ME 330 Engineering Materials

Lecture 4
Atomic Structure and Interatomic Bonding

 Chemistry review
 Interatomic bonding in solids
 Crystalline vs. Amorphous
 Crystals and crystallographic planes

Read Chapters 2 and 3

 Why Atomic Structure?
• Atomic level structure can strongly influence
material performance
– Modulus, melting point, coefficient of expansion all
depend on interatomic forces
• Will now demonstrate how to understand properties based
on bonding potentials
• Different bond types have different potentials

• Constructionist approach
– Look at most basic level to begin our understanding
– Today we’ll look (in detail) at:
• How atoms pack together
• How atoms bond together
• See how these effect macroscopic properties
 ATOMIC STRUCTURE AND BONDING

Why study it? Carbon (Diamond & Graphite)

Many properties of materials depend on

(i) bonds between atoms

(ii) atomic packing (arrangement)

NUCLEUS = PROTONS + NEUTRONS

ATOM +
ELECTRONS ( = no. of protons for neutrality)

Protons: + charge, neutrons: neutral charge, electrons: negative charge

Quantum mechanics – establishment of a set of
principles and laws that govern systems of atomic and
subatomic entities.

Models of atomic behavior:

Bohr atomic model – electrons revolve around

atomic nucleus in discrete orbitals.

Wave-mechanical model – electron exhibits both

wave-like and particle-like behavior. Position of
electron is defined by probability of electron’s being at
various locations around nucleus.
Nucleus 10-14m. diameter surrounded by electron
cloud. Atomic diameter  10-10 m
~ 99.98% of mass is in nucleus & most of volume is
electron cloud.

Bohr atomic model –

electrons are assumed to revolve
around nucleus in discrete orbitals
Electrons in ORBITALS or shells, characterized by four
QUANTUM numbers
– Size (K,L,M…) (shells; specified by a principal quantum
number n=1,2,3,…)
- Shape (s,p,d,f) (subshells – different shapes of electron orbits
in a shell; second quantum numbers)
- Spatial orientation (ml) (number of energy states for each
subshell; third quantum number)
- Spin (ms) (spin moment, oriented either up or down; fourth
quantum number)

See Table 2.1

Outermost shell contains VALENCE electrons (bonding,
chemical, electrical and thermal properties). These are of
most importance to us.

If outer shell is complete, i.e. the S &

P orbitals are full (S2P6 = 8 electrons)
then element is very stable and very
un-reactive - Noble gases (helium,
neon, argon, krypton). Some other
elements gain or lose electrons to try
and attain this stable configuration
through bonding.

Note: s, p, d, f subshells can accommodate

the total of 2, 6, 10, and 14 electrons, respectively
 SURVEY OF ELEMENTS
• Most elements: Electron configuration not stable.
Element Atomic # Electron configuration
Hydrogen 1 1s 1
Helium 2 1s 2 (stable)
Lithium 3 1s 2 2s 1
Beryllium 4 1s 2 2s 2
Boron 5 1s 2 2s 2 2p 1
Carbon 6 1s 2 2s 2 2p 2
... ...
Neon 10 1s 2 2s 2 2p 6 (stable)
Sodium 11 1s 2 2s 2 2p 6 3s 1
Magnesium 12 1s 2 2s 2 2p 6 3s 2
Aluminum 13 1s 2 2s 2 2p 6 3s 2 3p 1
... ...
Argon 18 1s 2 2s 2 2p 6 3s 2 3p 6 (stable)
... ... ...
Krypton 36 1s 2 2s 2 2p 6 3s 2 3p 6 3d 10 4s 2 4 6 (stable)

• Why? Valence (outer) shell usually not filled completely.

PERIODIC TABLE
Table of elements (types of atoms)

Atomic Number - number of protons

Hydrogen 1 proton
Helium 2 “ etc.

Atomic Mass - relative atomic mass

- mass of 6.023 x 1023 atoms of that element
(6.023 x 1023 of something is 1 mole)

1 mole of aluminium atoms (i.e. 6.023 x 1023

atoms) has a mass of 26.98 g etc.
 THE PERIODIC TABLE
• Columns: Similar Valence Structure
give up 1e

inert gases
give up 2e

accept 2e
accept 1e
Metal
give up 3e
Nonmetal
H He
Li Be Intermediate Ne
O F
Na Mg S Cl Ar Adapted from
Fig. 2.6,
K Ca Sc Se Br Kr Callister 6e.

Rb Sr Y Te I Xe
Cs Ba Po At Rn
Fr Ra

Electropositive elements: Electronegative elements:

Readily give up electrons Readily acquire electrons
to become +ve ions. to become -ve ions.
Most elements in Periodic Table are METALS. e.g.
Mg, Zn, Fe, Ti, Pd.
Few gases and non-metals and some in between.

CERAMICS are usually compounds based on mixtures

of elements Cr2O3 , Al2O3, Si3N4, SiC.

POLYMERS are usually based on CARBON chains /

networks.

Sizes of atoms can be important, i.e. Diffusion in Solids.

ELECTROPOSITIVE - metallic elements

give up outer electrons to form positive ions CATIONS
Mg  Mg2+ + 2e-
 ELECTRONEGATIVITY
• Ranges from 0.7 to 4.0,
• Large values: tendency to acquire electrons.
H He
2.1 -
Li Be F Ne
1.0 1.5 4.0 -
Na Mg Cl Ar
0.9 1.2 3.0 -
K Ca Ti Cr Fe Ni Zn As Br Kr
0.8 1.0 1.5 1.6 1.8 1.8 1.8 2.0 2.8 -
Rb Sr I Xe
0.8 1.0 2.5 -
Cs Ba At Rn
0.7 0.9 2.2 -
Fr Ra
0.7 0.9

Smaller electronegativity Larger electronegativity

ATOMIC BONDING
ATOMS bond to each other to reduce their overall
energy, i.e. to become more stable.

Everything tends towards a state of lower free

energy.

Bonding Forces and Energies

Inter-atomic spacing is caused by balance between
REPULSIVE and ATTRACTIVE forces.

Attractive force depends on type of bond trying to form

between atoms;
Repulsive force occurs when atoms get close together.

Net force between atoms is balance of two forces and

depends on inter-atomic distance.
FN = FA + FR
FN = 0 when FA = FR

FN : net force
 F N = FA + F R

Equilibrium is reached when: FA + FR = 0

Atoms happily sit this distance apart (r0) (often ro  0.3nm)

Also considered in energy terms: EN = EA + ER

In this case, equilibrium is reached when overall energy is

a minimum.

Bonding energy, E0 (binding energy) is the energy required

to break the bond (separate two atoms).
Higher bonding energy Stronger bonds  higher
strength & Melting point, (Tm)

Also Stiffness (slope (dF/dr) at r0 and thermal expansion

(trough of E curve)
PRIMARY ATOMIC BONDS
(Chemical) - STRONG

IONIC
COVALENT
METALLIC

SECONDARY BONDS
(Physical) - WEAK

Van Der Waals bonds/forces

Fluctuating + permanent dipoles
IONIC BONDS

Form between electropositive (metallic) and electronegative

(non-metallic) elements,
eg. CERAMICS NaCl, Al2O3,MgO

• Na looses outer electron to be more stable  Na+.

• Chlorine accepts extra electron to be more stable  Cl-
(Note: there is a size change when atoms form ions.)

After such a transfer, the chlorine atom has net negative

charge, while sodium atom has net positive charge. In sodium
chloride (NaCl), all sodium and chloride atoms exist as ions.
IONIC BONDS (cont)

Form between electropositive (metallic) and electronegative

(non-metallic) elements,

• Na loses outer electron to be more stable  Na+.

• Chlorine accepts extra electron to be more stable  Cl-

Opposite charges attract so get:

ELECTROSTATIC (coulombic) BONDING

A B
EA   and E R  n
r r

EA = Attractive energy, ER = Repulsive energy,

A, B and n are constants that depend on system (n  8).
 THE PERIODIC TABLE
• Columns: Similar Valence Structure
give up 1e

inert gases
give up 2e

accept 2e
accept 1e
Metal
give up 3e
Nonmetal
H He
Li Be Intermediate Ne
O F
Na Mg S Cl Ar Adapted from
Fig. 2.6,
K Ca Sc Se Br Kr Callister 6e.

Rb Sr Y Te I Xe
Cs Ba Po At Rn
Fr Ra

Electropositive elements: Electronegative elements:

Readily give up electrons Readily acquire electrons
to become +ve ions. to become -ve ions.
Example: NaCl

Na (metal) Cl (nonmetal)
unstable unstable
electron

Na (cation)
+ - Cl (anion)
stable Coulombic stable
Attraction
 Ionic Bonding
Metal Nonmetal
Na Cl
Atomic
Structure

Na+ Cl-

Ions

NaCl
Ionic
Bond
 Notes on Ionic Bonding
Repulsion Curve

• To be stable, all positive ions must be ER 
near negative ions r8 Separate Ions
• Bond strength is equal in all directions
(nondirectional) 0 Atoms
• Energy considerations

Energy (r)
– Coulombic attractive force
Electrostatic Attraction

1 EA  
Repulsiveforce:
r
– A  “Stable”
r
r
 
• 1
Generally, very high    8 Cl- Na+ Cl-

R  n (nbonding
 8) energies r r Na+ Cl- Na+
• r
Typically hard, brittle, thermally and
electrically insulative Cl- Na+ Cl-

• Ceramics
 EXAMPLES: IONIC BONDING

• Predominant bonding in Ceramics

NaCl
MgO
H He
2.1 CaF 2 -
Li Be O F Ne
1.0 1.5 Cs Cl 3.5 4.0 -
Na Mg Cl Ar
0.9 1.2 3.0 -
K Ca Ti Cr Fe Ni Zn As Br Kr
0.8 1.0 1.5 1.6 1.8 1.8 1.8 2.0 2.8 -
Rb Sr I Xe
0.8 1.0 2.5 -
Cs Ba At Rn
0.7 0.9 2.2 -
Fr Ra
0.7 0.9

Give up electrons Acquire electrons

NON-DIRECTIONAL, electrical neutrality is most
important.
IONS pack to maintain neutrality.

Eg. For NaCl

For each Na+ ion there must also be a Cl- ion.

Likewise, for MgCl2 there must be two Chlorine

ions for every magnesium ion.

Ionic bonds tend to be strong bonds - high

bonding energy. (Table 2.3)

Ceramics are usually ionically bonded and have

high melting points, high hardness, brittle and
electrically and thermally insulative (atoms and
electrons cannot move easily).
COVALENT BONDING
Atoms SHARE outer electrons with each other to attain noble
gas electron configurations.
Atoms close to each other in periodic table and in
electronegativities (X) tend to form covalent bonds

Covalent bonds do not distort very easily - so can be very

strong (Diamond) but appear in "weak" materials as well
(polyethylene - covalently bonded carbon chain)

Some materials show mixed Ionic/Covalent bonding.

 
% ionicity  1  exp  0.25( X A  X B ) 2 x100
XA, XB electronegativities for respective elements
 COVALENT BONDING

• Requires shared electrons

• Example: CH4 shared electrons
H
C: has 4 valence e, from carbon atom
CH 4
needs 4 more
H: has 1 valence e, H C H
needs 1 more
shared electrons
Electronegativities H from hydrogen
are comparable. atoms

Because atoms in covalent bonds have to share electrons

with other atoms, Direction is very important.
DIRECTIONAL BONDING e.g. DIAMOND
 Covalent Bonding
Cl Cl2
H C H CH4

H H

• Two atoms share electrons - extra electron belongs to both

• Bonding is directional - between atoms being bonded
• Many interatomic bonds are partially ionic and covalent
– Wider separation in periodic table  more ionic
• Ceramics, Metals, Polymer backbones
 EXAMPLES: COVALENT BONDING
H2 O

column IVA
H2 F2
C(diamond)
H He
2.1
Si C - Cl 2
Li Be C O F Ne
1.0 1.5 2.5 2.0 4.0 -
Na Mg Si Cl Ar
0.9 1.2 1.8 3.0 -
K Ca Ti Cr Fe Ni Zn Ga Ge As Br Kr
0.8 1.0 1.5 1.6 1.8 1.8 1.8 1.6 1.8 2.0 2.8 -
Rb Sr Sn I Xe
0.8 1.0 1.8 2.5 -
Cs Ba Pb At Rn
0.7 0.9 1.8 2.2 -
Fr Ra
0.7 0.9 GaAs
• Molecules with nonmetals
• Molecules with metals and nonmetals
• Elemental solids (RHS of Periodic Table)
• Compound solids (about column IVA)
 Covalent Potential

Repulsion Curve

Energy


(r)
ER 
• Widely variable rn
properties
0
– Diamond Atoms
• Hardest substance known
Electron Overlap Attraction
• Very stiff, strong 
EA  
• Tmelt = 3550 ºC rm
– Bismuth
r
• Very soft
 
• Weak  m
 n m  n 
r r
• Tmelt = 270 ºC
– Based on m & n
METALLIC BONDING- Found in metals and alloys.

Atoms of metal pack relatively closely together in

ordered arrangement - Ion cores
Valence electrons form "sea" in between cores -
"electron gas or cloud"
These electrons can move/drift - thermal/electrical
conduction. FREE electrons.

+ + + • Arises from a sea of

donated valence electrons
(1, 2, or 3 from each atom).
+ + +

+ + +
 METALLIC BONDING

Non-directional
Not many restrictions on metallic bond (no charge
neutrality - ionic, or electron-pair sharing -
covalent) so if metal deformed, atom positions
can move relatively large amounts without
breaking bonds. (Ductility)

Bonding energies affect melting points and vary

from low (-39 C) to high (3410C) values.

• Primary bond for metals and their alloys

 Metallic Bonding

e - e-
M+
M+
M+ M+ M+ M+

e- e
-
e-
e -
M+ M+ M+ M+ M+ M+
e- e- e-
M+ M+ M+ M+ M+ M+
e -
e-

• Ion Cores (M+)- net positive charge equal to total valence

• Valence electrons (e-) drift through metal in “electron cloud”
– Electrically shield ion cores
– Physically hold cores together
• Nondirectional bond
• Metallic bonding potential similar to covalent (use same eqn.)
• Wide variety of bonding energies and hence properties
• Excellent conductors due to mobility of electron cloud
• Metals and metallic alloys
SECONDARY BONDING - Van Der Waal's
forces (in biological systems)

Low energy - weak bonds 4 - 40 kJmol-1

(Primary 100  1500 kJmol-1)

Based on DIPOLES

When -ve and +ve charges are separated, an

electric dipole moment is set up.

FLUCTUATING INDUCED DIPOLE BONDS

Asymmetrical distribution of electron cloud
(vibrations etc)
e.g. noble gases - boiling, melting.
 Van der Waals Bonding
O  
H H    6  n (n  12)
O O
r r
H H H H

• Sometimes called physical bonds to contrast with chemical (primary)

• Much lower energy than primary bonds
• Arise from electric dipoles -
– Separation of + and - portions of atom - much weaker than ions
– Bonding from attraction of + from one dipole to - of other dipole
• Hydrogen bonding is special case when hydrogen is present
– Strongest secondary bonding type
• Polymeric interchain bonds
POLAR MOLECULE-INDUCED DIPOLE BONDS
Asymmetric charge distribution in some molecules
(polar)
Eg. HCl molecule. Can attract non-polar molecules.

PERMANENT DIPOLE BONDS

Van der Waals forces will also exist between adjacent
polar molecules.
H-F, H-O and H-N bonds
Hydrogen end becomes very +ve.

One of strongest secondary bonds.

eg. H2O. (Hydrogen bonding - Reason for high boiling
point of water.)
Also between carbon chains in polymeric materials.
• Fluctuating dipoles

asymmetric electron ex: liquid H 2

clouds H2 H2

+ - secondary + - H H H H
secondary
bonding bonding

• Permanent dipoles-molecule induced

secondary
-general case: + - bonding
+ -

secondary
-ex: liquid HCl H Cl bonding H Cl
second
-ex: polymer ary bondin
g
 PROPERTIES FROM BONDING: TM
• Bond length, r • Melting Temperature, Tm
F
F Energy (r)
r

• Bond energy, Eo ro
r
Energy (r)
smaller T m
unstretched length
ro larger T m
r
Eo = Tm is larger if Eo is larger.
“bond energy”
 PROPERTIES FROM BONDING: E
• Elastic modulus, E cross
sectional
length, Lo
area A o
Elastic modulus
undeformed F L
L =E
Ao Lo
deformed F

• E ~ curvature at ro
Energy

unstretched length
ro E is larger if Eo is larger.
r
smaller Elastic Modulus

larger Elastic Modulus

 PROPERTIES FROM BONDING: 
• Coefficient of thermal expansion, 
length, Lo coeff. thermal expansion
unheated, T 1
L L
= (T2 -T1 )
heated, T 2 Lo

•  ~ symmetry at ro
Energy

ro
r  is larger if Eo is smaller.
larger 

small er 
 SUMMARY: PRIMARY BONDS
Ceramics Large bond energy
(Ionic & covalent bonding): large Tm
large E
small 

Metals Variable bond energy

(Metallic bonding): moderate Tm
moderate E
moderate 

Polymers Directional Properties

(Covalent & Secondary): Secondary bonding dominates
small T
second
ary bondin small E
g
large 
Summary - Atomic Bonding in Solids
• Primary
– Ionic
– Covalent
– Metallic
• Secondary Force ro
– Van der Waals Energy
– Hydrogen d (r)
• Interatomic potential energies F( r ) 
dr
– Function of separation, r r
– Attractive - depends on bond
– Repulsive - atomic scale overlap
• Bonding energy (Eo) is strongly dependent
on bond type
– Effect on modulus ??? Eo
– Effect on thermal expansion ???
 Atomistic Origins of Properties
Force dF 
E   
F(r)  dr  r  r•o Modulus
– Proportional to slope of force-
separation curve at
Atomic separation, r equilibrium separation
distance
• Melting Temperature
– Large Eo leads to high Tmelt
Energy • Coefficient of thermal
(r) expansion
– Large Eo leads to small 
– Deep narrow trough forces
r
E0 large energy change for small
dimensional change
 Potentials & Properties
Atomic Separation
0 Material E (GPa)
0 2 4 6 8 10

-0.1
Silicon Carbide 475
Ceramics Alumina 375
-0.2 Glass 70
Ionic
n=2,m=8
Steel 210
~~Potential

-0.3
n = 2, m = 6 Metals Brass 97
Secondary

-0.4
Aluminum 69
PVC 3.3
-0.5
Polymers Epoxy 2.4
LDPE 0.23
-0.6

Material  (C-1x10-6)
-0.7

Relative differences in potential curves Silicon Carbide 4.1-4.6

Ceramics Alumina 7.6
Assumes  & are 1 - comparitive purposes only! Glass 9.0
Steel 12.0
Metals Brass 20
Aluminum 23.
PVC 90-180
Polymers Epoxy 81-117
LDPE 180-400

From Callister, p. 22
 Atomic Packing
• Crystalline:
– 3-D arrangement of atoms in which every atom has the same
geometrical arrangement of neighbors
– Long-range, periodic array over large length scales
– Most solids are crystalline (metals, most ceramics, some polymers)

• Amorphous
– Arrangement over which no long range order exists
– Often clear - not enough order to diffract light
– Rarely purely amorphous - have regions of crystallinity
– Many polymers and some ceramics
 Crystal Structure Definitions
• Unit cell: Smallest
repeating unit of the
crystal.

• Lattice: 3–D framework of

a crystal where atoms are
located

• Lattice parameters:
Dimensions (a,b,c) and
angles (,,) of the lattice c  


b
a
 Bravais Lattices

• French Crystallographer Bravais (1848)

• 7 crystal systems using primitive unit cells
• Primitive - one lattice point at origin
• 14 distinguishable point lattices
– P - simple
– F - face centered
– I - body centered Tetragonal

– C - base centered
• For now, interested in BCC, FCC, HCP FCC Monoclinic
– Metallic crystal structures Rhombohedral
– Metallic bond is non-directional
– No restriction on nearest neighbors
– Very dense packing
BCC
• First need to collect some definitions Cubic Hexagonal Orthorhomic Triclinic
HCP
 Crystallographic Directions
• Determining direction indices z
– Start vector at crystal axis [001]
– Draw to any point in the 3-D crystal
[111]
– Project vector on each xyz axes
• measure a in x-direction
• measure b in y-direction a=1
• measure c in z-direction y
b=½
– Multiply by common factor to achieve smallest
c=0 [010]
integer value
– Enclose in [ ] without commas x [100] [110]
[210]
• Negative directions indicated with -
• Family of directions indicated by < >
• Hexagonal crystals have 4 indices

In a cubic crystal, [100],[ 1 00],[010],[0 1 0],[001],[00 1 ] are

all in the <100> family.
 Crystallographic Planes
• Determining Miller indices
– Look at plane in unit cell which z
does not pass through the origin [001]

– Determine length of planar

intercept with each axes (again,
a,b,c) c = 1/3
– Take reciprocal of a,b,c b = 1/2 y
a=1 [010]
– Reduce to smallest integer value
– Enclose in ( ) without commas x [100] (123)
• Any parallel planes are equivalent
z z
1
Intercepts :  ,  1,
2
z
Re ciprocals : 0 ,  1, 2
Plane : 01 2 
y
x x (012)
-1