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Development of a Predictive Shielding Effectiveness Model for
Carbon Fiber/Nylon Based Composites
By
Nicholas B. Janda
Bachelor of Science, Case Western Reserve University, 2003
A Thesis
Submitted to the Graduate Faculty
of
Michigan Technological University
In partial fulfillment of the requirements
For the degree of
Master of Science
In
Chemical Engineering
Houghton, Michigan
August 2004
Nicholas B. Janda
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This thesis, Development of a Predictive Shielding Effectiveness Model for
Carbon Fiber/Nylon Based Composites, is hereby approved in partial fulfillment ofthe requirements for the degree of MASTER OF SCIENCE in the field of Chemical
Engineering.
DEPARTMENT Chemical Engineering
Signatures:
Thesis Advisor: ____________________________
Dr. Julia A. King
Thesis Co-advisor: ___________________________
Dr. Jason M. Keith
Department Chair: ______________________________
Dr. Michael Mullins
Date: ________________________________________
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i
Abstract
Development of a Predictive Shielding Effectiveness Model for Carbon
Fiber/Nylon Based Composites
The need for electromagnetic interference (EMI) shielding materials has increased recentlydue to the more prevalent use of personal communications devices (cell phones, pdas).
Metals have typically been used the material of choice for shielding applications. Designweight limitations for highly portable devices, however, has limited the applicability of metalsfor these applications. A need for light weight materials capable of providing EMI shielding
exists.
Through the addition of conductive fillers to normally electrically insulating polymer resins,electrically conductive composites can be used for shielding applications, providing lightweight shielding materials. Shielding theory for composite materials, however, is largely
undeveloped, unlike for metals. These models developed for metals cannot be used toaccurately predict the shielding effectiveness provided by a composite containing a wide range
of conductive fillers .
The shielding effectiveness (SE) of two different carbon fiber/nylon based composites was
studied over the radio frequency range (300 to 1000 MHz). The effects of incidentelectromagnetic wave (EM) frequency, filler volume percent, filler size (radius), and filler
orientation on the measured SE were examined.
The objective of this analysis is to characterize the factors involved in determining the SE of a
composite from first principles. From this analysis, a model predicting shielding effectivenessfor carbon fiber/nylon based composites is developed. The model is expected to perform well
at low filler loadings, but also can be used to accurately predict shielding effectiveness at fillerloadings above the percolation threshold, as seen from comparisons of the model to
experiments with ThermalGraph and Fortafil carbon fibers in nylon 6,6.
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Acknowledgements
I must first acknowledge the Michigan Technological University Graduate School for
providing me the opportunity to further pursue my degree. I have traveled a long road to get
to this point and feel fortunate to complete my degree at Michigan Tech.
I also need to thank Dr. Warren Perger for providing the guidance, knowledge and focus
needed to complete the project. Thank you for teaching me how to think/see like a wave
and providing such truisms as in the land of the blind the one eyed man is king.
I must thank my co-advisors, Dr. Julia King and Dr. Jason Keith. Your input and aid was
always valued. I truly appreciated your consistent enthusiasm and support when the project
left the realm of typical Chemical Engineering. Thank you for your patience in dealing with
an atypical situation, project and student.
I would be remiss if I did not acknowledge the MATLAB assistance of Troy Oxby and
Dr. Jason Keith. Thank you for helping me rediscover my programming skills and showing
me that the program has more to offer than just Simulink.
The generosity of the National Science Foundation must be acknowledged. The funding
provided through Award Number DMI-9973278 allowed for prior fabrication of the samples
investigated in this study.
I must thank Brian Ott and Chris Copeland for providing a nearly endless amount of
distractions. Carrie Majkrzak, you deserve a medal of honor for sharing an office with me for
the past year.
Finally, I need to thank my Mom and Dad. Thank you for putting up with my educational
pursuits and never losing faith when things did not go smoothly. Your help and support along
the way has never gone unappreciated. A special thank you to my Mom: thanks for never
letting your level of frustration reach a point to where you felt it was necessary to strangle
your son.
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Table of Contents
Abstract.......................................................................................................................... i
Acknowledgements ...................................................................................................... ii
Table of Contents........................................................................................................ iiiList of Figures................................................................................................................v
List of Tables .............................................................................................................. vii
CHAPTER 1: Introduction..........................................................................................11.1 Electromagnetic Radiation and Interference...................................................1
1.2 Polymer Based Composite Materials..............................................................31.3 Predicting Shielding Effectiveness in Composite Materials...........................3
1.4 Project Outline ................................................................................................5
CHAPTER 2: Project Materials and Sample Formulation......................................6
2.1 Introduction.....................................................................................................62.2 Materials .........................................................................................................62.3 Sample Preparation .........................................................................................8
2.3.1 Extrusion.................................................................................................8
2.3.2 Injection Molding..................................................................................102.4 Formulations .................................................................................................12
CHAPTER 3: Experimental and Characterization Methods.................................13
3.1 Introduction...................................................................................................133.2 Electrical Resistivity.....................................................................................13
3.2.1 Transverse Electrical Resistivity Test Method .....................................13
3.2.2 Longitudinal Electrical Resistivity Test Method ..................................133.3 Shielding Effectiveness.................................................................................15
3.4 Balance of Power Analysis ...........................................................................18
3.5 Fiber Volume Fraction, Fiber Length and Aspect Ratio...............................193.6 Orientation ....................................................................................................20
3.6.1 Fiber Orientation...................................................................................20
3.6.2 Transmission Orientation Dependence .................................................20
CHAPTER 4: Experimental Results.........................................................................224.1 Introduction...................................................................................................22
4.2 Shielding Effectiveness Results....................................................................22
4.2.1 Pure Nylon 6,6 ......................................................................................224.2.2 ThermalGraph DKD X......................................................................23
4.2.3 Fortafil 243............................................................................................254.3 Balance of Power Results .............................................................................26
4.4 Orientation Results........................................................................................29
CHAPTER 5: Electromagnetic Theory ....................................................................325.1 Introduction...................................................................................................32
5.2 Shielding Theory...........................................................................................32
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5.3 Scattered Field Theory..................................................................................33
5.4 Scattered Field Equation Derivation.............................................................35
5.4.1 Maxwells Equations ............................................................................355.4.2 Permittivity - Absorption Loss..............................................................36
5.4.3 Phasor Notation.....................................................................................38
5.4.4 Wave Equation Solution Incident Field.............................................395.4.5 Wave Equation Solution Scattered Field ...........................................42
5.5 Scattering Width ...........................................................................................49
CHAPTER 6: Shielding Effectiveness Model Design..............................................526.1 Introduction...................................................................................................52
6.2 Review of Problem Description and Focus ..................................................52
6.3 Analysis of Scattering Equations ..................................................................536.3.1 Dependence on Frequency, Optical Radius and Distance From Scatterer
to Observer............................................................................................................53
6.3.2 Deterministic Nature of Scattering Equations ......................................55
6.4 Accounting for Collision Probability............................................................576.5 Scaling Factor Analysis ................................................................................59
6.6 Shielding Effectiveness Model Results ........................................................626.7 Scaling Factor - Linear Fit ............................................................................65
6.8 White Model Comparison.............................................................................68
CHAPTER 7: Conclusions and Future Work..........................................................717.1 Thesis Goal ...................................................................................................71
7.1.1 Conclusions from Electrical Resistivity/Conductivity Experiments ....71
7.1.2 Conclusions from Shielding Effectiveness Experiments......................717.1.3 Conclusions from Power Balance Analysis..........................................72
7.1.4 Conclusions from Fiber Orientation Studies.........................................72
7.1.5 Conclusions from Model Development and Analysis ..........................727.2 Future Work..................................................................................................74
CHAPTER 8: References ...........................................................................................76
Appendix A: Formulation Summary .......................................................................78
Appendix B: Shielding Effectiveness Experiment Results.....................................81
Appendix C: Balance of Power Results (mW) ........................................................88
Appendix D: Reflected, Absorbed and Transmitted Signal Results in dB...........95
Appendix E: Scaling Factor Analysis.....................................................................102
Appendix F: Shielding Effectiveness Model Results ............................................109
Appendix G: Shielding Effectiveness Model Results............................................116
Appendix H White Model Derivation ..................................................................123H.1 Introduction ....................................................................................................123H.2 Absorption Term Derivation ..........................................................................124
H.2 Reflection Loss Term Derivation ...................................................................125
Appendix I: Proposed Model Comparison to White Model ................................128
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List of Figures
Figure 1.1-1: IEEE Standard for Safety Limits on Human Exposure to RF Fields (3) .2Figure 2.3-1: Leistritz Extruder Used for Compounding of Composites ......................8
Figure 2.3-2: Extruder Screw Design, Note Flow is From Right to Left ......................9
Figure 2.3-3: Niigata Injection Molder........................................................................10Figure 2.3-4: Four Cavity Mold...................................................................................11
Figure 2.3-5: Shielding Effectiveness Disk .................................................................11
Figure 3.2-1: Bar From Which Longitudinal Electrical Resistivity Samples Were Cut...............................................................................................................14
Figure 3.2-2: (A) Experimental Set-up for Four Probe Test Method, .........................15
Figure 3.3-1: Shielding Test Fixtures With Support................................................16Figure 3.3-2: Transmission Holder Without Sample...................................................16
Figure 3.3-3: Cross Sectional View of Transmission Holder (24) ..............................17
Figure 3.3-4: Reference and Load Shielding Effectiveness Disks (24).......................17
Figure 3.3-5: Reference Disk Alignment on Trasmission Fixture (25).......................18
Figure 3.4-1: Shielding Test Apparatus Schematic (25)..............................................19Figure 3.6-1: Dipole Antenna and Sample Holder ......................................................21
Figure 4.2-1: Shielding Effectiveness for Pure Nylon 6,6...........................................23Figure 4.2-2: Shielding Effectiveness As a Function of Filler Volume Percent At
Select Frequencies ................................................................................24
Figure 4.2-3: Shielding Effectiveness Results for ThermalGraph DKD X .............25Figure 4.2-4: Shielding Effectiveness Results for Fortafil 243 ...................................26
Figure 4.3-1: Balance of Power Results (mW) for NCN20 (ThermalGraph)..........28
Figure 4.3-2: Balance of Power Results (mW) for NDN20 (Fortafil 243)..................28
Figure 4.4-1: NDN40 Fiber to Incident Wave Orientation Dependence forTransmitted Signal Strength ...............................................................29
Figure 4.4-2: Depictions of Perpendicular and Parallel Fiber to Wave Orientations ..30Figure 4.4-3: Carbon Fiber/Epoxy Sheet Fiber to Incident Wave Orientation
Dependence for Transmitted Signal Strength ......................................31
Figure 5.2-1: Representation of Shielding Phenomena for Plane Waves Passing
Through a Homogeneous Barrier (10)..................................................33Figure 5.3-1: A cylinder Impinged by a Uniform Plane Wave....................................34
Figure 5.4-1: Electromagnetic Frequency Spectrum (25)............................................37
Figure 5.4-2: Cross Sectional View of Transmission Holder (24) ..............................39
Figure 5.4-3: Cylindrical Coordinate System ..............................................................41Figure 5.4-4: Block diagram Depicting the Two Step Process for Solving for the
Radiated Fields Given a Current and Charge Source (32)..............43
Figure 5.4-5: Diagram of the Position Vectors. The vector potential A at is
obtained by integrating the current Jat '. (3) .................................46
Figure 5.4-6: Uniform Plane Wave of TMz
Orientation Impinging a Single
Cylindrical Scatterer With Radius a (32)............................................48Figure 5.5-1: Cross Sectional View of Transmission Holder (24) ..............................50
Figure 6.3-1: Near Zone ( = 1.0 x 10-4
m) Scattering Width for Both Fibers............54
Figure 6.3-2: Far Zone ( = 50 m) Scattering Width for Both Fibers .........................55
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vi
Figure 6.3-3: Theoretical Shielding Effectiveness of a Single Carbon Fiber Scattering
an Incident Wave ..................................................................................56
Figure 6.4-1: Sample Wavelength Sized Window For Shielding Disk .......................58Figure 6.5-1: Scaling Factor Analysis for NCN05 ......................................................59
Figure 6.5-2: Scaling Factor Analysis for NDN05 ......................................................60
Figure 6.6-1: Model Predicted and Experimentally Determined ShieldingEffectiveness for NCN05......................................................................63
Figure 6.6-2: Model Predicted and Experimentally Determined Shielding
Effectiveness for NDN05......................................................................63Figure 6.6-3: Model Fit Quality Analysis for ThermalGraph Based Composites...64
Figure 6.6-4: Model Fit Quality Analysis for Fortafil Based Composites ..................65
Figure 6.7-1: Linear Fit Applied to ThermalGraph Scaling Factor Data.................66
Figure 6.7-2: Linear Fit Applied to Fortafil Scaling Factor Data................................66Figure 6.7-3: Model Predicted and Experimentally Determined Shielding
Effectiveness for ...................................................................................67
Figure 6.7-4: Model Predicted and Experimentally Determined Shielding
Effectiveness for ...................................................................................68Figure 6.8-1: White Model and Proposed Model Comparison and Experimentally
Determined Shielding Effectiveness for NCN10..................................69Figure 6.8-2: White Model and Proposed Model Comparison for NDN40 ................70
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CHAPTER 1:Introduction
1.1 Electromagnetic Radiation and Interference
In todays electronic age, electromagnetic (EM) fields are radiated from numerous
sources. EM waves with frequencies in the range of approximately 0.3 to 1000 MHz (Radio
Frequency - RF range) are used for communications signals (radio, television, cellular
telephones). The emitted fields from these communications devices can interfere with the
operation of other nearby electronic equipment. This situation is known as electromagnetic
interference (EMI). Some adverse effects of EMI are connectivity problems in cellular
phones, interrupted television signals and even data corruption on computer hard drives.
Along with interfering with the operation of electronic devices, EMI in the RF band may have
harmful biological effects. Some studies have found a correlation between length of exposure
time to the EM fields emitted from power lines and leukemia occurrences (1). There is also
increasing concern that EMI might adversely affect the operation of biological devices such as
pacemakers (2). IEEE currently provides a standard for safety limits on exposure to RF
electromagnetic waves, shown in Figure 1-1.
As the number of communications devices in use has drastically increased over the recent
past decades, stringent regulations controlling the field strength emitted by electronic devices
have been instigated by the Federal Communications Commission, producing a need for EMI
controlling materials (4). Along with external interference concerns, the trend of personal
electronics miniaturization has resulted in devices containing densely packed electronic
components. Due to the close proximity, the EM fields generated by the internal components
may interfere with each other, resulting in electromagnetic incompatibility problems. A
material capable of controlling the amount of EMI radiated between the components is
essential.
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Figure 1-1: IEEE Standard for Safety Limits on Human Exposure to RF Fields (3)
EMI radiation control is known as shielding. Materials with known shielding ability are
used to encase an electronic product to prevent it from emitting or receiving unwanted
electromagnetic energy. The ability of a material to resist the passage of an EM signal is
quantified as shielding effectiveness (SE). The SE of a material is ratio of the power received
with and without a material present for the same incident signal power. It is expressed in units
of decibels (dB), as shown in Equation 1.1-1
2
1
10 P
P
logSE 10dB = [1.1-1]
Where:
1P = received power with the material present (watts)
2P = received power without the material present (watts)
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1.2 Polymer Based Composite Materials
Until recently, electrically conductive metals were most commonly used to provide EM
shielding. For example, plastic computer cases are usually lined with a thin metal shroud to
control EMI emissions. Weight considerations decrease the viability of metal shields in
portable electronics. For example it is disadvantageous to use internal metal shrouds lap-top
computer cases. The demand for low cost, low weight shielding materials has shifted the
focus to plastics. Most polymer resins are electrically insulating, and therefore, typically
incapable of providing EM shielding. Through the addition of conductive fillers, such as
conductive metal fibers or carbon fibers, the electrical conductivity of these resins is increased
and acceptable shielding ability is obtained (4-7). An electrically conductive composite can
be used for computer cases and cell phone housings without the need for an extra metallic
shield. These devices retain the light weight desired by consumers and meet the FCC
guidelines.
1.3 Predicting Shielding Effectiveness in Composite Materials
The utility of different types of fillers for shielding applications has been thoroughly
researched (2,4-7). Bigg has experimentally studied composites based on: carbon black,
carbon fibers, metal fibers, metal flakes and metal-coated glass fibers (4-7). The long standing
reliance on shielding metals has produced a void in shielding theory for composite materials.
In contrast, the shielding behavior for metals is well understood (8). The work of White is
typically referenced in the EMI composite shielding literature (2,4-7,9) as a viable model for
shielding effectiveness in composites. The White model, however, is predicated on
assumptions that decrease its applicability and validity for composite materials. The model
proposed by White was derived for a homogeneous planar metallic barrier (10). Applying it
to a composite assumes that the filler particles behave similar to a uniform pure metal. Unlike
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a pure metal, the composite are not homogeneous and the material will not present uniform
resistance to the EMI signal.
Theoretical work focusing on shielding effectiveness of composite materials is currently
limited to the plane wave shielding characteristics of periodic, anisotropic laminated
composites (11-14). Both Chen and Krohn developed theory to model the shielding behavior
of laminated composites made of several plies of fiber-reinforced panels with various fiber
orientation patterns. The panels were assumed to be composed of regularly spaced,
unidirectional collimated fibers imbedded in a polymer resin. The resin was modeled as a
dielectric material, translucent to electromagnetic waves. Chen and Krohn both predicted a
direct relationship between impinging wave frequency and shielding effectiveness (11-14).
Also noticed was a preferred fiber orientation for the strength of signal reflected from the
composite. Fibers oriented parallel with the electric field were predicted to reflect 20 dB more
than fibers oriented perpendicular (13).
The complexity of non-periodic, non-laminar composites has typically discouraged
researchers from focusing on composites formed via injection molding. The works of Chen
and Krohn provide some insight into the shielding behavior of composite materials but the
usefulness of their proposed models for non-laminar composites is quite limited. The
materials analyzed by Krohn consisted of only 5 large filaments, spaced widely apart
(Filament radius: 317 m, Spacing: 5.69 cm) (14). These conditions are unrealistic for
injected molded parts typical used in personal communications devices. Injected molded parts
commonly utilize densely packed conductive fillers with radii several orders of magnitude
smaller.
Because of the simplistic nature of the analyses conducted by Chen and Krohn, the
probability of the incident electric field colliding with a fiber was not investigated. Since the
unwanted signal will only be impeded when it encounters a conductive filler within the
composite, determining the probability of a signal/filler interaction is key. Both researchers
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CHAPTER 2:Project Materials and Sample Formulation
2.1 Introduction
This chapter discusses the methods used in the fabrication of the polymer composite
samples. These samples were produced by Quinton Krueger (25) and Jessica Heiser (20).
The properties of both the matrix and filler materials are also given.
2.2 Materials
The thermoplastic matrix used was DuPont Zytel 101 NC010, an unmodified semi-
crystalline nylon 6,6 polymer of medium viscosity. The properties are listed in Table 2.2-1
below.
Table 2.2-1: Properties of DuPont Zytel 101 NC010 (15)
Melting Point 262C
Tg (Glass Transition Temp, DAM) 60C-70C (approx.)
50% Relative Humidity 23C (approx.)
Melt Flow Rate 12.35 g/10 min
Shear Viscosity at 1000 sec1 shear rate and 280C 137 Pa-sec
Tensile Strength at 23C (DAM) 82.7 MPa
Flexural Modulus at 23C (DAM) 2,827.0 MPa
Tensile Elongation at Break at 23C (DAM) 60%
Notched Izod Impact, 23C 53.0 J/m
Density at 23C 1.14 g/cm3
Electrical Conductivity at 23C 10-15 S/cm
Electrical Resistivity at 23 oC 1015 ohm-cm
Thermal Conductivity at 23C 0.25 W/mK
DAM = Dry As Molded
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Two different carbon fiber fillers were employed in this project: BP/Amocos
ThermalgraphTM
DKD X and Akzo Nobels Fortafil 243 PAN (polyacrylonitrile) based fiber.
ThermalGraph DKD X is a milled, 200 m long, petroleum pitch-based carbon fiber that is
both highly anisotropic and graphitized. This particular fiber was used due to its ability to
improve thermal and electrical conductivity of the conductive resin. Table 2.2-2 lists the
properties below. Akzo Nobels Fortafil 243 PAN based 3.2 mm chopped, surface treated and
pelletized carbon fiber was also used to improve the electrical and thermal conductivity of the
resin. A proprietary polymer was used as a binder for the pellets that also promoted adhesion
with nylon. Table 2.2-3 lists the properties for this fiber.
Table 2.2-2: Properties of BP/Amoco ThermalGraph DKD X (16)
Tensile Strength >1.39 GPa
Tensile Modulus 687-927 GPa
Electrical Resistivity 2.2 ohm-mThermal Conductivity 400-700 W/m K
Fiber Density 2.15 to 2.25 g/cm3
Bulk Density 0.25 to 0.55 g/cm3
Fiber Diameter 10 microns
Filament Shape Round
Average Filament Length 200 microns
Filament Length Distribution
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2.3 Sample Preparation
For this project, the fillers were used as received. The Zytel 101 NC010 was dried in an
indirect heated dehumidifying drying oven (dewpoint of the recirculating air = -40oC). After
drying, the polymer was stored in moisture barrier bags.
2.3.1 Extrusion
An American Leistritz Extruder Corporation Model ZSE 27 was used for all polymer
extrusion throughout the course of the project. The extruder, shown in Figure 2-1, has a 27
mm co-rotating intermeshing twin screw with 10 zones and a length/diameter ratio of 40. The
screw design used produced minimal filler degradation while still providing adequate dispersal
of the filler within the polymer. This screw design is shown in Figure 2-2
Figure 2-1: Leistritz Extruder Used for Compounding of Composites
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The polymer pellets (Zytel) were introduced in Zone 1. The second side stuffer, located
at Zone 7, was used to introduce the carbon fibers into the polymer melt. Two Schenck
AccuRate gravimetric feeders were used to accurately control the amount of each material
added to the extruder. A complete list of all formulations extruded is provided in Appendix A
and the extrusion conditions for each are discussed in detail by Weber (18), Clingerman (19)
and Heiser (20). Typical extrusion conditions are listed in Table 2.3-1.
AtmosphericVent
AtmosphericBack VentSide Stuffer Side Stuffer Main Feed
40D 36D 32D 28D 24D 20D 16D 12D 8D 4D
GFA
2-30-30
GFA
2-30-90
GFA
2-40-90
GFA
2-30-60
GFA
2-40-90
KB5-2-30-60
KB5-2-30-30
KB5-2-30-90
KB5-2-30-60
KB5-2-30-30
KS1-2-10E
GFA
2-30-60
GFA
2-40-90
GFA
2-40-90
KB5-2-30-90
KB5-2-30-60
KB5-2-30-30
KB5-2-30-60
GFA
2-20-30
GFA
2-30-90
GFA
2-40-90
KS1-2-10A
0D
For Screw Type Elements
GFA-d-ee-ff
G = co-rotating
F = conveying
A = Free-Meshing
d = number of threads
ee = pitch (length in millimeters for one
complete rotation)
ff = length of screw elements in millimeters
Kneading disks
KBj-d-kk-llKB = kneading block
J = number of kneading segments
d = number of threads
k = length of kneading block in millimeters
l = twisting angle of the individual kneading
segments
Kneading disks
KS1-d-hh-i
KS1 = Kneading disc
d = number of threads
h = length of kneading disc in millimeters
i = A for initial disc and E for end disc
Zones
0D to 4D is Zone 1 (water cooled, not
heated)
4D to 8D is Zone 2/Heating Zone 1
8D to 12D is Zone 3/Heating Zone 2
12D to 16D is Zone 4/Heating Zone 3
16D to 20D is Zone 5/Heating Zone 4
20D to 24D is Zone 6/Heating Zone 5
24D to 28D is Zone 7/Heating Zone 6
28D to 32D is Zone 8/Heating Zone 7
32D to 36D is Zone 9/Heating Zone 8
36D to 40D is Zone 10/Heating Zone 9
Nozzle is Heating Zone 10
Figure 2-2: Extruder Screw Design, Note Flow is From Right to Left
Table 2.3-1: Extrusion Conditions for Nylon 6,6 (19)
Zone 1 Temperature (by feed hopper) 210oC
Zone 2 Temperature 250oC
Zone 3 to Zone 5 Temperature 270oC
Zone 6 to Zone 7 Temperature 275oC
Zone 8 to Zone 10 Temperature 280oC
Total Throughput 19.0 kg/hr
Screw rpm 300 rpm
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2.3.2 Injection Molding
The test specimens were molded using a Niigata injection molding machine, model
NE85UA4, (Figure 2-3). Implementing a 40 mm diameter single screw with a length/diameter
ratio of 18, the lengths of the feed, compression and metering sections of the single screw
were 396 mm, 180 mm and 144 mm, respectively. Two different molds were used for this
project. The four cavity mold shown in Figure 2-4 was used to produce 3.2 mm thick ASTM
Type I tensile bars (end gated) and 6.4 cm diameter disks of 3.2 mm thickness. The tensile
bars were used for longitudinal electrical conductivity measurements while the disks were
used for transverse electrical conductivity tests. Figure 2-5 shows the mold from which the
shielding disks of 130 mm diameter and 3.2 mm thickness were created. The molding
conditions for each formulation using the four-cavity mold are discussed in detail in
Clingerman, Weber and Heiser (18-20). The typical operating conditions for the injection
molding machine can be found in Table 2.3-2.
Figure 2-3: Niigata Injection Molder
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Table 2.3-2: Injection Molding Conditions for Conductive Nylon (20)
Zone 1 Temperature (by feed hopper) 285oC
Zone 2 Temperature 290oC
Zone 3 Temperature 299o C
Zone 4 Temperature (die nozzle heater) 310 oC
Mold Temperature 88oC
Screw rpm 54 rpm
Injection Pressure 154 MPa
Hold Pressure 109 MPa
Back Pressure 3 MPa
Injection Time 15 seconds
Cooling Time 15 seconds
Interval Time 2 seconds
Figure 2-4: Four Cavity Mold Figure 2-5: Shielding Effectiveness Disk
Mold
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2.4 Formulations
Test specimens were labeled according to the material, weight percent filler, and the
order that the specimen came out of the injection molder using the following nomenclature:
N W X Y - ##
N = National Science Foundation Project
W = Filler used
X = Polymer
Y = Weight percent of conductive fiber
## = Sample Number, indicating the order that the sample came out of the injection molder
All formulations were designated with anNas the first letter to denote that they were
from a previous NSF project (Award Number DMI-9973278). Following was a multi-letter
combination to denote the filler (W). C denoted the ThermalGraph carbon fiber, while D
referred to Fortafil 243. Xwas used to designate the polymer matrix used, with N referring
to nylon 6,6. The Yin the above formula was the weight percent of the conductive filler.
Following the above naming convention, a sample labeled NCN15-3, refers to the third
composite sample from the mold containing 15 wt% ThermalGraph DKD X carbon fiber in a
nylon 6,6 matrix.
Table 2.4-1 shows the concentrations of the resins produced for use in this project.
Table 2.4-1: Loading Levels for Composite Samples Studied
Fiber Loading Levels, wt%
ThermalGraph DKD X 5.0, 10.0, 15.0, 20.0, 30.0, 40.0
Fortafil 243 5.0, 7.0, 10.0, 15.0, 20.0, 30.0, 40.0
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CHAPTER 3:Experimental and Characterization Methods
3.1 Introduction
In this section, the techniques used to determine the properties of the composite samples,
which were used to test the shielding effectiveness model developed in this thesis, are
discussed. These properties include: transverse (through-plane) and longitudinal (in-plane)
electrical resistivity (inverse of the electrical conductivity), shielding effectiveness, filler
volume fraction, filler orientation, filler length and aspect ratio.
3.2 Electrical Resistivity
3.2.1 Transverse Electrical Resistivity Test Method
For samples with an electrical resistivity greater than 104 ohm-cm, a through-plane (also
called transverse), volumetric electrical conductivity test was conducted on the as molded test
specimen. In this method, a constant voltage (typically 10 V or 100 V) was applied to the test
specimen and the resistivity was measured according to ASTM D257 using a Keithley 6517A
Electrometer/High Resistance Meter and an 8009 Resistivity Test Fixture (21). The Keithley
6524 High Resistance Measurement Software was used to automate the conductivity
measurement. For each formulation, a minimum of six specimens were tested. Each test
specimen was an injection molded disk that was 6.4 cm in diameter and 3.2 mm thick. Since
the presence of water can affect a samples conductivity, all samples were tested dry as
molded (DAM).
3.2.2 Longitudinal Electrical Resistivity Test Method
The volumetric longitudinal electrical resistivity (in-plane) was measured on all samples
with an electrical resistivity less than 104
ohm-cm. Test specimens cut from the center gauge
portion of a tensile bar, Figure 3-1, were surface ground on all sides and cut into sticks 2 mm
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2mm
2mm
25mm
V from
center 6mm
Constant current
in through sample
Constant
current out of
sample
Sample
Volt
Meter
Current
Source
(A) (B)
2mm
2mm
25mm
V from
center 6mm
Constant current
in through sample
Constant
current out of
sample
Sample
Volt
Meter
Current
Source
2mm
2mm
25mm
V from
center 6mm
Constant current
in through sample
Constant
current out of
sample
2mm
2mm
25mm
V from
center 6mm
Constant current
in through sample
Constant
current out of
sample
Sample
Volt
Meter
Current
Source
Sample
Volt
Meter
Current
Source
(A) (B)
Figure 3-2: (A) Experimental Set-up for Four Probe Test Method,
(B) Sample Dimensions and Longitudinal Current Flow (19)
3.3 Shielding Effectiveness
The electromagnetic shielding effectiveness of each formulation was measured
according to ASTM D 4935-89 (Reapproved 1994), for planar materials using a plane-wave,
far-field EM wave. Although it provides a method of measuring far-field SE, the nature of
the shielding test apparatus used in this study allowed for measurement of near-field shielding
effectiveness values (23). To be able to measure near-field power values, one must be able to
fully characterize the impinging wave directly before it collides with the shielding media.
The method is valid over a frequency range of 30 MHz to 1.5 GHz.
An Electro-Metrics, Inc. shielding effectiveness test fixture (model EM-2107A) was
used to hold the sample with a HP 8752C network analyzer generating and receiving the EM
signals. Figure 3-3 and Figure 3-4 show the shielding test apparatus and sample holder.
Figure 3-5 shows a cross-sectional view of the test fixture.
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Figure 3-3: Shielding Test Fixtures With
Support
Figure 3-4: Transmission Holder
Without Sample
For each formulation, one reference sample and at least 5 load samples were tested over a
frequency range of 30 MHz to 1.0 GHz. A reference sample consists of a large ring and a smaller
inner disk as shown in Figure 3-6. The shielding effectiveness (SE) of a material is the ratio of
the power received with and without a material present for the same incident power. For these
experiments, therefore, it is the difference ratio of the load sample to the reference sample. It is
expressed in units of decibels (dB), as shown in Equation 3.3-1 (4).
2
110P
PlogSE 10dB = [3.3-1]
Where:
1P = received power with the material present (watts)
2P = received power without the material present (watts)
The input power used was 0 dBm, corresponding to 1 mW. The dynamic range (difference
between the maximum and minimum signals measurable by the system) of the system was 80 dB.
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Figure 3-5: Cross Sectional View of Transmission Holder (24)
Figure 3-6: Reference and Load Shielding Effectiveness Disks (24)
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Figure 3-7: Reference Disk Alignment on Trasmission Fixture (25)
Figure 3-7 shows the placement of the reference sample on the transmission fixture. The
small disk and larger outer ring must be precisely aligned on the fixture to obtain accurate
readings. The nylon 6,6-based samples were tested DAM. The results from the analysis are found
in Appendix B and discussed in Chapter 4.
3.4 Balance of Power Analysis
The shielding effect test apparatus was also used to determine the contribution of reflection
(scattered) and absorption to the overall shielding effectiveness of a sample. The HP 8752C
Network analyzer is capable of measuring the transmitted power from test fixture and reflected
power from the top of the sample holder. Accounting for cable loss for both the input and output
cables from the fixture, as seen in Figure 3-8, the amount of signal reflected and transmitted
through the sample can be directly measured. The absorbed signal power can then be calculated
using a conservation of power analysis:
Absorbed(W) = I ncident(W) Ref lected(W) Leakage(W) [3.4-1]
Transmission Fixture
Small Reference Disk
Large Reference Ring
Hollow Coupling Area
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Figure 3-8: Shielding Test Apparatus Schematic (25)
The transmitted and reflected power was measured for at least 6 load samples for each
formulation. Results from the balance of power analysis are discussed in Chapter 4 and listed in
Appendices C and D.
3.5 Fiber Volume Fraction, Fiber Length and Aspect Ratio
A solvent digestion method was used to determine the weight percent of the filler in the
composite sample. As described in ASTM Standard D5226, this method completely dissolves the
polymer, leaving only clean filler particles (26). A 0.2 g sample cut from the center of a
transverse ER disk was used. Formic acid was used to dissolve the nylon 6,6 based composites at
23 oC. The filler was separated from the solvent/polymer mixture through vacuum filtration. The
mass of the dried filler particles was then compared to the weight of the original mass of the
composite/filler sample to determine the weight percent of the filler within the sample. 2 to 4
samples were tested per formulation. These filler volume fraction results are shown in detail
elsewhere (20,25). In all cases, the actual filler content of each formulation matched the target
amount within acceptable tolerances.
HP Analyzer
Shielding Apparatus
Regulator
Input Cable
Output Cable
Air Cylinder
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After the fibers were extracted frm the nylon 6,6 matrix, they were dispersed onto a glass
slide and viewed using an Olympus SZH10 optical microscope with an Optronics Engineering
LX-750 video camera. The images (at 60x magnification) were collected using Scion Image
version 1.62 software and then processed using Adobe Photoshop 5.0 and the Image Processing
Tool Kit v. 3.0. The length of each fiber was measured and the aspect ratio (AR),Diameter
LengthAR = ,
was calculated. For each formulation, between 200 and 3000 individual fibers were measured
(18-19,27). These results are shown in Appendix A.
3.6 Orientation
3.6.1 Fiber Orientation
The orientation of the carbon fibers within the composite was determined by viewing a
polished sample with an optical microscope. For each formulation, a 12.7 mm x 12.7 mm section
was cut from a SE test disk. The sample was mounted in epoxy and positioned such that the
depth of the sample could be viewed (3.2 mm). The samples, in the epoxy plug, were polished
and then viewed via an Olympus BX60 reflected light microscope at a magnification of 200x.
Scion Image version 1.62 software was used to collect the images, which were later processed in
Adobe Photoshop 5.0 using Image Processing Kit v. 3.0. The average orientation of 1000 to 4000
fibers per formulation was determined (28). Appendix A shows the results of this analysis.
3.6.2 Transmission Orientation Dependence
The effect of fiber orientation on the transmitted signal strength was investigated using the
fixture shown below in Figure 3-9. A large circular metal plate was affixed to a sheet of
plexiglass that was held in place by slots cut into a PVC pipe. The plate reduced the possibility of
wave diffraction around the composite sample interfering with the measured transmitted signal
strength. The shielding disk samples were placed in the remaining slot on the sample holder, in
front of the antenna and large metal plate. A dipole antenna was positioned directly behind the
metal plate with a transmission cable connecting it to the HP 8752C network analyzer. An
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electric field of known orientation (parallel to the plane of the floor) and strength was sent
through the sample to be received by the antenna. The transmitted signal strength was measured
over a frequency range of 500 to 2000 MHz. The shielding disk sample was then rotated 90
degrees and the process repeated. All measurements were conducted in an anechoic chamber to
reduce the error inducing effects of outside interference and incident field reflection.
Figure 3-9: Dipole Antenna and Sample Holder
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CHAPTER 4:Experimental Results
4.1 Introduction
The results from the balance of power analysis and shielding effect experiments, described
in Chapter 3, are discussed in this chapter. Also presented are the results from the fiber
orientation studies.
4.2 Shielding Effectiveness Results
From the data measured using the techniques discussed in Chapter 3, the shielding
effectiveness for each individual sample for each formulation was calculated using Equation
4.2.1.
2
110P
PlogSE 10dB = [4.2-1]
Where:
1P = received power with the material present (watts)
2P = received power without the material present (watts)
The SE results compiled in this investigation compared favorably to the work of Krueger (25) and
Heiser (20).
4.2.1 Pure Nylon 6,6
As expected, the pure matrix of only nylon 6,6 showed essentially no ability to shield
electromagnetic fields due to its dielectric nature. Ideally, an impinging electromagnetic wave
should encounter no resistance when passing through a dielectric material. Assuming the
material exhibits no conversion of the incident energy into heat while the wave travels through
the dielectric (condition known as a non-lossy dielectric), the shielding effectiveness should be
zero. Figure 4.2-1 shows the pure nylon 6,6 matrix following this behavior. Little shielding
effectiveness was measured, approximately 0.1 dB at the higher frequencies. This corresponds to
shielding only 2% of the incident field strength.
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Figure 4-1: Shielding Effectiveness for Pure Nylon 6,6
The solid line in Figure 4.2-1 represents the mean shielding effectiveness for the
formulation. For each formulation, the mean was calculated from at least 4. Typically, 6 samples
were measured. The upper dashed line corresponds to the highest SE value recorded for any of
these samples. Similarly, the lower dashed line refers to the lowest SE value recorded. It is
possible for a single trial to produce a maximum at one frequency and a minimum at another.
This, however, was frequently not the case. A single specific specimen of a formulation typically
would produce SE values that were either high, average or low.
4.2.2 ThermalGraph DKD X
The introduction of ThermalGraph carbon fiber into the nylon 6,6 matrix resulted in
enhanced EM shielding characteristics. Increasing the amount of filler within the sample resulted
in decreased electrical resistivity (ER) and increased shielding effectiveness. Also observed was
the effect of increased frequency on the measured SE values. This trend is expected and has been
300 400 500 600 700 800 900 1000-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
Frequency (MHz)
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reported elsewhere (9-14). As frequency is increased, the wavelength of the EM wave decreases
and becomes for comparable to the size of the fiber. Thus, higher frequency waves are more
likely to encounter fiber embedded in the polymer matrix. Similarly, as the weight percent of
fiber is increased, there is an improved probability that the wave will collide with a fiber. The
fibers, as opposed to the polymer rich areas, are more likely to scatter or absorb the wave, as the
nylon is virtually invisible to the wave. Hence, SE increases as frequency increases.
For all formulations studied, listed in Table 2.4-1, SE increased at higher frequencies. The
SE results for the ThermalGraph DKD X composites at 300, 500 and 800 MHz are shown in
Figure 4-2. Figure 4-3 shows both how shielding effectiveness directly increased as a function of
filler weight percent and frequency.
Figure 4-2: Shielding Effectiveness As a Function of Filler Volume Percent At Select
Frequencies
0 5 10 15 20 25 300
2
4
6
8
10
12
14
Volume Percent Fiber (%)
300 MHz500 MHz800 MHz
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Figure 4-3: Shielding Effectiveness Results for ThermalGraph DKD X
4.2.3 Fortafil 243
The addition of Fortafil 243 fibers into the matrix produced similar SE trends. Like the
ThermalGraph DKD X, the SE for the Fortafil samples increased with both frequency and filler
weight percent. The Fortafil samples, however, showed markedly better shielding behavior. For
example, NCN40 was found to have the best shielding effect performance among the
ThermalGraph samples, approximately 14 dB at 1.0 GHz. In comparison, NDN40 was found
to have a maximum shielding effectiveness of 72 dB at 1.0 GHz. This disparity between the
behaviors of the two fillers tracked with the ER results (Appendix A). As shown in Tables 2.2-2
and 2.2-3, both the Fortafil and ThermalGraph fibers have similar electrical resistivities. When
both fibers, however, were introduced into the nylon 6,6 matrix in equal weight percents, the
Fortafil sample was found to be two orders of magnitude more conductive than the
ThermalGraph. Thus, improved shielding for the Fortafil samples was observed.
300 400 500 600 700 800 900 10000
2
4
6
8
10
12
14
Frequency (MHz)
5 wt%10 wt%15 wt%20 wt%30 wt%40 wt%
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A corresponding trend was noticed in the percolation thresholds for Fortafil and
ThermalGraph electrical resistivity. A prior investigation determined the thresholds to be 9.5
and 3.4 volume percent, respectively (19). Previous work has suggested that the increased
shielding effectiveness afforded by the Fortafil 243 filler may be due in part to the increased
heteroatoms present on the surface of the individual fibers. Fortafil 243 results in improved
adhesion with the nylon matrix material which might explain increased composite SE (20,28-29).
The Fortafil based formulations are listed in Table 2.4-1. The SE results for the Fortafil based
composites are shown in Figure 4-4. Again, as frequency and filler weight percent were
increased, shielding effectiveness increased.
300 400 500 600 700 800 900 10000
10
20
30
40
50
60
70
80
Frequency (MHz)
ShieldingE
ffectiveness
(dB)
5 wt%
7 wt%
10 wt%
15 wt%
20 wt%
30 wt%
40 wt%
Figure 4-4: Shielding Effectiveness Results for Fortafil 243
4.3 Balance of Power Results
From the frequency dependent transmitted and reflected power data accumulated from the
balance of power experiments, the relative effects of electric field reflection (scattering) and
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absorption on the SE performance of a the composite samples was determined. Although the
experimental apparatus does not allow for direct measurement of the absorption power loss, a
simple power balance accounting for all methods of signal degradation allows for indirect
calculation of the absorption term, Equation 4.3-1.
Absorbed(W)= Incident (W) Reflected(W) Leakage(W) [4.3-1]
No noticeable change in transmitted or reflected signal strength was noticed when the
flanges of the shielding test fixture were wrapped with an EM insulator (aluminum foil).
Therefore, the effect of leakage on the behavior of the system was assumed to be negligible. The
HP 8752C network analyzer did not provide consistent incident signal power over the frequency
range investigated. The source EM signal was found to decrease monotonically with increased
frequency (from 1.0 mW at 30 MHz to approximately 0.9 mW at 1.0 GHz). To normalize the
input power at 1.0 mW across the frequency range, the measured transmitted and reflected power
was scaled-up according to the discrepancy between the desired set value and actual applied
signal.
Figure 4-5 and Figure 4-6 show the results of the analysis for NCN20 and NDN20.
Graphs for the remaining formulations can be found in Appendix C. The dashed lines again
indicate the maximum and minimum measured value during the course of the experiment. The
graphs are also expressed in dBm in Appendix D using the following equation to convert:
( )PdBm 10log10P = [4.3-2]Where:
PdBm = Power (dBm)
P = Reflected or Absorbed Power (mW)
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Figure 4-5: Balance of Power Results (mW) for NCN20 (ThermalGraph)
Figure 4-6: Balance of Power Results (mW) for NDN20 (Fortafil 243)
0 100 200 300 400 500 600 700 800 900 10000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Frequency (MHz)
Absorbed
Reflected
Transmitted
0 100 200 300 400 500 600 700 800 900 10000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Frequency (MHz)
Absorbed
Reflected
Transmitted
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The reflected power was equal to or greater than the absorbed power for each formulation
(with the exception of the pure nylon 6,6 sample). As the weight percent of both fillers was
increased, the reflection term became more dominant, indicating that it is the prevailing form of
signal loss. The reflection term also showed significantly more frequency dependence. Over the
frequency range under investigation, the absorbed power was relatively constant while the
reflected power varied greatly.
4.4 Orientation Results
From the transmission orientation dependence analysis, it was determined that for the
injected molded shielding disks the transmitted signal strength from an incident plane wave is
independent of disk orientation. Figure 4.4-1 shows the result from the analysis of NDN40.
500 1000 1500 2000-95
-90
-85
-80
-75
-70
-65
-60
-55
Frequency (MHz)
Transmission(dB)
Figure 4-7: NDN40 Fiber to Incident Wave Orientation Dependence for Transmitted Signal
Strength
Perpendicular
Parallel
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This result, however, did not conclusively prove that transmitted signal strength was
independent of fiber orientation. To further investigate fiber orientation dependence, the
experiment was repeated replacing the injection molded samples with a thin sheet of
unidirectional carbon fiber/epoxy (Hexcel Carbon Fiber/Epoxy AS4/3501-5A - 35 wt% carbon
fiber). The carbon fiber/epoxy sheet was analyzed with the fibers oriented in the plane of the
electric field and again with the fibers oriented transversely to the field. Figure 4.4-2 shows these
orientations.
Figure 4-8: Depictions of Perpendicular and Parallel Fiber to Wave Orientations
With this material, definite fiber orientation dependence was observed as the transmitted
signal strength differed by an average of 10 dB. Figure 4.4-2 shows this directional dependence.
The results are in partial agreement with the work of Chen. Chen also noticed that the best
shielding occurred when fibers were aligned in the plane of the impinging field (11-13). Casey
has suggested that the tensor constitutive parameters of the system can be estimated to
mathematically model the system (30). Casey, however, analyzed time-domain behavior while
this project has focused on the frequency domain response of the polymer composites.
E E
Perpendicular
OrientationParallel
Orientation
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500 1000 1500 2000-80
-75
-70
-65
-60
-55
-50
-45
-40
Frequency (MHz)
Transmission(dB)
Figure 4-9: Carbon Fiber/Epoxy Sheet Fiber to Incident Wave Orientation Dependence for
Transmitted Signal Strength
Although the orientation image analysis, results in Appendix A, found a general orientation
to the carbon fibers within the disk, the fibers were not sufficiently oriented to show any
significant dependence. Therefore, sample orientation with the electric field is not a dominant
factor in determining the shielding effectiveness of the composite. When theoretically modeling
the system, a fiber/wave orientation, however, must be chosen. Because parallel alignment of the
electric field and fiber was found to produce the greatest shielding, this orientation will be
selected for use in later analyses.
Perpendicular
Parallel
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possible, especially in the case of non-homogeneous materials, but typically play only a minor
role in determining the extent of transmission power loss (10).
Figure 5-1: Representation of Shielding Phenomena for Plane Waves Passing Through a
Homogeneous Barrier (10)
5.3 Scattered Field Theory
Although the results from the electrical resistivity and shielding effectiveness experiments
show correlation, it has been argued that resistivity tests alone cannot provide enough information
to predict SE (9). The resistivity experiments do not take into account all of the filler present,
only the fibers aligned in a conductive network. The fibers not connected in the network,
however, still have the potential of scattering or absorbing electromagnetic fields.
From the balance of power analysis described in Chapter 3 and discussed in Chapter 4, it
was determined that absorption played only a minor role in determining the shielding
effectiveness of the composite disks, leaving field scattering (reflection) as the largest SE
contributor. Because the pure nylon samples showed an inability to shield, it is proposed that the
scattering behavior of the system can be attributed singularly to the presence of the fibers. Thus,
a single cylindrical fiber will serve as the focus for the derivation of the relevant equations.
A
B
Scattered
FieldsTransmitted
Fields
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5.4.2 Permittivity - Absorption Loss
Permittivity is a measure of how much a medium changes to absorb energy when subject
to an electric field. As seen in Equation 5.4-5, it is defined as the ratio
E
D. Complex permittivity
is further defined by:
j=
[5.4-7]
Where:
= permittivity (real part) (Farads/meter) = conductivity (Siemens/meter) = angular frequency (radians/second)
j = 1 (imaginary number)
The following equation relates angular frequency to frequency, f :
f 2= [5.4-8]
fc = [5.4-9]
Where:
c = speed of light (Faradays/meter)
= wavelength (meters)
The imaginary part of Equation 5.4-7 describes the absorption loss at a given frequency. The
relation
is termed the loss factor (31). The ratio of the loss factor to the real part of the
permittivity indicates whether or not a material will exhibit large absorption losses. If the ratio is
large for a given frequency, the material is regarded as a good conductor (31).
1 [5.4-10]
Although the real part of the permittivities for ThermalGraph and Fortafil are not known,
both have electrical conductivity values within two orders of magnitude of copper
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5.4.3 Phasor Notation
Substituting the constitutive relations, Equation 5.4-5 and 5.4-6, into the original Maxwell
expressions, Equations 5.4-1 - 4, yields the following modified Maxwells equations:
t
HE
= [5.4-11]
t
EJH
+=
[5.4-12]
0= H [5.4-13]
vE =
)( [5.4-14]
The time derivatives in Equations 5.4-11 and 5.4-12 can be placed into phasor notation by using
the rule of equivalence for time-harmonic quantities. A phasor is a complex quantity that
represents a time-harmonic physical quantity. Phasor notation is a more convenient method of
representing the equations associated with electromagnetics.
The following sinusoidal, time-harmonic real physical quantity, )(tV ,
)cos()( 0 += tVtV [5.4-15]
can be expressed as a complex quantity using Eulers Identity.
xjxejx sincos += [5.4-16]
In phasor notation )(tV can be written as:
)Re{)( 0tjj
eeVtV= [5.4-17]
Where:
Re{ } = denotes taking the real part
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For convenience, the Re{ } symbol and frequency-time dependence term,tje , are generally
omitted in the literature and will not be written for the phasors listed in the remainder of this
chapter.
jeVtV 0)( = [5.4-18]
Therefore, j can be used to replace a time derivative when representing a time-harmonic
function as a complex quantity.
)Re{)Re{)( 00tjjtjj eeVjeeV
ttV
t
=
=
[5.4-19]
jeVjtV
t
0)( =
[5.4-20]
Finally, Equations 5.4-11 and 5.4-12 can be expressed as:
HjE = [5.4-21]
EjJH
+= [5.4-22]
5.4.4 Wave Equation Solution Incident Field
Although the plane waves created by the HP 8752C network analyzer and transmitted
through the shielding apparatus do not propagate through free space, it was assumed the conical
wave guide of the test fixture (Figure 3-5) allowed the EM fields to behave as if they are in a
source free media. This assumption was made to simplify the mathematics required to
characterize the incident electrical field.
Figure 5-4: Cross Sectional View of Transmission Holder (24)
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0 = permeability of free space (7104 Henrys/meter)
0 = permittivity of free-space (0
2
1
c = 910
36
1
Farads/meter)
20 12 cc
= [5.4-30]
20 = [5.4-31]
Since the geometry of the system in question is cylindrical in nature (fiber shape) the electric
equation should be solved in cylindrical coordinates of the following form:
),,( zE [5.4-32]
Where , and zare cylindrical coordinates diagrammed in Figure 5-5.
Figure 5-5: Cylindrical Coordinate System
External to the fiber, the plane wave introduced in Section 5.3 and shown in Figure 5-2 travels in
thex-plane with an electric field oscillating in thez-plane. Therefore, only the partial derivatives
with respect to the x-direction for the z component of the field, given by equation 5.4-28, are of
concern.
0002
2
2
=+
z
z Ex
E [5.4-33]
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Finally, Equation 5.4-34, describing the electric field of the simple plane wave can be found as a
solution to the simplified wave equation, Equation 5.4-33.
cos
0000 jxj eEzeEzE
== [5.4-34]
Where:
z = unit vector pointing in the direction of increasing z
The incident field,i
E , can further be expressed as:
=
=0
00 )cos()()(n
nn
ni
nJjEzE [5.4-35]
Where:
n = { 0201
=n
n
The summation arises from the representation of the plane wave as an infinite sum of cylindrical
wave functions (32).
5.4.5 Wave Equation Solution Scattered Field
A solution for the incident electric field can be found with relative ease because of the no
source/ free space assumption made at the beginning of the derivation. For an electromagnetic
signal to propagate through or interact with an object, however, oscillating currents must exist
within the object. The creation of a scattered electric field requires the induction of a current
source on the scattering object. Thus, the Maxwells equations must include the current density
term,J.
HjE = [5.4-21]
EjJH
+= [5.4-22]
Solving the two differential equations is challenging when the value ofJis not known. Because
of the presence of the curl operator, the electric field wraps around the current source. Thus,
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the math required for solving Equations 5.4-21 and 5.4-22 can be quite challenging. Introduction
of two intermediate auxiliary functions A and allows for easier determination of a solution.
Figure 5-6 illustrates this concept. A is defined as having the same vector direction as
J(traveling in the same direction). The solution forA can then be used to determineE.
A and are defined by the following relationships (3):
AB = (definition ofA ) [5.4-36]
= AjE (definition of ) [5.4-37]
Figure 5-6: Block diagram Depicting the Two Step Process for Solving for the Radiated Fields
Given a Current and Charge Source (32)
Using the tensor identity described in 5.4-25 with Equation 5.4-36 gives a second order
differential equation forA in terms of magnetic flux. The scattered fields are produced by a
current source, J. Therefore, the equations must be solved for in terms ofJ.
AAAB 2)()( == [5.4-38]
The curl of the magnetic flux density, B , can be found using the constitutive relation given in
Equation 5.4-6.
HB = [5.4-6]
Sources
J, v
Vector Potentials
A ,
Radiated Fields
E, H
Integration Path 1
Integration Path 2 Differentiation Path 2
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)()( HHB == [5.4-39a]
Inserting this new expression, Equation 5.4-39a into Equation 5.4-38 gives:
AAH 2)()( = [5.4-39b]
The divergence ofA is defined by the Lorentz condition (3):
0=+
jA (Lorentz Condition) [5.4-40]
AjH 2)()( =
[5.4-39c]
The curl of the magnetic field, expressed in terms of a current source J and electric field E,
(Equation 5.4-22) produces the following equation:
AjEjJ 2)()( =+
[5.4-39d]
Rearranged, Equation 5.4-39d becomes:
)(2 +=+
jEjAJ [5.4-39e]
The definition of can then be used to express the electric field:
)()(2
+=+
jAjjAJ [5.4-39f]
A simple rearrangement of Equation 5.4-39f produces the second order differential equation for
the vector potential,A , in terms of a current source (J).
)()(22 +=+
jjAAJ [5.4-39g]
JAA =+
22[5.4-41]
Similarly, a second-order differential equation for can be found using the definition
of , Equation 5.4-14 and the Lorentz condition. The divergence of the definition of ,
Equation 5.4-37, can be used to represent the expression in known terms.
= AjE [5.4-38]
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Taking the divergence gives:
)( = AjE [5.4-42a]
)()( = AjE [5.4-42b]
The divergence of the electric field has previously been related in Equation 5.4-14.
)()( = Aj
v
[5.4-42c]
The Lorentz condition again defines the quantity A :
)()j(jv =
[5.4-42d]
Upon rearrangement, a second order differential equation for in terms of surface charge ( v )
is realized.
=+ v
22 [5.4-43]
An infinitesimal antenna is an extremely short and thin wire driven by a current source (3).
This theoretical antenna is a good approximation for the tiny antenna produced from the induced
oscillating charge on the surface of the fiber. Assuming that the antenna oscillates in the z-plane
over an infinitesimal length ( z ), a current density (J) multiplied by the cross-sectional area
( A ) equal to I with the origin is set at the center of the antenna ( ' = 0), the vector potential
generated by the antenna is given by (3):
4
jzeIzA
= [5.4-44]
One can clearly see the similarity of the above equation and the equation for a scalar potential of
a point charge, Equation 5.4-45.
Scalar Potential =04
q [5.4-45]
Where:
q = point electric charge (Coulombs)
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Using this definition for an infinitesimal antenna, the general solutions forA and can be
found and are listed in Equations 5.4-46 and 5.4-47.
=
v
jeJdVA
'
)'('
4)(
'
[5.4-46]
=
v
j
v edV'
)'('
4
1)(
'
[5.4-47]
Where:
= vector indication the position of the potentials
' = position vector of the sources
' = distance between observation point and '
Figure 5-7 shows the position vectors and ' . Equations 5.4-46 is integrated over all points
where the source, )'(J , is not zero (3).
Figure 5-7: Diagram of the Position Vectors. The vector potential A at is obtained by
integrating the current Jat '. (3)
'
'
)'(J
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Through use of the solutions forA and and their associated definitions, an expression
for the scattered field,i
E , can be developed. Since the scattered waves travel outward from the
cylindrical fibers, the E-field solution must be expressed by cylindrical wave functions (32).
=
=n
nn
s
HcEzE )( )2(0 [5.4-48]
Where:
nc = unknown amplitude coefficients
)2(
nH = Hankel Fuction of the second kind
given by:
nn
)(
n jYJH =2
[5.4-49]
Where:
nJ = Bessel Function of the first kind
nY = Bessel Function of the second kind
Equation 5.4-48 includes only thezcomponent of the scattered field and ignores the and
directions. The results of the orientation analysis, discussed in Chapter 4, made this
simplification possible. From the fiber orientation shielding preference study, it was found that
aligning a length of a carbon fiber parallel to the electric field produced the most shielding.
Therefore, to model the maximum amount of scattering, we can narrow our focus to this
orientation, known as TMz mode. Figure 5-8 shows a wave traveling in the x-plane with the
electric field (E) pointing in the z-plane and the magnetic field pointing in the y-plane. This
orientation allows for the electric field to oscillate charges over the greatest distance in the fiber,
(the length of the fiber) and thus produce the largest scattered field intensity.
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Where:
= radial distance from center of target to the observer (meters)s
E = scattered electric field strength (volts/meter)i
E = incident electric field strength (volts/meter)
Substitution of Equation 5.4-45 into Equation 5.5-1 yields the following expression for the
scattering width:
+
= =
2
0
)2(
)2(2)cos()(
)(
)()(2lim
nHaH
aJj
n
n
n
nn
n
D [5.5-2]
Application of the limit,
, produces the far-field scattering width. Because of the
design of the shielding test apparatus, near-field measurements were made, instead. Although it
appears that the source and receiver are 34.4 cm apart, as shown in Figure 5.4-2, they are actually
very close together. Since the test fixture is essentially a conical wave guide, it behaves as a
transmission line, conducting the signal right up to the shielding disk sample. Therefore, the
source and receiver are very close to the scattering event.
Figure 5-9: Cross Sectional View of Transmission Holder (24)
Because the investigation was concerned with the near-field scattering width (small values
of ), the limit was removed. Further simplification of the equation was also made by selecting a
specific phase angle. When using the shielding apparatus to measure the intensity of the scattered
signal the orientation of the incident wave and fibers in the shielding disks was such that the
back-scattered wave ( = 180) was measured. Equation 5.5-2 then reduces to:
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CHAPTER 6:Shielding Effectiveness Model Design
6.1 Introduction
This chapter will present the methods and supporting logic utilized in the development of the
shielding effectiveness model. The equations for the scattering width and scattered electric field
strength developed in Chapter 5 will be applied to the data accumulated during the shielding
experiments described in Chapter 3, resulting in the creation of a predictive model for the
shielding effectiveness of nylon 6,6 composites containing ThermalGraph and Fortafil carbon
fibers.
6.2 Review of Problem Description and Focus
From the orientation analysis described in Chapter 4, it was determined that the composite
shielding disks are comprised of somewhat uniformly oriented cylindrical carbon fibers in a
nylon 6,6 matrix. Thus, the disks are complex non-homogeneous, non-isotropic systems.
Because of the dielectric nature of the nylon 6,6 matrix, the impinging wave sees only a
collection of fibers, some of which are in a conductive network arrangement. It is the interaction
of the wave with these fibers that determines the shielding effectiveness of the composite.
As discussed in Chapter 5, the scattered electric field produced by a plane wave impinging a
carbon fiber can be modeled with Equation 5.4-55.
=
=n
)(
n
)(
nnn
ns
)ncos()a(H
)(H)a(J)j(EzE
2
2
0 [5.4-55]
Where:
a = optical width of object, fiber diamter (meters)
= 0 =
2(meters-1)
= distance from scatterer to observer (meters)
n = {02
01
=
n
n
From this equation, the scattering width of the fiber can be calculated using Equation 5.5-3:
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The actual distance between the scatterer (fiber) and observer should be a fraction of the sample
thickness, 3.2 mm. Nevertheless, the choice of will not affect the quality of the final model
results. It will, however, influence the numerical values of the derived model parameters.
Figure 6-1: Near Zone (= 1.0 x 10-4
m) Scattering Width for Both Fibers
0
100 200 300 400 500 600 700 800 900 1000
2.7
2.8
2.9
3
3.1
3.2
3.3
3.4
3.5
3.6
3.7x 10-4
Frequency (MHz)
ThermalGraph
Fortafil
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Figure 6-2: Far Zone (= 50 m) Scattering Width for Both Fibers
6.3.2 Deterministic Nature of Scattering Equations
The deterministic nature of the scattered electric field and scattering width equations is a
major drawback in the applicability of the equation for non-homogeneous materials. The
equations include no prediction of whether or not the wave actually hits the object in
question. They simply give an indication of the power of the field scattered when impinged
with a plane wave. The wavelengths investigated ranged from 10 to 0.3 m (30 MHz 1.0
GHz). The fibers are on average 6 orders of magnitude smaller than the impinging wave.
This huge discrepancy in size produces a high probability that the wave will never see a fiber.
Thus, the reflected power portion of shielding effectiveness cannot be directly modeled with
the scattered field equation. This can best be seen by plotting the shielding effectiveness of a
single fiber of ThermalGraph due solely to scattering by using the solution for the scattered
electric field, Equation 5.4-55 and the definition of SE.
0 100 200 300 400 500 600 700 800 900 10000
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
Frequency (MHz)
ThermalGraph
Fortafil
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effectiveness. Increased frequency, however, results in decreased scattered field strength, and
therefore, decreased shielding effectiveness for a single fiber and decreased scattering width
(Figure 6-1). One would correctly assume that since a larger object produces a stronger back-
scattered field, it would be a better shielding material. This fact must be accounted for in the
shielding effectiveness model.
6.4 Accounting for Collision Probability
The probability of a wave collision with a fiber is dependent on a multitude of factors: the
apparent size of the fiber (radar cross section), the wavelength of the incident wave and the
volume fraction of fibers within the nylon 6,6 matrix. Prior assumptions in this analysis have
eliminated the influence of other factors such as fiber length and fiber orientation on the
probability of a collision. Because of the multiplicative behavior of the factors, it is quite
challenging to single out the direct effect of each component. This research will focus on
quantifying the cumulative effect of the factors.
As previously mentioned and shown in Figure 6-1 and Figure 6-3, the scattering width of
the cylindrical carbon fibers reduces in size as frequency in increased. The chance of a collision
with a fiber, however, increases with increased frequency (reduced wavelength). Both effects can
be accounted for by dividing the scattering width ( D2 ) by wavelength () to form a new term,
D2 , known as the bistatic scattering width. This ratio gives an indication of the size of the
fiber in a window one wavelength long. It shows the relative importance of the scattering width
and fiber visibility due to the incident wavelength. The scattering width of a ThermalGraph
fiber varies from 3.7 to 3.0 x 10-4 m from 30 MHz to 1.0 GHz. The size of the incoming wave,
however, shows greater frequency dependence (10 to 0.3 m). Therefore, even though the
scattering width of the fiber decreases slightly, the relative size of the fiber ( D2 ) in a unit cell of
length increases greatly with respect to frequency. Figure 6-4 shows the unit cell/window.
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6.5 Scaling Factor Analysis
The constantwas calculated for each formulation by using Equation 6.4-1 and the respective
D2 (calculated from Equation 5.5-3) for either ThermalGraph or Fortafil and is called the
Scaling Factorin this work. The Scaling Factor is defined as the average of the Shielding
Effectiveness data divided by the ratio
D2 over the range from 300 MHz to 1000 MHz. A
separate Scaling Factor was determined for each material formulation. Figure 6-5 and Figure 6-6
show the analysis for NCN05 and NDN05. Following the same convention from the shielding
effectiveness plots, the upper and lower dotted lines indicate the maximum and minimum
constants calculated at that given frequency. The solid line represents this average value. The
remaining scaling factor graphs can be found in Appendix E.
Figure 6-5: Scaling Factor Analysis for NCN05
300 400 500 600 700 800 900 1000
500
1000
1500
2000
2500
3000
Frequency (MHz)
Scaling Factor = 861.503
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Fortafil
DFit FactorScaling.Approx(dB)SE
= 2 [6.7-1]
3007.1)%Vol.(4085.7(dB)FactorScaling += [6.7-3]
0 5 10 15 20 25 30-0.5
0
0.5
1
1.5
2
2.5x 10
4
Volume Percent Filler
Sca
lingFactor(dB)
Figure 6-11: Linear Fit Applied to ThermalGraph Scaling Factor Data
0 5 10 15 20 25 300
2
4
6
8
10
12
14 x 10
4 Linear Scaling Factor Fit
Volume Percent Filler
ScalingFactor(dB)
Figure 6-12: Linear Fit Applied to Fortafil Scaling Factor Data
R2 = 0.9737
R2 = 0.9348
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Figure 6-14: Model Predicted and Experimentally Determined Shielding Effectiveness for
NDN05 using Linear Scaling Factor Fit Equation
6.8 White Model Comparison
This new model is a significant improvement over models proposed by White and Bushko
for predicting shielding effectiveness in composite materials having low electrical conductivities.
As shown in Appendix H, the White model equation was derived for homogeneous, isotropic
materials (9-10).
+=
r
rrrdB
fft
10log1016834.3SE [6.8-1]
Where:
t = thickness of material (inches)
f = frequency (Hertz)
r = conductivity relative to copper
r = magnetic permeability relative to copper
300 400 500 600 700 800 900 10002
4
6
8
10
12
14
16
Frequency (MHz)
Model
SE Data
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The assumption of homogeneity produces the greatest error when applying the White model
to a complex composite system. As discussed in Section 6.3.2, a shielding effectiveness model
for a media containing a collection of both shielding (fiber) and non-shielding (nylon 6,6)
materials must include a method for predicting the occurrence of shielding material/wave
collisions to be capable of accurately predicting shielding effectiveness.
The White model relies on effective electrical conductivity of the sample to determine the
shielding effectiveness