Modeling and Experimental Characterization of Metal

1

R. Kempers

Department of Mechanical and Manufacturing Engineering, Trinty College, Dublin, Ireland; Alcatel-Lucent, Blanchardstown, Dublin 15, Ireland e-mail: [email protected]

A. M. Lyons Department of Mechanical and Manufacturing Engineering, Trinity College, Dublin, Ireland; City University of New York, College of Staten Island, Staten Island, NY

A. J. Robinson Department of Mechanical and Manufacturing Engineering, Trinity College, Dublin, Ireland

Modeling and Experimental Characterization of Metal Microtextured Thermal Interface Materials A metal microtextured thermal interface material (MMT-TIM) has been proposed to address some of the shortcomings of conventional TIMs. These materials consist of arrays of small-scale metal features that plastically deform when compressed between mating surfaces, conforming to the surface asperities of the contacting bodies and resulting in a low-thermal resistance assembly. The present work details the development of an accurate thermal model to predict the thermal resistance and effective thermal conductivity of the assembly (including contact and bulk thermal properties) as the MMT-TIMs undergo large plastic deformations. The main challenge of characterizing the thermal contact resistance of these structures was addressed by employing a numerical model to characterize the bulk thermal resistance and estimate the contribution of thermal contact resistance. Furthermore, a correlation that relates electrical and thermal contact resistance for these MMT-TIMs was developed that adequately predicted MMT-TIM properties for several different geometries. A comparison to a commercially available graphite TIM is made as well as suggestions for optimizing future MMT-TIM designs. [DOI: 10.1115/1.4024737] Keywords: thermal interface materials, thermal contact resistance, electrical contact resistance, thermal management, heat transfer

1

Introduction and Background

The mitigation of thermal contact resistance is essential to the performance of conduction-based electronic thermal management solutions. Typically, the most feasible strategy to reduce thermal contact resistance is to insert a thermal interface material (TIM) of higher thermal conductivity between the mating surfaces to conform to the contacting surface asperities and displace any micro and macroscopic air voids, thereby providing a path of improved heat conduction. To work effectively, TIMs must physically conform to the mating surfaces under reasonable assembly pressures and exhibit low contact resistance with adequate bulk thermal conductivity. The bond-line thickness values are kept as low as possible to help reduce bulk thermal resistance; however, the thickness must be sufficiently large to enable the TIM to comply with surface irregularities and nonplanarities. Many different TIMs are commercially available that attempt to meet these requirements in different ways. These include a range of adhesives, greases, elastomeric pads, phase-change, carbon, and nano-structured materials [1–3]. However, the main weakness of many commercially available TIMs is their relatively poor thermal performance. Often the TIM consists of a low-conductivity organic phase, such as silicone grease, interspersed with higher conductivity metal (e.g., silver and copper) or ceramic particles (e.g., aluminum oxide, zinc oxide, or boron nitride) to enhance the overall effective thermal conductivity of the material. However, the effective thermal conductivity of these materials is limited by multiple point-to-point contacts between adjacent particles. Despite using extremely high conductivity filler materials, such as silver (k 420 W/m K), the effective thermal conductivity of the best commercially available 1 Corresponding author. Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received July 17, 2012; final manuscript received May 14, 2013; published online October 25, 2013. Assoc. Editor: Bruce L. Drolen.

Journal of Heat Transfer

TIMs is on the order of 5 to 10 W/mK, which is considerably lower than the thermal conductivities of typical mating components. In addition, dispensing and flow of the particle-matrix composite can result in voids being trapped within the bond. Another type of thermal interface material is graphite pads. These consist of graphite flakes that are exfoliated by thermally vaporizing an intercalant ion inserted between its layers, generating an internal pressure that causes the intercalated graphite to expand in the direction perpendicular to the layers. The expanded graphite flakes are then mechanically consolidated together with a matrix into a cohesive sheet of flexible graphitic material [4,5]. The results are a flexible sheet that has a reasonable through-plane thermal conductivity (3–5 W/m K) [4]. However, due to the highly anisotropic nature of the structure, the in-plane conductivity is significantly higher, ranging from 140 to 500 W/mK according to Smalc et al. [5]. As a result, they are a viable option for heat spreading applications [5]. However, despite being a flexible sheet, graphite pads exhibit very little compliance as they are compressed. Even at compressive pressures of 1 MPa, graphite pads exhibit only approximately 5% bulk strain [4]; compliance is more limited when high aspect ratio surface features are present. In many high thermal energy dissipating systems, the TIMs can account for up to 50% of the available thermal budget of the package [6]. With the implementation of high-performance liquid and phase-change cooling strategies, this percentage becomes even greater. If the thermal management of an electronic device is inadequate, unacceptable temperature levels may be reached which can adversely affect device performance, reliability, and lifespan [7]. These thermal issues have spawned a global effort towards the development of exotic TIMs with complex formulations [1–3,6–9]. 1.1 Metal Microtextured Thermal Interface Materials. To address the performance limitations of conventional thermal interface materials, particularly graphite pads used in radio frequency

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Fig. 1

Metal microtextured thermal interface material (MMT-TIM) concept

power amplifier applications, a metal microtextured thermal interface material (MMT-TIM) has been developed [10,11]. These materials consist of a thin metal substrate into which an array of small-scale (0.1 mm to 1 mm) features has been embossed. When this structure is compressed between two mating surfaces, the features plastically deform and conform to the contacting bodies as illustrated in Fig. 1. This approach reverses the conventional TIM paradigm by creating an interface composed of high-conductivity plastically deformable metal features and so the constraint on effective thermal conductivity imposed by multiple point-to-point contacts in conventional TIMs is eliminated. Plastic deformation of the features into the asperities of the contacting surfaces ensures intimate contact with the contacting bodies. In addition, the possibility of large void formation is significantly reduced since these microtextured contact points are fixed in an array. The result is an array of conformable, yet continuous solid metal features of high effective thermal conductivity that are in intimate contact with the mating surfaces due to the plastic deformation of the raised features. Furthermore, by employing pure metals such as copper, silver or aluminium, the features of the MMT-TIM are both good thermal conductors and mechanically compliant. Given the nature of this concept, there is clearly a large design space within which the materials and geometries of the MMTTIMs could be optimized to minimize thermal resistance and compressive pressure while maximizing mechanical compliance and effective thermal conductivity whilst minimizing thermal resistance. The objective of this investigation is to develop accurate modeling and characterization techniques that capture the underlying physics of this complex thermomechanical problem in order to predict the mechanical and thermal performance of these materials as they undergo the large-scale plastic deformations during compression. These models will ultimately serve as design tools in order to design and optimize MMT-TIMs for specific practical applications constrained by thermomechanical and cost considerations.

2

Modeling

As MMT-TIMs are compressed, the raised features undergo significant strains as they compress and conform to the mating surfaces. Depending on the MMT-TIM geometry, several possible modes of large plastic deformation such as bending, shearing, and buckling can occur that change the geometry to such a degree that it significantly influences the overall thermal resistance and effective thermal conductivity of the material. From a design

standpoint, it is important to characterize the compressive force required to deform a MMT-TIM to a given extent since, for example, there are usually practical limitations to the amount of pressure allowable in an assembled joint of an electronic package. Furthermore, since the topography of these features can undergo significant changes, the influence of contact area and self-contact (where portions of the MMT-TIM structure come into contact with other initially noncontacting parts) during compression on the effective thermal resistance and thermal conductivity of the MMT-TIMs must also be characterized. A commercial finite element (FE) package called DEFORM (Scientific Forming Technologies Corporation) was used to simultaneously model the large plastic deformations as well as the bulk thermal response of the MMT-TIMs as they undergo large topographic changes during compression. This package was designed for modeling high-strain operations such as forging, rolling, and extrusion and thus is well suited towards modeling the large plastic deformations inherent to this study [12–14]. While an entire MMT-TIM may consist of an array of many features, here the numerical model simulated a unit-cell consisting of a single MMT-TIM feature with the appropriate boundary conditions. A representation of a MMT-TIM unit-cell bounded by the two contacting bodies is shown in Fig. 2. The geometry of the MMT-TIM unit-cell and contacting bodies were discretized using a sufficient number of tetrahedral elements, and the deformation of the structures was calculated using an updated Lagrangian finite element formulation [12]. An example mesh for an idealized conical MMT-TIM feature is shown in Fig. 2(c). The structure was loaded by controlling the downward displacement of the upper surface, while the lower surface remained stationary. The total deformation of the moving die was applied in small steps to maintain the accuracy of the solution. The self-contact algorithm of DEFORM was employed to ensure the collapsing MMT-TIMs did not intersect themselves and give physically unrealistic solutions. In problems involving large deformations, the quality of the mesh in many FE packages can quickly degrade, causing poor predictions of element stresses, and possible numerical instabilities. This issue is handled automatically in DEFORM by remeshing areas of the geometry where excessive element distortion has been detected. The software constructs an optimum mesh in these areas, based upon the curvature of the area and its previous solution behavior [13]. This local remeshing tool avoids the need to remesh the entire structure, thus maintaining simulation accuracy while being computationally efficient. During the deformation analysis, the changing geometry leads to different levels of contact within the model for a given step. As

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Fig. 2 (a) Unit cell model for an arbitrary MMT-TIM geometry, (b) the equivalent thermal circuit, and (c) finite element formulation of unit-cell

the number of nodes in contact is constantly changing, unsteady thermal solutions would be predicted. Since it is desirable to extract only steady-state temperatures for accurate comparison with experimental results, the mechanical simulation was run to completion first. The resultant geometries, boundary and contact conditions from the mechanical simulation were then imported into a separate simulation for steady-state heat transfer computations which were carried out at the desired deformation step. The development of accurate mechanical predictions of MMT-TIM response has been detailed previously by Ref. [15]. Here SEM stereomicroscopic surface reconstruction techniques were developed to create realistic geometries that capture the small-scale geometric variations and stress concentrations found in the actual materials. Compared with idealized geometries, this resulted in very accurate predictions of the compressive response in terms of both compressive pressure and geometric topology during compression for the same MMT-TIM geometries examined here [15]. From a thermal standpoint, by imposing a thermal boundary condition on the system and assuming adiabatic conditions on the side surfaces, the system can be simplified to the thermal resistance network shown in Fig. 2(b). Here, the thermal resistance due to the MMT-TIM consists of its bulk resistance, RTIM, the contact resistances that occur where it touches the upper and lower bodies, Rcontact, and the constriction/spreading resistances that occur in the contacting bodies themselves due to the differences in contact area with the MMT-TIM, Rconstriction. Thus, the apparent or measured thermal resistance of the MMT-TIM, Rmeas, can be calculated as the sum of these five resistances in series. Numerically, the three-dimensional temperature field is calculated throughout the contacting bodies and the TIM, and the bulk resistances of the upper and lower contacting bodies, RU and RL, are subtracted from the end-to-end thermal resistance to render Rmeas. The temperature distribution in the MMT-TIM and upper and lower bodies is obtained in DEFORM by solving the energy balance equation, written here using the weighted residual method, as ð V

kTj dTj dV þ

ð

ð ð _ qcTdTdV a re_ dTdV ¼ qn dTdS

V

Journal of Heat Transfer

S

(1)

where V and S are volume and surface, k is the thermal conductivity, T is the temperature, q is the density, c is the specific heat, a is the fraction of mechanical energy that is converted into heat, r is the effective stress, e_ is the effective strain rate, and qn is the heat flux normal to the surface [13,14]. Here, the first term represents diffusion, the second is the energy storage term, the third term represents mechanical heat generation, and the last term is the surface integral of boundary heat flux. For the purposes of the simulations carried out here, a is set equal to zero since the quantity of generated heat from the mechanical deformation is irrelevant over time in comparison to the results obtained from the steady-state thermal solution of the thermal boundary conditions imposed here. A uniform heat flux boundary condition was imposed on the top-most surface of the upper body, while a uniform temperature was imposed on the bottom surface of the lower body as illustrated in Figs. 2(a) and 2(b). Adiabatic (symmetrical) boundary conditions were imposed on the sides of the unit-cell domain. The efficacy of the numerical simulation result was confirmed by tracking convergence of various parameters including energy conservation and grid refinement studies were performed to ensure grid independence [15]. Nodal positions and temperatures within the MMT-TIM unitcell and contacting bodies were exported to ASCII data files and MATLAB was used to perform calculations of the MMT-TIM thermal resistance and effective thermal conductivity of the MMT-TIM structure as a function of deformation.

3 Characterization of Thermal Contact Resistance in MMT-TIMs An initial investigation into modeling the thermal resistance of silver MMT-TIMs demonstrated that the thermal contact resistance of the features with the contacting bodies played a significant role in the total thermal resistance of the overall interface [10]. The results predicted the thermal resistance and effective thermal conductivity of the bulk MMT-TIM as well as its mechanical response. However, to achieve these predictions, an empirical fit parameter was used for the thermal contact resistance [10]. JANUARY 2014, Vol. 136 / 011301-3


Significant effort has been put forth in order to characterize and model the contact thermal resistance or conductance for a variety of contacting surfaces [16,17]. Thermal contact resistance is typically modeled by analytically calculating the thermal constriction resistance of a single contact spot using various geometries and boundary conditions. The number of contact spots and the mean contact spot radii (or sizes) are then estimated using statistical surface characterizations and mechanical deformation analysis and models [18]. The deformation analysis can take several forms depending on the length-scales of the joint in question: Smooth and nonconforming joints require macrogap resistance models, rough and conforming joints require microgap resistance models and rough, nonconforming joints require both microgap and macrogap resistance models [17]. Furthermore, the estimate of the size and number of contact spots will depend on the whether the deformation is elastic or plastic which itself depends on the geometry, loading and hardness of the materials and asperities [16]. For the conforming plastic deformation case, the Cooper, Mikic, Yovanovich (CMY) model is well known [17]. Here, thermal contact conductance is modeled as [17] hc ¼

2nakeff wðeÞ

(2)

where keff is the effective thermal conductivity of the joint given by 2k1 k2 (3) ks ¼ k1 þ k2 The thermal constriction parameter, w(e), is approximated as [17] wðeÞ ¼ ð1 eÞ1:5

for 0 < e < 0:3

(4)

where e is the relative contact spot size given by rffiffiffiffiffi Ar e¼ Aa

(5)

where Ar and Aa are the real and apparent contact areas, respectively. The CMY model goes on to relate the geometric contact parameters n, a, and Ar/Aa to the relative mean plane separation, the effective RMS surface roughness and the effective absolution mean asperity slope [17]. For nonconforming rough surfaces, the analysis increase in complexity due to the additional length-scales involved [16]. These models generally produce good predictions of thermal contact conductance for a range of engineering surfaces; however, they require detailed knowledge of the properties of the contacting surfaces including the microhardness, effective RMS surface roughness and the effective absolute mean asperity slope a priori. Furthermore, these models typically assume that there is little or no overall large length-scale deformation of the contacting structures or significant changes in apparent contact area as pressure is applied. Thus, for the purposes of modeling, the contact between deforming MMT-TIMs and rigid surfaces, they are not easily implementable due to the hollow nature of the structures which buckle and collapse at much larger length-scales. The significant overall deformation also serves to highlight the main challenge in characterizing the thermal performance of MMT-TIMs experimentally: specifically distinguishing the bulk thermal resistance of the MMT-TIM from the contact thermal resistance. Typically, for conventional, homogeneous TIMs, the contact resistance can be characterized experimentally by measuring several thicknesses of TIM, plotting the thermal resistance as a function of thickness and extrapolating contact resistance as the y-intercept where the thickness is zero [18]. For MMT-TIMs,

however, this method cannot be used due to the nonuniform behavior of the bulk TIM. In this work, the problem of MMT-TIM contact resistance characterization was addressed by leveraging aspects of the numerical model and experimental techniques: By first demonstrating that the mechanical behavior of the deforming MMT-TIM is simulated correctly, it could be assumed that the model predictions of their bulk thermal behavior are correct as well. In this way, the thermal contact resistance could be estimated by subtracting the bulk thermal resistance of the MMT-TIM predicted by the model from the measured total thermal resistance. Furthermore, for metallic TIMs such as these, the thermal contact resistance and electrical contact resistance are qualitatively similar, as both of these phenomena depend on the ratio of actual, intimate contact area to the apparent contact area [16,19]. However, the thermal and electrical resistance differ in subtle but significant ways. Whereas thermally the bulk resistance values are of similar orders of magnitude as the contact resistance values, electrically the resistances of the bulk MMT-TIM are extremely small compared to the resistance at the contact surfaces. We have previously proposed that a relatively straightforward electrical resistance measurement could be employed to characterize the thermal contact resistance [10]. The use of a straight-forward electrical resistance measurement to characterize actual contact area has been considered before, particularly in the areas of mechanical wear and tribological applications [20–22]. Mizuhara and Ozawa [23] represent the first study to develop a direct correlation between thermal and electrical contact resistance for the expressed purpose of estimating thermal contact resistance using a relatively simple electrical measurement. They examined the influence of surface roughness and interstitial oil for steel and cast iron contacts for pressures up to 1.53 MPa. As expected, they demonstrated a reduction in both thermal and electrical contact resistance with pressure and attributed this reduction to the increase in actual contact area. Thermal contact resistance was correlated to electrical contact resistance as a power-law relationship and asserted by the authors to be independent of contact pressure, surface roughness, or contact conditions [23]. Despite the data being somewhat scattered, the resulting relationship was correlated as RA ¼ 0:585ðRe AÞ0:74

(6)

where RA is the specific thermal contact resistance in m2K/W and ReA is specific electrical resistance in m2X. In our study, the estimated thermal contact resistance was obtained by subtracting the idealized thermal model from the experimental results and was further correlated with an electrical resistance measurement in order to develop a straight-forward method of characterizing MMT-TIM contact resistance. This approach was used to predict overall MMT-TIM performance during deformation.

4

Experimental Apparatus and Data Reduction

The design of the experimental apparatus was based upon a popular implementation of ASTM D5470 where wellcharacterized meter-bars are used to extrapolate surface temperatures and measure heat flux through a sample under test. In the present apparatus, measurements of thermal resistance, effective thermal conductivity, and electrical resistance can be made simultaneously as functions of pressure and sample thickness. This apparatus is unique in that it takes advantage of small, wellcalibrated thermistors for precise temperature measurements (60.001 K) using a Lakeshore 370 AC Resistance Bridge. Careful implementation of instrumentation to measure thickness and force also contribute to a low overall uncertainty and a robust error analysis provides uncertainties for all measured and calculated quantities [24].

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The temperatures at the meter-bar contact surfaces, Ta and Tb, and the heat flux, Q, for each meter-bar were obtained by performing least squares regression fit of the axial temperature distribution to a straight line and computing the resulting y-intercept and slope at the contact surfaces. As a result, the uncertainty of Ta, Tb, and Q depend on both the thermal and spatial uncertainties of each thermistor. Details regarding the uncertainty propagation through the least squares regression are presented in Ref. [24]. The heat transfer rate through each meter-bar is computed by dT kmb A (7) Qmb ¼ dx bestfit The measured thermal resistance of the TIM is then calculated as Rmeas ¼

ðTa Tb Þ Q

(8)

The effective thermal conductivity of the TIM can then be calculated using keff ¼

QL L ¼ AðTa Tb Þ ARmeas

experimental test area, an array 40 mm 40 mm was manufactured using the process outlined previously. Two silver plating thicknesses were tested resulting in the samples having a total mass of 0.781 g and 0.601 g, respectively. Upon completion of the surface reconstruction process outlined in Ref. [15], these masses translate to estimates of metal thickness of approximately 36 lm and 28 lm. Results from the mechanical deformation simulation illustrating the progressive compression of a MMT-TIM feature are shown in Fig. 4. To improve the accuracy of the mechanical deformation simulation, the geometry of the cones was derived from 3D SEM reconstructions; idealized cone models resulted in significantly less accurate predictions. Initially the peak of the cone is flattened followed by a nonuniform buckling and collapse in areas where stress concentrations in the geometry existed. Additional analysis of the mechanical aspects of these results is discussed by Kempers et al. [15]. At this point, it is worthwhile to discuss the range of lengthscales that exist in these MMT-TIMs. In the present investigation, there are three different length scales to consider with respect to these MMT-TIMs as illustrated in Fig. 5. First there exist the designed MMT-TIM features themselves (illustrated in Fig. 4 as

(9)

where L is the thickness of the specimen bond line which is measured with an optical micrometer. Four-wire electrical resistance measurements were made by mounting leads on the sides of the meter-bars. A Keithley model 2400 Sourcemeter was used to provide a constant current of up to 100 mA, while a Keithley model 2182 A Nanovoltmeter was used to measure the corresponding voltage drop. A current-reversal method was used to minimize thermoelectric voltage offsets [25]. The bulk electrical resistance of each meter bar was calculated to be 2.72 lX and contributes negligibly to the measured electrical contact resistance values. Additional details regarding the design of this apparatus and uncertainty analysis are provided in Kempers et al. [24].

5

Results and Discussion

5.1 Baseline Experimental Results. The MMT-TIM geometries studied here were fabricated using the following process: First, the 3D geometry of the desired surface was modeled in a conventional CAD package (PTC Pro/ENGINEER). This form or template was then printed from wax directly using a 3D Systems CPX-3000 3D printer in high-resolution mode. Silver was electroplated onto the wax template to a desired thickness and the wax subsequently removed. The resultant metal structure was appropriately annealed in order to promote grain growth and remove any internal stresses. This annealing step not only softened the MMT-TIM, but ensured the metal had known and reproducible mechanical properties that allow for its systematic modeling. The resulting MMT-TIM structure consists of an array of hollow metal features. This process allows for the creation of a wide range of relatively detailed geometries rendering it well suited for the study of geometrical effects on the mechanical and thermal performance of the MMT-TIMs. Furthermore, these hollow features potentially lend themselves to low-cost, high-volume manufacturing techniques such as microstamping or embossing, making this an attractive option for a low-cost TIM. A relatively simple MMT-TIM design is presented in this section to provide what will be considered a baseline case for subsequent modeling and experimental comparisons. An illustration of the approximate geometry is shown in Fig. 3 along with a photograph of the corresponding MMT-TIM and the 3D reconstructed geometry employed in the finite element simulation. This design consisted of an array of hollow silver cones approximately 0.95 mm tall and 1 mm in base diameter on a 2 mm pitch and is designated here as MMT-TIM Sample A. To correspond to the Journal of Heat Transfer

Fig. 3 (a) Nominal feature geometry for baseline silver MMT-TIM Sample A, (b) photograph and SEM image of MMT-TIM Sample A, (c) representative 3D reconstructed geometry used in FE model

Fig. 4 Experimental values (o) and model prediction results (—) of compressive deformation of reconstructed MMT-TIM Sample A geometry

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Fig. 5

The range of length-scales present in MMT-TIMs

the hollow cone from Fig. 3) which are on the order of 0.1 to 1 mm in size. Superimposed on the MMT-TIM features is the various asperities and surface imperfections that are a result of their fabrication process, whose dimensions are on the order of 10 to 20 lm. These smaller-scale asperities are captured in the finite element geometry using the 3D reconstruction technique described in Ref. [15]. As shown in Fig. 4 and Ref. [15], the inclusion of these imperfections and stress concentrations in the finite element geometry are essential to obtaining an accurate prediction of the mechanical response of the MMT-TIMs. Finally there exists roughness at a submicron level in both the MMT-TIM and the contacting bodies that is not represented in the finite element geometry. Upon completion of the mechanical simulation, subsequent thermal simulations were carried out on the MMT-TIM geometries at various stages of deformation, as described previously. Initial thermal simulations neglected the contact resistance between the MMT-TIMs and the contacting surfaces and only considered the bulk resistance of the MMT-TIM and the constriction resistances occurring in the upper and lower contacting bodies. A plot of specific thermal resistance measurements for both thicknesses of MMT-TIM Sample A along with these associated model predictions are shown in Fig. 6. Initially, the thermal resistance drops significantly as contact between all of the features is established. As the features are compressed and the large-scale plastic deformation takes place, the thermal resistance continues to drop due both albeit more slowly. As the features become fully compressed and begin to densify, the thermal resistance decreases further.

Here, the predicted specific thermal resistance of both MMTTIMs is lower than the experimentally measured value at all stages of its deformation. These results are plotted in terms of effective thermal conductivity in Fig. 7 calculated using Eq. (9). While the model prediction demonstrates a similar shape and captures the effect of the change in foil thicknesses between the two samples, the model predicts higher values for the effective thermal conductivity of the MMT-TIM over the entire range of thickness values. The correctly simulated mechanical behavior indicates that the salient physical features of the structure changes emulate reality so it is correct to assume that the corresponding finite element thermal simulation is accurate: i.e., the discrepancy between the thermal simulations and measured results shown in Figs. 6 and 7 is not an artefact of the simulation and must be attributed to the submicron scale contact resistance of MMT-TIM shown in Fig 5 which is neglected in the simulation. Attempts were made to relate the contact area predicted by the deformation simulation to estimated thermal contact resistance, however, no clear relationship could be established. This may be due in part to the submicron roughness and asperities on the MMT-TIM surface at lengths scales smaller (