Part V Thermal Properties of MaterialsPart V Thermal Properties of Materials Chap. 18 Introduction...

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

BE k T=

Average energy per atom ( three-dimensional harmonic oscillator )

32 BE k T=

Average kinetic energy of vibrating atom

3 BE 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

03 BE N k T=

Finally, the molar heat capacity is

v 0v

3 BEC N kT∂ = = ∂

Inserting numerical values for N0 and kB

v 25 J/mol K 5.98 cal/mol KC = ⋅ = ⋅

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

12i i hvε = +

= Planck's constant of action = frequency of harmonic oscillator

hv

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

1( ) /2

1( ) /2

1( ) /1( ) / 2 /2

1 1 /( ) / ( ) /2 2

/

13 32

112 3 32

23 12

hv i kT

i ihv i kT

hv i kThv i kT hvi kT

hvi kThv i kT hv i kT

hvi kT

neU n i hve

eie ienhv nhv

ee e

ienhv

e

ε− +

− +

− +− + −

−− + − +

−

′ = = +

= + = +

= +

∑ ∑∑

∑∑ ∑∑∑ ∑

∑/hvi kT−

∑

( P : partition function )i

inen

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 iie ix− =∑ ∑

22(1 2 3 )

(1 )xx x xx

+ + + =−

/ hv kTx e−=where gives,

and / 2 11

1hvi kT ie x x x

x− = = + + + =

−∑ ∑

in which case

/

/

/

3 2 3 21 12 1 2 13 3 2 1

hv kT

hv kT

hv kT

x eU nhv nhvx e

nhvnhve

−

−

′ = + = + − −

= +−


Heat capacity at a constant volume

/ 2 /v 2

v2 /

/ 2

3 ( 1)

3( 1)

hv kT hv kT

hv kT

hv kT

U hvC nhv e eT kT

hv enkkT e

−′∂ = = − ∂

= − Einstein temperature is

Ehvk

θ =

So, 2 /

v / 23( 1)

E

E

TE

Tec R

T e

θ

θ

θ = −


Total energy of vibration for the solid

( )oscE E D dω ω= ∫ 2

2 3

= the energy of one oscillator

3( )2

osc

s

E

VD ωωπ υ

=3

2 3 /0

32 1

D

kTs

VE de

ω

ω

ω ωπ υ

=−∫

debye frequencyDω =

Heat capacity at a constant volume 2 4 /

v 2 3 2 / 20

32 ( 1)

DkT

kTs

V eC dkT e

ωω

ω

ω ωπ υ

=−∫

Or indicating with Debye temperature

/

3 4phv 20

9 , ( 1)

TDx

D DDx

D

hvT x e hvC kn d xe kT kT k k

θ ωωω θθ

= = = = = − ∫

Dθ

20.2.2 Debye Model


Thermal energy at given temperature

kin B F B3 3 ( )2 2 BE k TdN k TN E k T= =

The Heat capacity of the electrons

el 2v B F

v

3 ( )EC k TN ET∂ = = ∂

For E < EF

*

FF

3 electrons( ) 2 JNN EE

= *

F number of eletrons which have an energy equal to or smaller than N E=* 2*

el 2 Bv B

v F F

3 9 J3 2 2 K

N k TE NC k TT 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, * 22 2

el *Bv B

F F

= 2 2

N k T TC N kE 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.

22 2el 0 Bv 0 B

F F

J= 2 2 K mol

N k T TC N kE T

π π = ⋅

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

tot el ph 3v v v

tot2v

C C C T T

C TT

γ β

γ β

= + = +

∴ = +

/

2

3 4

20

3 ( )

19( 1)

TD

B

x

xD

k N EF

x ekn de

θ

γ

β ωθ

=

= −

∫


Thermal effective mass

*th

0

(obs.)(calc.)

mm

γγ

=


From the kinetic theory of gases

0 2 13 (2 )

6 2 2V V

B Bn v n vdT dTJ 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.

2V Bn vk lK =

Relation between the heat conductivity and elvC

,

13

elvK C vl=

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

*3f BN k T

Km

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.

13

phvK C vl=

phvC

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

2aAfter 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

Lα : 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

Part V Thermal Properties of MaterialsPart V Thermal Properties of Materials Chap. 18 Introduction...

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