Elements of Materials Science and Engineering

Elements of Materials Science and Engineering

ChE 210

Chapter 2

Chapter 2

Chapter Outline Chapter Outline

Individual atoms and ions Molecules

Macromolecules

Three dimensional bonding Interatomic distances

Generalization based on atomic bonding

Understanding of interatomic bonding is the first

step towards understanding/explaining materials

properties

Individual atoms and ions Individual atoms and ions

Atoms = nucleus (protons and neutrons) + electrons Charges:

Electrons and protons have negative and positive charges of the same magnitude, 1.6 × 10^-19 Coulombs. Neutrons are electrically neutral.

Masses:

Protons and Neutrons have the same mass, 1.67 × 10-27 kg. Mass of an electron is much smaller, 9.11 × 10^-31 kg and can be neglected in calculation of atomic mass.

# protons gives chemical identification of the element

# protons = atomic number (Z)

# neutrons defines isotope number

The atomic mass (A) = mass of protons + mass of neutrons

Atomic mass units. Atomic weight.

The atomic mass unit (amu) is often used to express atomic weight. 1 amu is defined as 1/12 of the atomic mass of the most common isotope of carbon atom that has 6 protons (Z=6) and six neutrons (N=6).

M_proton ≈ M_neutron = 1.66 x 10^-24 g = 1 amu.

The atomic mass of the ¹²C atom is 12 amu.

The atomic weight of an element = weighted average of the atomic masses of the atoms naturally occurring isotopes.

Atomic weight of carbon is 12.011 amu.

The atomic weight is often specified in mass/mole.

Atomic mass units. Atomic weight.

A mole is the amount of matter that has a mass in grams equal to the atomic mass in amu of the atoms (A mole of carbon has a mass of 12 grams).

The number of atoms in a mole is called the Avogadro number, Nav = 6.023 × 10

²³

.

N

_av

= 1 gram/1 amu.

Example:

Atomic weight of iron = 55.85 amu/atom = 55.85 g/mol

Calculations

The number of atoms per cm³, n, for material of density ρ (g/cm³) and atomic mass M (g/mol):

n = Nav x ρ / M

Graphite (carbon): ρ = 2.3 g/cm³, M = 12 g/mol

n = 6×10²³ atoms/mol × 2.3 g/cm³/12 g/mol = 11.5 x 10²² atoms/cm³

Diamond (carbon):

ρ

= 3.5 g/cm

³

, M = 12 g/mol

n = 6×10

²³

atoms/mol x 3.5 g/cm

³

/12 g/mol = 17.5 x10

²²

atoms/cm

³

Water (H

₂

O) d = 1 g/cm

³

, M = 18 g/mol

n = 6×1023 molecules/mol x 1 g/cm

³

/18 g/mol = 3.3 x

10

²²

molecules/cm

³

Calculations

For material with n = 6 × 10

²²

atoms/cm

³

we can

calculate the mean distance between atoms L = (1/n)1/3

= 0.25 nm.

the scale of atomic structures in solids – a fraction

of 1 nm or a few A.

The Periodic Table

Elements in the same column (Elemental Group) share similar

properties. Group number indicates the number of electrons available for bonding.

0: Inert gases (He, Ne, Ar...) have filled subshells: chem. Inactive IA: Alkali metals (Li, Na, K…) have one electron in outermost occupied s subshell - eager to give up electron – chem. Active VIIA: Halogens (F, Br, Cl...) missing one electron in outermost occupied p shell - want to gain electron - chem. active

Periodic Table - Electronegativity

Electronegativity - a measure of how willing atoms are to accept electrons Subshells with one electron - low

electronegativity.

Subshells with one missing electron –high electronegativity.

Electronegativity increases from left to right

Metals are electropositive – they can give up their few

valence electrons to become positively charged ions

Bonding Energies and Forces

Bonding Energies and Forces

The repulsion between atoms, when they are

brought close to each other, is related to the Pauli principle: when the electronic clouds surrounding the atoms starts to overlap, the energy of the system increases abruptly.

The origin of the attractive part, dominating at large distances, depends on the particular type of

bonding.

Bond Angles

• Hybridized orbitals are formed by linear combination of pure atomic orbitals

• The type of hybridization determines the bond angle in the molecule:

• s + 3p 4 Sp3 BA = 109.28

^o

• s + 2p 3 sp2 BA = 120

^o

• s + p 2 sp2 BA = 180

^o

c

H

H H

H

109.5^o

C

H

Cl H

H

110.1^o

(a) (b)

Bond angles: (a) methane CH₄ is symmmetrical with bond angle 109.5^o

(b) chloromethane is distorted (bond angles are not the same)

(a) (b)

Bond angles: (a) Ammonis NH₃ and (b) water H₂O have bond angles

lower than 109.5^o because ammonia has lone pair of electron while water has two.

N

H

H H

107.3^o

O

H

H

104.5^o

• In symmetrical molecules, e.g. CH

₄

– Atoms are equally spaced around the central atom.

– ~ placing the carbon at the cube centre and pointing the orbitals towards four of the 8 corners.

– Distortion occurs if some of electrons are

present as lone pair, e.g. H

₂

O and NH

₃

Delocalized electrons

• σ bond is a single covalent bond and is formed by sharing of two electrons

C H

H

H

H C

H

C H

H

H H C C H

σ π

σ

Not all the valence electrons are localized e.g. benzene

How much energy is required or released if 2.6 Kg of acetylene C₂H₂ react with hydrogen to form C₂H₄?

C H

C H

H C C H

H H H₂

One triple and one H—H bond are eliminated Two C—H and one double bonds are formed H—H 435 KJ/mol C—H 370 KJ/mol

C=C 680 KJ/mol C ≡C 890 KJ/mol

2 2 21

- 24

24 -375 x10 J/C H

bonds 10

x 6 . 0

KJ 2(-435) 680

- bonds

10 x 6 . 0

KJ 890

435 + =

+ + +

Example 2-2.1, p 30

Example 2-2.2 p 31

• How many delocalized electrons in naphthalene molecule

• How many wave patterns (electron states Formula: C₁₀H₈

Total valence electrons = (4/C)(10 C) + (1/H)(8 H) = 48 Electrons in bonds: C—H 8x2 = 16

C—C 11x2 = 22

Reminder = 48-(16+22)= 10 delocalized electrons

HC

HC C H

C C H C

C H

CH CH

H C

Electron states = number of carbon atoms

Macromolecules (polymers)

• Large number of small repeated units called mers

• High molecular weight compounds

• Natural and synthetic

• Linear or branched

• Thermoplastic or thermosetting

• Melting range and not melting points

• No boiling points

Linear Molecules

Linear polymeric chains such as polyvinyl chloride (PVC)

n = degree of polymerization:

the number of mers per molecule

* C C

H H

R H

* n

C C H

H H

Cl

C C H

H H

Cl

C C H

H H

Cl

C C H

H H

Cl

C C C

H H

Cl H

C H

Cl

C C

H

H H

H

C H

Cl C

H

Cl

H

H

Macromolecules (polymers)

• Large number of small repeated units called mers

• High molecular weight compounds

• Natural and synthetic

• Linear or branched

• Thermoplastic or thermosetting

• Melting range and not melting points

• No boiling points

Linear Molecules

Linear polymeric chains such as polyvinyl chloride (PVC)

n = degree of

polymerization: the number of mers per molecule

Secondary bonds

• Covalent bonds are very strong and called primary bonds

• Intermolecular forces are called secondary bonds

• it exists because of the local electric fields within and around uncharged atoms.

• Three types are known:

a) Polarization due to electron oscillations in symm molecules

b) dipole-dipole interactions in asymmetric molecules

c) Hydrogen bridge

Network structure

HC HC C

H

CH C CH OH

HC HC C

H

CH C CH OH C

H H O

HC HC C

H

CH C C

OH H₂ C

HC C H

CH CH HC

Phenol formaldehyde reaction

Polyfunctional monomers network structure, e.g Phenolformaldehyde resin (Bakalite)

Heating: thermoplastic thermosetting polymer

Three dimensional bonding

Three dimensional bonding

Silicate glasses

Fused silica (SiO₂), each Si atom is linked to four adjacent oxygen atoms which inturn bridge between two silicons VERY STABLE network

Ionic Bonding

- Electrostatic attraction between cations and anions

- Unlimited numbers of ions can be bonded to produce solid materials

- Attraction and repulsion forces: F

_c

F

_c

= K

₀

(Z

₁

q)(Z

₂

q)/x

²

q is electron charge = 0.16 x 10

^-18

A.s.

Z is the valence

NaCl

coordination numbers

1- The strength of materials exhibit under applied stresses is related to the type of bonds that held the atoms together.

2- Interatomic attractions are caused by the electronic structure of atoms. Inert gases such as He, Ne, Ar, etc.

have a very stable arrangement of eight electrons (2 for He) in either outer electron orbitals. As a result they have no electrical charge.

3- Most other elements achieve the stable configuration of having 8 electrons in their outer orbital by:

- Receiving extra electrons to form anions, or - Releasing electrons to form cations or

- Sharing electrons.

coordination numbers

Naturally, ions of unlike charges are attracted to one another by electrostatic forces, while electrons sharing requires intimate contact between atoms. Thus, in both instances strong bonding is established between neighbouring atoms.

4- In addition to ionic and covalent bonds a third type of primary attractive mechanics is offered by delocalized electrons or an electron cloud able to move throughout the metal structure. This gives rise to the formation of the so- called metallic bond.

5- It is interesting to note that certain materials exhibit mixed bonding characteristics.

coordination numbers

6- Molecules on the other hand, may be defined as groups of atoms strongly bonded together.

7- Most engineering materials possess coordinated groups of many atoms. Therefore, when analyzing the bonding of atoms within materials, we speak of a coordination

number.

8- The Coordination Number (CN) can be simlly defined as number of first nearest neighbors surrounding an atom within a given material. Let us consider, for example, the

case of methane, CH₄

C

H

H H

H

coordination numbers

As it is already seen, the coordination number for carbon is 4 whereas the hydrogen atoms have only one nearest

neighbour.

9- The coordination number of an atom is controlled by 2 factors:

A- the number of valence electrons of the atom B- efficiency of atomic packing.

10- the halides which are situated in group VII of the periodic table and have 7 valence electron each, form only one bond and hence have one coordination number when bonded

covalently.

coordination numbers

Likewise, members of the oxygen group (VI) have a maximum coordination number of 2. (Note this is in the

gaseous state) In solids, efficient atomic packing is concerned, since energy is released as ions of opposite charges approach each other, ionic compounds have generally higher coordination numbers, without introducing the strong mutual repulsion

forces between ions of the charges.

This may be illustrated with MgO. Mg²⁺ ions are surrounded by O^2- ions. The Mg²⁺ ions has a radius r = 0.66 Ă, which is large enough to allow 6 O^2- ions with R = 1.40 Ă to surround it without direct of negative ions with one another.

CN Ionic Bonding (3-D)

a) A maximum of 6 O^2- surrounding Mg2+

b) CN of Si⁴⁺ among O^2- is only 4 (r/R < 0.4)

Ionic coordination numbers

• Covalent solids are loosely packed and posses

a large free space

• Ionic solids are colsely packed and contains less

free space. This due to the columbic attractions are omnidirectional.

•

+ve ions are smaller than –ve ions

R = radius of anion, r = radius of cations Ionic Coordination (2-Dimensional).

a) Coordination with r/R > 0.41. The smaller cation is coordinated with 4 anions (CN = 4 for 2-dimnsions).

b) Coordination with r/R < 0.4. The (+ve) ion does not have max.

conact with all 4 neighbouring (-ve) ions. There is repulsion between the contacting ions

c) When r/R < 0.4, then CN =3 is favoured (2-D).

R = radius of anion, r = radius of cations

Example 2-4.2

Show the origin of 0.41 as the minimum ratio for a coordination number of 6.

Procedure The minimum ratio of possible sizes to permit a CN = 6 is sketched below:

Note that the 5^th & 6^th ions sit above and below the central ion

Coordination Calculations:

Minimum r/R for 6-fold coordination (a Minimum r/R for 4-fold coordination

0.414 1

- 1.414 R

; r 414 .

1 1

2 2 2

2 45 2

cos

=

=

= +

= + =

= +

=

R r R

r R

r R

R

0.155 1

3 - 2 R

; r 3 1 2

3 2

866 .

2 0 30 3

cos

=

=

= +

+ =

+ =

=

=

R r R

r R

r R

R

coordination numbers

A coordination number CN =6 is widely come across in ionic compounds. Silicon in SiO₂ on the other hand has a CN = 4 because Si⁴⁺ ion is too small to accommodate 6 oxygen ions.

r/R for silicon is approximately 0.3.

To calculate r/R ratio for a four-fold coordination (i.e. when CN = 4)

The larger atoms are located at the four corners of a regular tetrahedron. The small atoms sits in the body centre of the tetrahedron and touches each of the four atoms at the corners. The distance between any two corner atoms = 2R

See below

The four corner atoms may be visualized as occupying the corners of a cube

Hence, it may be written that the body diagonal d = 2R + 2r, also d = a √3

But a √2 = 2R or a = 2R/√2

∴2(R+r) = a √3 = 2R/√2

Or R+r/R = √3/2

Hence, 1 + r/R = 1.224 And r/R = 0.224

Coordination of 12:

all the atoms are the same size,

each atom have 12 immediate neighbors.

Solid circles: four neighbors in the same plane as the central atom. Dashed circles: four

neighbors above, and four neighbors below. Each

neighbor also will be

coordinated with 12 neighbors.

Metallic Bonding

- Unlike ionic and covalent solids where the valence electrons are localized

- The nature of bonding is different from both types.

- like the bi electrons in benzene valence electrons are delocalized, it can easily move under the effect of electric field.

- it is best described as positive ions surrounded by a sea of delocalized electrons.

- CN can be >12 it can exceed 100

Interatomic Distances

• In solids the attraction forces is maximum because the attraction forces is maximum

• Interatomic forces F_c α x^-2

+Ve -Ve

Distance < 1 nm

Columbic forces

Electronic repulsion

• Very strong if x ~ 2 nm

– Due to valence and subvalence electrons – The repulsion forces F

_R

= -b/x

ⁿ

• b is the proportionality constant &

• n = 9 or 10

• i.e. it operates at a much closer range than the columbic forces

An equilibrium spacing is a natural results when

FF_cc + F+ F_{R R} = 0= 0

The eqm distance o-x`

Is the distance at which the net columbic attraction

forces = electronic repulsive forces

The lowest potential energy When O—x` is the

interatomic distance Since

The shaded area in (a) = depth of the energy well in (b)

∫^Fdx

E = ∫ F dx

-A tension force is required to overcome the attraction forces.

- A compressive force is required push the atom closer together.

- Equilibrium spacing is a very specific distance for a given pair of atoms.

- It can be measured by x-ray diffraction

Bonding Energy

dx ) F F

(

E

_R

x

c

+

= ∫

0 ∞

E reference,

Energy _x=∞ =

Strong Bond Weak Bond

Atomic and Ionic Radii

• The eqm distance is the sum of their ionic radii.

• In metallic iron, the mean distance is 0.2482 nm and the radius is 0.1241 nm.

• The distance between is atoms depends on the temperature and the ionic valence

• r_Fe > r_Fe²⁺ > r_Fe³⁺

• -ve ion is larger than the atom

• The interatomic distance also depends on the number of adjacent atoms.

• The case is different in covalent compounds, they are not spherical (see Table)

Properties and atomic bonding

• Density: determined at at. Wt., at. Radius and the coordination number (Sig)

• Melting & boiling: depth of the energy well = bond energy

• Hardness: the height of the total force curve

• Elasticity: the slope of the sum curve, where the net force is zero. It s also related to the bond energy.

• Thermal expansion: inversely related to the melting temperature. See Figure

Conductivity of Metals

• Electrical conductivity depends on the nature of atomic bons.

• Ionic and covalent materials are poor conductors in the solid state.

• In metals the delocalized electrons can free move along potential gradient.

• Thermal conductivity is high in metallic bonds, due to the delocalized electrons are efficient carriers of thermal as well as electrical energy.