Part V Thermal Properties of MaterialsPart V Thermal Properties of Materials Chap. 18 Introduction...
Transcript of 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
−
−
′ = + = + − −
= +−
20.2 Quantum Mechanical Considerations-The Phonon
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 = −
20.2 Quantum Mechanical Considerations-The Phonon
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
20.2 Quantum Mechanical Considerations-The Phonon
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
=
= −
∫
20.2 Quantum Mechanical Considerations-The Phonon
Thermal effective mass
*th
0
(obs.)(calc.)
mm
γγ
=
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 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
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
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