A Review of Thermal Conductivity of Polymer Matrix ... · thermal expansion (CTE) ... literature is available on thermal conductivity of solid-particle-ﬁlled composites.13 ... theoretical

A Review of Thermal Conductivity of Polymer Matrix SyntacticFoams—Effect of Hollow Particle Wall Thickness and VolumeFraction

NIKHIL GUPTA1,2 and DINESH PINISETTY1

1.—Composite Materials and Mechanics Laboratory, Mechanical and Aerospace EngineeringDepartment, Polytechnic Institute of New York University, 6 MetroTech Center, Brooklyn,NY 11201, USA. 2.—e-mail: [email protected]

Hollow-particle-filled composites called syntactic foams are lightweightparticulate composites that are useful in weight-sensitive applications such asaerospace and marine structures. Extensive literature is now available on themechanical properties of syntactic foams. The upcoming applications forsyntactic foams in aerospace structures require understanding of their ther-mal properties, such as the thermal conductivity. The present review articlesummarizes the available experimental results and theoretical models relatedto the thermal conductivity of syntactic foams. Experimental results areavailable for only a few compositions of syntactic foams. Basic understating ofthe relationship between thermal conductivity of syntactic foams and thematerial parameters, such as hollow particle volume fraction and wall thick-ness, is not available through experimental results at this point. Four theo-retical models are tested with the experimental data and found to provideclose predictions. These models are used to conduct parametric studies. It isobserved that the thermal conductivity of syntactic foams decreases as thevolume fraction of thin-walled particles is increased. An inverse relationship isobserved for thick-walled, hollow-particle-filled syntactic foams. These modelscan help in designing syntactic foams with required thermal conductivity.

INTRODUCTION

Hollow-particle-filled polymer matrix composites,called syntactic foams, are used in weight-sensitiveapplications. Syntactic foams are now used in anumber of applications, including in underwatervehicles, boat structures, and aircraft structures.Extensive literature is available on the mechanicalproperties of syntactic foams.1–3 Recent efforts havebeen focused on developing correlations between themechanical properties of syntactic foams and thewall thickness and volume fraction of hollow parti-cles used in their structure.4,5

Applications of syntactic foams are increasingwhere their thermal properties are of interest. Theiruse in deep sea oil pipelines as insulation materialrequires control over their thermal conductivity andthermal expansion.6,7 Use of syntactic foams inaircraft structures as core materials in sandwich

composites requires matching of the coefficient ofthermal expansion (CTE) of the core and skins tominimize interfacial stresses, which has motivatedcharacterization of their CTE.8,9 Four parametersare important in tailoring the thermal properties ofsyntactic foams: matrix resin material, hollow par-ticle material, ratio of wall thickness to radius ofhollow particles, and particle volume fraction.Available results show that several compositions ofsyntactic foams can have the same CTE value.8 Thisfinding enables the possibility of selecting a com-position that also has the required mechanicalproperties suitable for the application.

Temperature and frequency dependence ofmechanical behavior of syntactic foams is also foundstudied.10–12 Storage modulus, loss modulus, andloss tangent have been characterized with respectto temperature in these studies, and structure–property correlations have been developed. Such

JOM, Vol. 65, No. 2, 2013

DOI: 10.1007/s11837-012-0512-0� 2012 TMS

234 (Published online December 8, 2012)

information is useful for a wide range of applica-tions where complex environmental and loadingconditions are encountered.

Use of syntactic foams as insulation material isnow increasing, which requires understanding oftheir structure–property relations for thermal con-ductivity. The present review summarizes the avail-able experimental results and theoretical models forthermal conductivity of syntactic foams. Extensiveliterature is available on thermal conductivity ofsolid-particle-filled composites.13–15 The modelsapplicable to solid-particle-filled composites requiremodification to account for the void present inside thehollow particle filler. Models are also available forfoams containing closed-cell gas porosity. Applicationof these models to syntactic foams requires account-ing for the hollow particle that reinforces each pore ofthe foam. Five models capable of estimating thethermal conductivity of syntactic foams are dis-cussed. Four of these models are selected for experi-mental validation and parametric studies.

MICROSTRUCTURAL ASPECTS

Schematic representation of a syntactic foammicrostructure is shown in Fig. 1a. It mainly con-sists of three phases: matrix resin, hollow particles,and porosity. A typical syntactic foam microstruc-ture is shown in Fig. 1b. In this figure, a glass hol-low-particle-filled vinyl ester matrix syntactic foamis shown.16 Some air voids may be present in thematrix. However, the presence of these voids isneglected in the present discussion. Most availablereferences classify syntactic foams as two-phasemicrostructures comprising matrix resin and hollowparticles. Studies related to thermal conductivitymay benefit from classifying them as three-phasematerial, where hollow particles themselves areconsidered two-phase materials comprising theparticle shell and the gas inside the particle.

Hollow particles, also called microballoons ormicrospheres in various references, may have alarge distribution in diameter and wall thickness.Figure 2a shows a scanning electron micrograph ofa batch of widely used glass particles that arecommercially available. The diameter of particles isobserved to vary in the range of 1–250 lm.Figure 2b schematically illustrates the possible distri-bution in the particle size, which can be determinedusing a particle size analyzer. Narrow size ranges canbe separated using a sieve shaker. Since these parti-cles are hollow, their wall thickness may also showvariation within one batch of particles. Figure 2cillustrates the possibility that even the particles of thesame diameter may have different wall thick-nesses.17–20 Therefore, only size analysis is not enoughto characterize their geometrical properties. The wallthickness makes a difference in the true particle den-sity of the same size particles. The particles of differentwall thicknesses can be separated by density-basedclassification methods.

Thin-walled particles are used in synthesizingsyntactic foams to obtain benefit of weight saving.Figure 3 shows a typical glass hollow particle,where more than 90% volume is the void. Thepresent review will focus on analyzing the availabletheoretical models to understand the effect of par-ticle wall thickness and volume fraction on thethermal conductivity of syntactic foams.

Particle size is another important parameter inparticulate composites. The effect of particle size isnot straightforward to understand. Several theo-retical models are based on the unit cell approach,which takes a single particle in a unit cell of thematrix. Change in the size is associated with changein the particle volume fraction (U) in these modelsso these parameters are not independent in thesemodels. In reality, the change in the particle size atthe same volume fraction level is associated withtwo parameters: (I) particle–matrix interfacial areaand (II) curvature effects leading to stress concen-tration. The effect of particle size and interfacial

(a)

(b)Fig. 1. (a) Schematic representation of the microstructure of syn-tactic foam and (b) micrograph of a vinyl ester-glass hollow particlesyntactic foam.

A Review of Thermal Conductivity of Polymer Matrix Syntactic Foams—Effect of Hollow ParticleWall Thickness and Volume Fraction

235

bonding are also modeled in several studies.21–23 Inthe case of syntactic foams, the ratio of particle wallthickness to radius is an important parameter that

has been widely studied. The ratio of internal (ri) toouter (ro) radius of particle is termed the radiusratio g. The particle wall thickness is defined asd = ro(1 � g). The thermal conductivity of syntacticfoams will be analyzed with respect to g and U.

EXPERIMENTAL TRENDS

Polypropylene Matrix Glass Hollow ParticleSyntactic Foams

The thermal conductivity of polypropylene-glasssyntactic foams was found to decrease withincreasing particle volume fraction.9 The thermalconductivity of the neat resin (0.24 W/m K) washigher than any composition of the syntactic foam.This study tested syntactic foams containing up to20 vol.% of two types of hollow particles. Theseparticles, designated as TK35 and TK70, had aver-age sizes of 35 lm and 70 lm and true particledensities of 0.68 g/cc and 0.21 g/cc, respectively. Thecharacteristics of these particles are presented inTable I, considering the glass density of 2.21 g/cc.Both types of hollow particle have values of g over0.88.

Within this range of compositions, the decrease inthermal conductivity of syntactic foams with respectto particle volume fraction is nearly linear and thefoam containing 20 vol.% particles had thermalconductivity of 0.191 W/m K.

APO-Bismaleimide Matrix Carbon HollowParticle Syntactic Foams

APO-bismaleimide (APO-BMI) resin is taken asthe matrix material and carbon hollow particles aretaken as filler.24 The thermal conductivity of APO-BMI and carbon are given as 0.307 W/m K and4.6 W/m K, respectively. The particles have ther-mal conductivity 15 times higher than that of thematrix. In this case, it is expected that incorpora-tion of carbon particles in the APO-BMI matrix willresult in syntactic foams with thermal conductivityhigher than that of the matrix resin. However, thecontribution of particle–matrix interfacial bondingon the effective thermal conductivity of syntacticfoam is also important and poor bonding will resultin lower thermal conductivity than the expectedvalue.

The experimental results of this study show thatthe room temperature thermal conductivity of syn-tactic foams containing 59–70 vol.% carbon particles

Fig. 2. (a) A sample of widely used glass hollow particles. (b) Illus-tration of possible distribution in the (b) diameter and (c) wall thick-ness of the same diameter hollow particles used in syntactic foams.

Fig. 3. A broken hollow particle showing thin wall and the void spacethat was enclosed inside the glass shell..

Table I. Properties of glass hollow particles used inRef. 9

HollowParticle Type

True ParticleDensity (g/cc)

Diameter(lm) g d (lm)

TK35 0.68 35 0.885 2.02TK70 0.21 70 0.967 1.15

Gupta and Pinisetty236

ranges from 0.137 W/m K to 0.284 W/m K. Thethermal conductivity of all compositions is found tobe lower than that of the resin, despite havinghigher thermal conductivity filler particles. Thisreinforces the importance of particle–matrix inter-facial bonding. This study also found that thethermal conductivity increases with temperature.

MODELING

To model the thermal conductivity, syntacticfoams must be considered to have a three-phasemicrostructure comprising the matrix resin, hollowparticle shell, and the gas inside the hollow particle.Composite sphere or core–shell sphere based modelsdeveloped for particulate composites can be easilyapplied to syntactic foams.25 The gas-filled void andthe hollow particle shell can be considered as twoconcentric spheres, and most of the available modelscan be modified to suit syntactic foams. Before dis-cussing the analytical approaches, a preliminaryinquiry can be conducted in the thermal conductiv-ity of syntactic foams based on the properties of theconstituting materials.

A primary insight into the potential trends can beobtained by evaluating the typical values of thermalconductivities as 0.2 W/m K, 0.179 W/m K, and0.022 W/m K for polypropylene resin, glass particleshell, and the gas inside the particles, respec-tively.26 In this case, the thermal conductivity of theparticle material is lower than that of the resin, butthe difference is very small. On the contrary, thethermal conductivity values for vinyl ester, glassmaterial, and gas are reported as 0.25 W/m K,1.1 W/m K, and 0.01 W/m K, respectively, wherethe thermal conductivity of glass is 4.4 times that ofthe resin.27 In these two cases, the mechanism ofheat transfer inside the syntactic foam microstruc-ture will be different because the particles will forma preferred path when their thermal conductivity ishigher than that of the matrix. It is also previouslyreported that the convection effects within the

hollow particles are expected to be negligiblebecause of their small size.28

Available Theoretical Models for ThermalConductivity

Liang Model

Liang proposed a unit-cell–based model (Fig. 4)containing a single particle embedded in the resin

to estimate the thermal conductivity of syntacticfoams.26,28 This model is based on rule of mixturesapproach. As a result of the linear distribution oftemperature, the average thermal conductivity oftwo regions of only the matrix, marked as Zone 1 inFig. 4, is given as

�k1 ¼ kp (1)

where kp is the thermal conductivity of the matrixpolymer. The average thermal conductivity of Zone2, where matrix, particle shell, and gas are present,can be given as

�k2 ¼1

h2SkpVp þ kgVg þ kaVa

� �(2)

where Vp, Vg, and Va are the volumes of the polymermatrix, hollow particle shell, and gas void, respec-tively, and S is the surface area of the entire crosssection. In addition, kg and ka are the thermal con-ductivities of the hollow particle shell material andgas phase, respectively. The thermal conductivitiesfor Zones 1 and 2 can be used in rule of mixtures toobtain the effective thermal conductivity of syntac-tic foams as28

where qg, qa, and qs are the effective densities of thehollow particle shell, gas phase, and hollow particle,respectively. The single particle models in a cubicunit cell suffer from the disadvantage that themaximum possible particle volume fraction is about0.52. Also, the predictions start to deviate fromthe experimental values at high particle volumefractions. In syntactic foams, the weight-savingconsideration commonly leads to particle volume

k ¼ 1

kp1� 6U

p

� �1=3 !

þ 2 kp4p3U

� �1=3

þp2U9p

� �1=3

kgqs � qa

qg � qa

!

þ ka

qg � qs

qg � qa

!

� kp

! !�12

4

3

5

�1

(3)

Fig. 4. Unit cell geometry and notations used in Refs. 26 and 28.


237

fractions in the range of 40–65%, and the utility ofsuch models may be limited without their furtherextension.

Felske Model

Models based on a self-consistent approach arealso available.29 In this approach, two concentricparticles, corresponding to the hollow particle shelland gas void, are assumed to be embedded in thematrix resin. A temperature difference is imposedacross the composite slab. The ratio of the averageflux to the average temperature gradient in the unitcell defines the effective thermal conductivity of thehomogeneous medium. Considering the unit celland the notations shown in Fig. 5, the effectivethermal conductivity of syntactic foam is found as29

k ¼ 2 1� Uð ÞF � 3b2n2þ Uð ÞF � 3b2n

� �kp (4)

where functions F, n, and b2 are given as

F � ka

kg� 1

� �2� b2 2

kg

kpþ 1

� �� þ ka

kgþ 2

� �

1� b2 2kg

kp� 1

� �� 1

g

� �3 (5)

n ¼ 2 1� 1

g3

� �kg

kp

� �� 2þ 1

g3

� �ka

kp

� �(6)

b2 ¼k21r2

kg(7)

The term k21 is defined as contact conductance atthe particle–matrix interface. For ideal thermal

contact at the interface, the term b2 !1 and theequations can be further simplified. Since the ther-mal conductivity of air is much lower than that of theparticle material (glass or carbon), further simplifi-cation of Eq. 4 can be conducted by consideringkg � ka and kp � ka. In this case, we can assume

ka

kg¼ ka

kp¼ 0 (8)

The modified form of Eq. 4 with these conditions is

k ¼ 1þ 2XU1� XU

� �kp (9)

where parameters X and c are defined as X ¼ c�1cþ2

�

and

c ¼ 2kg

kp

� �1� g3

2þ g3

� �(10)

Equation 9 has a simple form and can be applied tosyntactic foams. This model can provide reasonablepredictions for syntactic foams containing a low-volume fraction of hollow particles.

As the volume fraction of particles in syntacticfoam increases, the particle-to-particle contact andcreation of a preferred path for heat conduction maybecome a significant issue, especially if the thermalconductivity of particle material is higher than thatof the matrix. This is where the model predictionsstart to deviate from the experimental results.

Pal Model

Application of differential schemes can be usefulin developing approaches that can predict thethermal conductivity of syntactic foams containinghigh-volume fractions of hollow particles. Thedifferential effective medium approach iterativelyadds a small volume fraction of hollow particles inthe matrix, finds the effective thermal conductivityof the solution, and then uses this value as theproperty of the matrix in the next iteration.25 Theiterations can be continued until the desired volumefraction of hollow particles is reached in the struc-ture. This approach provides two important benefitsfor modeling syntactic foams: (I) the maximumallowed particle volume fraction can be built in themodel and (II) hollow particles of different size andwall thickness can be incorporated in differentiterations. The second advantage is particularlyimportant for syntactic foams where polydispersionin the particle size and wall thickness are routinelyobserved and need to be accounted for in order toobtain realistic model predictions.17 The polydis-persivity in the properties of commercially availablehollow particles that have been widely used inmanufacturing syntactic foams have been studiedpreviously and experimental data are available.16

r1

r2

r3

1: Matrix2: Particle shell3: Gas void1

2

3

z

y

Fig. 5. Unit cell geometry and notations used in Ref. 29.


A significant issue in particulate composites is thepacking factor. Generally, the spheres of the samesize in an ideal packing arrangement can have vol-umetric packing efficiency as high as 74%. In ran-domly arranged particles of the same size, thepacking efficiency is close to 64%.30,31 The modelsdiscussed earlier are not able to place a limit auto-matically on the particle volume fraction.

The model presented by Eq. 4 was modified by Palto apply the differential scheme and introduce themaximum particle packing factor Umax.25 In thisscheme, Eq. 4 was converted to a differential equa-tion assuming that the existing hollow particle vol-ume fraction in the matrix is U and the incrementalincrease in the volume fraction in each iteration is

dU1�U=Umax

� . Upon integration with the limits k! km

and U! 0, the final form is obtained as

k

kp

� �1=3 b� 1

b� k

kp

� �

" #

¼ 1� UUmax

� ��Umax

(11)

where function b is defined as

b ¼ kg

kp

2g3 þ 1� �

ka � 2 g3 � 1� �

kg

g3 þ 2ð Þ � g3 � 1ð Þka(12)

This modified model successfully captured theexperimental trends observed in solid-particle-filledcomposites at high-volume fractions.15,25 Experi-mental data on thermal conductivity of syntacticfoams are very limited; hence, detailed validation ofthis model is not yet conducted. It should also benoted that this model presents an implicit rela-tionship and that it needs to be solved numerically.

Porfiri Model

Homogenization techniques are attractive in find-ing the effective properties of composite materials.A closed-form solution based on homogenizationtechniques is available for syntactic foams containingthin-walled particles in low-volume fractions.27 Thesame three-phase microstructure is assumed in thismodel with hollow particles modeled as core–shellparticles. The thermal conductivity of syntacticfoams is given as

k

kp¼1�

3U kp � ka

� �

2kp þ ka � U ka � kp

� �

þ 9Ukp

kg

2k2g � kakg � k2

a

2kp þ Ukp þ ka 1� Uð Þ� �2

1� gð Þ(13)

The first two summands in Eq. 13 correspond to theMaxwell–Garnett formula to calculate the thermalconductivity of a two-phase composite with solidparticle inclusions, and the last summand is thecorrection term to account for the presence of a thinparticle shell.

Park Model

Estimation of effective thermal conductivity ofhollow particles is useful for syntactic foams.Equations developed for coated solid particles canbe used for hollow particles.32 Based on thisapproach, the thermal conductivity of a hollowparticle can be determined by33

kmb ¼ kg2ð1� g3Þkg þ ð1þ 2g3Þka

ð2þ g3Þkg þ ð1� g3Þka

� �(14)

The prediction of Eq. 14 can be very useful insimplifying the computation of thermal conductivityof syntactic foams. In this case, once the effectivethermal conductivity of hollow particles is deter-mined, they can be replaced in the matrix byequivalent solid particles. Thus, the problemreduces to having solid particles in a polymericmatrix. Numerous theoretical models are availablefor solving the problem of solid-particle-filled com-posites. The thermal conductivity of syntactic foamsusing Eshelby’s method can be given as33

k ¼ 2ð1� UÞkp þ ð1þ 2UÞkmb

ð2þ UÞkp þ ð1� UÞkmb

� �kp (15)

Equation 14 is used to determine the effectivethermal conductivity with respect to the radius ratiofor hollow glass particles in Fig. 6. The parametersused in calculation of kmb are kg = 1.1 W/m K andka = 0.1 W/m K. Thin-walled particles are widelyused in manufacturing syntactic foams. It can beobserved in the figure that the thermal conductivityof thin-walled hollow particles changes rapidly withg. The effective thermal conductivity of hollow par-ticles can be as low as 10% of the thermal conduc-tivity of the glass material.

Fig. 6. Prediction of relative thermal conductivity of glass hollowparticles


239

Model Validation with Experimental Data

Experimental values of the thermal conductivityof syntactic foams are available in the publishedliterature.9 Available experimental data on poly-propylene-glass syntactic foams are used to validatethe predictions obtained from Liang, Felske, Porfiri,and Park models. The experimental data are avail-able on syntactic foams containing two types ofparticles, the details of which are presented inTable I and other input parameters are listed in

Table II. The experimental values of thermal con-ductivities of eight compositions of syntactic foamsand theoretical predictions obtained from the fourmodels are presented in Table III. The differencebetween experimental results and theoretical pre-dictions is presented in Table IV. It can be notedthat predictions obtained from all models are close.Except for Liang’s model for one composition, theo-retical predictions of all models show less than 10%difference from the experimental values. The limi-tation of this validation scheme is that the experi-mental data are available only for one matrix andparticle material. Although it is desirable to vali-date the models for data from thick-walled particles,such data are not available in the literature becausethe weight-saving benefit is lost due to the highdensity of the thick-walled particles. Most availablestudies have used hollow particles in the range0.85< g< 1.34–36

Parametric Studies on Thermal Conductivity

A general trend in the thermal conductivitywith respect to g and U can be observed through

Table II. Input parameters used for the validationof theoretical models and for the parametric study

ParameterValidation with

Liang and Li Data9 Parametric Study

kg (W/m K) 0.179 1.1ka (W/m K) 0.0228 0.1kp (W/m K) 0.24 0.25qa (kg/m3) 0.09 1.2qg (kg/m3) 2210 2540qs (kg/m3) 680 qg(1 � g3)

Table III. Comparison of experimental values of thermal conductivity with theoretical predictions

Syntactic Foam U (%)

k (W/m K)

(Liang and Li)a9 Liang Model Felske Model Porfiri Model Park Model

PP 0 0.24 – – – –PP/TK35 5 0.23 0.231 0.227 0.229 0.228

10 0.23 0.222 0.214 0.217 0.21715 0.21 0.213 0.201 0.207 0.20620 0.18 0.204 0.189 0.196 0.195

PP/TK70 5 0.22 0.229 0.224 0.226 0.22610 0.21 0.218 0.208 0.212 0.21215 0.21 0.208 0.193 0.199 0.19920 0.19 0.197 0.179 0.187 0.186

aExperimental values.

Table IV. Difference between experimental values of thermal conductivity with theoretical predictions

Syntactic foam U (%) Liang Model (%) Felske Model (%) Porfiri Model (%) Park Model (%)

PP 0 – – – –PP/TK35 5 0 1 0 1

10 4 7 6 615 1 4 1 220 13 5 9 8

PP/TK70 5 4 2 3 310 4 1 1 115 1 8 5 520 4 6 2 2


parametric studies. Liang, Felske, Porfiri, and Parkmodels are used to plot the thermal conductivityvalues with respect to g and U, as shown in Figs. 7and 8, respectively. These graphs are plotted forvinyl ester matrix-glass hollow particle syntacticfoams. The input parameters for this study aregiven in Table II.27 The theoretical values are plottedup to U = 0.5 because some of the models use a par-ticle inside a cubic unit cell.

It can be noted in Fig. 7 that the trends obtainedfrom the Liang, Felske, and Park models are verysimilar. Note that the y-axis is different in each partof this figure. The Porfiri model shows notably dif-ferent trends, especially for thick-walled, particle-filled syntactic foams. The thermal conductivityvalues predicted from this model are significantlyhigher compared with the other three models forsuch syntactic foams. However, neither experimen-tal results are available for syntactic foams

containing such particles, nor is there interest insuch syntactic foams due to their heavy weight.Therefore, these differences have only a little prac-tical significance. It can also be noted in Fig. 7 thatthe thermal conductivity of syntactic foams con-taining solid glass particles can be 2.5–7 timeshigher than that of the matrix resin.

It can also be observed in Fig. 7 that the thermalconductivity increases with volume fraction of thick-walled particles. However, the thermal conductivityof syntactic foams containing thin-walled particlesdecreases with increasing U because the volume ofgas is sufficiently high in their structure. The trendsof decrease and increase in the thermal conductivityof syntactic foams with respect to U for thin- andthick-walled hollow particles are well captured byall models.

The thermal conductivity data are plotted inFig. 8 to clearly observe the relation with respect to

Fig. 7. Variation of normalized thermal conductivity of syntactic foams as a function of hollow particle volume fraction at various radius ratiosobtained from (a) Liang, (b) Felske, (c) Porfiri, and (d) Park models.


241

g. Apart from the observations already made inFig. 7, it is interesting to note that the thermalconductivity versus g graphs show that the curvesfor various U values cross over. It should be notedthat the crossover does not happen at a single gvalue but over a narrow range. The approximate gvalues (gc) for crossover are:

Liang model: gc: 0.946Felske model: gc: 0.883Porfiri model: gc: 0.935Park model: gc: 0.925The gc values predicted by the Liang, Porfiri,

and Park models are close to each other. Fig-ure 9 shows a close-up of the region g = 0.85 � 1,which belongs to thin-walled particles commonlyused in syntactic foams. In the region g< gc, thethermal conductivity of syntactic foams in-creases with U. This behavior reverses in theregion g > gc, and syntactic foams with lower U

have higher thermal conductivity. Knowing thiskind of behavior is important because if the U ismonotonically increased to lower the syntacticfoam density, then the thermal conductivity ofthin- and thick-walled particle reinforced syn-tactic foams will first become equal and thenreverse the trend, which may be undesired bythe application.

Weight saving is the most important considerationin using syntactic foams. Figure 10 shows the profileof specific thermal conductivity of syntactic foamswith respect to g and U obtained from the fourselected theoretical models. The figure is plotted inthe region g = 0.8 � 1 to show the range relevant tothe syntactic foams parameters used in actualapplications. It can be observed that the profilespredicted by various models are different. Validationof these profiles with experimental results canfurther help in selecting the model that is the best in

Fig. 8. Variation of normalized thermal conductivity of syntactic foams as a function of hollow particle radius ratio at various volume fractionsobtained from (a) Liang, (b) Felske, (c) Porfiri, and (d) Park models.


describing the thermal conductivity of syntacticfoams.

SUMMARY

Five existing theoretical models that can predictthe thermal conductivity of syntactic foams arereviewed. The basic assumptions, applicability, andlimitations of these models are discussed. Four ofthese models are validated with available experi-mental results on polypropylene matrix glass hollowparticle syntactic foams. The predictions of allmodels for this available data set are in closeagreement. The models are further analyzed byconducting parametric studies on vinyl ester-glasssyntactic foams to determine thermal conductivitytrends with respect to hollow particle wall thickness

and volume fraction. The following trends areidentified from the results:

� The thermal conductivity of syntactic foamsdecreases with

– Increasing volume fraction of thin-walled par-ticles

– Decreasing volume fraction of thick-walledparticles

� A critical radius ratio of hollow particles can befound where the thermal conductivity of syntacticfoams is nearly the same as that of the matrix atall volume fraction levels

The availability of these models can help indesigning syntactic foams with a desired set ofthermal and mechanical properties.

Fig. 9. The region 0.85< g< 1 magnified from Fig. 8 for (a) Liang, (b) Felske, (c) Porfiri, and (d) Park models.


243

ACKNOWLEDGEMENTS

This work is supported by the Office of NavalResearch grant N00014-10-1-0988 and the U.S.Army Research Laboratory Cooperative WorkingAgreement W911NF-11-2-0096 with NYU-Poly. Theviews and conclusions are those of the authors andshould not be interpreted as presenting the officialpolicies or position, either expressed or implied, ofthe ONR, the ARL, or the U.S. Government unlessso designated by other authorized documents. Ron-ald L. Poveda is thanked for preparing solid modelsand illustrations, and Vasanth ChakravarthyShunmugasamy is thanked for help with figurepreparation.

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