1. Thermal Properties of Materials
Department of Physics
K L University

2. “Thermal Property”
It is the response produced by the material when subjected to Thermal Energy (Heat).
Response
Change in Temperature
Thermal Capacity
Change in Dimensions
Thermal Expansion
Transmission of Thermal Energy
Thermal Conduction

3. Thermal Properties: Heat Capacity Atomic vibrations, phonons
Temperature dependence
Contribution of electrons
2. Thermal Expansion
Connection to anharmonicity of interatomic potential
Linear and volume coefficients of thermal expansion
Thermal Conductivity
Heat transport by phonons and electrons

4. Introduction
The atoms in a solid are executing oscillations about their equilibrium positions with thermal energy.
Such oscillations in crystals are called LATTICE VIBRATIONS.
The lattice vibrations are responsible for the characteristic properties of matter such as specific heat, thermal conductivity, optical.

5. Thermal energy = kinetic energy of atomic motions + potential energy of distortion of interatomic bonds.
Vibrations of individual atoms in solids are not independent from each other. The coupling of atomic vibrations of adjacent atoms results in waves of atomic displacements.
Each wave is characterized by its wavelength and frequency. For a wave of a given frequency ν, there is the smallest “quantum” of vibrational energy, hν, called phonon.
Thus, the thermal energy is the energy of all phonons (or all vibrational waves) present in the crystal at a given temperature.

6. Atomic Vibrations

7. Heat Capacity
A solid material’s potential energy is stored as its heat energy.
Temperature of a solid is a measure its potential energy.
Heat capacity is a property that is indicative of a material’s ability to absorb heat from the external surroundings.
It is defined as the amount of energy required to produce a unit temperature rise.
Mathematically, it is expressed as:
Where dQ is the energy required to produce a temperature change equal to dT.
Heat capacity has units as J/mol-K or Cal/mol-K.

8. Specific Heat
For comparison of different materials, heat capacity has been rationalized.
Specific heat is heat capacity per unit mass. It has units as J/kg-K or Cal/kg-K.
With increase of heat energy, dimensional changes may occur. Hence, two heat capacities are usually defined.
Heat capacity at constant pressure, Cp, is always higher than heat capacity at constant volume, Cv.
Cp is ONLY marginally higher than Cv.
Heat is absorbed through different mechanisms: lattice vibrations and electronic contribution.

9. Classical Theory of specific heat
(Dulong- Petit Law)
Assumptions:
Vibrating atoms can be considered as linear harmonic oscillators
All the atoms are vibrating with the same frequency
The constant value of the heat capacity of many simple solids is sometimes called Dulong – Petit law
In 1819 Dulong and Petit found experimentally that for many solids at room temperature,
cv ≈ 3R = 25 J - K-1mol-1
This is consistent with equi partition theorem of classical mechanics: energy added to solids takes the form of atomic vibrations and both kinetic and potential energy is associated with the three degrees of freedom of each atom.

10. Merits & De-Merits:
This law of Dulong and Petit (1819) is approximately obeyed by most solids at high T ( > 300 K). But by the middle of the 19th century it was clear that CV  0 as T  0 for solids.

11. Quantum Theory of specific heat
(Einstein’s Theory)
The Law of Dulong and Petit assumed that Maxwell-Boltzmann statistics and equipartition of energy could be applied even at low temperatures.
Einstein recognized that for a quantum harmonic oscillator at energies less than kT, the Einstein-Bose statistics must be applied.
Einstein (1907): model a solid as a collection of 3N independent 1-D oscillators, all with constant , and use Planck’s equation for energy levels
[later QM showed is actually correct]

12. Quantum Theory of specific heat
(Einstein’s Theory)
This was the same conclusion that was drawn about blackbody radiation. The statistical distribution of energy in the vibrational states gives average energy:
For high temperatures, this expression approaches agreement with the Law of Dulong and Petit.

13. Debye’s Theory of specific heat
Einstein's oscillator treatment of specific heat gave qualitative agreement with experiment and gave the correct high temperature limit (the Law of Dulong and Petit).
The quantitative fit to experiment was improved by Debye's recognition that there was a maximum number of modes of vibration in a solid.
He pictured the vibrations as standing wave modes in the crystal, similar to the electromagnetic modes in a cavity which successfully explained blackbody radiation.
The density of states for these modes, which are called "phonons", is of the same form as the photon density of states in a cavity.

14. To impose a finite limit on the number of modes in the solid, Debye used a maximum allowed phonon frequency now called the Debye frequency uD. In the treatment of specific heat, we define a Debye temperature by
For low temperatures, Debye's treatment led to a specific heat
The dependence upon the cube of the temperature agreed with experimental results for nonmetals, and for metals when the electron specific heat was taken into account.

15. Debye’s Theory of specific heat
Heat capacity has a weak temperature dependence at high temperatures (above Debye temperature θD) but decreases down to zero as T approaches 0K.
The low-T behavior can be explained by quantum theory.
The first explanation was proposed by Einstein in He considered a solid as an ensemble of independent quantum harmonic oscillators vibrating at a frequency ν.
Debye advanced the theory by treating the quantum oscillators as collective modes in the solid (phonons) and showed that

16. At low temperatures, vibrational heat contribution of heat capacity varies with temperature as follows:
The above relation is not valid above a specific temperature known as Debye temperature. The saturation value is approximately equal to 3R.

17. The electron contribution to cv is proportional to temperature,
cvel = γT
and becomes significant (for metals only) at very low temperatures
(remember that contribution of phonons cv ~ AT3 at T → 0K).

18. directions translational invariance is assumed
Crystal vibrations with mono atomic basis
Schematic illustration of displacements of planes along one spatial direction. In the two orthogonal
directions translational invariance is assumed

19. Schematic illustration of transverse displacements.

20. The force resulting from the displacements un; um is then given by:
If we consider nearest neighbours, the total force acting upon an atom within a plane

21. Newton’s second law then yields,
(1)
This differential equation can be solved by assuming a wavelike solution with wave vector ~k and with an additional phase factor which scales linearly with the plane position, i.e.
After substituting the un un+1 un-1 values in eq’n (1) and simplifying the above equation, we will get
This is the dispersion relation for an elastic wave of wave vector k and of frequency ω

23. Fig : Illustration of the dispersion relation

24. Phonon momentum
A lattice vibration is characterized by its frequency ω and its wave vector k
What is the physical meaning of k ?
Is the wave vector ?
Related to the momentum ?
No it is not, because the motion we are describing with k, is a motion of the relative coordinates.
The physical momentum is given by the following relation:

25. This quantity is known as the crystal momentum.
We therefore conclude that phonons carry no real physical momentum but they do behave (in their interactions with other particles) as if they carried a definite momentum
This quantity is known as the crystal momentum.
Energy momentum

26. Applications of Specific Heat
1. Substances having a small specific heat capacity can be quickly heated up, it also experience a big change in temperature even though only small amount of heat is supplied. 2. Substances having a small specific heat capacity, are very useful as material in cooking instruments such as frying pans, pots, kettles and so on, because, they can be quickly heated up even when small amount oh heat is supplied.
Substances that have a high specific heat capacity is suitable as a material for constructing kettle handlers, insulators and oven covers, because, a high amount of heat will cause only a small change in temperature aka the material won't get hot too fast!

27. 4. Sensitive thermometers also must be made from materials with small specific heat capacity so that it can detect  and show a change of temperature rapidly and accurately.
5. Heat storage instruments are very useful and they are usually made of substances with a high specific heat capacity.
6. Water as a cooling agent acts excellent as a cooling agent in engines. Water is also used in houses in cold climate countries because as it is heated up (boiled) it tends to retain heat and warm the house due to its high specific heat capacity.

28. Characteristics of an object with low specific heat:
- Fast heated up: have a faster temperature increase
- Fast cooled down: have a faster temperature decrease
- Sensitive to temperature changes
Ex: Aluminium, copper etc
Frying pans, pots, kettles- Low Specific Heat.
Characteristics of an object with high specific heat:
Heats up and cools down at a slower rate
Requires more heat to rise its temperature by a specific
amount
- Can absorb a great amount of heat
Ex: plastic, water, concrete etc.
kettle handlers, insulators and oven covers- high specific heat capacity

29. Thermal Expansion
Increase in temperature may cause dimensional changes.
Linear coefficient of thermal expansion (α) defined as the change in the dimensions of the material per unit length.
T0 and Tf are the initial and final temperatures (in K)
l0 and lf are the initial and final dimensions of the material and ε is the strain.
α has units as (°C)-1.
α values: for metals 5-25x10-6 , for ceramics x10-6 ,for polymers x10-6

30. A volume coefficient of thermal expansion, αv (=3α) is used to describe the volume change with temperature.
Where Δv and vo are the volume change and the original volume.
An instrument known as dilatometer is used to measure the thermal expansion coefficient.
At microscopic level, because of asymmetric nature of the potential energy trough, than increase in vibrational amplitude. changes in dimensions with temperature are due to change in inter-atomic distance, rather than increase in vibrational amplitude

32. Thermal Expansion in Metals:
Linear coefficients of thermal expansion for some of the common metals range between about 5 x 10-6 and 25 x 10-6 (°C) -1.
iron-nickel and iron-nickel-cobalt alloys that have αl values on the order of
1 x 10-6 (°C)-1.
Thermal Expansion in Ceramics:
Relatively strong inter-atomic bonding forces are found in many ceramic materials
Comparatively low coefficients of thermal expansion range between about
0.5 x10-6 and 15 x 10-6 (°C)-1.
For non-crystalline ceramics and also those having cubic crystal
structures, αl is isotropic otherwise it is anisotropic.
For inorganic glasses, the coefficient of expansion is dependent on composition.

33. Thermal Expansion in Polymers
polymeric materials experience very large thermal expansions upon heating.
coefficients that range from 50 x 10-6 to 400 x 10-6 (°C)-1.
Expansion is higher in linear and branched polymers because the secondary intermolecular bonds are weak and minimum of cross linking.
With increased cross linking,expansion coefficient diminishes; the lowest coefficients are found in the thermosetting network
polymers such as phenol-formaldehyde, in which the bonding is almost entirely covalent.

34. Thermal Expansion

36. Applications of Thermal Expansion
Fixing of Iron Rim to a Wooden Wheel
Blacksmiths use the principle of expansion on heating and contraction on cooling, to fix the iron rim onto the wooden wheel of a bullock cart. The radius of the iron rim is slightly less than that of the wooden wheel. The iron rim is heated to red hot, so that it expands and its radius increases. It is then slipped over the wooden wheel and cooled by pouring water. The iron rim contracts on cooling and gets tightly fixed to the wooden wheel. Gap in Railway Tracks
While laying railway tracks, a small gap is left between adjacent rails. This is because the iron rails expand in summer, and the gap allows space for the expansion.
Opening a Tightly Fixed Lid of a Bottle 
When the lid or the cork on a bottle is too tight and can’t be opened, immerse the mouth of the bottle in hot water. The lid undergoes thermal expansion and becomes slightly loose, and then opens easily.
Thermometers
A thermometer also works on the principle of expansion and contraction of matter on heating and cooling. Depending on the type of material used in thermometers, they are classified into solid, liquid and gas thermometers.

37. Thermal Conductivity Thermal conductivity:
Thermal conductivity is ability of a material to transport heat energy through it from high temperature region to low temperature region.
The heat energy, Q, transported across a plane of area A in presence of a temperature gradient ΔT/Δl is given by
where k is the thermal conductivity of the material.
It has units as W/m.K.
It is a microstructure sensitive property.
Its value range
o for metals W/m.K
o for ceramics 2-50 W/m.K
o for polymers order of 0.3 W/m.K

38. Mechanisms - Thermal conductivity
Heat is transported in two ways – electronic contribution, vibrational (phonon) contribution.
In metals, electronic contribution is very high. Thus metals have higher thermal conductivities. It is same as electrical conduction. Both conductivities are related through Wiedemann-Franz law:
where L – Lorentz constant (5.5x10-9 cal.ohm/sec.K2)
As different contributions to conduction vary with temperature, the above relation is valid to a limited extension for many metals.

39. Mechanisms - Thermal conductivity
With increase in temperature, both number of carrier electrons and contribution of lattice vibrations increase. Thus thermal conductivity of a metal is expected to increase.
k=kl+ke
Where kl and ke represent the lattice vibration and electron thermal conductivities
However, because of greater lattice vibrations, electron mobility decreases.
The combined effect of these factors leads to very different behavior for different metals.
Eg.: Thermal conductivity of iron initially decreases then increases slightly; thermal conductivity decreases with increase in temperature for aluminium; while it increases for platinum

40. Thermal conductivity in Metals:
Common metals values generally range between about 20 and 400 W/m-K.
Here free electrons are responsible for both electrical and thermal conduction in pure metal.
Wiedemann–Franz law
Alloying metals with impurities results in a reduction in the thermal conductivity
copper–zinc alloys- Plot
Thermal Conductivity in Ceramics:
Nonmetallic materials are thermal insulators
Amorphous ceramics have lower conductivities than crystalline ceramics, since the phonon scattering is much more effective when the atomic structure is highly disordered and irregular

41. Thermal Conductivity in Polymers:
Thermal conductivities for most polymers are on the order of 0.3 W/m-K
Highly crystalline and ordered structure will have a greater conductivity than the equivalent amorphous material.
due to the more effective coordinated vibration of the molecular
chains for the crystalline state..

42. Thermal Conductivity in Metals

43. Applications of Thermal Conductivity:
In everyday life.
Some objects may feel cold to the touch if they are good conductors because they carry away heat from the bodyrapidly, so a concrete or tiled floor feels much colder to stand on than a carpeted one.
On the other hand, in a very hot room (e.g. Turkish bath), metal objects can feel very hot to the touch and may actually burn the skin. In a block of hot metal the atoms/molecules may vibrate rapidly, perhaps thousands of times each second. If one touches it with one’s finger,
the rapidly vibrating atoms cause the molecules of the skin to go into sudden and violent motion, resulting in the sensation of pain.

44. Heat Treatment Methods

45. Definition of heat treatment
Heat treatment is an operation or combination of operations involving
Heating at a specific rate,
Soaking at a temperature for a period of time and
Cooling at some specified rate.
The aim is to obtain a desired microstructure to achieve certain predetermined properties (physical, mechanical, magnetic or electrical).

46. Objectives of heat treatment
To increase strength, harness and wear resistance
(bulk hardening, surface hardening)
To increase ductility and softness
(tempering, recrystallization annealing)
To increase toughness
(tempering, recrystallization annealing)
To obtain fine grain size
(recrystallization annealing, full annealing, normalising)
To remove internal stresses induced by differential deformation by cold working, non-uniform cooling from high temperature during casting and welding
(stress relief annealing)

47. To improve machineability
(full annealing and normalising)
To improve cutting properties of tool steels
(hardening and tempering)
To improve surface properties
(surface hardening, corrosion resistance-stabilising treatment and high temperature resistance-precipitation hardening, surface treatment)
To improve electrical properties
(recrystallization, tempering, age hardening)
To improve magnetic properties
(hardening, phase transformation)

48. Heat treatment processes
Hardening
Tempering
Quenching
Nitriding

49. Hardening Definition:
Hardening is the process of heating a piece of steel to a temperature within or above its critical range and than cooling it rapidly (quenching)“ Or
Hardening is that property of a material that enables it to resist plastic deformation, penetration, indentation, scratching"

50. Hardening
Hardening is carried out by quenching a steel, that is cooling it rapidly from a temperature above the transformation temperature.
Steel is quenched in water or brine for the most rapid cooling, in oil for some alloy steels, and in air for certain higher alloy steels.
With this fast cooling rate, the transformation from austenite to pearlite cannot occur and the new phase obtained by quenching is called marten site.
Martensite is a supersaturated metastable phase and have body centered tetragonal lattice (bct) instead of bcc.
After steel is quenched, it is usually very hard and strong but brittle.
Martensite looks needle-like under microscope due to its fine lamellar structure.

51. Hardening means making a material, particularly a metal, physically harder, and includes particular cases such as: Hardening (metallurgy), the strengthening of metal alloys by heat treatment

52. Hardening

53. Hardening

54. Tempering
Tempering (formerly called drawing), consists of reheating a quenched steel to a suitable temperature below the transformation temperature for an appropriate time and cooling back to room temperature.
Freshly quenched marten site is hard but not ductile.
Tempering is needed to impart ductility to martensite usually at a small sacrifice in strength.

55. Tempering The effect of tempering may be illustrated as follows.
If the head of a hammer were quenched to a fully marten­sitic structure, it probably would crack after the first few blows.
Tempering during manufacture of the hammer im­parts shock resistance with only a slight decrease in hard­ness.
Tempering is accomplished by heating a quenched part to some point below the transformation temperature, and holding it at this temperature for an hour or more, depending on its size.

56. Tempering
The micro structural changes accompanying tempering include loss of acicular marten site pattern and the precipitation of tiny carbide particles.
This micro structural is referred to as tempered marten site.

57. Quenching
In materials science, quenching is the rapid cooling of a work piece to obtain certain material properties.
It prevents low-temperature processes, such as phase transformations, from occurring by only providing a narrow window of time in which the reaction is both thermodynamically favorable and kinetically accessible. For instance, it can reduce crystallinity and thereby increase toughness of both alloys and plastics (produced through polymerization).
In metallurgy, it is most commonly used to harden steel by introducing martensite, in which case the steel must be rapidly cooled through its eutectoid point, the temperature at which austenite becomes unstable. In steel alloyed with metals such as nickel and manganese, the eutectoid temperature becomes much lower, but the kinetic barriers to phase transformation remain the same. This allows quenching to start at a lower temperature, making the process much easier. High speed steel also has added tungsten, which serves to raise kinetic barriers and give the illusion that the material has been cooled more rapidly than it really has. Even cooling such alloys slowly in air has most of the desired effects of quenching.

58. Nitriding
It is the introduction of the nitrogen in to the surface of certain type of steels by heating it and holding it at a suitable temperature in contact with particularly dissociate (distance) ammonia or other suitable media.
This produces a hard case with out quenching or any other heat treatment.