Material Science I
Ceramic Materials
Chapter 3: Bond Energy and Properties
F. Filser & L.J. Gauckler
ETH-Zürich, Departement Materials
frank.filser@mat.ethz.ch
HS 2007
Ceramics: Bond Energy and Properties, Chap 3
1
Material Science I
Goal of this Chapter is …
to develop semiquantitative relationships between
• the properties of a ceramic material and
• the depth and shape of the energy well
Ceramics: Bond Energy and Properties, Chap 3
2
Material Science I
The Bond Energy and the Physical Properties
•
Bond forces / energy between ions or atoms composing a
solid determine a lot of its physical properties
Hence we can use the bond energy as a means to predict
physical properties
Examples: melting temperature, modulus of elasticity,
strength, hardness
•
•
•
•
This prediction works in a lot of cases but not in all.
Refinement is required for crystallized solids, i.e. effect of
Madelung, and for solids made up of mixed ionicconvalent bondings.
Ceramics: Bond Energy and Properties, Chap 3
3
Material Science I
Contents
• potential well & bond energy for ionic bonding, the equilibrium distance
• bond force as a function of the inter-ionic distance, max. force, inflexion
point.
• melting temperature and hardness for ionic bonded compounds
• limitation of the prediction by potential well (example of MgO / Al2O3)
-> introduction of covalency (of an ionic bond)
• thermal expansion explained with the potential well
• elastic modulus
• theoretic strength of compounds
Ceramics: Bond Energy and Properties, Chap 3
4
Material Science I
The Bond Energy for Ionic Type of Bonding
Enet  Eatt  Erep
z1  z2  e 2
Eatt  r  
4 0  r
repelling
Sum
E repulsion
attracting
-
Ion’s Distance
r0
+
E attraction
r0 = equilibrium distance
B
Erep  r   n
r
z1  z2  e 2 B
Enet  r  
 n
4 0  r r
Potential
Ebond
Ceramics: Bond Energy and Properties, Chap 3
z1  z2  e2  1 

1  
4 0  r0  n 
5
Material Science I
Potential and Force
as Function of Inter-Ionic Distance
150
40
20
50
0
0
x2
-50
Force [nN]
Potential [eV]
100
-20
x1
-100
x1 x2
-40
-150
0
100
200
300
400
500
600
Inter-Ionic Distance r [pm]
z1  z2  e2 B
Enet  r  
 n
4 0  r r
Ceramics: Bond Energy and Properties, Chap 3
700
800 0
100
200
300
400
500
600
700
800
Inter-Ionic Distance r [pm]
Fnet  r  
dEnet  r 
dr
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Material Science I
Comparison of Potential – Inter-Ionic Distance Curves
for NaCl, MgO, LiF
40
• MgO potential well is
much deeper than for LiF
and NaCl (ca 4x deeper)
NaCl
LiF
MgO
• LiF potential well is a bit
deeper than for NaCl.
Potential [eV]
20
r0
• Same crystal structure
(Rocksalt)
0
• Inter-Ionic Equilibrium
Distances
- NaCl r0=283 pm
- LiF r0= 209 pm
- MgO r0=212 pm
-20
Ebond
-40
0
0.5
1
1.5
2
z1  z2  e2  1 

1  
4 0  r0  n 
2.5
3
3.5
Relative Inter-Ionic Distance r/r0 [-]
Ceramics: Bond Energy and Properties, Chap 3
4
• Valencies are different
9
Material Science I
The Melting Temperature
The Bond strength Ebond
-> depends strongly on the valency and the ionic radii/distance (lattice distance).
• The bond strength Ebond of ionic bonded compounds is directly proportional the
multiplication of its ionic charges z1 and z2 and inverse proportional the
equilibrium ionic distance r0.
• The higher the valency the stronger the bond strength.
• The compounds MgO, NaCl and LiF crystallize in same lattice (fcc lattice),
and ionic character of the bond is prevailing (>60 %).
MgO
NaCl
LiF
Crystal Structure
2852°C
801°C
848°C
Rocksalt
Ceramics: Bond Energy and Properties, Chap 3
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Material Science I
Melting Temperature of some Compounds
Ionic
Distance [Å]
z1=+1, z2=-1
inter-ionic distance
increasing due to anion
radius increasing
Melting
Temperature [°C]
NaF
2.31
988
NaCl
2.81
801
NaBr
2.98
755
NaI
3.23
651
MgO
2.1
2800
CaO
2.4
2580
SrO
2.57
Comparable
BaO
2.76
!!!
LiF
2.01
824
NaF
2.311
988
KF
2.67
846
RbF
2.82
775
melting temperature
decreasing
z1=+2, z2=-2
inter-ionic distance
increasing due to cation
radius increasing
2430 decrease
1923
melting temperature
decreasing
!!!
z1=+1, z2=-1
inter-ionic distance
increasing due to cation
radius increasing
melting temperature
decreasing
The melting temperature increases as the ionic distance decreases within the lattice.
The melting temperature increases for increasing valency given about same ionic distance
Ceramics: Bond Energy and Properties, Chap 3
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Material Science I
Hardness
as function of the inter-ionic distance and the ionic charge
Compound
Ionic Distance
[Å ]
Hardness
[Mohs]
BeO
1.65
9
MgO
2.3
6.5
CaO
2.4
4.5
SrO
2.57
3.5
BaO
2.76
3.3
+F Na
NaF
2.01
3.2
2+O2Mg
MgO
2.3
6.5
3+N3ScN
Sc
2.67
7-8
4+C4TiC
Ti
2.82
8-9
z1=+2, z2=-2
inter-ionic distance
increasing due to cation
radius increasing
valency of ions
increasing & despite
inter-ionic distance
increasing
hardness
decreasing
hardness
increasing
The hardness increases with decreasing ionic distance, assuming constant ionic charges.
The hardness increases for increasing valency, despite ! increasing ionic distance.
Ceramics: Bond Energy and Properties, Chap 3
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Material Science I
The Melting Temperature of Al2O3 and MgO
Al2O3: 2054 °C
MgO: 2852 °C
}
Presumption: MgO has the lower melting temperature.
Why?
Criteria of Analysis:
• Ionic Distance
• Valency
• Bond Energy
• Lattice Energy
Ceramics: Bond Energy and Properties, Chap 3
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Material Science I
The Melting Temperature of Al2O3 and MgO
Al2O3: 2054 °C
MgO: 2852 °C
}
Presumption: MgO has the lower melting temperature.
Why?
Criteria of Analysis:
• Ionic Distance
-> r0Al2O3 = 193.5 pm, r0MgO = 212 pm
• Valency
-> (z1 x z2)Al2O3= -6, (z1 x z2)MgO= -4
• Bond Energy
E
• Lattice Energy
E
Al2O3
bond
Al2O3
Lattice
MgO
E bond  1.64
MgO
E Lattice  23.54
The analysis based on the potential well of an ionic bonded solid is often good and correct,
however not all the time!!!
Ceramics: Bond Energy and Properties, Chap 3
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Material Science I
The Melting Temperature of Al2O3 and MgO
Al2O3: 2054 °C
MgO: 2852 °C
}
We need other and better criteria !!!
Further Criterium of Analysis:
-> Type of Bond: amount of covalency in the bonds for Al2O3 is higher
than for MgO.
A measure for covalency is, for example, the difference in
electronegativity of the ions. DENAl2O3 = 1.83, DENMgO = 2.13
Ceramics: Bond Energy and Properties, Chap 3
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Material Science I
The Covalent Character of a Bond
TiO2 idealized Rutile
Tm = 1857°C
CdI2 layer structure
Tm = 387°C
CO2 molecule lattice
Tm = -57°C
MX2 stoichiom., DEN = 1.9
MX2 stoichiom., DEN = 0.97
MX2 stoichiom., DEN = 0.89
Tm = melting temp.
• The covalent character of a bond increases from the left to right.
• The network structure of the bonds changes: from a 3D structure of TiO2 (Rutile), to a
layered structure of CdI2, to a molecule lattice of CO2. The melting temperature
decrease in this direction, too.
Ceramics: Bond Energy and Properties, Chap 3
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Material Science I
What issues influence the amount of covalency
in an ionic bond?
MgO vs Al2O3
• Polarizing power of the cation
fAl3+ = 60 1/nm; fMg2+ = 31 1/nm
• Polarizibility of the anion
aeO2- equal for both cases
• Elektron configuration of the cation
no d-electrons in both cases
ideal pair of ions
(no polarization)
Ceramics: Bond Energy and Properties, Chap 3
polarized
pair of ions
high amount of polarizing sufficient
to form a covalent bond
17
Material Science I
The Thermal Expansion Coefficient
1  l 


l0  T  p
Potential Energy
a
rmin r0
rmax
max
X
ionic distance r
maximum potential energy
energy level of the thermal vibration
= mean ion density (location) for increasing temperature
Ceramics: Bond Energy and Properties, Chap 3
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Material Science I
Thermal Expansion of Chosen Ceramic Materials
• Metals possess a higher thermal
expansion than ceramic materials
• a is a function of the temperature
• The higher T the higher a
• Loosely packed, non-dense structures
(higher amount of bond covalency) may
have very small a  changement of
angle of the the bonds
Temperature °C
Ceramics: Bond Energy and Properties, Chap 3
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Material Science I
Thermal Expansion Coefficient
in case of phase transformation
Cristobalit
Cristobalite
Ceramics: Bond Energy and Properties, Chap 3
Quarz
Quartz
• a is a function of temp.
• Quartz shows one
transformation temperature.
Q. is a single crystal - the
other materials are
polycrystals.
• a of b-quartz has a negative
slope, i.e. increasing temp.
leads to smaller a(see also
ZrO2)
• Quartz has a lower a than
cristobalite because quartz
bonding can change angles,
and cristobalite bond angles
are already more straight
• SiO2 vit. : bond angles change
in all spatial directions.
20
Material Science I
Anisotrope thermal Expansion Coefficients
b-Eucryptite (LiAlSiO4) = Glass Ceramic
cold
kalt
heiss
hot
Ceramics: Bond Energy and Properties, Chap 3
21
Material Science I
Glass ceramics: Zerodur
Astro Space: Mirrors of future x-ray satelites
Micro lithography: Zerodur® components are used as movable elements in wafer-stepper and wafer-scanners.
Metrology: Because of its very low thermal expansion and its long-term stability, components made of Zerodur® will show
excellent precision in measurements instruments and metroloy.
Mechanic: Excellent machinability of Zerodur® in combination with the modern high-tech manufacturing technologies enables
complex shapes.
Further Applications: Zerodur® has good transmission properties in visible and infrared spectrum and a very good optical
homogeneity. Because of these properties Zerodur® is often used in optical systems.
http://www.schott.com/optics_devices/german/products/zerodur/?c=mL
Ceramics: Bond Energy and Properties, Chap 3
22
Material Science I
Thermal Expansion and Melting Temperature
of chosen chemical Elements
Ceramics: Bond Energy and Properties, Chap 3
23
Material Science I
Thermal Expansion and Melting Temperature
of chosen chemical Elements
0
Potential [eV]
-40
-80
-120
-160
-200
-240
0
100
200
300
400
500
600
700
800
Abstand [pm]
Ceramics: Bond Energy and Properties, Chap 3
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Material Science I
Thermal Expansion and Melting Temperature
of chosen chemical Elements
0
• The higher the
melting temperature
the deeper the
potential well.
Potential [eV]
-40
-80
• The deeper the
potential well the
more symmetric it
appears.
-120
-160
• The more symmetric
the less thermal
expansion
-200
-240
0
100
200
300
400
500
600
700
800
Abstand [pm]
Ceramics: Bond Energy and Properties, Chap 3
25
Material Science I
The Elastic Modulus of Materials
150
Fmax
100
20
50
r0
0
0
r0
Force [nN]
Potential [eV]
40
Hook’s
law
rfail
-50
Epot
-20
-100
-40
-150
0
100
200
300
400
500
600
Ionic Distance r [pm]
Enet
z1  z2  e2 B

 n
4 0  r r
Ceramics: Bond Energy and Properties, Chap 3
700
800 0
100
200
300
400
500
600
700
800
Ionic Distance r [pm]
dEnet z1  z2  e2 n  B
Fnet 

 n 1
2
dr
4 0  r
r
26
Material Science I
The Elastic Modulus of Materials
8
Kraft
Fmax
F
6
Force
4
F (r )  Enet / r 
Dx/Dy = elastic modulus,
linear elastic portion of
Hook’s law
Kraft
Force
2
0
z1 z2 e 2
4 0 r 2

nB
r n1
-2
-4
-6
-8
-10
0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8
r00
Abstand
Distance
The force – inter-ionic distance curve.
In the equilibrium point r0 a tangential line exists which in a first approximation describes
good the linear elastic behaviour of a solid under tensile force.
Ceramics: Bond Energy and Properties, Chap 3
27
Material Science I
The Elastic Modulus of Materials
  E 
F  S0  (r  r0 )
 F 
S0  


r

r  r0
S0
E
ro
1  F 
1   2 Enet 
E 
  
2 
r0  r r r0 r0  r r r
0
This result is important:
1. the stiffness (elastic modulus) of a solid is directly related to the curvature of its potential –
ionic distance curve. The curvature is inverse of the curvature radius.
2. compounds with stronger bonds have a higher elastic modulus (stiffness) than weak bonded
compounds, and
3. compounds with a high melting temperature, i.e. ceramic materials, (deep potential well) are
very stiff solids.
Ceramics: Bond Energy and Properties, Chap 3
28
Material Science I
The Elastic Modulus
Ceramics: Bond Energy and Properties, Chap 3
29
Material Science I
The Elastic Modulus
Ceramics: Bond Energy and Properties, Chap 3
30
Material Science I
Force-Distance-Curve
Ceramics: Bond Energy and Properties, Chap 3
31
Material Science I
The Theoretic Strength of Solids
- simple approximation assuming that generally bonds in solids fail at 25% elongation,
which calculates to 1.25 x r0.
2 Fmax
2 Fmax
S0 

rBruch  r0 1.25r0  r0
The (tensile) strength of an ionic bonded solid should be ~ 1/8 of the elastic modulus.
Ceramics: Bond Energy and Properties, Chap 3
32
Material Science I
The Theoretic Strength of Solids
- more sophisticated approximation generalized form of the potential – distance function
with n > m and max » Fmax/(r0)2
E
typical values for n and m, in case of ionic bonds (n = 9, m = 1) leads to
 max
E

15
The (tensile) strength of an ionic bonded solid should be ~ 1/15 of the elastic modulus.
Ceramics: Bond Energy and Properties, Chap 3
33
Material Science I
The Theoretic Strength of Solids
simple approximation:
more sophisticated approximation:
Examples:
Al2O3: bend = 330 MPa, E = 300 GPa (/ 910) (http://www.accuratus.com/)
SiC: bend = 550 MPa, E = 410 GPa (/ 745)
BN: bend = 75.8 MPa, E = 46.9 GPa (/ 620)
• The tensile strength of ionic bonded solids should be about ~ 1/10 of the elastic modulus E.
• However, we find experimentally that the strength of these materials is about
1/100 to 1/1000 x E. That is much less than our approach using the potential well predicts !!!
• There must be other issues determining the low strength than the potential well!
Ceramics: Bond Energy and Properties, Chap 3
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Material Science I
Summary
1.) the bond energy / force determines
2.) the deeper the potential well
covalency in ionic bonds
3.) the thermal expansion
deeper potential well
loose packed structures
 many physical properties of a solid, i.e.
melting temperature Tm
thermal expansion a
elastic modulus E
theoretical strength 
 the stronger the bonds
 the higher the melting temperature.
 stabilizes discrete structure elements
 lowers melting temperature lower.
 anharmonic potential well.
 smaller thermal expansion.
 smaller thermal expansion.
4.) stiffness / elastic modulus
 proportional to the curvature of the potential
Solids with stronger bonds are stiffer than solids with weaker bonds.
5.) theoretical strength should be
 ~1/10 of the elastic modulus E.
However, experimentally measured strength values are about 1/100 to 1/1000 of this value.
Ceramics: Bond Energy and Properties, Chap 3
35
Material Science I
Additional Slides
Ceramics: Bond Energy and Properties, Chap 3
36
Material Science I
Ratio of the Bond Energy of Al2O3 to MgO
Ebond
z1  z2  e2  1 

1  
4 0  r0  n 
Al2O3
Al2O3
Ebond
MgO
Ebond
 z1  z2 
 r 
  0  MgO  1.64
 z1  z2 
 r 
 0 
Al2O3: r0 = 193.5 pm, n = 7
MgO: r0 = 212 pm, n = 7
Ceramics: Bond Energy and Properties, Chap 3
37
Material Science I
Ratio of the Lattice Energy of Al2O3 to MgO
z1  z2  e
 N Av 
4 0  r0
2
ELattice
Al2O3
Lattice
MgO
Lattice
E
E
N Av  E

N Av  E
Al2O3
Ebond
 1.64
MgO
Ebond
Ceramics: Bond Energy and Properties, Chap 3
Al2O3
bond
MgO
bond
Al2O3:
MgO:
 1
1  a
 n
a
 23.54
MgO
a
Al2O3
aAl2O3 = 25.0312
aMgO = 1.7475
38
Material Science I
Determination of
„B“ (Born Constant) and „n“ Born Exponent
• at equilibrium
 Elattice 

 0
 r r  r0
r0 can be measured
N Av  z1  z2  e 2  a n  B

 n 1  0
2
4 0  r0
r0
N Av  z1  z2  e 2  a r0 n 1
B

2
4 0  r0
n
N Av  z1  z2  e 2  a  r0 n  2
B
4 0  n
• n is still unknown!
• To find n, we need to move away from equilibrium, i.e. compress the solid and
measure its compressibility
Ceramics: Bond Energy and Properties, Chap 3
39
Material Science I
Compressibility
• compressiblity is measured
• then we can calculate n
1  V 
    
V0  P T
4
4


18
r

0
0

2
a  e   n  1
• Examples: NaCl 4.18 x 10-11 1/Pa -> n = 7.7
Ceramics: Bond Energy and Properties, Chap 3
40
Material Science I
Sample calculation for NaCl
 = 8.854×10-12 SI units
e = 1.602×10–19 coulombs
a = 1.74756 (NaCl structure)
d = 5.628×10-10 m giving r0 = 2.814×10–10 m
 = 4.18×10–11 SI
n is found by
This compares to 769.4 kJ/mole experimental (2.4% error)
Ceramics: Bond Energy and Properties, Chap 3
41