Bond Energy and Physical Properties - ETH - NONMET - Nonmetallic ...

Material Science I

Ceramic Materials Chapter 3: Bond Energy and Properties

F. Filser & L.J. Gauckler ETH-Zürich, Departement Materials [email protected]

HS 2007

Ceramics: Bond Energy and Properties, Chap 3

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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


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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.


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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


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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


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 8

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 [-]


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


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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


!!!

z1=+1, z2=-1 inter-ionic distance increasing due to cation radius increasing


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


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Material Science I


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


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


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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



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)


polarized pair of ions

high amount of polarizing sufficient to form a covalent bond

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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


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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


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.

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Material Science I

Anisotrope thermal Expansion Coefficients b-Eucryptite (LiAlSiO4) = Glass Ceramic cold kalt

heiss hot


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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


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Material Science I

Thermal Expansion and Melting Temperature of chosen chemical Elements


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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]


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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]


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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


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

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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

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Material Science I

The Elastic Modulus


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Material Science I

The Elastic Modulus


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Material Science I

Force-Distance-Curve


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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.


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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

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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.


35

Material Science I

Additional Slides


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


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