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THERMAL PROPERTIES
Heat Capacity
The ability of a material to absorb heat
• Quantitatively: The energy required to produce a unit rise in
temperature for one mole of a material.
heat capacity
(J/mol-K)
dQ
C
dT
energy input (J/mol)
temperature change (K)
• Two ways to measure heat capacity:
Cp : Heat capacity at constant pressure.
Cv : Heat capacity at constant volume.
Cp usually > Cv
J
Btu




• Heat capacity has units of
mol  K  lb  mol  F 
Chapter 17 -
1
Dependence of Heat Capacity on
Temperature
• Heat capacity...
-- increases with temperature
-- for solids it reaches a limiting value of 3R
R = gas constant 3R
= 8.31 J/mol-K
0
Cv = constant
0
qD
• From atomic perspective:
T (K)
Adapted from Fig. 17.2,
Callister & Rethwisch 3e.
Debye temperature
(usually less than T room )
-- Energy is stored as atomic vibrations.
-- As temperature increases, the average energy of
atomic vibrations increases.
Chapter 17 - 2
Atomic Vibrations
Atomic vibrations are in the form of lattice waves or phonons
Adapted from Fig. 17.1,
Callister & Rethwisch 3e.
Chapter 17 - 3
Specific Heat: Comparison
increasing cp
Material
• Polymers
Polypropylene
Polyethylene
Polystyrene
Teflon
cp (J/kg-K)
at room T
1925
1850
1170
1050
• Ceramics
Magnesia (MgO)
Alumina (Al2O3)
Glass
940
775
840
• Metals
Aluminum
Steel
Tungsten
Gold
900
486
138
128
cp (specific heat): (J/kg-K)
Cp (heat capacity): (J/mol-K)
Selected values from Table 17.1,
Callister & Rethwisch 3e.
Chapter 17 - 4
Other heat capacity contributions
• Electrons absorbing energy by increasing their kinetic
energy
– Only for free elections
– Very small for insulating/semiconducting materials
• Randomization of electron spin at some specific
temperature
• These two factors are (very) small compared to the
contribution from the vibrations
Chapter 17 - 5
Thermal Expansion
Materials change size when temperature
is changed
 initial
 final
Tinitial
Tfinal
Tfinal > Tinitial
 f   0 

 α (T finalT initial
)
0

linear coefficient of
thermal expansion (1/K or 1/°C)
V
V0
 v T
For isotropic materials
αv = 3αl
Chapter 17 -
Atomic Perspective: Thermal
Expansion
Thermal expansion arises from an increase in the average distance
between the atoms
Asymmetric curve:
-- increase temperature,
-- increase in interatomic
separation
-- thermal expansion
Symmetric curve:
-- increase temperature,
-- no increase in interatomic
separation
Chapter 17 -- no thermal expansion
Coefficient of Thermal Expansion:
Comparison
Material
increasing 
• Polymers
Polypropylene
Polyethylene
Polystyrene
Teflon
• Metals
Aluminum
Steel
Tungsten
Gold
• Ceramics
Magnesia (MgO)
Alumina (Al2O3)
Soda-lime glass
Silica (cryst. SiO2)
 (10-6/C)
at room T
145-180
106-198
90-150
126-216
23.6
12
4.5
14.2
Polymers have larger 
values because of weak
secondary bonds
• Q: Why does 
generally decrease
with increasing
bond energy?
A: deeper and narrower
energy “trough”
13.5
7.6
9
0.4
Chapter 17 - 8
Thermal Expansion: Example
Ex: A copper wire 15 m long is cooled from
40 to -9°C. How much change in length will it
experience?
• Answer: For Cu
  16.5 x 106 ( C)1
rearranging Equation 17.3b
     0 T  [
16.5 x 10 6 (1/ C)](15 m)[ 40C  ( 9C)]
  0.012 m  12 mm
Chapter 17 - 9
Thermal Conductivity
The ability of a material to transport heat.
Fourier’s Law
heat flux
(J/m2-s)
dT
q  k
dx
temperature
gradient
thermal conductivity (J/m-K-s)
T2
T1
x1
heat flux
T2 > T1
x2
• Atomic perspective: Atomic vibrations and free electrons in
hotter regions transport energy to cooler regions.
Chapter 17 - 10
Mechanisms of Heat Conduction
Chapter 17 - 11
increasing k
Thermal Conductivity: Comparison
Material
k (W/m-K)
• Metals
Aluminum
247
Steel
52
Tungsten
178
Gold
315
• Ceramics
Magnesia (MgO)
38
Alumina (Al2O3)
39
Soda-lime glass
1.7
Silica (cryst. SiO2)
1.4
• Polymers
Polypropylene
0.12
Polyethylene
0.46-0.50
Polystyrene
0.13
Teflon
0.25
Energy Transfer
Mechanism
atomic vibrations
and motion of free
electrons
(electron motion is
more efficient)
atomic vibrations
(phonons are primarily
responsible)
vibration/rotation of
chain molecules
Selected values from Table 19.1, Callister & Rethwisch 3e.
Chapter 17 - 12
Thermal Stresses
• Occur due to:
-- restrained thermal expansion/contraction
-- temperature gradients that lead to differential
dimensional changes
Thermal stress 
 E (T0 Tf )  E T

Chapter 17 - 13
Example Problem
-- A brass rod is stress-free at room temperature (20°C).
-- It is heated up, but prevented from lengthening.
-- At what temperature does the stress reach -172 MPa?
Solution:
T0
Original conditions
0
Step 1: Assume unconstrained thermal expansion
0


Tf
 thermal   (Tf T0 )
room
Step 2: Compress specimen back to original length
0




compress 

 thermal
room
Chapter 17 -
Example Problem (cont.)
0


The thermal stress can be directly
calculated as
  E(compress)
Noting that compress = -thermal and substituting gives
  E(thermal )  E
 (Tf T0 )  E (T0 Tf )
Rearranging and solving for Tf gives

20ºC
Tf  T0 
Answer: 106°C

100 GPa

E
-172 MPa (since in compression)
20 x 10-6/°C
Chapter 17 - 15
Thermal Shock Resistance
• Occurs due to: nonuniform heating/cooling
• Ex: Assume top thin layer is rapidly cooled from T1 to T2
rapid quench
tries to contract during cooling
T2
resists contraction
T1

Tension develops at surface
  E (T1 T2 )
Critical temperature difference
for fracture (set  = f)
Temperature difference that
can be produced by cooling:
(T1  T2 ) 
quench rate
k

(T1 T2 ) f racture 
f
E
set equal
• (quench rate) f or f racture  Thermal
Shock Resistance ( TSR) 
• Large TSR when
f k
E
f k
is large
E
Chapter 17 - 16
Summary
The thermal properties of materials include:
• Heat capacity:
-- energy required to increase a mole of material by a unit T
-- energy is stored as atomic vibrations
• Coefficient of thermal expansion:
-- the size of a material changes with a change in temperature
-- polymers have the largest values
• Thermal conductivity:
-- the ability of a material to transport heat
-- metals have the largest values
• Thermal shock resistance:
-- the ability of a material to be rapidly cooled and not fracture
-- is proportional to
f k
E
Chapter 17 - 17
Magnetic Properties
Generation of a Magnetic Field -- Vacuum
• Created by current through a coil:
B0
H
I
N = total number of turns
 = length of each turn (m)
I = current (ampere)
H = applied magnetic field (ampere-turns/m)
B0 = magnetic flux density in a vacuum
(tesla)
• Computation of the applied magnetic field, H:
H
NI
(A / m)
• Computation of the magnetic flux density in a vacuum, B0:

B0 = 0H
permeability of a vacuum
(1.257 x 10-6 Henry/m)
Chapter 17 - 18
Magnetic field vector etc.
• Electric field can be defined in terms of the force acting on a
charge
– Ie, if a charge q at rest is experiencing electric force FE from
the electric field E,
– FE = qE
– where E is electric field.
• Magnetic field could be defined in a similar way, if there is a
monopole, but there’s none
• Instead, we can use the moving charge q with velocity v.
• The force acted on the moving charge by a magnetic field B is
– FB = qv x B
– Then the unit of B is
N / (coulomb m / s) = N/(A m) = T = Tesla
Chapter 17 - 19
Magnetic field vector etc.
Chapter 17 - 20
Finally
B0 = 0H
Chapter 17 - 21
H&B
• B : magnetic field in the material
– From external magnetic field + “internal” magnetic
field (magnetization)
– Magnetic induction, magnetic flux density
• H: “Driving” magnetic influence from external field
– Magnetic field strength
• In vacuum, B & H are essentially same (up to 0)
• H can be defined as
H = B/ 0 - M
Chapter 17 - 22
Generation of a Magnetic Field -within a Solid Material
• A magnetic field is induced in the material
B
applied
magnetic
field H
current I
B = Magnetic Induction (tesla)
inside the material
B = H
permeability of a solid

• Relative permeability (dimensionless) r 
0
Chapter 17 - 23
Generation of a Magnetic Field -within a Solid Material (cont.)
M = cmH
• Magnetization
Magnetic susceptibility
(dimensionless)
B = 0H + 0M
• B in terms of H and M
• Combining the above two equations:
B
B = 0H + 0 cmH
= (1 + cm)0H
cm > 0
vacuum cm = 0
cm < 0
H
permeability of a vacuum:
(1.26 x 10-6 Henry/m)
cm is a measure of a material’s
magnetic response relative to a
vacuum
Chapter 17 -
Origins of Magnetic Moments
• Magnetic moments arise from electron motions and the
spins on electrons.
magnetic moments
electron
electron
spin
nucleus
electron orbital
motion
Adapted from Fig. 18.4,
Callister & Rethwisch 3e.
electron
spin
• Net atomic magnetic moment:
-- sum of moments from all electrons.
-- in atom with completely filled shells, there is total
cancellation of orbital & spin moments -> cannot be
permanently magnetized
Chapter 17 - 25
B (tesla)
Types of Magnetism
(3) ferromagnetic e.g. Fe3O4, NiFe2O4
(4) ferrimagnetic e.g. ferrite(), Co, Ni, Gd
( cm as large as 106 !)
(2) paramagnetic ( cm ~ 10-4)
e.g., Al, Cr, Mo, Na, Ti, Zr
vacuum (cm = 0)
(1) diamagnetic (cm ~ -10-5)
e.g., Al2O3, Cu, Au, Si, Ag, Zn
H (ampere-turns/m)
Plot adapted from Fig. 18.6, Callister & Rethwisch 3e.
Values and materials from Table 18.2 and discussion in
Section 18.4, Callister & Rethwisch 3e.
Chapter 17 - 26
Diamagnetic / Paramagnetic
random
aligned
Applied
Magnetic Field (H)
(2) paramagnetic
opposing
No Applied
Magnetic Field (H = 0)
none
(1) diamagnetic
Chapter 17 -
27
Ferromagnetism
Chapter 17 - 28
Antiferromagnetism
• Antiferromagnetism
– Another form of magnetic moment
coupling between the adjacent
atoms/ions
– Forms antiparallel alignment of
magnetic dipoles
– MnO (Manganese Oxide)
– O2- ion has no net magnetic moment
– Mn2+ has net magnetic moment (due to
spin)
– They align so that the adjacent ions are
antiparallel, resulting in no net magnetic
moment overall
Chapter 17 - 29
Ferrimagnetism
• Like ferromagnets, permanent
– Consider the mineral magnetite Fe3O4
– Can be written Fe2+O2--(Fe3+)2(O2-)3
Chapter 17 - 30
Effect of temperature
• The atomic thermal motion counteract
the coupling forces between the
atomic dipole moments -> Decrease in
saturation magnetization
• At Curie temperature Tc, it becomes 0
Chapter 17 - 31
Domains
• Domain: small region in
ferro/ferrimagnetic solid where there is a
alignment along the same direction
– Very small, smaller than grain
– Magnetization of entire solid is the
vector sum of the magnetization of all
domains
Chapter 17 - 32
Domains & Hysteresis
• B does not increase linearly with M
– Recall B = H
– So  changes with H
– Initial permeability i.
– Starting with initially randomly
oriented domain, the domains
aligning with the magnetic fields
grows
– At maximum saturation, a single
domain remains.
Chapter 17 - 33
Domains & Hysteresis
•
•
•
•
•
From saturation, reduce H
The B-H curve does not retraces the original path –
HYSTRTERESIS
B decreases “slower” than H and does not become
0 when H = 0 -> remanence (remanent flux density
Br)
– Single domain rotates
– New domain forms
– Some domain with original direction remains
(remanence)
Can be explained in terms of domain
– Resistance to domain wall motion
Hc: coercivity
– Magnitude need to make B=0
Chapter 17 - 34
Hysteresis and Permanent
Magnetization
• The magnetic hysteresis phenomenon
B
Stage 3. Remove H, alignment
remains! => permanent magnet!
Stage 2. Apply H,
align domains
H
Stage 4. Coercivity, HC
Negative H needed to
demagnitize!
Stage 5. Apply -H,
align domains
Adapted from Fig. 18.14,
Callister & Rethwisch 3e.
Stage 1. Initial (unmagnetized state)
Stage 6. Close the
hysteresis loop
Chapter 17 - 35
Hard and Soft Magnetic Materials
B
-- large coercivities
-- used for permanent magnets
-- add particles/voids to
inhibit domain wall motion
-- example: tungsten steel -Hc = 5900 amp-turn/m)
Soft
Hard magnetic materials:
H
Soft magnetic materials:
-- small coercivities
-- used for electric motors
-- example: commercial iron 99.95 Fe
Adapted from Fig. 18.19, Callister & Rethwisch
3e. (Fig. 18.19 from K.M. Ralls, T.H. Courtney,
and J. Wulff, Introduction to Materials Science
and Engineering, John Wiley and Sons, Inc.,
1976.)
Chapter 17 - 36
Superconductivity
Found in 26 metals and hundreds of alloys & compounds
Mercury
Copper
(normal)
4.2 K
Fig. 18.26, Callister &
Rethwisch 3e.
• TC = critical temperature
= temperature below which material is superconductive
Chapter 17 - 37
Critical Properties of
Superconductive Materials
TC = critical temperature - if T > TC not superconducting
JC = critical current density - if J > JC not superconducting
HC = critical magnetic field - if H > HC not superconducting
 T 2 
HC (T )  HC (0)1  2 
 TC 

Fig. 18.27, Callister &
Rethwisch 3e.
Chapter 17 - 38
Summary
• A magnetic field is produced when a current flows
through a wire coil.
• Magnetic induction (B):
-- an internal magnetic field is induced in a material that is
situated within an external magnetic field (H).
-- magnetic moments result from electron interactions with
the applied magnetic field
• Types of material responses to magnetic fields are:
-- ferrimagnetic and ferromagnetic (large magnetic susceptibilities)
-- paramagnetic (small and positive magnetic susceptibilities)
-- diamagnetic (small and negative magnetic susceptibilities)
• Types of ferrimagnetic and ferromagnetic materials:
-- Hard: large coercivities
-- Soft: small coercivities
• Magnetic storage media:
-- particulate g-Fe2O3 in polymeric film (tape)
-- thin film CoPtCr or CoCrTa (hard drive)
Chapter 17 - 39
ANNOUNCEMENTS
Reading:
Core Problems:
Self-help Problems:
Chapter 17 - 40
(3) Ferromagnetism
• Saturation magnetization Ms:
– Maximum possible magnetization
– Magnetization when all the dipoles are
mutually aligned with the external field
Chapter 17 - 41
Chapter 17 - 42
Chapter 17 - 43