1. ELECTRICAL, THERMAL AND MECHANICAL PROPERTIES OF RANDOM MIXTURES MATERIALS RESEARCH CENTRE DEPARTMENT OF MECHANICAL ENGINEERING UNIVERSITY OF BATH, UK

2. ELECTRICAL PROPERTIES – POWER LAW DISPERSIONS AND UNIVERSAL DIELECTRIC RESPONSE THERMAL PROPERTIES MECHANICAL PROPERTIES

3. Log frequency Log POWER LAW DISPERSIONS CONDUCTORS Slope n ( )= dc + A n 0<n<1 Log frequency Log 0<n<1 Slope (n-1)

4. 110 C 50 C 80 C 170 C 140 C 200 C 230 C EXAMPLES: Al 2 O 3 -TiO 2 Yttria doped ZrO 2 (0) n 200ºC 700ºC

5. 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

6. ANOMALOUS POWER LAW DISPERSIONS ARE UBIQUITOUS THE UNIVERSAL DIELECTRIC RESPONSE A SATISFACTORY EXPLANATION MUST ACCOUNT FOR THIS UBIQUITY

7. THEORETICAL INTERPRETATIONS 1-DISTRIBUTIONS OF RELAXATION TIMES 2-EXOTIC MANY-BODY RELAXATION MODELS STRETCHED EXPONENTIALS POWER LAW RELAXATION 3-ELECTRICAL NETWORK MODELS

8. 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

9. Microstructure of a real technical ceramic. Alumina 3%Titanium oxide

10. EXAMPLE OF AN ELECTRICAL NETWORK OF RANDOMLY POSITIONED RESISTORS AND CAPACITORS CHARACTERISED USING CIRCUIT SIMULATION SOFTWARE.

11. 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 n=capacitor proportion = 0.4 n-1 = -0.6

12. Ac conductivity of 256 2D networks randomly filled with 512 components 60% 1 k resistors & 40% 1 nF capacitors POWER LAW ( ) n NETWORK INDEPENDENT PROPERTY PERCOLATION DETERMINED DC CONDUCTIVITY Network type (%R:%C) Power law fit, n 60:400.399 50:500.487 40:600.594

13. NETWORK CAPACITANCE POWER LAW DECAY ( ) n-1

14. ORIGIN OF THE POWER LAW RC NETWORK CONDUCTIVITY AND PERMITTIVITY ARE RELATED TO COMPONENT VALUES BY THE LOGARITHMIC MIXING RULE – LICHTENECKERS RULE: * NET =(i C) n (1/R) 1-n Network complex conductivity Capacitor conductivity (admittance) Resistor proportion Capacitor proportion Re. * NET = C n (1/R) 1-n cos(n /2) n AC Conductivity Resistor conductivity

15. NETWORK CAPACITANCE C net = Im. * net /i C net = C n (1/R) 1-n sin(n /2) n-1 system = ( ins 0 ) n ( cond ) 1-n cos(n /2) n system =( ins 0 ) n ( cond ) 1-n sin(n /2) n-1 Real Heterogeneous Materials

16. FREQUENCY RANGE OF POWER LAW CHARACTERISTIC FREQUENCY R -1 = C Resistor conductivity = R -1 frequency independent Capacitor ac conductivity = C frequency dependent

17. EXPERIMENTAL INVESTIGATION MATERIALS REQUIREMENTS: TWO-PHASE CONDUCTOR-INSULATOR SYSTEM WITH A RANDOM MICROSTRUCTURE CONDUCTIVITIES OF THE TWO PHASES SIMILAR, IN THE RADIO FREQUENCY RANGE 0 <10 7 <2000 8.854x10 -12 10 -1 Sm -1 (metals 10 7 Sm -1 )

18. SYSTEM CHOSEN INSULATING PHASE: 22% POROUS PZT CERAMIC 1500 CONDUCTING PHASE: WATER 10 -1 Sm -1 = 0 at <1MHz

19. COMPONENT CHARCTERISTICS BOTH PHASES RELATIVELY FREQUENCY INDEPENDENT

20. SYSTEM CHARACTERISTICS system =( PZT 0 ) n ( water ) 1-n sin(n /2) n-1 system = DC +( PZT 0 ) n ( water ) 1-n cos(n /2) n DC PZT = 1500 water = 0.135 Sm -1 n = 0.78 (PZT %density)

21. 78% dense PZT + Methanol 10% water Conductivity 3.6x10 -3 S/m 0 at <0.1MHz EFFECT OF REDUCING CONDUCTIVITY Characteristic frequency

22. EFFECT OF SAMPLE POROSITY ON RELATIVE PERMITTIVITY 36% 28% 22% 16%

23. COMPARISON OF SYSTEM AND COMPONENT CHARACTERISTICS x20

24. TEST OF OTHER MATERIALS (estimation of characteristic frequency from component data) ~ 20 DC [Archies Law] At the characteristic frequency = 0 f ch = /2 0 ~ 20 DC /2 0

25. TEST OF OTHER MATERIAL SYSTEMS estimation 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, f ch AC ~10x DC 10x DC f 10DC Log frequency Log

26. Theoretical f ch ~ 20 DC /2 0 Experimental f ch ~ f 10DC [ AC ~10x DC ]

27. TEST CORRELATION Saltwater high Whitestone low High frequency 0 =

28. DRYING n 1 WET saturated, n=0.78 DRY, n 1 gradient=0.98 ZIRCONIA COOLING

29. 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.

30. k 2 (constant) k 1 (variable, low to high) log(k 1 /k 2 ) Thermal conductivity equivalent Network thermal conductivity

31. K eff (W/ m K)

32. T= 0ºC Base constrained to same temperature Apply constant heat flux Measure steady state T to calculate effective conductivity 50% k 1, 50% k 2 mixture

33. K(k 1,k 2 ) = k 1 0.5. k 2 0.5 log effective conductivity

34. k 2 (blue) constant k 1 (purple variable)

35. K(k 1,k 2 ) = k 1 0.7. k 2 0.3

36. K(k 1,k 2 ) = k 1 0.3. k 2 0.7

37. Mechanical Network A truss made from random mix of springs k 1 and k 2 with volume fractions 1 and 2

40. Rapid protoype: Polyamide Infiltrate: Epoxy 50vol.% Polyamide 50vol.% Epoxy dynamic modulus (E 1 ) loss modulus (E 2 ) tan delta (E 2 /E 1 ) from -70 to 70°C

41. E 1,composite = (E 1 amide ) n (E 1 epoxy ) 1-n

43. Gradient of log(E composite /E epoxy ) vs. log(E amide /E epoxy ) = n log(E composite /E epoxy )

44. Conclusions