PHY1039

Properties of MatterMacroscopic (Bulk) Properties:

Thermal Expansivity, Elasticity and Viscosity

20 & 23 February, 2012

Lectures 5 and 6

Thermal Expansivity, b

+ dTVo To

+F

+ F

+F

3-D +F

Vo

+F

+F+ dV

T + dTConstant P (dV and dT usually have the same sign)

1-D

+F +FLo

ATo

To + dT

Lo+dL+F+F

Linear Expansivity, a

Constant F

(dL is usually the same sign as dT)

AP = F

Potential Energy of a Harmonic Oscillator

Stretching or compressing the spring raises the potential energy.

Extension = r – roro

uo

PE =

K is a spring constant

Figure from “Understanding Properties of Matter” by M. de Podesta

At equilibrium, the spring length (atomic spacing) is ro

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)

Atomic Origins of Thermal Expansion: Anharmonic Potential

Thermal energy is the sum of the kinetic and potential energies.

ro

Increasing T raises the thermal energy.

r

(DL/L)*100%

T

a increases slightly with temperature.

Thermal Expansivity of Metals and Ceramics

Substance Linear expansivity, a (K-1) (room T)

Invar steel 1 x 10-6

Pyrex glass 3 x 10-6

Steel 11 x 10-6

Aluminium 24 x 10-6

Ice 51 x 10-6

Water* 6 x 10-4

Mercury* 6 x 10-4

Steel

SiC

* Deduced from b (b 3a)

bliquid >> bsolid

C.A. Kennedy, M.A. White, Solid State Communications 134, (2005) 271.

Negative Thermal ExpansivityThe volume of these materials decreases when they are heated!

Science, 319, 8 February (2008) p794-797

Low THigh T

Low T High T

Vo

AP = F3-D

T Vo+dV

+dF

+dF

+dF

T

Bulk Modulus, K

(dV is usually negative when dP is positive)

Constant T

1-D

Lo

AT

Young’s Modulus, Y

+dF

Lo+dL+dF

T

Constant T

(dL is usually positive when dF is positive)

+F

+ F

+F

Initial pressure could be atmospheric pressure.

Increased pressure: dP

P-V Relation in an Ideal Gas

Volume, V

Pre

ssur

e, P

𝑃=𝑛𝑅𝑇𝑉

= -

Po

ten

tial E

ner

gy, u

, fo

r P

air

of M

ole

cule

s

Separation between molecules (r/s)

r

s

Potential Energy for a Pair of Non-Charged Molecules

=0 Equilibrium spacing at a temperature of absolute zero, when there is no kinetic energy.

Figure from “Understanding Properties of Matter” by M. de Podesta

F=−𝒅𝒖𝒅𝒓

F

-

+

Elastic (Young’s) modulus is a function of how the macro-scale force of compression or tension, , varies with distance, L.

Relating Molecular Level to the Macro-scale Properties

Considering the atomic/molecular level, the slope of this curve around the equilibrium point describes mathematically how the force will vary with distance.

Compression

Tension

Figure from “Understanding Properties of Matter” by M. de Podesta

u r/s

Strain, eA

pplie

d S

tres

s, s

Elastic (Young’s) Modulus, Y

Length, L

For

ce,

x

Brittle solids will fracture

Y𝜎=

F𝐴

Stress: Strain: 𝜀=Δ𝐿𝐿𝑜

𝑌=𝜎𝜀

Lo

+F +FL

AT

Young’s and Bulk Moduli of Common Solids and LiquidsMaterial Y (GPa) K (GPa)Polypropylene 2Polystyrene 3Lead 16 7.7Flax 58 --Aluminium 70 70Tooth enamel 83 --Brass 90 61Copper 110 140Iron 190 100Steel 200 160Tungsten 360 200Carbon Nanotubes ~1000 --

Diamond 1220 442Mercury -- 27Water -- 200Air -- 10-4

A0

F

F

L

dL

L

bb

db

Poisson’s Ratio

Poisson’s ratio =

LdLb

db

StrainAxial

StrainLateral

_

_

Therefore, usually n is positive. Solids become thinner when pulled in tension.

b usually decreases when L increases.

If non-compressible (constant V), then n = 0.5.

http://www.product-technik.co.uk/News/news.htm

Auxetic Materials have a Negative Poisson’s Ratio!

http://www.azom.com/details.asp?ArticleID=168

http://data.bolton.ac.uk/auxnet//action/index.html

( )PT

VV ∂

∂1=

Summary of Bulk Properties

PropertyVolume expansivity

(3-D)

Equation of State

f(P,V,T) =0

Formula SI Units

K-1

Linear expansivity

(1-D)f(F,L,T) =0 ( )

FTL

L ∂∂1=

K-1

Isothermal Bulk modulus (3-D)

f(P,V,T) =0 ( )TV

PVK ∂∂= Pa = Nm-2

Young’s modulus

(1-D)f(F,L,T) =0 ( )

TLF

AL

Y ∂ ∂= Pa = Nm-2

Isothermal compressibility

(3-D)

f(P,V,T) =0 ( )TP

V

VK ∂∂1

=1

= Pa-1 = N-1m2

A

A

y

FDx

tx

v

=

There is a velocity gradient (v/y) normal to the area. The viscosity h relates the shear stress, ss, to the velocity gradient.

y

v

yt

xs

The top plane moves at a constant velocity, v, in response to a shear stress:

v

h has S.I. units of Pa s.

Definition of Viscosity

=

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 thermally-activated.

Viscosity is exponentially dependent on 1/T

n

Viscosity, h, of an Ideal Gas

Viscosity varies as T ½ but is independent of P.

=

m𝜈=√ 3𝑘𝑇

𝑚

Figure from “Understanding Properties of Matter” by M. de Podesta