PHY1039
Properties of Matter
Macroscopic (Bulk) Properties:
Thermal Expansivity, Elasticity and Viscosity
20 & 23 February, 2012
Lectures 5 and 6
3-D
+
Thermal Expansivity, b
F
Vo To
P = FA
+ dT
Vo+ dV
+F
T + dT
Constant P
+F
1-D
+F
+F
(dV and dT
usually have
the same sign)
Linear Expansivity, a
To
+F
+F
A
+F
Lo
Constant F
(dL is usually the same sign as dT)
To + dT
+F
+F
Lo+dL
Potential Energy of a Harmonic Oscillator
PE = 𝑢𝑜 +
1
𝐾
2
𝑟 − 𝑟𝑜
2
Stretching or
compressing the
spring raises the
potential energy.
uo
K is a spring constant
At equilibrium, the
spring length (atomic
spacing) is ro
Extension = r – ro
ro
Figure from “Understanding Properties of
Matter” by M. de Podesta
Atomic Origins of Thermal Expansion:
Anharmonic Potential
Increasing T raises
the thermal
energy.
Thermal energy is
the sum of the
kinetic and potential
energies.
ro
r
r = r0; Potential energy is at
minimum. Kinetic energy is
maximum.
Potential energy is at maximum.
Kinetic energy is minimum (or zero
for an instant)
Thermal Expansivity of Metals and Ceramics
Substance
(DL/L)*100%
Steel
SiC
T
a increases slightly with
temperature.
Invar steel
Pyrex glass
Steel
Aluminium
Ice
Water*
Mercury*
Linear expansivity,
a (K-1) (room T)
1 x 10-6
3 x 10-6
11 x 10-6
24 x 10-6
51 x 10-6
6 x 10-4
6 x 10-4
* Deduced from b (b 3a)
bliquid >> bsolid
Negative Thermal Expansivity
The volume of these materials decreases when they are heated!
Low T
High T
C.A. Kennedy, M.A. White, Solid State Communications 134, (2005) 271.
Low T
High T
Science, 319, 8 February (2008) p794-797
+
F
3-D
Vo
T
Bulk Modulus, K
P = FA
+F
Increased pressure: dP
+dF
Vo+dV
T
Constant T
+F
Initial pressure could be atmospheric
pressure.
Young’s Modulus, Y
1-D
T
Lo
A
+dF
+dF
(dV is usually
negative when
dP is positive)
(dL is usually positive
when dF is positive)
T
+dF
+dF
Lo+dL
Constant T
P-V Relation in an Ideal Gas
Pressure, P
𝑛𝑅𝑇
𝑃=
𝑉
𝜕𝑃
=
𝜕𝑉 𝑇
Volume, V
-
𝑛𝑅𝑇
𝑉2
Potential Energy, u, for Pair of Molecules
Potential Energy for a Pair of Non-Charged Molecules
s
r
𝒅𝒖
=0
𝒅𝒓
Equilibrium spacing at a
temperature of absolute
zero, when there is no
kinetic energy.
Separation between molecules (r/s)
Figure from “Understanding Properties of Matter” by M. de Podesta
Relating Molecular Level to the Macro-scale Properties
u
r/s
Elastic (Young’s) modulus
is a function of how the
macro-scale force of
compression or tension, F,
varies with distance, L.
𝒅𝒖
F=−
𝒅𝒓
+
F
Tension
Compression
-
Considering the
atomic/molecular level, the
slope of this curve around the
equilibrium point describes
mathematically how the force
will vary with distance.
Figure from “Understanding Properties of
Matter” by M. de Podesta
Elastic (Young’s) Modulus, Y
T
+F
+F
Force,
Applied Stress, s
L
Brittle solids will
fracture x
Lo
A
Length, L
F
Stress: 𝜎 =
𝐴
Δ𝐿
Strain: 𝜀 =
𝐿𝑜
Y
Strain, e
𝜎
𝑌=
𝜀
Young’s and Bulk Moduli of Common Solids and Liquids
Material
Y (GPa)
K (GPa)
Polypropylene
Polystyrene
Lead
Flax
Aluminium
Tooth enamel
Brass
Copper
Iron
Steel
Tungsten
Carbon Nanotubes
2
3
16
58
70
83
90
110
190
200
360
~1000
7.7
-70
-61
140
100
160
200
--
Diamond
Mercury
Water
Air
1220
----
442
27
200
10-4
Poisson’s Ratio
db
b
b
F
dL
A0
L
L
F
b usually decreases
when L increases.
db
Lateral _ Strain
 b
Poisson’s ratio = 
dL
Axial _ Strain
L
Therefore, usually n is positive. Solids become
thinner when pulled in tension.
If non-compressible (constant V), then n = 0.5.
Auxetic Materials have a Negative Poisson’s Ratio!
http://www.product-technik.co.uk/News/news.htm
http://data.bolton.ac.uk/auxnet/
/action/index.html
http://www.azom.com/details.asp?ArticleID=168
Summary of Bulk Properties
Property
Volume expansivity
Equation of
State
f(P,V,T) =0
(3-D)
Linear expansivity
f(F,L,T) =0
(1-D)
Formula
V)
b = V1 (∂
∂
T P
∂L )
a = L1 (∂
T F
∂P )
(
V ∂V T
SI Units
K-1
K-1
Isothermal Bulk
modulus (3-D)
f(P,V,T) =0
Young’s modulus
f(F,L,T) =0
L ∂F
Y = ( ∂L )T
A
Pa = Nm-2
f(P,V,T) =0
1
1 ∂V
(∂P )T
= =
K
V
Pa-1 = N-1m2
(1-D)
Isothermal
compressibility
(3-D)
K=
Pa = Nm-2
Definition of Viscosity
The top plane moves at a constant velocity, v, in response to a shear
stress:
F
A
Dx
v
y
A
Dx
v=
Dt
F
𝜎𝑠 =
𝐴
There is a velocity gradient (v/y) normal to the area. The viscosity h
relates the shear stress, ss, to the velocity gradient.
Dx
v
s s h
h
Dt y y
h has S.I. units of Pa s.
Viscosity describes the resistance to flow of a fluid.
Inverse Dependence of the Viscosity of
Liquids on Temperature
Thermal energy is
needed for molecules
to “hop” over their
neighbours.
Viscosity of liquids increases with pressure, because molecules are
less able to move when they are packed together more densely.
Temperature Dependence of Viscosity
Flow is thermallyactivated.
Viscosity is
exponentially
dependent on 1/T
Viscosity, h, of an Ideal Gas
𝜂=
n
3
2
1 𝑚𝜈
3 𝜋𝜎 2
1
2
𝑘𝑇 = m𝜈
𝜈=
2
3𝑘𝑇
𝑚
Viscosity varies as T ½ but is independent of P.
Figure from “Understanding Properties of Matter” by M. de Podesta