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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