Effective Thermophysical Properties Of Thermal Interface .
THERMAL TOCProceedings of IPACK03International Electronic Packaging Technical Conference and ExhibitionJuly 6-11, 2003, Maui, Hawaii, USAProceedings of InterPACK2003:International Electronic Packaging Technical Conference and ExhibitionMaui, Hawaii, USA, July 6-11, 2003IPACK2003-35088InterPack2003-35088EFFECTIVE THERMOPHYSICAL PROPERTIES OF THERMALINTERFACE MATERIALS: PART I DEFINITIONS AND MODELSI. Savija, J.R. Culham, M.M. YovanovichMicroelectronics Heat Transfer LaboratoryDepartment of Mechanical EngineeringUniversity of WaterlooWaterloo, Ontario, Canada N2L 3G1http://www.mhtlab.uwaterloo.caABSTRACTc 1 , c2 The conductivity of thermal interface materialsare typically determined using procedures detailed inASTM D 5470. The disadvantages of using these existing procedures for compliant materials are discussedalong with a proposed new procedure for determiningthermal conductivity and Young’s modulus.EHhkks The new procedure, denoted as the Bulk Resistance Method, is based on experimentally determinedthermal resistance data and an analytical model forthermal resistance in joints incorporating thermal interface materials. Two versions of the model are presented, 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 parameters such as surface characteristics and thermophysical properties of the contacting solids. Both methodscan be used to predict material in situ thickness as afunction of load.mmpn PP CMPV CRT VQRRM SrTT IMt TjYλ νσ NOMENCLATUREAAST M aBRMb area of contact (m2 )American Society for Testing andMaterialslinear fit coefficient (slope)Bulk Resistance Methodlinear fit coefficient (intercept)11coefficients for VickersmicrohardnessYoung’s modulus (M P a)hardness (M P a)thermal conductance (W/m2 K)thermal conductivity (W/mK)harmonic mean thermalconductivity (W/mK)asperity mean absolute slope (rad)mean planenumber of data points inBulk Resistance Methodcontact pressure (M P a)phase change materialPolyvinyl chlorideroom temperature vulcanizingheat transfer rate (W )thermal resistance (K/W )root mean squarespecific thermal resistance (m2 K/W )temperature (K)thermal interface materialthickness (m)joint temperature drop (K)mean plane separation (m)dimensionless mean planeseparationPoisson’s ratioRMS roughness (m)Copyright 2003 by ASME
Subscripts1, 2 indices for contacting surfacesa apparentb bulkB Brinellc contacte elasticf finalg gapi indices for interfacesj jointl lowerm thermal interface materialo initialp polymerr radiationu upperSuperscripts dimensionless effectiveINTRODUCTIONWhen two surfaces are in mechanical contact, theresulting 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 overcome the thermal joint resistance. The thermal jointresistance depends on many geometric, mechanical andthermal parameters, including contact pressure, characteristics of interface surfaces, thermophysical properties of the contacting solids, and if present, the interstitial 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 critical, part of the thermal network established betweenthe heat source and the heat sink. The need to enhance heat transfer across the interface becomes animportant issue in the design of high thermal energydissipating systems. Therefore, the interest in interstitial materials that provide better thermal contactbetween the surfaces is increasing rapidly. These materials are known as thermal interface materials (TIMs)and include greases, oils, gases, thin metallic and nonmetallic 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, Kapton22film or nylon mesh. Materials are manufactured in theform of thin sheets varying from a fraction of a millimeter to a few millimeters in thickness and can haveadhesive on one or both sides. The performance depends on the material conductivity generally found tobe 1 - 10 W/mK, and the compliance to the contactingsurface which is a function of the material hardness. Inaddition, there is a similar group of flexible graphitebased materials, which are very compliant and showgood thermal performance. The thermal interface materials are generally assessed by considering thermalconductivity and thermal resistance as the main properties.Thermal conductivity and thermal resistance areusually reported in the material specifications and aremainly determined following the ASTM D 5470 procedure. 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 thermal resistance since many geometrical, mechanical andthermophysical parameters such as thermal conductivity, Young’s modulus, hardness, compliance, surfaceroughness and waviness effect the TIM thermal performance. Therefore, more detailed material data sheetsshould be available while the procedures for thermophysical 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 thermal 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 relationship between thermal joint resistance and the jointparameters such as contact pressure, surface parameters and thermophysical properties of interface material and contacting solids.The main objective of the first part of this studyis to propose a method for determining material thermophysical properties. The method incorporates ananalytical model and determines thermal conductivity and effective Young’s modulus from thermal resistance data. Also the disadvantages of the widely usedexisting standard ASTM D 5470 method will be discussed and recommendations for its improvement willbe presented. In the second part of this study (Savija et al. [1]) an extensive experimental invesigation ofCopyright 2003 by ASME
Grafoil GTA interface material will be conducted andfrom the obtained thermal resistance data the material properties will be determined applying the methodproposed in the first part of the study.REVIEW OF PREVIOUS WORKSummary of studies on thermal interface materialsby Savija et al. [2] shows that new commercially available polymeric and graphite-based sheets are mainlycharacterized experimentally. There are few analyticalstudies related to the thermal behavior of sheet materials based on the material properties and the mechanical and geometrical joint characteristics. However, 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 studies of the thermal conductance for joints with polymers. One of the first experimental investigations onthermal conductance of metal-to-polymer joints wasconducted by Fletcher and Miller [3]. The joint conductance values for tested elastomers were lower thanthe conductance of a bare aluminum junction. It wasalso concluded that elastomers with metallic or oxide fillers yielded higher conductance values than plainelastomers. Fletcher et al. [4] conducted an experimental investigation of polyethylene materials to determinethe effect of additives on their thermal characteristics.The samples were tested at load pressures ranging from0.41 M P a to 2.76 M P a and mean junction temperatures of 29 57 oC. The thermal conductance increasedwith increasing temperatures and content of additivessuch as carbon.Ochterbeck et al. [5] conducted an experimental investigation on thermal conductance of polyamide filmscombined with paraffin, commercial-grade diamondsand metallic foils. The experimental data indicatedthat polyamide films coated with paraffin-based thermal compound showed the best thermal performanceand improved contact conductance seven to ten timescompared to bare joints. The hardness of the interstitial material was seen to be the most important parameter in the selection of an interface material.Marotta and Fletcher [6] presented experimentalconductance data for several polymers. The conductance of the materials tested were shown to be independent of pressure (0.51 2.76 M P a) except for polyethylene, Teflon and polycarbonate which are relativelysoft and ductile thermoplastic polymers. The apparentthermal conductivity of the materials was measured attemperatures between 10 o C and 100 o C and a pressureof 1.38 M P a. Almost all materials tested had thermalconductivity values independent of temperature.Parihar and Wright [7] performed a detailed ex33periment measuring the thermal contact resistance ofstainless steel SS 304-to-silicone rubber joint in air, under light pressures (0.02 0.25 M P a). 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 softness.An experimental study conducted by Mirmira etal. [8] showed that thermal contact conductance ofsome commercial elastomeric gaskets and graphitebased materials become less dependent on the contactpressure as the load increased, with the bulk conductance becoming predominant in the high pressure range(around 1000 kP a 1500 kP a). The authors observedthat the change in the mean interface temperature didnot significantly effect the thermal conductance valuesfor the gasket materials. Materials with fiberglas reinforcement showed poorer thermal performance thanmaterials without reinforcement. Also, these materials 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 metals and thermoplastic or elastomeric polymers by assuming optically flat surfaces at uniform pressures invacuum. The mode of the deformation between themetal and the softer polymer was assumed to be elastic under light to moderate load based on experimentalstudies conducted by Parihar and Wright [7]. Incorporating 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 contact conductance was obtained:hc σ 1.49ks m 2.3PEp m 0.935(1)By defining the thickness in terms of strain, the following expression for the bulk conductance was derived:hb kptp Pand tp tpo 1 Ep(2)where kp and tp are the polymer conductivity andthickness under load, respectively and tpo is the poly-Copyright 2003 by ASME
Table 1: Research Related to Polymer-based Interstitial Materials Tested in VacuumAuthorsFletcher and Miller [3]Contact MaterialAl 2024-T4Fletcher et al. [4]Al 2024-T4Ochterbeck et al. [5]Al 6061-T6Marotta and Fletcher [6]Al 6061-T6Parihar and Wright [7]Mirmira et al. [8]SS 304Al 6061-T6Fuller and Marotta [9]Marotta et al. [12]Al 6061, SSAl 6061Interstitial MaterialsSilicone elastomers, Fluocarbonelastomer, Nitrile elastomerEthylene vinyl acetate copolymers,Ethyl vinyl acetate copolymer,Polyethylene homopolymerPolyamide in combinations withfoil, paraffin, diamonds, copperPolyethylene, PVC, Polypropylene,Teflon, Delrin, Nylon,Polycarbonate, PhenolicSilicone rubber (elastomer)Elastomeric gaskets(Cho-Therm, T-pli, Grafoil)Delrin, Teflon, Polycarbonate, PVCeGraf, Furon (graphite-based)mer thickness at zero load. Therefore, the joint con- D 5470 method or “Standard Test Method for Therductance was defined as:mal Transmission Properties of Thin Thermally Conductive Solid Electrical Insulation Materials,” is the1hj (3) method that manufacturers and distributors generally1tpo [1 (P/Ep )]1refer to when reporting the material conductivity mea hc,1kphc,2surements.Experimental data from Marotta and Fletcher [6] andFuller and Marotta [9] were compared to the joint conductance model and good agreement was found.The most recent experimental and analytical studies on TIMs were conducted by Marotta et al. [12]. Ananalytical model for the resistance across the joint incorporating a sheet of interstitial elastic material and agap filler such as gas or phase change material, was developed. The proposed joint model is compared withthe experimental data obtained for several commercially available graphite materials. The specimens wereplaced under 0.34 1.03 M P a pressure, between aluminum 6063 contacting surfaces of 1 µm roughness.The joint resistance ranged from 65.8 mm2K/W to10.3 mm2 K/W showing a good agreement with theproposed model. A reduction in thermal performanceof the tested TIMs was observed as the joint temperature increased from 40 oC to 80 o C. This was explainedby degradation in material thermal conductivity.Thermal Conductivity Measurement MethodsThere are a number of ASTM standard test procedures for quantifying thermal conductivity of thin material 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 ASTM44The test method is designed for measuring jointthermal resistance of homogeneous or composite thermally conductive layers having a thickness rangingfrom 0.02 mm to 10 mm (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 exclude air from the interfaces so that thermal contactresistance is negligible. In the case of high-modulusmaterial, test specimens are combined with the lowmodulus layers. The tested material can be stacked toobtain results for different thickness specimens whereit is assumed that the layers coalesce resulting in insignificant 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 finish within 0.4 µm. The specimen thickness is measuredbefore test using ASTM D 374 - Method C. The singlematerial layer or stacked layers are subjected to 3 M P apressure so that the thermal contact resistance is reduced to an insignificant level. The insulating materialis placed around the calorimeter sections and the jointtemperature is maintained at 50 o C while the tests areconducted in air.Copyright 2003 by ASME
Specific Joint Resistance (m2K/W)and lead to unreliable conductivity values. Also rjmeasured at low pressures can not be used as a reference value since rc plays a significant role and dependson surface characteristics and thermophysical properties of the contacting solids.Figure 1:rj rc to/krcto,1Test Apparatus for ASTM D 5470Method (ASTM D 5470-01 [13])to,2to,nInitial Thickness (m)Defining the bulk resistance as rb to /k, whereto is thickness of the material before loading, and plotting the measured resistance of the single and stackedsheets as shown in Fig. 2, the apparent thermal conductivity is obtained as a reciprocal of the slope whilethe value at the y-intercept is the value of thermal contact resistance.Using this method, the effective conductivity ofthe incompressible tested material is well predicted,while the effective conductivity of a compressible material is overestimated. The enhancement of the heattransfer across the joint due to the thickness reductionis incorrectly related to the conductivity. Determining rc as y-intercept assumes constant rc for all material 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 0.14 M P a and thereforethermal resistance and conductivity are mainly determined with ASTM D 5470 procedure at this low pressure. 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. 255Figure 2:Determining Thermal ConductivityUsing ASTM D 5470 MethodMODELING OF THERMAL JOINTRESISTANCEA 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 contact with a sheet of softer, non-flat and rough interfacematerial. The vertical dimensions in Fig. 3 a) are exaggerated for clarity and the material thickness tm ismuch greater than the surface roughness σ1 and σ2 sothat tm σ σ12 σ22 . k1 and k1 are conductivities 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.Copyright 2003 by ASME
Figure 3:The thermal resistance is affected by a great number of geometric, mechanical and thermophysical parameters: surface roughness, mean asperity slope, apparent contact pressure, contact or elastic hardness, compressive modulus of elasticity of the TIM, thickness ofthe layer and thermal conductivities of all bodies incontact. If the radiation at the joint is neglected (reasonable assumption for most applications where thejoint temperature is below 600 o C) the corresponding thermal resistance network can be presented withFig. 5.a) Real Joint b) ConformingSurfaces Joint with TIMAt 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 5:Figure 4:Joint with Thermal Interface MaterialThermal Resistance NetworkIn the general case, the thermal joint resistanceand conductance are defined as follows (Yovanovich etal. [15]):Conforming Rough SurfacesThermal Interface Material Joint 1 11111 Rb (7)Rc1Rg1Rc2Rg2111 hc1 hg1hbhc2 hg2Rj When steady heat transfer is present the following heatflow paths are possible: conduction through the micro1 contacts (Qc ), conduction through the gaps (Qg ), andhjradiation (Qr ) if the interstitial substance is transparent to radiation. The heat transfer rate across the joint where Rc , Rg and Rb are the contact, gap and bulkresistance respectively and hc , hg and hb are the correis (Yovanovich [14], Yovanovich et al. [15]):sponding thermal conductances. The two interfacesgenerally have different contact and gap resistancesQj Qc Qg Qr(4)since contacting surfaces can have different propertiesand gaps can be occupied with different substances.andThis problem can be simplified for some specialcases.If there is no substance present in the gaps Tjand Qj hj Aa Tj(5) and the thermal joint is in vacuum, then Rg1 ,Qj RjRg2 , hg1 0 and hg2 0, and:where Tj is the temperature drop, Rj is the thermalRj Rc1 Rb Rc2joint resistance, hj is the thermal joint conductance,and Aa is the apparent contact area. The joint con(8)1111 ductance and resistance are related as follows:hjhc1hbhc2At relatively high contact pressure, where the bulk(6) resistance dominates over the contact resistance, i.e.hc1 hb and hc2 hb , a further simplification iswhere rj is the specific joint resistance introduced to possible:define thermal joint resistance for the apparent contactarea.(9)Rj Rb and hj hbhj 11 Aa Rjrj66Copyright 2003 by ASME
The relations for contact and bulk resistancesSong and Yovanovich [17] showed that the relativeand corresponding conductances can be obtained from contact pressure is:models that are based on the following simplifying as1sumptions: PP1 0.071c2 (16)(a) nominally flat rough surfaces with a Gaussian asHcc1 (1.62σ/m)c2perity height distributionThe Vickers microhardness correlation coefficients(b) random distribution of surface asperities over theapparent area(c) thermal conductivity of interface material doesnot vary with pressurec1 and c2 , are related to the Brinell hardness by therelationships (Sridhar and Yovanovich [18]):c13178c2(d) thickness change is linear under an applied loadPlastic Contact Conductance ModelIf the surface asperities of the softer material, i.e.TIM, experience plastic deformation, the contact conductance can be defined using the relation for conforming rough surfaces and plastic deformation of contacting asperities (Cooper et al. [16]): exp( λ2 /2)2ks mhc 1.5 4 π σ 1erfc(λ/ 2)1 22 ki kmki kmElastic Contact Conductance ModelFor conforming rough surfaces and elastic deformation of contacting asperities, Mikic [10] obtained thefollowing relations for contact conductance:(10)hc 2σi2 σmm2i m2m(18)orks mhc 1.55σ 2PE m0.94and rc 1hc(19)(12)The effective gap thickness is obtained from thefollowing approximation (Yovanovich [14]):and absolute asperity slope is:m exp( λ2 /2)ks m 1.5 4 π σ 1erfc(λ/ 2)1 4(11)The effective surface roughness is:σ 3 HB /3178where HB is the Brinell hardness, and HBfor a Brinell hardness range from 1300 M P a to7600 M P a.where ks is the harmonic mean thermal conductivity:ks 2 4.0 5.77HB 4.0 (HB) 0.61 (HB) HB 0.370 0.442(17)c1(13)λ 2 erfc 14PHe(20)where i 1,2 refers to the interface i.e. contacting solid where the elastic contact hardness is defined as:and subscript m refers to the interface material. ThemE effective gap thickness λ is given by the theoretical re(21)He lation (Yovanovich [14]):2λ 2P2 erfc 1HcThe effective Young’s modulus of the solid(14) interface material interface, E , is:2where P is the apparent contact pressure and Hc is the11 νi21 νm (22)contact hardness of the softer material.E EiEmEquation (10). is reduced to a simpler form where ν , E and ν , E are Poisson’s ratio andiimm(Yovanovich [14]):Young’s modulus of the contacting solid and thermalinterface material respectively. 0.95m P1and rc (15)hc 1.25 ksσ Hchc77Copyright 2003 by ASME
Bulk Conductance ModelThe specific bulk resistance of the material itselfis directly related to its thickness and thermal conductivity:Generally, the waviness of the TIM and contacting solids is significant which makes mechanical andthermal modeling difficult due to the lack of analytical models for random non-conforming surfaces. Thethermal joint resistance for conforming surfaces can betmkmaccepted as the lower bound of the thermal joint reand hb (23)rb sistance for wavy surfaces. The waviness of TIM dikmtmminishes at higher pressures while the bulk resistanceMaterial thickness can vary with load depending on dominates over the contact resistance.the Young’s modulus. If the material is compressibleand shows linear deformation under load, the bulk reBULK RESISTANCE METHODsistance becomes:A new method for determining the TIM’s thermal conductivity from thermal resistance data incorporatPtm o 1 ing the above analytical model is developed by SavEmrb (24) ija [19]. The method is named the Bulk ResistancekmMethod (BRM) since it is applied to the data in thewhere tmo is the initial thickness at zero load and Em bulk resistance region. Unlike the ASTM D 5470 prois the effective Young’s modulus of the material. The cedure, the BRM considers material thickness changessmaller the elastic modulus, the larger the material under load and in addition to thermal conductivity predeformation. As a direct consequence of the material dicts a value of the effective Young’s modulus and inthickness decrease under a load, the bulk and over- situ thickness.In order to apply the BRM to the thermal reall thermal joint resistance is reduced. Although bulkresistance is an additional term in the resistance net- sistance data, the following experimental parameterswork, the overall joint resistance generally decreases need to be known:when TIM is present at the joint since the contact re(a) interface temperature,sistance is greatly reduced.(b) material initial thickness,(c) surface characteristics of contacting solids andSummary of Conductance Modelsthermal interface materials:From the presented contact and bulk resistancemodels for joints incorporating TIMs in the form of(i) RMS roughness,sheets, it can be concluded that a TIM’s thermal con(ii) mean asperity slope andductivity, hardness and Young’s modulus are impor(iii) waviness;tant parameters in the thermal resistance network. As(d) material properties of contacting solids:expected, a TIM with higher thermal conductivity enhances the heat transfer through the joint. The hard(i) conductivity (at the interface temperature),ness and Young’s modulus define the material’s com(ii) Young’s modulus andpliance to the surface and effect on contact resistance.(iii)Poisson’s ratio;If the material is soft enough to completely occupy thegaps then the contact resistance is greatly reduced.(e) properties of thermal interface materials:A plot of modeled thermal joint resistance versus(i) contact hardness andcontact pressure (Fig. 6) shows the three resistance(ii)Poisson’s ratio.regions. The first region, referred to as the contactresistance region, appears at relatively low pressuresIf the surface characteristics and thermophysicalwhere the contact resistance dominates the trend of the properties are not known or difficult to measure, thejoint resistance. The second region is the transition re- Simple BRM is applicable as the above properties afgion where the contact resistance becomes significantly fect contact resistance and are assumed to be insignifsmaller. The third region, occurring at higher pres- icant in the bulk region. If all of the above propertiessures, is the bulk resistance region, where contact resis- are known or measured then the General BRM cantance is significantly reduced. If the material thickness be applied. The thermal interface temperature doesreduction under load is linear, the linear trend of the not directly influence this method, but it should bebulk resistance region can be observed. The position of recorded since thermal conductivity of some materialthese three regions on such a plot depends on material properties can show significant dependence on temperhardness, Young’s modulus, and the characteristics of ature.all surfaces in contact.88Copyright 2003 by ASME
Development of Simple Bulk Resistancetm 1Method(30)Em okm ajFrom the thermal resistance data obtained at various loads the bulk resistance region is identified as theDiscretizing Eq. (29), the values of conductivityregion where data points fit a linear trend as shown in are obtained for each pressure point, P , and correiFig. 6.sponding thermal joint resistance, rji :Thermal Joint Resistance 104 (m2K/W)6kmi ContactResistanceRegion5nkm TransitionRegion3(31)where i 1, 2, ., n. The resulting thermal conductivity is calculated by averaging kmi values:BulkResistanceRegion4tm orji Pi aj1kmin i 1(32)rj aj P bj2rci1rb ab P bb001234567Contact Pressure (MPa)Figure 6:Specific Thermal Joint Resistance forAl 2024 - Grafoil GTA - Al 2024 JointA 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: Ptm o 1 Emrj rb (26)kmDifferentiating with respect to contact pressure,the following expression results:drjtm 1 odPkm Em(27)Solving Eqs. (26) and (27) for km and Em , andintroducing the slope of the linear fit aj :drj aj(28)dPthe expressions for thermal conductivity and effectivemodulus of elasticity become:km tm orj P aj(29)99Figure 7:Simple Bulk Resistance Method-FlowChartCopyright 2003 by ASME
Knowing km and going back to Eq. (30), the effective Young’s modulus, Em , is easy to determine. Thein situ thickness of the material can be obtained as afunction of pressure: Ptm tm o 1 (33)EmThe uncertainty in the obtained values of materialthermophysical properties depends on the experimental data and the slope of the linear fit - the more dataobtained in the bulk resistance region, the more accurate the slope and the less erroneous final result. TheSimple Method is presented in the flow chart in Fig. 7.Development of General Bulk ResistanceMetho
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