properties of materials

Tao Deng, [email protected] 1

1896 1920 1987 2006

Properties of Materials

Chapter One

Thermal Properties of Materials

Tao Deng


1 Thermal properties of materials

Heat Capacity

Thermal Conduction

Thermal expansion


The physics of thermal property

For solid materials, the physics of their thermal

properties is related to the thermal vibration of

lattice.

nnnn xxx

dt

xdm 2112

2

The sum of thermal vibrational energy Uk is the

quantity of thermal energy ( Q ) ：

N

iikUQ

1

Xn Xn+1 Xn-1


Lattice wave

• The low frequency lattice wave: the difference of phase

between adjacent particles is very small -- acoustic branch。

• The high frequency lattice wave: the difference of phase

between adjacent particles with almost opposite moving

direction is very big; its frequency is often in the infrared

region -- optical branch.

The lattice wave in one-dimensional diatomic lattice

(a) acoustic branch (b) optical branch


1. Thermal properties of materials

• Application examples of thermal properties

– Thermal/Heat capacity

Low： foam package, building materials

High：thermal battery, thermal storage

– Thermal expansion

Low：high precision balance, standards (ruler, capacitor etc)

High：temperature sensors

– Thermal conduction

Low：Industry furnace inner coating; construction materials;

hypersonic vehicle protection layer

High: steam turbine blade, electronic heat spreader


1.1 Heat capacity

2.1.1 The concept of heat capacity

T

TT

QC

Specific heat capacity (mass)

mT

Qc

1

Heat capacity at T


Other concept of heat capacity and expression

Molar heat capacity Cm－ The heat capacity of 1

mol of material，J / (mol.K)

mTT

Qcaverage

1

12

Mean specific heat capacity －The absorbed heat

averaged by mass with the material’s temperature

increases from T1 to T2：

mT

Qc

V

V

1

Specific heat capacity

at constant volume：

mT

Qc

P

P

1

TVcc mVVP

2

VdT

dVV

VdP

dV

Specific heat capacity

at constant pressure：


1.1.2 The change of heat capacity with temperature

At high temperature, specific heat capacity changes very slowly,

and it can be described by Dulong-Petit‘s law

(R is the gas constant (J/Kmol) and M is the molar mass (Kg/mol)).

At low temperature, heat capacity is proportional to the cube of

temperature, i.e. 3TC

When temperature→0 ， heat capacity→0

Molar heat

capacity of

NaCl vs

temperature

C = 3R/M


1.1.3 Heat capacity theory of crystal

1.1.3.1 The classical theory of the heat capacity

Dulong-Petit Law

－heavy atom’s molar heat capacity under constant volume is 25J/(K.mol).

For lighter elements, their heat capacities need to be adjusted to the following

values：

element： H B C O F Si P S Cl

CV, m : 9.6 11.3 7.5 16.7 20.9 15.9 22.5 22.5 20.4

Neumann-Kopp Law

－The molar heat capacity of a compound is the sum of the atomic

heat capacity of all the elements in the compound.

• Diatomic compounds：CV,m = 2×25 J / (K.mol)；

•Triatomic compounds： CV,m = 3×25 J / (K.mol)；

• …


1.1.3.1 The classical theory of heat capacity

KmolJRkNT

kTN

T

EC A

V

A

V

molmV /2533

3,

There are three degrees of

freedom in an atom, and the sum

of the average kinetic energy and

potential energy is thus 3kT (k is

the Boltzmann constant). There

are NA atoms in 1mol solid, so the

total energy Emol = 3NAkT.

According to the definition of the

heat capacity, there are:

For diatomic and triatomic solid compound, the number of atoms in 1mol

substance is 2 NA and 3 NA respectively, so the molar heat capacity is 2 ×

25 and 3 × 25 J / mol.K, respectively.


1.1.3.2 The quantum theory of heat capacity

ii hnE

2

1Basic starting point－Quantization

of lattice vibration energy

The number of harmonic oscillator with

vibrational energy Ei in crystal NEi is：

i

i

i

i h

kT

h

hE

2

1

1exp

Average energy of the harmonic

oscillator：

}2

1

1exp

{3

1

3

1

i

i

iN

i

N

i

i h

kT

h

hEE

AA

The total vibrational energy of

harmonic oscillator in 1 mol of

solid is ：

2

3

1

2

,

1exp

exp

kT

h

kT

h

kT

hk

T

EC

i

i

N

i

i

V

mV

A

Constant-volume

heat capacity：

kT

EAN i

EiexpThe number of harmonic oscillator with

vibrational energy Ei in crystal NEi is：


（1）Einstein model

2

2

,

1exp

exp

3

kT

h

kT

h

kT

hkNC AmV

k

hE

TRf

T

T

TRC E

E

E

E

EmV

3

1exp

exp

32

2

,

Basic assumptions: every harmonic oscillator is independent

and non-interacting, and the vibrational frequencies are all ν.

Constant-volume

heat capacity：

Assumping (Einstein Temperature)


Einstein model

, 3 e xp 3EV mC R R

T

TTRC EE

mV

exp3

2

,

•If T ＞＞ θE ，There is ：

•If T ＜＜ θE ， There is ：

CV exponentially decreases as temperature decreases, not the

same as the experimental curve ( T3).

Causes: The atomic vibrations are not independent in the

actual crystal; there are the coupling effects among different

vibrations; In addition, the vibration frequency is not

exactly the same.


（2）Debye model

TRfC D

DmV

3,

D

kT

hd

kT

h

kT

h

kT

h

T

Tf

D

DD

0

2

4

3

1exp

exp

3

k

hD

max

• Include the interaction of atoms in a crystal.

• Consider only the contribution from acoustic wave

(continuous wave) to the heat capacity.

Debye function of

specific heat

capacity

Debye Temperature


Debye model

• If T ＞＞θD ,

--Dulong-Petit Law

• If T ＜＜ θD

– Show that when T → 0, CV is proportional to T3 , (Debye

temperature cubic law), in good agreement with the

experimental results.

– Reasoning: at very low temperatures, the long-

wavelength excitation is dominant.

34

,5

12

D

mV

TRC

RC mV 3,


1.1.4 Heat capacity of engineering materials

1.1.4.1 Heat capacity of the metals

TTCCC VeVlV 3

Where，CVl and CVE is the

contribution of the lattice vibrations

and free electron to the heat

capacity, respectively. α and γ are

the heat capacity coefficient。

（1）Free electron’s contribution to the heat

capacity


1.1.4 Heat capacity of engineering materials

Temp Range Trend Contribution

I： 0-5K CV T Free electron （ lattice vibration can be

neglected）

II: 5K～ D CV T3 Lattice vibration （ free electron can be

neglected ）

III：T>>D CV > 3R Free electrons and lattice vibrations (free

electron contributes to the parts that more than

3R)

1.1.4.1 Heat capacity of the metals

For the transition metal, the contribution from the outer layer (d

layer, the f layer) electrons to the heat capacity is also important.

In general the constant-volume specific heat capacities of transition

metals are higher than non-transitional metal.


(2)Metals: Debye temperature and the melting point

At the melting point Tm, the vibration reaches maximum

frequency (νm) that damages the lattice:

32

12108.2mr

mm

VA

T

where Ar as relative atomic mass of an element ; Vm is

the molar volume with unit of cm3/mol. Since Debye

temperature:

(Lindemann Equation)

k

hD

max

(h - Planck's constant)

(k- Boltzmann constant)


(2)Metals: Debye temperature and the melting point

32137

mr

mD

VA

T

Debye temperature may reflect the binding force

between atoms. Higher melting point (Tm) means that

the binding force is strong and θD is higher. For

example, the θD of diamond is 2230K, but the θD of

lead is 105K.


（3）Heat capacity of metal alloys

iiCmC

Naiman - Kopp's Law: the heat capacity of alloy is

the sum of the heat capacity of individual

composition in proportional to its mass percentage:

式中，Ci －第 i 相热容；mi －第 i 相质量分数。

Where Ci is the heat capacity of i-composition; mi is

the mass fraction of i-composition.

How to calculate your heat capacity?


1.1.4.2 The heat capacity of the ceramic material

• If T ＞θD，CV → 25 J/(K.mol)；

• If T ＜θD， CV ∝T3

material graphite BeO Al2O3

ΘD（K） 1973 1173 923

Heat capacity of some inorganic materials

at different temperatures Heat capacity of the different forms Cao+Sio2

with the molar ratio of 1:1.


1.1.4.3 The heat capacity of the polymer material

Polymers have semi-crystalline structure or amorphous structure so we will

not be able to apply the theory of crystalline heat capacity to them. The

specific heat capacity of the majority of polymers are relatively small when the

temperature is below the glass transition temperature. When the temperature

rises to the glass transition temperature, heat capacity will have a stepped

changes due to the increased thermal motion.

JOURNAL OF CHEMICAL PHYSICS 119 (7),3590-3593


Heat capacity


Heat capacity of engineering materials

Category Materials Heat capacity Cp,m（J/kg·K）

Ceramics aluminium oxide Al2O3

775

beryllium oxide BeO 1050

Magnesium oxide MgO 940

Spinel MgAl2O4 790

Fused silicon oxide SiO2 740

Nano calcium glass 840

Metal Aluminum 900

Gold 129

Nickel 443

316 stainless steel 502

Polymer Polyethylene 2100

Polypropylene 1880

Polystyrene 1360

Polytetrafluoroethylene 1050


1.1.5 The effect of phase change

Due to discontinuity in heat absorption at phase

change, there is also a sudden change in the heat

capacity：

• Solidification

• Polycrystalline transformation

• Solid solution, precipitated

• Melt

• Ferroelectric transition

• Ferromagnetic transition

• Order - disorder transition The change of enthalpy and heat capacity with

temperature: (a) First order phase transition ;

(b) Second order phase transition


1.1.6 The measurement of the heat capacity

1.1.6.1 The mixture method

The method of mixture based on the fact that when

a hot substance is mixed with a cold substance, the

hot body loses heat and the cold body absorbs heat

until thermal equilibrium is attained. At equilibrium,

final temperature of mixture is measured. The

specific heat of the substance is calculated using the

law of heat exchange.

QLOST BY SUBSTANCE = QGAINED BY LIQUID + QGAINED BY CALORIE METER

Ms cs (Ts – T) = ml cl (T – Tl) + mc cc (T – Tc)


1.1.6.2 Electric method

The material under investigation is heated by an electrical

immersion heater and the input energy (Q) and the rise in

temperature that this produces are measured. If the mass of

the specimen (solid or liquid) is m and its specific heat

capacity c, then:

Q = mc(T2-T1) + Qloss




– Heat capacity











1.2 Thermal expansion

1.2.1 Definition

Thermal expansion: the increase of volume or length with the increase of

temperature. The change of length with temperature follows the following

equation: 1212 1 TTLL

1112

12 11

LT

L

LTT

LL

LdT

dLT

1

The coefficient of average

linear expansion ：

The coefficient of linear expansion at

any point of time：

The coefficient of

volume expansion ： 0

1

VT

VV

For anisotropic crystals：

For isotropic cube ： 3V

cbaV


The coefficient of thermal expansion (linear, ppm)

)10( 6 co

th

)10( 6 co

th

12ASTM B160 Nickel alloy

26AZ31B Magnesium alloy

20-40Silicone resin24Pb-Sn Solder(50-50)

60-80Phenolic Resin18Brass

45-90Epoxy resin17ASTM152 Copper alloy

50-90Plexiglass22.92017 Aluminum alloy

80-160Cellulose acetate23.23003 Aluminum alloy

80Nylon17Stainless steel

66Polycarbonate121020 Steel

100PTFE35Zn

60-80Polystyrene5W

58-100Polypropylene9Ti

100-200Polyethylene20Sn

Macromolecule

without Orientation7Ta

0.55Silica rock(99.9%)70Na

0.8Silica rock(96%)19Ag

4.6Borosilicate glass83K

9.2Soda-lime-silica

Glasses9Pt

7.6Spinel(MgAl2O4)13Ni

2.3-Si3N45Mo

2.9-Si3N425Mg

2.2Si29Pb

4.8SiC12Fe

13.5MgO14Au

9BeO17Cu

6.5-8.8Al2O212Co

Ceramics6Cr

5Kovar(Fe-Ni-Co)25Al

8.8Business TiMetal

MaterialMaterial

12ASTM B160 Nickel alloy

26AZ31B Magnesium alloy

20-40Silicone resin24Pb-Sn Solder(50-50)

60-80Phenolic Resin18Brass

45-90Epoxy resin17ASTM152 Copper alloy

50-90Plexiglass22.92017 Aluminum alloy

80-160Cellulose acetate23.23003 Aluminum alloy

80Nylon17Stainless steel

66Polycarbonate121020 Steel

100PTFE35Zn

60-80Polystyrene5W

58-100Polypropylene9Ti

100-200Polyethylene20Sn

Macromolecule

without Orientation7Ta

0.55Silica rock(99.9%)70Na

0.8Silica rock(96%)19Ag

4.6Borosilicate glass83K

9.2Soda-lime-silica

Glasses9Pt

7.6Spinel(MgAl2O4)13Ni

2.3-Si3N45Mo

2.9-Si3N425Mg

2.2Si29Pb

4.8SiC12Fe

13.5MgO14Au

9BeO17Cu

6.5-8.8Al2O212Co

Ceramics6Cr

5Kovar(Fe-Ni-Co)25Al

8.8Business TiMetal

MaterialMaterial


1.2.2 The physics of the thermal expansion

Lattice vibration

Attractive -repulsive force between the

particles and the potential energy curves in Crystal

The diagram of the asymmetric vibration of particle in crystal


1.2.3 Grüneisen law

Thermal expansion and heat capacity are related through

lattice vibration. Usually, when the temperature is not too high,

both the thermal expansion and heat capacity increase with

temperature, but the ratio of thermal expansion over heat

capacity is a constant.

VK

rCVV

0

VK

rCVl

03

The comparison of

the curves of

thermal expansion

and the heat

capacity


Pure Metals

Usually, when pure metal is heated

from 0 K to Tm(the melting point ),

the relative expansion is about 6%.

cV

VVT mT

Vm

0

0

where，V0 and VTm are the volume

of metal in the absolute zero and the

melting point, respectively; c is the

constant (about 0.06 to 0.076 for

cubic and hexagonal structure of the

metal).

022.0mlT

The relationship between Melting point

and the expansion coefficient


The effect of binding energy

For most metals, thermal expansion coefficient(α) is

inversely proportional to the binding energy (Em). ：

bE

a

m

Where, a and b are constants related to lattice type:

The type of

lattice a b

fcc 1270 5.0

bcc 973 -100

hcp 885 0


1.2.4 Factors that affecting thermal expansion

1.2.4.1 Crystal defects and crystal density

Usually, The expansion coefficient of the crystal

with a denser structure is larger than that of crystals

with smaller density. Less dense structures can absorb

more vibrational energy and adjust the bond angle to

absorb more vibrational energy.

Crystal defects, especially point defects, will cause the

volume change and affect the thermal expansion.

Generally, the concentration of the point defects is

strongly related to the temperature. The additional

volume change of the crystal caused by the increase

of point vacancy due to the rise of temperature is:

0 expQ

V BVkT

Where B is a constant; V0 is the volume of the crystal at 0K; Q is the energy of

vacancy formation; k is Boltzmann's constant, T is the thermodynamic

temperature.


1.2.4 Factors that affecting thermal expansion

When the temperature gets close to the melting point,

the effect of thermal defects became more apparent.

The change of the volume expansion coefficient caused

by the vacancy is

2expV

Q QB

T kT


1.2.4.2 Crystal anisotropy

1133 23

1 average

iiCos 2

3322113

1 average

cubic α11 α11 α11 0 0 0

Tetragonal α11 α11 α33 0 0 0

Hexagonal α11 α11 α33 0 0 0

Orthogonal α11 α22 α33 0 0 0

Monocline α11 α22 α33 0 α5 0

Triclinic α11 α22 α33 α4 α5 α6

3V• Cubic system：

•Hexagonal 、 trigonal

system：

• Except the monoclinic and triclinic, the coefficient of linear

expansion of all single crystals in any direction is ：

• Orthorhombic system ：


1.2.4.3 Alloy

• In the case of a solid solute in the matrix

– αS is larger than αB ，αalloy

increases

– αS is smaller than αB ，αalloy

decreased

• When the intermetallic compound is formed

– The trend is more

complicated. For example,

when carbon is added in the

α--Fe , if carbides is formed,

α-steel is reduced.

The effect of different solute elements on α-Iron


1.2.4.4 Multiphase (composite) materials

i

ii

i

iii

WE

WE

Turner Equation：

Ei: Young’s modulus; Wi: mass fraction; i: density


1.2.4.6 Anomalous expansion of ferromagnetic metal

The expansion curve of the iron, nickel and

cobalt in the magnetic transition region

Negative anomalous expansion

curve of Fe-35% Ni alloy

T/0C

For ferromagnetic metals and alloys, change of the coefficient

of expansion with temperature does not follow the normal trend.

They have an extra peak along the normal curve. These extra

peaks are called anomalous expansion peaks.


1.2.5 Measurement of coefficient of thermal expansion

1.2.5.1 Optical methods (interferometry) - Fizeau interferometer

The sample S is covered by a thin plate P2

and is placed in between a bottom plate P3

and a wedge-shaped plate P1. The upper

surface of P1 reflects the incident beam (i) to

a reflected beam (r) so that it is removed

from the interference process. The relevant

interference takes place between ray (1)

reflected by the lower surface of P1 and ray

(2) reflected by the upper surface of P2. A

cylindrical tube T, which defines the distance

between P1 and P3. P2, is usually made of

fused silica, a material of low and well

known thermal expansion. The measured

dilatation is caused, therefore, by the

difference between thermal expansion of the

sample and a portion of the fused silica tube

of equal length. The whole apparatus is

mounted in a thermostat.


1.2.5 Measurement of coefficient of thermal expansion

1.2.5.2 Optical methods (interferometry) - Michelson interferometer

The reference mirror M and the beam-

splitter B are placed outside the thermostat.

The upper face of the sample S is one

interference plane and the upper surface of

the bottom plate is the other. The

interference pattern IP is divided into two

fields corresponding to the two ends of the

sample. The difference of fringe

movements within these two fields yields

the absolute thermal expansion of the

sample.




– Thermal capacity











1.3 Thermal Conduction


1.3 Thermal Conduction

1.3.1 Category of Thermal Conduction

1.3.1.1 Steady Conduction

tAdx

dTQ

Thermal Conduction:

The conduction of thermal

energy from hot side to

cold side in a solid

material. For steady state

conduction:

dx

dT

tA

Qq

Q: heat T: temperature x: dimension

A: area λ: Thermal conductivity

q: heat flux

Heat flow

Low T High T

dx

Area A


1.3.1.2 Transient Conduction

Transient conduction: the temperature in a heat conduction process varies

with time：

2

2

dx

Td

cdt

dT

p

(ρ density ；cp heat capacity at constant pressure)

Under transient conduction, the rate of temperature change is in

proportional relationship with thermal conductivity, but inverse

proportional to the volume heat capacity ( ). Thermal diffusivity is

thus defined as:

pc

Thermal diffusivity is the measure of thermal inertia; It has the SI unit of m²/s

pc

http://en.wikipedia.org/wiki/SI


1.3.1.2 Transient Conduction

Thermal diffusivity of selected materials and substances

Pyrolytic graphite, parallel to layers 1.22 × 10−3

Pyrolytic graphite, normal to layers 3.6 × 10−6

Gold 1.27 × 10−4

Copper at 25C 2.55 × 10−4

Aluminium 8.418 × 10−5

Steel, stainless 304A 4.2 × 10−6

Silicon 8.8 × 10−5

Quartz 1.4 × 10−6

PC (Polycarbonate) at 25°C 0.144 × 10−6

PTFE (Polytetrafluorethylene) at 25°C 0.124 × 10−6

Water at 25°C 0.143 × 10−6

Air (300 K) 1.9 × 10−5

Glass, window 3.4 × 10−7

Rubber 1.3 × 10−7

Wood (Yellow Pine) 8.2 × 10−8


1.3.2 Physics of Thermal Conduction

• Free electrons - electronic thermal conduction ；

• Lattice vibrational wave－ phononic thermal conduction ；

• Electromagnetic radiation － photonic thermal conduction 。

Solid thermal conductivity:

j

jjj LVC3

1

j －carrier type；Cj －unit volume heat capacity for j carrier

Vj －velocity for j carrier；Lj －mean free path for j carrier。

Thermal conduction is an energy transport process.

There are three different carriers in a solid, which

represent three different ways of thermal conduction:


1.3.2. Mean Free Path

D

D

Total Length Traveled = L

Total Collision Volume

Swept = D2L

Number Density of Molecules = n

Total number of molecules encountered in

swept collision volume = nD2L

Average Distance between

Collisions, mc = L/(#of collisions)

Mean Free Path

nLDn

Lmc

12

: collision cross-sectional area



Number Density of

Molecules from Ideal

Gas Law: n = P/kBT

kB: Boltzmann constant

1.38 x 10-23 J/K

Mean Free Path:

P

Tk

nB

mc 1

Typical Numbers:

Diameter of Molecules, D 2 Å = 2 x10-10 m

Collision Cross-section: 1.3 x 10-19 m2

Mean Free Path at Atmospheric Pressure:

m0.3or m103103.110

3001038.1 7

195

23

mc

At 1 Torr pressure, mc 200 m; at 1 mTorr, mc 20 cm



Length Scale

1 m

1 mm

1 m

1 nm

Human

Automobile

Butterfly

1 km

Aircraft

Computer

Wavelength of Visible Light

MEMS

Width of DNA

MOSFET, NEMS

Blood Cells

Microprocessor Module

Nanotubes, Nanowires

100 nm

l


1.3.2.1 Electronic Thermal Conduction

Free electrons play major role in electronic thermal

conduction. In metals, free electrons can be view as electronic

gases so the energy transfer between free electrons is similar

to the energy transfer between gas molecules. In gas state, the

gas molecules transfer energy through collision:


1.3.2.1 Electronic Thermal Conduction

eeee LVC3

1

For Gas thermal conduction

Cg－gas unit volume heat capacity

Vg－gas molecue velocity

Lg －mean free path for gas molecule。

gggg LVC3

1

Same for electronic thermal conduction:

Ce－ unit volume heat capacity for free electrons

Ve－free electron velocity

Le －mean free path for free electrons。


1.3.2.2 Phononic Thermal Conduction

Thermal conduction through lattice vibration; treated as particles:

Similar expression:

aaaa LVC3

1

Ca－ unit volume heat capacity for phonons

Va－phonon velocity

La－mean free path for phonons。

Phonon Scattering Mechanisms

• Boundary Scattering

• Defect & Dislocation Scattering

• Phonon-Phonon Scattering

phononboundarydefecta LLLL

1111


1.3.2.3 Photonic Thermal Conduction

At high temperature, materials emit high frequency

electromagnetic wave. Thermal conduction through thermal

radiation of such electromagnetic wave, similar to photon

transport in a media:

dLVC rrrr3

1

rr LTn 32

3

16

σ－The Stefan-Boltzmann Constant；n －refraction index；T －temperature, Lr – mean free path。


1.3.3 Thermal Conductivity vs Electrical Conductivity

Wiedeman-Franz law：The ratio of metal’s thermal

conductivity(electron portion) over electrical conductivity

(λ/σ ) has approximately the same value for different metals at

the same temperature

LT

The ratio of metal’s thermal conductivity over electrical

conductivity (λ/σ ) is proportional to the temperature (T)

L －Lorentz constant： 28

22

1045.23

KW

e

kL

k －Boltzman constant；e －electron net charge。


1.3.4 Thermal Conductivity of Common Materials


1.3.4.1 Temperature Effect

For crystalline materials, thermal resistance (ω) is the sum of lattice thermal

resistance (ωP ) and defect thermal resistance (ω0 )

Low T dominated by defect resistance

High T dominated by lattice resistance

Low T： T

0

High T： .Constp

Medium T： 2

0 TT

p

2

1

TT

So：


Non-Crystalline Materials and Polymers

For non-crystalline materials, thermal conductivity often increases with

temperature.

g’

F

h’

h g

λ

T/K

O

OF：400-600K

Dominated by

phonon conduction;

as T increases, heat

capacity increases

and phonon

conductivity

increases

Fg’：600-900K，The increase of

heat capacity

gets small, but

photon

conduction

increases。

g’h’：higher

than 900K，phonon

conduction is

stable, but the

photon

conduction

increase with

T3.

For non-transparent inorganic materials,

photon conduction is small. Conduction mostly is

phonon conduction and is stable.


1.3.4.2 Materials Composition and Structure

• Atomic structure

- Monovalent elements tend to have better thermal conductivity, such as

alkali metals , Cu, Ag, Au etc.

- Normally metals that have high electronic conductivity tend to

conduct heat better, and follow Widemann – Franz law。

• Composition

– Impurities and alloying elements will interfere with electron

transport and decrease thermal conductivity；

– Ordered solid solutions tend to have larger lattice constant, increase

the electron free path, and thus thermal conductivity.


1.3.4.2 Materials Composition and Structure

• Crystalline structure

– The more complex the crystalline structure, the higher probability

phonons get scattered and the thermal conductivity gets reduced

• Grain size

– Electrons are scattered by grain boundaries, so smaller grains will

decrease thermal conductivity

• Multiphase composite materials

– Thermal conductivity can be calculated with linear mixing.

• Porosity (p)

– λ=λs（1－P）(assuming thermal conductivity close to zero for pores

filled with air)

λs is the thermal conductivity without pores.


1.3.5 Measure of the thermal conductivity

l

TTd

l

TTSQ 21

2

21

4

The quantity of heat traveled through

area S in time τ：

34 TTmCQ If water cooling is used at the low-temperature side, within

time τ, the temperature of water increases from T3 to T4 (mass:

m, capacity: C), then：

21

2

344

TTd

mTTlC

From two equations above, we have：

The schematic diagram of thermal conductivity detector The measuring principle of thermal conductivity detector


1.3.6 Other ways of thermal conduction


1.4 Method of thermal analysis

• Common approach

– Measure the T-t curve（The curve of heat analysis）

• Differential approaches

– DTA－Differential Thermal Analysis

Measure the ΔT-T or ΔT-t curve

– DSC－Differential Scanning Calorimetry

Measure ΔQ-T curve

1.4.1 Thermal analysis


Extended methods of thermal analysis

Physical properties Name of thermal analysis technique Abbreviation

mass

Thermogravimetry TG

The detection of isobaric mass change

The detection of escaping gas

Analysis of escaping Gas EDG

Analysis of thermal radiation EGA

Analysis of hot particle s

Temperature

Determination of heating curve

Differential thermal analysis DTA

Heat Differential scanning calorimetry DSC

Size Thermal expansion method

Mechanical properties Thermo-mechanical analysis TMA

Dynamic thermo-mechanical analysis DMA

Acoustic performance Hot vocalizations

Thermal acoustic

Optical properties Thermo-optical

Electrical properties Thermoelectric

Magnetic properties Thermal-magnetic


1.4.1.1 DTA

Control sample requirement：

• Should be inert，without decomposition, phase transition or

damage during the whole range of temperature measurement;

•Can not react with the sample to be tested;

•The heat capacity and thermal conductivity of the control

sample should be as close to that of sample as possible.

Endothermic

Exothermic


DTA curve

• Endothermic transition Tsample<Tcontrol

• Exothermic transition Tsample> Tcontrol

T

Time Time Temperature

Control

Sample

ΔT = TS-TC ΔT


1.4.1.2 DSC


The key points in this chapter

• The heat capacity – The basic concept of heat capacity; the classical theory of

heat capacity; the quantum theory of heat capacity

– Method of thermal analysis（DTA、DSC）

• Thermal expansion – Basic concept and physics

– The influencing factors of thermal expansion

– Measure of the thermal expansion coefficient

• Heat conduction – The basic concept of thermal conductivity

– The physics of thermal conductivity: electronic thermal conductivity, phonon thermal conductivity, photon thermal conductivity

– Weidmann - Franz law


The key concepts in this chapter

Heat capacity Specific heat capacity

Specific heat capacity at constant

pressure Specific heat capacity at constant volume

Molar heat capacity Dulong-Petit law

Neumann-Kopp law Debye Temperature

Coefficient of thermal expansion Thermal conductivity

Anomalous expansion Coefficient of thermal diffusivity

Coefficient of thermal conductivity Electronic thermal conductivity

Coefficient of thermal resistivity Photonic thermal conductivity

Phononic thermal conductivity Thermal stability

Wiedemann-Franz Law Differential therml analysis

Therml analysis Differentlial scanning calorimetry


Homework

Pick either heat capacity, thermal expansion or thermal

conduction as your topic area.

Read one paper in your topic area that’s published in

Science, Nature, Advanced Materials, Nano Letters or

other top English journals.

Write a paragraph of your learning after reading the

paper.

properties of materials

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