High Thermal Conductivity Polymer Nanocomposite Encapsulants for Undersea Acoustic Transducers

©2010 John V. Costa
The University of Massachusetts Dartmouth
Mechanical Engineering Department
High Thermal Conductivity Polymer Nanocomposite Encapsulants for
Undersea Acoustic Transducers
A Thesis in
Mechanical Engineering
by
John V. Costa
Copyright ©2010 by John V. Costa
Submitted in Partial Fulfillment of the
Requirements for the degree of
Master of Science
December 2010

Reproduction, translation, or transmittal of any part of this work, by any means beyond
that permitted by Section 107 or 108 of the 1976 United States Copyright Act without
permission of the copyright owner is unlawful. Requests for permission or further
information should be addressed to the author, John V. Costa.

ABSTRACT
The goal of this research is to develop a high thermal conductivity polymer
nanocomposite encapsulant to replace low thermal conductivity elastomer encapsulants
currently used to protect heat producing electronic systems from seawater.
Past attempts to increase thermal conductivity of polyurethane entailed adding a high
mass fraction (Mf > 25%) of ceramic micropowder filler, which impaired crucial
encapsulant properties, such as acoustic transparency, strength, fracture toughness and
water absorption.
The nanocomposites developed in this project included low mass fractions (Mf < 5.0%)
of hexagonal boron nitride and aluminum nitride in polyurethane. Composites were made
using both untreated and functionalized particles dispersed in the polyurethane with mass
fractions of 0.01, 0.03, and 0.05. Specifically, 70nm boron nitride and 40nm aluminum
nitride particles were functionalized with alkoxysilane, nitric acid, or acetone to alter the
wetting characteristics and inhibit particle agglomeration as a means to probe the effect
of thermal contact on composite thermal conductivity. Particle dispersion with extended
ultrasonication was found to further enhance thermal conductivity, separately and in
combination with 12 atm pressure during cure, which was also found to increase thermal
conductivity by improving interfacial wetting and possibly adhesion while reducing the
volume fraction of residual air by 92%. Scanning electron microscopy was used to
observe particle dispersion and evidence of trapped air between particle and matrix. All
particles exhibited good wetting and normally present, light aggregation was observed at

0.03 and 0.05 mass fractions. As a control, high mass fraction composites using 1 micron
hexagonal boron nitride particles were added to polyurethane to form composites with
mass fractions of 0.05, 0.1, 0.2, and 0.3 of silane treated and untreated particles.
The results show that the thermal conductivity of polyurethane containing 0.3 mass
fraction of untreated 1.0 micron h-BN showed the highest overall thermal conductivity,
0.62 W/m-K, a 170% increase over neat polyurethane; however, both the mechanical
properties and moisture characteristics of this system are unsatisfactory for the intended
application. The highest thermal conductivity obtained from the low mass fractions of
interest to this study was 0.41 W/m-K from a composite made of 0.05 mass fraction,
untreated 1µm hexagonal boron nitride particle filled urethane, cured with 12 atm
pressure, a 73% thermal conductivity increase over that of neat polyurethane.
What may be the most important outcome of the project was the discovery of a non-
contact percolation threshold for low mass fraction polymer composites in the 0.01 to
0.02 mass fraction region, where the highest thermal conductivity measurement was 0.37
W/m-K, a 54.2% increase of TC using 0.01 mass fraction 1µm hexagonal boron nitride
filler in 3140 PU cured at 12 atm pressure. In all, three 0.01 mass fraction composites
were made that exceeded the thermal conductivity predictions of all effective medium
theory models, including the Hamilton-Crosser model, which includes thermal
conductivity enhancements due to particle surface area.
The discovery of a non-contact PT can lead to optimized polymer composites with
optimized sized particles at optimized weight fractions from 0.01 to 0.02 with higher
thermal conductivity being developed in the future.

iv
Dedication
I would like to thank the many scientists and staff members I was privileged to work
with at the Naval Undersea Warfare Center in Newport RI, as well as the Office of
Naval Research and the Navy's University Laboratory Initiative program which
sponsored this work.

v
TABLE OF CONTENTS
Page
Dedication....................................................................................................................... iv
LIST OF TABLES......................................................................................................... xv
LIST OF FIGURES .....................................................................................................xvii
CHAPTER 1: INTRODUCTION.................................................................................... 1
1.1. Motivation of Work............................................................................................... 1
1.2. Objectives .............................................................................................................. 2
1.3 Significance ............................................................................................................ 3
1.4. Organization of Thesis........................................................................................... 4
CHAPTER 2: BACKGROUND...................................................................................... 6
2.1. Matrix and Materials.............................................................................................. 6
2.2. Filler Materials and the Nano-Effect..................................................................... 9
Boron Nitride............................................................................................................ 12
Aluminum Nitride..................................................................................................... 16
2.3 Particle Coatings, Surface Treatments and Additives .......................................... 19
Silane Surface Coatings............................................................................................ 21
Surface Treatments ................................................................................................... 22
Apparent Contact Angle ........................................................................................... 23
Additive Method....................................................................................................... 24

vi
2.4 Thermal Conductivity and Analytic Prediction.................................................... 24
Conduction across Interfacial Boundaries................................................................ 27
Effective Medium Theories of Thermal Conductivity ............................................. 36
Percolation Models ................................................................................................... 40
Effect of Interfacial Resistance on Percolation......................................................... 42
2.5 Dispersion and Percolation................................................................................... 44
Non-Contact Percolation .......................................................................................... 48
Achieving Effective Dispersion................................................................................ 50
2.6 Verification of Dispersion .................................................................................... 51
CHAPTER 3: Experimental Work and Procedures....................................................... 55
3.1 Experimental Approach........................................................................................ 55
3.2 Materials ............................................................................................................... 55
3.3 Composite Fabrication.......................................................................................... 56
Batch Size and Mixing Procedure Variations........................................................... 60
Entrapped Air............................................................................................................ 61
3.4 Pouring and Molding the Composite.................................................................... 62
3.5 Silane Surface Coating Application and Measurement........................................ 65
Silane Coating of AlN .............................................................................................. 68
Matrix Additive Method for Silane .......................................................................... 70
Differential Scanning Calorimetry ........................................................................... 70
Thermogravimetric Analysis .................................................................................... 71
3.6 Surface Treatments............................................................................................... 71
Acid Treatment ......................................................................................................... 72
Acetone Treatment.................................................................................................... 72

vii
3.7 Cryo-Ultramicrotomy and FE-SEM Imaging....................................................... 73
3.8 Cryo-Ultramicrotomy Procedure.......................................................................... 75
3.9 FE-SEM Sample Mounting and Sputtering.......................................................... 78
3.10 Sessile Drop Testing Procedure.......................................................................... 79
3.11 Tensile Testing Polyurethane and Composites................................................... 82
3.12 Thermal Conductivity Testing............................................................................ 84
Chapter 4: Results and Discussion................................................................................. 88
4.1 Experimental Results............................................................................................ 88
4.2 Sessile Drop Tests................................................................................................. 88
4.3 Differential Scanning Calorimetry ....................................................................... 91
4.4 Thermogravimetric Analysis of Functionalized Particles .................................... 92
4.5 FE-SEM Imaging of Dispersion and Interfacial Properties.................................. 95
4.6 Tensile Testing Data and Plots ........................................................................... 106
4.7 Thermal Conductivity Prediction Using Mixing Rule Models .......................... 110
Mass Fraction to Volume Fraction Conversion...................................................... 110
Maxwell Garnett Model Predictions....................................................................... 111
Hamilton Crosser Model Predictions...................................................................... 113
Renovated Maxwell Garnett Model (R-MG) Predictions ...................................... 115
4.8 Thermal Conductivity Measurements Using the Modified Transient Plane Source
Technique ................................................................................................................. 119
Thermal Conductivity Measurements of Polymer Matrix Materials...................... 119

viii
Thermal Conductivity Measurements of h-BN Plate and Various Powders .......... 121
Thermal Conductivity Measurements of 1µm h-BN in 3140 PU Cured at Standard
Conditions............................................................................................................... 123
Thermal Conductivity Measurements of 137nm h-BN in 3140 PU and Hapflex® PU
Cured at Standard Conditions................................................................................. 125
Thermal Conductivity Measurements of 70nm h-BN in 3140 PU Cured at Standard
Conditions with Various Treatments...................................................................... 130
Thermal Conductivity Measurements of AlN in 3140 PU cured at Standard
Conditions............................................................................................................... 135
Comparison of Composites Cured at Standard Conditions to Effective Medium
Theory Predictions.................................................................................................. 138
Thermal Conductivity of h-BN and AlN Pressure Cured in 3140 PU ................... 141
Comparison of 12 atm and 1 atm Pressure Cured h-BN and AlN Filled 3140 PU
Composites ............................................................................................................. 156
Chapter 5: Conclusions and Future Work.................................................................... 158
5.1 Summary, Conclusions and Contributions ......................................................... 158
Summary................................................................................................................. 159
Conclusions............................................................................................................. 160
Contributions .......................................................................................................... 161
5.2 Future Work........................................................................................................ 164
Optimized Composite Procedure and Materials ..................................................... 165
Elastic Fatigue Testing ........................................................................................... 167
Epilogue.................................................................................................................. 168
REFERENCES ............................................................................................................ 169
Appendix A.................................................................................................................. 178

ix
Appendix B.................................................................................................................. 183
Epon Epoxy Composites.............................................................................................. 185
Stycast Epoxy Composites........................................................................................... 187
Appendix C.................................................................................................................. 220
Appendix D.................................................................................................................. 230
Appendix E .................................................................................................................. 237

x
NOMANCLATURE
ABS: Acrylonitrile butadiene styrene, a thermoplastic material
ACS: American Chemical Society
Ak: Kapitza radius, defines volume where the temperature gradient occurs, having SI
units of m-1
, and related to G by way of k1, see Eqn. 22
AlN: Aluminum nitride
atm: Atmosphere (unit) an international reference pressure defined as 101,325 Pa
c: Celsius, a unit of measurement for temperature
c-BN: Cubic boron nitride
cp: Specific heat of a material
D: Thermal diffusivity, D = k /ρcp
DSC: Differential scanning calorimetry
dT/dx: The temperature gradient in the x-direction
eqn: References a mathematical equation
Fig: references a figure
g: Gram, a metric unit of measure for mass equal to a thousandth of a Kilogram
G: Interfacial thermal resistance with units of MW/m2
-K, related to Ak by way of k1; see
eqn. 22
h: The interfacial layer thickness
h-BN: Hexagonal boron nitride
H-C: Hamilton-Crosser thermal conductivity model

xi
hrs: Hour(s), a unit of measure for time
IC: An integrated electronic circuit of micrometer scale components set on a silicon chip
id: Inner diameter, the sectional measurement of a round hole or feature
K: Kelvin, measure of absolute temperature
k: Thermal conductivity, W/m-K
k1: Thermal conductivity of the matrix
k2: Thermal conductivity of the particulate filler
k3: The thermal conductivity of the complex particle, which is a combination of particle
and affected interfacial layer
Kapitza: A resistance to heat flow across the interface between two adjacent materials
keffective: Thermal conductivity of theoretical composite
kHapflex: the thermal conductivity of pure Hapflex® PU; kHapflex = 0.270W/m-K
l: Length of the phonon mean free path as determined by Debye's formula
LS: Liquid-solid interface
LV: Liquid to vapor interface
m: Meter, the international standard unit of length
M: Ceramic reactive group
Mf: Mass fraction and weight fraction
MFP: The mean free path of phonon propagation in a solid.
M-G: Maxwell-Garnett thermal conductivity model
min: Minute, a unit of measure for time
ml: Milliliter, a unit of volume measure equal to a thousandth of a liter

xii
mm: Millimeter, a metric unit of length equal to a thousandth of a meter
MTPS: Modified transient plane source technique of thermal conductivity measurement
mV: Millivolt, a unit of potential difference equal to one thousandth of a volt
n: Hamilton Crosser Shape Factor; 3/n 
nm: Nanometer, a metric unit of length equal to 10-9
meters
od: Outer diameter, the sectional measurement outside of a round object or feature
Pa-s: Pascal-seconds, the SI unit of viscosity
Pg: Designation of a page
pH: Potential of hydrogen
Phase 1: Matrix material
Phase 2: Filler material
PMMA: Poly (methyl methacrylate), a thermoplastic material
PT: Percolation threshold
PTFE: Polytetrafluoroethylene
PU: Polyurethane
Q1: The first quartile of data in a data set
Q3: The third quartile of data in a data set
"
q : Heat flux, or heat transfer rate, with units of 2
W
m
"
xq : Heat flux, or heat transfer rate in the x-direction with units of 2
W
m

xiii
r: Ratio of a roughened surface area divided by a theoretically smooth surface area
R: Polymer reactive group
Rayls: A measure of acoustic impedance with units of; Pa-s/m
R-MG: Renovated Maxwell-Garnett thermal conductivity model
rp: Radius of a spherical particle used in Equations 19, 22, and 30
rpm: Revolutions per minute
s: seconds, a unit of measurement for time
sect: Reference to a section
SI: The International System of Units which specifies a set of unit prefixes, otherwise
known as metric prefixes
sonic: ultrasonic energy
SV: Solid-vapor interface
T: The temperature of the material of interest in Kelvin
t: Time
TC: Thermal conductivity
TGA: Thermogravimetric analysis
Tm: The melting temperature of a solid material in Kelvin
TPS: Transient plane source technique of thermal conductivity measurement
V: Volts, the standard unit of potential difference
V2: Volume fraction of filler
Vcritical: Volume fraction of filler at which percolation threshold, PT, is achieved
Vf: Volume fraction

xiv
z: The lattice distance constant, z ≈ 0.5nm
ρ: Density of a material
λ: Complex particle ratio;
p
h
r
 
υ: A dimensionless parameter used in the Unified Mixing Rule to correlate M-G ( 0  ),
H-C ( 0  ), Bruggeman ( 2  ) Models for spherical inclusions
θ: Contact angle: angle of liquid on a surface after settling used to measure wetting
K : Dielectric constant of a material
K : Dielectric constant parallel to crystal plane
K : Dielectric constant perpendicular to crystal plane
ψ: Sphericity Ratio;
Area of a Sphere of Specific Mass
Surface Area of Specific Mass of Particles
 
0 : Initial contact angle: contact angle before settling occurs.
γ: Surface energy
 : The Gruneisen parameter, 1 
 : The TC ratio parameter derived from TC ratios of 1 2 3, , andk k k
T : The temperature gradient in three dimensions; , ,
T T T
T
x y z
   
   
   
#: Number sign, the designation of a number
1.5IQR: The Interquartile Rule used to reject outlier data; IQR=Q3-Q1
3140 PU: Uralite® FH-3140 Polyurethane

xv
LIST OF TABLES
Table I: Matrix Materials and Manufacturers Specifications ............................................. 8
Table II: Particulate Fillers with Published Specifications as Noted................................ 10
Table III: Contact Angle Data Measured on Hot Pressed h-BN Plate, Wetted with Water
and Treated As Noted ....................................................................................................... 88
Table IV: Silane Coating Applied to Particles as Measured by TGA .............................. 95
Table V: Particle Dimension and Shape Data Obtained Through SEM Microscopy..... 105
Table VI: Linearized Stress-Strain Slopes for 3140 PU vs. Ultrasonicated PU and 1µm h-
BN Composites of 3140 PU............................................................................................ 109
Table VII: Mf to Vf Conversion for h-BN and AlN in 3140 PU..................................... 110
Table VIII: TC Results of Pure Polymer Matrix Materials Degassed and Cured at
Standard Conditions........................................................................................................ 119
Table IX: TC Data from Hot Pressed h-BN Plate Compared to Nano and Micropowders
......................................................................................................................................... 121
Table X: TC Data from Silane Treated and Untreated 1µm h-BN in 3140 PU.............. 123
Table XI: TC Data from 137nm h-BN in 3140 PU Cured at Standard Conditions........ 125
Table XII: TC Data from 137nm h-BN in Hapflex® PU Cured at Standard Conditions127
Table XIII: TC Data from 70nm h-BN in 3140 PU with 5% Mf and Treatments Noted 131
Table XIV: TC Data from 70nm h-BN in 3140 PU with 3% Mf and Treatments Noted132
Table XV: TC Data from 70nm h-BN in 3140 PU with 1% Mf and Treatments Noted 132

xvi
Table XVI: TC Data from 2.5% Mf 70nm h-BN and 2.5% Mf 25A Clay in 3140 PU with
Treatments Noted............................................................................................................ 133
Table XVII: TC Data from 40nm AlN in 3140 PU Cured In Standard Conditions ....... 135
Table XVIII: TC Data from 12 atm and 1 atm Cured 5% Mf 70nm h-BN in 3140 PU.. 142
Table XIX: TC Data from 12 atm Cured and 1 atm Cured 5% and 1% Mf 137nm h-BN in
3140 PU .......................................................................................................................... 148
Table XX: TC Data from 12 atm Cured and 1 atm Cured 5% and 1% Mf 1µm h-BN in
3140 PU .......................................................................................................................... 151
Table XXI: TC Data from 12 atm and 1 atm Cured 5% Mf 40nm AlN Untreated and
Silane Treated in 3140 PU.............................................................................................. 153
Table XXII: Summary of Results for the Ten Highest TC Low Mf 3140 PU Composites
......................................................................................................................................... 158
Table XXIII: TC Data from Silane Treated and Untreated 8µm h-BN in Stycast Epoxy
......................................................................................................................................... 178
Table XXIV: TC Data from PT180S h-BN in Epon R-813 Epoxy Cured with Pressure as
Indicated.......................................................................................................................... 180

xvii
LIST OF FIGURES
Figure Page
Figure 1: SEM Image of 1 Micron h-BN Particles Set in Epoxy ..................................... 11
Figure 2: Thermal Conductivity of a Single Hexagonal Boron Nitride Crystal [19] ....... 13
Figure 3: X-Ray Diffraction from Three Different h-BN Samples .................................. 15
Figure 4: Transmission Electron Microscopy-Energy Dispersive Spectroscopy of 137nm
h-BN (NanoAmor Batch 2)............................................................................................... 16
Figure 5: FE-SEM Image of Gold Sputtered Cryo-Ultramicrotome Section of 5% Mf Z-
6020® Silane Coated Wurtzite AlN in 3140 PU.............................................................. 17
Figure 6: X-Ray Diffraction Analysis of 40nm AlN as Compared to a Known Peak
Profile................................................................................................................................ 18
Figure 7: Three Different Conditions of Wetting [8]........................................................ 20
Figure 8: Plueddemann’s Reversible Bond Associated with Hydrolysis [8].................... 21
Figure 9: Increased Thermal Conductivity Enhancement due to Increased Effective
Volume of Highly Conductive Clusters............................................................................ 32
Figure 10: Single Spherical Particle with Interfacial Layer of Affected Matrix [28]....... 34
Figure 11: Average Surface to Surface Distance for Spherical Particles in Composites
[35].................................................................................................................................... 35

xviii
Figure 12: Comparison of the effective medium theory predictions for randomly oriented
long fibers (solid curves) and platelets (dashed curves) for k2/kl = 100 at 0.5% volume
fraction showing the effects of increasing interfacial resistance [25]............................... 43
Figure 13: PT and TC Predicted by the M-G Model for Spherical h-BN Filler [27] ....... 47
Figure 14: Methodology Used to Fabricate Polyurethane Microparticle or Nanoparticle
Filled Composites ............................................................................................................. 57
Figure 15: Impeller Showing that Extended Shear Mixing (1hr) Caused Nanoparticle
Aggregation....................................................................................................................... 58
Figure 16: Specially Designed Molds and Mold Rotation Device ................................... 64
Figure 17: Thermogravimetric Analysis Comparison of 70nm h-BN Treated in 2% Sol of
Z-6040® and Untreated Particles ..................................................................................... 66
Figure 18: Micro-Star Cryo-Ultramicrotome.................................................................... 75
Figure 19: Micro-Magnetic Vices with PU Nanocomposites Surfaced by Cryo-
Ultramicrotomy................................................................................................................. 76
Figure 20: FE-SEM Image Comparison of Dull Knife vs. Sharp Knife Examples.......... 78
Figure 21: Sessile Drop Test Images on Hot Pressed Boron Nitride Plate....................... 81
Figure 22: Wet Sanding Samples for Thermal Conductivity Testing............................... 85
Figure 23: Example Contact Angle Report for Untreated h-BN Plate ............................. 90
Figure 24: Differential Scanning Calorimetry Used to Detect Silane on 70nm h-BN ..... 91
Figure 25: TGA of Six Separate Batches 0.25% Sol Z-6040® on 70nm h-BN Particles 92
Figure 26: TGA of 1% Sol Z-6020® on 40nm AlN Particles vs. Untreated.................... 93

xix
Figure 27: Comparison of Untreated h-BN to Water Treated and Various Volumes of 2%
Sol Z-6040® Treated h-BN .............................................................................................. 94
Figure 28: 1% Mf of Untreated 70nm h-BN in 3140 PU Displaying Poor Dispersion with
Aggregation and Agglomeration....................................................................................... 96
Figure 29: 1% Mf of Acetone Treated 70nm h-BN in 3140 PU Displaying Good
Dispersion and Minimal Aggregation............................................................................... 97
Figure 30: 3% Mf of 2% Sol Z-6040® Treated 70nm h-BN in 3140 PU Displaying Minor
Clustering of Randomly Dispersed Particles .................................................................... 98
Figure 31: 5% Mf of Acid Etched 70nm h-BN in 3140 PU Exhibiting Pronounced
Clustering of Particles....................................................................................................... 99
Figure 32: 5% Mf of Untreated 137nm h-BN in 3140 PU made with 1/2 hr Shear and 1/2
hr Ultrasonic Dispersion Showing air Pockets and Interfacial Debonding .................... 100
Figure 33: 3% Mf of 1% Sol Z-6020® Treated 40nm AlN in 3140 PU Displaying Good
Bonding with Failure and Rupture of the Particle .......................................................... 102
Figure 34: Contact Percolation Network in 30% Mf of 1µm h-BN Platelets in 3140 PU
......................................................................................................................................... 103
Figure 35: FE-SEM Microscopy Used to Determine Particle Shape and Dimensions... 104
Figure 36: Stress vs. True Strain Averaged From Six Data Curves for Each Sample
Shown ............................................................................................................................. 107
Figure 37: Linearized Curves with Slopes Noted........................................................... 108
Figure 38: M-G Model Predicts the Percolation Threshold to Occur at Vf =1 for h-BN
and AlN Composites....................................................................................................... 111

xx
Figure 39: M-G Model up to 2.44% Vf (5% Mf of h-BN) Filler for h-BN and AlN
Respectively.................................................................................................................... 112
Figure 40: H-C Model up to 100% Filler for 70nm to 1µm h-BN and 40nm AlN
Respectively.................................................................................................................... 113
Figure 41: H-C Model up to 2.44% Vf (5% Mf of h-BN) Filler for 70nm to 1µm h-BN
and 40nm AlN................................................................................................................. 114
Figure 42: R-MG Model up to 100% Filler for 70nm to 1µm h-BN and 40nm AlN with
an Interfacial Layer Thickness of 1 nm .......................................................................... 116
Figure 43: R-MG Model up to 2.44% Vf (5.0% Mf of h-BN) Filler for 70nm to 1µm h-
BN and 40nm AlN Respectively with an Interfacial Layer Thickness of 1 nm ............. 117
Figure 44: R-MG Model up to 2.44% Vf (5% Mf of h-BN) Filler for 70nm to 1µm h-BN
and 40nm AlN with an Interfacial Layer Thickness of 2nm .......................................... 118
Figure 45: TC Results of Polymer Matrix Materials Cured at Standard Conditions...... 120
Figure 46: TC Results of Hot Pressed h-BN Plate and Various Powders ...................... 122
Figure 47: TC Results of Silane Treated and Untreated 1µm h-BN in 3140 PU ........... 124
Figure 48: TC Results from 137nm h-BN in 3140 PU Cured at Standard Conditions with
Dispersion Method and Mass Fractions As Noted ......................................................... 126
Figure 49: TC Results from 137nm h-BN in Hapflex® PU Cured at Standard Conditions
......................................................................................................................................... 128
Figure 50: TC Results from 70nm h-BN in 3140 PU with Mf and Treatment Noted..... 133
Figure 51: TC Results from 40nm AlN in 3140 PU Cured In Standard Conditions ...... 136

xxi
Figure 52: Comparative Plot of 0% to 5% Mf h-BN and AlN of Various Size and
Treatment in 3140 PU to M-G, H-C and R-MG Models................................................ 138
Figure 53: Split Section of 12 atm Pressure Cured Composite and Mold...................... 141
Figure 54: TC Results from 12 atm and 1 atm Cured 5% Mf 70nm h-BN in 3140 PU.. 143
Figure 55: TC Results from 12 atm Cured and 1 atm Cured 5% and 1% Mf 137nm h-BN
in 3140 PU ...................................................................................................................... 149
Figure 56: Results from 12 atm Cured and 1 atm Cured 5% and 1% Mf 1µm h-BN in
3140 PU .......................................................................................................................... 152
Figure 57: TC Results from 12 atm and 1 atm Cured 5% Mf 40nm AlN Untreated and
Silane Treated in 3140 PU.............................................................................................. 155
Figure 58: Comparison of Low Mass Fraction PU Composite Responses to 1 atm vs. 12
atm Curing ...................................................................................................................... 156
Figure 59: TC Results from Silane Treated and Untreated 8µm h-BN in Stycast 1264
Epoxy.............................................................................................................................. 178
Figure 60: TC Results from PT180S h-BN in Epon Epoxy Cured with Pressure as
Indicated.......................................................................................................................... 181
Figure 61: Ultrasonic Particle Dispersion with Nitrogen Cooling.................................. 182

1
CHAPTER 1: INTRODUCTION
1.1. Motivation of Work
The U. S. Navy has an ever increasing need for more powerful and sensitive acoustic
transducers and sensing electronics. A major impediment to the deployment of new, more
powerful, hotter operating electronics has been the encapsulants used to protect such
systems from seawater. Currently used encapsulants, such as polyurethane (PU), are poor
conductors, with a thermal conductivity (TC) of about 0.2W/m-K, that have already
limited deployment of transducer elements because of heat management issues.
Piezoceramic materials, used in the active elements, have a TC of about 2.0W/m-K and
require a high TC encapsulant that can better conduct heat into the seawater.
Past attempts have produced PU micro-composites with high TC particulate fillers, only
to find that the high weight fraction of filler needed reduced the elasticity of PU and
made it unusable for sound transmission in seawater [1, 2, 3].
Nanocomposites may solve this issue by taking advantage of the increased specific
surface area and enhanced (as compared to the same bulk material) [2, 3] interfacial
catalytic processes typical of nanostructured materials. These nano-effects may enable
higher TC composites than are possible with micro-powders and the same or greater TC
may be achieved at much lower weight fractions, allowing the PU to retain its acoustic
transparency.

2
Composite fillers used in this study were high TC, platelet-shaped, hexagonal boron
nitride (h-BN) and spherical aluminum nitride (AlN) particles. The particulate fillers
were added in weight percentages ranging from 1.0 to 5.0. The PU matrix was Uralite®
FH-3140, a typical encapsulant, which, future testing will likely show, retains its acoustic
transparency and low water absorption qualities up to 5.0 mass percent filler.
1.2. Objectives
The purpose of this project is to develop a new elastomeric nanocomposite encapsulant
that is highly thermally conductive, acoustically transparent, highly dielectric, and
electrically insulating. It can replace current elastomeric materials now in use by the U.S.
Navy, allowing a better match with the 2.0W/m-K TC of piezoelectric ceramics
commonly used in the s acoustic generators being encapsulated.
The major technical objectives of this work were:
(a) Successfully fabricate high thermal conductivity polyurethane-BN and
polyurethane-AlN nanocomposites with well-controlled dispersion.
(b) Determine a thermal conductivity threshold, if one exists at or below 5.0 mass
percent, to achieve the maximum thermal conductivity value.
(c) Maintain the “acoustic transparency” of the nanocomposite: The acoustic
impedance must remain the same as that of seawater, approximately 1.50 x 106
metric Rayls [4].

3
1.3 Significance
The immediate benefit of such an encapsulant is that it would allow increased power
output, and longer duty cycles for current acoustic generators and transducers. It would
also allow the deployment of more powerful and more sensitive next-generation
transducer arrays that cannot yet be deployed because of heat management issues.
Other undersea instruments, such as sensors for measuring the thermocline, would benefit
from a high TC encapsulant, allowing more rapid thermal equilibrium, producing a more
accurate and rapid response to temperature change.
Another advantage of a high TC encapsulant would be the ability to use non-destructive
thermal-based sensing methods, such as thermal effusivity probing to look within and
underneath the encapsulant to detect debonded regions, corrosion, water infiltration, and
air cavities. Currently, suspect systems are inspected by destructive methods. With the
current procedure, an obvious external flaw such as visible corrosion would necessitate
the total breakdown of the sensor array. But many times there is no internal damage
found on a unit that otherwise could have continued in service. Also, other systems can
fail for no obvious reason until a total breakdown reveals hidden internal damage after
the fact.
A high TC encapsulant would allow inspectors to determine the extent of corrosion or
damage on a suspect unit with non-destructive means, allowing quick re-deployment of

4
usable systems. It would also allow inspection and detection of other systems that may
have hidden internal problems.
Ultimately, the significance of a high TC encapsulant would be a more able and ready
U.S. Naval undersea defense system, while achieving cost savings for the U. S.
taxpayers.
1.4. Organization of Thesis
This thesis consists of five chapters with an Introduction
 Chapter 1 – Outlines the motivation, objectives and the significance of the work.
 Chapter 2 – Gives the background for the work which includes Matrix Materials,
Filler Materials and the Nano-Effect, Particle Coatings, Surface Treatments and
Additives, Thermal Conductivity and Analytic Prediction, and Dispersion and
Imaging.
 Chapter 3 – Summarizes Experimental Work and Procedures, including
Composite Fabrication, and Microscopy and Imaging and Analytic Prediction
sections.
 Chapter 4 – Results and Discussion gives an overview of all meaningful
experimental and analytic results with a comprehensive discussion explaining the
results, ending in an overview of the highest TC composites.

5
 Chapter 5 – Summary, Conclusion, and Future Work concludes the thesis by a
summation of work done with final conclusions, recapping project goals,
explaining how the results relate to the goals and describes the future work
needed to further the project goals.
 Appendix A – gives an overview of results obtained making epoxy matrix
composites
 Appendix B – illustrates all useful raw data obtained from thermal conductivity
testing organized in descending order of particle size and mass fraction.
 Appendix C – Shows all MathCAD calculations used to derive all mixing rule
thermal conductivity predictions and displays alternate plots of the data not used
in the text.
 Appendix D – displays the individual force vs. elongation plots for each tensile
testing specimen followed by the averaged stress vs. true strain plot for each
composite tested.
 Appendix E – displays figures showing the distance between spherical particles of
various size and mass fraction in a two-phase composite.

6
CHAPTER 2: BACKGROUND
2.1. Matrix and Materials
The composites made for this project were two-phase particulate-filled, having a single
matrix material (phase 1) and a single particulate filler (phase 2) in each composite
specimen.
The literature relating to the fabrication of high TC polymer composites usually describes
composites made with polymer matrices such as epoxy with particulate additions
typically ranging from 20 to 60% by weight, or more [1, 5, 6]. In most of these cases the
composite was not required to retain any particular matrix quality or property such as
high strain and flexibility, low moisture, absorption etc. The goal of those projects was
solely to obtain a polymer composite with high TC and, for some, high electrical
resistivity [5, 6, 7].
It is a fact that most man-made composites in use today are reinforced by filler materials
that are intended to toughen, stiffen or strengthen the matrix. But the toughening effect of
the filler is detrimental to the required engineering properties of the elastomeric
composite developed in this work [8-12].
For this project, the matrix materials used were dielectric, conformal, amorphous, and
isotropic polyurethane encapsulants, such as those now in use by the U. S. Navy to keep
seawater away from electronic components. Urethane encapsulants for undersea

7
applications are required to be elastomeric with the purpose being that they act as an
acoustically transparent interphase between transducer and seawater [4, 10-12].
To retain the high elasticity, acoustic transparency and low water absorption, as well as
other important encapsulant properties, particulate mass loadings did not exceed 5.0% by
weight.
In the initial stages of the project three matrix materials were considered; as shown in
Table I the materials were Conathane® EN-7, Uralite® FH-3140, and Hapflex® 566 [4,
10, 11, 12]. All three materials are 2-part thermosetting elastomeric polyurethanes (PU).
EN-7 PU, an often used transducer encapsulant, was a natural first choice for
consideration. FH-3140, an encapsulant used for similar marine applications by the U. S.
Navy, was also considered, along with the lower viscosity Hapflex® PU [4].
Note the viscosities of the matrix materials listed in Table I; they span about one order of
magnitude, from 0.670 Pa-s to 5.550 Pa-s [10-12]. The reason for including materials in
this range is that matrix viscosity during mixing, particulate dispersion and pouring is a
major factor affecting dispersion and possibly the alignment of particulates [1-3, 5-7, 9].

8
Table I: Matrix Materials and Manufacturers Specifications
The 2-part urethane made by H. B. Fuller Corporation, commercially known as Uralite®
FH-3140, has a medium viscosity Part-A component. The part-A weight percentage of 82
allows the remaining 18%, Part-B, to be added without particles, thus simplifying the
process [11].
By contrast and comparison, some composites were made using Hapflex® 566. The
Hapflex® was tested to determine if its lower viscosity eased mixing and dispersion, but
this in fact was not the case. Hapflex® 566, a two part urethane with 50% part-A and
50% part-B by weight, has a high viscosity part-A necessitating part-B to have a low
viscosity, too low in which to mix and suspend particles. In the high-viscosity part-A it
was also more difficult to disperse particles, requiring more shear energy and ultrasonic
energy than part-A of the Uralite® product. Finally, the 1:1 mass ratio of Hapflex® parts
A and B mixture necessitated both parts to contain particles prior to final mixing, thus
making the Hapflex® PU impractical and more difficult to prepare as a matrix material
[12].
Urethane Type
Conathane® EN-7
[10]
Uralite® FH-3140
[11]
Hapflex® 566 [12]
Viscosity @25C, Pa-s 5.550 3.800 0.670
Specific gravity @25C 1.01 1.07 1.05
Young’s modulus, MPa Not Available 20.0 4.14
Percent Elongation > 400 700 500
Tensile strength, MPa > 13.79 7.07 6.21

9
Testing on the other urethanes suggested that a higher-viscosity Conathane® EN-7
composite would have been more difficult to manufacture [4, 10] so we decided not to
use it.
Consequently all PU composites (with the exception of one Hapflex® experiment) in this
project used Uralite® 3140 as the matrix material.
2.2. Filler Materials and the Nano-Effect
Fillers used in particulate composites are classified, amongst other ways, according to
their size, which is a particle’s average characteristic dimension. This would be the
diameter for spherical shaped particles and planar thickness for platelet-like particles.
Fillers typically used in particulate composites are micrometer-sized, so they have a
higher specific surface area than larger particles. Smaller, still, are nanoparticles, which
are described as having at least one average characteristic dimension of less than 100nm,
giving them a marked increase in specific surface area as compared to microparticles
[13].
The importance of filler size and particulate specific surface area should be made clear;
surface area determines the magnitude of the catalytic processes and the effects of
interfacially driven phenomena such as wetting, adhesion and heat conductance between
matrix and filler. The large surface area of nanostructured particles maximizes the
interfacial contact area, producing a proportionally greater property-altering effect per

10
volume fraction of particles than is possible with micropowders; this phenomenon is
often called the nano-effect [5, 13, 14].
Because of the nano-effect, the increase in TC of a composite, brought about by the
addition of high TC particles, is expected to increase with decreasing particle size at the
same volume or weight fraction. In fact, a low volume fraction of nanoparticles may
preserve desired matrix characteristics while achieving a significantly larger increase in
TC compared to much higher volume fraction loadings of micro-particles of the same
material [2, 13, 14].
The filler materials used in this project were hexagonal boron nitride (h-BN) in various
sizes and 40nm aluminum nitride (AlN) particulates, as shown in Table II.
Table II: Particulate Fillers with Published Specifications as Noted
Description
Boron
Nitride
Micropowder
[15]
Boron
Nitride
Micropowder
[16]
Boron
Nitride
Nanopowder
[17]
Boron
Nitride
Nanopowder
[16]
Aluminum
Nitride
Nanopowder
[18]
Trade name None None
B-N-02-
NP.H135
None None
Average
particle size
nm
1,000 500 137 70 40
Specific area
m2/kg
5,600 20,000 19,400 45,000 78,000
Density
kg/m3 2,290 2,300 2,250 2,300 3,260
Thermal
Conductivity,
k, W/m-K at
298K
Perpendicular to Basal Plane; k < 30 [19]
Parallel to Basal Plane; k < 600 [19]
Directional Average; k ~ 33.5 [16]
285
Crystal
Structure
Hexagonal Hexagonal Hexagonal Hexagonal
Hexagonal
(Wurtzite)

11
The particle dimensions, as provided by the manufacturer in Table II, are the smallest
average dimension, the mean size, of each batch of particles used. The particles’ size and
shape were verified by SEM and FE-SEM, which showed that the h-BN particles were
platelet-shaped whereas the AlN particles were essentially jagged, elongated spheres. Fig.
1 shows an edge-on platelet from the 1,000 nm h-BN batch. The scale confirms the mean
dimension to be approximately 1,000 nm.
Figure 1: SEM Image of 1 Micron h-BN Particles Set in Epoxy
The high TC and highly dielectric properties of h-BN and AlN make them good
candidates for creating dielectric, high TC particulate composites. For this reason, h-BN
1 micron particle,
edge-on

12
and AlN nanoparticles were the fillers used in all nanocomposites developed for this
project.
Boron Nitride
Boron nitride, BN, is a man-made ceramic in which the outer shell electrons of boron are
bound to the nitrogen, making it a stable compound even at temperatures as high as
900°C in air. It is nonreactive with water, ostensibly an important trait for the proposed
undersea application [20].
The two allotropes of boron nitride are cubic, c-BN, and hexagonal, h-BN. Although c-
BN has a very high TC, it is expensive and difficult to obtain in nanopowder form. It is
also a very hard and abrasive material able to damage and shorten the working-life of
molds and machine tools during composite manufacture.
Hexagonal boron nitride, on the other hand, is a comparatively soft ceramic with a
graphite-like structure, endowing it with wet and dry lubricating properties much like
those of graphitic carbon. h-BN also has a low dielectric constant, K K ≈4 [16], and
unlike the isotropic TC of c-BN, the TC of h-BN varies directionally as shown in Table I
and in Fig. 2 [7, 20].

13
k < 600 W/m-Kk < 600 W/m-K
k < 30 W/m-K (Out of Plane)
Basal Plane
c
a1
a2
Figure 2: Thermal Conductivity of a Single Hexagonal Boron Nitride Crystal [19]
The anisotropy in the thermal conductivity of h-BN is greater than that in aluminum
nitride and other ceramics such as silica, alumina, etc., but the average thermal
conductivity of h-BN, as listed in Table II, is significantly lower, making the use of h-BN
appear impractical. In actual use, however, the thermal conductivity of h-BN-polymer
composites exceeds the others [5, 7]. The reason is still unclear, but the literature clearly
shows that epoxy particulate composites made with h-BN have superior thermal
conductivity over epoxy matrix composites made with other particulate additives [5, 7].
One theory explaining the successful use of h-BN in epoxy composites is that relatively
soft h-BN reacts at the interface with the matrix creating a deep interfacial zone or
interphase region formed by the interdiffusion of both materials [13, 14, 21]. The effect is
thought to produce ordered, multi-layered, zones of high thermal conductivity that
interconnect particles along basal planes to create a non-contact network, a pathway for
increased thermal transfer throughout the composite system [13, 14, 21].

14
Another theory is that h-BN, being a solid lubricant, alters the rheological properties of
the polymer matrix, which aligns particles in the direction favoring increased mobility.
As a result, h-BN crystals align along basal planes within the matrix, creating a network
of thermal bridges throughout the composite [2, 3].
For the reasons mentioned, and the fact that h-BN nanoparticles, now commonly used in
lubricants, are readily available from many sources at low cost, h-BN filled
nanocomposites can be a financially viable product suitable for use in a host of common
applications beyond the motivations responsible for this work.
The h-BN materials listed in Table II were purchased from three sources. All h-BN
materials displayed a platelet-like, jagged-edged morphology. The chemical content for
all nanoparticles was confirmed by XRD and TEM-EDS, as shown below in Figs. 3 and
also Figs. 4 and 6.

15
Figure 3: X-Ray Diffraction from Three Different h-BN Samples
Note: NanoAmor Batch 1 was NOT boron nitride!
The peak at 2θ = 27° for batch 2 of the 137nm h-BN is lower and broader than the peak
for 1µm h-BN; this is consistent with the expected effect of the smaller sized
nanoparticles.
In Fig. 3 the NanoAmor, batch 1 material was clearly not h-BN. It was returned to the
supplier. This problem occurred two other times with another supplier whose material
was ultimately rejected. Fig. 4 displays additional efforts to confirm the composition of
the composite filler materials purchased.
0
1000
2000
3000
4000
5000
6000
7000
20 30 40 50 60
Differential Angle (2-Theta) Degrees
Intensity(Counts)
Aldrich 1.0 micron h-BN
NanoAmor h-BN (batch 1)
NanoAmor h-BN (batch 2)

16
Figure 4: Transmission Electron Microscopy-Energy Dispersive Spectroscopy of
137nm h-BN (NanoAmor Batch 2)
Fig. 4 shows that the second batch of 137nm h-BN was composed of only boron and
nitrogen; copper is from the grid used to support the h-BN material in the TEM.
Aluminum Nitride
Aluminum nitride, synthesized by nitridation of aluminum [20], is a ceramic material that
first found commercial applications in microelectronics. This highly thermally conductive
material is stable in air up to 700°C, and has a high dielectric constant, 9.14K at
26.85°C. Nano-sized AlN particles are available in bulk quantity, making it practical for
large scale applications.
from BN nanoparticles
from TEM sample grid

17
The hexagonal-wurtzite AlN nanopowder, shown in Fig. 5 appears as jagged particles,
having crystal morphology similar to that of the h-BN structure, shown in Fig. 2.
The wurtzite structure of AlN is the form most commonly available, and, having very
high thermal conductivity, is the material of choice when making high TC AlN
particulate composites [20].
Figure 5: FE-SEM Image of Gold Sputtered Cryo-Ultramicrotome Section of 5%
Mf Z-6020® Silane Coated Wurtzite AlN in 3140 PU

18
Notice in Fig. 5 the large (>1µm) particles aggregated at center and the smaller and
fainter particles surrounding the aggregate. The large aggregate may be oxide of
aluminum or some other impurity while the smaller particles are likely AlN. As with h-
BN, the AlN particles were analyzed using XRD, as shown in Fig. 6.
Figure 6: X-Ray Diffraction Analysis of 40nm AlN as Compared to a Known Peak
Profile
Note extra peaks at positions 2θ=42°, 44° and 51°.
The results show additional peaks between 2θ = 40° and 52° in Fig. 6. This implies a
somewhat lower than specified “99% minimum purity” from the supplier, and could have
been the result of poor packaging and subsequent environmental reaction, or it could
simply be poor quality control by the manufacturer. In either case, XRD and FE-SEM
imaging proved the material, a late addition to the project, to be suitable for this project’s
requirements. Therefore, we proceeded to use it, keeping in mind the impurity issue.
Intensity
Counts
Position, 2-Theta
Degrees

19
One drawback of AlN is that it hydrolyzes slowly in water potentially causing some
technical problems when making composites. And another possible drawback is that the
AlN filler may react with seawater, making it unusable for undersea applications;
underwater testing of the resulting AlN nanocomposites will be required to determine the
validity of this concern. Yu, Chung, and Mroz found AlN-epoxy micro-composite
degradation in wet applications at the high filler Vf of 0.6 [6].
In spite of possible water degradation issues when AlN is encased in epoxy, AlN
nanoparticles may not be attacked by seawater when encased in a PU matrix. Also, if
AlN does provide a TC benefit, perhaps other applications will be more forgiving. In any
case, the AlN nanocomposite TC results were useful as a comparison to the h-BN
composites [6, 19, 20].
2.3 Particle Coatings, Surface Treatments and Additives
The interfacial region between particle and matrix is generally a major source of thermal
resistance in a composite system. Alleviating this problem in order to improve thermal
conductivity in composites has been a subject of interest for more than a decade [1-3, 5-7,
14, 19, 22-25]. Some scientists have fabricated high thermal conductivity polymer
(epoxy) composites with surface treated h-BN and AlN having TC as high as 11.0 W/m-
K [5, 6]. These studies concluded that thermal conductivity of composites was
significantly improved with certain surface treatments [2, 3, 5-7].

20
Surface treatment of the particulate filler, known as functionalization, can improve
wetting at the particle-matrix interface which may reduce the number and size of
interfacial gaps and flaws which would increase the area of contact and available thermal
pathways between particle and matrix [2]. It is important to mention that wetting and
bonding are not synonymous; wetting is necessary but, alone, is insufficient to insure
proper bonding [2, 8]. Good chemical bonding, beyond good wetting, may also decrease
thermal resistance at the interface since good bonding is thought to offer more pathways
at the atomic level for heat transport between particle and matrix and particle to particle
[2, 13].
The relationship between contact angle and surface energy is summed-up by Young’s
equation [8]:
cosSV LS LV     , (1)
in which is the specific surface energy and subscripts SV, LS and LV represent
interfaces of solid to vapor, liquid to solid, and liquid to vapor respectively.  is the
contact angle of the liquid matrix on a plate of the solid particle surface [8]. Wetting is
depicted in Fig. 7.
Liquid
0o
 
solid
No Wetting Wetting
180o
 
Self Spreading
Liquid
Liquid
solid
Vapor SV
LV
LS
Vapor
solid
0 90o o
 
Figure 7: Three Different Conditions of Wetting [8]

21
Increased particle wetting in a composite may increase interfacial heat transfer between
particle and matrix. Some silanes improve only wetting; others improve wetting and
adhesion between the inorganic particle and organic matrix PU [2, 26].
Silane Surface Coatings
In theory, a silane coupling agent works by creating reversible silanol bonds between the
polymer matrix and ceramic filler where the inorganic group bonds to the oxygen on the
surface of the ceramic while the organic reactive group, R, bonds to or entangles with the
polymer, as shown in Fig 8:
Ceramic
Particle
Polymer
Ceramic
Particle
R
O Si O
O
H H
O
M
Polymer
R
H2O + O Si O
O
M
Ceramic
Particle
R
O Si O
O
H H
O
M
Polymer
Figure 8: Plueddemann’s Reversible Bond Associated with Hydrolysis [8]
In Plueddemann’s theory, as illustrated in Fig. 8, hydrated silanols form hydrogen bonds
on the ceramic surface at the M-O-H segment, where M is the base material of a glass
(ceramic) surface, such as Si, Al, B, Fe, etc. The reversible, hydrolyzed bond in the
presence of water, which diffuses in from the resin, allows the particle and matrix
surfaces to slide past one another during strain without rupturing the bond. After strain

22
has occurred, the bond re-forms, maintaining intimate contact and adhesion at the
interface. The bonds which form and re-form in the presence of water exist in a state of
dynamic equilibrium; making and breaking bonds allows relaxation of interfacial stresses
on a molecular level [2, 8].
Research determined that an appropriate silane for functionalizing h-BN in PU was Dow
Corning’s Z-6040® and for AlN in PU, Dow Corning’s Z-6020® was well suited [26].
Z-6040® is a bifunctional silane containing a glycidoxy reactive organic group and a
trimethoxysilyl inorganic group.
 -Glycidoxypropyltrimethoxysilane is especially reactive with elastomeric urethanes
such as FH-3140 and readily bonds to ceramics such as h-BN, increasing interfacial
surface wetting [5, 26].
Z-6020® is a diaminofunctional silane similar to Z-6040® in that it has organic and
inorganic groups especially suited to bond elastomers to ceramics. Its chemical properties
make it better suited to coating hydro-reactive AlN particles than is Z-6040® [5, 6, 26].
Surface Treatments
In contrast to surface coatings such as silanes, surface treatments are usually intended to
modify the particle surface so as to increase surface area or roughness, which can
enhance mechanical bonding and often improve physical (chemical) bonding. Surface
treatments may also modify surface chemistry, improving interfacial wetting and/or
bonding. Surface treatments, such as oxidative etching with nitric acid and chemical

23
treatment with acetone can increase the surface area of h-BN to benefit mechanical
bonding and possibly create reactive sites which can improve chemical bonding,
interfacial zone reactivity, or both. Such improved bonding may increase the number of
thermal pathways and overall rate of heat transfer through the composite [2, 5, 6]. For
these and other reasons, nitric acid and acetone-treated h-BN composites were fabricated
and the results were compared to those of untreated and surface coated particulate
composites [2, 5, 8].
Apparent Contact Angle
According to Wenzel, surface treatments can also decrease or increase the apparent
contact angle determined by sessile drop wetting tests similar to that shown in Fig. 7 [8].
To explain the relationship between surface area and contact angle, Wenzel discussed the
effect of surface roughness by stating that within a measured unit on a rough surface,
there is actually more surface and therefore more surface energy than in the same
measured unit on a smooth surface. Therefore if apparent contact angle, r, is put in terms
of surface area; r/real p ojectedr A A , Young’s equation can be used to find the increase (or
decrease) in wetting according to the ratio of surface areas [8]:
 
0cos SV LS
LV
r  



 (2)
In Eqn. 2, 0 is the static contact angle that is measured after all phases have achieved
natural equilibrium and the 3-phase line or contact angle no longer changes.

24
Additive Method
The additive method is another means of functionalizing inorganic particles so that
wetting with the organic matrix is improved [26]. Silane added to the matrix may diffuse
throughout the composite where it could bloom at the h-BN and AlN interface [2, 26].
Attempts to use the additive method are described in the Experimental Procedure.
2.4 Thermal Conductivity and Analytic Prediction
Numerous authors have documented attempts to analytically predict the properties of
two-phase composites, suspensions and nanofluids, such as electrical conductivity,
permittivity or thermal conductivity [14, 21, 24, 27, 28]. Recently, models for the
properties of 2-phase dielectric isotropic composites that include an interfacial layer, or
interphase, have been developed. They are usually based on earlier macroscopic theories
of two-phase models with the addition of interfacial resistance [7, 25, 29].
A major issue with macroscopic thermal conductivity models is that the thermal gradient
between the matrix and particle is, without experimental data, unknown. This is due to
numerous factors that combine to determine the interfacial properties and interfacial
resistance of a composite. For example, different size particles, even of the same
materials, can have vastly different energy level structures that vary greatly with small
size differences [13]. Even composites made of the same materials with identical
intensive properties, using the same volume fractions, can have significantly different TC
simply because of differences in the method, procedure or conditions in which it was

25
made. The interphase and interfacial thermal resistance in a composite is a product of
both intensive and extensive properties, which can make for considerable variability of
TC, from one composite batch to another. Therefore, any equation that factors in
interfacial resistance must be partially derived from the experimental results of the
particular composite in question. Nevertheless, a good model should be able to give a
reasonable prediction of the TC of a composite under ideal conditions, providing the
researcher with a realistic target.
Analytic models used to predict heat transfer for this work are all based on the process of
conduction which is described as the diffusion of energy due to the random motion of
molecules in a system, and is, in general, modeled by the rate equation of heat also
known as Fourier’s law of Heat Conduction [30]:
"
x
dT
q k
dx
  (3)
The 1-D rate equation defines heat flux "
xq , with units, 2
W
m
, representing the rate of heat
transfer in the X-direction, per unit area perpendicular to the direction of heat transfer at
temperature T.
The proportionality constant, k, is the transport property known as thermal conductivity
with SI units of Watts per meter-Kelvin, or W/m-K.

26
The thermal conductivity constant k is an intensive property of a material whether it is a
single substance such as boron, a compound such as boron nitride or a composite of two
or more phases. The minus sign indicates that heat transfer occurs in the direction of
decreasing temperature [30].
The three dimensional case of Eqn. 3 can be expressed as [30]:
"
q k T   , (4)
where
, ,
T T T
T
x y z
   
   
   
(5)
is the temperature gradient with SI units K/m [30].
In the case of this work using an isotropic matrix having randomly dispersed particles,
the result is a composite with isotropic TC therefore the gradient components become
equal, such that:
T T T
x y z
  
 
  
. (6)
And as a result, Eqn. 3 becomes valid for use on any surface of an isotropic composite.
For conduction in a 2-phase composite, D. J. Jeffrey [14] added to the rate equation by
taking into account the average dipole strength of neighboring particles [14]:
"
q k T nS    (7)
Wheren is the number of particles per unit volume in a composite; S is the average
dipole strength for a single particle in a composite. S is the average dipole strength of all

27
particles, per unit volume, as a whole. The average dipole strength is determined by the
volume fraction of particles in the matrix as well as how they are dispersed within the
matrix [14].
Conduction across Interfacial Boundaries
From a macro-scale perspective, it is well known that thermal conduction across
interfacial boundaries can be a problem. For example, the surfaces of an integrated circuit
and a heat sink are not smooth enough to eliminate air pockets and gaps at the interface
when the heat-sink is pressed to the IC under low pressure. High pressure would damage
the delicate IC [19]. Even though the two components’ surfaces’ are highly conductive,
they will not efficiently conduct heat across the interface unless there is a conformal, high
thermal conductivity material joining them. Highly conductive, highly viscous grease or
paste, usually made of silicone with a high volume fraction of h-BN, is often added to the
interface. It can flow under low enough pressure to avoid damage to the IC and still
create a voidless interfacial system with intimate contact. From applied pressure, the
paste becomes a thin interphase layer that excludes air and bridges gaps between the IC
and cooling fin surface [23]. It also should be mentioned that a certain level of adhesion
between boundaries is created in such a system; perhaps interfacial adhesion is required
for heat conduction to occur across interfacial boundaries in solid materials [23, 31].
Surfaces of solid materials will not readily adhere together or come into intimate contact
unless pressure is high enough to cause permanent deformation and hence, interdiffusion
to occur. Without deformation and interdiffusion, the two surfaces are never perfectly

28
conformal, having hills and valleys that result in voids and non-contact regions over most
of the interfacial surface area [31, 32]. In the case of bi-clad and tri-clad metals, such as
commonly seen in metal pans, the clad layers, usually stainless steel and copper or
aluminum, are rolled together under very high pressure. High pressure rolling pushes air
from between layers which deform, flow, and adhere together, making for very good
conduction across the interface [32]. As part of experimental work conducted for this
project, the concept that pressure improved interfacial boundaries and TC was tested by a
series of experiments which produced 12 atm, pressure cured composites to compare with
1 atm cured composites.
Certainly, interfacial thermal resistance plays a major role in the thermal conductivity of
composites. And so does the mechanism by which heat crosses the interface [5, 6, 25, 33-
35]. The question to be answered is: Can a relatively hard, highly dielectric and thermally
conductive substance such as h-BN, even with functionalization, form an intimate contact
network, on a nanostructured scale, with a compliant, organic polymer such as
polyurethane, and maintain the necessary contact pressure and/or adhesion needed at the
interface to efficiently conduct heat between particle and matrix?
An explanation of the heat transport mechanism in a dielectric composite may help
illuminate the answer.
According to most microscopic theories, virtually all heat transport in glassy and
crystalline dielectric solids is due to thermal conduction, which is described as elastic

29
vibration of the lattice structure. In dielectric solids, the elastic vibration due to thermal
transport can be attributed to phonons, which are described as quasiparticles
characterized by the quantization (energy transfer) of the periodic modes of vibration
[30]. Thermal transport is viewed by macroscopic models to be randomly diffuse and
created at random. And in isotropic materials such as amorphous solids, heat is conducted
evenly in all directions. Amorphous solids include materials such as glasses and
polymers, like PU. They are described as substances with atoms held apart at equilibrium
spacing with no long range order [30]. The nearly random atomic order of amorphous
solids causes them to have a low level of elastic vibration and therefore low thermal
conductivity. The low thermal conductivity typical of all amorphous solids is unlike the
highly variable thermal conductivities of various crystals [30, 25, 35]. The macroscopic
view of diffuse thermal transport is described by the Diffusion Equation:
2T
D T
t

 

, (8)
where t is time, D is thermal diffusivity, 2
 is the second degree temperature gradient.
/ pD k c (9)
And ρ is the density with SI units of kg/m3
, while cp is the specific heat of the material
with SI units of J/K [30].
The Diffusion equation, derived from Fourier's Law (Eqn. 3), being the basis of the
macroscopic approach to thermal conduction, is used by all effective medium theories to
predict TC. In effective medium theory, particle size plays no role; only particle shape,
surface area, volume fraction and TC determine the effective TC of a composite [14, 21,
24, 25, 27-30, 33-37]. Effective medium theory predicts a variation of temperature across
the particle, causing heat transport and a temperature gradient between particle and

30
matrix. Microscopically, temperature gradients in a crystalline particle are thought to be
caused by phonon scattering which can occur when phonons collide with other phonons,
material defects, and interfacial boundaries where changes in atomic structure occur [30,
35]. While acknowledging that changes in atomic structure at the particle to matrix
interface can greatly increase phonon scattering and thermal resistance, good interfacial
adhesion can allow for increased elastic vibration to propagate across the interface to
another particle, thereby increasing heat transport. Therefore interfacial thermal
conduction is strictly limited by the quality of interfacial adhesion, the individual phonon
transport properties of the particle, and the proximity of neighboring particles, in
accordance to Eqn. 7 [14, 30, 33-35].
Several researchers have concluded that effective medium theories and diffusion models
are unable to account for the high thermal conductivities realized from experimental
results. Consequently, other mechanisms of conduction have been postulated in an
attempt to rationalize the performance of high TC composites that are beyond the
phenomenological predictions of effective medium theories [14, 21, 25, 33-35].
One such theory involves the calculation of the phonon mean free path (MFP), which is
the distance phononic energy can travel in a medium before it encounters another wave,
causing it to scatter. And it is the measure of the rate at which energy is exchanged
between phonon modes [30].
10 mzT
l
T
 (10)

31
Where Tm is the melting point of the solid, T is the temperature of the material of interest
in Kelvin, z ≈ 0.5nm, is the atomic lattice constant, and 1  is the Gruneisen parameter
which in Debye's formula render the length of the MFP,l , in nanometers [30, 35].
Using Eqn. (10) for h-BN, having Tm = 3,246 K [20] at a temperature of T = 298 K, gives
Tm/T= 10.9, rendering the MFP, h BNl  = 54.5nm. And likewise, the MFP at 298K for AlN
is AlNl = 36.9nm [20]. Consequently, phonons cannot diffuse in particles smaller than
their computed MFP value because phonon wavelengths are too large to exist there. In
such a case, phonons are scattered by the particle and heat transport is effectively
reduced.
Another issue concerning phonon transport in a polymer composite is the MFP
calculation for the matrix, which has been calculated by Debye's formula to be in the
10ths of nanometers range for amorphous polymers at normal temperatures [30, 35].
The overly small MFP of the matrix is the cause of its low TC and the reason that simple
phonon transport theory alone cannot explain the TC of composites with non-contact
particle structures. Other theories have been developed to explain how phonon transport
can apply to low mass fraction polymer composites.
Ballistic Phonon Theory predicts that elastic lattice vibration cannot propagate in
particles smaller than the Debye size limit and therefore phonons or thermal energy
moves ballistically across the particle, leaving the temperature within the particle

32
essentially constant. But the same boundary conditions for heat flow can be duplicated by
a very fast diffusion model. However, other ballistic phonon effects lead to higher value
TC estimates that accurately predict conductivity of high TC composites that cannot be
explained or predicted by any diffusion theory. In particular, the low rate of phonon
transport in the PU matrix, due to a MFP limited to tenths of nanometers, may be
overcome by ballistic transport. And, it is theorized that an interfacial matrix layer, 3nm
or less, between adjacent particles, may support ballistic thermal transport between
particles [25, 33-35]. This effect could be helpful at high mass fractions where randomly
spaced particles are close to one another. But, it can also be helpful in low mass fraction
composites where particles aggregate close together into clusters having fully wetted
particles that are separated by a 3nm, or less, of voidless matrix material. For this theory
to work, such clusters must form a network with other clusters, each separated by 3nm or
less of voidless matrix.
Figure 9: Increased Thermal Conductivity Enhancement due to Increased Effective
Volume of Highly Conductive Clusters [35]

33
Clustering as illustrated in Fig. 9, taken from the work of Keblinski et al [35], can be too
close, as in l, where a high Vf of agglomerated filler does little to increase TC, or
optimally separated, as in lV where clustering with proper spacing can achieve high TC
values with low a Vf of filler [35].
If ballistic thermal transport is possible, a properly spaced network of such clusters in a
low Mf composite can form ballistic thermal pathways rendering a higher TC than
diffusion models can predict.
Another theoretical effect, known as the liquid layering effect, has been postulated to be
responsible for an improved MFP and TC in composites. This is an interfacial effect
where it is thought that particle surface energy and density causes liquid layering of the
surrounding matrix to a theorized depth of 1 nm to 2.5 nm.
Liquid layering is thought to give order to the atomic structure of the matrix at the
interphase, increasing MFP and TC to the same value as the crystalline particle [25, 28,
33, 35]. It is also suggested that interfacial atomic order, MFP, and TC decrease as the
layer thickness increases [28, 35].

34
Particle
Tinfinity
Interfacial
Layer
1.0nm-2.5nm
thick
Unaffected
Surrounding
Matrix
Zone of peak
atomic order and
peak TC
Zone of
lowest
affected order
and
lowest TC
Figure 10: Single Spherical Particle with Interfacial Layer of Affected Matrix [28]
If the liquid layering theory is correct, good interfacial adhesion would minimize phonon
scatter at the interface, allowing phonon transport to continue at the same rate through the
ordered matrix into the closest neighboring particle, if it is close enough. This effect is
claimed to be the equivalent of doubling the volume fraction of filler in some
nanocomposites and may be the mechanism by which a low mass fraction nanocomposite

35
may achieve high TC. This effect is also thought to work together with ballistic phonon
transport and particle clustering and may account for the high TC of some
nanocomposites [25, 33, 35].
On the other hand, if particles are considered to be spherical and evenly dispersed in the
matrix, as simulated by effective medium theories, spacing between the particles would
be such that enhanced interfacial conduction as described by microscopic theories would
either not occur, or have less of an impact.
0
10
20
30
40
50
60
70
80
90
100
110
120
130
140
150
160
170
180
190
200
210
220
0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150
SurfacetoSurfaceDistance,nm
Particle Diameter, nm
1% h-BN Mass Fraction 3% h-BN Mass Fraction 5% h-BN Mass Fraction
1% AlN Mass Fraction 3% AlN Mass Fraction 5% AlN Mass Fraction
Figure 11: Average Surface to Surface Distance for Spherical Particles in
Composites [35]
Derived from the work of Keblinski et al [35], Fig. 11 is an example of average surface to
surface distances between spherical particles of known diameter and Mf in a composite.
Fig. 11 shows surface distances between 40nm AlN spherical particles to be about 13 nm

36
at 5% Mf and about 220nm surface to surface distance at 1% Mf. Clearly, the large
distance between surfaces in evenly dispersed systems would not allow the above
described microscopic heat transfer theories to operate with particles of diameter as small
as 40 nm. Alternate views of data from Fig. 11 are shown in App. E.
Effective Medium Theories of Thermal Conductivity
Maxwell Garnet Mixing Rule:
A literature search for macroscopic conduction models of heat transfer in composites
showed that the earliest relevant model still in use is the Maxwell-Garnett (M-G) Mixing
Rule (1904) [27], which is derived from James Clerk Maxwell's 1873 text "a Treatise on
Electricity and Magnetism"1
[21, 24, 27]:
2 1 2 1 2
1
2 1 2 1 2
2 2 ( )
2 ( )
effective
k k V k k
k k
k k V k k
   
     
(11)
k1 and k2 are the thermal conductivities of the matrix and the filler and V2 is the volume
fraction of the filler. In this model, the fillers particles are spherical and the filler's surface
area and particle size are not considered.
Under appropriate conditions, such as 2-phase composites with spherical particles, this
model has been found to give a good approximation of thermal conductivity [14, 24].
1
Despite published credit for the M-G model going to J. C. Maxwell and J. C. M. Garnett (See [23]), some authors credit J. C. M.
Garnett alone for the M-G Model, since his full name was James Clerk Maxwell Garnett and worked independently of J. C. Maxwell,
but a comparison of the M-G model and equations published in Maxwell's 1873 Treatise show an unmistakable correlation. See
reference [43] for more information on this matter.

37
When nanoparticles are used in a dielectric composite, the M-G equation, in some cases,
predicts lower TC than experimentally measured values, possibly because it does not
allow for the increased interfacial area and higher reactivity between nano-sized fillers
and matrix [14].
In Appendix C, a MathCAD example sheet is shown, making clear that only the thermal
conductivity, density, and volume fraction of each material is needed to solve this
analytical model. Also, note the surface area of a sphere is the lowest surface area
possible for a given volume of material. Therefore, this model represents the lowest area
of interfacial interaction possible for a specified particle mass in a 2-phase composite
system [14, 24]. The obvious disadvantage of this model for our work is that it will not
accurately model TC because it ignores the shape and size of the filler materials which
are known to affect the TC of a composite [2, 7, 13, 14, 24, 25, 33-35, 36, 37].
The Hamilton-Crosser Equation:
Another widely used model for predicting the TC of two-phase composite materials is the
Hamilton equation (1960) [24];
1 1 1 2 2 2
1 1 2 2
( / ) ( / )
( / ) ( / )
effective
kV dT dx k V dT dx
k
V dT dx V dT dx



. (12)
Wherein (dT/dx)1 and (dT/dx)2 are the average temperature gradients for each phase in a
two-part system. The advantage of this model is that the average gradient ratio in a
system can be determined and used to include the particle shape and therefore, particle

38
surface area so that small, high surface area nanoparticles with varying geometry can be
factored into the model [24].
2 1
1 2 1
( / )
Temperature gradient ratio between matrix (1) and particle (2);
( / ) ( 1)
dT dx nk
dT dx k n k

 
(13)
in which the shape factor for dispersed particles; 3/ .n  (14)
Surface Area of a Sphere of one Particle Mass
And the sphericity ratio;
Actual Particle Surface Area
  (15)
If the sphericity ratio equals 1, the temperature gradient ratio becomes:
2 1
1 2 1
( / ) 3
( / ) 2
dT dx k
dT dx k k


(16)
The temperature gradient ratio is given in two forms, Eqn. (13) and Eqn. (16). Eqn. (13)
uses shape factor, n, to include the sphericity ratio (or surface area ratio) of a single
particle vs. a sphere of equal mass. This function will allow the average particle surface
area to be modeled analytically using effective medium theory. The value of 3 used for
the shape factor is an experimentally derived parameter taken from the work of Hamilton
and Crosser [24].
Substituting the gradient temperature equation ratio, (16), into Hamilton's equation gives
the Maxwell-Garnett mixing rule, since it uses a sphericity ratio of 1.
Substituting the temperature gradient ratio, equation (13), into Hamilton's equation gives
the Hamilton-Crosser model for thermal conductivity [24]:
2 1 2 1 2
1
2 1 2 1 2
( 1) ( 1) ( )
( 1) ( )
effective
k n k n V k k
k k
k n k V k k
     
      
. (17)

39
Testing has shown that results do not depend strongly on the shape factor, n, unless there
is a thermal conductivity ratio ≥ 100:1 for k2, (filler): k1, (matrix) in the composite [24].
The thermal conductivity of PU is just over 0.2W/m-K, so the thermal conductivity of h-
BN and AlN exceeds the criteria determined by Hamilton and Crosser. This model shows
promise for predicting the thermal conductivity of our nanocomposites.
Thermal Conductivity Models Considering the Interfacial Layer:
More recent models incorporate thermal interfacial resistance between filler and matrix
[14, 21, 28]. One example [14] is the Renovated Maxwell-Garnett (R-MG) model by Yu
and Choi:
3
2 1 2 1 2
1 3
2 1 2 1 2
2 2( )(1 )
2 ( )(1 )
effective
k k k k V
k k
k k k k V


    
      
(18)
Where the symbol, , is the ratio of layer thickness to particle radius;
p
h
r
  (19):
h is the interfacial layer thickness and rp is the particle radius. This model inserts the
term 3
(1 ) into the numerator and denominator of the M-G model which accounts for
the interfacial layer thickness and particle size. While the particle size is taken into
consideration, a caveat with this model is that the interfacial layer is given the same
thermal conductivity as the particles making this model a combination of macroscopic
and microscopic theories [21, 35]. Varying the interfacial thickness, h, in the model is the
only way to alter the effect of the interfacial region [21].

40
In a recent article, Murshed et al. [14], used the renovated M-G model and factored in
Jeffrey's particle average dipole strength to obtain a model for the thermal conductivity of
a two-phase "static" composite:
     
   
3 3 3 3
2 2 1 2 3 2 1 2 2 3
1 3 3 3
2 2 1 2 1 2 2 3
2 3 2
32 6 2
2 1
3
( ) 2 1 2 1 1
2 2 1
23 9 3
3 ... , where
4 16 2 3 4
effective
f
f
V k k k k V
k k
k k k k V
k k
V k
k k
       
    

           
        
     
           
(20)
 is an empirical parameter of interfacial layer orderliness that must be determined
experimentally, but is always greater than unity. There are three thermal conductivity
values in this equation, 1 2 3k k k  , where 3k is the complex particle, or a particle with an
interfacial layer, 3k must be determined by considering Brownian motion and other
kinetic model theory that does not apply to static models.  is a TC ratio parameter
derived from TC ratios of 1 2 3, , andk k k and the differences between 2 3andk k as well as
differences between 1k in a solid state and 1k in a liquid state of dynamic motion. The
determination of the value for  is made by experimentally determining the empirical
value of 3k and 1k from kinetic fluid-flow models which cannot be used or applied to static
materials. Hence, the Murshed et al., "static" model for TC, is not a static model and
cannot be used in our study.
Percolation Models
Other models based on macroscopic and microscopic theory, which were developed to
determine the thermal conductivity of two-phase composites, incorporate fractal

41
morphology of nanoparticle cluster formations and particle chains that span the entire
composite. One such model combining clusters, chains and randomly placed independent
particles is the Three-Level Homogenization model by Evans et al [25]. In the Evans
model, the first layer is concerned with individual, non-clustered particles. The second
level models "dead end" particle groups or clusters. Level-Three models the long chains
that connect throughout the composite. This type of model and any others that consider
interconnecting contact networks are usually based on the Bruggeman Mixing Rule and
are best suited for predicting thermal conductivity in composites with a high particle Vf
(greater than 10% for spherical particles) [25, 36]:
  1 2
2 2
1 1
1 0
2 2
Effective Effective
Effective Effective
k k k k
V V
k k k k
 
  
 
. (21)
Interconnectivity between randomly situated filler particles in a composite is known to
cause percolation. Interconnectivity between filler particles in a composite does not
necessarily require physical contact as predicted by microscopic theories [14, 28, 33-35,
36]. To "percolate" means to flow through; essential to percolation is the process of
flowing of some physical property such as heat, electricity, ferromagnetism, permittivity
and many other phenomena [36]. Generally, the lowest volume or weight fraction of filler
that achieves a peak value of a particular bulk property is called the percolation threshold
(PT) [36].
At a particular mass or volume fraction, the percolation network reaches a point, after
which, more added particles have little or no effect on the composite's desired bulk
property. It should be mentioned that the bulk property sought here, maximum TC, could,

42
very likely have a different PT than another bulk property such as fracture toughness,
permittivity, etc.
For other particle shapes, such as high aspect ratio whiskers or nanotubes, a contact
network can form and cause percolation to occur at lower volume fractions than spheres
and therefore the Bruggeman model and all three levels of Evan's model can be applied to
volume fractions below 10% [25].
Effect of Interfacial Resistance on Percolation
Interfacial resistance poses a barrier to heat flow that can greatly inhibit the effects of a
high TC filler material below the PT of a composite. In our case of low volume fractions
with well dispersed nanoparticles, and considering particles as spherical, all effective
medium theories including M-G and Bruggeman models predict [25, 36]:
1
2 2
1 1
/ 1 / 1
1 3 3
/ 2 / 2
effective p k p
p k p
k r A r G k
V V
k r A r G k
 
  
 
(22)
Where rp is the particle radius and Ak is the Kapitza radius which represents the radial
thickness of interphase layers over which the temperature drop, in a planar geometry, is
the same as at the interface. The Kapitza radius is related to interfacial resistance, G by
way of the matrix k1 value where Ak = k1/G as shown in Eqn. (22) [25, 36, 37]. From
Eqn. (22), it can be seen that when the particle radius becomes equal to the Kapitza
radius, there is no TC enhancement from the filler particles. For larger interfacial
resistance, the addition of particles actually decreases the effective TC of the composite
[25, 37]. This diffusion theory effect is paralleled (for certain particle types, sizes, and

43
temperatures) by microscopic theory in the form of the phonon MFP and the Debye
equation.
Equation (22) shows that interfacial resistance becomes larger as the particles decrease in
size since the ratio rp / Ak decreases with smaller particles.
The Kapitza radius for a low thermal conductivity matrix, such as PU, where k1~
0.2W/m-K, ranges from 20 nm to a fraction of a nanometer. Therefore, Eqn. 22 precludes
the possibility that the majority of nanoparticles added to our matrix will not significantly
increase the thermal conductivity [25].
The theory also predicts that for small diameter particles with large interfacial resistance,
G, where reportedly a high resistance, G ~ 10MW/m2
-K [25, 37], as with single wall
carbon nanotubes in a low TC matrix, keffective may be significantly reduced over the TC
of the matrix alone [25].
Figure 12: Comparison of the effective medium theory predictions for randomly
oriented long fibers (solid curves) and platelets (dashed curves) for k2/kl = 100 at
0.5% volume fraction showing the effects of increasing interfacial resistance [25].

44
Fig. 12, taken from the work of Evans et al, shows keffective/k1 for randomly oriented
fibers and platelets as a function of aspect ratio and also displays the effect of increased
interfacial thermal resistance for fibers and platelets. For a fixed aspect ratio and volume
fraction, as in Fig. 12, the TC enhancement decreases more with increased interfacial
resistance from platelets, than for long fibers [30]. For our work, the h-BN platelets and
spherical AlN fillers cannot be used at the high volume fractions that form contact
percolation networks because other crucial mechanical properties of the matrix will be
compromised [25, 29].
2.5 Dispersion and Percolation
Achieving uniform spatial distribution of individual nanoparticles in a composite has
been a significant challenge. And it has been a topic of interest in the scientific and
research community for many years [2, 3, 5, 6, 9, 13, 21, 22, 24, 27, 28, 31, 36, 38-40].
Arguably, a homogeneous non-contact dispersion of particles of a given size affects the
greatest volume of matrix material possible for the least amount of filler [8, 9, 29].
Arguments against this theory are reflected in models such as the Three-Level
Homogenization model proposed by Evans et al [25], where contact percolation is
required, and the non-contact particle clustering model of Keblinski et al [35].
Some thermal conductivity models, such as the Evans [25] and Keblinski [35] models,
draw distinctions between the types of dispersions found in a composite. These include:
1) Randomly dispersed individual particles with no aggregation or network system.

45
2) Dead-end structures having two or more particles that do not form a percolation
network are called aggregate structures.
3) Interconnecting percolation networks throughout the composite, called
agglomeration.
4) Clusters are defined by Keblinski et al [35] as grouped particles either in physical
contact or in close proximity wherein percolation occurs. Clusters may connect,
physically or not, and form a percolation network.
In spite of low weight fractions and deliberate procedures intended to form a
homogeneous dispersion, there are always some second-level or dead-end structures
consisting of two or more particles that cluster but do not form a percolation network. In
diffusion models, random non-contact dispersion with the least aggregated structures
achieves the greatest effect per volume fraction of filler; hence, to satisfy effective
medium theories, a good dispersion is necessary to achieve the maximum effect at lowest
possible filler fractions.
Models, such as H-C and M-G which describe random disorder of the guest phase can be
used to gain some insight into the parameters of percolation, most especially PT, using
the M-G Model and others through the Unified Mixing Rule [36]:
   
1 2 1
2
1 1 2 1 12 2
effective
effective effective effective
k k k k
V
k k k k k k k k 
 

     
(23)
The dimensionless analytical parameter  is used to convert the Unified Mixing Rule to
one of several mixing rule models for spherical inclusions: 0  applies to M-G and H-

46
C Models and 2  applies to the Bruggeman Model [36]. The Unified Mixing Rule, like
Eqn. 22, is derived from all effective medium theories and represents the macroscopic
view of diffuse thermal transport in an isotropic 2-phase composite.
Considering the above model in the search for a PT, one should let the TC ratio, 2 1/k k
become very large giving the relation:
 
1
2
1 12
effective
effective effective
k k
V
k k k k


  
(24)
However, the above relation is only effective if 2 1/k k is very small, where percolation
only begins to appear. Therefore, the effective conductivity in this region is obtained:
 
 
2
1
2
1 2
1 2
Effective
V
k k
V


 

 
. (25)
Clearly, if the denominator is allowed to approach the value zero, Effectivek   where
2
1
1
CriticalV PT V

  

. (26)
M-G and H-C, with
1 1
1
1 1 0
CriticalV

  
 
, has the PT occurring at V2 = Vf =1 [36].
For Bruggeman's theory, where 2  , PT occurs at a volume fraction of 1/3, or 33.3%
[36].
Why the big difference between Bruggeman and M-G? It is because the M-G Model
considers the TC of a particular volume fraction of spherical inclusions as evenly
distributed within a homogeneous matrix, while Bruggeman considers the TC of the
inclusions and matrix in a symmetrical fashion. For Bruggeman's model, there is no
difference between phases; it balances both matrix and filler with respect to the unknown

47
effective medium (the composite), using the volume fraction of each component as a
weight. Also, the strength of Bruggeman's model is its better accuracy predicting
properties of higher volume fractions, which is conversely the area where M-G is
considered weakest [21, 29, 36].
0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 100
0
3.5
7
10.5
14
17.5
21
24.5
28
31.5
35
Volume Fraction, Percent
ThermalConductivity,k,W/m.K
BN x( )
x
Figure 13: PT and TC Predicted by the M-G Model for Spherical h-BN Filler [27]
The reader will observe that the M-G model of Fig. 13 is extreme when considering PT
behavior since it pushes PT to the limit of Vf = 1 but this, however, is not unknown
experimentally [36].
Experimental studies show that PT covers a broad range. And Landauer [36] holds that
the PT for a particular composite and particular properties can range anywhere between 0
< Vf < 1 depending on the property being studied [36].

48
It is often true that real-life PT values fall below a Vf = 0.5 and it is a weakness
mentioned by detractors of M-G. On the other hand, the M-G model accurately predicts
the Mossotti catastrophe and Fröhlich frequencies, real effects that are difficult to explain
using Bruggeman's model [36]. Also, more recent models, such as H-C, based on M-G,
but with improvements brought about by considering particle surface area and shape,
may better predict TC and PT.
Non-Contact Percolation
For this work, a non-contact, non-agglomerated percolation network is sought that will
form overlapping interfacial regions, regions that are presumed to be higher in thermal
conductivity than the matrix alone. As already mentioned, how this occurs is still debated
and there are at least a few theories that have already been discussed [2, 3, 13, 14, 21, 25,
33-35]. One such theory commonly applied to nanofluids, discussed in detail in Sect. 2.4
under the subheading Conduction across Interfacial Boundaries, asserts that the high
surface energy of inorganic fillers aligns the molecules of the organo-fluid matrix in the
vicinity of the particle, creating enhanced pathways for phonon travel, resulting in
increased conduction in the interfacial zone surrounding the particle [28, 35, 38]. Since
such theoretical phenomena could also occur in the curing organic PU, it should be
considered as a possible component of the interphase, validating use of Eqn. 18, Yu and
Choi's R-MG model [21], and microscopic theory by Keblinski et al [35], for our work.
A similar theory, as mentioned under the Boron Nitride subheading, attributes increased
TC of the interphase to the relatively soft h-BN spalling at the interface which mixes with

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