Electrical properties of hollow glass particle filled vinyl estermatrix syntactic foams

Vasanth Chakravarthy Shunmugasamy •

Dinesh Pinisetty • Nikhil Gupta

Received: 25 June 2013 / Accepted: 21 August 2013 / Published online: 10 September 2013

� Springer Science+Business Media New York 2013

Abstract Low dielectric constant materials play a key

role in modern electronics. In this regard, hollow particle

reinforced polymer matrix composites called syntactic

foams may be useful due to their low and tailored dielectric

constant. In the current study, vinyl ester matrix/glass

hollow particle syntactic foams are analyzed to understand

the effect of hollow particle wall thickness and volume

fraction on the dielectric constant of syntactic foams. The

dielectric constant is found to decrease with increase in the

hollow particle volume fraction and decrease in the wall

thickness. Theoretical estimates are obtained for the

dielectric constant of syntactic foams. Parametric studies

are conducted using the theoretical model. It is found that a

wide range of syntactic foam compositions can be tailored

to have the same dielectric constant, which provides pos-

sibility of independently tailoring density and other prop-

erties based on the requirement of the application.

List of Symbols

U Volume fraction (subscripts m:

matrix, g: glass, a: air, p: matrix

porosity, mb: glass microballoon)

e Dielectric constant (subscripts 0:

vacuum)

g Radius ratio of hollow particle

ri, ro Inner and outer radii of hollow

particle

w Wall thickness of hollow particle

re Radius of surrounding medium

qth, qexp Theoretical and experimental

densities of syntactic foams

Z Impedance

R Resistance

Xc Reactance

/ Phase angle

f Frequency

C Capacitance

t Thickness of specimen

A Contact area

W Electrical potential

ee Dielectric constant of surrounding

medium

an, bn, Dn, En, Gn,

Hn, In, and Jn

Constants

Pn(cosh) Legendre polynomial

r Radial distance

dm;n Kronecker delta

K, L, and S Constants

e0 External applied field

a Polarization parameter

Introduction

Very large-scale integration of electronic circuits has

drastically reduced the size of circuit boards used in elec-

tronic devices. This has created a challenge to develop

materials with low dielectric constant, high specific

strength, low density, low moisture absorption, and high

durability. Integrated circuit boards, which form the heart

V. C. Shunmugasamy � N. Gupta (&)

Composite Materials and Mechanics Laboratory, Mechanical

and Aerospace Engineering Department, Polytechnic Institute of

New York University, Brooklyn, NY 11201, USA

e-mail: ngupta@poly.edu

D. Pinisetty

The California Maritime Academy, 200 Maritime Academy

Drive, Vallejo, CA 94590, USA

123

J Mater Sci (2014) 49:180–190

DOI 10.1007/s10853-013-7691-0

of computers, require electrical insulators with low and

preferably tunable dielectric properties [1]. Polymers and

polymeric composites have found applications in such

fields due to their low dielectric properties [2, 3].

Epoxy resins, which are often used as matrix materials

for composites, are also used in electrical and electronic

fields as insulators, dielectrics, and as underfills in circuit

boards [4, 5]. One of the desirable ways of decreasing the

dielectric constant is by introducing porosity in the poly-

mer [6]. Since air has a low dielectric constant of 1, the

dielectric constant of polymer foams is low but it also

accompanies low strength and stiffness, which are unde-

sirable. In addition, irregular size and distribution of gas

voids in polymer foams can lead to mechanical property

variation within the material.

Composite materials prepared by embedding hollow

filler particles into a continuous matrix are called syntactic

foams [7–9]. This approach ensures low density of the

composite with significant volume fraction of porosity but

without the penalty on mechanical properties. Hollow

particles made of glass, carbon, polymers, and ceramics

have been used in syntactic foams [10]. The mechanical

and thermal properties such as modulus, coefficient of

thermal expansion (CTE), and thermal conductivity of

syntactic foam can be tailored [11–15]. The variables in

designing syntactic foams include: particle and matrix

materials, volume fraction of particles, and wall thickness

of particles. In most existing applications, weight saving is

an important consideration in using syntactic foams.

Existing studies have shown that a combination of hollow

particle wall thickness and volume fraction can be used to

independently tailor the CTE and the density of syntactic

foams to achieve weight saving in structural applications

[11].

The available studies on electrical properties of syn-

tactic foams characterized with respect to frequency and

temperature are summarized in Table 1, whereas Table 2

contains a summary of studies that have included the effect

of environmental exposure. This information sets the

context for the present work. It can be observed in both

tables that all the existing studies have used epoxy resin as

the matrix material. In addition, glass hollow particles (also

called glass microballoons or GMBs) are the most common

type of particles used in syntactic foams. A wide range of

material compositions, temperatures, and frequencies have

been covered. The results that are summarized in these

tables show that the dielectric constant.

• decreases with increasing Umb of GMBs

• decrease with increasing test frequency

• increases with increasing temperature

Impedance is also found to have a behavior similar to

that of dielectric constant with respect to Umb and

frequency. It is clear from the summary of the available

literature that there is a lack of studies on syntactic foams.

• with any other matrix except epoxy resin

• understanding the effect of hollow particle wall

thickness

• that develop theoretical relations of dielectric constant

with Umb and microballoon wall thickness. The avail-

able studies are mainly experimental.

The present work is aimed at filling this gap by char-

acterizing vinyl ester/GMB syntactic foams for dielectric

properties with specific focus on understanding the relation

of Umb and hollow particle wall thickness with dielectric

constant of the syntactic foams. In addition, theoretical

models are developed to predict the dielectric constant of

syntactic foams. Maxwell–Garnett [16] and Jayasundere–

Smith (J–S) [17] equations, applicable to solid particle

filled composites, are modified to include the hollow par-

ticle wall thickness. The theoretical predictions are vali-

dated with experimental results. The models are used to

conduct parametric studies to understand the weight saving

potential of syntactic foams in applications, where the

dielectric constant is the primary consideration.

Materials and methods

Glass microballoons (3 M, MN) and vinyl ester resin (U.S.

Composites, FL) are used to fabricate syntactic foam slabs.

The neat vinyl ester resin and GMBs are measured in

appropriate proportions and mixed in a beaker. To the

uniform mixture, hardener is added and continuously stir-

red. The slurry is poured into aluminum molds coated with

a lubricant (Dow Corning, MI) and allowed to cure at room

temperature for at least 24 h. The detailed manufacturing

procedure is explained in the published literature [18].

The electrical impedance was measured using a CH

Instruments (Austin, TX) 700D potentiostat by the AC

impedance method, as schematically represented in Fig. 1.

The experiments were conducted in a frequency range of

10-2–106 Hz, with applied AC wave amplitude of 500 mV.

A specimen size of 18 9 14 9 1 mm3 was used in per-

forming the experiments. The specimens were cut using a

low speed precision diamond blade saw (Isomet�; Buehler

Ltd, Lake Placid, NY) to ensure that the surfaces were

parallel to each other. Five specimens were tested for each

composition type of syntactic foams and the average values

along with the standard deviations are reported.

GMBs of three different nominal true particle densities

(220, 320, and 460 kg/m3) are used in four different vol-

ume fractions (30, 40, 50, and 60 %) to fabricate 12 types

of syntactic foams. The scanning electron micrographs of

460-type microballoon reinforced vinyl ester matrix

J Mater Sci (2014) 49:180–190 181

123

syntactic foams containing 30 and 60 % is shown in Fig. 2.

The GMBs are characterized by radius ratio g as

g ¼ ri=ro ð1Þ

where the radius ratio can be related to the GMB wall

thickness as

w ¼ ro 1� gð Þ ð2Þ

Increasing value of g refers to decreasing hollow particle

wall thickness. The GMB properties, including radius ratio,

are given in Table 3. These GMBs have been extensively

characterized in previous studies and information on the

measured average diameter, size distribution, and density

are available in previous studies [13, 18]. These

experimental values are used in the present work for

determining the GMB parameters.

The specimen nomenclature starts with VE repre-

senting vinyl ester resin, followed by three digit true

particle density and two digits of microballoon volume

fraction. Apart from the porosity that exists inside

GMBs, some air is entrapped in the matrix during the

composite material fabrication and is termed as matrix

porosity. Presence of matrix porosity may also affect the

composite properties. The volume fraction of matrix

porosity is calculated by

Up ¼qth � qexp

qth

� �ð3Þ

The theoretical density calculated using the rule of

mixtures and the experimentally measured density of

syntactic foams are reported in Table 3. The estimated

Table 1 Existing studies on impedance and dielectric properties of syntactic foams

Reference Material used Test condition Results

Shahin et al. [26] Matrix: epoxy, hollow particles:

glass (0–55 wt%)

f = 100 Hz–30 kHz 1. The impedance of the composite decreases

with the frequency and the increases with

Umb.

2. The dielectric constant decreases with

frequency and Umb.

Shahin et al. [27] Matrix: epoxy, hollow particles:

glass (0–55 wt%)

f = 100 Hz and 10 kHz

Temperature = 20–125 �C

1. The syntactic foam impedance decreases

with increasing temperature.

2. The dielectric constant increases with

increasing temperature.

Park et al. [28] Matrix: epoxy, hollow particles:

glass (0–2 wt%)

f = 1–10 GHz Dielectric constant decreased with increasing

frequency and Umb.

Gupta et al. [29] Matrix: epoxy, hollow particles:

glass (Umb = 0.3–0.65)

f = 1–100 kHz

Temperature = 40–120 �C

1. The dielectric constant decreases with

increasing Umb.

2. The dielectric constant increases 5-10 %

with increase in the temperature.

3. The impedance decreased with increasing

frequency and the phase angle remained

close to -90�, suggesting capacitive

behavior.

Andritsch et al. [30] Matrix: epoxy, hollow particles:

glass (Umb = 0.5) filled with

SO2

f = 0.5 Hz–1 MHz

Temperature = -140 to

?120 �C

1. The syntactic foams had higher dielectric

loss than the neat resin at the tested

frequency and temperature.

2. SO2 gas inside the microspheres play an

important role at \1000 Hz and between

-140 and ?20 �C.

Yung et al. [24] Matrix: epoxy, hollow particles:

glass (Umb = 0–0.51)

f = 1 kHz–1 MHz 1. At 1 MHz, the dielectric constant

decreased from 3.98 to 2.84 as Umb

increased from 0–0.51.

2. A maximum decrease of 44 % is observed

in the dielectric loss at Umb = 0. 51

compared to neat resin.

Zhu et al. [22] Matrix: epoxy, hollow particles:

glass (Umb = 0.1–0.6)

f = 1 MHz

Temperature = 10 �C

1. Maximum 51 decrease % was observed in

the dielectric constant at Umb = 0.60

compared to neat resin.

2. Maximum 54 % decrease was observed in

the dielectric loss at Umb = 0.6 compared

to neat resin.

182 J Mater Sci (2014) 49:180–190

123

matrix porosity is low and is between 1 and 4.5 vol% for

most syntactic foam slabs.

Results and discussion

The impedance obtained from the experiment is a complex

quantity, containing the real (resistance), and the imaginary

(reactance) parts and is defined as

Zj j ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2 þ X2

c

qð4Þ

The potentiostat provides measurements of these

quantities. The phase angle is given as

u ¼ tan�1 Xc

R

� �ð5Þ

The phase angle is found to be around -90�, which

indicates the capacitive nature of the neat resin. The

capacitance can be obtained as

C ¼ 1

2p fXc

ð6Þ

The dielectric constant is obtained from the calculated

capacitance as

e ¼ Ct

e0Að7Þ

The values of t, A, and e0 are taken as 10-3 m,

25.2 9 10-5 m-2, and 8.854 9 10-12 F/m [19],

respectively.

Table 2 Existing studies on impedance and dielectric properties of syntactic foams, along with environmental effects

Reference Material used Test condition Results

Ansermet et al. [31] Matrix: epoxy, hollow particles:

glass (Umb = 0.37–0.57) and

organic (phenolic and copolymer

of acrylonitrile-vinylidene

chloride) (Umb = 0.34–0.5)

f = 100 Hz–10 MHz (dried

specimens) f = 40 Hz–1 GHz)

Water absorption at 85 �C and

85 % humidity

1. The dielectric constant of resin is

lowered with both glass and phenolic

microsphere addition. Water absorption

deteriorated the dielectric properties of

the composites.

2. Microspheres of acrylonitrile-

vinylidene chloride copolymer provide

both low and water-resistant dielectric

properties.

Andritsch et al. [32] Matrix: epoxy, hollow particles:

glass (Umb = 0.5)

f = 10-4–10-1 Hz

temperature = 40 and 70 �C

(wet: 80 % relative humidity and

80 �C for 7 days; normal: used

as obtained; dry: 200 �C for

2 days and then maintained at

80 �C)

1. The difference between the specimens

containing microspheres subjected to

wet and normal conditions were

minimal.

2. A conductive layer is formed due to the

moisture absorption in wet specimens

which result in DC-conductivity.

Strauchs et al. [20] Matrix: epoxy, hollow particles:

glass (Umb = 0.4) (Type A:

US–Al, 60 lm; Type B: silane

coated, 60 lm; Type C: US–Al,

40 lm; Type D: US, 40 lm)

f = 50 Hz temperature = 20 �C

water immersion for 0–50 days

1. Variation in dielectric properties

caused by the water absorption had no

relation to microsphere diameter.

2. Specimens with Type A and C

microspheres show an increase in the

dielectric constant with increasing

duration of water storage.

3. Specimens with Type B show

negligible difference, while specimen

with Type D show increase for the first

10 days of testing, before becoming

saturated.

Roggendorf et al. [33] Matrix: epoxy, hollow particles:

acrylonitrile copolymer coated

with CaCO3 (Umb = 0.4,

R0 = 40 and 95 lm)

f = 50 Hz Aging condition:

climatic chamber (1500 h) and

pressure cooker (100 h).

Syntactic foam reinforced with 95 lm

radius showed lower dielectric constant

across the entire aging time, under

climatic chamber aging.

US untreated surface, Al Alkali

Fig. 1 The setup used in the measurement of impedance of syntactic

foams

J Mater Sci (2014) 49:180–190 183

123

In the first step, the impedance of neat vinyl ester resin

used as the matrix in syntactic foams is measured. The

variation of impedance with respect to frequency for the

vinyl ester resin is shown in Fig. 3. The impedance is

found to decrease with increasing frequency across the

selected range. The impedance values are used to calculate

dielectric constant at various frequencies, which are com-

pared with the results obtained for syntactic foams.

In the next step, syntactic foams are tested in a similar

manner and their dielectric constants are calculated from

the experimental results. The impedance-frequency plots of

syntactic foams show characteristics similar to that of the

neat resin. These plots are used to determine the dielectric

constant at various frequencies of interest.

The dielectric constants of the neat resin and the syn-

tactic foams at a representative frequency of 100 kHz are

presented in Fig. 4. As a general trend, the dielectric

constant of syntactic foams is lower than that of the neat

resin. It is also observed that GMB volume fraction has a

prominent effect on the dielectric constant of syntactic

foams. The dielectric constant of syntactic foams decreases

with increasing GMB volume fraction. This observation is

Table 3 Theoretical and experimental densities, along with matrix porosity of the syntactic foams used in the study

Specimen type Mean particle

diameter (lm)aWall thickness

(lm)aRadius ratio

(g)aTheoretical

density (kg/m3)

Experimental

density (kg/m3)

Matrix porosity

(vol.%)

VE220-30 35 0.52 0.970 878 839 4.5

VE220-40 784 774 1.2

VE220-50 690 676 2.0

VE220-60 596 570 4.4

VE320-30 40 0.88 0.956 908 888 2.2

VE320-40 824 787 4.5

VE320-50 740 712 3.8

VE320-60 656 633 3.6

VE460-30 40 1.29 0.936 950 937 1.4

VE460-40 880 843 4.3

VE460-50 810 782 3.4

VE460-60 740 716 3.3

a Data taken from [18]

Fig. 2 Scanning electron micrographs of vinyl ester syntactic foams containing 460-type microballoons in a Umb = 0.3 and b Umb = 0.6

Fig. 3 The variation of the impedance of neat vinyl ester resin with

respect to frequency

184 J Mater Sci (2014) 49:180–190

123

expected because the air porosity volume fraction increases

with GMB volume fraction and air has lower dielectric

constant than that of matrix resin and GMB material

(glass). The plot of dielectric constant with respect to

syntactic foam density, Fig. 5, is illustrative because both gand GMB volume fraction affect the density. The neat resin

has the highest density in this data set. The dielectric

constant varies almost linearly with respect to the syntactic

foam density. The figure includes dielectric constant at two

different frequencies of 100 Hz and 100 kHz. In both cases

the slope of the line is the same. Since the syntactic foam

density is dependent on both GMB volume fraction and g,

the dependence of dielectric constant on syntactic foam

density requires further investigation to understand the

individual effect of these parameters. The relations

between these parameters can be better understood by

theoretical means. Although the interfacial bonding

between GMB and vinyl ester is expected to play a role in

the measured dielectric constant, this parameter is not

included in the study. All syntactic foams use GMBs of the

same type of glass material, therefore, the interfacial

bonding characteristics are expected to be the same for all

syntactic foams. In addition, the mean radius is nearly the

same for all types of GMBs used in syntactic foams, which

implies that the syntactic foams having the same volume

fraction of different types of GMBs will have the same

interfacial area and the effect of bonding characteristics

will also be the same in these cases. Hence, the attention is

focused on evaluating the effect of GMB volume fraction

and g in the theoretical models. In addition, including

interfacial bonding as a parameter in mathematical models

requires quantitative information, but quantitative infor-

mation on interfacial bonding is not available. The inter-

facial bonding includes combined effects of chemical

bonding, mechanical interlocking, and interfacial friction.

Moreover, the work by Strauchs et al. [20] studied the

electrical properties of both untreated and silane treated

particle reinforced syntactic foams. The silane treatment

does not appear to have significantly changed the dielectric

constant of syntactic foams (water immersion time = 0 h

in Fig. 4 of the reference), which was found to be strongly

dependent on the properties of the matrix resin [20].

Estimation of the dielectric constant of hollow particles,

which comprise a thin shell filled with air, is the first step in

obtaining the effective dielectric constant of syntactic

foams. In order to find a closed form expression for the

dielectric constant of GMBs, Maxwell’s theory can be

utilized. The polarization of GMBs is equated to the

polarization of an equivalent sphere and the dielectric

constant of GMB is obtained as follows. The potential w of

the GMB is obtained by solving the Laplace equation in

polar coordinates (Fig. 6), given by

r2w ¼ 0 ð8ÞThe solution of the Laplace equation is obtained in the

form [21]

wðr; hÞ ¼X1n¼0

ðanr�ðnþ1ÞþbnrnÞPnðcos hÞ ð9Þ

Fig. 4 Experimentally measured values (at 100 kHz) of the dielectric

constant for the neat resin and 12 types of syntactic foams

Fig. 5 Variation of the experimentally measured dielectric constant

at 100 Hz and 100 kHz with respect to the syntactic foams density

Fig. 6 Schematic of hollow glass microballoon to determine the

dielectric constant

J Mater Sci (2014) 49:180–190 185

123

where Pnðcos hÞis the nth order Legendre polynomial and

an and bn are constants. Taking into account the axial

symmetry of the sphere, the potential is given by

w1ðr; hÞ ¼X1n¼1

ðDnrnÞPnðcos hÞ ðr� riÞ ð10Þ

w2ðr; hÞ ¼X1n¼1

ðEnr�ðnþ1Þ þ GnrnÞPnðcos hÞ ðri� r� roÞ

ð11Þ

w3ðr; hÞ ¼X1n¼1

ðHnr�ðnþ1Þ þ InrnÞPnðcos hÞ ðro� r� reÞ

ð12Þ

w4ðr; hÞ ¼ �e0rP1ðcos hÞ þX1n¼1

Jnr�ðnþ1ÞPnðcos hÞ

ðre� rÞð13Þ

The constants Dn, En, Gn, Hn, In, and Jn are found by

applying the following boundary conditions and utilizing

the orthogonality property, given by

w1ðri; hÞ ¼ w2ðri; hÞ ð14aÞw2ðro; hÞ ¼ w3ðro; hÞ ð14bÞw3ðre; hÞ ¼ w4ðre; hÞ ð14cÞeaðorw1Þr¼ri

¼ egðorw2Þr¼rið15aÞ

egðorw2Þr¼ro¼ eeðorw3Þr¼ro

ð15bÞ

eeðorw3Þr¼re¼ ðorw4Þr¼re

ð15cÞZp

h¼0

Pnðcos hÞPmðcos hÞ sin hdh ¼ 2

ð2mþ 1Þ dm;n ð16Þ

where dm;n is the Kronecker delta function. Applying the

above boundary conditions the value of the constant J1 is

found to be

J1 ¼ðee � 1Þðee þ 2Þ r

3e K � ð2ee þ 1Þr3

oL

ðee þ 2Þð2ee þ egÞ

� �Se0 ð17Þ

where the constants K, L, and S are represented by

K ¼ 1þ 2ðee � egÞðeg� eaÞð2ee þ egÞð2egþ eaÞ

r3i

r3o

� �

L¼ ðee� egÞ þðeeþ 2egÞðeg� eaÞð2egþ eaÞ

r3i

r3o

� �

S¼ 1þ 2ðee � egÞðeg� eaÞð2ee þ egÞð2egþ eaÞ

r3i

r3o

� 2ðee� 1Þðee� egÞðeeþ 2Þð2eeþ egÞ

r3o

r3e

�

�2ðee � 1Þðeeþ 2egÞðeg� eaÞðee þ 2Þð2eeþ egÞð2egþ eaÞ

r3i

r3e

��1

The dipole moment induced in the sphere due to the

applied external field e0 is 4pe0J1. In addition, the

polarization of the system, a, is the ratio of the induced

dipole moment to the external applied field e0. Hence, the

polarization can be written as

a ¼ ð4pe0Þr3e

ðee � 1Þðee þ 2ÞK �

ð2ee þ 1ÞLðee þ 2Þð2ee þ egÞ

r3o

r3e

� �S ð18Þ

The effective dielectric constant of GMB is found by

equating the polarization of the system consisting of a

hollow glass shell containing the air void to the

polarization of an equivalent solid sphere. The dielectric

constant of GMBs can be written as

emb ¼1� 2g3 ðeg�1Þ2

ðegþ2Þð2egþ1Þ þ 2ð1� g3Þ ðeg�1Þðegþ2Þ

1� 2g3 ðeg�1Þ2ðegþ2Þð2egþ1Þ � ð1� g3Þ ðeg�1Þ

ðegþ2Þ

24

35 ð19Þ

where eg and ea are taken as 5.6 and 1, respectively [22].

The calculated effective dielectric constants of the three

types of GMBs used in the experimental study is plotted

with respect to their density in Fig. 7. The figure also

contains upper and lower bounds obtained by the parallel

and the series rule of mixtures, respectively. The values

calculated from the model are between the two bounds.

Although the syntactic foams are assumed to be filled

with GMBs that are identical in all respects, a distribution

exists in their size and radius ratio. Experimental data are

available on the size and radius ratio distribution of GMBs

used in the present work and can be used in the models to

account for size and radius ratio variability range [13]. The

radius ratio values calculated from the experimentally

measure density values of GMBs are 0.958, 0.948, and

0.928 for the 220, 320, and 460 type microballoons [13].

The effective dielectric constant for GMBs is now used

in the Maxwell–Garnett and the J–S equations to predict

Fig. 7 Dielectric constant calculated for the glass microballoon using

Eq. (19), presented along with values obtained from the parallel and

series model

186 J Mater Sci (2014) 49:180–190

123

the dielectric properties of particulate reinforced compos-

ites [22, 23]. The Maxwell–Garnett equation is given by

e ¼ em 1þ3Umb

emb�em

embþ2em

1� Umbemb�em

embþ2em

" #ð20Þ

The J–S equation is based on the Kerner’s equation

taking into account the particle-to-particle interaction

between GMBs [17] and is given by

e ¼Umem þ Umbemb

3em

embþ2em

� �1þ 3Umbðemb�emÞ

embþ2em

� �

Um þ Umb3em

embþ2em

� �1þ 3Umbðemb�emÞ

embþ2em

� � ð21Þ

The validation of these models is conducted with data

obtained from a study published by Yung et al. [24].

Figure 8 shows that both the J–S and the Maxwell–Garnett

model predictions closely match with the experimental

results with only ±6 and ±5 % deviation, respectively.

This experimental data was obtained on syntactic foams

containing GMB of 600 kg/m3 density tested at 1 MHz

frequency and using eg = 5.6. The value of eg was not

experimentally measured in this study and was taken based

on the general range for the sodalime borosilicate glass.

The dielectric constants predicted by the modified

Maxwell–Garnett and J–S equations for the syntactic foams

characterized in the present study are given in Table 4.

These predictions use the experimentally measured

dielectric constant of the neat resin as an input parameter

and eg is taken as 5.6, consistent with Yung et al. [24].

Since the dielectric constant is dependent on the frequency,

selection of em at appropriate frequency helps in obtaining

predictions for syntactic foams at that frequency. The

modified Maxwell–Garnett model predictions show a lar-

ger deviation from experimental values (±22 %) than

those obtained from the J–S model (±12 %), as listed in

Table 4. Although the modified J–S model predictions are

closer to the experimental results, both models consistently

under-predict the syntactic foams dielectric constant. To

better understand this trend, the J–S model is used for

conducting reverse calculations using the experimentally

measured dielectric constant of syntactic foams for calcu-

lating the dielectric constant of glass eg, while taking

em = 3.87. Experimentally measured values of g are used

in the calculation, obtained from [13]. The calculated eg

value is obtained as 12.1 ± 1.8. This value is high com-

pared to the expected range for the sodalime borosilicate

glass. Therefore, a sensitivity analysis is conducted on the

model with respect to the two important input parameters:

eg and em. The results are shown in Fig. 9, where it is

observed that the model is much more sensitive to variation

in em than in eg. Changing the value of eg from 3.87 to 4.2

shifted the theoretical cures to match with the experimental

data. Small variations in a number of parameters such as

cure conditions and resin to hardener ratio can affect the

measured value of eg. This analysis shows that the modified

J–S model is capable of capturing the trend observed in the

dielectric constant of syntactic foams and provides pre-

dictions with reasonable accuracy. It is also noted from this

parametric study that the model has a strong sensitivity to

the dielectric constant of the resin, which is the continuous

phase on syntactic foams.

After validation with experimental results obtained from

the present study and also with the data obtained from

previously published study, the modified J–S model is used

for conducting a parametric study to understand the effect

of GMB volume fraction and g on the dielectric constant of

syntactic foams. The dielectric constant of the neat resin is

taken as 3.87 at 100 kHz frequency in the parametric study.

Fig. 8 Comparison of theoretically calculated e using J–S and

Maxwell–Garnett model’s with the experimental values (taken from

[24]), at 1 MHz, for epoxy/GMB syntactic foams

Table 4 Dielectric constant for the various types of syntactic foams

along with the prediction from Maxwell–Garnett equation and J–S

equation

Specimen

type

Average

experimental eMaxwell–

Garnett model

J–S model

e Error

(%)

e Error

(%)

VE220-30 3.36 3.01 10.4 3.15 6.0

VE220-40 3.05 2.75 10.1 2.98 2.6

VE220-50 3.04 2.50 17.9 2.81 7.6

VE220-60 2.84 2.26 20.3 2.65 6.5

VE320-30 3.46 3.05 12.0 3.18 8.2

VE320-40 3.27 2.80 14.5 3.00 8.2

VE320-50 3.10 2.56 17.5 2.84 8.4

VE320-60 2.99 2.33 22.1 2.68 10.4

VE460-30 3.49 3.12 10.4 3.23 7.3

VE460-40 3.30 2.90 12.1 3.06 7.0

VE460-50 3.19 2.68 16.1 2.91 9.0

VE460-60 3.13 2.47 21.2 2.75 12.4

J Mater Sci (2014) 49:180–190 187

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Figure 10a, b shows the variation of the dielectric con-

stant of syntactic foams with respect to the GMB volume

fraction and wall thickness, respectively. It can be observed

that with the appropriate selection of GMB volume fraction

and g, the dielectric constant of syntactic foams can be

tailored over a wide range of 2.6–4.9. Thin-walled GMBs,

especially at g[ 0.7, can result in syntactic foams with

dielectric constant lower than that of the neat resin. This

trend reverses for GMBs of g\ 0.7.

The specific dielectric constant normalized with respect

to the syntactic foam density is presented in Fig. 11. The

figure shows that the density decreases more rapidly than

the dielectric constant of syntactic foams. Therefore,

GMBs of lower wall thickness provide higher specific

dielectric constant. Figure 11b also shows that the effect of

wall thickness variation on specific dielectric constant

becomes negligible at g[ 0.7 for all GMB volume frac-

tions. This observation is very important in evaluating the

weight saving obtained by using certain combination of

GMB volume fraction and g. In syntactic foams, Umb and gcan be independently varied. Several combinations of Umb

and g can provide the same syntactic foam density.

Therefore, the relation between dielectric constant, Umb

and g is further evaluated in Fig. 12.

Figure 12a shows a 3D contour plot of dielectric constant

variation along the vertical axis, whereas the floor of the

graph contains a 2D contour plot of syntactic foam density

variation with respect to GMB volume fraction and gmarked on two different axes. The 2D contour plot is shown

again in Fig. 12b for clarity, where axes are marked in terms

of Umb and g to be more illustrative of the contribution of

these parameters on syntactic foam density. The scale bar in

Fig. 12a represents both dielectric constant and density of

syntactic foams in their appropriate units for parity.

Two representative values of the dielectric constant,

namely e = 2.75 and 3, are denoted by a solid and a dashed

line, respectively, in Fig. 12a. All the syntactic foam

compositions along a given line have the same dielectric

constant. These two lines are also projected on the density

contour plot. It is noted that syntactic foams having den-

sities in the range 630–740 kg/m3 can be selected to have

the same dielectric constant of 2.75. The syntactic foams

(a)

(b)

Fig. 9 Comparison of theoretically calculated e using J–S model.

Parametric study for the values of a eg ¼ 5:6; 8; 10; and 12.15 and

taking em ¼ 3:87 and b em ¼ 3:87; 4; 4:1; and 4.2 and taking eg ¼ 5:6:

(a) (b)

Fig. 10 Dielectric constant of syntactic foams as a function of a volume fraction and b radius ratios of GMBs

188 J Mater Sci (2014) 49:180–190

123

having this range of densities have GMB volume fraction

in the range of 0.5–0.6 and g in the range 0.987–0.936.

Since the mechanical properties of syntactic foams are also

related to Umb and g; the flexibility of selecting these

parameters in a range, instead of a fixed value, provides the

possibility of independently tailoring the dielectric constant

and modulus within the ranges given by such charts.

Similar observations can also be made for the second

representative value of dielectric constant of 3, where

syntactic foam density can vary in the range 790–930 kg/

m3, which can be obtained by GMB volume fractions of

0.35–0.5 and g of 0.987–0.898. Most widely used GMBs in

syntactic foams are within these ranges of Umb and g.

Appropriate combinations of Umb and g can be selected to

have the same syntactic foam density and dielectric con-

stant, while having the desired mechanical properties.

Extensive literature is also available for the variation of

mechanical properties such as the modulus and Poisson’s

ratio with respect to Umb and g [12, 18, 25]. This knowl-

edge of both mechanical and electrical properties will be

helpful in designing the syntactic foams based on the

desired application.

Conclusions

The dielectric properties of vinyl ester/glass microballoon

syntactic foams have been studied. The dielectric constant

of syntactic foams is found to decrease with increase in the

volume fraction of the microballoons and increase with

increase in the wall thickness of the microballoons. The

dependence of syntactic foam dielectric constant on mi-

croballoon volume fraction is much stronger compared to

the wall thickness.

(a) (b)

Fig. 11 Variation of specific dielectric constant (normalized with respect to the syntactic foam density) as a function of a volume fraction and

b the radius ratios of GMBs

(a)

(b)

Fig. 12 a Contour plot of variation of the dielectric constant with

respect to the syntactic foam density. The syntactic foam density is

varied with respect to both the volume fraction and the radius ratio of

the microballoons. The scale bar represents both the dielectric

constant and the foams density in their respective units. b The

syntactic foams density plotted as a function of volume fraction and

the radius ratio of the microballoons. The scale bar represents density

in kg/m3

J Mater Sci (2014) 49:180–190 189

123

A linear relation is found between the dielectric constant

and density of the syntactic foams at all testing frequencies.

Increase in the testing frequency leads to reduced dielectric

constant of the syntactic foams. An equivalent sphere

approach is used to obtain the dielectric constant of the

hollow glass microballoons. This closed form expression is

used in coherence with the Maxwell–Garnett and the J–S

models to obtain predictions of the dielectric constant of

the syntactic foams. The J–S model predictions were closer

to the experimental values. The J–S model was also vali-

dated with data obtained from another published study. The

J–S model was then used to perform a parametric study to

understand the relations between syntactic foam dielectric

constant with microballoon volume fraction and wall

thickness. The study shows that several compositions of

syntactic foams can be developed to obtain the same

dielectric constant. This approach helps in tailoring the

dielectric constant independent of other properties and

density of syntactic foams over a wide range.

Acknowledgements The work is supported by the Office of Naval

Research grant N00014-10-1-0988 with Dr. Yapa D.S. Rajapakse as

the program manager. The authors thank the Department of

Mechanical and Aerospace Engineering for providing facilities and

support. Mohammed Omar is thanked for help in specimen prepara-

tion. Youngsu Cha and Linfeng Shen are thanked for helping with the

experiment.

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