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NIST Technical Note 1 529

Carbon-Loaded Polymer

Composites Used as

Human Phantoms: Theoretical

Models for Predicting LowFrequency Dielectric Behavior

Richard G. Gayer

James R. Baker-Jarvis

Michael D. Janezic

Raian K.F. Kaiser

loO

^^ '-?.? nal Institute of Standards and Technology

^

y 6^^ ilogy Administration, U.S. Departnnent of Commerce

NIST Technical Note 1529

Carbon-Loaded Polymer

Composites Used as

Human Phantoms: Theoretical

Models for Predicting LowFrequency Dielectric Behavior

Richard G. Geyer

James R. Baker-Jarvis

Michael D. Janezic

Ralan K.F. Kaiser

Radio-Frequency Technology Division

Electronics and Electrical Engineering Laboratory

National Institute of Standards and Technology

325 Broadway

Boulder, Colorado 80305

June 2003

,^

/

U.S. Department of Commerce

Donald L. Evans. Secretary

Technology Administration

Phillip J. Bond, Under Secretary of Commerce for Technology

National Institute of Standards and Technology

Arden L. Bement. Jr.. Director
Certain commercial entities, equipment, or materials may be identified in this document in order to describe an experimental procedure or concept

adequately. Such identification is not intended to imply recommendation or endorsement by the National Institute of Standards and Technology, nor is it intended to imply that the entities, materials, or equipment are necessarily the

best available for the purpose.

National Institute of Standards and Technology Technical Note 1529

Natl. Inst. Stand. Technol. Tech. Note 1529, 43 pages (June 2003)

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Contents

1 Introduction 2

2 Lorentz-Lorenz, Clausius-Mossotti, Maxwell- Garnett, and Bruggeman

Effective Medium Formulations 6

2.1 Lorentz-Lorenz and Clausius-Mossotti Relations 6 2.2 Maxwell- Garnett and Bruggeman Relations 9 2.3 Modified Clausius-Mossotti Relation for Penetrable or Nonpenetrable

Dispersed Particles 10

3 Effective-Medium Formulations with Local Particle Aggregates 14

3.1 Clausius-Mossotti Adaptation to Aggregate Model 14

3.2 Bruggeman Adaptation to Aggregate Model 16

3.3 Maxwell- Garnett Adaptation to Aggregate Model 17

4 Measurements and Theory 21

5 Conductivity Temperature Dependence of Carbon-Black-Filled Polymers 27

5.1 General Effect of Temperature Changes 27

5.2 Variable- Temperature Measurements on Carbon- Black Conductive Poly- mers 32

6 Summary 35

7 References 37

111

Carbon-Loaded Polymer Composites

Used as Human Phantoms:

Theoretical Models for Predicting

Low-Frequency Dielectric Behavior

Richard G. Geyer, James R. Baker- Jarvis, Michael D. Janezic

Raian K.F. Kaiser

National Institute of Standards and Technology

Radio- Frequency Technology Division

Boulder, CO 80305

Conductive polymers loaded with carbon black are considered for use as semisolid

biologic phantom materials. Adapted forms of Clausius-Mossoti, Maxwell- Garnett, and

Bruggeman predictive dielectric mixing rules are considered that take into account dis- ordered cluster topologies of the conductive particles in a dielectric composite. The

disordered cluster suspensions can lead to rapid, nonlinear increases of permittivity and

conductivity at volume-loading fractions far below the close-packing limit, where the

percolation threshold would normally be approached in well- stirred metal-dielectric

composites.

The measured low-frequency electrical properties of fabricated carbon-black-loaded

silicone composites exhibit a threshold percolation at carbon loading fractions of onethird the close-packing Umit. This conductivity and permittivity behavior is predicted

by an adapted form of the classical Bruggeman rule that takes into account the cluster

topology of the carbon black particles in the fabricated composites.

Conductivity measurements of carbon-black- loaded silicone polymer composites from

10 to 25 °C exhibit a temperature behavior similar to that of an extrinsic, partially

compensated n-type semiconductor. The measured conductivity dependence with temperature can be related to the conduction-band density of states, as well as donor and

acceptor concentrations in the composite. A large positive temperature coefficient with

respect to resistance was observed at temperatures greater than 40 °C for the 7 % by

weight carbon black-loaded sihcone composite, which is consistent with nonpercolative

behavior at these temperatures.

Key words: Carbon-loaded polymer; conductivity; dielectric; disordered suspensions;

effective medium; mixing rules; nano- composite; percolation; permittivity; phantom
Chapter 1

Introduction

The development of materials that simulate the relevant electromagnetic properties of

the human body is crucial to studies of low-frequency electromagnetic interaction with

the body, as well as health effects of microwaves and interaction of wireless transmitters

with human tissue [1]. When such materials are used as phantoms, standardized methods for evaluating and assessing the interaction of personal medical electronic devices

(PMEDs) and metal weapons with magnetic fields generated by hand-held (HH) and

walk-through (WT) metal detectors can also be developed and improved. A review

of various phantom materials previously developed for use at low RF frequencies has

been given by Baker- Jarvis, Kaiser, Janezic, Paulter, and Stricklett [1]. Some of these

phantom materials are based on liquids and salts or various gelling agents and salts. Dielectric liquids generally work well at low frequencies because mobile ions from the salts can mimic the dielectric loss characteristics of human tissue. Examples of low-frequency

conductivity data for various human tissues are shown in Fig. 1.1. Semisolid materials, however, have several advantages over liquids in that they allow

test objects to be permanently embedded or encapsulated at particular positions. The

use of semi-solid phantom materials that simulate the electromagnetic properties of

the human body over the frequency range at which metal detectors operate allows

standardized testing for HH and WT metal sensors designed for detecting metal objects

concealed within human body cavities.

One semisolid composite that can potentially be used for phantom materials is a

carbon-black-loaded polymer. Carbon black is commonly chosen for its electrical con- ductivity and low cost, and the electrical conductivity of a polymer can be greatly

adjusted and improved by incorporating carbon-black powders. Understanding the

mechanisms of conduction and electromagnetic-wave propagation is critical to the development of these composite materials in phantoms that simulate the human body.

A complicating factor generally not taken into account in predictive dielectric mixing
-•- blood -i- bone cortical -k- grey matter -i- heart -9- liver -B- lung deflated

-t- lung inflated -6- muscle

skin dry

-^ skin wet -^

stomach

10' 10' 10 10" 10^ 10'

Frequency (Hz)

Figure 1.1: Measurements of conductivity of various body tissues as function of fre- quency. After Gabriel et al. [2] (no uncertainties assigned).
rules [3] is that carbon black may also form conducting clusters that depend on the

total loading. The existence of carbon-black clusters affects the threshold level at which

percolation occurs. A knowledge of how the volume fraction of the clusters depends

on the carbon-black loading in the polymer and how, in turn, the electric properties

of the composite phantom material depend on the topological structure of the clusters

is important. The development of appropriate electromagnetic models that account

for connective clustering of conductive inclusions in a dielectric medium permits predictive understanding of the electrical properties of carbon black-loaded polymers and

optimizes the fabrication process of acceptable phantom composite materials.

In general, the effective electrical properties of an inhomogeneous medium that is composed of particles of one substance embedded in a continuum of a different material

is a complicated function of the permittivities and conductivities of the constituents, the

particle shape and size distributions, the volume loading, and the spatial arrangement of

the distributions. The loading constituent may take the form of small spheres, ellipsoids,

platelets, rods, or other shapes. The bulk properties of the composite will also depend on

the alignment of the loading particles and therefore may be dielectrically or magnetically

anisotropic. In addition, the bulk properties of the composite will generally not be

a symmetric function of the volume-fraction of the matrix and filler. When any of

these parameters have nonzero variance, the predictive theoretical problem becomes

very difficult to solve. Permittivities and conductivities of disordered suspensions are

difficult to handle, even when the suspended particles are spheres with a uniform size

distribution.

If the composite medium is electrically isotropic in nature, the well-known ClausiusMossotti approximation may be utilized, provided that only dipole interactions are present. For regular particle array suspensions, the assumption of dipole interactions

is usually valid only in the limiting case of low volumetric loading. Higher multipole

interactions between the loading particles become more important when the particles approach contact, so that the Clausius-Mossotti approximation breaks down under higher

volumetric loading conditions. When there are random or disordered distributions,

higher multipole interactions can occur with any volumetric loading and must be considered. These higher-order multipole interactions depend inversely on increasing powers

of the interparticle distance and can often be observed as a strong nonlinear dependence

of the effective permittivity or conductivity when plotted as a function of the volumetric

filling factor. In this report we review the general effective permittivities predicted by the theoretical models of Clausius-Mossotti, Lorentz-Lorenz, MaxweU-Garnett, and Bruggeman,

highlighting some of their fundamental differences. We then consider the adaptation

of the Clausius-Mossotti, Bruggeman, and MaxweU-Garnett rules to effective media in

which the particle-size distribution has negligible variance, but in which aggregates or

local clusters can form where local contact exists between the loading particles. These
models have application to conductive carbon particle and insulating polymer composites, which are currently being tested for use in human phantoms. Lastly, we compare

the adapted aggregate forms of the Clausius-Mossotti, Bruggeman, and Maxwell- Garnet

theoretical effective permittivity predictions with room-temperature laboratory dielectric data obtained on carbon-silicone composites.
Chapter 2

Lorentz-Lorenz,

Clausius-Mossotti,

Maxwell-Garnett, and

Bruggeman Effective Medium

Formulations

2.1 Lorentz-Lorenz and Clausius-Mossotti Relations

The Lorentz-Lorenz relation [4-6] is given by

-^— =P= T:Na, (2.1)

which relates the effective permittivity e'^jj and the polarization /3 of a spherical sample

of the medium to the number density TV of a suspension of point dipoles, and the

polariz ability a of each dipole. The Lorentz-Lorenz relation was essentially developed

to describe point polarizable dipole particles embedded in vacuum. If the polarization of

a spherical sample of the medium expressed by eq (2.1) is equated to the volume filling

factor 1/ of a. distribution of metal spheres having individual isolated polarizabilities

a — R^, eq (2.1) reduces to the Clausius-Mossotti relation,

-^—- = /3 = -ttTVq = u. (2.2)

Equation (2.2) relates the polarization (5 to the in situ dipole polariz ability a and dipole

number density, which is related to the macroscopic permittivity of a spherical sample.

6

^i
If a includes the effects of all higher-order multipole interactions, eq (2.2) could be used

to calculate the exact permittivity. However, this has seldom been accomplished for atomic or molecular dipoles, since so few higher-order atomic multipole polarizabilities are known. We see that the well-known Lorentz "catastrophe" occurs when the number

density is such that 47r iVa/S = 1, for which a singularity in permittivity occurs, where

permittivity rapidly increases. Similarly, the Clausius-Mossotti relation erroneously

predicts that there is a permittivity singularity when u = 1. However, equal-sized

spheres can never meet this condition nor can they completely fill space. Generally,

the Lorentz catastrophe is prevented by a conductive transition which occurs when any

collection of conducting spheres begin to touch each other. The maximum packing

density of a suspension of spheres of equal size depends upon their spatial arrangement.

For cubic lattices the maximum possible volume filling factors Uc are 0.52, 0.68, and 0.74,

for sc, bcc, and fee lattices, respectively. For random packing of equal-sized spheres,

the maximum possible volume filling factor is Uc = 0.63 [7,8].

All regular or random or disordered distributions undergo a conductive transition

when spherical loading particles make contact. In disordered carbon suspensions in a

silicone matrix, we usually have no detailed control over interparticle carbon spacings. In this case, disordered samples contain local conductive regions (particle aggregates or

clusters) where the critical loading density has been reached, and other nonconductive

or dielectric regions with lower particle densities. At any filling factor below the critical

value, a disordered system acts like a mixture of dielectric regions and conductive particle aggregates. Suspensions of dispersed particles both with no aggregates or clusters

forming and with clusters forming are shown in Figures 2.1 and 2.2. With increasing

volume-filling factors, the conductive aggregates grow at the expense of the dielectric

regions. Hence, in contrast with the behavior or regular arrays of ordered suspensions,

conduction in disordered suspensions occurs at different times in different regions of the

sample and the dielectric anomaly is spread over an entire range of filling factors below

Uc- A critical filling factor still occurs at Uc, though, when the entire suspension exhibits a polarization anomaly. The Clausius-Mossotti equation for a suspension of spheres of

two constituents is given by

<// -' -,.4-1,4-1 ,231

where e'^jj = Ce/z/eo, f'l = fi/^o, and e'j = €2/60 are, respectively, the relative permittivities of the suspension and the two particle types with respect to the permittivity Cq

of free space (8.854 x 10"^^ F/m); the volume fractions of the two particles are given

by ui and 1^2.
Silicone Host

Phase

Carbon

Dispersed

Phase

Figure 2.1: Disordered suspension of carbon particles in a silicone matrix host phase

where no aggregates of the carbon particles exist.

C Clusters

Unclustered C particles (volume fraction V3)

(volume fraction V2)

Host Silicone Phase

(volume fraction V.,)

Figure 2.2: Disordered suspension of carbon particles in a silicone matrix where aggregates of the carbon particles exist.
2.2 Maxwell- Garnett and Bruggeman Relations

The Maxwell- Garnett effective medium approximation differs from the Clausius-Mossotti

relation in that the inclusions are in a matrix background other than vacuum. For a two-phase composite, the Maxwell-Garnett approximation is given by [9],

'•'''''"" -"4^, (2.4)

<// + 2ei, e'i+2e;,.

where e'^ = em/fo is the relative permittivity of the matrix (host), e[ is the relative

permittivity of the dispersed phase, and Ui is the volume fraction of the dispersed

phase. The Majcwell- Garnett rule works fairly well if the dispersed phase inclusions

make up a small fraction of the total volume. Note that the Maxwell-Garnett rule is not symmetric; that is, different effective permittivities are obtained when the roles of

the matrix host and particle inclusions are interchanged, even if the volume fractions

remain the same. We may write the effective permittivity of the composite for this

topology as

The Maxwell-Garnett dielectric mixing rule requires that

• the sizes of dispersed particles are much smaller than both the electromagnetic

wavelength and the skin depth within the particle material,

• a dilute suspension of spherical (carbon) particles in a continuous host matrix

(silicone).

Where there is significant aggregation or clustering of carbon particles, especially for

dense composites, the Maxwell-Garnett model, although convenient, becomes inappropriate.

The Clausius-Mossotti approximation cannot be employed for mixtures of two constituents when the topology of the composite consists of an intermingling of many

small, irregularly shaped regions having different complex permittivities. In this case, Bruggeman [10] proposed making the properties of the matrix or host the same as the

effective medium itself and derived the following two-phase composite effective medium

formulation for spherical elements,

where Vi and V2 are, respectively, the volume fractions occupied by phases of media

1 and 2. The Bruggeman model is a symmetric formulation, and the solution for the
effective permittivity for a two-phase composite leads to the solution of the following

quadratic,

.;4+[e;(2y:-V2) + 4(2^2 -Vi)]4/y-2«^^f = 0. (2.7)

Implicit to the Bruggeman topological model [10] is that

• the suspended inclusions are distributed randomly;

• all suspended inclusions are very small compared with the size of the composite

(and very large compared with atomic sizes);

• inclusions of the same type (material, size, form) are present in very large numbers

and phases are very close together with negligible interstices;

• the composite consists of homogeneous and isotropic elements of two materials;

and

• no embedding of one phase within the other is allowed.

One limitation of the Bruggeman formulation, as presently posed, is that it cannot be

applied to a monodisperse suspension of spherical elements because such a suspension

cannot fiU all sample space, even for closest-packing densities.

The Lorentz-Lorenz, Maxwell-Garnett, and Bruggeman topological models are special cases of an n-phase mixture rule of the form

f' — f' c' c' c' c' n' t:'

where fgyy,e^ represent the relative permittivities for the effective medium and host

medium, and e^, Cj, • • • , e^ represent the permittivities for suspended inclusions of types

1, 2, • • • , n. The factors vi, f2, • • • , t'n denote the respective volume fractions of inclusions

1,2, • • • ,n. The spherical inclusion geometry, which has minimum dipole moment and

depolarization, are common to the Lorentz-Lorenz, Maxwell-Garnett, and Bruggeman

models.

2.3 Modified Clausius-Mossotti Relation for Penetrable

or Nonpenetrable Dispersed Particles

A modified Clausius-Mossotti derivation is given in [9,11], in which spherical dispersed

particles of radius a, having arbitrary complex permittivity e* = €p(l — tan^e.p); a-nd magnetic permeability \i' = /i'(l — tancJ^^^p), are distributed on a uniform cubic lattice

10
o o o

o o o o

jO O O 0^

-^o o o o

Ep. ^^p

£ m. ^^ m

H,

Em.^^ m

Figure 2.3: Uniform lattice of dispersed dielectric permeable spheres in a dielectric

permeable host matrix.

with distance d between centers in a host medium that has arbitrary complex permittivity e'^ = e^(l - tan^e.m), a-nd complex permeability //^ = l^'mi^ ~ tan(5rn,m)- (See Fig.

2.3.) In the derivation, general excitation of electric and magnetic dipole sources in the

composite are considered and represented in terms of Hertz vectors. The scattered rera- diated fields from all the spherical particles are taken into account, which allows for the

examination of structural resonant effects (higher-order multipole interactions) between

the dispersed inclusions. When the radii of the spherical loading particles become small

relative to the host matrix propagation constant (| k^a |<C 1, where k^i = Uy/n'^e^

and u) = 2-K f is angular frequency), the dominant contributions to the electric and magnetic fields are those from the induced electric and magnetic dipoles, respectively. This

simplifies the formulation of the field components in the composite. In the analysis, an electromagnetic plane wave of unit amplitude and e-^'^* time dependence and with

electric field polarized in the x-direction and propagating in the z-direction is considered

to be incident upon the matrix loaded with spherical particles. Boundary conditions

that must be satisfied are the continuity of the normal electric displacement field and

normal magnetic induction field, as well as the tangential electric and magnetic field components at the surface of a spherical loading particle. Application of these boundary conditions to the electric- and magnetic-field components, as expressed in spherical

coordinates, yields two coupled integrai equations for the total x-directed electric and

11
y-directed magnetic fields at any point {x,y,z):

_£:£ Z!L _ Ij, / (nU _!f f* — f

q /-OO /r* _ /r* "^ ^ "^ ^e*„ + 2e* 2 7o ^-^ "e*„ + 2e* ^^ ^

+ (4i)ljy*(^)4^^A),-.^^k-lrf^ (2.9)

and

^iW - (i^)le-^-^-- + z/ffi(z) ^°-^ ^^^ - lu rijkm ^°'^ /""^ Hl{w)

+ C\)^El{w) f

-^ " '^ A)eife^l^-^U^. (2.10)

In eqs (2.9) and (2.10), :^ = A-Ka^ Ji^d^) is the volumetric loading of the dispersed particles, and e*_p and /i* p are the apparent complex permittivity and complex permeability

of the dispersed particles which are related to the true complex permittivity and true complex permeability by a complex penetration factor G{kpa); i.e., e*_p — G{kpa)e* and

H* = G{kpa)fi*, where

G(k a) = 2

sin(A;pa)-A:pacos(fcpa)

^ kpa cos{kpa) + {k^a"^ — 1) sin(A;pa)

and kp — u^/pi*€*. Note that the function modifying the true particle inclusion complex

permittivity and permeability is oscillatory and modulates the effective complex electri- cal and magnetic properties of the composite in a nonobvious manner. The penetration

function G{kpa) is shown as a function of the modulus | kpa \ in Fig. 2.4. We see that,

for various values of | kpa |, pecuhar structural resonant effects are evident in which

the effective electrical properties of the composite are not bounded by the individual

constituent properties of that composite. The three terms on the right side of eqs (2.9)

and (2.10) may be identified as the incident field at z, the scattered field from a spherical particle at the point z, and the scattered field from all the other particles. If the

coupled integral equations (2.9) and (2.10) are solved for the effective propagation constant and effective impedance, we obtain the following expression for effective complex

permittivity.

-e//

3K<,p-C)

(2.12)

Formally, eq (2.12) is at least consistent and correct in the dilute and dense dispersed

suspension extremes. For no loading, f = and 6*^^ = e^. When there is vanishing

12
25

J

I I I

I

I I I I

I

I I I I

I

I I I I I

I I I I

I

I I I I

I

I 1 I I

I

I I I I

I

i

I I I

I

I I I

20 -

-

15 J

10 J

'"" 5

-

ca - a _ — -

cs _

-5 -

-10 -

-15 —

-

-20

ri I I I I I I I I I 'I I I I I I ' I I I I I I I I I I I I I I I I I I I I I I j

I I I I I I I r10

kpH 1

Figure 2.4: Penetration function illustrating effects of particle inclusion size relative to

particle inclusion wavenumber on effective electrical properties of composite.

host-matrix material, r/ w 1 and | kpa |> 1, so that f*. . = e* Although this model

intrinsically accounts for higher-order multipole interactions between the dispersed in- rlusion particles, it is restricted to inclusion sizes that are small relative to a wavelength

in the host matrix and to ordered suspension topologies that are uniform and nonpercolating. Its application to composites, where local aggregates or conductive clusters can form together with dispersed suspensions, is questionable.

13
Chapter 3

Effective- Medium Formulations

with Local Particle Aggregates

3.1 Clausius-Mossotti Adaptation to Aggregate Model

The Clausius-Mossotti equation for a suspension of spheres having two differing constitutive electric properties is given in eq (2.3). This equation is valid for topologies

involving ordered island- or cermet-type suspensions. For the Clausius-Mossotti model,

the total volume u of all particles present is given by

1/ = //J + U2, (3.1)

and is less than the sample volume except when all particles are in aggregates or clusters

and the aggregates fill all space. In the latter case, the suspension as a whole would be

randomly close-packed. To adapt the "island" topology of the Clausius-Mossotti model

to one that includes the formation of aggregates, we define x as being equal to the total

conductive particle volume fraction and Xi and X2 representing the conductive particle

volume fractions in isolated spheres and aggregates, respectively. We then have

X = xi -\- X2- (3-2)

If the fraction of the conductor content in the aggregates is denoted by /, then

xi = (1 - f)x, X2 = fx. (3.3)

We now note that within each isolated spherical inclusion, the volume fraction is unity,

whereas the compact aggregates are have a conductor volume fraction equal to the close- packing limit Xc- Hence the volume fractions of the isolated spheres and the aggregates

may be written as

r/i = (l-/)x, U2 = fx{l/xc). (3.4)

14
The Clausius-Mossotti adaptation to a model that includes both dispersed sphere and

aggregate topologies is then [12]

'-efl - 1

'-eff + 2 ^177

fi-1

4+2 + 1^2-

e'+2

= (^-^)^^ + ^%..

4-1

(3.5)

In order to use eq (3.5) we must either know or make a reasonable statistical assumption

about the aggregate fraction / in relation to the total conductor volume fraction x and

the close-packing limit Xc- For x = 0, / = and for x = Xc, f = I. If we assume a

linear relation between these two limits, e.g., / = x/xc, we have

'-eff - 1

'-eff + 2

X 4-1

Solving for e'^j^,

xJ''e; + 2

, _ Nc-M

^eff -

^ ' xj e', +

2- (3.6)

(3.7) Dc-M

where

Nc-M = xl{€[ + 2)(e'2 + 2) - 2x^ [^(^(^^c - 1) + 2x, + 1) - ^(x, + 2) - 2{x, - 1)]

-2x2x(l-e;)(€'2 + 2) (3.8)

and

Dc-M ,2/,/(e'l + 2)(e'2 + 2) + x^ [4(4(^c - 1) + 2xc + 1) - e2(xc + 2) - 2(xc - 1)]

+x2x(l-6;)(6'2 + 2).

If we treat both e[ and €2 ^-s large relative to unity in eq (3.5), we may write

''^^^"^=(l-/)x + /x(-).

-eff + 2

;3.9)

(3.10)

Again, for weU-stirred samples, / =^ at x = and / = 1 at x = Xc, so that, with

/ — X / Xc,

<// - 1 X [x^ + x(Xc - 1)]

Hence

<// + 2 xl - X

\ 3; / 1

1 + — —

- 1

.> -

Zxl

-e// x2(xc — 1) — x^(x — r

(3.11)

(3.12)

15
3.2 Bruggeman Adaptation to Aggregate Model

The Bruggeman relation for a two-phase medium composed of spherical elements is

The volume fractions Vi and V^ differ from ui and u^ since they apply to suspensions

of different topologies. Because an implicit assumption of the Bruggeman model is that

the phases are very close together with negligible interstices, the sum of the volumes

must fill the sample volume so that

Vi = l-y2. (3.14)

The Bruggeman model is clearly not applicable to a monodisp'erse suspension of spherical elements, since such a suspension cannot fill sample volume space, even at closest

packing density. An artifact of the Bruggeman model is to use a heterodisperse set of suspensions of ever smaller spheres so that, in the limit, all sample volume space

is filled. This is not a useful artifact to employ when we are dealing with an actual

monodisperse suspension of spheres. Our suspension consists of monodisperse isolated

conductive spheres and compact conductive aggregates of spheres. The aggregates are not necessarily spherical in themselves, but may form irregular conductive regions that

are intermingled with similarly irregular dielectric regions containing the remaining unclustered spheres (see Fig. 2.2). However, both the dielectric insulating and conductive

regions individually do satisfy the assumptions of the Bruggeman's model. Hence we

can treat regions 1 and 2 of the (effective) suspension as an intermingled mixture of

dielectric (region 1) and conductor (region 2) phases. The aggregate fraction will still be / and the critical volume fraction for closest random packing will still be Xc- Instead

of being the volume fractions of particle types 1 and 2, as in the Clausius-Mossotti

model, the Vi and V2 in the adapted Bruggeman model will be taken as region volume

fractions. In this case, we stiU have

xi = {\ - f)x, X2 = fx, (3.15)

and

X2 = fx = XcV2, (3.16)

as long as / is the aggregate or clustered volume fraction and Xc represents the close- packed volume fraction of the conductive phase in terms of actual conductor volume

fraction. From eq (3.4),

V2 = fx(-). (3.17)

16
Because the conductor phase is close- packed, 2^2 = ^2- However, the dielectric phase is not close-packed, so that i^i ^ Vi; rather, from eq (3.14),

V, = l-/^(^)- (3.18)

Because we are using the Bruggeman model to model the regions of conductive dispersed

and clustered phases, we require the actual conductor volume fraction in region 1 relative to the volume of region 1. From eq (3.18) this is

Equation (3.19) may then be used in the (island) Clausius-Mossotti relation to determine

the effective permittivity of the unclustered dielectric region 1, after which the Bruggeman relation is applied to both regions 1 and 2 to determine ih.e effective permittivity

of the composite. The effective permittivity of region 1 is then given by

3x'

^eff, region! = 1 + -, _ ,

• (3.20)

regionl

If eq (3.20) is used in eq (3.13), a rather complex quintic polynomial results when at- tempting to solve for the effective permittivity in terms of e'2. However, this equation

simplifies considerably when the permittivity €2 is large relative to unity. The Bruggeman relation simplifies to

^ ~ J x^)\^eff, regionl ~ ^eff) r_^ _ q

/o 91 'J ^eff, regionl + ^^eff ^c

yielding

I _ eff,regionl /o oo^

'^^^ - 1 - 3x/x,

• ^^-^^^

3.3 Maxwell-Garnett Adaptation to Aggregate Model

The MaxweU-Garnett mixing rule, given in eq (2.4), is also known as the Rayleigh mixing

formula [13]. The Rayleigh rule does not contain information about an individual scatterer; rather, only the volume fraction u\ of the inclusion phase and the permittivities

of the inclusion phase e'^ and host e^ appear in the mixing rule. The MaxweU-Garnett mixing formula yields eq (2.5) for the effective permittivity of

the composite, which may be written in the following form:

f' - e'

17
or

-T^ = l + 3z/i-, '-^

-, . (3.24)

^ + 2-^i(p 2-ui{^-l)

We note that, when ui —> 0, the effective permittivity reduces to that of the matrix,

and when i^i —>• 1, the effective permittivity reduces to that of the inclusion phase. The

only way that the condition i/j si 1 can be achieved, however, is that a polydisperse

suspension of spherical inclusions be permitted. Sihvola [3] gives useful insight into the

average electric fields internal and external to spherical inclusions and their relation to the Maxwell- Garnett rule by defining the effective macroscopic permittivity of a 2- phase composite as a constant relating the volume-averaged electric field < E > and

displacement field flux density < D >; that is, <D>=f:y^<E>, (3.25)

where e*ry = ^e//(^

~" itan(5e//), e'^tt is the effective real permittivity, and tan 6e// is the effective dielectric loss tangent. The average electric field and displacement flux

density are then written in terms of the corresponding volume fractions and the electric

fields internal, Ej, and external, Eg, to the inclusions:

<E>=/E, + (l-/)Ee (3.26)

and < D >= fe.Ei + (1 - /)€eEe, (3.27)

where €i and e^ are the complex permittivities of the inclusions and the matrix host.

From eqs (3.26) and (3.27),

<D >

Af^ + (l-/)ge

/ft + (!-/)

(3.28)

For spherical, isotropic inclusions, only an equivalent dipole source exists for each in- clusion, and the polarizability is a scaJar that does not depend on the direction of the

electric field, but only on its amplitude. If we assume for this case that the ratio between

the internal electric field and external electric field is approximately the same as the

quasi-static case, Sihvola [3] has shown that

154 = ^'^ (3.29)

Ee C. + 2€e ^

^

18
Direct substitution of eq (3.29) into eq (3.28) yields

which is identical to the Maxwell- Garnett mixing formula, eq (3.23).

We may now adapt the Maxwell-Garnett mixing rule using the same technique as performed with the Bruggeman and Clausius-Mossotti topologies. This is done by ais- suming that the effective permittivity of the carbon-clustered region may be adequately

given by a carbon host phase with silicone inclusions, where the volume fraction of

the carbon in the clusters is the Bernal maximum random-packing limit for equal-sized

spherical inclusions. In the unclustered region, the Maxwell-Garnett rule is used with

the opposite topology. Then a composite permittivity is given by

'^f^- V,c, + V2C2 + V,c,

' ^^-^^^

where Vi = 1 - x(l - x/xc) - {x/xcf, V2 = x{l - x/xc), V3 - {x/xc^, ci = 1/3,

C2 = ei/(2ei + €2), C3 = ei/(2ei + 63), x is the total volume fraction of carbon, and

Xc = 0.63 is the maiximum volume filling factor. With the Maxwell-Garnett rule, the

permittivity of the carbon cluster region with a carbon host and silicone inclusions is given by

€i[€i{3-2Xc) + 2Xc€2] , .

[xcei + €2(3 - Xc)]

In Fig. 3.1 the respective volume fractions Vi, V2, and V3 for the host silicone, the

unclustered carbon, and the clustered carbon are illustrated as a function of total volume

fraction of the carbon in the composite. These volume fractions are plotted using the

packing fraction limit of 0.63 for randomly organized spheres.

19
0.8

c

o

"o 0.6

CO LL

CD£^ 0.4

o

>

0.2

•Host Silicone

•Unclustered Carbon

Clustered Carbon

X =0.63

c

0.1 0.2 0.3 0.4 0.5 0.6 0.7

Total Volume Fraction of Carbon in Composite

Figure 3.1: Volume fractions of host silicone, unclustered carbon, and clustered carbon

in composite as function of total volume fraction of carbon in composite.

20
Chapter 4

Measurements and Theory

The conductivities and permittivities of carbon-black-loaded polymer composites were measured at frequencies from 1000 Hz to 1 MHz with an open-circuit coaxial holder.

This fixture consists of an open-circuited 14 mm coaxial hne with the center conductor

shorter than the outer-conductor shield. The shielded open-circuited holder is quite

useful for measuring the dielectric properties of powders, hquids, and semisolids because

of the ease of sample insertion. Previous dielectric measurements performed using the

shielded open-circuited sample holder are described in the literature [14-22]. A rigorous

treatment of the fringing capacitance at the open-circuited termination has been given in

[1]. The reflection coefficient of the transmission line was measured using an automatic

network analyzer. Alternatively, the line capacitance and conductance were measured

using an LCR meter.

Spherical carbon-black pearls, having individual particle sizes of 12 nm, were mechanically stirred with silicone, and temperature annealing was required to stabilize the

carbon-black silicone composite [1]. A study of how percolation at thermodynamic equilibrium is correlated to dynamic percolation under differing experimental temperatures

and annealing times in carbon-black-filled polymer composites has been given by Wu,

Asai, Zhang, Miura, and Sumita [23]. Real relative permittivity and conductivity data

of this composite at 800 kHz for vaxious volume fraction carbon loading are shown in Figs. 4.1 and 4.2. Both the real permittivity and conductivity increase rapidly when

the carbon volume fraction exceeds 0.2, which is a volume fraction well below the close- packing limit. The predicted normalized composite permittivity from the weU stirred

Clausius-Mossotti cluster model and the Bruggeman cluster model as a function of car- bon loading are compared in Fig. 4.3. The Maxwell- Garnett adaptation that accounts

for aggregation or clustering of conductor particles in a conductor-insulator composite

has a percolation threshold similar to that of the Clausius-Mossotti aggregate model.

This can be seen in Fig. 4.4, where calculated real permittivities of carbon-black-loaded

21
500

0.04 0.08 0.12 0.16

Carbon Loading Volume

Figure 4.1: Real relative permittivity of carbon-black pearl and silicone composites as a function of cajbon loading. Frequency and temperature are 800 kHz and 22 °C. After

CO

o

D

a

coO

CD

'Zn O

Q.

E

oO

1.6

1.4

1.2

1

0.8

0.6

0.4

0.2

i J

-*-f = 800 kHz

1

/

/

'. '

'

:

' X

'

,

, , y

/

0.05 0.1 0.15 0.2

Volume Fraction Carbon Loading

Figure 4.2: Conductivity of carbon-black pearl and silicone composites as a function of

carbon loading. Frequency and temperature are 800 kHz and 22 °C. After [l].

22
CD>

-1—' ca(D ccu

(D >. N

Tri >

F

-1—

1

o E

^ CD

CD D-

-4—' cnO

CZL £

o

o

50

40

30 -

20 -

10

0.1 0.2 0.3 0.4 0.5 0.6 0.7

Carbon Volume Loading

Figure 4.3: Normalized composite real permittivity (or normalized composite conductivity) as a function of conductor-particle volume loading x for differing disordered

suspensions. The solid line is the reduced permittivity (or conductivity) given by the

Clausius-Mossotti aggregate model, reflecting well stirred composite samples, with clus- ter fraction / = x/xc- The dashed line is the normalized real permittivity (or normalized

conductivity) given by the Bruggeman aggregate cluster model, with cluster fraction

/ = 1. All calculations use Bernal's experimental value for random closest packing of

spherical particles, Xc = 0.63. Note that the percolation threshold for the ClausiusMossotti aggregate model occurs where x = Xc = 0.63, whereas for the Bruggeman

aggregate model, the percolation threshold occurs well below the close-packing limit at X z= xj3 = 0.21.

23
silicone composites using the Maxwell- Garnett aggregate model (eq (3.30)) are shown

as a function of total volumetric carbon loading. The Bruggeman aggregate predictive

model, showing the normalized permittivity (or conductivity) as a function of carbon

loading for differing parametric cluster fractions, is given in Figs. 4.5 and 4.6. The same

percolation threshold is evident for all total volumetric metal particle loadings, but the

normalized permittivities or conductivities can vary with carbon loading at values below

the percolation threshold, depending on the cluster volume fraction.

Comparison of these predictive model considerations, which incorporate aggregates

or localized clusters of the conductor (carbon) loading particles, indicate that the

adapted Bruggeman aggregate model appears to best fit our laboratory data on the

composite phantom materials, as presently fabricated (where percolation begins well

below the close-packing limit). A range of carbon black loading < a; < 0.15 in the sil- icone provides a suitable conductivity range for phantom materials that would simulate

most body tissues (see Fig. 1.1) so that we can, in principle, simulate all human body

tissues with various carbon-black-loaded silicone composites. Better reproducibility of

the low-frequency electrical properties of the carbon-black-loaded silicone composites

could be obtained by forcing the percolation threshold to occur at higher carbon volumetric loadings near the close-packing limit. This type of carbon suspension topology

requires well-stirred sample fabrication.

24
450 —

4nn

'_ — Maxwell-Garnett Cluster Model

Permittivity

CJ

CO

O

en

o

o

'-

^ 250 -

ra

1^ 200 -

r

•^ 150 o1 100

: 1

; /

50

n

; ^

''

—,—i ',-, 1 , . , , 1 , , . , 1 , , . . 1 , , . ,

0.1 0.2 0.3 0.4 0.5 0.6 0.7 Carbon Volume Loading

Figure 4.4: Predicted real permittivity of caxbon-black-loaded silicone composite with

Maxwell-Garnett aggregate model. Real relative permittivities of 3.37 for silicone and

788 for carbon black were used in calculations. Note that with this model the percolation

threshold occurs at the close-packing limit, x = Xc=0.63.

45

40

03>

ra 35

a

11

B

'55 O

Q.

E

oo

30

25

20

15

10

Bruggemann Cluster (f = 1)

-t- Bruggemann Cluster (f =0.3)

••Bruggemann Cluster (f=0.6)

•*• Bruggemann Cluster (f=0.4)

T- Bruggemann Cluster (f=0.2)

-t- Bruggemann Cluster (f = x/x )

Q 0.05 .0.1 0.15 0.2 Carbon Volume Loading

Figure 4.5: Composite normalized permittivity (or conductivity) as a function of

conductor-particle loading and cluster fraction with Bruggeman aggregate model.

25
(D>

'-—

'

_C0CD CC

ID >^ N

oi >

E

t^

1—o^ CD

03

CI. 4—

>

w

o

Q_

E

o

CJ

45

40

35

30

25

20

15

10

Bruggemann

Bruggemann

Bruggemann

Bruggemann

Bruggemann

Bruggemann

Cluster (f = 1)

Cluster (f = 0.8)

Cluster (f = 0.6)

Cluster (f = 0.4)

Cluster (f = 0.2)

Cluster (f = x/x )

0.1 0.15

Carbon Volume Loading

0.2

Figure 4.6: Windowed normalized permittivity (or conductivity) as a function of con- ductive particle loading and cluster fraction using Bruggeman aggregate model.

26
Chapter 5

Conductivity Temperature

Dependence of

Carbon-Black-Filled Polymers

Polarization processes occurring in materials can be strong functions of both frequency

/ and absolute temperature T (K). The dependence of conductivity on frequency for various carbon- black-loaded silicone composites that are useful for simulating biologic

phantom materials has been reported previously from 1000 Hz to 1 MHz [1]. The roomtemperature experimental conductivity data demonstrate little frequency dependence

over the frequency range of most metal detection systems, which is an important crite- rion when selecting phantom biologic models to be used for assessing the sensitivity of

metal detectors.

5.1 General Effect of Temperature Changes

In order to gain insight into temperature-dependent variations of the effective complex

dielectric permittivity over the frequency range where dipolar or orientation phenomena

occur, we can usefully consider concepts borrowed from statistical thermodynamics [24]. If the polarizability of a material at a given temperature and under the influence of a time-varying electric field can be related to the known (measured) activation energy

U of that material, then we can derive a method for predicting the dielectric behavior

of that material as a function of temperature. Of course, this predictive ability will

be constrained by the assumptions implicit in the physical model and must be vali- dated by measurements of the dielectric permittivity and conductivity as a function of

temperature.

Consider a bistable model of an elementary dipole within a dielectric whose molecular

27
POTENTIAL ENERGY

y^qdE [-^

Figure 5.1: Potential energy of a bistable model having two minima, depicted by states

1 and 2. The depths of these minima are modified by an applied electric field, and

charged particles can occupy either state [26].

groupings can be characterized by well defined permanent dipole moments. In this type

of model, we assume that a charge q may be in one of two states that are separated

be a distance d. These states are portrayed in Fig. 5.1. In this model, an electric field acting on the dielectric causes movement of charge from the minimum of state 1 to the

minimum of state 2. This charge movement is equivalent to a 180° rotation of a dipole

of moment

\v\=\qd. (5.1)

The potential difference due to the applied electric field E is simply

Tp-^ - il;2 = 2'p E = qdE cos 9, (5.2)

where 9 is the angle between the direction of the electric field and the dipole moment.

The interaction between individual dipoles within the dielectric may be neglected

by assuming that the total number N of bistable dipoles per unit volume is small. This

28
would correspond to levels of carbon-black loading in the polymer below the percolation

threshold. For dipoles we may also generally assume that 9 — and that equal potential

energy exists for states 1 and 2 in the absence of an electric field. We can visualize this

microscopic assembly in which the bistable dipoles are in a heat reservoir which consists

of spontaneously active particles that exchange energy through thermal fluctuations

with each other and the dipoles. Therefore, the directions of the dipoles fluctuate.

Occasionally, thermal fluctuations allow a charge located in the minimum of state 1 to acquire sufficient energy to go "over" the potential hill U and drop into the minimum of

state 2. In state 2, the energy of the charge is returned to the heat reservoir and stays

in state 2 until it acquires enough energy from the reservoir to return to state 1. The probability that a charge may jump in a double potential well (state) has been

derived and is well known in classical statistical thermodynamics. Here, we can state that the number of dipoles jumping per unit of time from state 1 to state 2 is given in

terms of the difference of potential energy between the two wells as: uu = Ae-(^±P^)/^^, (5.3)

where k is Boltzmann's constant (1.3807xl0~'^"^ J/K). The constant A may or may

not depend on temperature. Generally, for ionic solids it does not, whereas for organic

polymers, A is inversely proportional to T. The ± signs within the exponential of eq

(5.3) dictate whether the minimum in state 2 is lower or higher than that of state 1. Generally, U > kT.

For dipoles in normal dielectrics, p is of the order 10"'^^ C-m, whereas E for fields

below breakdown strength is always less than 10^ V/m. Therefore, in general,

§ « 1. (5.4)

Note that pE/{kT) is dimensionless and that, as T —> oo, thermal fluctuations dominate

polarization, whereas when T -> 0, polarization (pE) dominates. Equation (5.3) may

approximately be written as vE

ui2 ~ u{l - —), (5.5)

u = Ae-^/^^ (5.6)

is the frequency of jumps in the absence of an applied electric field. Similarly, the

frequency of jumps from state 2 to state 1 can be written in linear form as

«2i«<l +

|f).

(5.7)

Provided that the number of charges Ni jumping from state 1 to state 2 per unit of

time is the same as that jumping from state 2 to state I, N2, the average population

29

where
of charges in the wells of states 1 and 2 will not change with time. In other words, at equilibrium

NiUi2 = N2U2i. (5.8)

The total number of bistable dipoles is constant or

iVi + A^2 = N. (5.9)

Equations (5.8) and (5.9) allow us to calculate iVi and N2 in equilibrium. Since the

polarization per unit volume P is defined as that number of dipoles acting in one

direction which is not compensated by dipoles acting in the opposite direction, or P = P, = (iVi - N2)p. (5.10)

we have a basis from statistical thermodynamics for determining the variation of polarization with temperature when the dielectric is in equilibrium; that is, from eq (5.8),

A^l^l-^) = ^2<1 + ^)

or N,-N2 = {N, + iV2)^ = N^^. (5.11)

Hence eq (5.10) may be written as

Ps = N^. (5.12)

We are now in a position to derive the frequency-dependent properties of the model.

We note that the change in number of dipoles in state 1 is equal to the outflow number

to state 2 less the inflow number from state 2; or ^= -N,uu + N2U2i. (5.13)

at Since the total number of bistable dipoles is constant,

dNi dN2

(5.14)

dt dt

'

or d{Ni-N2) JNi

dt " dt so that eq (5.13) becomes

dNi 1 d{Ni - N2)

dt 2 dt = -NMl-^) + N2u{l + ^). (5.15)

30
Equation (5.15) can be simplified to read

lfc^ = -„(JV,-iV,) + „(^. + iV,)|f (5.16)

or

We now have a differential equation for the dipolar polarization of a dielectric medium

as a function of temperature; that is

1 dP ^ Np^E

This equation is very similar to that for a dielectric medium that has a single relaxation

time [24]; in other words, eq (5.18) is a relaxation equation with a relaxation time

r=^ = ^e""/'^, (5.19)

2u 2A ' ^ ^

so that we may generalize eq (5.18) to ^dPo

dt

+ Pd = aoE, (5.20)

where Pd is the dipolar polarization and a^ is the dipolar polarizability of the dielectric

material. Using simple Fourier-transform methods we obtain the following expressions

for the complex permittivity €*{u,T) = e'{uj,T) — J€"{u,T) = fo4(l "" tan 6):

.•Kr)-.„(r) = ^-^, (5.21)

or ,'(u,T) = ,^(T)+jf^, (5.22)

and

tan<5(a;,T) = ^^""^ , (5.24)

where u = 2irf, too is the optical dielectric constant defined by the simple relation

€^ - I z= n? — I = P^/E, n is the refractive index of the material, and Poo is the optical

polarization. The dependence of conductivity on frequency is given by

a = ueoe';. (5.25)

31
-1.2

-1.4 -

[to] c

-2 -

-2.2

-2.4

/ 1

"""]

j I I

•1.6 -•- -7-

-1.8

3.1 3.2 3.3 3.4

10Q0/T (K-^)

3.5 3.6

Figure 5.2: Log conductivity versus 1000/T for a carbon black-loaded silicone composite

with loading of 7 % by weight. Data shown are average log- conductivity values for the

frequency range 1 kHz to 1 MHz.

Equations (5.21-5.25) describe dispersive dielectric behavior for a bistable dielectric

as a function of temperature and frequency. The temperature-dependent permittivity

and conductivity are given in terms of the dipolar polarizability, activation energy, and

high-frequency (optical) permittivity at temperature T. The derivation is limited to

dielectric materials where interaction between individual dipoles can be neglected, and

for conditions where pE -C kT (nonsuperconducting states).

5.2 Variable-Temperature Measurements on Carbon- Black

Conductive Polymers

Vanable- temperature measurements of conductivity were performed with the shielded

open-circuited sample holder. The shielded open-circuited measurement system was

placed in an environmental chamber which operates from —150 °C to 350 °C. This

chamber has feed-through bulkhead adapters on the sides for passage of coaxial rf feeds

to the measurement system being used [25]. In addition, the chamber has purging ports

for injecting nitrogen gas, either to reduce water vapor or to reduce oxidation in the

chamber. The derived conductivity data obtained from these reflection measurements

are shown as an Arrhenius plot in Fig. 5.2.

32
In general, the conductivity a (S/m) for an Arrhenius process is

a = a^e-^l^'^, (5.26)

where ctq is a reference conductivity. Equation (5.26) can be written as Incr — Ino-Q =

— UI{kT). We may suppress the reference value In ctq and simply plot In a versus 1/T to determine the activation energy U, since In Cq acts only to shift functional values of In a

on the ordinate. An Arrhenius plot allows us to predict the conductivity temperature

dependence expected for the carbon black conductive polymer composite under the

influence of a time- varying electric field. Plots of measured temperature-dependent conductivity can also allow us to evaluate

conductivity mechanisms associated with differing high- and low-temperature processes

in the composite. For example, two Arrhenius processes with different temperaturedependent activation energies may be summed as

a:=aie-^^/(^^) + c72e-^^/(^^); (5.27)

the respective slopes of ln(cr/cri) or ln(a"/cr2) versus 1/T would yield the sum of the

two activation energies for the two processes associated with two differing conduction

mechanisms. Activation energies for high-temperature conduction processes are greater

than those for low-temperature processes. Knowledge of the conduction mechanisms

can, within practical limits, permit the engineering of conductivity of the composite, or

at least constrain the temperature variability of conductivity in the composite.

The data shown in Fig. 5.2 were taken over a limited temperature range; i.e., from

10 °C to 60 °C (283 to 333 K), which, in general, exhibit conductivity variations asso- ciated with high- temperature processes. The temperature behavior of the conductivity

for the carbon-black composite from 10 to 25 °C (283 to 298 K) is similar to that

of an extrinsic, partially compensated n-type semiconductor [27]. For an extrinsic n- type semiconductor, the high-temperature process is associated with conductivity in the

conduction band. If measurements were taken at much lower temperatures, we would

expect a second Arrhenius conduction process having lower activation energy associated

with nearest-neighbor hopping of electrons from occupied to unoccupied sites. However,

a very strong reversal of this conductivity increase with increasing temperature is apparent at temperatures greater than 40 °C (313 K). The carbon-black conductive polymer

composite has changed from one that has a negative temperature coefficient of resistance

to one that has a very large positive temperature coefficient (PTC) of resistance, or one which exhibits a very large increase of resistance at temperatures above 40 °C. The

large PTC of carbon-black-filled polymers above room temperature has been observed

by Carmona and Ravier [28].

This phenomenon can be explained physically in the following way. The 7 % by

weight carbon-black-loaded polymer is close to the expected percolation threshold, as

33
predicted by the aggregate adaptation of the Bruggeman rule. As temperature is in- creased above 40 °C, the thermal expansion of the composite itself induces changes in

the correlation lengths of the clustered carbon. In other words, the carbon particles are likely disconnected, causing the composite to exhibit non-percolative behavior and

a very large PTC.

Materials having high PTC can be used in important applications such as self- regulated heaters and electrical safety devices. For biologic phantom model applications,

temperatures should be kept below 40 °C in order to maintain required conductivity

values that simulate body tissues.

34
Chapter 6

Summary

Carbon- black-loaded silicone composites have been fabricated for use as semisolid phantom materials. The electrical properties of various carbon loadings have been measured

as a function of frequency (1000 Hz to 1 MHz) and compared to reported electrical

properties of various biological tissues. Unavoidable clustering of the carbon black in

the composite occurs, which leads to disordered suspension topologies in the resulting

composite. Clustering or aggregation of metal particles in a dielectric host affect the

electrical (both permittivity and conductivity) properties. These effects are generally

not taken into account in classical predictive mixing rules for dielectrics. In addition,

such cluster topologies lead to rapid, nonlinear increases of conductivity and permittivity of the phantom material as volume fractions of the carbon loading are increased.

The conductor loading volume fraction, for which very rapid increases in permittivity

and conductivity of the composite occur, is usually at the percolation-threshold close- packing limit X = Xc = 0.63 for random suspensions of spherical conductor particles.

This is predicted by adapted Clausius-Mossotti and Maxwell-Garnett mixing rules that

take into account the effects of clustering in the well- stirred but disordered conductorparticle suspensions.

The measured electrical properties of fabricated carbon-black-loaded silicone composites exhibit a threshold percolation at carbon loading fractions well below the close- packing limit expected in well stirred samples; e.g., at a; = Xc/3 = 0.21. This is predicted

by an adapted form of the classical Bruggeman rule reported here that takes into account

the cluster topology of the carbon-black particles in the fabricated composites.

We have presented effective cluster models for two broad classes of permittivity and

conductivity enhancements observed in low-frequency measurements on monodisperse

suspensions of conductive spheres in a dielectric insulating host. The enhancements are attributed to higher multipole interactions in the carbon particle clusters. These multipole interactions also produce the particle aggregations. Nonpercolating suspensions are

35
treated as mixtures of isolated conductive (carbon) spheres and closed-packed conductive aggregates of spheres using the Clausius-Mossotti formulation, whereas percolating

suspensions are treated as mixtures of randomly intermingled conductive and dielectric

insulating regions using the symmetric Bruggeman formulation. In the cluster-adapted

Bruggeman formulation, the conductive regions are close-packed with clusters of carbon

spherical particles, while the dielectric regions contain only isolated spheres.

We observed that the only difference in conductivity and permittivity enhancement

between the predictive models is the cluster topology. Any chemically inert conductor and insulator can be used as inclusion particle and host. In well-stirred samples,

the permittivity increases rapidly but smoothly with carbon loading up to the close- packing limit; conductivity varies linearly up to the close-packing limit. The other type

of suspension is more difficult to prepare reproducibly because such suspensions exhibit

percolation and rapidly increasing conductivity and permittivity at volume fractions

well-below the close-packing limit. For biologic phantom materials, carbon-black con- centrations should be below the expected percolation threshold. Fabrication methods

affecting the cluster topology of the incorporated carbon black that result in percolation

thresholds near the close-packing limit need to be explored. In addition, the relation of

percolation at thermodynamic equilibrium with differing annealing temperatures and

annealing times needs to be studied. The type of cluster topology and percolation

threshold behavior will be reflected in low-frequency electrical property measurements.

Conductivity measurements of carbon-black-filled silicone polymer composites from

10 °C to 25 °C exhibit a temperature behavior similar to that of an extrinsic, partially

compensated n-type semiconductor. The measured conductivity-temperature dependence can be related to the conduction-band density of states, as well as donor and

acceptor concentrations in the composite. The carrier density hats a temperature dependence for a compensated semiconductor different from that of the carrier mobility,

whose temperature dependence depends on the scattering mechanism. This complicates

the conversion of cr vs. T into carrier density vs. T in predictive models of temperature dependence. A large positive temperature coefficient with respect to resistance was observed at temperatures greater than 40 °C for the 7 % carbon- black-filled silicone

composite, consistent with nonpercolative behavior at these temperatures. For phantom

biologic materials using carbon-black-loaded composites, temperatures should be kept

below 40 °C to avoid unacceptable conductivity variations.

Financial support from the DOJ National Institute of Justice and the NIST-EEEL

Office of Law Enforcement Standards is gratefully acknowledged.

36
Chapter 7

References

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[2] Gabriel, C; Gestblom, B., "Microwave absorption in aqueous solutions," Nature,

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9, pp. 70-103, 1880.

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des Lichtes und der Korperdichte," Annalen der Physik und Chemie, IX, 4, pp. 641-

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[7] Bernal, J.D., "Geometry of the structure of monatomic liquids," Nature, 185, pp.

68-70, 1960.

[8] Coxeter, H.S.M., Introduction to Geometry, John Wiley: New York, 1962.

37
[9] Geyer, R.G.; Majitese J.; Baker-Jarvis, J. /'Effective medium theory for ferrite-loaded

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39

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