Thermal Properties of Engineering Materials

Brief summary of topic but will expand over time

What governs macroscopic thermal properties?

heat capacity, thermal expansions and thermal conductivity

Thermal stresses lead to …

catastrophic failure

Heat capacity is …

the ability to absorb heat from external surroundings.

MORE SPECIFICALLY heat capacity is…

amount of energy required to produce a unit temperature rise

Heat capacity in mathematical terms…

C = dQ/dT

What are the two ways of measuring heat capacity?

At constant volume or at constant pressure

What is usually slightly bigger at room temp and above, C_v or C_p?

C_p is slightly greater - BUT difference usually negligible

How is heat primarily stored in solids?

Vibrational energy

Atoms in solid crystal lattice constantly vibrating.

When thermal energy added to material…

there’s an increase in amplitude and frequency of vibrations

Effect of adjacent atoms being coupled due to atomic bonding when thermal energy applied…

vibrations travel through the crystal lattice in the form of elastic waves

How is energy is energy in the form of elastic waves quantised?

Single quantum of vibrational energy is called a phonon

What does thermal energy absorption generate more of ?

Phonons

Total vibrational energy of a solid is …

sum of energy of all its phonons

Heat capacity is not constant.

It is dependent on …

Temperature

At absolute zero, vibrational energy at minimum and C_v is 0.

As T rises, what happens to C_v

it increases rapidly, then plateaus at higher temperatures

What is C_v like at very low temperatures, mathematically?

C_v proportional to T³

Change in dimensions with temperature quantified by …

coefficients of thermal expansion, a_l

What is a_l

material property which indicates the extent to which a length of material expands upon heating

Define a_l mathematically

change in length/original length = a_l * change in T

volume coefficient of thermal expansion a_v describes what?

fractional change in volume with temperature

For isotropic materials, approximate a_v

a_v = 3*a_l

What is shown by the potential energy versus interatomic distance curve?

the relationship between the potential energy of a system of atoms and their relative positions

What happens, in terms of potential energy, when atoms move closer together?

their potential energy decreases due to attractive forces between them What happens, in terms of potential energy, when atoms move closer together

What happens, in terms of potential energy, when atoms reach equilibrium bond length?

potential energy reaches a minimum and the atoms are most stable

Effect of increase in energy on a perfectly symmetric energy trough …

most energy troughs for solids are not symmetric

would not change the average interatomic distance - SO no thermal expansion would occur

Describe the typical energy trough for solid materials

asymmetric, repulsive force curve rises much more steeply than the attractive force curve

What do very deep, narrow energy troughs indicate about a material?

strong interatomic bonding and likely a lower coefficient of thermal expansion

Describe expansion in metals

Coefficient of thermal expansion: 5×10^-6 to 25×10^-6

intermediate a_l, mid expansion

strong metallic bonding keeps expansion moderate

metals// higher melting point: stronger bonds, deeper potential energy troughs, lower a_l values

a_l dependent on melting point

Describe expansion in ceramics

Coefficient of thermal expansion: 0.5×10^-6 to 15×10^-6

very low a_l, least expansion

very strong interatomic bonds: ionic or covalent

highly resistant to thermal shock

fused silica has a_l close to 0

Describe expansion in polymers

Coefficient of thermal expansion: 50×10^-6 to 400 ×10^-6

high a_l, highest rates of thermal expansion

weaker van der waals bonds holding polymer chains together

leads to significant macroscopic expansion - T increases and weak bonds allow chains to easily move further apart

Define thermal conductivity

the phenomenon by which heat is transported from high temperature regions to low temperature regions within a solid

Define thermal expansion macroscopically, mathematically

Fourier’s Law: q = -k(dT/dx)

q is the heat flux

dT/dx is the temperature gradient within the medium

k is the thermal conductivity of the medium

What is the significance of the negative sign in Fourier’s law

Heat flows down the temperature gradient - hot to cold

The two primary mechanisms by which heat is transported through a solid are …

phonons: lattice vibration waves

translational motion of free electrons

Mathematically, the total thermal conductivity is …

the sum of the lattice/phonon conductivity and the electronic conductivity

k = k_l + k_e (respectively)

Atomically, what two factors increase when a region of a solid is heated?

the amplitude and frequency of atom vibrations

How do enhanced vibrations, in a solid when heated, travel through the lattice?

as phonons, carrying thermal energy to cooler regions

What additional feature of metals carries heat to cooler regions?

(not phonons)

free electrons in the hot region gain kinetic energy and migrate to cooler regions

here energy is transferred to the lattice via collisions

What is the Wiedemann-Franz law?

The Wiedemann-Franz law states that the ratio of the electronic contribution of the thermal conductivity (κ) to the electrical conductivity (σ) of a metal is proportional to the temperature (T).

What is the formula of the Wiedemann-Franz law?

$$\frac{\kappa}{\sigma} = LT$$

thermal conductivity (κ)

electrical conductivity (σ)

proportionality constant, Lorenz number (L)

In what materials do the electron mechanism heavily dominate the phonon mechanism

high-purity metals

Effect of alloying metals on the thermal conductivity and why does this occur?

decreases thermal conductivity

impurity atoms introduce severe scattering centers

• impede on both electron and phonon motion

What materials have practically no electron contribution?

non metallic materials such as glass and ceramics

lack a significant concentration of free electrons

(also polymers)

What mechanism are ceramics dependent on for thermal conductivity, atomically?

almost entirely dependent on phonons (lattice vibrations)

What are phonons and how are they scattered in ceramics?

elastic waves

scattered by imperfections within the crystal lattice

in ceramics: scattered by lattice defects, grain boundaries, and other phonons

as T increases, more atomic vibrations and more frequent phonon-phonon collisions (Umklapp scattering)

What happens when heat energy can no longer travel through a ceramic due to Umklapp scattering?

Thermal conductivity drops dramatically

Heat conduction in ceramics primarily relies on phonon transport.

If Umklapp scattering becomes so strong that phonons lose their ability to carry heat efficiently, the material’s thermal conductivity plummets.

could lead to structural or functional consequences

How does porosity effect thermal conductivity of ceramics

pores are gas filled voids

they have exceptionally low thermal conductivity

engineers exploit this by designing highly porous ceramic structures for use as a thermal insulator

What mechanism are polymers dependent on for thermal conductivity, atomically?

scattering of vibrational energy

caused by complex, intertwined and largely amorphous structure of polymer chains

What atomic movements does heat transfer within polymers rely heavily on?

translation, rotation and vibration of molecular chain segments

BUT movements highly restricted so heat transfer is very slow

How can thermal conductivity be increased (slightly)?

increase the crystallinity of the polymer

more ordered lattice allows more efficient vibrational energy transfer compared to an amorphous structure

Define thermal stresses

stresses induced on a body as a result of temperature change

What can thermal stresses cause in materials?

plastic deformation

catastrophic fracture - especially in brittle materials

What two sources usually lead to thermal stresses?

mechanical constraint of uniform thermal expansion/contraction

presence of temperature gradients across a material

Effect of constrained expansion is …

internal compressive stresses

Effect of cooling a constrained body so that it cannot contract is …

internal tensile stresses

What is the magnitude of internal stresses caused by constraining thermal expansion/contraction, mathematically?

σ = E*a_l*change in T

magnitude of stress (σ)

modulus of elasticity (E)

coefficient of thermal expansion (a_l)

At what point will permanent damage or failure occur due to thermal expansion/contraction?

when stress exceeds yield strength or fracture strength

When can thermal stresses occur in material when body is completely unconstrained?

when temperature varies from one part of the body to another

Effect of rapid heating or cooling is …

surface changes T quicker than core

one wants to expand/contract whilst the other is restricting it

Effect: massive tensile stresses on surface whilst core experiences compressive forces to balance the load

Effect of rapid cooling on materials such as ceramics:

ceramics: no capacity for plastic deformation to relieve stress

rapid cooling induces surface tensile stresses that exceed fracture strength

leads to rapid crack propagation and explosive failure AKA thermal shock

What is the capacity of a material to withstand failure caused by thermal stress?

thermal shock resistance

High thermal shock resistance favored by …

high fracture strength

high thermal conductivity

low modulus of elasticity

low coefficient of thermal expansion

Example of material with good thermal shock resistance and why?

fused silica

near to zero coefficient of thermal expansion, a_l