electrical, thermal and mechanical properties of random mixtures materials research centre...

ELECTRICAL, THERMAL AND MECHANICAL PROPERTIES OF RANDOM MIXTURES

MATERIALS RESEARCH CENTRE

DEPARTMENT OF MECHANICAL ENGINEERING

UNIVERSITY OF BATH, UK

• ELECTRICAL PROPERTIES – POWER LAW DISPERSIONS AND UNIVERSAL DIELECTRIC RESPONSE

• THERMAL PROPERTIES

• MECHANICAL PROPERTIES

Log frequency

Log

POWER LAW DISPERSIONS CONDUCTORS

Slope n

()= dc + An

0<n<1

Log frequency

Log ’

0<n<1

Slope (n-1)

10-1

100

101

102

103

104

105

106

10-11

10-10

10-9

10-8

10-7

10-6

10-5

10-4

Frequency (Hz)

Conducti

vit

y (

Sie

mens/

m)

110C

50C

80C

170C

140C

200C

230C

EXAMPLES:

Al2O3-TiO2 Yttria doped ZrO2

100

101

102

103

104

105

106

10-8

10-7

10-6

10-5

10-4

10-3

10-2

Frequency (Hz)

Conduct

ivit

y (

Sie

mens/

m)

(0)n

200ºC

700ºC

ANOMALOUS POWER LAW DISPERSIONS HAVE BEEN FOUND IN

ALL CLASSES OF MATERIALS

SINGLE CRYSTALS

POLYCRYSTALLINE MATERIALS

POLYMERS

GLASSES

CERAMICS AND COMPOSITES

CONCRETE & CEMENTS

IONIC & ELECTRONIC CONDUCTORS

ANOMALOUS POWER LAW DISPERSIONS ARE

UBIQUITOUS

“THE UNIVERSAL DIELECTRIC RESPONSE”

A SATISFACTORY EXPLANATION MUST ACCOUNT

FOR THIS UBIQUITY

THEORETICAL INTERPRETATIONS

1-DISTRIBUTIONS OF RELAXATION TIMES

2-EXOTIC MANY-BODY RELAXATION MODELSSTRETCHED EXPONENTIALSPOWER LAW RELAXATION

3-ELECTRICAL NETWORK MODELS

THE ANOMALOUS POWER LAW DISPERSIONSARE NOT CAUSED BY

UNCONVENTIONAL ATOMIC LEVEL RELAXATION EFFECTS

THEY ARE MERELY THE AC ELECTRICAL CHARACTERISTICS OF THE ELECTRICAL NETWORKS

FORMED INSAMPLE MICROSTRUCTURE

Microstructure of a real technical ceramic.

Alumina 3%Titanium oxide

10m

RTiO2

CAl2O3

EXAMPLE OF AN ELECTRICAL NETWORK OF RANDOMLY POSITIONED RESISTORS AND

CAPACITORS CHARACTERISED USING CIRCUIT SIMULATION SOFTWARE.

102

103

104

105

106

1E-7

1E-6

1E-5

1E-4

1E-3

(b)

(a)

slope -0.6

slope 0.4Network conductivity

Network capacitance (F)

1E-9

1E-8

Con

duct

ivity

(S

)

Frequency (Hz)

Simulations of (a) ac conductivity and (b) capacitance of a 2D square network containing 512 randomly positioned components,

60% 1k resistors and 40% 1nF capacitors.

POWER LAW FREQUENCYDEPENDENCES

n=capacitor proportion

= 0.4

n-1 = -0.6

Ac conductivity of 256 2D networks randomly filled with 512 components 60% 1 k resistors

& 40% 1 nF capacitors

POWER LAW () n NETWORK INDEPENDENTPROPERTY

PERCOLATION DETERMINED DCCONDUCTIVITY

Network type (%R:%C) Power law fit, n

60:40 0.399

50:50 0.487

40:60 0.594

NETWORK CAPACITANCE

POWER LAW DECAY () n-1

ORIGIN OF THE POWER LAW

RC NETWORK CONDUCTIVITY AND PERMITTIVITY ARE RELATED TO COMPONENT VALUES BY THE

LOGARITHMIC MIXING RULE – LICHTENECKER’S RULE:

*NET=(iC)n(1/R)1-n

Networkcomplex

conductivity

Capacitorconductivity(admittance)

Resistorproportion

Capacitorproportion

Re. *NET = Cn(1/R)1-n cos(n/2) n

ACConductivity

Resistorconductivity

NETWORK CAPACITANCE

Cnet = Im. *net /i

Cnet= Cn (1/R)1-n sin(n/2) n-1

system = (ins0)n(cond)1-n cos(n/2) n

system =(ins0)n(cond)1-n sin(n/2) n-1

Real Heterogeneous Materials

FREQUENCY RANGE OF POWER LAW

1 10 100 1000 10000 100000 1000000 1E7 1E8 1E91E-3

0.01

0.1

1

10

60% R, 40% C

Nor

mal

ised

Con

duct

ivity

Frequency (Hz)

1 10 100 1000 10000 100000 1000000 1E7 1E8 1E91E-9

1E-8

1E-7

1E-6

1E-5

1E-4

1E-3

0.01

0.1

1

10

C

R-1

AC

Con

duct

ance

(oh

m-1)

Frequency (Hz)

CHARACTERISTICFREQUENCY

R-1 = C

Resistor conductivity = R-1

frequency independent

Capacitor ac conductivity = C

frequency dependent

EXPERIMENTAL INVESTIGATION

MATERIALS REQUIREMENTS:

•TWO-PHASE CONDUCTOR-INSULATOR SYSTEMWITH A RANDOM MICROSTRUCTURE

•CONDUCTIVITIES OF THE TWO PHASES SIMILAR,IN THE RADIO FREQUENCY RANGE

0

<107

<2000

8.854x10-12

10-1 Sm-1 (metals 107 Sm-1)

SYSTEM CHOSEN

INSULATING PHASE: 22% POROUS PZT CERAMIC1500

CONDUCTING PHASE: WATER 10-1 Sm-1

= 0 at <1MHz

COMPONENT CHARCTERISTICS

102

103

104

105

106

(a)

(b)

PZT rel. permittivity

100

1000

Rel. P

ermittivity

1.0

0.1

Water conductivity

Con

duct

ivity

Sm

-1

Frequency (Hz)

BOTH PHASES RELATIVELY FREQUENCY INDEPENDENT

SYSTEM CHARACTERISTICS

102

103

104

105

106

slope -0.22

(b)

(a)

PZT + water rel. permittivity

1000

10000

Rel. P

ermittivity

0.1

0.01

PZT +water conductivity

Con

duct

ivity

Sm

-1

Frequency (Hz)

system =(PZT0)n(water)1-n sin(n/2) n-1

system = DC +(PZT0)n(water)1-n cos(n/2) nDC

PZT = 1500water = 0.135 Sm-1

n = 0.78 (PZT %density)

100 1000 10000 100000 1000000 1E71E-4

1E-3

0.01

0.1

water/methanol conductivity

Con

duct

ivity

(S

/m)

Frequency (Hz)

100 1000 10000 100000 1000000 1E7

1000

10000

slope -0.22

Rel

ativ

e P

erm

ittiv

ity

Frequency (Hz)

78% dense PZT+

Methanol 10% waterConductivity 3.6x10-3 S/m

0 at <0.1MHz

EFFECT OF REDUCINGCONDUCTIVITY

Characteristic frequency

EFFECT OF SAMPLE POROSITY ONRELATIVE PERMITTIVITY

36%

1000 10000 100000 1000000 1E7

1000

10000

Rel

ativ

e P

erm

ittiv

ity

Frequency (Hz)

28%

22%16%

COMPARISON OF SYSTEM AND COMPONENT CHARACTERISTICS

102

103

104

105

106

(d)

(a) PZT rel. permittivity

(b) water conductivity

slope -0.22

PZT + water rel. permittivity

1000

10000

Rel. P

ermittivity

0.1

0.01 (c) PZT + water conductivity

Con

duct

ivity

(S

iem

ens/

m)

Frequency (Hz)

x20

TEST OF OTHER MATERIALS(estimation of characteristic frequency from component data)

~ 20DC [Archie’s Law]

At the characteristic frequency = 0

fch = /20 ~ 20DC/20

TEST OF OTHER MATERIAL SYSTEMSestimation of characteristic frequency from experimental data

AC=(0)n()1-n cos(n/2) n

At the characteristic frequency where 0=

AC=cos(n/2)~ /2

Conduction phase conductivity ~20x DC

Thus at the characteristic frequency, fch AC ~10x DC

10x DC

f10DC

Log frequency

Log

Theoretical fch ~ 20DC/20

Experimental fch ~ f10DC [AC ~10x DC]

TEST CORRELATION Saltwater

high

Whitestone

low

High frequency

0=

102 103 104 105 106

10-7

10-6

10-5

10-4

10-3

10-2

Con

duct

ivit

y (S

iem

ens/

m)

Frequency (Hz)

DRYING

n1

10-1

100

101

102

103

104

105

106

10-11

10-10

10-9

10-8

10-7

10-6

10-5

10-4

Frequency (Hz)

Con

duct

ivit

y (S

iem

ens/

m)

110C

50C

80C

170C

140C

200C

230C

110C

50C

80C

170C

140C

200C

230C

WET saturated, n=0.78

DRY, n1gradient=0.98

ZIRCONIA COOLING

ELECTRICAL NETWORKS

•ANOMALOUS POWER LAW FREQUENCY DEPENDENCES ARE AC CHARACTERISTICS OF RANDOM ELECTRICAL NETWORKS FORMED BY SAMPLE MICROSTRUCTURE.

•THERE IS NO NEED TO INTRODUCE ANY “NEW PHYSICS” TO EXPLAIN THE ANOMALOUS POWER LAW FREQUENCY DEPENDENCES.

APPLICATIONS: DESIGN OF COMPOSITES WITH SPECIFIC DIELECTRIC/CONDUCTION PROPERTIES.

1 10 100 1000 10000 100000 1000000 1E7 1E8 1E91E-9

1E-8

1E-7

1E-6

1E-5

1E-4

1E-3

0.01

0.1

1

10

C

R-1

AC

Con

duct

ance

(oh

m-1)

Frequency (Hz)

k2 (constant)

k1 (variable, low to high)

log(k1/k2)

Thermal conductivity equivalentN

etw

ork

ther

mal

con

duct

ivity

Kef

f (W

/ m

K)

102

103

104

105

106

slope -0.22

(b)

(a)

PZT + water rel. permittivity

1000

10000

Rel. P

ermittivity

0.1

0.01

PZT +water conductivity

Con

duct

ivity

Sm

-1

Frequency (Hz)

T= 0ºC

Base constrained to same temperature

Apply constant heat flux

Measure steady state T to calculate effective conductivity

50% k1, 50% k2 mixture

-4

-3

-2

-1

0

1

2

3

-5 -4 -3 -2 -1 0 1 2 3 4 5

log component conductivity, k1

log e

quiv

alen

t co

nduct

ivity,

K

k2 = 1

50% k1 , 50% k2

12 randomised cases30 x 30 array

Slope = 0.5 line for reference

Slope = 1 line for reference

Slope = 1 line for reference

K(k1,k2) = k10.5. k2

0.5

lo

g ef

fect

ive

cond

uctiv

ity

-4

-3

-2

-1

0

1

2

3

-5 -4 -3 -2 -1 0 1 2 3 4 5

log component conductivity, k1

log

eq

uiv

alen

t co

nd

uct

ivit

y, K

k2 = 1

50% k1 , 50% k2

12 randomised cases30 x 30 array

Slope = 0.5 line for reference

Slope = 1 line for reference

Slope = 1 line for reference

k2 (blue) constant

k1 (purple variable)

-5

-4

-3

-2

-1

0

1

2

3

4

-5 -4 -3 -2 -1 0 1 2 3 4 5

log component conductivity, k1

log

eq

uiv

ale

nt

co

nd

uc

tiv

ity

, K

k2 = 1

70% k1 , 30% k2

12 randomised cases30 x 30 array

Slope = 0.7 line for reference

K(k1,k2) = k10.7. k2

0.3

-1

-0.5

0

0.5

-5 -4 -3 -2 -1 0 1 2 3 4 5

log component conductivity, k1

log

eq

uiv

ale

nt

co

nd

uc

tiv

ity

, K

k2 = 1

30% k1 , 70% k2

12 randomised cases30 x 30 array

Slope = 0.3 line for reference

K(k1,k2) = k10.3. k2

0.7

Mechanical Network

A truss made from random mix of springs k1 and k2 with volume fractions 1 and 2

Rapid protoype: Polyamide

Infiltrate: Epoxy

50vol.% Polyamide

50vol.% Epoxy

• dynamic modulus (E1)

• loss modulus (E2)

• tan delta (E2/E1)

from -70 to 70°C

E1,composite = (E1amide)n (E1

epoxy)1-n

)}1(sin{)()(

)}1(cos{)()(

][][

1,2

1,1

1)()(*

1

212*

1

nnEEE

nnEEE

eEeEE

iEEiEEE

amideamidenepoxynamide

composite

amideamidenepoxynamide

composite

nepoxyiepoxynamideiamidecomposite

nepoxyepoxynamideamidecomposite

1.E+07

1.E+08

1.E+09

1.E+10

1.E+11

-100 -50 0 50 100

Tem perature(°C)

E1 (

GP

a)

polyamide

epoxy

composite

model

1.E+07

1.E+08

1.E+09

1.E+10

-100 -50 0 50 100

Temperature (°C)

E2 (

GP

a)

polyamide

epoxy

composite

model

0

0.2

0.4

0.6

0.8

1

1.2

1.4

-100 -50 0 50 100

Temperature (°C)

tan

del

ta

polyamide

epoxy

composite

model

y = 0.4862x - 0.0082

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-1 -0.5 0 0.5 1 1.5 2

log(Epolyamide/Eepoxy)

)}1(cos{)()( 1,1 nnEEE amideamide

nepoxynamidecomposite

Gradient of log(Ecomposite/Eepoxy) vs. log(Eamide/Eepoxy) = n

log(

Eco

mpo

site/E

epox

y)

Conclusions

nepoxyepoxynamideamidecomposite iEEiEEE

1

212*

1

1

/1Re RCi

1

2121, kkkkK

1

2121, ssssS