Proceedings of InterPACK2003:International Electronic Packaging Technical Conference and Exhibition

Maui, Hawaii, USA, July 6-11, 2003

InterPack2003-35088

EFFECTIVE THERMOPHYSICAL PROPERTIES OF THERMALINTERFACE MATERIALS: PART I DEFINITIONS AND MODELS

I. Savija, J.R. Culham, M.M. YovanovichMicroelectronics Heat Transfer LaboratoryDepartment of Mechanical Engineering

University of WaterlooWaterloo, Ontario, Canada N2L 3G1http://www.mhtlab.uwaterloo.ca

Proceedings of IPACK03 International Electronic Packaging Technical Conference and Exhibition

July 6-11, 2003, Maui, Hawaii, USA

IPACK2003-35088

ABSTRACT

The conductivity of thermal interface materialsare typically determined using procedures detailed inASTM D 5470. The disadvantages of using these ex-isting procedures for compliant materials are discussedalong with a proposed new procedure for determiningthermal conductivity and Young’s modulus.

The new procedure, denoted as the Bulk Resis-tance Method, is based on experimentally determinedthermal resistance data and an analytical model forthermal resistance in joints incorporating thermal in-terface materials. Two versions of the model are pre-sented, the Simple Bulk Resistance Model, based onthe interface material thickness prior to loading anda more precise version denoted as the General BulkResistance Model, that includes additional parame-ters such as surface characteristics and thermophysi-cal properties of the contacting solids. Both methodscan be used to predict material in situ thickness as afunction of load.

NOMENCLATURE

A = area of contact (m2)ASTM = American Society for Testing and

Materialsa = linear fit coefficient (slope)BRM = Bulk Resistance Methodb = linear fit coefficient (intercept)

11

c1, c2 = coefficients for Vickersmicrohardness

E = Young’s modulus (MPa)H = hardness (MPa)h = thermal conductance (W/m2K)k = thermal conductivity (W/mK)ks = harmonic mean thermal

conductivity (W/mK)m = asperity mean absolute slope (rad)mp = mean planen = number of data points in

Bulk Resistance MethodP = contact pressure (MPa)PCM = phase change materialPV C = Polyvinyl chlorideRTV = room temperature vulcanizingQ = heat transfer rate (W )R = thermal resistance (K/W )RMS = root mean squarer = specific thermal resistance (m2K/W )T = temperature (K)TIM = thermal interface materialt = thickness (m)∆Tj = joint temperature drop (K)Y = mean plane separation (m)λ = dimensionless mean plane

separationν = Poisson’s ratioσ = RMS roughness (m)

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Subscripts1, 2 = indices for contacting surfacesa = apparentb = bulkB = Brinellc = contacte = elasticf = finalg = gapi = indices for interfacesj = jointl = lowerm = thermal interface materialo = initialp = polymerr = radiationu = upper

Superscripts∗ = dimensionless′

= effective

INTRODUCTIONWhen two surfaces are in mechanical contact, the

resulting interface consists of numerous microcontactsand gaps that separate the surfaces. Because the realcontact area is a small fraction of the apparent contactarea, heat flow from one surface to another must over-come the thermal joint resistance. The thermal jointresistance depends on many geometric, mechanical andthermal parameters, including contact pressure, char-acteristics of interface surfaces, thermophysical prop-erties of the contacting solids, and if present, the in-terstitial substance in the gap. Due to this resistance,a temperature drop across the interface is observable.

In nuclear power generating systems, aerospaceand microelectronics applications, the interface formedbetween two surfaces is a very important, if not crit-ical, part of the thermal network established betweenthe heat source and the heat sink. The need to en-hance heat transfer across the interface becomes animportant issue in the design of high thermal energydissipating systems. Therefore, the interest in inter-stitial materials that provide better thermal contactbetween the surfaces is increasing rapidly. These ma-terials are known as thermal interface materials (TIMs)and include greases, oils, gases, thin metallic and non-metallic coatings, metallic foils, polymeric compositesand phase-change materials (PCMs).

A wide group of polymeric materials is commonlyused in microelectronics cooling. Usually, silicon-basedcompounds with highly conductive filler particles suchas ceramic powder, boron nitride and aluminum oxideare reinforced with fiberglass, aluminum foil, Kapton

22

film or nylon mesh. Materials are manufactured in theform of thin sheets varying from a fraction of a mil-limeter to a few millimeters in thickness and can haveadhesive on one or both sides. The performance de-pends on the material conductivity generally found tobe 1 - 10W/mK, and the compliance to the contactingsurface which is a function of the material hardness. Inaddition, there is a similar group of flexible graphite-based materials, which are very compliant and showgood thermal performance. The thermal interface ma-terials are generally assessed by considering thermalconductivity and thermal resistance as the main prop-erties.

Thermal conductivity and thermal resistance areusually reported in the material specifications and aremainly determined following the ASTM D 5470 pro-cedure. With no information on this test procedure,a thermal design engineer would choose the materialwith the lowest reported resistance. With such designpractice, however, in most engineering applications,the reported resistance obtained from the ASTM D5470 procedure does not agree with the true resistanceof the interface. Not surprisingly, the highest thermalconductivity does not necessarily imply the lowest ther-mal resistance since many geometrical, mechanical andthermophysical parameters such as thermal conductiv-ity, Young’s modulus, hardness, compliance, surfaceroughness and waviness effect the TIM thermal perfor-mance. Therefore, more detailed material data sheetsshould be available while the procedures for thermo-physical characterization should be selected carefullyin order to properly identify material’s thermophysicalproperties and reduce thermal joint resistance in thereal application.

The present work is confined to the study of ther-mal interface materials in the form of thin sheets.An analytical model for the thermal resistance ofjoints formed by the mechanical contact of conforming,rough, nominally flat surfaces with interface materialswill be developed here. The model will give a rela-tionship between thermal joint resistance and the jointparameters such as contact pressure, surface parame-ters and thermophysical properties of interface mater-ial and contacting solids.

The main objective of the first part of this studyis to propose a method for determining material ther-mophysical properties. The method incorporates ananalytical model and determines thermal conductiv-ity and effective Young’s modulus from thermal resis-tance data. Also the disadvantages of the widely usedexisting standard ASTM D 5470 method will be dis-cussed and recommendations for its improvement willbe presented. In the second part of this study (Sav-ija et al. [1]) an extensive experimental invesigation of

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Grafoil GTA interface material will be conducted andfrom the obtained thermal resistance data the mater-ial properties will be determined applying the methodproposed in the first part of the study.

REVIEW OF PREVIOUS WORKSummary of studies on thermal interface materials

by Savija et al. [2] shows that new commercially avail-able polymeric and graphite-based sheets are mainlycharacterized experimentally. There are few analyticalstudies related to the thermal behavior of sheet ma-terials based on the material properties and the me-chanical and geometrical joint characteristics. How-ever, thermal properties of the materials used in theabove studies are obtained by standard methods foundto have certain limitations.

Table 1 is a summary of various experimental stud-ies of the thermal conductance for joints with poly-mers. One of the first experimental investigations onthermal conductance of metal-to-polymer joints wasconducted by Fletcher and Miller [3]. The joint con-ductance values for tested elastomers were lower thanthe conductance of a bare aluminum junction. It wasalso concluded that elastomers with metallic or ox-ide fillers yielded higher conductance values than plainelastomers. Fletcher et al. [4] conducted an experimen-tal investigation of polyethylene materials to determinethe effect of additives on their thermal characteristics.The samples were tested at load pressures ranging from0.41MPa to 2.76MPa and mean junction tempera-tures of 29−57 oC. The thermal conductance increasedwith increasing temperatures and content of additivessuch as carbon.

Ochterbeck et al. [5] conducted an experimental in-vestigation on thermal conductance of polyamide filmscombined with paraffin, commercial-grade diamondsand metallic foils. The experimental data indicatedthat polyamide films coated with paraffin-based ther-mal compound showed the best thermal performanceand improved contact conductance seven to ten timescompared to bare joints. The hardness of the intersti-tial material was seen to be the most important para-meter in the selection of an interface material.

Marotta and Fletcher [6] presented experimentalconductance data for several polymers. The conduc-tance of the materials tested were shown to be inde-pendent of pressure (0.51−2.76MPa) except for poly-ethylene, Teflon and polycarbonate which are relativelysoft and ductile thermoplastic polymers. The apparentthermal conductivity of the materials was measured attemperatures between 10 oC and 100 oC and a pressureof 1.38MPa. Almost all materials tested had thermalconductivity values independent of temperature.

Parihar and Wright [7] performed a detailed ex-

3

periment measuring the thermal contact resistance ofstainless steel SS 304-to-silicone rubber joint in air, un-der light pressures (0.02 − 0.25MPa). The authorsobserved that the resistance at the hot interface was1.3 to 1.6 times greater than the resistance at the coldinterface. The resistances were different due to therubber conductivity dependence on temperature. Thejoint resistance Rj decreased with increasing load. Ingeneral, contact resistance was shown to be a strongfunction of temperature due to the large temperaturedependence of the rubber thermal conductivity and toa lesser extent of pressure P due to the elastomer soft-ness.

An experimental study conducted by Mirmira etal. [8] showed that thermal contact conductance ofsome commercial elastomeric gaskets and graphite-based materials become less dependent on the contactpressure as the load increased, with the bulk conduc-tance becoming predominant in the high pressure range(around 1000 kPa− 1500 kPa). The authors observedthat the change in the mean interface temperature didnot significantly effect the thermal conductance valuesfor the gasket materials. Materials with fiberglas re-inforcement showed poorer thermal performance thanmaterials without reinforcement. Also, these materi-als demonstrated hysteresis effects - the conductancein the loading cycle was lower than in the unloadingcycle.

The most comprehensive analytical study based onpolymer experimental data was conducted by Fullerand Marotta [9]. They obtained an analytical modelfor predicting thermal joint resistance between met-als and thermoplastic or elastomeric polymers by as-suming optically flat surfaces at uniform pressures invacuum. The mode of the deformation between themetal and the softer polymer was assumed to be elas-tic under light to moderate load based on experimentalstudies conducted by Parihar and Wright [7]. Incor-porating the elastic model of Mikic [10] and defininga new polymer elastic hardness from the Greenwoodand Williamson [11] definition of the elastic contacthardness, a simple correlation for dimensionless con-tact conductance was obtained:

hcσ

ksm= 1.49

(2.3PEpm

)0.935

(1)

By defining the thickness in terms of strain, the follow-ing expression for the bulk conductance was derived:

hb =kp

tpand tp = tpo

(1− P

Ep

)(2)

where kp and tp are the polymer conductivity andthickness under load, respectively and tpo is the poly-

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Table 1: Research Related to Polymer-based Interstitial Materials Tested in Vacuum

Authors Contact Material Interstitial MaterialsFletcher and Miller [3] Al 2024-T4 Silicone elastomers, Fluocarbon

elastomer, Nitrile elastomerFletcher et al. [4] Al 2024-T4 Ethylene vinyl acetate copolymers,

Ethyl vinyl acetate copolymer,Polyethylene homopolymer

Ochterbeck et al. [5] Al 6061-T6 Polyamide in combinations withfoil, paraffin, diamonds, copper

Marotta and Fletcher [6] Al 6061-T6 Polyethylene, PVC, Polypropylene,Teflon, Delrin, Nylon,Polycarbonate, Phenolic

Parihar and Wright [7] SS 304 Silicone rubber (elastomer)Mirmira et al. [8] Al 6061-T6 Elastomeric gaskets

(Cho-Therm, T-pli, Grafoil)Fuller and Marotta [9] Al 6061, SS Delrin, Teflon, Polycarbonate, PVCMarotta et al. [12] Al 6061 eGraf, Furon (graphite-based)

mer thickness at zero load. Therefore, the joint con-ductance was defined as:

hj =1

1hc,1

+tpo [1− (P/Ep)]

kp+

1hc,2

(3)

Experimental data from Marotta and Fletcher [6] andFuller and Marotta [9] were compared to the joint con-ductance model and good agreement was found.

The most recent experimental and analytical stud-ies on TIMs were conducted by Marotta et al. [12]. Ananalytical model for the resistance across the joint in-corporating a sheet of interstitial elastic material and agap filler such as gas or phase change material, was de-veloped. The proposed joint model is compared withthe experimental data obtained for several commer-cially available graphite materials. The specimens wereplaced under 0.34 − 1.03MPa pressure, between alu-minum 6063 contacting surfaces of 1µm roughness.The joint resistance ranged from 65.8mm2K/W to10.3mm2K/W showing a good agreement with theproposed model. A reduction in thermal performanceof the tested TIMs was observed as the joint tempera-ture increased from 40 oC to 80 oC. This was explainedby degradation in material thermal conductivity.

Thermal Conductivity Measurement MethodsThere are a number of ASTM standard test proce-

dures for quantifying thermal conductivity of thin ma-terial sheets. Some comparative test procedures areASTM E 1530, E 1225, F 433 and C 518, while thereis only one direct method, ASTM D 5470. The ASTM

44

D 5470 method or “Standard Test Method for Ther-mal Transmission Properties of Thin Thermally Con-ductive Solid Electrical Insulation Materials,” is themethod that manufacturers and distributors generallyrefer to when reporting the material conductivity mea-surements.

The test method is designed for measuring jointthermal resistance of homogeneous or composite ther-mally conductive layers having a thickness rangingfrom 0.02mm to 10mm (ASTM D 5470-01 [13]). Theobtained thermal resistance data are used to determinethe effective thermal conductivity. The test method isapplied to materials with low Young’s modulus thatare malleable enough to comply to the surface and ex-clude air from the interfaces so that thermal contactresistance is negligible. In the case of high-modulusmaterial, test specimens are combined with the low-modulus layers. The tested material can be stacked toobtain results for different thickness specimens whereit is assumed that the layers coalesce resulting in in-significant contact resistance at the interfaces.

The ASTM D 5470 apparatus is shown in Fig. 1.Meter bars are manufactured from a highly conductivematerial such as aluminum with contacting surface fin-ish within 0.4µm. The specimen thickness is measuredbefore test using ASTM D 374 - Method C. The singlematerial layer or stacked layers are subjected to 3MPapressure so that the thermal contact resistance is re-duced to an insignificant level. The insulating materialis placed around the calorimeter sections and the jointtemperature is maintained at 50 oC while the tests areconducted in air.

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Figure 1: Test Apparatus for ASTM D 5470Method (ASTM D 5470-01 [13])

Defining the bulk resistance as rb = to/k, whereto is thickness of the material before loading, and plot-ting the measured resistance of the single and stackedsheets as shown in Fig. 2, the apparent thermal con-ductivity is obtained as a reciprocal of the slope whilethe value at the y-intercept is the value of thermal con-tact resistance.

Using this method, the effective conductivity ofthe incompressible tested material is well predicted,while the effective conductivity of a compressible ma-terial is overestimated. The enhancement of the heattransfer across the joint due to the thickness reductionis incorrectly related to the conductivity. Determin-ing rc as y-intercept assumes constant rc for all ma-terial thicknesses. This assumption is acceptable onlyat high pressures where the bulk resistance dominatesand total rc is negligible. Difficulties validating thisassumption arise if data at only one pressure point areknown so domination of the bulk resistance becomesquestionable.

In industrial applications TIMs are mainly usedat contact pressures of about 0.14MPa and thereforethermal resistance and conductivity are mainly deter-mined with ASTM D 5470 procedure at this low pres-sure. However, contact resistance at this pressure issignificant for most materials and can not be assumedto be constant for all thicknesses. This can cause agreat non-linearity of thermal resistance data in Fig. 2

55

and lead to unreliable conductivity values. Also rj

measured at low pressures can not be used as a refer-ence value since rc plays a significant role and dependson surface characteristics and thermophysical proper-ties of the contacting solids.

Initial Thickness (m)

Spe

cifi

cJo

intR

esis

tanc

e(m

2 K/W

)

rj = rc+to/k

rc

to,nto,1 to,2

Figure 2: Determining Thermal ConductivityUsing ASTM D 5470 Method

MODELING OF THERMAL JOINTRESISTANCE

A typical joint incorporating a thermal interfacematerial in the form of a sheet is presented in Fig. 3.Two industrial surfaces, non-flat and rough, are in con-tact with a sheet of softer, non-flat and rough interfacematerial. The vertical dimensions in Fig. 3 a) are ex-aggerated for clarity and the material thickness tm ismuch greater than the surface roughness σ1 and σ2 sothat tm � σ =

√σ2

1 + σ22 . k1 and k1 are conductiv-

ities of contacting bodies and km is the conductivityof the TIM. Because of the flatness deviations and thesurface roughness, contact between the surfaces occursat only a few points. The distribution of these pointsis hard to determine, making geometrical and thermalproblems difficult to solve. Referring to Fig. 3 b), ifthe contact surfaces of the solids numbered 1 and 2are machined to be flat and sufficient load flattens theinterface material then a thermal joint which is easierto analyze is created in the form of two conforminginterfaces.

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Figure 3: a) Real Joint b) ConformingSurfaces Joint with TIM

At the microscopic level, the interface between thesheet and the contacting solid, or another TIM sheet,consists of numerous discrete microcontacts and gapswhich separate the two surfaces as shown in Fig. 4.

Figure 4: Conforming Rough Surfaces-Thermal Interface Material Joint

When steady heat transfer is present the following heatflow paths are possible: conduction through the micro-contacts (Qc), conduction through the gaps (Qg), andradiation (Qr) if the interstitial substance is transpar-ent to radiation. The heat transfer rate across the jointis (Yovanovich [14], Yovanovich et al. [15]):

Qj = Qc +Qg +Qr (4)

and

Qj =∆Tj

Rjand Qj = hjAa∆Tj (5)

where ∆Tj is the temperature drop, Rj is the thermaljoint resistance, hj is the thermal joint conductance,and Aa is the apparent contact area. The joint con-ductance and resistance are related as follows:

hj =1

AaRj≡ 1

rj(6)

where rj is the specific joint resistance introduced todefine thermal joint resistance for the apparent contactarea.

66

The thermal resistance is affected by a great num-ber of geometric, mechanical and thermophysical para-meters: surface roughness, mean asperity slope, appar-ent contact pressure, contact or elastic hardness, com-pressive modulus of elasticity of the TIM, thickness ofthe layer and thermal conductivities of all bodies incontact. If the radiation at the joint is neglected (rea-sonable assumption for most applications where thejoint temperature is below 600 oC) the correspond-ing thermal resistance network can be presented withFig. 5.

Figure 5: Joint with Thermal Interface MaterialThermal Resistance Network

In the general case, the thermal joint resistanceand conductance are defined as follows (Yovanovich etal. [15]):

Rj =[

1Rc1

+1Rg1

]−1

+ Rb +[

1Rc2

+1Rg2

]−1

(7)

1hj

=1

hc1 + hg1+

1hb

+1

hc2 + hg2

where Rc, Rg and Rb are the contact, gap and bulkresistance respectively and hc, hg and hb are the corre-sponding thermal conductances. The two interfacesgenerally have different contact and gap resistancessince contacting surfaces can have different propertiesand gaps can be occupied with different substances.

This problem can be simplified for some specialcases. If there is no substance present in the gapsand the thermal joint is in vacuum, then Rg1 → ∞,Rg2 → ∞, hg1 → 0 and hg2 → 0, and:

Rj = Rc1 +Rb +Rc2

1hj

=1hc1

+1hb

+1hc2

(8)

At relatively high contact pressure, where the bulkresistance dominates over the contact resistance, i.e.hc1 >> hb and hc2 >> hb, a further simplification ispossible:

Rj = Rb and hj = hb (9)

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The relations for contact and bulk resistancesand corresponding conductances can be obtained frommodels that are based on the following simplifying as-sumptions:

(a) nominally flat rough surfaces with a Gaussian as-perity height distribution

(b) random distribution of surface asperities over theapparent area

(c) thermal conductivity of interface material doesnot vary with pressure

(d) thickness change is linear under an applied load

Plastic Contact Conductance ModelIf the surface asperities of the softer material, i.e.

TIM, experience plastic deformation, the contact con-ductance can be defined using the relation for conform-ing rough surfaces and plastic deformation of contact-ing asperities (Cooper et al. [16]):

hc =√2ks

4√π

m

σ

exp(−λ2/2)[1−

√12erfc(λ/

√2)

]1.5 (10)

where ks is the harmonic mean thermal conductivity:

ks =2 kikm

ki + km(11)

The effective surface roughness is:

σ =√σ2

i + σ2m (12)

and absolute asperity slope is:

m =√m2

i +m2m (13)

where i=1,2 refers to the interface i.e. contacting solidand subscript m refers to the interface material. Theeffective gap thickness λ is given by the theoretical re-lation (Yovanovich [14]):

λ =√2 erfc−1 2P

Hc(14)

where P is the apparent contact pressure and Hc is thecontact hardness of the softer material.

Equation (10). is reduced to a simpler form(Yovanovich [14]):

hc = 1.25 ksm

σ

(P

Hc

)0.95

and rc =1hc

(15)

77

Song and Yovanovich [17] showed that the relativecontact pressure is:

P

Hc=[

P

c1(1.62σ/m)c2

] 11 + 0.071c2 (16)

The Vickers microhardness correlation coefficientsc1 and c2, are related to the Brinell hardness by therelationships (Sridhar and Yovanovich [18]):

c13178

= 4.0− 5.77H∗B + 4.0 (H∗

B)2 − 0.61 (H∗B)3

c2 = −0.370 + 0.442(HB

c1

)(17)

where HB is the Brinell hardness, and H∗B = HB/3178

for a Brinell hardness range from 1300MPa to7600MPa.

Elastic Contact Conductance ModelFor conforming rough surfaces and elastic defor-

mation of contacting asperities, Mikic [10] obtained thefollowing relations for contact conductance:

hc =ks

4√π

m

σ

exp(−λ2/2)[1−

√14erfc(λ/

√2)

]1.5 (18)

or

hc = 1.55ksm

σ

(√2P

E′m

)0.94

and rc =1hc

(19)

The effective gap thickness is obtained from thefollowing approximation (Yovanovich [14]):

λ =√2 erfc−1 4P

He(20)

where the elastic contact hardness is defined as:

He =mE′√2

(21)

The effective Young’s modulus of the solid-interface material interface, E′, is:

1E′ =

1− ν2i

Ei+

1− ν2m

Em(22)

where νi, Ei and νm, Em are Poisson’s ratio andYoung’s modulus of the contacting solid and thermalinterface material respectively.

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Bulk Conductance ModelThe specific bulk resistance of the material itself

is directly related to its thickness and thermal conduc-tivity:

rb =tmkm

and hb =km

tm(23)

Material thickness can vary with load depending onthe Young’s modulus. If the material is compressibleand shows linear deformation under load, the bulk re-sistance becomes:

rb =tmo

(1− P

Em

)km

(24)

where tmo is the initial thickness at zero load and Em

is the effective Young’s modulus of the material. Thesmaller the elastic modulus, the larger the materialdeformation. As a direct consequence of the materialthickness decrease under a load, the bulk and over-all thermal joint resistance is reduced. Although bulkresistance is an additional term in the resistance net-work, the overall joint resistance generally decreaseswhen TIM is present at the joint since the contact re-sistance is greatly reduced.

Summary of Conductance ModelsFrom the presented contact and bulk resistance

models for joints incorporating TIMs in the form ofsheets, it can be concluded that a TIM’s thermal con-ductivity, hardness and Young’s modulus are impor-tant parameters in the thermal resistance network. Asexpected, a TIM with higher thermal conductivity en-hances the heat transfer through the joint. The hard-ness and Young’s modulus define the material’s com-pliance to the surface and effect on contact resistance.If the material is soft enough to completely occupy thegaps then the contact resistance is greatly reduced.

A plot of modeled thermal joint resistance versuscontact pressure (Fig. 6) shows the three resistanceregions. The first region, referred to as the contactresistance region, appears at relatively low pressureswhere the contact resistance dominates the trend of thejoint resistance. The second region is the transition re-gion where the contact resistance becomes significantlysmaller. The third region, occurring at higher pres-sures, is the bulk resistance region, where contact resis-tance is significantly reduced. If the material thicknessreduction under load is linear, the linear trend of thebulk resistance region can be observed. The position ofthese three regions on such a plot depends on materialhardness, Young’s modulus, and the characteristics ofall surfaces in contact.

88

Generally, the waviness of the TIM and contact-ing solids is significant which makes mechanical andthermal modeling difficult due to the lack of analyti-cal models for random non-conforming surfaces. Thethermal joint resistance for conforming surfaces can beaccepted as the lower bound of the thermal joint re-sistance for wavy surfaces. The waviness of TIM di-minishes at higher pressures while the bulk resistancedominates over the contact resistance.

BULK RESISTANCE METHODA new method for determining the TIM’s thermal

conductivity from thermal resistance data incorporat-ing the above analytical model is developed by Sav-ija [19]. The method is named the Bulk ResistanceMethod (BRM) since it is applied to the data in thebulk resistance region. Unlike the ASTM D 5470 pro-cedure, the BRM considers material thickness changesunder load and in addition to thermal conductivity pre-dicts a value of the effective Young’s modulus and insitu thickness.

In order to apply the BRM to the thermal re-sistance data, the following experimental parametersneed to be known:

(a) interface temperature,(b) material initial thickness,(c) surface characteristics of contacting solids and

thermal interface materials:

(i) RMS roughness,(ii) mean asperity slope and(iii) waviness;

(d) material properties of contacting solids:

(i) conductivity (at the interface temperature),(ii) Young’s modulus and(iii) Poisson’s ratio;

(e) properties of thermal interface materials:

(i) contact hardness and(ii) Poisson’s ratio.

If the surface characteristics and thermophysicalproperties are not known or difficult to measure, theSimple BRM is applicable as the above properties af-fect contact resistance and are assumed to be insignif-icant in the bulk region. If all of the above propertiesare known or measured then the General BRM canbe applied. The thermal interface temperature doesnot directly influence this method, but it should berecorded since thermal conductivity of some materialproperties can show significant dependence on temper-ature.

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as

wi

Development of Simple Bulk ResistanceMethod

From the thermal resistance data obtained at var-ious loads the bulk resistance region is identified as theregion where data points fit a linear trend as shown inFig. 6.

Region

Contact Pressure (MPa)

The

rmal

Join

tRes

ista

nce⋅

104

(m2 K

/W)

0 1 2 3 4 5 6 7

ResistanceBulk

ContactResistance

Region

TransitionRegion

rj = aj P + bj

rci

rb = ab P + bb

0

1

2

3

4

5

6

Figure 6: Specific Thermal Joint Resistance forAl 2024 - Grafoil GTA - Al 2024 Joint

A linear fit in the form:

rj = aj P + bj (25)

is applied to n available data points in the bulk region.For the Simple BRM, the thermal joint resistance inthe bulk region is assumed to be the same as the bulkresistance:

rj = rb =tmo

(1− P

Em

)km

(26)

Differentiating with respect to contact pressure,the following expression results:

drj

dP= − tmo

km

1Em

(27)

Solving Eqs. (26) and (27) for km and Em, andintroducing the slope of the linear fit aj :

drj

dP= aj (28)

the expressions for thermal conductivity and effectivemodulus of elasticity become:

km =tmo

rj − P aj(29)

99

Em = − tmo

km

1aj

(30)

Discretizing Eq. (29), the values of conductivityre obtained for each pressure point, Pi, and corre-ponding thermal joint resistance, rji :

kmi =tmo

rji − Pi aj(31)

here i = 1, 2, ..., n. The resulting thermal conductiv-ty is calculated by averaging kmi values:

km =1n

n∑i=1

kmi (32)

Figure 7: Simple Bulk Resistance Method-FlowChart

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Knowing km and going back to Eq. (30), the effec-tive Young’s modulus, Em, is easy to determine. Thein situ thickness of the material can be obtained as afunction of pressure:

tm = tmo

(1− P

Em

)(33)

The uncertainty in the obtained values of materialthermophysical properties depends on the experimen-tal data and the slope of the linear fit - the more dataobtained in the bulk resistance region, the more accu-rate the slope and the less erroneous final result. TheSimple Method is presented in the flow chart in Fig. 7.

Development of General Bulk ResistanceMethod

Similar to the Simple BRM, the bulk resistance re-gion is identified. Based on the surface characteristicsand properties of the contacting solids and TIM, thethermal contact resistance is calculated as a functionof pressure using Eqs. (15) or (19) The type of asperi-ties deformation can be determined from the differencebetween loading and unloading cycle. If the asperitydeformation is elastic, the contact resistance remainsthe same in unloading cycle, so the discrepancy be-tween loading and unloading data is actually the dif-ference between the bulk resistances in these two cyclescaused by the material permanent bulk deformation, ifpresent. If the asperities are plastically deformed, ther-mal contact resistance becomes smaller in the unload-ing cycle and therefore, the difference between the load-ing and unloading thermal joint resistances is greaterthan the difference between the corresponding bulk re-sistances.

For the thermal contact resistance calculation, it isnecessary to know the conductivity and Young’s modu-lus of the tested material as well as Poisson’s ratio andcontact hardness. It should be noted that Poisson’sratio and contact hardness are usually not available incommercial data sheets and therefore should be mea-sured or estimated. Thermal conductivity and effectivemodulus of elasticity are not known, but can be ini-tially guessed and iteratively solved using the GeneralBRM. To reduce the total number of required itera-tions, the Simple BRM can be applied first to obtainkm and Em values to be used as the initial guess in theGeneral BRM.

Once the contact resistance is calculated as a func-tion of pressure, it is subtracted from the measuredthermal joint resistance in the bulk region, yielding thebulk resistance data points which can also be approxi-mated with a linear fit (Fig. 6). The further steps arethen the same as for the Simple BRM - instead of rj

1010

Figure 8: General Bulk Resistance Method-FlowChart

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and drj/dP , calculated rb and drb/dP are used in the-calculation of the thermal conductivity and effectiveYoung’s modulus. The value of thermal conductivityis slightly larger compared to the result of the SimpleBRM since rb < rj and |drb/dP | < |drj/dP | due tothe contact resistance dependence on pressure.

This method is applicable at low pressures butsince the TIM surfaces are generally non-flat, the plas-tic and elastic analytical models underpredict thermalcontact resistance. Therefore, it is recommended to usethe bulk region data. The General Method is presentedin the flow chart in Fig. 8.

There are two possible procedures for BRM vali-dation:

(a) The material thickness calculated using themodel is compared to the measured in situ thick-ness determined during testing using an experi-mental set up;

(b) The final thickness after the loading-unloadingcycle, denoted as tmf

, can be determined usingthe unloading cycle data. A procedure similar tothe Simple BRM is applied if the material thick-ness change is linear during unloading (unload-ing bulk resistance region has linear trend). Thethermal joint resistance in the bulk region of theunloading cycle is approximated with the ther-mal bulk resistance (rj ≈ rb). The y-intercept ofthe approximated linear fit of the bulk resistancedata represents the thermal bulk resistance atzero load, referred to as the final bulk resistancerbf

. Using the value of the thermal conductivityfound with the BRM applied to the loading cy-cle, the final thickness can be determined fromthe following relation:

tmf= rbf

km (34)

The obtained value can be compared to the mea-sured final thickness in order to validate themethod.

The two Bulk Resistance methods proposed aboveare also applicable to a stack of TIM sheets. If theSimple BRM is applied, the additional contact resis-tance between the sheets is ignored while in the caseof General BRM, contact resistance is calculated. Tofind the average thermal conductivity of different thick-ness samples, a procedure similar to the ASTM D 5470method can be applied. While the ASTM D 5470 stan-dard recommends a plot of joint resistance versus ini-tial material thickness, joint resistance and bulk re-sistance versus in situ thickness are required for theSimple and General BRM, respectively. The slope of

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the linear approximation is the reciprocal of averageconductivity for the tested material.

SUMMARY AND CONCLUSIONSThe principal disadvantage of using ASTM D 5470

for determining thermal conductivity in compressiblematerials is the inherent overprediction of km due tothe deformation of the sample during loading. TheBulk Resistance method eliminates this problem, pro-viding a more precise estimation of the thermal con-ductivity.

The method was developed primarily for materialstested in a vacuum, but it can be easily extended tojoints in air or some other gaseous environment. Inthe case of non-vacuum conditions, the gap resistanceterm should be integrated into the proposed methodif the method is to be applied to more complex joints.The method should be tested on a wider range of TIMsand if the change in the material thickness is non-linearunder an applied load, the proposed method should bemodified to accommodate non-linear material deforma-tion. In this case, the measurements of in situ thicknesswould be of great benefit.

This study considered smooth and nominally flatsurfaces The current method could be extended to in-clude contacts of non-flat surfaces, however, initiallyit would be necessary to develop a thermal joint resis-tance model for such contacts.

ACKNOWLEDGMENTSThe authors gratefully acknowledge the financial

support of the Centre for Microelectroncis Assemblyand Packaging and Graftech Inc. of Parma, OH forproviding the test samples used in this study.

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