www.elsevier.com/locate/carbon

Carbon 45 (2007) 695–703

The effect of a CNT interface on the thermal resistanceof contacting surfaces

Shadab Shaikh a,*, Khalid Lafdi a, Edward Silverman b

a University of Dayton, 300 College Park, Dayton, OH 45469, United Statesb Northrop Grumman Space Technology, One space Park, 01/2040, Redondo Beach, CA 90278, United States

Received 4 October 2006; accepted 5 December 2006Available online 19 January 2007

Abstract

Experimental and theoretical analyses were used to study the effect of thermal contact resistance in two materials, aluminum andgraphite. Experimental investigation included the use of a modern laser flash device to measure the effective thermal conductivity of eachmaterial for three different cases: in direct contact, with a graphite coating and with a thin sheet of carbon nanotube (CNT) thermalinterface material (TIM). For both materials total thermal resistance values were determined corresponding to different cases for samecontact pressure. Results showed that the CNT TIM produced the minimum thermal contact resistance. A theoretical study was carriedout to compare the experimental results with thermal resistance models from the literature. Based on the surface roughness of the mate-rials tested, two models were used. Both models showed reasonable agreement with the experimental results with an error of less than6.5%. The results demonstrate the effectiveness of CNT materials in improving the thermal conductance of contacting surfaces.� 2006 Elsevier Ltd. All rights reserved.

1. Introduction

Heat dissipation is the most critical problem that limitsthe performance, power, reliability and further miniaturi-zation of microelectronics. The thermal performance ofthese devices is greatly affected by the thermal resistanceassociated with interface between the heat sink and heatsource. The improvement of a thermal contact is associatedwith a decrease of the thermal resistance imposed by thisinterface as the heat flows across it [1–3]. As surfaces arenever perfectly flat, the interface comprises point contactsat asperities and air pockets. Some heat is conductedthrough the physical contact points, but much more istransmitted through the air gaps. Since air is a poor con-ductor of heat, it should be replaced by a more conductivematerial to increase the joint conductivity and thusimprove heat flow across the thermal interface. Use of

0008-6223/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.doi:10.1016/j.carbon.2006.12.007

* Corresponding author. Tel.: +1 937 229 4363; fax: +1 937 229 3433.E-mail address: shaikhsa@notes.udayton.edu (S. Shaikh).

suitable interface materials can thus have a significant rolein lowering thermal contact resistance.

The main objective of thermal management in manyapplications is the efficient removal of heat from the semi-conducting device to the ambient environment. In general,this heat removal involves the following four major stages:(a) heat transfer within the device itself; (b) heat transferfrom the device to a heat sink; (c) heat transfer throughthe heat sink; and (d) heat transfer from the heat sink tothe ambient environment. While each of the four heat trans-fer stages affects the rate of heat removal from a semicon-ducting device, the heat transfer between the contactingsurfaces of the device and the heat sink is generally consid-ered the rate limiting process. This is because the contactingsurfaces between the device and the heat sink are generallyneither fully conforming nor smooth.

Rather, the surfaces are somewhat non-conforming andrough making the real contact area between the packageand the heat sink is substantially smaller than the corre-sponding nominal contact surface area. Consequently, theheat transfer across the device/heat sink interface takes

Microgaps filled with air

Microcontacts

Interface

Fig. 1. Enlarged view of an interfacial joint for any two contactingsurfaces.

696 S. Shaikh et al. / Carbon 45 (2007) 695–703

place through surface-asperity micro-contacts and throughair-filled micro-gaps (Fig. 1), and is, hence, associated witha significant thermal resistance or thermal contact resis-tance [4].

To reduce the thermal resistance between contactingsurfaces different variety of materials commonly referredas thermal interface materials have been developed [5–9].These materials are used to reduce or completely eliminatethe air gaps from the contact interfaces by conforming tothe rough and uneven mating surfaces. Because the thermalinterface materials generally possess a higher thermal con-ductivity than the interfacial gas (air) they replace, theinterfacial thermal resistance is reduced giving rise to alower temperature of the electronic component junction.The efficiency of the thermal interfacial materials in reduc-ing the interfacial thermal resistance depends on a numberof factors among which thermal conductivity of the mate-rial and its ability to wet the mating surfaces appear to bethe most significant. There are several classes of thermalinterface materials. Some of the materials used to eliminateair gaps from a thermal interface include greases, reactivecompounds, elastomers and pressure sensitive adhesivefilms. All are designed to conform to surface irregularities,thereby eliminating air voids and improving heat flowthrough the thermal interface. Some have secondary prop-erties and functions, as well.

The design of a thermal interface material for fillinggaps differs from that of one for filling the valleys in thesurface topography of the mating surfaces that are in con-tact. These valleys contain air which should be displaced bya thermal interface material that is highly conformable andthat is very thin (ideally just thick enough to fill the valleys,as the thermal resistance increases with thickness) [10,11].Thus, a thermal interface material for filling the valleys istypically a paste of low viscosity. (The higher the viscosity,the less is the conformability [8].) However, for gap fill-ing, the thermal interface material must be thick enoughto fill the gap and be able to maintain its geometry. Thus,a gap-filling material typically involves a thixotropic paste(such as silicone filled with thermally conductive particles[5–7]), which is in contrast to the fluidic pastes (such as

polyethylene glycol filled with thermally conductive parti-cles [10,11]) that are effective for filling the valleys in sur-faces that are in contact. Another type of gap-fillingmaterial is a solid sheet such as a metal foil. Ideally thesheet exhibits resiliency and hence conformability. Metalsare ductile but have low values of the elastic limit. There-fore, metals are limited in resiliency or conformability. Incontrast, ‘‘flexible graphite’’ [12–15] is a graphite sheet thatis flexible and is resilient in the direction perpendicular tothe sheet. The resiliency is made possible by the microstruc-ture, which involves the mechanical interlocking of exfoli-ated graphite in the absence of a binder [16]. In general,a gap-filling material in the form of a solid sheet may bemade more effective by coating both sides of the sheet witha thermal paste. Thixotropic pastes have been used for thispurpose. Due to the relatively large thickness of a gap-fill-ing material, high thermal conductivity is a more importantattribute for a gap-filling material than for a material forfilling valleys in surfaces that are in contact.

Carbon nanotubes (CNT) with their light weight andhigh thermal conductivity value have the potential to beused as thermal interface material. The high intrinsic ther-mal conductivity of CNT [17–22] suggests many heat trans-fer enhancement applications. Choi et al. [23] measured theeffective thermal conductivity of nanotube-in-oil suspen-sions and found that, with 1 vol% of nanotubes, the effec-tive thermal conductivity increased significantly (morethan twice the value of the base oil) though not nearly asmuch as a simple, above-threshold percolation modelwould predict [24]. Biercuk et al. [25] used single-wallednanotubes (SWNTs) to augment the thermal transportproperties of industrial epoxy and found that epoxy loadedwith 1 wt% unpurified SWNTs exhibited a 70% increase inthermal conductivity at 40 K and 125% at room tempera-ture. Now the CNT possess the suitable properties so thatwhen used as interface material can be effective in filling thelow thermal conductivity micro-gaps at the same timeretaining the micro-asperity contact. Thus the CNT canplay a significant role in reducing the thermal resistanceby lowering or eliminating the micro-gaps at the same timeproviding a high thermal conductivity path through them.

Recently, Xu and Fisher [26] have shown that the ther-mal contact resistance between silicon wafers and copperwith a CNT array interface can be as low as 23 mm2 K/W. They also have shown that CNT arrays with differentproperties produce measurable variations in thermalenhancement. Fisher and Xu [26] concentrated on improv-ing the CNT array height and coverage on samples with anew catalyst configuration. In addition, they also studiedthe performance of other thermal interface materials(phase change material and indium sheet) and combina-tions of these materials with CNT arrays. Their experimen-tal findings showed that free-standing CNT arrays can bevery good thermal interface materials under moderate loadcompared to indium sheet and phase-change thermal inter-face materials. Further, combinations of CNT arrays andexisting thermal interface materials can improve these

S. Shaikh et al. / Carbon 45 (2007) 695–703 697

materials’ thermal contact conductance. According to theirresults under a load of 0.35 MPa, the PCM–CNT arraycombination produced a minimum thermal interface resis-tance of 5.2 mm2 K/W. However, variation of CNT arrayparameters including array density, diameter, and align-ment, deserves further study to explore their effects onenhancement of thermal contact conductance.

The twofold focus of this letter is first, to study the effect ofthermal contact resistance on heat transfer capability of highthermal conductivity materials and second is to demonstratethe effectiveness of CNT as thermal interface materials.

Fig. 3. Raman spectrum of vertically aligned single-walled and multi-walled CNTs.

2. Materials fabrication (aligned carbon nanotubes)

Using quartz as the substrate, we managed to growaligned CNTs with thickness ranging from several microm-eters to about 200 lm with a narrow diameter distributionaround 15 nm (Fig. 2). The thickness was monitored pre-cisely by the growth time. The growth time varies between10 and 30 min.

Scanning electron microscopy characterization showsthat CNT film is made of nanotube with homogeneouslength and diameter about 10 nm (Fig. 2). However, useof Raman spectroscopy shows that aligned CNT film ismade of mixture of single and multi-walled carbon nano-tubes (Fig. 3).

Even though our process is focused on making 100%multi-walled carbon nanotubes (MWNTs) it produced amixture of single-walled carbon nanotubes (SWNTs) aswell (Fig. 3). Raman spectra of these tubes are quite inter-esting because of resonance phenomena and sensitivity totube structure. There is very strong excitation wavelengthdependence of the spectra resulting from the electronicband structure. The most important feature in the Ramanspectrum of CNTs is the radial breathing mode (RBM),which is usually located between 75 and 300 cm�1 fromthe exciting line; an illustration of the spectrum resultingfrom this mode is displayed in Fig. 3. Since SWNTs are

Fig. 2. Scanning electron microscopy image of low density aligned CNT.

subject to inter-tube interactions which increase the fre-quency of the RBM, the D mode (disorder band locatedbetween 1330 and 1360 cm�1 when excited with a visiblelaser) is expected to be observed in (MWNTs). However,when it is observed in SWNTs, one assumes that it is duedefects in the tubes. The G mode or tangential mode corre-sponds to the stretching mode in the graphite plane. Thismode is located around 1580 cm�1.

3. Experimental analysis

The experimental work involved the use of a modernlight flash apparatus (LFA 447) in measuring the thermaldiffusivity of different samples analyzed with and withoutthermal interface material. Fig. 4 shows the simple princi-ple of operation of the LFA device. A xenon flash lampfires a pulse at the sample’s lower surface, while the infra-red detector measures the temperature rise of the sample’stop surface. A homogeneous illumination of the entiresample surface is achieved. The sophisticated software thendetermines the sample’s thermal diffusivity. The LFA 447 is

Testsample Furnace

Xenon lamp

Infra red Detector

Fig. 4. Principle of operation of light flash apparatus used for experiment.

698 S. Shaikh et al. / Carbon 45 (2007) 695–703

a modern contact-free laser flash method used for the mea-surement of thermal diffusivity of both solids and liquids.

The samples are located in an automatic sample changer,with which up to four samples can be measured in one mea-suring cycle. There are standard sample holders for testinground and square samples from 8 mm to 25.4 mm width.The furnace (room temperature to 300 �C) is directly inte-grated into the sample changer, creating a small thermalmass and therefore fast heating and cooling times are possi-ble. The non-contact measurement of the temperature riseguarantees an easy sample change and a short response timefor the signal acquisition system. There are 15 different eval-uation models, with and without correction, which can takeinto consideration heat losses from the side and from thefront surfaces. The thermal diffusivity measuring range is0.01–1000 mm2/s with reproducibility of approximately±3%. In addition to the thermal diffusivity, by employinga comparative method, the specific heat can also be deter-mined with this apparatus by using a known sample asthe reference. For the specific heat, a reproducibility of±5% is achieved. If the bulk density is known, a directdetermination of the thermal conductivity is possible. Thethermal conductivity range is 0.1–2000 W/mK.

Two types of materials were analyzed for thermal con-tact analysis:

• Aluminum (high thermal conductivity material).• Graphite (very high thermal conductivity material).

The value of density was first determined for all thematerials by measuring the weight of sample and its size.Now for each type of material four different tests were car-ried out. In the first case a single sample piece for eachmaterial with size 10 mm · 10 mm and 1.6 mm thicknesswas measured for its thermal diffusivity on the LFA. Forthe second case two similar sized pieces for the same mate-rial were kept in surface to surface contact (Fig. 5a) and thethermal diffusivity of the combined sample was measured.For the third case a graphite coating was provided betweenthe two sample pieces and the thermal diffusivity of thecombined sample was measured (Fig. 5b). Finally, for thefourth case a thin sheet of CNT TIM was sandwichedbetween the two sample pieces and once again the thermaldiffusivity of the combined sample was measured (Fig. 5c).A few drops of acetylene were added on the CNT sheet tohave better wetting at the interface and then dried off. In asimilar manner specific heat was measured for all the four

Directcontact

Fig. 5. Schematic of contacting surfaces: (a) in direct contact; (b) separated

type of cases for each material. All the measurements werecarried out at a temperature of 25 �C.

When any two surfaces are brought in contact the inter-face between them builds up a thermal resistance. The rateof conductive heat transfer, Q, across the interface is givenby

Q ¼ kAðT s1 � T s2Þ

Lð1Þ

where k is the thermal conductivity of the interface, A is theheat transfer area, L is the interface thickness and Ts1 andTs2 are the surface temperatures of the two bodies in con-tact. The thermal resistance of the joint, Rj, is given by

Rj ¼ðT s1 � T s2Þ

Qð2Þ

and on rearrangement,

Rj ¼L

kAð3Þ

Thus, the thermal resistance of the joint is directly pro-portional to the joint thickness and inversely proportionalto the thermal conductivity of the medium making up thejoint and to the size of the heat transfer area. Thermalresistance is minimized by making the joint as thin as pos-sible, increasing joint thermal conductivity by eliminatinginterstitial air and making certain that both surfaces arein intimate contact.

Thermal conductivity K for each material for all the allcases was estimated using Eq. (4)

K ¼ aqCp ð4Þwhere a denotes the thermal diffusivity, q the density andCp the specific heat in general for the different materials.

Thus, finally the total thermal resistance for the com-bined pieces for each materials was estimated for usingEq. (3) by substituting the values for thermal conductivity(as calculated above), nominal contact area, and the pre-cisely measured thickness of the samples. This total thermalresistance was a combination of the thermal resistance ofthe two solid pieces plus the thermal contact resistance attheir interface.

4. Measurements, results and discussion

The following tables give the thermal diffusivity andthermal conductivity for aluminum and graphite respec-tively with the percent enhancements (see Table 1).

Graphitecoating

CNT TIM

by graphite coating; (c) separated thin sheet of CNT interface material.

Table 1Measured thermal conductivity values for aluminum and graphite sampleswith their percentage enhancements

Sample type Thermal conductivity(W/m K)

Enhancement(%)

Aluminum solid 95.73 –Aluminum pieces in direct

contact8.957 –

Aluminum pieces withgraphite interface

37.983 324.06

Aluminum pieces with CNTinterface

43.457 385.16

Graphite solid 102.066 –Graphite pieces in direct

contact13.475 –

Graphite pieces with graphiteinterface

50.033 271.80

Graphite pieces with CNTinterface

62.278 362.79

Table 2Experimentally determined values of total thermal resistances for alumi-num and graphite samples

Sample type Thermal resistance (K/W)

Aluminum pieces in direct contact 3.573Aluminum pieces with graphite interface 0.842Aluminum pieces with CNT interface 0.736

Graphite pieces in direct contact 2.523Graphite pieces with graphite interface 0.679Graphite pieces with CNT interface 0.562

S. Shaikh et al. / Carbon 45 (2007) 695–703 699

The samples used for graphite materials were finely pol-ished pieces on the other hand the aluminum samples werepolished with sand paper. All the combined samples weresubjected to a same load creating a contact pressure of0.025 MPa.

For the aluminum samples a huge drop in thermal con-ductivity was observed for aluminum pieces in direct con-tact. With the coating of graphite a rise of over 300% wasobserved. Finally, with the introduction of CNT interfacematerial an enhancement of more than 350% was obtained.

For the graphite samples also considerable drop in ther-mal conductivity was observed for surfaces in direct con-tact. With the introduction of graphite coating anincrease of over 250% was obtained and by the insertionof CNT interface material an even higher enhancement ofmore than 350% was observed.

It was obvious from results that without any interfacematerial air trapped in the micro-gaps created a thermalresistance bringing down the thermal conductivity. Theeffect of micro-airgaps was more pronounced for the caseof aluminum with relatively rough surface as comparedto graphite. The finely polished surface of graphite materi-als created a better contact at the interface. From the ther-mal conductivity values obtained from the experiments forall three materials the CNT TIM produced the greatestenhancement. The thin CNT sheet TIM helped in provid-ing a high conductivity heat transfer path thus minimizingthe effect of micro-airgaps resistance. However, it was clearthat by combining the CNT sheet with a material with highwetting characteristic (phase change thermal interfacematerial) a better contact can be achieved especially forthe aluminum samples with rough surface.

The next task was to estimate the total thermal resis-tances corresponding to the conductivity values obtainedfor the aluminum and graphite samples. Thus by usingEq. (3) thermal resistances were determined as shown inTable 2 below. From this table, it can be seen that for allthe materials tested the inclusion of first the graphite coat-

ing and then the CNT sheet interface material helped inreducing the thermal contact resistance. Both the high ther-mal conductivity materials (aluminum and graphite) expe-rienced a greater thermal interface resistance without theinclusion of interface material. This may be due to the hugedrop in heat flux at the interface consisting of air gaps.Also, the relatively rough surface of the aluminum samplecaused a slightly greater resistance which means that forthe same contact pressure the use of smooth mating sur-faces in place of rough ones increases the thermal contactconductance. This is consistent with previous work ofLuo et al. [12], and is due to the relatively small size ofair pockets for the case of smooth surfaces. However, thecase of rough surfaces is closer to the situation inmicroelectronics.

5. Theoretical model

In order to compare the experimental results with someof the thermal resistance models from the literature a the-oretical study was carried for the two materials tested.The theoretical models were selected based on the rough-ness characteristics of the sample surfaces. Accordinglythe total thermal resistance of relatively rough aluminumsamples (0.8 lm) was determined using the model basedon micro-contacts Savija et al. [27] and Teertstra et al.[28]. For the case of polished graphite samples with surfaceroughness values of 0.121 lm the general theoretical modelbased on the work of Veziroglu [29] was implemented.

5.1. Thermal resistance calculations for aluminum

Thermal joint resistance at the interface is a function ofseveral geometric, physical and thermal parameters such assurface roughness and waviness, surface micro-hardness,thermal conductivity of the contacting solids, propertiesof the interstitial materials, and the contact pressure. Thethermal resistance of a joint formed by two nominally flatrough surfaces (Fig. 6) can be obtained from a model thatis based on following simplifying assumptions:

• Surfaces are nominally flat and rough with Gaussianheight distributions.

• The load is supported by the contacting asperities only.• The load is light; nominal contact pressure is small; P/

Hc � 10�3–10�5.

Yσ

mp

Fig. 6. Equivalent rough surface and smooth plane contact.

Fig. 7. Equivalent thermal resistance circuit for: (a) contacting surfaces indirect contact and separated by graphite coating; (b) contacting surfacesseparated by thin sheet of TIM (CNT).

700 S. Shaikh et al. / Carbon 45 (2007) 695–703

In general, the joint thermal conductance hj and jointthermal resistance Rj depends on the contact and gapcomponents.

The joint conductance is modeled as

hj ¼ hc þ hg ð5Þ

The joint resistance is modeled as

1

Rj

¼ 1

Rc

þ 1

Rg

ð6Þ

where Rc is the micro-contact resistance and Rg is the gapresistance. The expressions for both these terms are ob-tained using Eqs. (7)–(13) [27,28].

Rc can be obtained from the contact conductance rela-tionship for conforming rough surfaces and plastic defor-mation of contacting asperities:

hc ¼1

RcAa

¼ 1:25ks

mr

� � PH c

� �0:95

ð7Þ

where Aa is the apparent area of contact of two joining sur-faces and ks is the harmonic mean thermal conductivity ofthe two contacting surfaces of thermal conductivity ks andks and expressed as

ks ¼2k1k2

k1 þ k2

ð8Þ

The effective rms surface roughness r of the two contact-ing surfaces with roughness r1 and r2 can be expressed as

r ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2

1 þ r22

qð9Þ

The effective absolute mean asperity slope m can beobtained from the individual absolute mean asperity slopesof two contacting materials m1 and m2.

If the absolute mean asperity slopes m1 and m2 areunknown, they can be obtained from the approximate cor-relation equation of Antonetti et al.:

mi ¼ 0:1259ðri � 106Þ0:402 ð10ÞThe contact pressure is P and Hc is the surface micro-hard-ness of the softer of the two contacting solids.

Based on the assumptions given above, the gap conduc-tance is modeled as an equivalent layer of thickness t = Y

filled with TIM having thermal conductivity kg. The joint

resistance obtained from gap conductance can be expressedas

hg ¼1

RgAa

¼ kg

Y þMð11Þ

Rg ¼Y þMkgAa

ð12Þ

M = 0 if the gap is filled with thermal interface material, ifthe gap is filled with air, M = abK.

For air, gas parameters a = 2.4, b = 1.7 and molecularfree path K = 0.06 lm.

The mean plane separation Y shown in Fig. 6 is givenapproximately by the simple power law relation:

Y ¼ 1:53rP

H c

� ��0:097

ð13Þ

Using above relations the thermal contact resistance Rj canbe expressed as

Rj ¼1

Rc

þ 1

Rg

� ��1

ð14Þ

Rj ¼ 1:25Ks

mr

� � PH c

� �0:95

þ kg

Y þM

( )Aa

" #�1

ð15Þ

The above relation was used to calculate the thermal con-tact resistance Rj for the combined aluminum samples.

Now the total thermal resistance RT was calculatedbased on Fig. 7 for the aluminum samples using the rela-tions below.

With a direct contact and with graphite coating

RT ¼ Rb1 þ Rj þ Rb2 ð16Þ

With CNT interface material

RT ¼ Rb1 þ Rj þ 2RTIM þ Rb2 ð17Þ

S. Shaikh et al. / Carbon 45 (2007) 695–703 701

where Rb1 and Rb2 are the thermal resistances through thebulk solid pieces in contact.

5.2. Thermal resistance calculations for graphite

Various theories have been formulated to predict ther-mal conductance for a variety of conditions. Agreementwith experiments has been variable [30–36] especially whenvery smooth surfaces are involved. In order to correlate theanalytical and measured contact conductance it is first nec-essary to measure and quantify the finish of the opposingsurfaces. For rougher surfaces and if one or both of thesurfaces is composed of a material that can deform easilyat either the microscopic or macroscopic level then thepressure can affect the contact conductance. While discrep-ancies are often due to lack of precise input parametersWolff and Schneider [37] showed that a general theorycould be modified to give a good estimation of the thermalresistance with a variety of interface materials.

Now the general model for interface thermal conduc-tance is based on [29]. An effective gap thickness isdescribed by

d ¼ Nðd1 þ d2Þ ð18Þ

where N equals the slope of a best fit line through theempirical data points of the sum of surface roughness ver-sus the effective gap thickness Veziroglu determinedN = 3.56 for (d1 + d2) < 7 lm and 0.46 if >7 lm.

The effective fluid thermal conductivity is determined bykf = k0 if the interstitial fluid is liquid and

kf ¼k0

1þ 8cðm=t0Þða1þa2�a1a2Þ100dPra1a2ðcþ1Þ

þ 4rdE1E2T 3m

E1 þ E2 � E1E2

ð19Þ

if it is a gas.Now the conduction number K is found from relation

K ¼ kf

k1 þ k2

2k1k2

� �ð20Þ

The constriction number C is defined as

C ¼ffiffiffiffiffiPM

rð21Þ

while the interface number S is

S ¼ffiffiffiAp

dð22Þ

The gap number B is the relationship between the constric-tion number and the interface size number given by

B ¼ 0:335C0:315S0:137 ð23Þ

The conductance number U can be found by the iterationof the following transcendental equation:

U ¼ 1þ BC

K tan�1fð1=CÞð1� ð1� UÞÞ0:5 � 1gð24Þ

or graphically from the chart for thermal contact conduc-tance using C and B/K.

The thermal contact conductance u is thus given byrelation

u ¼ hA¼ Ukf

dð25Þ

Thus, the joint thermal contact resistance for the polishedsurface can be obtained from the relation

Rj ¼1

u¼ d

Ukf

ð26Þ

Wolff and Schneider [37] used the above model for ther-mal contact conductance and noticed the discrepancybetween their measured and predicted values based onthe expression for the effective gap thickness given by Eq.(18). They compared the data for three size ranges for0.2–0.7 lm, 0.7–7 lm, and 7–90 lm by regression analyses.The results of their analyses suggested a value of aboutN = 84 (for d1 + d2 = 0.1158 lm).

Based on the measured values of surface roughness forgraphite (0.121 lm), we selected a suitable value for N,by using extrapolation technique for the data from [37].Taking the value of other parameters in Eq. (19) for air,the joint thermal resistance was calculated for graphitesamples using Eq. (26). Finally, the total thermal resistancefor this material was calculated using Eqs. (16) and (17).

6. Computations results and discussion

6.1. Aluminum samples

The total thermal resistance estimated for the aluminumsample using Eq. (17) for different cases is shown in Table 3below. For all the cases studied using the theoretical modelthe thermal resistance model slightly overestimates thevalues.

For the case of aluminum pieces in direct contact as themicro-contact resistance (Rc) is much greater than themicro-gap resistance (Rg), the joint resistance Rj is primar-ily dependent on Rg. Since the micro-gap is filled with airwith a low thermal conductivity value a high value of ther-mal contact resistance is obtained in this case. For the casewith graphite coating, it was assumed some of the graphiteparticles fill the micro-gaps, which lowered down Rc andalso due the presence of graphite particles in the micro-gap the conductivity value kg was taken as the harmonicmean of the air and graphite particle conductivity. This fur-ther lowered down Rg which resulted in the reduction in thejoint resistance Rj. For the case of aluminum pieces withCNT TIM the micro-contact at the aluminum and TIMmaterial produced a low thermal resistance due to the highthermal conductivity of the CNT sheet. Also it wasassumed that the retention of CNT in the micro-gapsaffects the value kg, which reduced the value of Rg andeventually the value of Rj. The total thermal resistancevalues obtained for all cases using the theoretical model

Table 3Theoretically estimated values of total thermal resistances for aluminumsamples

Sample type Thermal resistance(K/W)

Error (%)

Aluminum pieces in direct contact 3.751 4.9Aluminum pieces with graphite

interface0.889 5.6

Aluminum pieces with CNTinterface

0.778 5.8

Table 4Theoretically estimated values of total thermal resistances for graphitesamples

Sample type Thermal resistance(K/W)

Error (%)

Graphite pieces in direct contact 2.658 5.4Graphite pieces with graphite

interface0.720 6.0

Graphite pieces with CNT interface 0.598 6.4

702 S. Shaikh et al. / Carbon 45 (2007) 695–703

were within 6% error as compared with experimentalvalues.

6.2. Graphite samples

The total thermal resistances values for graphite samplesfor different cases are shown in Table 4 below. The valuesfor gap coefficients N to calculate the effective distancebetween surfaces d were estimated by extrapolating thedata from regression analysis of [37]. Accordingly the val-ues of N were taken as 92 for graphite.

For graphite samples the model for polished surfacespredicted an overestimation in total thermal resistance val-ues. The thermal contact resistance from Eq. (26) dependson the conductance number U, the fluid thermal conductiv-ity kf, and the effective distance between surfaces d. Theimportance of having good conforming surfaces is clearlyemphasized by the fact that this can lead to the minimizingof d. Also with such surfaces a high thermal conductivityTIM can become more effective by providing a greater heattransfer path. Since most of the data for calculating theabove parameters was gathered from empirical relationsin Eqs. (20)–(23) the predictions of the theoretical modelwith less than 6.5% error shows a reasonable agreementwith the experimental values.

7. Conclusions

An experimental analysis was conducted using a modernlaser flash device to measure the effective thermal conduc-tivity of aluminum and graphite for different cases: in directcontact, with a graphite coating and with a CNT TIM. Theinclusion of graphite coating and thin sheet of CNT TIMshowed considerable reduction in thermal resistance values

for both materials. In order to compare the experimentalresults with the theoretical models from literature two ther-mal resistance models were selected based on the surfaceroughness of materials analyzed. The values of total ther-mal resistances using both models showed reasonableagreement with the experimental results with an error ofless than 6.5%. From the experimental and theoreticalstudies it was established that two factors namely the pres-ence of conforming surfaces and high thermal conductivitypath are extremely important for lowering the thermal con-tact resistance. A good surface-to-surface contact at theinterface improves the efficiency of the high conductivitycoating or TIM in transferring heat across the interfaceand thereby lowering the thermal interface resistance. Forboth aluminum and graphite the presence of micro-gapswas the main factor in the high thermal contact resistance.The graphite spray coating filled the gaps to certain extentand also created a greater conductivity at the micro-contactthereby lowering down the thermal resistance. However,the CNT TIM proved to be more effective by further low-ering down the thermal contact resistance and improvingthe thermal contact conductance for both the materialsanalyzed. An important factor which can further improvethe performance of the CNT TIM is its wettability. Theuse of phase change materials with high wetting character-istic can be used along with the CNT sheet to form an evenbetter conforming contact. This can further lower down thethermal interface resistance value by reducing or even elim-inating the micro-airgaps at the interface.

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