nick janda thesis aug2004 ctc

7/27/2019 Nick Janda Thesis Aug2004 CTC

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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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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


k = length of kneading block in millimeters

l = twisting angle of the individual kneading

segments

Kneading disks

KS1-d-hh-i

KS1 = Kneading disc


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









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



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:



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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19

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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20

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:



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.



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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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43

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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44

)()( 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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46

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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47

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.


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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54

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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55

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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57

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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69

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

nick janda thesis aug2004 ctc

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