 Å = 10−10 m
tant points, which we can consider as a group
6
Albert Marin
2.3. CRYSTALLINE STRUCTURES
Finally, when we talk about a threedimensional distribution, it is just a projection
of 2D motifs. However, there are a few rules that
apply for everything:
• Homogeneity: all the nodes are identical.
• Anisotropy: the properties of the net
changes with the direction of consideration.
• Symmetry: it exists.
Figure 2.7: Potential energy
• Similarity: in each motif there is the same
bond length and energy diminish. Dense, ornumber of atoms chemically bonded.
dered packed structures tend to have lower energies. When materials are packed periodically
2.2.1 Main crystallographic con- they are called crystalline materials whereas
cepts
amorphous or non crystalline is used when
To be able to classify the crystals, there are a they are not. The simplest crystalline material is
SiO2 (Quartz) which is transparent3 and has its
few parameters that one has to keep in mind:
amorphous homologue, the f used sillica4 , not
1. Unit cell: structure that it is repeated in a transparent at all with “air spaces” in between
crystalline network filling all the space. The its atoms. The atom packing factor, AP F , can
vertex of the unit cell might not coincide be determined as follows:
with atoms in these positions.
atoms
× Vatoms in unit cell
(2.2)
AP F = unit cell
Vunit cell
2. Primitive cell: the smallest unit cell. For
example, a cube’s unit cell is the cube itself In addition, another useful quantitative descripand the primitive cell an atom (linear net tion of the material is the coordination number,
projected in 3D). Sometimes, the primitive which is defined as the number of nearest atom
cell can be the same as the unit cell.
neighbours that any atom chosen at random can
have. There is no formula for this number, just
3. Bravais lattice: (or crystal lattice) is the
a little of visuospatial imagination and schemes.
classification of an infinite array of discrete
There are four types of structures (3 cubic +
points generated by a set of discrete trans1 hexagonal) that are of particular interest to
lation operations. There are 14 Bravais latstudy.
tices and any 3D periodic structure textbfmust be one of them although not all the
atoms fall in each vertex. The classification
is given by the formula: R = n1 a+n2 b+n3 c,
where ni ∈ Z and a,b,c are the lattice constant or lengths of the sides of the unit cell.
4. Crystal: a crystal is the sum of the net
(cells and those concepts) and a base. The
base is the set of atoms inside the cell, which
repeat in the crystal.
5. Miller index: form a notation system in
crystallography for planes in crystal lattices.
2.3
Figure 2.8: Cubic structures
Crystalline structures
It is useful to remember the electron potential
1. Simple Cubic Structure (SC). It is a
energy as a function of the radius, Pe (R). We
very rare structure due to the low packcan find the radius, R0 , in which Pe (R0 ) =
ing density. The close-packed directions are
min(Pe (R)). Taking this parameter into accube edges. The coordination number is 6,
count, we can consider the packing grade and
3 Typical of metals, many ceramics and some polymers
4 Occurs in complex structures and due to rapid coolthe energy levels: when packed, there is more
room for atoms so both the typical neighbour ing
7
Biomedical Engineering
CHAPTER 2. CRYSTALS AND AMORPHOUS STRUCTURES
which makes the AP F ≈ 0.68 approximately. Notice how larger is this one with
respect to the previous packing. If the central atom was of a different specie than the
rest, it would have a different radius to be
considered as well as a different mass (in
case we wanted to find the density of the
material) that should be considered in further calculus.
Figure 2.9: SC coordination number
3. Face Centred Cubic Structure (FCC).
Starting with the SC, we put an atom in
each face centre so atoms touch each other
along the face diagonals. As the previous
one, different colours are just for ease of
viewing. The coordination factor of this
one is a bit tricky but the right number is
12. The ratio is the atom of the SC plus 6
halves of atom, so the total is 4 atoms per
unit cell. The relation of the radius with
the side of the unit cell is found with a simple Pythagorean relation as shown in the
bottom figure 2.11.
√
√
We can see that 4R = 2a =⇒ R = 42 a.
Again, we are set to find the APF:
as we can see the scheme in the figure 2.9.
If we look at the ratio of atoms per unit
cell, we can clearly see that if a side of the
cube is a, then the radius is R = 0.5a and
it contains 8 times 1/8 of atom, so 1 atom
per unit cell. The APF can be calculated
as follows:
1 43 π( a2 )3
π
=
3
3
a
a
6
(2.3)
which makes the AP F ≈ 0.52 approximately.
AP F =
atoms
unit cell
× 43 πR3
=
2. Body Centred Cubic Structure
(BCC). Similar as the previous one but
with an atom in the center so atoms touch
each other along the cube diagonals. In the
figure 2.8 we can see that there are different
colours but all atoms are identical; the
centre atom is shaded differently only for
ease of viewing. The coordination number
is 8 and the ratio of atoms per unit cell is
easily calculated: if a SC had 1 atom and
we only added a whole atom at the center
of the unit cell, then the atoms per unit
cell ratio is 2. The radius it requires to use
trigonometrical relations to find a relation
between a & R, seen in the diagrams
2.10.
√
√
3
So then: 4R = 3a =⇒ R = 4 a. Now,
we are ready to discover the APF of this
structure:
√
√
3 3
2×4
2 × 43 πR3
3
3 π( 4 a)
AP F =
=
=
π
3
3
a
a
8
(2.4)
4 × 43 πR3
AP F =
=
a3
√
2 3
42
3 π( 4 a)
3
a
√
2
π
6
(2.5)
which makes the AP F ≈ 0.74 approximately. This is the maximum achieable
APF.
=
Figure 2.11: FCC relations and HCP
4. Hexagonal Close-Packed Structure
(HCP). It is mostly found in minerals and
it has a coordination number of 12, 6 atoms
per unit cell and its APF is also approximately 0.74. It can also be found in Cd,
Mg, Ti and Zn.
Another useful feature is the theoretical density,
ρ. It is obtained applying the formula:
M ass of atoms in unit cell
=
T otal volume of unit cell
nA
=
V c NA
(2.6)
Density = ρ =
Figure 2.10: Relation in BCC
8
Albert Marin
2.4. STRUCTURE GEOMETRY
3. Adjust to smallest integer values. Negative
where n is the number of atoms per unit cell,
numbers are symbolised with a bar upon
A the atomic weight, VC the volume of unit
the number: 1̄ = −1.
cell (a3 for cubic environment) and NA the
Avogadro’s number. To make an example, we
can take the chrome (Cr), which has an A =
4. Enclose in square brackets without commas:
52.00 g/mol, R = 0.125 nm and is BCC. With
[uvw].
that, if we remember the relation of a and R we
found just above the equation (2.4) we can find For example: if we take a cubic unit cell with
an atom in the position (x, y, z) = (1, 0, 21 ),
a and find the density5 :
the vector which goes from the origin to this
√
4
atom is the same as the coordinates. Thus,
4R = 3a =⇒ a = √ R
3
a = 1, b = 0 and c = 12 , readjusted to smallest
nA
3
integers and enclosed in square brackets will give
ρ= ≈ 7.18 g/cm
3
us the direction [201]. If, due to symmetry, some
√4 R
N
A
3
directions are equivalent to some others, we will
It doesn’t seem wrong, as water density is ρ = put them between brackets huvwi. For example,
1 g/cm3 and knowing that if we leave a chrome if we consider the direction [010], we will see that
7
bar on a bucket with water, the bar will sink. . . is equivalent to [100], [001], [1̄00], [01̄0]. . . So we
6
can
simplify
the notation by saying h010i. Once
it seems consistent . In general, as we can
we
have
this
defined, we can compute the linsee in the figure 9.2 from the previous chapear
density
of atoms, LD which is just the
ter, ρmetals > ρceramics > ρpolymers and this
number
of
atoms
in a direction divided by the
is so because metals have close-packing (metalunit
lenght
of
its
oriented
vector. For example,
lic bonding) and often large atomic masses, cethe
linear
density
of
Al
in
[110], knowing that
ramics lower dense packing and made of lighter
a
=
0.405
nm
equals
3.5
atoms
per nm. A parelements, polymers low packing density (often
amorphous) and only made of (C, H, O) and
composites have intermediate values. The variation of densities is of less than 2 magnitude
orders.
2.4
Miller indices and crystallographic planes
The Miller indices are the ones that define the
shape of the unit cell, using directions and
planes.
2.4.1
Directions
The way of defining directions is by putting axis
somehow and the easiest way to define them in a
crystal is using a cubic distribution: put a vertex as the origin’s coordinates (0,0,0) and define the three axis along the edges. This way
we have Cartesian axis as crystallographic axis.
However, when the angles between these lines
are not 90◦ , we will have a different system. To
calculate the crystallographic directions: there
is a simple algorithm:
Figure 2.12: Hexagonal coordinates.
ticular case is the hexagonal symmetry in which
we need to define an extra direction. If we look
into a hexagonal distribution, the atoms won’t
follow Cartesian coordinates. So although the
maximum number of linearly independent vectors in 3D are 3, we need to add an extra vector and redefine the planar angles between the
non-z axis so as to make angles range from 90◦
to 120◦ . Thus, we can have equivalent vectors:
[0010] = [112̄0]. However, if we want to transform from Cartesian to the 4 parameter MillerBravais lattice coordinates, it can be done with
1. Vector from the origin to one atom.
2. Read off projection in terms of unit cell dimensions a, b and c.
5 The
ρ =
values one should plug into the equation are:
2×52 g/mol
in order
3
23
( √4 ×0.125×10−9
3
m) ×6.023×10
7 Equivalent
means that is a vector of modulo 1 in
only one axis. [011]
√ would not be equivalent because its
modulus equals 2 and it is the bisector vector between
planes y and z.
atom/mol
to cancel out everything except for g/m3 .
6 Real density: ρ
3
Cr = 7.19 g/cm .
9
Biomedical Engineering
CHAPTER 2. CRYSTALS AND AMORPHOUS STRUCTURES
the function:
[u0 v 0 w0 ] → [uvtw]
1
u = (2u0 − v 0 )
3
1
v = (2v 0 − u0 )
3
t = −(u + v)
(2.7)
w = w0
2.4.2
Planes
To determine a plane, they have to contain at
least an atom. They are defined by the normal
vector to them: (100) is the plane yz,(271̄)...
The easiest planes to compute are the ones corresponding to the faces of a cube. The algorithm
to find the plane to any shape:
Figure 2.14: Plane example (2): hexagonal
1. Read off the intercepts of the plane with 1 0 − 1 1. So then, as we cannot reduce the
axes in terms of a, b and c. No intercept terms, the Miller indices = (101̄1).
has to be considered as ∞.
Similarly to the linear atom density, we can
compute the planar density of atoms, PD
2. Take the reciprocals of the intercepts.
as:
3. Reduce them to the smallest integer values.
atoms per 2D repeat unit
PD =
(2.8)
area per 2D repeat unit
4. Enclose in parentheses without commas:
(hkl)
It gives a slightly different information if we
Like directions, the bar upon a number indicates measure different planes of the same structure,
a negative number and there exist also plane generally the higher Miller number, the lower
families, which are denoted with the numbers the planar density and the closer the planes are
between curly brackets: {100} defines all the between them. If we compute the planar den◦
planes with only one component, {037} all the sity of (100) and (111) iron below 912 , which
has
a
BCC
structure,
we
will
see
that
the
PD of
planes with that components, etc. For example:
the (100) plane gives 1.2 × 1019 atoms/m2 while
the (111) results in 0.92 × 1019 atoms/m2 . If we
compute the distance between
planes in (100) is
√
a 2
just a, while in (111) is 2 .
2.4.3
X-Ray diffraction
Another interesting feature of planes is the one
related to x-rays. As a reminder: one should
remember that the inverse of the energy is the
wavelength. Speaking of the rays we study, we
can consider that x-rays mean λ = 1 Å, which
is important if we remember that the atoms are
usually separated by this distance. It is because
of the wavelength that x-ray cannot resolve spacFigure 2.13: Plane example (1): cubic
ings lower than λ. The interaction of them with
the layers of the material (planes) can be simThe figure intercepts in (a, b, c) = (1, 1, ∞) =⇒ ply modelled as a reflection of a wave against a
1
reciprocals: 11 11 ∞
=⇒ 1 1 0. So then, as we surface. Measuring the angle with which the ray
cannot reduce the terms, the Miller indices = goes to the Miller plane, we can determine the
(110).
interplanar space, d:
In hexagonal unit cells, the same idea is
used: The figure intercepts in (a1 , a2 , a3 , z) =
nλ
a
=
(2.9)
d= √
1
(1, ∞, −1, 1) =⇒ reciprocals: 1 ∞
− 1 1 =⇒
2 sin θc
h2 + k 2 + l2
10
Albert Marin
2.6. AMORPHOUS MATERIALS
(a) X-ray diffraction
(b) α-iron pattern
Figure 2.15: X-rays effects
where n = 1 (unless another number is said),
λ is the wavelength (1 Å) and θc is the critical
angle.
This is the simple way to define d, in a beautiful cubic distribution. However, the general
formula to compute the plane distance in any
shape is given by the triclinic formula, which we
will rarely use. In addition, when we measure
the diffraction we can plot a continuous graphic
to search for a pattern (as seen in the figure
15.b) and, computing some measurements, define Miller planes of the material to be able to
identify its structure. When plotting the results,
peak points will be extremely visible (in terms
of the number in the intensity scale axis) and
that will mean that we have found a plane.
2.5
ferent configuration. When this happens, it is
called polymorphisms. The clearest example is
C: it can be graphite, carbon or diamond. Another example is the iron system, it is α−F e below 912◦ with BCC configuration, γ − F e when
temperature ranges from 912 to 1394◦ with FCC
structure and δ − F e when temperature ranges
from 1394 to 1538 ◦ with a different BCC configuration. After that temperature it becomes liquid. As we can imagine, depending on the temperature, the iron will have different planes and
thus their x-ray diffraction patterns will vary
too.
2.6
Amorphous materials
As previously said, an amorphous material
is such that does not have a crystal structure.
However, we could have a phase with different
small crystalline units. When this occurs, it is
called a polycrystal and it has both good and
Polimorphism
Sometimes the same material can have more
than one phase, and each one can have a dif11
Biomedical Engineering
CHAPTER 2. CRYSTALS AND AMORPHOUS STRUCTURES
bad properties at a mechanical level because we
could have some planes with low strength but
also others other ones very closely bonded. That
would lead us to make a good analysis of fractures, for example. Most of engineering materials are polycrystals. In single crystals the properties (such as the modulus of elasticity, E, in
BCC iron) vary with the direction we consider
(anisotropic) whereas in polycrystals they might
or might not vary depending on if the grains
are randomly oriented (isotropic) or if they are
textured (anisotropic). Amorphous or non crystalline materials are materials that lack on longrange order in their atomic structure. They can
have different degrees of order but nevertheless,
they tends to achieve a crystalline state because
that is the most stable one and corresponds to
the lowest energy configuration.
12
CHAPTER
THREE
SOLIDIFICATION AND CRYSTALLINE IMPERFECTIONS
3.1
Introduction
When molten alloys are cast, solidification starts
at its edges as it is being cooled. This process
takes place not at a specific temperature but
over a range of temperatures and the perfection
of the crystals will be better if we cool it slower
rather than faster and cooler.
3.2
Solidifation of metals
Below this value the material can either be in
liquid or non-crystal solid state. Two kinds of
energy are to be considered: the volume free
energy released by the liquid-to-solid transformation and the surface energy required to form
the new solid surfaces of the solidified particles. When a pure liquid metal is cooled below
its equilibrium freezing temperature, the driving energy for the liquid-to-solid transformation
is the difference in the volume free energy ∆Gv
of the liquid and that of the solid. The upper
In general, the solidification can be divided into
two steps:
1. Formation of stable nuclei in the melt (nucleation).
2. Growth of the nuclei into crystals and the
formation of a grain structure.
The shapes that each grain acquires depends on
many factors, of which thermal gradients are important: if we keep the same temperature in the
whole process, we will get a more homogeneous
material. The two main mechanisms by which
Figure 3.2: Energy vs. radius
Figure 3.1: Crystal solidification
the nucleation of solid particles in liquid metal
occurs are homogeneous and heterogeneous nucleation.
3.2.1
Homogeneous nucleation
It is the simplest case: the liquid metal itself provides the atoms needed to form a nuclei. This
can be modelled thermodynamically: there is a
minimum radius by which the nuclei are stable.
13
line is the surface free energy that makes the
system to melt down whereas the lower line is
the volume free energy that makes the union of
the system. There is the critical radius, r∗ , that
is the point which determines the minimum size
that a solid particle has to get to prevent redissolving or making imperfections. To find out the
value of that r we must derivate the free energy
with respect to the radius:
d(∆GT )
d 4 3
=
πr ∆Gv + 4πr2 γ
dr
dr 3
12 ∗2
max(r) ≡
πr ∆Gv + 8πr∗ γ = 0
3
2γ
2γTm
∴ r∗ = −
→ r∗ =
∆Gv
∆Hf ∆T
Biomedical Engineering
CHAPTER 3. CRYSTALLINE SOLIDIFICATION
The greater degree of undercooling ∆T below the equilibrium melting temperature, the
greater change in ∆Gv . However, the surface
change will not be significant = thus the critical
nucleus size is determined mainly by ∆Gv .
be well organized although there will be differences between them. When we have a substrate,
the directions will be lead by it. To build crystals in a determined direction, it has to be cooled
very slowly, with a temperature very near to the
fusion one. In addition when solidifying a material we can obtain equiaxed grains (they follow
an axis) or columnar ones (appiled lines of material).
3.3
Solidification of single
crystals (monocrystals)
In growing single crystals solidification must
take place around a single nucleus so that no
other crystals are nucleated and grow: the interface temperature between the solid and liquid
must be slightly lower than the melting point
of the solid, and the liquid temperature must
Figure 3.3: Stability
increase beyond the interface. To achieve this
temperature gradient, the latent heat of solidification must be conducted through the solidify3.2.2 Heterogeneous nucleation
ing solid crystal. The growth rate of the crysIf we introduce a seed (or impurity), which is a tal must be slow so that the temperature at the
solidification external nucleus, the material will liquid-solid interface is slightly below the meltstart the process in there. It can also be created ing point of the solidifying solid.
In electronics, silicium is used with very pure
in a solid surface (substrate) so it is created a
crystals
to cut them in fine slices. It is called
continuation of the initial system. This is so bethe
Czocharlski
method.
cause they lower the critical free energy required
to form a stable nucleus. Heterogeneous nucle-
3.4
Crystalline
tions
imperfec-
It can be random imperfection or organized.
The classification of imperfections is:
Figure 3.4: Heterogeneous nucleation
1. Point defects
ation takes place on the nucleating agent because the surface energy to form a stable nucleus
is lower on this material than in the pure liquid
itself. So, since the surface energy is lower for
heterogeneous nucleation, the total free-energy
change for the formation of a stable nucleus will
be lower, the critical size of the nucleus will be
smaller and the amount of undercooling will be
much lower too.
3.2.3
Grain formation
After stable nuclei have been created, grains
form1 . The crystallographic directions of each
of them might be different but each grain will
1 The crystals in the solidified metal are called grains,
and the surfaces between them are called grain boundaries.
14
(a) Vacancy atoms: there is a hole instead
of an atom, so there is a free bond and
the lattice is distorted (extensive local
tension).
(b) Interstitial atoms: the opposite case
of the vacancy, there is an extra atom
(compressive local tension, less space
= more compression).
(c) Substitutional atoms: the same as the
previous one but with atoms substituting the material ones (e.g. Cu in Ni,
whereas an interstitial solid would be
C in Fe). If the percentage of the impurity is high, it is not an imperfection
but an alloy.
2. Line defects: dislocations. This are the line
efects (can be a straight or a curved line). It
Albert Marin
3.4. CRYSTALLINE IMPERFECTIONS
usually produces plastic deformation (permanent) and we can see the types in the
picture.
3. Area defects: grain boundaries. Crystals
with different directions but with a point in
which the atoms coincide.
Figure 3.5: Line defects
There are criteria to determine that an atom
is substituting another one or not:
1. Atomic radius RA < 15%.
2. Electronegativity similar (proximity)
3. Metallic crystal structure similarity
4. Same ionic number (valence): preferred a
higher one.
Example 1:would one predict more Al or Ag to
dissolve in Zn? Al accomplishes the item 1 and
is better in the 2nd and 4th while Ag gets only
the 1st. Then we choose Al.
Example 2: more Zn or Al in Cu? If we do
the same, we will see that Zn is in everything
positive (except for item 3) so then we have Zn.
We can quantify that with weigh or atom percentage:
C1 =
m1
× 100%
m1 + m2
(3.1)
where we can change m (mass) for n (number of
moles).
3.4.1
Identification of crystalline
imperfections
There are three main types of identification:
1. Optical metallography. These are used to
study the features and internal make-up of
materials at the micrometer level. Qualitative and quantitative information pertaining to grain size, boundary, existence of various phases, internal damage, and some defects may be extracted using optical metallography techniques.
15
2. Scanning Electron Microscopy (SEM).The
SEM impinges a beam of electrons in a pinpointed spot on the surface of a target specimen and collects and displays the electronic
signals given off by the target material.
3. Transmission Electrom Microscopy (TEM).
Useful in studying defects and precipitates
(secondary phases) in materials. Defects
such as dislocations can be observed on the
image screen of a TEM. Specimens to be analyzed using a TEM must have a thickness
of several tens of nanometers or less
CHAPTER
FOUR
DIFFUSION IN SOLIDS
4.1
Introduction
As seen below, the scheme is very representative
of interdiffusion and self-diffusion.
Rate processes in solids: the system needs an
increment of energy to overcome an activation
energy barrier that leads the process. The additional energy require above the average energy
of the atoms is called the activation energy ∆E ∗
(J/mol or cal/mol). The probability of finding a
molecule or atom at an energy level E* greater
than the average energy E of all the molecules or
atoms of the system at a particular temperature
T is give by the Maxwell-Boltzmann distribution:
E ∗ −E
(4.1)
P ∝ e− kT
Where k is the Boltzmann constant. Defects are
created randomly because of the thermic vibration of the atoms. If we want to compute the
fraction of molecules with a greater E than the
average, we van use the previous equation and
obtain:
E∗
ndef f ects
= Ce− kT
(4.2)
NT otal
Where C is an arbitrary constant and N the
number of atoms per m3 .
Figure 4.1: Diffusion with vacancies
4.2.1
Self-diffusion
Atoms can move in crystal lattice from one
atomic site to another if there is enough acti4.2 Atomic
diffusion
in vation energy provided by the thermal vibration
of the atom and if there are vacancies or other
solids
crystal defects in the lattice for atoms to move
into. Vacancies in metals and alloys are equiThere are four main concepts of diffusion:
librium defects, and therefore there are a few
1. Diffusion: mass transport by atomic mo- always present to enable substitutional diffusion
tion.
of atoms to take place. In addition, we can as1
2. Mechanisms: in not solids is done by Brow- sume that if we increase the temperature, there
nian motion whereas in solids is done by will be a higher number of vacancies and thus a
greater diffusion rate.
vacancy diffusion or interstitial diffusion.
3. Self-diffusion: in a simple solid, atoms also
migrate. The activation energy for this effect is equal tot he sum of the activation energy to form a vacancy and the one needed
to move it.
4.2.2
Interdiffusion
It is common with the previous type of diffusion
but it is useful to define it here. The mechanism
is the vacancy diffusion although is the atom
who is being displaced in this effect. The
4. Interdiffusion: in alloy, atoms tend to mirate of the diffusion depends on the number of
grate from regions of high concentrations to
1 From the equation (4.2)
regions of low concentration.
16
Albert Marin
4.4. TEMPERATURE INFLUENCE
vacancies and the activation energy needed to
exchange or also called the frequency of jumping.
4.4
Effect of temperature on
diffusion in solids
When we have different temperature values, the
diffusion coefficient increases with absolute temOne particular case of interdiffusion is the perature, T:
interstitial diffusion: smaller atoms can diffuse
Qd
D
=
D
exp
−
(4.4)
0
between the elements of the lattice and it is
RT
faster than the vacancy diffusion.
Where D0 is a pre-exponential constant, and Qd
the activation energy in J/mol or eV/atom.
The utilities of this are:
When the diffusion is interstitial, the diffusion
1. Case hardening: diffusing C atoms into Fe vs temperature is much higher than the substitutional case, as we can see in the plot.
at the surface makes iron (steel) harder.
Interstitial diffusion
2. Doping: in n-type semiconductors, we can
deposit a P sections on a Si surface to heat it
and then get a doped semiconductor region
(it is very used in electronics). This process
would lead to a steady state diffusion
4.3
Quantification
diffusion
of
rate
Figure 4.2: Effect of temperature on types of
diffusion
This effect is computed through the flux. In
steady-state diffusion we can use Flick’s first law
which is:
dC
(4.3)
J = −D
dx
• Example 1. At 300◦ C the diffusion coefficient and activation energy for Cu in Si
are:
– D(300◦ C) = 7.8 × 10−11 m2 /s
Which means that the flux is negatively pro– Qd = 41.5kJ/mol
portional to the concentration gradient by a
What is the diffusion coefficient at 350◦ C?
constant, or diffusion coefficient D. The negative sign is to indicate that the diffusion flux
We can convert the expression (4.4) to isolate
goes from the higher concentration region to the D2 :
lower one. When the displacement is lineal, so
∆C
∼
Qd 1
as x is small, we can simplify dC
.
=
dx
∆x
lnD2 = lnD0 −
R T2
• Example:
chemical protective clothing.
Qd 1
lnD1 = lnD0 −
Methylene chloride is a common ingredient
R T1
of paint removers. Besides being an irrilnD2
Qd 1
1
tant, it also may be absorbed through skin.
lnD2 − lnD1 =
=−
−
lnD1
R T2
T1
When using this paint remover, protective
Q
1
1
d
gloves should be worn. If butyl rubber
D2 = D1 · exp −
−
R T2
T1
gloves (0.04 cm thick) are used, what is the
diffusive flux of methylene chloride through
D2 = 15.7 × 10−11 m2 /s
the glove? (D = 110 × 10−8 cm2 /s, C1 =
In non-steady state diffusion, the concentra0.44 g/cm3 , C2 = 0.02 g/cm3 )
tion of diffusing species is a function of both time
The solution will just simply plug in the values and position: C = C(x, t). So we need to use
Flick’s Second Law :
in the Flick’s first law:
∂2C
∂C
0.02 g/cm3 − 0.44 g/cm3
=D 2
(4.5)
J = 110 × 10 cm /s ·
∂t
∂x
0.04cm
If we analyse this function in a practical exam−5 g
= 1.16 × 10
cm2 s
ple, for example the diffusion of Cu into a bar of
−8
2
17
Biomedical Engineering
CHAPTER 4. DIFFUSION IN SOLIDS
• Example 3. Methylene chloride is a common ingredient of paint removers. Besides
being an irritant, it also may be absorbed
through skin. When using this paint remover, protective gloves should be worn.
If butyl rubber gloves (0.04 cm thick) are
used, what is the breakthrough time (tb ),
i.e., how long could the gloves be used before methylene chloride reaches the hand?
D = 110 × 10−8 cm2 /s
Figure 4.3: Diffusion in space and time
As x is very small, we can assume linear concentration gradient. If we keep in mind that the
Al, we can plot the concentration of Cu atoms diffusion is cm2 /s, the breakthrough time should
as a function of space and time as follows: The be:
analytical solution of the equation (4.5) is given
2
0.04cm2
by the error function2 . The values for this equa- tb = ` =
= 240s = 4min
6D
6 · 110 × 10−8 cm2 /s
tion in defined points have to be extracted from
official tables.:
So as to summarize the diffusion speeds:
x
C(x, t) − C0
√
= 1 − erf
(4.6)
• It is faster for:
Csurf ace − C0
2 Dt
1. Open crystal structures
• Example 2. An FCC iron-carbon alloy ini2. Materials with secondary bonding
tially containing 0.20 wt% C is carburized
at an elevated temperature and in an atmo3. Smaller diffusion atoms
sphere that gives a surface carbon concen4. Lower density materials
tration constant at 1.0 wt%. If after 49.5 h
the concentration of carbon is 0.35 wt% at
• It is slower for:
a position 4.0 mm below the surface, deter1. Close-packed structures
mine the temperature at which the treatment was carried out.
2. Materials with covalent bonding
Firstly, we should find the value of z of erf.
x
C(x, t) − C0
0.35 − 0.20
√
1 − erf
=
=
Csurf ace − C0
1.0 − 0.20
2 Dt
= 1 − erf (z) ∴ erf (z) = 0.8125
We search the closest values in a table and we
find out that there is not the z corresponding to
0.8125 so we have to make an linear interpolation:
erf (zn ) − erf (zn−1 )
zn − zn−1
=
zn+1 − zn−1
erf (zn+1 ) − erf (zn−1 )
z − z0.90
0.8125 − 0.7970
=
∴ z = 0.93
z0.95 − z0.90
0.8209 − 0.7970
Now we have the value, we solve for D:
x
x2
z= √
D= 2
4z t
2 Dt
−11 2
∴ D = 2.6 × 10 m /s
Now we simply isolate T from the equation (4.4)
and we obtain:
T =
Qd
= 1300K = 1027◦ C
R(lnD0 − lnD)
2 erf (z)
=
√2
π
Rz
0
2
e−y dy
18
3. Larger diffusion atoms
4. Higher density materials
CHAPTER
FIVE
MECHANICAL PROPERTIES OF METALS
5.1
Introduction
The mechanical properties of metals are those
that reveal the reaction, either elastic or plastic, of a metal to an applied stress. Tensile
and yield strength, elongation, reduction of area,
hardness, impact strength, and bend ability are
mechanical properties. The plastic deformation
in a metal is easy to schematize:
1. The crystals are ordered: initial status.
2. We apply a small load: bonds stretch and
planes shear. We have an elongation of the
metal (δelastic+plastic) which is the difference between the new length and the initial
one.
3. We take off the load: the planes are still
sheared but reorganize so as to try to get
the same structure as before. However, the
metal can have a plastic deformation (permanent, δplastic) so it will not recover its
original length.
5.2
Stress and strain in metals
We can define two types of stress:
Ft
A0 :
where Ft is the
• Tensile stress, σ =
applied perpendicular force, and A0 is the
original area before loading.
Fs
A0 :
Figure 5.1: Tension and shear stress
• Simple compression: such as any surface on
which we leave a weight.
• Bi-axial tension: such as a pressurized tank
in which the walls of the tank are being
pressed to the exterior in the direction of
the cylinder θ and z.
• Hydrostatic compression: such as the one
that a fish suffers underwater.
The strain, or deformation, has three types:
• Tensile strain, ε = Lδ0 : the elongation in
the direction of the applied force.
δL
• Lateral strain, εL = − W
: the decrease of
0
the material width due to the force. It is
the narrowing.
• Shear strain, γ =
deformation.
∆x
y
= tanθ: the angle
• Shear stress, τ =
where Fs is the component of the applied force which goes par- Strain is always dimensionless.The common
allel to the surface.
way of testing this property is in a machine that
uses a load to deform the material and its exBoth have units of N/m2 . In addition, we can
tension is what is measured. Another method
define a few states of stress:
is with the diamond anvil cell (DAC), which is
• Simple tension: such as a cable being pulled the compression of a specimen with two opposite
diamonds.
by both sides.
If we plot the stress vs strain we will see that
• Torsion (a form of shear): such as a drive there is a lineal region: this is the elastic deshaft of a ski lift.
formation and its slope it is called the elasticity
19
Biomedical Engineering
CHAPTER 5. MECHANICS
the x direction:
−2.5 × 10−30 mm
= −2.5 × 10−4
10 mm
εx
−2.5 × 10−4
εz = − = −
= 7.35 × 10−4
ν
0.34
σ = E · εz = 7.35 × 10−4 · 97 × 103 =
εx =
= 71.3 M P a
2
d0
π=
F = σA0 = σ
2
2
10 × 10−3 m
= 71.3 × 106 N/m2 ·
π
2
Figure 5.2: Strains
≈ 5600 N
The slope of stress strain plot (which is proportional to the elastic modulus, and means the
force that one should apply [tension] to a deform
a material) depends on bond strength of metal.
If the substance has atoms strongly bonded, the
slope will be very high whereas it will be lower
if the bonds are weak. Other interesting properties are the elastic shear modulus, G1 , elastic
Bulk modulus, K2 and the relations for isotropic
materials:
Figure 5.3: Strain testing
G=
modulus, E or Young’s modulus. From here, we
can apply the general Hooke’s Law :
E
2(1 + ν)
K=
E
3(1 − 2ν)
We can compare the Young’s moduli of different
species and we will see that metal alloys is in the
σ = Eε
(5.1)
range of hundreds of GPa, graphite/ceramics the
wide range of (10,1200) GPa, polymers below 6
with E units of GPa or psi. The Poisson’s GPa and composites/fibres (0.4,400) GPa.
ratio,ν, is defined as the slope of the graph of
Finally, useful linear elastic relationships are
the strain vs longitudinal strain (εL vsε):
the simple tension:
ν=−
εL
ε
(5.2)
F L0
EA0
F w0
δL = −ν
EA0
δ=
(5.3)
(5.4)
As one can imagine, ν has no dimension. The
common values for it is 0.33 in metals, 0.25 in
ceramics and 0.40 for polymers. In addition, if it and simple torsion:
is above 0.50 it means that the density is increas2M L0
α=
(5.5)
ing in the deformation whereas if it is below, it
πr04 G
is a clue of that it is decreasing (voids form).
where M is the momentum and α the angle of
twist. Material, geometric and loading parame• Example 1. A tensile stress is to be applied ters contribute all to deflection. From the equaalong the long axis of a cylindrical brass tions, the larger the elastic module is, the minirod that has a diameter of 10 mm. Deter- mum elastic deflection it has.
mine the magnitude of the load required to
produce a 2.5 × 10−3 mm change in diamPlastic deformation of
eter if the deformation is entirely elastic. 5.3
ν = 0.34, E = 97 GP a
metallic single crystal
When the force F is applied, the specimen will When speaking of plastic (permanent deformaelongate in the z direction and at the same time tion), one can plot the engineering stress vs
1 τ = Gγ. It is measured in simple torsion test
experience a reduction in diameter ∆d = 2.5 ×
2 P = −K ∆V . It is used to make preassure tests.
10−30 mm in the x direction. For the strain in
V0
20
Albert Marin
5.3. DEFORMATION
the strain and we can see the plotted result are two types of fracture: the brittle fracture
for low stress. The stress at which noticeable (elastic energy) and the ductile fracture (with
both elastic and plastic energy).
Figure 5.6: Toughness
The resilience, Ur , is the ability of a material
to store energy (similar to elasticiy). This is best
Figure 5.4: Simple tension test
stored in elastic region and can be calculated as:
Z εy
plastic deformation has occurred is called the
σdε
(5.8)
Ur =
yield strength σy and happens at a strain
0
εp = 0.002. At 300 K, the yield strength of metbut if we assume a linear curve, it can be simals range from 10 to 2000 MPa, polymers from
plified to:
10 to 100 MPa and other materials is hard to
1
(5.9)
Ur u σy εy
measure as they break before yield.
2
There is also another concept called the tenThe elastic strain recovery is the amount of
sile strength, TS, which is the maximum srtess
elongation that a deformed material can revert.
on a engineering stress-strain curve. In metals,
It is better understood by analysing the plotted
it occurs when noticeable necking3 starts and in
graph:
polymers it happens when the backbone chains
are aligned and about to break. The TS in different materials range from 100 to 2000 MPa in
metals, 10 to 1000 MPa in ceramics, 10 to 100
MPa in polymers and fibres vary much from 2
to 5000 MPa.
The ductility is the ability of a material to
deform under tensile stress and can be measured
as the plastic tensile strain at failure or considering the necking process:
Lf − L0
· 100
L0
A0 − Af
· 100
%RA =
A0
%EL =
(5.6)
(5.7)
Figure 5.7: Elastic strain recovery
5.3.1
Dislocations
When one defines the material dislocation
classes, it can be classified as:
Figure 5.5: Ductility
The toughness is the energy to break a unit
volume of material and can be approximate by
the area under the stress-strain curve. There
3 Process
by which a narrowing at the center of a specimen occurs.
21
1. Metals (Cu, Al): the easiest dislocation motion. They have non-directional bonding
and close-packed directions for slip.
2. Covalent ceramics (Si, diamond): its motion is difficult because they have directional (angular) bonding.
Biomedical Engineering
CHAPTER 5. MECHANICS
3. Ion ceramics (NaCl): motion difficult be- the relation between the applied σ and τR can
cause they need to avoid nearest neighbours be defined as:
of similar sign (reppulsion).
τR = σ cos λ cos φ
(5.10)
The dislocation motion in metals is linked to
plastic deformation and it happens by slipping: The condition for start the dislocation motion is
an edge dislocation (extra half plane of atoms as that τR > τCritical Resolved Shear Stress , which is
seen in previous chapters) suffers a shear stress typically 10−4 to 10−2 GPa. Obviously, the ease
and slides over adjacent half-planes of atoms un- of dislocation motion depends on the crystallographic orientation: it is maximum when both
til a step is left aside.
angles are 45◦ (due to the cosine).
• Example 2: deformation of a single crystal. If σ = 45 M P a, φ = 60◦ , λ =
35◦ and τCRSS = 20.7 M P a, will crystal
yield? If not, what stres is needed?
We just have to simply plug in the values in the
equation:
Figure 5.8: Dislocation motion
A particular point to keep in mind is that
if dislocations cannot move, plastic deformation
does not occur. The slip direction is the same
as the Burgers vector one and it can be origined
by an edge or screw dislocation.
The mecanism of the slip is defined by:
τ = σ cos λ cos φ = 45 cos 35◦ cos 60◦ ≈
≈ 18.4M P a
τ <τCRSS = not enough stress to yield
τCRSS = σy cos λ cos φ ≈ σy · 0.41
∴ σy = 50.5 M P a
It needs at least 50.5 MPa of stress.
1. Slip plane: the plane on which easiest slipTo finish this subsection, we have to take into
page occurs. It usually happens in highest
account the anisotropy in deformation: if we
planar densities with large interplanar spachave a rolled cylinder from a machine and we
ing.
fire it to a target, we will get a deformed cylinder
2. Slip direction: the direction of the move- with the impact zone wider than the ‘tail’. The
ment. It happens in the highest linear den- non circular end view shows the anisotropic deformation of the material because the direction
sities.
of impact makes the deformation non circular (it
For example, in FCC slip occurs on {111} planes has an ellipsoid shape).
(close-packed) in h110i directions (close-packed)
so this means a total of 12 slip systems. For BCC 5.3.2 Strategies for Strengthening
and HCP there are other slipping systems4 .
There are four strategies for strengthening:
1. Reduce gran size. Grain boundaries are barriers to slip movement. Its strength increases with a higher angle of misorientation and thus having suddenly smaller grain
size makes more barriers to slip. It can be
quantified with the Hall-Petch equation:
1
σy = σ0 + ky d(− 2 )
Figure 5.9: Resolved shear stress
The resolved shear stress, τR results from
applied tensile stresses so first we apply a tensils
stress (σ = F
A ) and a slip occurs. Then, the
resolved shear stress is in the direction of the
movement (normal to the plane created) and so
4 In BCC: {110} and h1̄10i; in HCP: (0001) and
h112̄0i.
22
(5.11)
2. Form solid solutions. Impurity atoms distort de lattice and generate lattice strains
against the movement. These strains can
act as barriers too. Small impurities concentrate at dislocations so the can reduce
their mobility and large impurities tend to
go at dislocations with tensile strains so
they block the movement.
Albert Marin
5.3. DEFORMATION
After cold working Cu we have a decrease in diameter of 3 mm (seen in tables on the internet).
Figure 5.10: Lattice strains around dislocations
%CW =
πD02
4
−
πDd2
4
πD02
· 100 =
D02 − Dd2
· 100
D02
3. Precipitation strengthening. Hard precipi4
tates are difficult to shear, for example ce2
15.2
− 12.22
ramics in metals. They are used to pin sites,
%CW =
· 100 ≈ 35.6%
15.22
with distance S, of slipping so σy ∼ S1 . This
is used in aeronautics.
Now we must look up at tables the values for a
Cu with %CW = 35.6%
4. Cold work (strain hardening). Deformation
at room temperature by forging, rolling,
drawing or extrusion to reduce the crosssectional area (and increase the condensation of the material). It can be quantified
as:
%CW =
A0 − Af
· 100
A0
5.3.3
(5.12)
Heat treatment
An hour treatment at at annealing5 temperaFor example, after cold working Ti, disloca- ture revert the effects of cold work. This process
tions entangle between themselves so as to has three stages: recovery, recrystallization and
grain growth.
make motion more difficult.
The cold work has important implications.
The dislocation density can be computed with
the total dislocation length divided by the unit
volume. This cold work makes yield stress increase as ρd increases. In addition, TS also increases and ductility decreases.
1. Recovery. In this stage the point is to reduce the dislocation density by annihilation.
• Scenario 1: the heating makes diffusion of atoms so they modify tension
sites which make dislocations annihilate and form a perfect atomic plane.
• Scenario 2: dislocation can not move
so atoms move to leave vacancies
which will be used to make dislocation
‘climb’ towards a plane to distress.
2. Recrystallization. New grains are formed
with low dislocation densities, are small in
size and consume and replace all parent
cold-worked grains eventually.
3. Grain growth. After a few time, average
grain size increases if we keep the temperature at the same value. Small
grains shrink and dissapear and larger ones
continue to grow. There is an empirical relation for that:
Figure 5.11: Cold work effects
• Example 3. What are the values of yield,
tensile strength and ductility after cold
working a 15.2mm rod of Cu?
23
dn − dn0 = Kt
5 “Re-cooking”
(5.13)
Biomedical Engineering
CHAPTER 5. MECHANICS
where d is the grain diameter at time t, n
is usually 2, K is a coefficient dependent on
material and T and t is the elapsed time.
Figure 5.13: True curve
Metals can be hardened through a process
called hardening, which cause an increase in σy
so higher resistance to plastic deformation. This
process can be done in many ways but the result
is always the same, just to increase the hardness
of the material. There exist variability in ma-
Figure 5.12: Heating effects
In the figure 5.12 we can see that there is a recrystallization temperature and it is the temperature at which recrystallization just reaches
completion in 1 h. It follows the rule 0.3Tm <
TR < 0.6Tm and it depends always inversely on
%CW and purity of the metal.
5.4
Hardness and hardness
testing
Figure 5.14: Hardening
terial properties: E is a material property that
depend largely on sample flaws (defects, impurities, etc.). As one can imagine, larger samples,
more probability to have more flaws. This is
the reason for which the design or safety factors exist: the stress changes with this factor,
N , which usually ranges between 1.2 and 4. The
relation is the yield stress divided by N.
• Example 1: Calculate the diameter, d, to
The concept of hardness is the resistance to
ensure that yield does not occur in the 1045
permanently indenting the surface. Large hardcarbon steel rode applying a F = 220000 N .
ness means resistance to plastic deformation or
Use a factor safety of 5.
cracking in compression and better wear properties. The hardnesses of the materials range from A 1045 plain carbon steel rod, has a σy =
low to high: most plastics, Al alloys, steels, file 310 M P a, T S = 565 M P a so then:
hard, cutting tools, nitrided steels and diamond.
σy
310 · 106
The way of measuring is by the rockwell or Brinσworking =
=
N
5
nell Hardness systems: they differ in the compuF
220000
tations to understand the hardness and the type
σworking =
=
2
A0
π d4
of penetrator used. We can realise that stress
and strain are not the true values we can see
d = 0.067 m = 6.7 cm
in reality. This happens because of the necking
effect, and there is a way to find out the true
stress-strain curve:
5.5 Fracture of metals
F
σT =
= σ(1 + ε)
Ai
`i
εT = ln
= ln(1 + ε)
`0
(5.14) There are two main types of fracture mecanisms:
the ductile fracture (accompanied by a signifi(5.15) cant plastic deformation) ad the brittle fracture
(little or no plastic deformation and with catastrophic outcome). The ductile failure gives us a
24
Albert Marin
5.6. FATIGUE OF METALS
Figure 5.16: Stress concentration factor
Figure 5.15: Ductile vs brittle failure
where γS is the specific surface energy and a is
one half length of internal crack. However, for
ductile materials, there must be added a plastic
deformation energy so γ = γS + γP .
The largest and most highly stressed cracks,
then, grow first. There are specific rules of design against the crack growth. One has to consider the fracture toughness, KIC , which is the
minimum stress that has to be applied to a crack
1. Necking: a neck starts to form.
to make it fracture.
2. Void nucleation: small air bubbles start to
√
(5.17)
K ≥ KIC = Y σ πa
form within the core of the cross-section
neck. Particles serve as void nucleation sites
where Y is an adimensional geometric factor and
too.
a is the length of a surface crack or half the
3. Void growth and coalescence: the bubbles length of an internal one. We can have two scenarios:
grow and fusion to each others.
warning that the material is going to break by
showing a large deformation of the material and
breaking into one piece whereas the brittle fracture gives many little pieces and small or none
deformations (so then no warning). If we speak
of a moderately ductile failure we can distinguish
five stages:
4. Shearing at surface: they start to reach the
surface.
1. Maximum flaw size dictates design stress:
σdesign <
5. Fracture: broken!
An important feature of brittle fracture surfaces
is the point that it can be intergranular, between grains leaving them intact, or transgranular, through grains breaking them.
The main point is that the TS of engineering materials is greatly smaller than the perfect ones. Da Vinci observed that the longer
the wire of a pendulum, the smaller the load for
provoking a failure. This is because the flaws
cause premature breaking as well as larger samples contain longer flaws. Another important
thing is to avoid sharp corners. Why is it so?
Because cracks having sharp tips propagate easier than cracks having blunt tips. If we compute
and
the stress concentration factor Kt = σmax
σ0
we plot it with a few relations we can clearly see
that for the same initial stress, the maximum
stress decreases. The relation in energy for this
process is that the energy is stored as the material is elastically deformed until it is released
when the crack propagates because the creation
of new surfaces require energy. So the criterion
for crack propagation is that it will propagate if
crack-tip stress, σm , exceeds a critical stress, σc ,
defined by:
r
2EγS
(5.16)
σc =
πa
25
Y
KIC
√
πamax
2. Design stress dictates maximum flaw size:
2
KIC
1
amax <
π Y σdesign
5.6
Fatigue of metals
The definition of the fatigue is a failure under
applied cyclic stress. A typical experiment is
to put a specimen in a machine which will apply a cyclic stress and analyse the failure. The
Figure 5.17: Fatigue experiment
stress varies with time and it can be measured
with the cycling frequency, the mean σ (σ̄) and
S (S = σmax − σ̄). This is a very important feature as almost 90% of mechanical engineering
failures are due to it and can cause the break of
a material although σmax < σy !
There are two behaviours of the fatigue: the
case for typical steels is that there exist a fatique
Biomedical Engineering
CHAPTER 5. MECHANICS
limit Sf at so values of stress amplitude, S, below
this point cause no fatigue. However, there are
some materials which do not have fatigue limit.
The rate of fatigue crack growth follows a differential equation:
da
= (∆K)m
dN
(5.18)
where m ranges typically between 1 and 6 and
∆K can be isolated
√ from equation 5.17 or ap- Figure 5.19: Temperature dependence of creep
proximate to ∆σ a. A crack grows faster as
∆σ increases, crack gets longer and loading frehigh temperatures, such as T > 0.4Tm . In the
quency increases.
To improve fatigue life, there are two typical secondary creep, where the strain rate is constant at a given T and σ, the strain hardening
methods:
is balanced by recovery and its value is:
1. Impose compressive surface stresses to supQc
press surface cracks from growing by shoot(5.19)
ε̇S = K2 σ n e(− RT )
ing metallic balls (shot peening) or carburwhere K2 is a material constant, the exponent n
izing a sample.
is a material parameter and the Qc is the acti2. Removing stress concentrators (sharp an- vation energy for creep (also a material paramgles and/or corners!).
eter).
The failure of creep is due to cavities formed
along
grain boundaries. The prediction of this
5.7 Creep and stress ruprupture lifetime can be estimated, tr as:
ture of metals
T (20 + log tr ) = L
The creep, or cold flow, is the tendency of
a solid material to deform slowly and permanently under the influence of constant mechanical stresses and at high temperature. If we analyse a plot of stress vs time we will not see anything but a straight line parallel to x-axis (as
stress is constant) but it we check the strain vs
time we will see a different thing. In the pri-
where T is temperature and L is the function of
applied stress.
• Example 4. Find the time of rupture of S590 iron at T = 800◦ , σ = 2 × 104 psi.
One has to find the value in a chart data and
then plug in the values:
1073(20 + log tr ) = 24 × 103
tr = 233 h
Figure 5.18: Creep
mary creep, the slope decreases with time; in
the secondary one, we will find a steady state
creep (slope is constant of ∆ε
∆t ) and in the tertiary creep, there is an acceleration rate until
the failure. The effect of temperature in this
process does not make any big influence until
26
(5.20)
CHAPTER
SIX
PHASE DIAGRAMS
6.1
Phase diagrams of pure
substances
When we combine two elements, they will get
to an equilibrium state. In particular, when we
specify the composition (i.e. in weight percent)
and the temperature, we will be able to find out
how many phases are formed in the compound,
the composition of each phase and the amount
of each one. Firstly one should know the typical
vocabulary:
Figure 6.1: Cu-Ni system with wt%Ni
• System: the universe and any part of it.
• Phase: a region in the system that has a
variables T and C. In the diagram shown of the
different structure and/or composition.
Cu-Ni (binary) system we can see that there are
• Structure: how the atoms or molecules of 2 phases (L = liquid and α = FCC solid solution)
the components are physically arranged in that hold 3 different phase fields: L, L + α and
α. If we look the water phase diagram we will
space.
see a easy and unary phase diagram. However,
• Composition: the relative amounts of dif- we can find more complexity for example with
ferent components.
the pure iron diagram.
• Components: chemically distinct species,
generally pure elements or compounds.
• Phase diagram: a graphical representation
of the influence of various factors, such as
temperature, pressure and composition on
the phases that exist in a system.
• System types:
– Unary: a system with only one component.
– Binary:...two.
– Ternary:...three.
– Quaternary:...four.
Figure 6.2: Pure iron phase diagram.
• A, B, C...: generic names of components.
There is a equilibrium phase line ih the diagrams which separate the solution (solid, liquid
Now we are ready to show a phase diagram. It or gas solution, a single phase) from the mixture
indicates the phases as a function of T, C and P. (more than one phase). The easiest case is the
Usually as P is fixed, we can see the independent sugar and water phase diagram.
• L, α, β, γ...: generic names of phases.
27
Biomedical Engineering
CHAPTER 6. PHASE DIAGRAMS
ditional lattice. These phases are called ordered phases.
• Substitutional solid solution: solute atoms
will randomly substitute original atoms.
The solubility in this case depends on several conditions seen in chapter 4: the HumeRothery rules 1 .
Figure 6.3: Sugar-water phase diagram
6.2
Gribbs phase rule
The solubility limit shows the maximum con- This rule tells us how many phases can exist
centration for which only a single phase solution under a given set of circumstances:
exists. If we are interested in finding the soluP +F =C +2
(6.1)
bility limit for sugar at 20◦ C we sill see that is
65wt% sugar. If the concentration is above that where:
limit, we will have syrup and sugar whereas if
we have it below that point we will only have
• P is the number of phases.
syrup. If we changed the pressure too, it would
• F is the number of degrees of freedom (numhave a different value for instance.
ber of independent variables).
Once we have a mixture, we can microscopically see clear differences. For example, in the
• C is the number of components
Al-Cu alloy, we can see an α phase (darker
phase) and a β one, the lighter one. If we look at
• The number 2 indicates the ability to
change temperature and pressure.
However, there is the modified Gibbs phase
rule: as most engineering systems function at
a pressure of 1 atmosphere, we have picked the
pressure as one of our degrees of freedom and
thus the number 2 will have to be reduced as
pressure will not change. Then:
P +F =C +1
Figure 6.4: Different phases of Al-Cu
(6.2)
the effect of temperature and composition, one
6.3 Cooling curves
can realise that if we modify temperature we can
move from one phase to another one vertically. A schematic representation of the process of
On the other hand, if we move along the concen- cooling can be drawn as seen in the following
tration variable, we can change phases by sliding diagram. The solidification occurs at a conhorizontally.
As a review, there are three types of solid
phases:
Figure 6.6: Solidification curve
Figure 6.5: Solid phases
• Interstitial solid solution: atoms with a
small atomic radii dissolve in the lattice.
Usually N, C, H and B. In this case, it will
be a limit of solubility.
stant temperature while latent heat of fusion is
released. One has to think that even the digraph phase shows us the concentration of the
allow and the phase in which the material is,
the process of moving from one to another is
1 Radii difference below 15%, similar crystal structure,
• Superlattice: solute atoms will occupy specific positions of the lattice, creating an ad- same or higher valence and same electronegativity.
28
Albert Marin
6.4. ALLOY SYSTEMS
not instantaneous. As an example: Ni-Cu cool• At TD = 1190◦ C there is ony α so then
ing curves. The solidus locus is the temperatures
Cα = C0 = 35 wt% N i.
below which all compositions are solid (the start
• At TB = 1250◦ C there will be both phases:
of solidification during cooling) and the liquidus
CL = Cliquidus = 32 wt% N i, Cα =
locus the same but with liquid (start of melting
Csolidus = 43 wt% N i.
during heating). This happens as it is a binary
isomorphous alloy: it has complete solubility of
Rule 3: if we know T and C0 then we can deone component in another.
termine the weight fraction, Wi of each phase.
Considering the same example as before, at the
temperatures with only one element we will have
the 100% of each phase. However, at temperature B we will have to use the lever rule:
Figure 6.7: Ni-Cu cooling curves
6.4
Binary isomorphous alloy systems
6.4.1
Determination of phases(s)
present
Figure 6.9: Lever rule
Rule 1: if we know T and C0 then we know which
phase(s) is (are) present. Just move along the
ML
Cα − C0
S
WL =
=
=
(6.3)
axis. If we look back to the figure 6.1, a comM
+
M
R
+
S
C
L
α
α − CL
◦
pound A at 1100 C and 60wt% Ni, we have only
C0 − CL
Mα
R
=
Wα =
=
(6.4)
α phase whereas if we look a for a compound B
M
+
M
R
+
S
C
L
α
α − CL
at 1250◦ and 35wt% Ni, we have a L + α mixture.
Then the fraction of liquid is 0.73 and the other
one 0.27.
The tie line that is plotted in the diagram is
6.4.2 Determination
of
phase
called sometimes the isotherm. It connects the
compositions
phases in equilibrium with each other. In addiRule 2: if we know T and C0 then we can de- tion, if we look at the micrsostructural changes
termine the composition of each phase. If we that accompany the cooling of a Cu-Ni alloy, we
would see clear differences: There can happen
Figure 6.8: Interested area zoomed in
Figure 6.10: Cooling microstructures
consider C0 = 35 wt% N i:
• At TA = 1320◦ C there is only liquid, so
CL = C0 = 35 wt% N i.
two things: if we have a slow rate of cooling we
will get an equilibrium structure with uniform
29
Biomedical Engineering
CHAPTER 6. PHASE DIAGRAMS
Cα . However, if we use a fast rate of cooling, a
cored structure will appear in which the first α
to solidify will be the inner concentration of Ni
and the last one will be the outer region (which
will be in < 35 wt% N i).
6.5
Ternary phase diagrams
The ternary phase diagrams can be quite very
complex. In general, the diagrams only show the
phases in a fixed temperature and pressure so
each three compound material has to be checked
in all temperatures and all pressures. Two examples are shown:
Figure 6.11: Al-Cu-Fe diagram
Figure 6.12: Hypothetical diagram
30
CHAPTER
SEVEN
ENGINEERING ALLOYS
7.1
Introduction
• Can be classified as steels or cast irons.
Most engineering metallic materials are alloys.
Metals are alloyed to enhance their properties,
such as strength, hardness or corrosion resistance, and to create new properties, such as
shape memory effect. Engineering alloys can
be broadly divided into ferrous alloys and nonferrous alloys as seen below:
7.1.1
Steels
7.1.2
Cast irons
These irons are ferreous alloys with under 2.1
weight percent of C (more commonly below 3
and 4.5). They have a low melting grade (easy
to cast) and they are usually brittle. The types
of cast iron are:
The iron-based alloys:
• Contain Fe as the main element.
• The most important ferrous alloy system:
Fe-C system.
• Gray iron
– Graphite flakes
• Alloys of this system can be further divided
into steels and cast irons.
– Weak and brittle in tension
– Stronger in compression
• Steels
contain
less
C
(generally
< 1.4 wt%C) than do cast irons (generally 2.4 ∼ 4.3 wt%C).
– Excellent vibrational dampening
– Wear resistant
• Then, all steels solidify into a single γ-Fe
structure (FCC) first and then experience
the complex eutectoid reaction. Therefore,
heat treatment processes, which alter the
eutectoid reaction, are vitally important for
controlling microstructure and properties of
steels.
• Cast irons experience complex eutectic reaction during solidification, due to the formation of graphite or cementite. Solidification control is the most important single
factor for properties of cast irons.
31
• Ductile iron
– Add Mg and/or Ce
– Graphite as nodules not flakes
– Matrix often pearlite: stronger but less
ductile
• White iron
– < 1 wt% Si
– Pearlite + cementite
– Very hard and brittle
Biomedical Engineering
CHAPTER 7. ENGINEERING ALLOYS
7.2.1
• Malleable iron
– Heat treat white iron at 800-900◦ C
Fe-C phase diagram
If we analyse the phase diagram: The plain car-
– Graphite in rosettes
– Reasonably strong and ductile
• Compacted graphite iron
– Relatively high thermal conductivity
– Good resistance to thermal shock
– Lower oxidation at elevated temperatures
7.2
Production of iron and
steel
The first step is to produce pig iron: we put ore
(F e2 O3 ) together with coke (3CO) in a blast
furnace to react and obtain 2F e + 3CO2 . Then
we are ready to make steel:
1. Pig iron and 30% steel crap is fed into refractory furnace to which oxygen lane is inserted.
Figure 7.2: Fe-C phase diagram
bon steel has from 0.03% to 1.2% C, 0.25% to
1% Mn and other impurities. We can find three
main compositions:
1. α-ferrite: very slow solubility of carbon.
2. Oxygen reacts with liquid bath to form iron
oxide.
3. F eO + C −→ F e + CO
2. Austentite: interstitial solid of carbon in γ
iron.
3. Cementite: intermetallic compound.
4. Slag forming fluxes are added.
There are a series of invariant reactions that are
schemed below:
5. Carbon content and other impurities are
lowered.
6. Molten steel is continuously cast and
formed into shapes.
The process of refinement can be seen in the
following diagram: Once we have the material,
That makes the phase diagram to differ from the
theoretic. Here is the true equilibrium diagram:
Figure 7.1: Refinement of steel
we can blacksmith it (forged), cast molten metal
into a mould or with forming operations which
may be hot working (creates large deformations
to make recrystallization) or cold working (small
deformations created by recrystallization at low
temperatures and where strain hardening occurs).
32
Figure 7.3: True Fe-C equilibrium diagram
Albert Marin
7.2.2
7.3. BIOMEDICAL ALLOYS
7.3
Stainless steel
Titanium and biomedical alloys
This material has excellent corrosion resistance
due to high (at least 12%) chromium oxide
There are three main types of alloys: the ones
formed on the surface.
made by magnesium, by titanium and by nickel.
7.3.1
Mg alloys
• Low density metal, high cost, low castability, low strength, poor creep/fatigue and
wear resistance.
• There are two types: wrought alloys (sheet,
plate, extrusion) and casting alloys (casting).
• Designated by two capital letters and 2-3
numbers which indicate the two major alloying elements and the wt% of alloying elements.
Figure 7.4: Fe-Cr diagram
The point is that the ferrite stainless steel:
• Limited cold working due to the HCP structure, so it is usually hot worked.
• Contains between 12-30% Cr
• Al and Zn are added to increase strength.
• It has a structure mainly ferritic (BCC α).
• Alloying it with rare earth elements (i.e.
cerium) produces rigid boundary network.
• Cr extends that α region and suppresses a
γ region, creating a γ loop.
• Is a low cost material with high strength
(517 MPa).
• Tensile strength = 179-310MPa and elongation: 2-11%.
7.3.2
7.2.3
• Low density and high strength
Martensitic stainless steel
It is a very hard form of steel crystalline structure which is composed of Cr (12-17%) and C
(0.15-1%). It is formed by quenching1 the austentite region. However, it has poor corrosion resistance and tensile strength 517-1966 MPa. It is
greatly used for machine parts, pumps, bearings
and valves. When carbon exceeds the normal
composition, an α loop is enlarged.
7.2.4
Ti alloys
• EXPENSIVE: used for aircraft applications.
• Superior corrosion resistance.
• Special technique needed to work with
metal.
• HCP at room temperature that changes to
BCC at 883◦ C.
• Al and O increase transformation temperature.
Austentitic region
This is the γ-Fe phase region. The composi• Tensile strenght = 662-862MPa.
tion is a ternary alloy of Fe-Cr(16-25%)-Ni(7But despite its cost, it has important medical
20%) which stands in a FCC γ structure at all
temperatures due to nickel. It has better cor- utilities:
rosion resistance than other steel with a ten• Easily formed, outstanding corrosion resissile strength range 559-759 MPa. It is mostly
tance
used for chemical equipment, pressure vessels,
etc. Alloying columbium (quaternary alloy) pre• Low elastic modulus, highly biocompatible
vents intergranular corrosion if the alloy is to be
• Pure Ti is used in low strength applications
used for welding.
1 Rapidly
cooling the material from very high temperatures to room temperature so as to maintain stable the
hot condition crystalline structure at ∼ 25◦ C.
33
• Alpha-beta alloys of Ti like Ti-6Al-4V
(F1472) are strengthened by solution heat
treatment.
Biomedical Engineering
CHAPTER 7. ENGINEERING ALLOYS
• Poor wear resistance and notch sensitivity
• Beta alloys have low elastic modulus
• Ion implantation improves wear resistance
7.3.3
Ni alloys
• Expensive, good corrosion resistance and
high formability.
• Commercial Nickel and Monel alloys: good
weldability, electrical conductivity and corrosion resistance.
• Nickel + 32% Cu −→ Monel alloy (strengthens nickel).
• Nickel based super alloys: High temperature creep resistance and oxidizing resistance for gas turbine parts.
Figure 7.5: Screws and other biocompatible elements
• 50 -60% Ni + 15-20% Cr + 15-20% Co +
1-4% Al + 2-4% Ti.
as bone screws to be removed after healing (i.e.
• 3 phases: gamma austenite, gamma prime, a broken long bone).
On the other hand, Co with Cr (for long term
carbide particles.
corrosion resistance), Ni and W (machinability
and fabrication easiness) is used in permanent
7.3.4 Intermetallic alloys
fixation devices as a total knee replacement prosThey offer unique combination of properties: for thesis.
example Ni-Al + Fe-Al + Ti-Al have high temperature applications. The intermetallic alloys:
• Low density, good high temperature
strength, less corrosion but brittle.
• 0.1% Boron and 6-9% Cr added to reduce
embrittlement and to increase ductility.
• Applications: Jet engine, pistons, furnace
parts, magnetic applications(F e3 Si) and
electronic applications (M oSi2 ).
7.3.5
Biomedical applications
Biometals come in direct contact with human
body fluids: they are used to replace tissues,
to support damaged ones while healing and to
fill spaces. The high problem is the biocompatibility as the internal environment of human body is highly corrosive. This makes metals to be degraded and release harmful ions.
The chemical stability, corrosion resistance noncarcinogenity and non-toxicity are called biocompatibility. General ideas are that they must
have high fatigue strength, Pt/Ti/Zr have good
biocompatibility and Co/Cu/Ni are toxic.
For instance, 316 L stainless steel (cold worked
with a grain size of minimum 5) is used most
often because it is cheap, easily shaped, has a
limited corrosion resistance and it can be used
34
Figure 7.6: Knee replacement prosthesis
The main issues in orthopaedic applications
are:
• High yield strength, fatigue strength and
hardness of implants are desired. The implant should support a healing bone.
• Low elastic modulus: it should carry proportionate amount of load, it should not
shield the bone from load (it would stop
remodelling the bone and thus weakening
it).
• Wear causes metallic toxicity. However, CoCr alloys have good wear resistance to be
taken into consideration.
CHAPTER
EIGHT
POLYMERIC MATERIALS
8.1
Introduction
A polymeric solid is a material which has a repeating unit many times. They are made of hydrocarbons, which are organic compounds consisting entirely of H and C. The polymers can
be:
• Fibres (e.g. silk): chains aligned parallel to
fibre axis.
• Plastics: which can be reheated to form new
materials (thermoplastics) or no modifiable
(thermosetting plastics)
• Elastomeres (e. g. rubber): loosely crosslinked network
8.2
Polymerization
tions
2. Propagation.
Once the process has
started, extend the polymer by successive
addition of monomers. It happens spontaneously because the energy of the chemical
system is lowered by polymerization.
3. Termination. Achieved by the addition
of terminator free radicals, combining two
chains or by inserting impurities.
The molecular structure for polymers can be
of four types, but the most common ones are
shown in red. The advantage of it is that chain
reac-
Double and triple bonds which are a bit unstable
can form new bonds with simple reactions. The
starting phase consist in a free radical polymerization by an initiator and then a propagation.
Figure 8.1: Polymer structures
bending and twisting are possible by rotation of
carbon atoms around their chain bonds without
the need of breaking their bondings. The bulk
polymers are the ones which have a repeating
unit over and over again lineally. In addition,
this can be used to create a copolymer which
An example of an initiator is the benzoyl perox- can be as shown in the scheme. An interesting
feature is the crystallinity in polymers: they are
ide which gives 2 free radicals.
ordered atomic arrangements involving molecular chains so it is possible to identify crystal
structures in terms of unit cells. This happens
because of the attractive and repulsive forces
that appear in between the chains forming the
So then, the steps are:
polymer. They usually happen to order them1. Initiation. Beginning the proces of turn- selves in a zig-zag way (chain folded structure).
ing monomers into polymers with a catalyst It usually has a distance of 10 nm which is equiv(free radical).
alent to 60 C bonds.
35
Biomedical Engineering
CHAPTER 8. POLYMERIC MATERIALS
the condensation (step polymerization) method.
The process starts by taking the raw materials
(natural gas, petroleum and coal) and polymerizate them to obtain granules, pellets, etc. The
addition polymerization consists of two methods:
• Bulk polymerization: monomer and activator are mixed in a reactor and
heated/cooled as desired.
• Solution polymerization: monomer is dissolved in a non-reactive solvent and catalyst.
The other way consists of:
• Suspension polymerization: monomer and
catalyst are suspended in water.
• Emulsion polymerization: monomer and
catalyst are suspended in water along with
emulsifier.
Figure 8.2: Copolymers
Some additives can be putted into the containers so as to improve mechanical properties,
processability, durability among others:
• Fillers: added to improve tensile strength,
abrasion resistance, toughness and decrease
costs. For example: carbon black, silica gel,
glass...
Figure 8.3: Crystal polymer
8.3
Processing of plastic
materials: thermoplastic and thermosets materials
There is an indicator of the average number of
repeat units per chain called degree of polymerization, DP, which can be obtained by:
DP =
M̄
m̄
(8.1)
where M̄ is the mean of the amount of atoms
present in the sample and m̄ is the average
molecular weight of the repeat unit. If we have
a copolymer, it should be assessed by:
X
m̄ =
fi mi
(8.2)
• Plasticizers: added to reduce the glass transition temperature below room temperature. It transforms a brittle polymer into
a ductile one. The additive is usually PVC.
• Stabilizers: antioxidants, UV protectants...
• Lubricants: added to allow easier processing so as the polymer slides through dies
easier. Example: sodium stearate.
• Colourants: dyes and pigments.
• Flame retardants: substances with chlorine,
fluorine and boron.
Thermoplastic materials can be reversibly
cooled and reheated (recycled) so the main way
to work with them is to heat them until it is a
soft material, shape as desired and then cooled
down. Examples are polyethylene, polypropylene, polystyrene...
In contrast, thermoset plastics form a molecuwhere f is the chain fraction and m the molecular
network by chemical reactions when heated.
lar weight of the repeating unit. So for example
This
means it degrades when temperature is
the following molecule has a DP = 6.
high (does not melt down). One has to have a
prepolymer moulded into the desired shape and
then make the chemical reaction. This gives the
material a high thermal and dimensional stabilWe can find two types of polymerization: by ity, rigidity, resistance to creep and light weight.
addition (also known chain polymerization) or For example: urethane, epoxy...
36
Albert Marin
8.4. MECHANICAL PROPERTIES
To work this two types of polymers, it has
to be done in a special compression moulding
machine, as pictured below...
Figure 8.7: Blown-film extrusion
deformation strains can go up to more than the
1000% of the initial length1 .
Figure 8.4: Compression moulding of polymers
...or an injection moulding (the ram retracts
to let plastic pellets drop from the hopper into
the barrel – the ram forces plastic into the heating chamber, around the spreader where the
plastic melts as it moves forward – molten plastic is forced under pressure (injected) into the
mould cavity where it assumes the shape of the
mould)...
Figure 8.8: Stress-strain curves for polymers
As said before, thermoplastics have little
cross-linking which makes it ductile and soft
when heated, in comparison with thermosets,
which have a significant crosslinking (10 to 50%
of repeat units), are hard and brittle and does
not soften with heating. The effects of decreasing T in thermoplastics: increases E, TS and
decreases EL%. It has the same effects as increasing the strain rate.
Figure 8.5: Injection moulding
...or in a more general way for plastic materials: extrusion...
Figure 8.6: Extrusion
...or with a blown-film extrusion.
A special case of the injection moulding can
be used to make multiple parts at a time as well
as small and intricate parts.
Figure 8.9: Melting and glass transition temperatures
8.4
Mechanical properties of
polymers
If we increase the chain stiffness, we will get a
higher Tm and Tg . The stiffness is increased by
bulky or polar side-groups and by double bonds
The stress-strain behaviour is very different from and aromatic chain groups. However, the regmetals. Their E is much less than metals and ularity of the repeating unit arrangements only
1 Most metals have below the 10% of their initial
the fracture strength of polymers is around a
10% of those for metals. Despite of that, their length.
37
Biomedical Engineering
CHAPTER 8. POLYMERIC MATERIALS
affects to Tm .
should anneal it. Remember that in metals, the
Another important feature is the time- cold working makes the opposite effect.
Finally, elastomeres are extremely ductile,
dependent deformation. It can be done a stress
relaxation test to check the decrease in stress with long reversible deformation:
with time at a constant strain. The relaxation
modulus is a function of time:
Er (t) =
σ(t)
ε0
(8.3)
If we modify the temperature, we can dramatically decrease this modulus if T > Tg . This
effect of the decrease in stress is a feature very
useful. The craze is a formation of microvoids,
at the same time as fibrilar bridges that align
chains, that appears prior to cracking.
8.5
Mechanisms of deformation of polymeric materials
8.6
The plots for the fracture is quite beautiful to
see: for brittle crosslinked and network polymers
goes as follows:
Figure 8.12: Elastomeric deformation
Polymer types and advanced polymers
• Fibres – length/diameter ¿ 100.
– Primary use is in textiles.
– Fibre characteristics: high tensile
strengths, high degrees of crystallinity and structures containing polar groups.
– Formed by spinning: extrude polymer
through a spinneret (a die containing
many small orifices) – the spun fibres
are drawn under tension which leads
to highly aligned chains (fibrillar structure).
Figure 8.10: Brittle failure
A semicrystalline polymer (plastic) will break
in a different way: One can predeformate the
• Miscellaneous.
– Coatings – thin polymer films applied to surfaces – i.e., paints, varnishes – which protects from corrosion/degradation, is decorative (improves appearance) and can provide
electrical insulation
– Adhesives – bonds two solid materials
(adherents). Can bond secondarily by
Van der Waals forces or mechanically
by penetration into pores/crevices.
– Films – produced by blown film extrusion.
– Foams – gas bubbles incorporated into
plastic.
Figure 8.11: Plastic failure
material by drawing it: it stretches the polymer prior to use so as to align the chains in the
stretching direction. This increases E, TS and
decreases ductility. To reverse the changes, we
38
• Advanced polymers: ultrahigh molecular
weight polyethylene (UHMWPE). Have a
weight of around 4 × 106 g/mol with outstanding properties:
Albert Marin
8.6. POLYMER TYPES
1. High impact strength.
2. Resistance to wear/abrasion.
3. Low coefficient of friction.
4. Self-lubricating surface.
It is normally used in bullet-proof vests, golf
ball covers or hip implants (acetabular cup)
Summary
• Limitations of polymers:
– E, σy , Kc , Tapplication are generally
small.
– Deformation is often time and temperature dependent.
• Thermoplastics (PE, PS, PP, PC):
– Smaller E, σy , Tapplication .
– Larger Kc .
– Easier to form and recycle.
• Elastomers
strains!
(rubber):
large
reversible
• Thermosets (epoxies, polyesters):
– Larger E, σy , Tapplication .
– Smaller Kc .
• Polymer processing: compression and injection molding, extrusion and blown film extrusion.
• Poylmer melting and glass transition temperatures.
• Polymer applications: elastomers, coatings,
films, fibres, adhesives, foams and advanced
polymeric materials.
39
CHAPTER
NINE
CERAMICS AND COMPOSITE MATERIALS
9.1
Introduction
touching only two anions it will be unstable whereas a cation touching four anions
(even if they are touching between themselves) will be stable.
Ceramics have a certainly different properties
from the other materials:
• Inorganic and non-metallic.
• Consists of metallic and non-metallic elements bonded together primarily by ionic1
or covalent bonds.
• Good electrical insulators because of the absence of conduction electrons.
• Maintenance of charge neutrality. The net
charge should be 0 – this is already reflected
in the chemical formula.
To form a stable structure, one should look
because it increases
up the ratio between rrcation
anion
with the coordination number as we can see in
the figure2 .
• Good thermal insulators: same reason as
before.
• Hard and brittle.
• Less ductile and toughness than metal.
• High chemical stability because of the stability of their bonds and thus a high melting
temperature.
One has to keep in mind that the percentage
of ionic bonding can be calculated as we saw in
chapter 2.
Figure 9.1: Coordination number and ionic radii
If we would want to find the minimum cationanion radius ratio (i.e. in an octahedral site) one
should look for the geometry of the system and
in this example we would find:
√
9.2.1 Oxide structures
2ranion + 2rcation = 2a (a = 2ra )
√
√
These are interesting because O anions are larger
2ra + 2rc = 2 2ra → ra + rc = 2ra
√
than metal cations, they have the oxygen lose
rc
∴
= 2 − 1 = 0.414
packed in the lattice (usually FCC) and cations
ra
fit into interstitial sites among O ions.
There also exist a bond hybridization, which
The factors that determine the crystal struchappens when there is significant covalent bondture are:
ing: hybrid electron orbitals form. For example
• Relative sizes of ions – Form stable struc- the SiC ceramic has a 89% of covalent bonding
tures if the structure has the maximum and they prefer sp3 hybridization so then Si ocnumber of oppositely charged ion neigh- cupies tetrahedral sites.
bours. So if we have a cation very small
9.2
Simple ceramic crystal
and silicates structures
1 Cation
2 Tetrahedral = ZnS (zinc blende), octahedral = NaCl
and cubic = CsCl
= +, anion = −.
40
Albert Marin
9.2. STRUCTURES
• Example 1.: predict the crystal structure of
FeO.
The ionic radius of F e2+ is 0.077 and the oxygen
one is 0.140. If we compute the ratio we will find
0.550. If we look up the value in the table we
will see that is a coordination number of 6 and
therefore the crystal sctructure will be the same
as NaCl.
That structure is called the rock salt structure. In the example of NaCl, the ratio is 0.564
and therefore cations prefer octahedral sites (the
atom of the center is Na). MgO and FeO have
Figure 9.4: Fluorite structure
Figure 9.2: Rcok salt structure
Figure 9.5: Perovskite structure
also the same structure, so each Mg or Fe atom
has 6 oxygen neighbour atoms.
9.2.2 Silicate ceramics
The AX-Type crystal is a BCC with the cation
occupying the center position so it has 8 anions These are interesting as the most common elin the surroundings.
ements on Earth are Si & O. SiO2 (silica)
polymorphic forms are quartz, crystobalite and
tridymite and their strong bonds lead to a high
melting temperature (1710◦ C). Bonding adjacent SiO44− is possible by sharing common corners, edges or faces and the presence of cations
such as Ca, Mg and Al help maintain charge
neutrality as well as bond ionically the SiO44−
to one another.
Figure 9.3: AX-type crystal
The doubled previous structure, the AX2 Type crystal or Fluorite structure has an
antifluorite structure – positions of cations
and anions are reversed.
Examples are:
U O2 , T hO2 , ZrO2 , CeO2 . . . .
The ABX3 crystal structure or Perovskite
structure is one with three types of atoms, with
the structure shown below.
The density computation formula for ceramics
is the following one:
P
P
n( AC + AA )
(9.1)
ρ=
VC NA
where n is the number of formula units/unit
cell, the two sumatories are the sum of atomic
weights of all cations or anions in the formula
unit, the V is the volume of unit cell and N the
Avogadro’s number.
41
The glass is an amorphous fused silica to wich
no impurities have been added. Other common
glasses however may contain impurity ions (Na,
Ca, Al and B) to make for example soda glass
(for windows, imlk bottles...).
Figure 9.6: Quartz vs. soda glass
Biomedical Engineering
CHAPTER 9. CERAMICS AND COMPOSITES
There are also layered silicates (e.g. clays,
mica, talc...) in which the tetrahedra are connected together to form a 2-D plane. A net negative charge is associated with each (Si2 O5 )2−
unit but the negative charge is balanced by an
adjacent plane rich in positively charged cations.
This adjacent sheets are loosely bound by vdW
forces.
Figure 9.7: Layered kaolinite clay silicate
9.2.3
9.2.4
Point defects
They may be vacancies (both for cations and
anions) and interstitials which are mostly for
cations because anions are large in contrast to
the interstitial sites. There may also be a pair
of new defects: Frenkel defect which is a cation
vacancy-cation interstitial pair altogether and
the Shottky defect which is a paired set of cation
and anion vacancies. The equilibrium concenQD
tration of defects is proportional to e− kT .
When imperfections appear, electroneutrality
must be maintained. As an example we have
a cation vacancy impurity:
Polymorphic forms of carbon
Diamond: it is a tetrahedral bonding of carbon (hardest material known with a very high
Figure 9.8: Impurity correction
thermal conductivity). It has large single crystals – gemstones – or small crystals – used to
At some point, we can have a big mixture of
grind/cut other materials. The thin films of diimpurities which eventually we want to them to
amond are hard surface coatings used for cutting
be a mixture of phases. In this case, we can altools, medical devices...
ways get a phase diagram as we had with metals.
9.3
Mechanical and thermal
properties of ceramics
As ceramics are very brittle, one cannot measure
its strength by applying forces. As a solution,
there is a flexural test which consists in apGraphite: it has a layered structure of par- plying a force in the midpoint of the length of a
allel hexagonal arrays of carbon atoms held to- bar which is lying upon two scaffold points. Te
gether by weak vdW forces which makes it to measure is the amount of deflection between the
centre of mass without the force and with the
slide easily – good lubricant.
force. So then, elastic modulus can be determined by the linear-elastic behaviour quantified
by:
F L3
(b × d rect. cross section) (9.2)
δ 4bd3
F L3
E=
(circular cross section) (9.3)
δ 12πR4
E=
Furellenes and nanotubes: the furellenes
are like a soccer ball with 60 carbon atoms
whereas the nanotubes are graphite enrolled into
a tube ending with furellene hemispheres.
42
where F is the force applied, δ the midpoint deflection, L length and in rectangular cross section b and d are the lengths whereas R is the
radius in circular cross section. The flexural
Albert Marin
9.5. APPLICATIONS
strength is computed by:
3Ff L
(rect.)
2bd2
Ff L
=
(circ.)
πR3
σf s =
(9.4)
σf s
(9.5)
The location of maximum tension is in the centre
of the material, as one could have thought and
the typical values for the previous parameters
are about 350 GPa for E and from 100 to 1000
MPa for the flexural strength.
9.4
Intrachapter summary
• Bonding in ceramics: ionic mostly and or
covalent.
• Crystal structures are based on charge neutrality and cation-anion radii ratio.
• Imperfections can be:
– Atomic point: vacancy, interstitial
(cation), Frenkel or Schottky.
2. Cutting tools: grinding glass, cutting or oil
drilling, diamonds are inserted in a metal
or resin matrix to reshape other substances
by microfracturing along cleavage planes.
3. Sensors: ZrO2 is a oxygen sensor. The
principle is that it increases diffusion rate
of oxygen to produce rapid response of sensor signal to changes in oxygen concentration. Adding Ca2+ impurity to ZrO2 increases both O2− vacancies and diffusion
rate. The operation of this sensor is by
a voltage difference: it is produced when
oxygen ions diffuse from the external surface through the sensor to the reference gas
surface. The magnitude of voltage difference is proportional to the partial pressure
of oxygen at the external surface.
– Impurities: substitutional or interstitial.
– Maintenance of charge neutrality always.
• Room temperature mechanical behaviour:
should use flexural tests. If it has a linearelastic behaviour: E, brittle fracture: σf l .
9.5
9.5.1
Applications and processing of ceramics
Applications
Schematically, one should consider the uses of
ceramics as:
1. Drawing metals: the die banks are made of
ceramics because the die blanks need wear
(desgast) ressitant properties. The surface
of the dies are made of polycrystalline diamond particles cemented in tugsten carbide
because this way the diamond gives uniform
hardness in all directions to reduce wear.
43
4. Refractors: they are used when it is needed
to be at high temperatures, like up to
1400◦ C (e.g. in high temperature furnaces).
5. Advanced ceramics for automobile engines:
they have both advantages and disadvantages: high temperature efficiency, low frictional losses, operate without a cooling system and lower weights than current engines
but they are brittle, difficult to remove internal voids and are hard to form and machine. However, Si3 N4 , SiC, ZrO2 are
good materials to make engine blocks and
piston coatings, for example.
6. Advanced ceramics for ceramic armour :
they have an outer facing plates and a backing sheet because it needs different properties: on the outside it is needed a hard,
brittle and high-velocity projectile at fracture materials (Al2 O3 , B4 C, SiC, T iB2 )
whereas in the interior we need soft and
ductile materials so as to deform and absorb the remaining energy (Al, synthetic fibre laminates...).
Biomedical Engineering
9.5.2
CHAPTER 9. CERAMICS AND COMPOSITES
Fabrication
In addition to that, one can heat-treat the glass:
the annealing removes internal stresses caused
Glass forming: pressing, blowing and fibre
by uneven cooling whereas tempering puts the
drawing
outer surfaces into compression and the inner
The process of blowing blass bottles is rarely one into tension so, when broken, it crumbles
unknown, as well as the pressing (the same as into small granular chunks instead of splintering
the other but pushing with a presser instead of into jagged shards as the annealed glass.
air) or the fibre drawing. The way to produce
a sheet of glass is by a continuous casting as we Particulate forming: hydroplastic, slip
can see below:
casting, powder pressing and tape casting
Hola
Cementation
Hola
9.6
The properties of the glass can be compared with
crystalline materials:
1. Specific volume ( ρ1 ) vs. Temperature. Crystalline materials crystallize at melting temperature and have an abrupt change in specific volume at this temperature, whereas
glasses do not crystallize, have a glass transition temperature as slope and are transparent.
2. Viscosity. It relates the shear stress (τ ) with
dv
the speed gradient ( dy
). If one plots the
relation of ν vs. T one can easily see that
it decreases with T:
44
Composite materials
CHAPTER
TEN
COMODÍÍÍÍN LURU
Lorem ipsum dolor sit amet, consectetuer
adipiscing elit. Ut purus elit, vestibulum ut,
placerat ac, adipiscing vitae, felis. Curabitur
dictum gravida mauris. Nam arcu libero, nonummy eget, consectetuer id, vulputate a, magna.
Donec vehicula augue eu neque. Pellentesque
habitant morbi tristique senectus et netus et
malesuada fames ac turpis egestas. Mauris ut
leo. Cras viverra metus rhoncus sem. Nulla et
lectus vestibulum urna fringilla ultrices. Phasellus eu tellus sit amet tortor gravida placerat. Integer sapien est, iaculis in, pretium quis, viverra
ac, nunc. Praesent eget sem vel leo ultrices
bibendum. Aenean faucibus. Morbi dolor nulla,
malesuada eu, pulvinar at, mollis ac, nulla. Curabitur auctor semper nulla. Donec varius orci
eget risus. Duis nibh mi, congue eu, accumsan
eleifend, sagittis quis, diam. Duis eget orci sit
amet orci dignissim rutrum.
Nam dui ligula, fringilla a, euismod sodales,
sollicitudin vel, wisi. Morbi auctor lorem non
justo. Nam lacus libero, pretium at, lobortis
vitae, ultricies et, tellus. Donec aliquet, tortor sed accumsan bibendum, erat ligula aliquet
magna, vitae ornare odio metus a mi. Morbi
ac orci et nisl hendrerit mollis. Suspendisse ut
massa. Cras nec ante. Pellentesque a nulla.
Cum sociis natoque penatibus et magnis dis parturient montes, nascetur ridiculus mus. Aliquam
tincidunt urna. Nulla ullamcorper vestibulum
turpis. Pellentesque cursus luctus mauris.
Nulla malesuada porttitor diam. Donec felis erat, congue non, volutpat at, tincidunt tristique, libero. Vivamus viverra fermentum felis.
Donec nonummy pellentesque ante. Phasellus
adipiscing semper elit. Proin fermentum massa
ac quam. Sed diam turpis, molestie vitae, placerat a, molestie nec, leo. Maecenas lacinia. Nam
ipsum ligula, eleifend at, accumsan nec, suscipit
a, ipsum. Morbi blandit ligula feugiat magna.
Nunc eleifend consequat lorem. Sed lacinia nulla
vitae enim. Pellentesque tincidunt purus vel
magna. Integer non enim. Praesent euismod
nunc eu purus. Donec bibendum quam in tellus. Nullam cursus pulvinar lectus. Donec et
45
mi. Nam vulputate metus eu enim. Vestibulum
pellentesque felis eu massa.
Quisque ullamcorper placerat ipsum. Cras
nibh. Morbi vel justo vitae lacus tincidunt ultrices. Lorem ipsum dolor sit amet, consectetuer
adipiscing elit. In hac habitasse platea dictumst.
Integer tempus convallis augue. Etiam facilisis. Nunc elementum fermentum wisi. Aenean
placerat. Ut imperdiet, enim sed gravida sollicitudin, felis odio placerat quam, ac pulvinar
elit purus eget enim. Nunc vitae tortor. Proin
tempus nibh sit amet nisl. Vivamus quis tortor
vitae risus porta vehicula.
Fusce mauris. Vestibulum luctus nibh at lectus. Sed bibendum, nulla a faucibus semper, leo
velit ultricies tellus, ac venenatis arcu wisi vel
nisl. Vestibulum diam. Aliquam pellentesque,
augue quis sagittis posuere, turpis lacus congue
quam, in hendrerit risus eros eget felis. Maecenas eget erat in sapien mattis porttitor. Vestibulum porttitor. Nulla facilisi. Sed a turpis eu
lacus commodo facilisis. Morbi fringilla, wisi
in dignissim interdum, justo lectus sagittis dui,
et vehicula libero dui cursus dui. Mauris tempor ligula sed lacus. Duis cursus enim ut augue. Cras ac magna. Cras nulla. Nulla egestas. Curabitur a leo. Quisque egestas wisi eget
nunc. Nam feugiat lacus vel est. Curabitur consectetuer.
Suspendisse vel felis. Ut lorem lorem, interdum eu, tincidunt sit amet, laoreet vitae, arcu.
Aenean faucibus pede eu ante. Praesent enim
elit, rutrum at, molestie non, nonummy vel, nisl.
Ut lectus eros, malesuada sit amet, fermentum
eu, sodales cursus, magna. Donec eu purus.
Quisque vehicula, urna sed ultricies auctor, pede
lorem egestas dui, et convallis elit erat sed nulla.
Donec luctus. Curabitur et nunc. Aliquam dolor
odio, commodo pretium, ultricies non, pharetra
in, velit. Integer arcu est, nonummy in, fermentum faucibus, egestas vel, odio.
Sed commodo posuere pede. Mauris ut est.
Ut quis purus. Sed ac odio. Sed vehicula hendrerit sem. Duis non odio. Morbi ut dui. Sed accumsan risus eget odio. In hac habitasse platea
Biomedical Engineering
CHAPTER 10. YURU
dictumst. Pellentesque non elit. Fusce sed justo
eu urna porta tincidunt. Mauris felis odio, sollicitudin sed, volutpat a, ornare ac, erat. Morbi
quis dolor. Donec pellentesque, erat ac sagittis
semper, nunc dui lobortis purus, quis congue purus metus ultricies tellus. Proin et quam. Class
aptent taciti sociosqu ad litora torquent per
conubia nostra, per inceptos hymenaeos. Praesent sapien turpis, fermentum vel, eleifend faucibus, vehicula eu, lacus.
Pellentesque habitant morbi tristique senectus
et netus et malesuada fames ac turpis egestas.
Donec odio elit, dictum in, hendrerit sit amet,
egestas sed, leo. Praesent feugiat sapien aliquet
odio. Integer vitae justo. Aliquam vestibulum
fringilla lorem. Sed neque lectus, consectetuer
at, consectetuer sed, eleifend ac, lectus. Nulla
facilisi. Pellentesque eget lectus. Proin eu metus. Sed porttitor. In hac habitasse platea dictumst. Suspendisse eu lectus. Ut mi mi, lacinia
sit amet, placerat et, mollis vitae, dui. Sed ante
tellus, tristique ut, iaculis eu, malesuada ac, dui.
Mauris nibh leo, facilisis non, adipiscing quis, ultrices a, dui.
Morbi luctus, wisi viverra faucibus pretium,
nibh est placerat odio, nec commodo wisi enim
eget quam. Quisque libero justo, consectetuer
a, feugiat vitae, porttitor eu, libero. Suspendisse sed mauris vitae elit sollicitudin malesuada. Maecenas ultricies eros sit amet ante. Ut
venenatis velit. Maecenas sed mi eget dui varius euismod. Phasellus aliquet volutpat odio.
Vestibulum ante ipsum primis in faucibus orci
luctus et ultrices posuere cubilia Curae; Pellentesque sit amet pede ac sem eleifend consectetuer. Nullam elementum, urna vel imperdiet sodales, elit ipsum pharetra ligula, ac
pretium ante justo a nulla. Curabitur tristique
arcu eu metus. Vestibulum lectus. Proin mauris. Proin eu nunc eu urna hendrerit faucibus.
Aliquam auctor, pede consequat laoreet varius,
eros tellus scelerisque quam, pellentesque hendrerit ipsum dolor sed augue. Nulla nec lacus.
Suspendisse vitae elit. Aliquam arcu neque,
ornare in, ullamcorper quis, commodo eu, libero.
Fusce sagittis erat at erat tristique mollis. Maecenas sapien libero, molestie et, lobortis in, sodales eget, dui. Morbi ultrices rutrum lorem.
Nam elementum ullamcorper leo. Morbi dui.
Aliquam sagittis. Nunc placerat. Pellentesque
tristique sodales est. Maecenas imperdiet lacinia
velit. Cras non urna. Morbi eros pede, suscipit ac, varius vel, egestas non, eros. Praesent
malesuada, diam id pretium elementum, eros
sem dictum tortor, vel consectetuer odio sem sed
wisi.
Sed feugiat. Cum sociis natoque penatibus et
magnis dis parturient montes, nascetur ridiculus
46
mus. Ut pellentesque augue sed urna. Vestibulum diam eros, fringilla et, consectetuer eu, nonummy id, sapien. Nullam at lectus. In sagittis
ultrices mauris. Curabitur malesuada erat sit
amet massa. Fusce blandit. Aliquam erat volutpat. Aliquam euismod. Aenean vel lectus.
Nunc imperdiet justo nec dolor.
Etiam euismod. Fusce facilisis lacinia dui.
Suspendisse potenti. In mi erat, cursus id, nonummy sed, ullamcorper eget, sapien. Praesent
pretium, magna in eleifend egestas, pede pede
pretium lorem, quis consectetuer tortor sapien
facilisis magna. Mauris quis magna varius nulla
scelerisque imperdiet. Aliquam non quam. Aliquam porttitor quam a lacus. Praesent vel
arcu ut tortor cursus volutpat. In vitae pede
quis diam bibendum placerat. Fusce elementum
convallis neque. Sed dolor orci, scelerisque ac,
dapibus nec, ultricies ut, mi. Duis nec dui quis
leo sagittis commodo.
Aliquam lectus.
Vivamus leo.
Quisque
ornare tellus ullamcorper nulla. Mauris porttitor pharetra tortor. Sed fringilla justo sed mauris. Mauris tellus. Sed non leo. Nullam elementum, magna in cursus sodales, augue est
scelerisque sapien, venenatis congue nulla arcu
et pede. Ut suscipit enim vel sapien. Donec
congue. Maecenas urna mi, suscipit in, placerat
ut, vestibulum ut, massa. Fusce ultrices nulla et
nisl.
Etiam ac leo a risus tristique nonummy.
Donec dignissim tincidunt nulla. Vestibulum
rhoncus molestie odio. Sed lobortis, justo et
pretium lobortis, mauris turpis condimentum
augue, nec ultricies nibh arcu pretium enim.
Nunc purus neque, placerat id, imperdiet sed,
pellentesque nec, nisl. Vestibulum imperdiet
neque non sem accumsan laoreet.
In hac
habitasse platea dictumst. Etiam condimentum
facilisis libero. Suspendisse in elit quis nisl aliquam dapibus. Pellentesque auctor sapien. Sed
egestas sapien nec lectus. Pellentesque vel dui
vel neque bibendum viverra. Aliquam porttitor
nisl nec pede. Proin mattis libero vel turpis.
Donec rutrum mauris et libero. Proin euismod
porta felis. Nam lobortis, metus quis elementum commodo, nunc lectus elementum mauris,
eget vulputate ligula tellus eu neque. Vivamus
eu dolor.
Nulla in ipsum. Praesent eros nulla, congue
vitae, euismod ut, commodo a, wisi. Pellentesque habitant morbi tristique senectus et netus
et malesuada fames ac turpis egestas. Aenean
nonummy magna non leo. Sed felis erat, ullamcorper in, dictum non, ultricies ut, lectus. Proin
vel arcu a odio lobortis euismod. Vestibulum
ante ipsum primis in faucibus orci luctus et ultrices posuere cubilia Curae; Proin ut est. Aliquam
Albert Marin
odio. Pellentesque massa turpis, cursus eu, euismod nec, tempor congue, nulla. Duis viverra
gravida mauris. Cras tincidunt. Curabitur eros
ligula, varius ut, pulvinar in, cursus faucibus,
augue.
Nulla mattis luctus nulla. Duis commodo
velit at leo. Aliquam vulputate magna et leo.
Nam vestibulum ullamcorper leo. Vestibulum
condimentum rutrum mauris. Donec id mauris. Morbi molestie justo et pede. Vivamus
eget turpis sed nisl cursus tempor. Curabitur
mollis sapien condimentum nunc. In wisi nisl,
malesuada at, dignissim sit amet, lobortis in,
odio. Aenean consequat arcu a ante. Pellentesque porta elit sit amet orci. Etiam at turpis
nec elit ultricies imperdiet. Nulla facilisi. In hac
habitasse platea dictumst. Suspendisse viverra
aliquam risus. Nullam pede justo, molestie nonummy, scelerisque eu, facilisis vel, arcu.
Curabitur tellus magna, porttitor a, commodo
a, commodo in, tortor. Donec interdum. Praesent scelerisque. Maecenas posuere sodales odio.
Vivamus metus lacus, varius quis, imperdiet
quis, rhoncus a, turpis. Etiam ligula arcu, elementum a, venenatis quis, sollicitudin sed, metus. Donec nunc pede, tincidunt in, venenatis
vitae, faucibus vel, nibh. Pellentesque wisi. Nullam malesuada. Morbi ut tellus ut pede tincidunt porta. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Etiam congue neque id
dolor.
Donec et nisl at wisi luctus bibendum. Nam
interdum tellus ac libero. Sed sem justo, laoreet
vitae, fringilla at, adipiscing ut, nibh. Maecenas non sem quis tortor eleifend fermentum.
Etiam id tortor ac mauris porta vulputate. Integer porta neque vitae massa. Maecenas tempus libero a libero posuere dictum. Vestibulum
ante ipsum primis in faucibus orci luctus et ultrices posuere cubilia Curae; Aenean quis mauris
sed elit commodo placerat. Class aptent taciti
sociosqu ad litora torquent per conubia nostra,
per inceptos hymenaeos. Vivamus rhoncus tincidunt libero. Etiam elementum pretium justo.
Vivamus est. Morbi a tellus eget pede tristique
commodo. Nulla nisl. Vestibulum sed nisl eu
sapien cursus rutrum.
Nulla non mauris vitae wisi posuere convallis. Sed eu nulla nec eros scelerisque pharetra.
Nullam varius. Etiam dignissim elementum metus. Vestibulum faucibus, metus sit amet mattis rhoncus, sapien dui laoreet odio, nec ultricies
nibh augue a enim. Fusce in ligula. Quisque at
magna et nulla commodo consequat. Proin accumsan imperdiet sem. Nunc porta. Donec feugiat mi at justo. Phasellus facilisis ipsum quis
ante. In ac elit eget ipsum pharetra faucibus.
Maecenas viverra nulla in massa.
47
Nulla ac nisl. Nullam urna nulla, ullamcorper
in, interdum sit amet, gravida ut, risus. Aenean
ac enim. In luctus. Phasellus eu quam vitae
turpis viverra pellentesque. Duis feugiat felis ut
enim. Phasellus pharetra, sem id porttitor sodales, magna nunc aliquet nibh, nec blandit nisl
mauris at pede. Suspendisse risus risus, lobortis eget, semper at, imperdiet sit amet, quam.
Quisque scelerisque dapibus nibh. Nam enim.
Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Nunc ut metus. Ut metus justo, auctor at, ultrices eu, sagittis ut, purus. Aliquam
aliquam.