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
POWER LAW DISPERSIONS
Log 
CONDUCTORS
(w)= dc + Awn
Slope n
0<n<1
Log frequency
Log ’
Slope (n-1)
0<n<1
Log frequency
EXAMPLES:
Al2O3-TiO2
10
-2
10
(w)  wn
(0)
-3
10
700ºC
10
10
10
10
10
-4
-5
-6
10
10
10
10
-7
200ºC
10
-8
10
0
10
1
10
2
10
3
10
4
Frequency (Hz)
10
5
-4
230C
Conductivity (Siemens/m)
Conductivity (Siemens/m)
10
Yttria doped ZrO2
10
6
10
-5
200C
-6
170C
-7
140C
-8
110C
-9
80C
-10
50C
-11
10
-1
10
0
10
1
10
2
10
3
10
Frequency (Hz)
4
10
5
10
6
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 MODELS
STRETCHED EXPONENTIALS
POWER LAW RELAXATION
3-ELECTRICAL NETWORK MODELS
THE ANOMALOUS POWER LAW DISPERSIONS
ARE NOT CAUSED BY
UNCONVENTIONAL ATOMIC
LEVEL RELAXATION EFFECTS
THEY ARE MERELY THE AC ELECTRICAL
CHARACTERISTICS OF THE
ELECTRICAL NETWORKS
FORMED IN
SAMPLE MICROSTRUCTURE
Microstructure of a real technical ceramic.
Alumina 3%Titanium oxide
RTiO2
CAl2O3
10m
EXAMPLE OF AN ELECTRICAL NETWORK OF
RANDOMLY POSITIONED RESISTORS AND
CAPACITORS CHARACTERISED USING CIRCUIT
SIMULATION SOFTWARE.
Conductivity (S)
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
FREQUENCY
DEPENDENCES
1E-3
Network conductivity
slope 0.4
(a) n=capacitor proportion
1E-4
= 0.4
1E-8
1E-5
Network capacitance (F)
(b)
1E-9
1E-6
slope -0.6
n-1 = -0.6
1E-7
2
10
3
10
4
10
Frequency (Hz)
5
10
6
10
Ac conductivity of 256 2D networks randomly
filled with 512 components 60% 1 k resistors
& 40% 1 nF capacitors
PERCOLATION
DETERMINED DC
CONDUCTIVITY
POWER LAW (w) wn
NETWORK INDEPENDENT
PROPERTY
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
(w)  wn-1
ORIGIN OF THE POWER LAW
RC NETWORK CONDUCTIVITY AND PERMITTIVITY
ARE RELATED TO COMPONENT VALUES BY THE
LOGARITHMIC MIXING RULE – LICHTENECKER’S RULE:
*
Network
complex
conductivity
n(1/R)1-n
=(iwC)
NET
Capacitor
conductivity
(admittance)
Capacitor
proportion
Resistor
proportion
Resistor
conductivity
Re. *NET = Cn(1/R)1-n cos(n/2) wn
AC
Conductivity
NETWORK CAPACITANCE
Cnet = Im. *net /iw
Cnet= Cn (1/R)1-n sin(n/2) wn-1
Real Heterogeneous Materials
system = (ins0)n(cond)1-n cos(n/2) wn
system =(ins0)n(cond)1-n sin(n/2) wn-1
FREQUENCY RANGE OF POWER LAW
10
Normalised Conductivity
1
Resistor
conductivity = R-1
0.1
60% R, 40% C
frequency independent
0.01
1E-3
1
10
100
1000
10000
100000 1000000
1E7
1E8
1E9
Frequency (Hz)
10
1
frequency dependent
0.1
-1
AC Conductance (ohm )
Capacitor ac
conductivity = wC
0.01
-1
R
1E-3
1E-4
CHARACTERISTIC
FREQUENCY
R-1 = wC
1E-5
wC
1E-6
1E-7
1E-8
1E-9
1
10
100
1000
10000
100000 1000000
Frequency (Hz)
1E7
1E8
1E9
EXPERIMENTAL INVESTIGATION
MATERIALS REQUIREMENTS:
•TWO-PHASE CONDUCTOR-INSULATOR SYSTEM
WITH A RANDOM MICROSTRUCTURE
•CONDUCTIVITIES OF THE TWO PHASES SIMILAR,
IN THE RADIO FREQUENCY RANGE
w0
<107
8.854x10-12
<2000
10-1 Sm-1 (metals  107 Sm-1)
SYSTEM CHOSEN
INSULATING PHASE: 22% POROUS PZT CERAMIC
1500
CONDUCTING PHASE:
WATER
10-1 Sm-1
 = w0 at <1MHz
Conductivity Sm
-1
COMPONENT CHARCTERISTICS
1.0
(b)
Water conductivity
0.1
1000
PZT rel. permittivity
100
2
10
3
10
4
10
5
10
6
10
Frequency (Hz)
BOTH PHASES RELATIVELY FREQUENCY INDEPENDENT
Rel. Permittivity
(a)
Conductivity Sm
-1
SYSTEM CHARACTERISTICS
0.1
PZT +water conductivity
(a)
0.01
system = DC +(PZT0)n(water)1-n cos(n/2) wn
DC
10000
slope -0.22
1000
2
10
3
10
4
10
Frequency (Hz)
system =(PZT0)n(water)1-n sin(n/2) wn-1
5
10
Rel. Permittivity
PZT + water
rel. permittivity
(b)
6
10
PZT = 1500
water = 0.135 Sm-1
n = 0.78 (PZT %density)
EFFECT OF REDUCING
CONDUCTIVITY
w0 at <0.1MHz
Conductivity (S/m)
0.01
water/methanol conductivity
Characteristic
frequency
1E-3
1E-4
100
1000
10000
100000
1000000
1E7
Frequency (Hz)
10000
Relative Permittivity
78% dense PZT
+
Methanol 10% water
Conductivity 3.6x10-3 S/m
0.1
slope -0.22
1000
100
1000
10000
100000
Frequency (Hz)
1000000
1E7
EFFECT OF SAMPLE POROSITY ON
RELATIVE PERMITTIVITY
36%
10000
Relative Permittivity
28%
16%
22%
1000
1000
10000
100000
Frequency (Hz)
1000000
1E7
(b) water conductivity
0.1
x20
(c) PZT + water conductivity
0.01
PZT + water
(d)
rel. permittivity
10000
slope -0.22
1000
(a) PZT rel. permittivity
2
10
3
10
4
10
Frequency (Hz)
5
10
6
10
Rel. Permittivity
Conductivity (Siemens/m)
COMPARISON OF SYSTEM AND COMPONENT
CHARACTERISTICS
TEST OF OTHER MATERIALS
(estimation of characteristic frequency from component data)
 ~ 20DC
[Archie’s Law]
At the characteristic frequency  = w0
fch = /20 ~ 20DC/20
TEST OF OTHER MATERIAL SYSTEMS
estimation of characteristic frequency from experimental data
AC=(0)n()1-n cos(n/2) wn
At the characteristic frequency where
w0= 
AC=cos(n/2)~ /2
Conduction phase conductivity  ~20x DC
Thus at the characteristic frequency, fch AC ~10x DC
Log 
10x DC
f10DC
Log frequency
Theoretical
fch ~ 20DC/20
Experimental fch ~ f10DC [AC ~10x DC]
TEST CORRELATION
Saltwater
high 
log 20 (0)/2 o Hz [theortl.]
10
Whitestone-saltwater
Carbon blackthermoset
8
LiCl.H 2O (-114°C)
Polypyrole-polyoxyphenylene
Hydroge l
Whitestone
low 
Water-PZT
6
Na -alumina
High
frequency
Carbon nanotube-epoxy
4
w0= 
Na2O.3SiO2
Nylon 11
2
0
12 mol% yttria zirconia
-2
-2
0
2
4
log f(10(0)) Hz
6
[exptl.]
8
10
DRYING
WET saturated, n=0.78
10-2
10
10
10-3
ZIRCONIA COOLING
-4
230C
-5
200C
Conductivity (Siemens/m)
Conductivity (Siemens/m)
n1
10-4
DRY, n1
gradient=0.98
10-5
10
10
10
10
10
10-6
10
-6
170C
-7
140C
-8
110C
-9
80C
50C
-10
-11
10
-1
10
0
10
1
10
2
10
3
Frequency (Hz)
10-7
102
103
104
Frequency (Hz)
105
106
10
4
10
5
10
6
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.
Thermal conductivity equivalent
10
0.1
-1
AC Conductance (ohm )
Network thermal conductivity
1
0.01
-1
R
1E-3
k2 (constant)
1E-4
1E-5
wC
1E-6
k1 (variable, low to high)
1E-7
1E-8
1E-9
1
10
100
1000
10000
100000 1000000
Frequency (Hz)
log(k1/k2)
1E7
1E8
1E9
-1
Conductivity Sm
0.1
PZT +water conductivity
(a)
0.01
10000
slope -0.22
1000
2
10
3
10
4
10
Keff (W/ m K)
Frequency (Hz)
5
10
6
10
Rel. Permittivity
PZT + water
rel. permittivity
(b)
50% k1, 50% k2
mixture
T= 0ºC
Measure steady
state DT to
calculate
effective
conductivity
Base constrained to
same temperature
Apply constant heat flux
3
equivalent
conductivity, K
conductivity
effective
loglog
Slope = 1 line
for reference
2
1
Slope = 0.5 line
for reference
0
-1
-2
k2 = 1
50% k1 , 50% k2
12 randomised cases
30 x 30 array
Slope = 1 line
for reference
-3
-4
-5
-4
-3
-2
-1
0
1
2
log component conductivity, k1
K(k1,k2) = k10.5. k20.5
3
4
5
k2 (blue) constant
k1 (purple variable)
3
Slope = 1 line
for reference
log equivalent conductivity, K
2
1
0
Slope = 0.5 line
for reference
-1
-2
-3
k2 = 1
50% k1 , 50% k2
12 randomised cases
30 x 30 array
Slope = 1 line
for reference
-4
-5
-4
-3
-2
-1
0
1
2
log component conductivity, k1
3
4
5
4
log equivalent conductivity, K
3
2
1
0
Slope = 0.7 line
for reference
-1
-2
k2 = 1
70% k1 , 30% k2
12 randomised cases
30 x 30 array
-3
-4
-5
-5
-4
-3
-2
-1
0
1
2
log component conductivity, k 1
K(k1,k2) = k10.7. k20.3
3
4
5
log equivalent conductivity, K
0.5
0
Slope = 0.3 line
for reference
-0.5
k2 = 1
30% k1 , 70% k2
12 randomised cases
30 x 30 array
-1
-5
-4
-3
-2
-1
0
1
2
log component conductivity, k 1
K(k1,k2) = k10.3. k20.7
3
4
5
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 (E1epoxy)1-n
E
*
composite

 E1
amide
 iE
 E
amide n
2
epoxy
1
 iE

epoxy 1 n
2
*
Ecomposite
 [ E amide e i ( amide ) ]n [ E epoxyei ( epoxy) ]1 n
E1,composite  ( E amide ) n ( E epoxy )1 n cos{n amide   amide (1  n)}
E2,composite  ( E amide ) n ( E epoxy )1 n sin{n amide   amide (1  n)}
1.E+10
1.E+11
1.E+10
E2 (GPa)
1.E+09
polyamide
epoxy
1.E+08
composite
1.E+08
polyamide
model
epoxy
composite
model
1.E+07
-100
-50
0
50
1.E+07
-100
100
-50
Tem perature(°C)
0
Temperature (°C)
1.4
1.2
polyamide
1
tan delta
E1 (GPa)
1.E+09
epoxy
composite
0.8
model
0.6
0.4
0.2
0
-100
-50
0
Temperature (°C)
50
100
50
100
E1,composite  (E amide )n (Eepoxy )1n cos{namide  amide (1  n)}
Gradient of log(Ecomposite/Eepoxy) vs. log(Eamide/Eepoxy) = n
1
0.6
0.4
0.2
0
-0.2
log(Ecomposite/Eepoxy)
0.8
y = 0.4862x - 0.0082
-0.4
-1
-0.5
0
0.5
1
log(Epolyamide /Eepoxy)
1.5
2
Conclusions
 (w )  Re (iwC ) (1/ R )

K (k , k )  k k
1
2

1
1
2
S (s , s )  s s
1
*
composite
E
2

 E1
amide

1
1
2
 iE
1
 E
amide n
2
epoxy
1
 iE


epoxy 1 n
2