research article effect of micro- to nanosize inclusions upon the thermal...

Research ArticleEffect of Micro- to Nanosize Inclusions upon theThermal Conductivity of Powdered Composites withHigh and Low Interface Resistance

Muhammad Zain-ul-abdein1 Waqas S Awan1 Hassan Ijaz1 Aqeel A Taimoor2

Ayyaz Muhammad23 and Sami ullah Rather2

1Mechanical Engineering Department University of Jeddah Jeddah Saudi Arabia2Department of Chemical and Materials Engineering King Abdulaziz University Jeddah Saudi Arabia3Institute of Chemical Engineering and Technology University of Punjab Lahore Pakistan

Correspondence should be addressed to Muhammad Zain-ul-abdein mzainulabdeingmailcom

Received 17 June 2015 Accepted 28 September 2015

Academic Editor Christian Brosseau

Copyright copy 2015 Muhammad Zain-ul-abdein et al This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

Materials for thermal management application require better control over the thermophysical properties which has largely beenachieved by fabricating powdered composite There are however several factors like filler volume fraction shape morphologyinclusion size and interfacial thermal resistance that limit the effective properties of themediumThis paper presents amethodologyto estimate the effective thermal conductivity of powdered composites where the filler material is more conductive than the matrixOnly a few theoretical models such as Hasselman and Johnson (HJ) model include the effect of interfacial resistance in theirformulation Nevertheless HJ model does not specify the nature of the interfacial thermal resistance Although Sevostianov andKachanov (SK) method takes care of interface thickness they on the other hand have not taken into account the interfacialresistance due to atomic imperfections In the present work HJ model has been modified using SK method and the results werecompared with experimental ones from the literature It has been found that the effect of interfacial resistance is significant in highlyresistive medium at microscale compared to nanoscale such as Cudiamond system while in a highly conductive medium likebakelitegraphite system the effect of shape factor is more significant than interfacial thermal resistance

1 Introduction

Thermal conductivity is one of the fundamental propertiesof key importance in the energy applications where vari-able (fastslow) heat dissipationstorage is often requiredAlthough metals and alloys provide a large range of con-ductivities to be used in several electronic and mechanicalappliances the quest for the search of new composite materi-als with modifiable properties has still been unquenched Inaddition to fabricating the two-phase composites with differ-ent inclusions like metal matrix composite (MMC) polymermatrix composite (PMC) and so forth the researchershave shown growing inclination towards manipulating theinclusion size It is therefore of interest to be able to observe

the effective conductive properties of a given composite frommicro- to nanoregimes

In the current framework Clausius (1879) [1] andMaxwell (1884) [2] introduced one of the earliest mod-els based upon self-consistent scheme (SCS) Both modelsattempted to predict the effective properties of a two-phasemedium where the inclusions in dilute concentration weretreated as isolated particles immersed into a given matrixBruggeman (1935) [3] later on implemented an incrementalscheme called differential scheme (DS) or effective mediumscheme (EMS) The basic idea was to introduce the secondphase particle into a homogenousmatrix in small incrementsand estimate the overall properties of the resulting mediumNevertheless the very first upper and lower bounds based

Hindawi Publishing CorporationJournal of NanomaterialsVolume 2015 Article ID 843914 8 pageshttpdxdoiorg1011552015843914

2 Journal of Nanomaterials

only upon volume fractions of matrix and inclusion wereformulated by Wiener (1912) [4] The Wiener upper bound(UB) is still widely used by the researchers as the ldquorule ofmixturesrdquo Assuming macroscopically isotropic behavior ofthe composite Hashin and Shtrikman (1962) [5] defined amore conservative set of upper and lower bounds For acomposite with spherical inclusions the Hashin-Shtrikman(HS) lower bound (LB) coincides with Maxwell and Moriand Tanka (1973) [6] models Mori-Tanaka (MT) howeveralso introduced depolarization factors in their formulationto accommodate the prolate and oblate shape morphologiesSimilar to the Kanaun and Levin (1994) [7] scheme MTscheme is merely a variant of effective field (EF) methodwhich treats the inclusion as an isolated particle and assumesthe presence of an effective field over each inhomogeneity(see Markov (2000) [8] for a detailed review) Hamilton andCrosser (1962) [9] modified Maxwell model to incorporatethe shape factor (119899 = 3120595) for nonspherical particles wherethe sphericity (120595) is the ratio of the surface area of spherewith the same volume of the particle to the surface area of theparticle Using 120595 = 1 for a spherical inclusion regenerates theMaxwell model Likewise Zimmerman (1996) [10] exploitedMaxwell and Bruggeman methods and introduced an aspectratio and conductivity ratio dependent parameter (120573) forelliptical inhomogeneities

Most of the above mentioned models assume ideal con-tact between the inclusion and the matrix which impliesthat the heat flow across the interface remains continuousIn reality however the interface regardless of its natureis nonideal and leads to a discontinuous temperature fieldacross the boundary due to interfacial thermal resistancealso known as Kapitzarsquos resistance (after Kapitza (1941) [11])In order to deal with interface behavior Kanaun (1984) [12]introduced the concept of singular inclusion in the elasticitycontext From the perspective of thermal conductivity thenonideal contact may be assumed as either ldquosuperconduc-tiverdquo or ldquoresistiverdquo A superconducting or resisting singularinclusion requires that its conductivity (119896

119904) be 119896

119904rarr infin or

1119896119904

rarr infin respectively provided that one of the inclusionrsquosdimensions is of the order 120575 rarr 0 Torquato and Rintoul(1995) [13] also defined conductive and resistive contacts inrelation to the conductive properties of the interphase layer

between the matrix and inclusion Detailed discussions onconductiveresistive interfacesmay also be found in theworksof Benveniste and Miloh (1986) [14] Benveniste (1987) [15]and Chiew and Glandt (1987) [16]

In the present work a method based on Hasselmanand Johnson (1987) [17] and Sevostianov and Kachanov(2007) [18] models has been proposed to predict the thermalconductivity of particulate filled composites with high andlow interfacial thermal resistance as functions of filler volumefractions inclusion size shape factor and conductivity ofinterphase layer The effect of particle size ranging frommicro- to nanoscale has been investigated in the highlyconductive and resistive media It will be shownwith the helpof case studies that the inclusion size behaves differently inresponse to the interfacial resistance

2 Theoretical Model

21 Effects of Shape Factor and Interfacial Resistance In theirwork Hasselman and Johnson [17] addressed the nonidealcontacts in terms of interfacial thermal resistance (119877) andderived a mathematical model for spherical morphology ofparticles The model states

119896eff = 119896119898

sdot

2 (119896119891119896119898

minus 1 minus (119896119891119903119900) 119877) 120601

119891+ 119896119891119896119898

+ 2 + 2 (119896119891119903119900) 119877

(1 minus 119896119891119896119898

+ (119896119891119903119900) 119877) 120601

119891+ 119896119891119896119898

+ 2 + 2 (119896119891119903119900) 119877

(1)

where 119896119898and 119896

119891are thermal conductivities of matrix and

filler (inclusion) in Wsdotmminus1sdotKminus1 respectively 119903119900is the filler

radius inm 120601119891is the filler volume fraction119877 is the interfacial

resistance in Wminus1sdotm2sdotK and 119896eff is the effective thermalconductivity of the composite 119877 by definition is an inverseof thermal conductance (119862) that is 119877 = 1119862 Note thatHJ model is valid for any system of units the SI units arementioned here only for the readerrsquos convenience Jiajun andXiao-Su (2004) [19] modified the HJ model (1) in order toaccommodate the nonspherical morphology of the particlesThey actually used Hamilton and Crosser [9] shape factorparameter (119899 = 3120595) where 120595 is the sphericity as describedearlier The modified HJ therefore takes the following form

119896eff = 119896119898

119896119891

(1 + (119899 minus 1) (119896119898

119903119900) 119877) + (119899 minus 1) 119896

119898+ (119899 minus 1) 120601

119891(119896119891

(1 minus (119896119898

119903119900) 119877) minus 119896

119898)

119896119891

(1 + (119899 minus 1) (119896119898

119903119900) 119877) + (119899 minus 1) 119896

119898minus 120601119891

(119896119891

(1 minus (119896119898

119903119900) 119877) minus 119896

119898)

(2)

Note that (2) reduces to (1) for spherical inclusions thatis 120595 = 1 and also that the HJ model reproduces theMaxwell model for ideal contact when 119877 = 0 (or 119862 = infin)Conversely for nonideal contacts the interfacial resistancemust be nonzero and the conductance finite It should alsobe mentioned that increasing filler fractions yield eitherdecreasing 119896eff when 119896

119898gt 119862 sdot 119903

119900or increasing 119896eff when

119896119898

lt 119862 sdot 119903119900 Consequently the interfacial thermal resistance

would be substantially large in the former case as in resistive

composites and vanishingly small in the latter case similarto the superconductive composites

22 Effect of Inclusion Size Micro- to Nanoscale Sevostianovand Kachanov [18] recently modified an incremental schemeformulated by Shen and Li (2005) [20] so as to incorporatethe effect of interface layer thickness (ℎ sim 1-2 nm [21])upon the overall properties of the material at nanoscale Itis understood that in the microregime ℎ ≪ 119903

119900 however in

Journal of Nanomaterials 3

h

kf gt km

ki = const

ro

kf

km

ki

(a)

k = f(kf ki)

kmri

k

(b)

Figure 1 SK model (a) interface zone (b) equivalent homogeneous inclusion

case of a nanoinclusion the outer radius 119903119900is comparable to

the interface thickness that is ℎ asymp 119903119900 The basic idea was

to incrementally add the interface thickness to the sphericalinclusion and homogenize at each step until 119903

119900+ ℎ =

119903119894 where 119903

119894is the outer radius of the interface layer (see

Figure 1) Given that the current radius (119903) is 119903119900

lt 119903 lt 119903119894

the conductivity across the interface 119896119894

= 119896119894(119903) and the

conductivity of the equivalent homogeneous radius 119903119894is 119896 =

119896(119903119894) a differential equation with initial condition 119896(119903

119900) =

119896119891 of the following form may be written for the equivalent

homogeneous inclusion

119889119896

119889119903

= minus

1

119903

(119896 (119903) minus 119896119894(119903)) (119896 (119903) + 2119896

119894(119903))

119896119894(119903)

(3)

Setting 119903119900

+ ℎ = 119903119894

= 119903 for the simplest case of ahomogeneous interface zone 119896

119894(119903) = 119896

119894= const integration

of (3) yields

119896 (119903119894)

= 119896119894

[

[

2 + (1 + ℎ119903119900)3

sdot ((2 (119896119894119896119891) + 1) (1 minus (119896

119894119896119891)))

(1 + ℎ119903119900)3

sdot ((2 (119896119894119896119891) + 1) (1 minus (119896

119894119896119891))) minus 1

]

]

(4)

For a general solution of (3) in terms of hypergeometricfunctions the reader is referred to Sevostianov and Kachanov[18] The estimation of effective thermal conductivity of thecomposite requires placement of the equivalent homoge-neous inclusion in the effective field (EF) of heat flux (Levin[22] Kanaun [23] and Kanaun and Levin [7]) Based uponabove discussion Sevostianov-Kachanov (SK) modified the

HS bounds and EF method to calculate the 119896eff as in thefollowing

119896119898

[1 +

31206011015840

(119896 minus 119896119898

)

3119896119898

+ (1 minus 1206011015840) (119896 minus 119896

119898)

] le 119896eff

le 119896 [1 minus

3 (1 minus 1206011015840

) (119896 minus 119896119898

)

3119896 minus 1206011015840(119896 minus 119896

119898)

] (HS bounds)

(5)

119896eff = 119896119898

[1 +

31206011015840

(119896 minus 119896119898

)

3119896119898

+ (1 minus 1206011015840) (119896 minus 119896

119898)

]

(Kanaun-Levin EF method)

(6)

where 1206011015840 is the filler volume fraction including interface layer

Note that the Kanaun-Levin EF method coincides with theMaxwell model and HS lower bound

23 Proposed Scheme Modified HJ and SK Method It hasbeen assumed throughout that 119896

119891gt 119896119898 which in case of

ideal contact would lead to 119896119891

gt 119896eff gt 119896119898 Nonideal contact

however requires that 119896119894be a function of 119896

119898in the limit 119903

119900lt

119903 lt 119903119894 Subject to the condition of a homogeneous interface

zone (119896119894

= const) an interface with very high resistancemay acquire conductivity 119896

119894= 001119896

119898 Likewise for very

low interfacial resistance 119896119894

= 099119896119898 It has been pointed

out earlier that the interface thickness (h) is of the order of1-2 nm which at atomic level means placing almost 10sim20average sized atoms side by side in the thickness directionHere the interface would be comprised of not only filler andmatrix atoms but also the interatomic defects like vacanciesand so forth It further implies that a higher concentration ofdefects would result in an increased interfacial resistance andvice versa (Figure 2)This effectmay be incorporated in termsof 119877 as in modified HJ model (2) The SK method may nowbe employed to calculate 119896eff from (2) as follows

119896eff = 119896119898

119896 (1 + (119899 minus 1) (119896119898

119903119894) 119877) + (119899 minus 1) 119896

119898+ (119899 minus 1) 120601

1015840

(119896 (1 minus (119896119898

119903119894) 119877) minus 119896

119898)

119896 (1 + (119899 minus 1) (119896119898

119903119894) 119877) + (119899 minus 1) 119896

119898minus 1206011015840(119896 (1 minus (119896

119898119903119894) 119877) minus 119896

119898)

(7)


h

km

kf gt km

ki = const

ro

kf

ki

(a)

R

kmri

k = f(kf ki)

k

(b)

Figure 2 Current model (a) interface zone + interatomic defects (b) equivalent homogeneous inclusion + interfacial resistance

where 119896 is a function of 119896119894 119896119891 119903119900 and ℎ defined previously

in (4) Note that the above equation estimates effectiveconductivity considering volume fraction of filler with inter-face layer (120601

1015840

) shape factor (119899) filler matrix and interfaceconductivities (119896

119891 119896119898

119896119894) particle size including interface

thickness (119903119894) and interfacial resistance (119877) Also note that the

factor ℎ119903119900is negligibly small at microlevel while significantly

large at nanoscale It must be mentioned that a size of 01 120583mwas assumed to be the lower limit of micro- and upper limitof nanoinclusions that is 119898119894119888119903119900119904119888119886119897119890 gt 01 120583m gt 119899119886119899119900119904119888119886119897119890

3 Case Studies

The application of the above method will now be presentedin the context of two case studies (i) a metal matrixCudiamond composite with decreasing conductivity as afunction of diamond volume fraction (Hanada et al (2004)[24]) and (ii) a polymer matrix graphitebakelite compositewith increasing conductivity as a function of graphite volumefraction (Azeem and Zain-Ul-abdein (2012) [25]) The twocomposites were chosen as representatives of highly resistiveand conductive systems

31 CopperDiamond Composite High Interfacial ResistanceIn the literature several copperdiamond composites havebeen reported to yield decreasing thermal conductivity withincreasing diamond volume fraction due to high interfacialresistance (Hanada et al [24] Kang et al [26] Shen et al[27] and Raza and Khalid [28]) Given that the thermalconductivities of Cu and diamond are 119896

119898= 400Wmminus1Kminus1

and 119896119891= 2000Wmminus1Kminus1 respectively (4) can be used to

draw 119896119896119891as a function of ℎ119903

119900for different values of 119896

119894

while assuming ℎ = 2 nm (see Figure 3) The curve showsthat as 119903

119900varies from nano- to microsize 119896 approaches 119896

119891

Moreover increase in 119896119894from 001119896

119898to 099119896

119898results in a

significant improvement of 119896 values Since the interface ishighly resistive 119896

119894= 001119896

119898is more likely to yield realistic

0

02

04

06

08

1

0 005 01 015 02 025 03 035 04

Nanosize

Microsize

kk

f

hro

ki = 001kmki = 03kmki = 05km

ki = 07kmki = 099km

Figure 3 Cudiamond composite system 119896119896119891as a function of ℎ119903

119900

(4)

conductivity values for the Cudiamond system within thenano- and microregions

Applying the above condition to the HS lower boundin (5) would result in the effective thermal conductivityof composite Figure 4 illustrates the calculated thermalconductivity as a function of filler (diamond) volume frac-tion as well as the micro- and nanoregimes Note thatthe experimental conductivities for Cudiamond system asmeasured by Hanada et al [24] who actually used irregularshaped diamond particles with an average size of 77 120583m arealso presented Figure 4 however shows that the measuredconductivities lie within the nanoregime and contrary tothe experimental findings a filler size of 77 120583m should yieldincreasing thermal conductivity (see the dotted line labeled77 120583m) Reason for this discrepancy is that the HS lowerbound assumes spherical inclusions and does not take intoaccount the interfacial resistance (119877) It therefore requires


Microregime

02 04 06 08 1 120Filler volume fraction

Effec

tive t

herm

al co

nduc

tivity

(Wm

-K)

100120583m77 120583m01 120583m

1nmHanada et al 2004

0

400

800

1200

1600

2000

Nanoregime

Figure 4 Cudiamond composite system 119896eff using HS LowerBound ((5) 119896

119894= 001119896

119898)

0

02

04

06

08

1

0 005 01 015 02 025 03 035 04

Nanosize

Microsize

kk

f

hro


ki = 07kmki = 099km

Figure 5 Bakelitegraphite composite system 119896119896119891as a function of

ℎ119903119900(4)

modification to incorporate the effect of 119877 which will betreated in Section 4

32 BakeliteGraphite Composite Low Interfacial ResistanceIn a recent study AzeemandZain-Ul-Abdein [25] have inves-tigated effective thermal conductivity of a polymer matrixbakelitegraphite composite Graphite added in increasingvolume fractions was found to yield improved properties ofthe system A similar observation was also made by Ling etal [29] where they used expanded graphite to enhance thethermal conductivity of an organic phase change materialFigure 5 correlates 119896119896

119891with ℎ119903

119900against different 119896

119894values

using (4) and assuming ℎ = 2 nm 119896119898

= 14Wmminus1Kminus1 forbakelite and 119896

119891= 130Wmminus1Kminus1 for graphite Note that an

overall trend of the curves is identical to that of Figure 3

0

4

8

12

16

20

0 02 04 06 08 1 12

Effec

tive t

herm

al co

nduc

tivity

(Wm

-K)

Filler volume fraction

Microregime

Nanoregime

100120583m145 120583m01 120583m

1nmAzeem et al 2012

Figure 6 Bakelitegraphite composite system 119896eff using HS lowerbound ((5) 119896

119894= 099119896

119898)

that is the 119896 increases with increasing 119903119900and also when

119896119894varies from 001119896

119898to 099119896

119898 However it may also be

noticed that the slopes of the curves in Figure 5 are steeper atleast within the limits of microscale than the correspondingcurves in Figure 3 This is simply a result of relatively higher119896119891119896119898ratio in bakelitegraphite composite (Figure 5) than

in Cudiamond composite (Figure 3) Since the interface wasassumed to have very low resistance (or high conductance)it may be expected that 119896

119894= 099119896

119898would predict the

acceptable resultsHS lower bound may now be obtained using (5) and

conductivities of bakelite and graphite Figure 6 summa-rizes the effective thermal conductivities of the compositewithin the micro- and nanoregimes as a function of filler(graphite) volume fraction The experimental values of thebakelitegraphite system from Azeem and Zain-Ul-Abdein[25] are also shown for comparison The authors usedgraphite powder of irregular shapemorphology with an aver-age size of 145 120583m In comparison to experimental resultsit may be observed that HS lower bound underestimatesthe conductivity values Clearly the inherent assumptionsthat the inclusions are spherical and there is no interfacialresistance (119877) are insufficient to reproduce the experimentalfindings (see the dotted line labeled 145 120583m in Figure 6)The modification to incorporate the effects of interfacialresistance and inclusion shapemorphology is discussed in thesubsequent section

4 Results and Discussion

41 Effect of Particle Size HS Lower Bound Although the HSlower bound could not reproduce the experimental results ata given particle size it may however be observed that thenanoregime for the resistive (Figure 4) as well as conductive(Figure 6) media is overshadowed by the microregime Itwill be shown shortly that this behavior does not change


even with the introduction of shape factor (119899) and interfacialresistance (119877)

It should also be noticed that in a resistive medium theconductivity values span over a large range for the micro-sized inclusions while nanosize limits the thermal propertieswithin a very short range For instance consider the Cudia-mond composite with a volume fraction of 05 for each Herethe difference in conductivities was found to be Δ119896micro =

119896100 120583m minus 119896

01 120583m asymp 597Wmminus1Kminus1 and Δ119896nano = 11989601 120583m minus

1198961 nm asymp 117Wmminus1Kminus1 within the micro- and nanoregimesrespectivelyThis shows that the former differs from the latterby a factor of almost 5 This behavior is a result of very lowconductivity of the interface 119896

119894= 001119896

119898 which leads to

a rapid decrease in effective properties of the medium as 119903119900

varies from micro- to nanoscaleOn the contrary in a conductive medium the nanosize

inclusions offer large range of effective properties whereasmicrosize inclusions confine the conductivity values to avery small range For example for the bakelitegraphitecomposite with a volume fraction of 05 for each constituentthe conductivity difference within micro- and nanoregimeswas Δ119896micro = 119896

100 120583m minus 11989601 120583m asymp 042Wmminus1Kminus1 and

Δ119896nano = 11989601 120583m minus 119896

1 nm asymp 345Wmminus1Kminus1 respectively Herethe latter is 8 times more than the former Obviously arelatively conductive interface zone 119896

119894= 099119896

119898 would not

allow significant decrease in properties at least at microscaleWithin nanoregime however the more the 119903

119900decreases the

more the interface thickness (ℎ) becomes effective whichleads to a large span of properties

42 Effect of Interfacial Thermal Resistance Modified HJModel It has been pointed out earlier that the shape factor119899 = 3120595 where 120595 = 1 for spherical and 120595 = 09 forcuboctahedral particles The case studies under considera-tion however use randomparticles of graphite and diamondIn their work Azeem and Zain-Ul-Abdein [25] determinedthat 120595 = 04 for irregular shape of graphite inclusionsSince Hanada et al [24] did not mention the shape factor fortheir irregular diamond particles the sphericity was assumedidentical to that of graphite particles that is 120595 = 04

421 Case I High Interfacial Resistance Considering 119896119894

=

001119896119898and 120595 = 04 for highly resistive Cudiamond compos-

ite (7) may be used to calculate the effective conductivity ofthe composite as a function of filler volume fraction over theentire range of micro- to nanosized particles (see Figure 7)Note that (7) also requires definition of 119877 such that theexperimental values could be approximated A value of R= 3 times 10

minus7m2sdotKsdotWminus1 was found to reproduce experimentalvalues to a reasonable agreement It may be observed that (7)converges to HS lower bound for 119877 = 0 and 120595 = 1 that iswhen the interfacial resistance is ignored and the inclusionsare spherical In the typical Cudiamond composites the R-values range from 1sim3times 10minus8m2sdotKsdotWminus1 (Kang et al [26] Shenet al [27]) depending upon the diamond shape morphologyand interface characteristics The 119877-value identified in thiswork is almost 10 times more than the one found in theliterature This simply means that the Cudiamond system

0

50

100

150

200

250

300

350

400

450

0 02 04 06 08 1

Effec

tive t

herm

al co

nduc

tivity

(Wm

-K)


Microregime

Nanoregime

100120583m77 120583m01 120583m


Figure 7 Cudiamond composite system 119896eff using modified HJmodel ((7) 119877 = 3 times 10

minus7m2sdotKsdotWminus1 119896119894= 001119896

119898)

under consideration exhibits extremely high interfacial resis-tance Or in other words as 119877 rarr 0 the interface becomesmore conductive and less resistive Probable reasons for highresistance are the imperfections at the interface including airvoids and decohesion (see Raza and Khalid [28] for details)

A comparison of Figures 4 and 7 indicates that increasein 119877 leads to significant decrease in conductivity bothwithin micro- and nanoregimes It may be observed fromFigure 7 that the experimental conductivities are at the limitof nanoscale In addition at such a high value of 119877 (3 times

10minus7m2sdotKsdotWminus1) and a constant 120595 = 04 the effective

conductivity of the composites is almost independent ofthe inclusion size at least within the nanoregime Briefly aresistive interface dominates the effect of both the particle sizeand shape

422 Case II Low Interfacial Resistance Now considering119896119894

= 099119896119898

and 120595 = 04 for highly conductive bake-litegraphite composite (7) may be exploited to estimate theeffective thermal conductivity of the medium as a functionof filler volume fraction (see Figure 8) Here the value of119877 has been identified as 1 times 10minus10m2sdotKsdotWminus1 so as to yieldthe experimental values reported by Azeem and Zain-Ul-Abdein [25] The observation that 119877 is quite close to 0 andis almost 3 times 10

minus4 times smaller than that of Cudiamondcomposite indicates the presence of almost perfect interfacewith extremely low resistance In fact any further diminutionof 119877-value does not bring about any significant change in theconductivity values This also means that unlike the metalmatrix Cudiamond composite there is a very less probabilityof interfacial defects such as air gaps and decohesion in caseof polymer matrix bakelitegraphite composite

A comparative analysis of Figures 6 and 8 suggests thatalthough there is not much difference between the evolutionof conductivity values within the micro- and nanoregimesusing HS lower bound or modified HJ model the latter helpsin identifying the role of inclusion shape factor and interfacial


0

4

8

12

16

20

0 02 04 06 08 1Effec

tive t

herm

al co

nduc

tivity

(Wm

-K)


Nanoregime

Microregime

100120583m145 120583m01 120583m

1nmAzeem et al 2012

Figure 8 Bakelitegraphite composite system 119896eff using modifiedHJ model ((7) 119877 = 1 times 10


119898)

R = 3 times 10minus7 m2middotKWki = 001km

120601f = 01

120601f = 03

120601f = 05

400

350

300

250

200

150

10minus1010minus8

10minus610minus4

Log-scale inclusion radius (m)

MicroEffec

tive t

herm

al co

nduc

tivity

(Wm

middotK)

Nano

1

05

0

380

360

340

320

300

280

260

240

220

200

180Sphericity

Figure 9 Cudiamond composite system 119896eff as a function of 120601119891120595

and 119903119900

thermal resistance more precisely To summarize the effectof shape factor is more pronounced in composites with lowinterfacial resistance

43 Effect of Shape Factor Since shape factor (119899) is a functionof sphericity (120595) such that 0 lt 120595 le 1 it is relativelyconvenient to represent effective thermal conductivity as afunction of sphericity It is also of interest to observe simul-taneously the effect of inclusion radius (119903

119900) at micro- and

nanoscales Figures 9 and 10 illustrate the effects of both thesphericity and filler radius upon the thermal conductivity ofCudiamond and bakelitegraphite composites respectivelywith filler volume fractions of 120601

119891= 01 03 and 05

In case of thermally resistive medium (119896119894

= 001119896119898

119877 = 3 times 10minus7m2sdotKsdotWminus1) it may be noticed that the thermal

conductivity is less responsive to the change in shape factorat a given volume fraction within the microregime than thenanoregime (Figure 9) Consider for example the extreme


120601f = 01

120601f = 03

120601f = 05

MicroNano

10minus1010minus8

10minus610minus4


Effec

tive t

herm

al co

nduc

tivity

(Wm

middotK)

1

05

0

Sphericity

30

25

25

20

20

15 15

10

105

5

0

Figure 10 Bakelitegraphite composite system 119896eff as a function of120601119891 120595 and 119903

119900

case when 120601119891

= 05 for 119903119900

= 100 120583m the 119896eff varies from3398Wsdotmminus1sdotKminus1 (at 120595 = 1) to 3424Wsdotmminus1sdotKminus1 (at 120595 = 01)whereas for 119903

119900= 1 nm the 119896eff changes from 160Wsdotmminus1sdotKminus1

(at 120595 = 1) to 1966Wsdotmminus1sdotKminus1 (at 120595 = 01)Nevertheless the thermally conductive medium (119896

119894=

099119896119898 119877 = 1 times 10

minus10m2sdotKsdotWminus1) showed strong dependencyof effective properties upon the shape factor as well as fillervolume fraction (Figure 10) For instance it may be noticedthat when 120601

119891= 05 and 119903

119900= 100 120583m the change in 119896eff ranges

from 53 to 267Wsdotmminus1sdotKminus1 for the corresponding values of120595 from 1 to 01 while for 119903

119900= 1 nm the 119896eff does not change at

least up to two significant digits that is 119896eff = 14Wsdotmminus1sdotKminus1(at 120595 = 01sim1)

5 Conclusions

Based upon the above investigation it may be concluded thatthe composite material for thermal management applicationscan offer wide range of properties that depend upon factorslike volume fractions of the matrix and filler shape morphol-ogy and particle size In addition the properties of interfacezone play a decisive role in limiting the behavior of compositewithin the limits of both the micro- and nanoregimes Theproposed scheme in comparison to the published resultssuccessfully estimates the effective properties for the extremecases of binary composites with high and low interfacialresistance Compared to nanoparticles the contribution ofmicrosize inclusions is less affected by the interface zone dueto very small interface thickness to inclusion radius ratioHowever unlike the latter the effect of the former is lessinfluenced by the shape morphology or sphericity of theinclusions

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper


References

[1] R Clausius Die Mechanische Behandlung der ElectricitatVieweg Braunshweig Germany 1879

[2] J C Maxwell A Treatise on Electricity and Magnetism DoverPublications New York NY USA 3rd edition 1884

[3] D A G Bruggeman ldquoCalculation of various physical constantsin heterogeneous substances I Dielectric constants and con-ductivity of composites from isotropic substancesrdquoAnnalen derPhysik vol 24 no 5 pp 636ndash679 1935 (German)

[4] O Wiener ldquoDie theorie des Mischkorpers fur das feld desstationaaren stromung Erste abhandlung die Mttelswertsatzefur kraftrdquo Abhandlungen der Mathematisch-Physischen Klasseder Koniglich-Sachsischen Gesellschaft der Wissenschaften vol32 no 6 pp 509ndash604 1912

[5] Z Hashin and S Shtrikman ldquoA variational approach to thetheory of the effective magnetic permeability of multiphasematerialsrdquo Journal of Applied Physics vol 33 no 10 pp 3125ndash3131 1962

[6] T Mori and K Tanaka ldquoAverage stress in matrix and averageelastic energy of materials with misfitting inclusionsrdquo ActaMetallurgica vol 21 no 5 pp 571ndash574 1973

[7] S K Kanaun and V M Levin ldquoEffective field method inmechanics of matrix composite materialsrdquo in Advances inMathematical Modelling of Composite Materials K Z MarkovEd pp 1ndash58 World Scientific 1994

[8] K Z Markov ldquoElementary micromechanics of heterogeneousmediardquo in Heterogeneous Media Micromechanics ModelingMethods and Simulations K Z Markov and L Preziozi EdsModeling and Simulation in Science Engineering and Technol-ogy pp 1ndash162 Birkhauser Boston Mass USA 2000

[9] R L Hamilton and O K Crosser ldquoThermal conductivity ofheterogeneous two-component systemsrdquo Industrial and Engi-neering Chemistry Fundamentals vol 1 no 3 pp 187ndash191 1962

[10] R W Zimmerman ldquoEffective conductivity of a two-dimensional medium containing elliptical inhomogeneitiesrdquoProceedings of the Royal Society A Mathematical Physical andEngineering Sciences vol 452 no 1950 pp 1713ndash1727 1996

[11] P L Kapitza ldquoThe study of heat transfer in helium IIrdquo Journalof Physics-USSR vol 4 pp 177ndash181 1941

[12] S K Kanaun ldquoOn singular models of a thin inclusion in ahomogeneous elastic mediumrdquo Journal of Applied Mathematicsand Mechanics vol 48 no 1 pp 50ndash58 1984 (Russian)

[13] S Torquato and M D Rintoul ldquoEffect of the interface on theproperties of composite mediardquo Physical Review Letters vol 75no 22 pp 4067ndash4070 1995

[14] Y Benveniste and T Miloh ldquoThe effective conductivity ofcomposites with imperfect thermal contact at constituent inter-facesrdquo International Journal of Engineering Science vol 24 no9 pp 1537ndash1552 1986

[15] Y Benveniste ldquoEffective thermal conductivity of compositeswith a thermal contact resistance between the constituentscondilute caserdquo Journal of Applied Physics vol 61 no 8 pp2840ndash2843 1987

[16] Y C Chiew and E D Glandt ldquoEffective conductivity ofdispersions the effect of resistance at the particle surfacesrdquoChemical Engineering Science vol 42 no 11 pp 2677ndash26851987

[17] D P H Hasselman and L F Johnson ldquoEffective thermalconductivity of composites with interfacial thermal barrierresistancerdquo Journal of Composite Materials vol 21 no 6 pp508ndash515 1987

[18] I Sevostianov and M Kachanov ldquoEffect of interphase layerson the overall elastic and conductive properties of matrixcomposites Applications to nanosize inclusionrdquo InternationalJournal of Solids and Structures vol 44 no 3-4 pp 1304ndash13152007

[19] W Jiajun and Y Xiao-Su ldquoEffects of interfacial thermal barrierresistance and particle shape and size on the thermal conductiv-ity of AlNPI compositesrdquo Composites Science and Technologyvol 64 no 10-11 pp 1623ndash1628 2004

[20] L Shen and J Li ldquoHomogenization of a fibresphere with aninhomogeneous interphase for the effective elastic moduli ofcompositesrdquo Proceedings of the Royal Society A MathematicalPhysical and Engineering Sciences vol 461 no 2057 pp 1475ndash1504 2005

[21] T Mutschele and R Kirchheim ldquoHydrogen as a probe for theaverage thickness of a grain boundaryrdquo ScriptaMetallurgica vol21 no 8 pp 1101ndash1104 1987

[22] V M Levin ldquoDetermination of effective elastic-moduli ofcomposite-materialsrdquo Soviet PhysicsmdashDoklady vol 220 pp1042ndash1045 1975

[23] S K Kanaun ldquoEffective field method in the linear problems ofstatics of composite mediardquo Journal of AppliedMathematics andMechanics vol 46 pp 655ndash665 1982

[24] K Hanada K Matsuzaki and T Sano ldquoThermal properties ofdiamond particle-dispersed Cu compositesrdquo Journal of Materi-als ProcessingTechnology vol 153-154 no 1ndash3 pp 514ndash518 2004

[25] S Azeem and M Zain-Ul-Abdein ldquoInvestigation of ther-mal conductivity enhancement in bakelite-graphite particulatefilled polymeric compositerdquo International Journal of EngineeringScience vol 52 pp 30ndash40 2012

[26] Q Kang X He S Ren et al ldquoPreparation of copper-diamondcomposites with chromium carbide coatings on diamond par-ticles for heat sink applicationsrdquo Applied Thermal Engineeringvol 60 no 1-2 pp 423ndash429 2013

[27] W Shen W Shao Q Wang andM Ma ldquoThermal conductivityand thermal expansion coefficient of diamond5 wtSi-Cucomposite by vacuum hot pressingrdquo Fusion Engineering andDesign vol 85 no 10-12 pp 2237ndash2240 2010

[28] K Raza and F A Khalid ldquoOptimization of sintering parametersfor diamond-copper composites in conventional sintering andtheir thermal conductivityrdquo Journal of Alloys and Compoundsvol 615 pp 111ndash118 2014

[29] Z Ling J Chen T Xu X Fang X Gao and Z Zhang ldquoThermalconductivity of an organic phase change materialexpandedgraphite composite across the phase change temperature rangeand a novel thermal conductivity modelrdquo Energy Conversionand Management vol 102 pp 202ndash208 2015

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of



CeramicsJournal of


CompositesJournal of

NanoparticlesJournal of



International Journal of

Biomaterials


NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research



MetallurgyJournal of


BioMed Research International

MaterialsJournal of


Nano

materials


Journal ofNanomaterials


only upon volume fractions of matrix and inclusion wereformulated by Wiener (1912) [4] The Wiener upper bound(UB) is still widely used by the researchers as the ldquorule ofmixturesrdquo Assuming macroscopically isotropic behavior ofthe composite Hashin and Shtrikman (1962) [5] defined amore conservative set of upper and lower bounds For acomposite with spherical inclusions the Hashin-Shtrikman(HS) lower bound (LB) coincides with Maxwell and Moriand Tanka (1973) [6] models Mori-Tanaka (MT) howeveralso introduced depolarization factors in their formulationto accommodate the prolate and oblate shape morphologiesSimilar to the Kanaun and Levin (1994) [7] scheme MTscheme is merely a variant of effective field (EF) methodwhich treats the inclusion as an isolated particle and assumesthe presence of an effective field over each inhomogeneity(see Markov (2000) [8] for a detailed review) Hamilton andCrosser (1962) [9] modified Maxwell model to incorporatethe shape factor (119899 = 3120595) for nonspherical particles wherethe sphericity (120595) is the ratio of the surface area of spherewith the same volume of the particle to the surface area of theparticle Using 120595 = 1 for a spherical inclusion regenerates theMaxwell model Likewise Zimmerman (1996) [10] exploitedMaxwell and Bruggeman methods and introduced an aspectratio and conductivity ratio dependent parameter (120573) forelliptical inhomogeneities

Most of the above mentioned models assume ideal con-tact between the inclusion and the matrix which impliesthat the heat flow across the interface remains continuousIn reality however the interface regardless of its natureis nonideal and leads to a discontinuous temperature fieldacross the boundary due to interfacial thermal resistancealso known as Kapitzarsquos resistance (after Kapitza (1941) [11])In order to deal with interface behavior Kanaun (1984) [12]introduced the concept of singular inclusion in the elasticitycontext From the perspective of thermal conductivity thenonideal contact may be assumed as either ldquosuperconduc-tiverdquo or ldquoresistiverdquo A superconducting or resisting singularinclusion requires that its conductivity (119896

119904) be 119896

119904rarr infin or

1119896119904

rarr infin respectively provided that one of the inclusionrsquosdimensions is of the order 120575 rarr 0 Torquato and Rintoul(1995) [13] also defined conductive and resistive contacts inrelation to the conductive properties of the interphase layer

between the matrix and inclusion Detailed discussions onconductiveresistive interfacesmay also be found in theworksof Benveniste and Miloh (1986) [14] Benveniste (1987) [15]and Chiew and Glandt (1987) [16]

In the present work a method based on Hasselmanand Johnson (1987) [17] and Sevostianov and Kachanov(2007) [18] models has been proposed to predict the thermalconductivity of particulate filled composites with high andlow interfacial thermal resistance as functions of filler volumefractions inclusion size shape factor and conductivity ofinterphase layer The effect of particle size ranging frommicro- to nanoscale has been investigated in the highlyconductive and resistive media It will be shownwith the helpof case studies that the inclusion size behaves differently inresponse to the interfacial resistance

2 Theoretical Model

21 Effects of Shape Factor and Interfacial Resistance In theirwork Hasselman and Johnson [17] addressed the nonidealcontacts in terms of interfacial thermal resistance (119877) andderived a mathematical model for spherical morphology ofparticles The model states

119896eff = 119896119898

sdot

2 (119896119891119896119898

minus 1 minus (119896119891119903119900) 119877) 120601

119891+ 119896119891119896119898

+ 2 + 2 (119896119891119903119900) 119877

(1 minus 119896119891119896119898

+ (119896119891119903119900) 119877) 120601

119891+ 119896119891119896119898

+ 2 + 2 (119896119891119903119900) 119877

(1)

where 119896119898and 119896

119891are thermal conductivities of matrix and

filler (inclusion) in Wsdotmminus1sdotKminus1 respectively 119903119900is the filler

radius inm 120601119891is the filler volume fraction119877 is the interfacial

resistance in Wminus1sdotm2sdotK and 119896eff is the effective thermalconductivity of the composite 119877 by definition is an inverseof thermal conductance (119862) that is 119877 = 1119862 Note thatHJ model is valid for any system of units the SI units arementioned here only for the readerrsquos convenience Jiajun andXiao-Su (2004) [19] modified the HJ model (1) in order toaccommodate the nonspherical morphology of the particlesThey actually used Hamilton and Crosser [9] shape factorparameter (119899 = 3120595) where 120595 is the sphericity as describedearlier The modified HJ therefore takes the following form

119896eff = 119896119898

119896119891

(1 + (119899 minus 1) (119896119898

119903119900) 119877) + (119899 minus 1) 119896

119898+ (119899 minus 1) 120601

119891(119896119891

(1 minus (119896119898

119903119900) 119877) minus 119896

119898)

119896119891

(1 + (119899 minus 1) (119896119898

119903119900) 119877) + (119899 minus 1) 119896

119898minus 120601119891

(119896119891

(1 minus (119896119898

119903119900) 119877) minus 119896

119898)

(2)

Note that (2) reduces to (1) for spherical inclusions thatis 120595 = 1 and also that the HJ model reproduces theMaxwell model for ideal contact when 119877 = 0 (or 119862 = infin)Conversely for nonideal contacts the interfacial resistancemust be nonzero and the conductance finite It should alsobe mentioned that increasing filler fractions yield eitherdecreasing 119896eff when 119896

119898gt 119862 sdot 119903

119900or increasing 119896eff when

119896119898

lt 119862 sdot 119903119900 Consequently the interfacial thermal resistance

would be substantially large in the former case as in resistive

composites and vanishingly small in the latter case similarto the superconductive composites

22 Effect of Inclusion Size Micro- to Nanoscale Sevostianovand Kachanov [18] recently modified an incremental schemeformulated by Shen and Li (2005) [20] so as to incorporatethe effect of interface layer thickness (ℎ sim 1-2 nm [21])upon the overall properties of the material at nanoscale Itis understood that in the microregime ℎ ≪ 119903

119900 however in


h

kf gt km

ki = const

ro

kf

km

ki

(a)

k = f(kf ki)

kmri

k

(b)





119900+ ℎ =

119903119894 where 119903



lt 119903 lt 119903119894


= 119896119894(119903) and the



119900) =



119889119896

119889119903

= minus

1

119903

(119896 (119903) minus 119896119894(119903)) (119896 (119903) + 2119896

119894(119903))

119896119894(119903)

(3)

Setting 119903119900

+ ℎ = 119903119894


119894(119903) = 119896


of (3) yields

119896 (119903119894)

= 119896119894

[

[

2 + (1 + ℎ119903119900)3

sdot ((2 (119896119894119896119891) + 1) (1 minus (119896

119894119896119891)))

(1 + ℎ119903119900)3

sdot ((2 (119896119894119896119891) + 1) (1 minus (119896

119894119896119891))) minus 1

]

]

(4)



119896119898

[1 +

31206011015840

(119896 minus 119896119898

)

3119896119898

+ (1 minus 1206011015840) (119896 minus 119896

119898)

] le 119896eff

le 119896 [1 minus

3 (1 minus 1206011015840

) (119896 minus 119896119898

)

3119896 minus 1206011015840(119896 minus 119896

119898)

] (HS bounds)

(5)

119896eff = 119896119898

[1 +

31206011015840

(119896 minus 119896119898

)

3119896119898

+ (1 minus 1206011015840) (119896 minus 119896

119898)

]


(6)









119900lt


zone (119896119894


119894= 001119896





119896eff = 119896119898

119896 (1 + (119899 minus 1) (119896119898

119903119894) 119877) + (119899 minus 1) 119896

119898+ (119899 minus 1) 120601

1015840

(119896 (1 minus (119896119898

119903119894) 119877) minus 119896

119898)

119896 (1 + (119899 minus 1) (119896119898

119903119894) 119877) + (119899 minus 1) 119896

119898minus 1206011015840(119896 (1 minus (119896

119898119903119894) 119877) minus 119896

119898)

(7)


h

km

kf gt km

ki = const

ro

kf

ki

(a)

R

kmri

k = f(kf ki)

k

(b)




1015840


119891 119896119898





3 Case Studies







119894



119891


119898to 099119896

119898results in a


119894= 001119896


0

02

04

06

08

1

0 005 01 015 02 025 03 035 04

Nanosize

Microsize

kk

f

hro


ki = 07kmki = 099km


119900

(4)




Microregime


Effec

tive t

herm

al co

nduc

tivity

(Wm

-K)

100120583m77 120583m01 120583m


0

400

800

1200

1600

2000

Nanoregime


119894= 001119896

119898)

0

02

04

06

08

1

0 005 01 015 02 025 03 035 04

Nanosize

Microsize

kk

f

hro


ki = 07kmki = 099km


ℎ119903119900(4)



119891with ℎ119903


119894values





0

4

8

12

16

20

0 02 04 06 08 1 12

Effec

tive t

herm

al co

nduc

tivity

(Wm

-K)


Microregime

Nanoregime

100120583m145 120583m01 120583m

1nmAzeem et al 2012


119894= 099119896

119898)


119896119894varies from 001119896

119898to 099119896




119894= 099119896









119896100 120583m minus 119896



119894= 001119896






Δ119896nano = 11989601 120583m minus 119896


119894= 099119896

119898 would not


119900decreases the




=




0

50

100

150

200

250

300

350

400

450

0 02 04 06 08 1

Effec

tive t

herm

al co

nduc

tivity

(Wm

-K)


Microregime

Nanoregime

100120583m77 120583m01 120583m




119898)






= 099119896119898





0

4

8

12

16

20

0 02 04 06 08 1Effec

tive t

herm

al co

nduc

tivity

(Wm

-K)


Nanoregime

Microregime

100120583m145 120583m01 120583m

1nmAzeem et al 2012



119898)


120601f = 01

120601f = 03

120601f = 05

400

350

300

250

200

150

10minus1010minus8

10minus610minus4


MicroEffec

tive t

herm

al co

nduc

tivity

(Wm

middotK)

Nano

1

05

0

380

360

340

320

300

280

260

240

220

200

180Sphericity


and 119903119900





119891= 01 03 and 05


= 001119896119898




120601f = 01

120601f = 03

120601f = 05

MicroNano

10minus1010minus8

10minus610minus4


Effec

tive t

herm

al co

nduc

tivity

(Wm

middotK)

1

05

0

Sphericity

30

25

25

20

20

15 15

10

105

5

0


119900

case when 120601119891

= 05 for 119903119900




119894=

099119896119898 119877 = 1 times 10


119891= 05 and 119903





5 Conclusions





References





































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




h

kf gt km

ki = const

ro

kf

km

ki

(a)

k = f(kf ki)

kmri

k

(b)





119900+ ℎ =

119903119894 where 119903



lt 119903 lt 119903119894


= 119896119894(119903) and the



119900) =



119889119896

119889119903

= minus

1

119903

(119896 (119903) minus 119896119894(119903)) (119896 (119903) + 2119896

119894(119903))

119896119894(119903)

(3)

Setting 119903119900

+ ℎ = 119903119894


119894(119903) = 119896


of (3) yields

119896 (119903119894)

= 119896119894

[

[

2 + (1 + ℎ119903119900)3

sdot ((2 (119896119894119896119891) + 1) (1 minus (119896

119894119896119891)))

(1 + ℎ119903119900)3

sdot ((2 (119896119894119896119891) + 1) (1 minus (119896

119894119896119891))) minus 1

]

]

(4)



119896119898

[1 +

31206011015840

(119896 minus 119896119898

)

3119896119898

+ (1 minus 1206011015840) (119896 minus 119896

119898)

] le 119896eff

le 119896 [1 minus

3 (1 minus 1206011015840

) (119896 minus 119896119898

)

3119896 minus 1206011015840(119896 minus 119896

119898)

] (HS bounds)

(5)

119896eff = 119896119898

[1 +

31206011015840

(119896 minus 119896119898

)

3119896119898

+ (1 minus 1206011015840) (119896 minus 119896

119898)

]


(6)









119900lt


zone (119896119894


119894= 001119896





119896eff = 119896119898

119896 (1 + (119899 minus 1) (119896119898

119903119894) 119877) + (119899 minus 1) 119896

119898+ (119899 minus 1) 120601

1015840

(119896 (1 minus (119896119898

119903119894) 119877) minus 119896

119898)

119896 (1 + (119899 minus 1) (119896119898

119903119894) 119877) + (119899 minus 1) 119896

119898minus 1206011015840(119896 (1 minus (119896

119898119903119894) 119877) minus 119896

119898)

(7)


h

km

kf gt km

ki = const

ro

kf

ki

(a)

R

kmri

k = f(kf ki)

k

(b)




1015840


119891 119896119898





3 Case Studies







119894



119891


119898to 099119896

119898results in a


119894= 001119896


0

02

04

06

08

1

0 005 01 015 02 025 03 035 04

Nanosize

Microsize

kk

f

hro


ki = 07kmki = 099km


119900

(4)




Microregime


Effec

tive t

herm

al co

nduc

tivity

(Wm

-K)

100120583m77 120583m01 120583m


0

400

800

1200

1600

2000

Nanoregime


119894= 001119896

119898)

0

02

04

06

08

1

0 005 01 015 02 025 03 035 04

Nanosize

Microsize

kk

f

hro


ki = 07kmki = 099km


ℎ119903119900(4)



119891with ℎ119903


119894values





0

4

8

12

16

20

0 02 04 06 08 1 12

Effec

tive t

herm

al co

nduc

tivity

(Wm

-K)


Microregime

Nanoregime

100120583m145 120583m01 120583m

1nmAzeem et al 2012


119894= 099119896

119898)


119896119894varies from 001119896

119898to 099119896




119894= 099119896









119896100 120583m minus 119896



119894= 001119896






Δ119896nano = 11989601 120583m minus 119896


119894= 099119896

119898 would not


119900decreases the




=




0

50

100

150

200

250

300

350

400

450

0 02 04 06 08 1

Effec

tive t

herm

al co

nduc

tivity

(Wm

-K)


Microregime

Nanoregime

100120583m77 120583m01 120583m




119898)






= 099119896119898





0

4

8

12

16

20

0 02 04 06 08 1Effec

tive t

herm

al co

nduc

tivity

(Wm

-K)


Nanoregime

Microregime

100120583m145 120583m01 120583m

1nmAzeem et al 2012



119898)


120601f = 01

120601f = 03

120601f = 05

400

350

300

250

200

150

10minus1010minus8

10minus610minus4


MicroEffec

tive t

herm

al co

nduc

tivity

(Wm

middotK)

Nano

1

05

0

380

360

340

320

300

280

260

240

220

200

180Sphericity


and 119903119900





119891= 01 03 and 05


= 001119896119898




120601f = 01

120601f = 03

120601f = 05

MicroNano

10minus1010minus8

10minus610minus4


Effec

tive t

herm

al co

nduc

tivity

(Wm

middotK)

1

05

0

Sphericity

30

25

25

20

20

15 15

10

105

5

0


119900

case when 120601119891

= 05 for 119903119900




119894=

099119896119898 119877 = 1 times 10


119891= 05 and 119903





5 Conclusions





References





































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




h

km

kf gt km

ki = const

ro

kf

ki

(a)

R

kmri

k = f(kf ki)

k

(b)




1015840


119891 119896119898





3 Case Studies







119894



119891


119898to 099119896

119898results in a


119894= 001119896


0

02

04

06

08

1

0 005 01 015 02 025 03 035 04

Nanosize

Microsize

kk

f

hro


ki = 07kmki = 099km


119900

(4)




Microregime


Effec

tive t

herm

al co

nduc

tivity

(Wm

-K)

100120583m77 120583m01 120583m


0

400

800

1200

1600

2000

Nanoregime


119894= 001119896

119898)

0

02

04

06

08

1

0 005 01 015 02 025 03 035 04

Nanosize

Microsize

kk

f

hro


ki = 07kmki = 099km


ℎ119903119900(4)



119891with ℎ119903


119894values





0

4

8

12

16

20

0 02 04 06 08 1 12

Effec

tive t

herm

al co

nduc

tivity

(Wm

-K)


Microregime

Nanoregime

100120583m145 120583m01 120583m

1nmAzeem et al 2012


119894= 099119896

119898)


119896119894varies from 001119896

119898to 099119896




119894= 099119896









119896100 120583m minus 119896



119894= 001119896






Δ119896nano = 11989601 120583m minus 119896


119894= 099119896

119898 would not


119900decreases the




=




0

50

100

150

200

250

300

350

400

450

0 02 04 06 08 1

Effec

tive t

herm

al co

nduc

tivity

(Wm

-K)


Microregime

Nanoregime

100120583m77 120583m01 120583m




119898)






= 099119896119898





0

4

8

12

16

20

0 02 04 06 08 1Effec

tive t

herm

al co

nduc

tivity

(Wm

-K)


Nanoregime

Microregime

100120583m145 120583m01 120583m

1nmAzeem et al 2012



119898)


120601f = 01

120601f = 03

120601f = 05

400

350

300

250

200

150

10minus1010minus8

10minus610minus4


MicroEffec

tive t

herm

al co

nduc

tivity

(Wm

middotK)

Nano

1

05

0

380

360

340

320

300

280

260

240

220

200

180Sphericity


and 119903119900





119891= 01 03 and 05


= 001119896119898




120601f = 01

120601f = 03

120601f = 05

MicroNano

10minus1010minus8

10minus610minus4


Effec

tive t

herm

al co

nduc

tivity

(Wm

middotK)

1

05

0

Sphericity

30

25

25

20

20

15 15

10

105

5

0


119900

case when 120601119891

= 05 for 119903119900




119894=

099119896119898 119877 = 1 times 10


119891= 05 and 119903





5 Conclusions





References





































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




Microregime


Effec

tive t

herm

al co

nduc

tivity

(Wm

-K)

100120583m77 120583m01 120583m


0

400

800

1200

1600

2000

Nanoregime


119894= 001119896

119898)

0

02

04

06

08

1

0 005 01 015 02 025 03 035 04

Nanosize

Microsize

kk

f

hro


ki = 07kmki = 099km


ℎ119903119900(4)



119891with ℎ119903


119894values





0

4

8

12

16

20

0 02 04 06 08 1 12

Effec

tive t

herm

al co

nduc

tivity

(Wm

-K)


Microregime

Nanoregime

100120583m145 120583m01 120583m

1nmAzeem et al 2012


119894= 099119896

119898)


119896119894varies from 001119896

119898to 099119896




119894= 099119896









119896100 120583m minus 119896



119894= 001119896






Δ119896nano = 11989601 120583m minus 119896


119894= 099119896

119898 would not


119900decreases the




=




0

50

100

150

200

250

300

350

400

450

0 02 04 06 08 1

Effec

tive t

herm

al co

nduc

tivity

(Wm

-K)


Microregime

Nanoregime

100120583m77 120583m01 120583m




119898)






= 099119896119898





0

4

8

12

16

20

0 02 04 06 08 1Effec

tive t

herm

al co

nduc

tivity

(Wm

-K)


Nanoregime

Microregime

100120583m145 120583m01 120583m

1nmAzeem et al 2012



119898)


120601f = 01

120601f = 03

120601f = 05

400

350

300

250

200

150

10minus1010minus8

10minus610minus4


MicroEffec

tive t

herm

al co

nduc

tivity

(Wm

middotK)

Nano

1

05

0

380

360

340

320

300

280

260

240

220

200

180Sphericity


and 119903119900





119891= 01 03 and 05


= 001119896119898




120601f = 01

120601f = 03

120601f = 05

MicroNano

10minus1010minus8

10minus610minus4


Effec

tive t

herm

al co

nduc

tivity

(Wm

middotK)

1

05

0

Sphericity

30

25

25

20

20

15 15

10

105

5

0


119900

case when 120601119891

= 05 for 119903119900




119894=

099119896119898 119877 = 1 times 10


119891= 05 and 119903





5 Conclusions





References





































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials






119896100 120583m minus 119896



119894= 001119896






Δ119896nano = 11989601 120583m minus 119896


119894= 099119896

119898 would not


119900decreases the




=




0

50

100

150

200

250

300

350

400

450

0 02 04 06 08 1

Effec

tive t

herm

al co

nduc

tivity

(Wm

-K)


Microregime

Nanoregime

100120583m77 120583m01 120583m




119898)






= 099119896119898





0

4

8

12

16

20

0 02 04 06 08 1Effec

tive t

herm

al co

nduc

tivity

(Wm

-K)


Nanoregime

Microregime

100120583m145 120583m01 120583m

1nmAzeem et al 2012



119898)


120601f = 01

120601f = 03

120601f = 05

400

350

300

250

200

150

10minus1010minus8

10minus610minus4


MicroEffec

tive t

herm

al co

nduc

tivity

(Wm

middotK)

Nano

1

05

0

380

360

340

320

300

280

260

240

220

200

180Sphericity


and 119903119900





119891= 01 03 and 05


= 001119896119898




120601f = 01

120601f = 03

120601f = 05

MicroNano

10minus1010minus8

10minus610minus4


Effec

tive t

herm

al co

nduc

tivity

(Wm

middotK)

1

05

0

Sphericity

30

25

25

20

20

15 15

10

105

5

0


119900

case when 120601119891

= 05 for 119903119900




119894=

099119896119898 119877 = 1 times 10


119891= 05 and 119903





5 Conclusions





References





































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




0

4

8

12

16

20

0 02 04 06 08 1Effec

tive t

herm

al co

nduc

tivity

(Wm

-K)


Nanoregime

Microregime

100120583m145 120583m01 120583m

1nmAzeem et al 2012



119898)


120601f = 01

120601f = 03

120601f = 05

400

350

300

250

200

150

10minus1010minus8

10minus610minus4


MicroEffec

tive t

herm

al co

nduc

tivity

(Wm

middotK)

Nano

1

05

0

380

360

340

320

300

280

260

240

220

200

180Sphericity


and 119903119900





119891= 01 03 and 05


= 001119896119898




120601f = 01

120601f = 03

120601f = 05

MicroNano

10minus1010minus8

10minus610minus4


Effec

tive t

herm

al co

nduc

tivity

(Wm

middotK)

1

05

0

Sphericity

30

25

25

20

20

15 15

10

105

5

0


119900

case when 120601119891

= 05 for 119903119900




119894=

099119896119898 119877 = 1 times 10


119891= 05 and 119903





5 Conclusions





References





































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




References





































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials










CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials



research article effect of micro- to nanosize inclusions upon the thermal...

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