Part V Thermal Properties of Materials

Part V Thermal Properties of Materials

Chap. 18 Introduction

Chap. 19 Fundamentals of Thermal Properties Chap. 20 Heat Capacity

Chap. 21 Thermal Conduction

Chap. 22 Thermal Expansion

Average energy of one-dimensional harmonic oscillator

E = k T

B

Average energy per atom ( three-dimensional harmonic oscillator )

3 2

^B

E = k T

Average kinetic energy of vibrating atom

3

_B

E = k T

20.1 Classical (Atomistic) Theory of Heat Capacity

Fig 20.1. (a) A one-dimensional harmonic oscillator and (b) a three- dimensional harmonic oscillator.

Total internal energy per mole

3

0 _B

E = N k T

Finally, the molar heat capacity is

v 0

v

3

_B

C E N k

T

 ∂ 

=   ∂   =

Inserting numerical values for N₀ and k_B

v

25 J/mol K 5.98 cal/mol K

C = ⋅ = ⋅

Average potential energy of vibrating atom has the same average magnitude as the kinetic energy.

So Total energy of vibrating atom has the same average magnitude as the average per atom.

20.1 Classical (Atomistic) Theory of

Heat Capacity

The energy of the i th energy level of a harmonic oscillator

1

i

i 2 hv

ε = +    

 

= Planck's constant of action

= frequency of harmonic oscillator h

v

The energy of Einstein crystal ( which can be considered to be a system of 3n linear harmonic oscillator)

( 1) / 2 ( 1) /

2

( 1) /

( 12) / 2 /

1 1 /

( ) / ( ) /

2 2

/

3 3 1

2 1 2 1

3 3

2 3 2

1

2

hv i kT i i

hv i kT

hv i kT hv i kT

hvi kT hvi kT

hv i kT hv i kT

hvi kT

U n i hv ne

e

ie e ie

nhv nhv

e e e

nhv ie

e ε

− +

− +

− +

− + −

− + − + −

−

 

   

′ = =  +    

 

 

   

=  +  =  + 

= +

∑ ∑

∑

∑ ∑ ∑

∑ ∑ ∑

∑

/ hvi kT

−

 

 

 



∑



( P : partition function )

i

i

n ne

P

βε

= −

20.2 Quantum Mechanical

Considerations-The Phonon

20.2.1 Einstein Model

Fig 20.2. Allowed energy levels of a phonon: (a) average thermal energy at low temperatures and (b) average thermal energy at high temperatures.

Taking

/

hvi kT i

ie

⁻

= ix

∑ ∑

2

(1 2 3 )

2

(1 )

x x x x

+ + + = x

 −

/ hv kT

x = e

⁻

where gives,

and

/ 2

1

1 1

hvi kT i

e x x x

x

−

= = + + + =

∑ ∑  −

in which case

/ /

/

3 2 3 2

1 1

2 1 2 1

3 3

2 1

hv kT hv kT

hv kT

x e

U nhv nhv

x e

nhv nhv e

−

−

 

 

′ =  + −  =  + − 

= +

−

20.2 Quantum Mechanical

Considerations-The Phonon

Heat capacity at a constant volume

/ 2 /

v 2

v

2 /

/ 2

3 ( 1)

3

( 1)

hv kT hv kT

hv kT hv kT

U hv

C nhv e e

T kT

hv e

nk kT e

′

−

 ∂ 

=   ∂   = −

 

=     −

Einstein temperature is

E

hv θ = k

So,

2 /

v

3

/ 2

( 1)

E

E

T E

T

c R e

T e

θ θ

 θ 

=     −

20.2 Quantum Mechanical

Considerations-The Phonon

Total energy of vibration for the solid

osc

( )

E = ∫ E D ω ω d

²

2 3

= the energy of one oscillator ( ) 3

2

osc

s

E

D ω Vω

= π υ

3

2 3 0 /

3

2 1

D

kT s

E V d

e

ω

ω

ω ω

= π υ

∫

^

 − ω

_D = debye frequency Heat capacity at a constant volume

2 4 /

v 2 3 2 0 / 2

3

2 ( 1)

D

kT kT s

V e

C d

kT e

ω ω

ω

ω ω

= π υ

∫

^{ }

−



Or indicating with Debye temperature

/

3 4

ph

v

9

0 2

,

( 1)

D T

x

D D

x D D

hv

T x e hv

C kn d x

e kT kT k k

θ

ω ω θ ω

θ

 

=     ∫ − =  = =  =

θD

20.2.2 Debye Model

20.2 Quantum Mechanical

Considerations-The Phonon

Thermal energy at given temperature

kin B F B

3 3

( )

2 2 ^B

E = k TdN = k TN E k T The Heat capacity of the electrons

el 2

v B F

v

3 ( )

C E k TN E

T

 ∂ 

=   ∂   =

For E < E_F

* F

F

3 electrons

( )

2 J

N E N

E

 

=   N^* = number of eletrons which have an energy equal to or smaller than EF

* 2

*

el 2 B

v B

v F F

3 9 J

3 2 2 K

N k T

E N

C k T

T E E

 ∂   

=   ∂   = =    

So far, we assumed that the thermally excited electrons behave like a classical gas.

20.3 Electronic Contribution to

The Heat Capacity

In reality, the excited electrons must obey the Pauli principle. So,

* 2

2 2

el B *

v B

F F

2 = 2

N k T T

C N k

E T

π π

=

If we assume a monovalent metal in which we can reasonably assume one free electron per atom, N^* can be equated to the number of atoms per mole.

2

2 2

el 0 B

v 0 B

F F

= J

2 2 K mol

N k T T

C N k

E T

π π  

=   ⋅  

Below the Debye temperature, the heat capacity of metals is sum of electron and phonon contributions.

tot el ph 3

v v v

tot v 2

C C C T T

C T

T

γ β

γ β

= + = +

∴ = +

/

2

3 4

0 2

3 ( )

9 1

( 1)

D T

B

x x D

k N EF

kn x e d

e

θ

γ

β ω

θ

=

 

=  

∫

−

20.2 Quantum Mechanical

Considerations-The Phonon

Thermal effective mass

* th

0

(obs.) (calc.) m

m

γ

= γ

20.2 Quantum Mechanical

Considerations-The Phonon

Part V Thermal Properties of Materials

Chap. 18 Introduction

Chap. 19 Fundamentals of Thermal Properties Chap. 20 Heat Capacity

Chap. 21 Thermal Conduction

Chap. 22 Thermal Expansion

From the kinetic theory of gases

0 2 1

3 (2 )

6 2 2

V V

B B

n v dT n v dT

J E E k l k l

dx dx

= − = − = −

21.3 Thermal Conduction in Metals and Alloys Classical Approach

Fig 21.1. For the derivation of the heat conductivity in metals.

Note that (dT/dx) is negative for the case shown in the graph.

2

V B

n vk l K =

Relation between the heat conductivity and

C

_v^el

,

1 3

el

K = C vl

v

21.3 Thermal Conduction in Metals

and Alloys Classical Approach

Do all the electrons participate in the heat conduction?

No, Only those electrons which have an energy close to the Fermi energy participate in the heat conduction

21.2 Thermal Conduction in Metals and

Alloys-Quantum Mechanical Considerations

2 2

3

* f B

N k T

K m

π τ

=

(Lorentz number)

21.2 Thermal Conduction in Metals and

Alloys-Quantum Mechanical Considerations

Heat conduction in dielectric materials occurs by a flow of phonons.

At low temperatures

Only a few phonons exist, the thermal conductivity depends mainly on the heat capacity which increses with the low temperatures.

1 3

ph

K = C vl

v

ph

Cv

At higher temperature

The phonon-phonon interactions are

dominant, the phonon density increases with increasing T.

Thus the mean free path and the thermal conductivity decreases for temperatures

21.3 Thermal Conduction in Dielectric

Materials

Umklapp Process

When two phonons collide, a third phonon results in a proper manner to conserve momentum. Phonons can be represented to travel in k-space.

Outside the first Brillouin zone, Resultant phonon vector

a

2

After the collision in a direction that is almost opposite to

Fig 21.2. Schematic representation of the thermal conductivity in dielectric materials as a function of temperature.

21.3 Thermal Conduction in Dielectric

Materials

Part V Thermal Properties of Materials

Chap. 18 Introduction

Chap. 19 Fundamentals of Thermal Properties Chap. 20 Heat Capacity

Chap. 21 Thermal Conduction

Chap. 22 Thermal Expansion

α

L : Coefficient of linear expansion

The length L of a rod increases with increasing temperature

,

Atomistic point of view

For small temperature

: a atom may rest in its equilibrium position

As temperature increases

the amplitudes of the vibrating atom increases

Asymmetric

Because potential curve is not

symmetric, a atom moves farther apart from its neighbor

Fig 22.1. Schematic representation of the potential energy, U(r), for two adjacent atoms as a function of internuclear separation, r.

22 Thermal Expansion