Investigations on thermal contact resistance between filled polymer composites and solids using micro thermography

This article reports the use of a new measurement technique based on micro thermography for determining the thermal contact resistances (TCRs) between filled polymers and solids. The thermal conductivity of polymers can be significantly increased by using thermally conductive fillers. For numerous applications, not only is a high intrinsic thermal conductivity required but also a good thermal transfer between the filled polymer and an adjacent solid surface. The physical principles of thermal transport when considering this type of contact have not yet been investigated in detail, and only a few experimental results are available. The most common measurement techniques determine a macroscopic resistance and project it onto the contacting surface. However, the heterogeneous microstructure of a filled polymer causes the TCR to be a volumetric phenomenon in the overall boundary region. The utilized IR camera system takes thermal images with a spatial resolution of less than 15μm per pixel. The new method resolves the TCRs spatially and gives new insights into the microscale effects on the particle level. In addition to the common zero-gap extrapolation for the extraction of TCRs, we propose another evaluation method that considers all microscale effects of the boundary layers and evaluates TCR as a volumetric phenomenon. For the first systematic study, samples consisting of two aluminum substrates and a filled epoxy polymer were prepared and investigated. We studied the effects of filler size, filler material, filler volume fraction, and surface structure, focusing on monomodally filled polymers with filler amounts between 30 and 60v% . The obtained results and the uncertainties of the new method are discussed within this paper.

This article reports the use of a new measurement technique based on micro thermography for determining the thermal contact resistances (TCRs) between filled polymers and solids. The thermal conductivity of polymers can be significantly increased by using thermally conductive fillers. For numerous applications, not only is a high intrinsic thermal conductivity required but also a good thermal transfer between the filled polymer and an adjacent solid surface. The physical principles of thermal transport when considering this type of contact have not yet been investigated in detail, and only a few experimental results are available. The most common measurement techniques determine a macroscopic resistance and project it onto the contacting surface. However, the heterogeneous microstructure of a filled polymer causes the TCR to be a volumetric phenomenon in the overall boundary region. The utilized IR camera system takes thermal images with a spatial resolution of less than 15 µm per pixel. The new method resolves the TCRs spatially and gives new insights into the microscale effects on the particle level. In addition to the common zero-gap extrapolation for the extraction of TCRs, we propose another evaluation method that considers all microscale effects of the boundary layers and evaluates TCR as a volumetric phenomenon. For the first systematic study, samples consisting of two aluminum substrates and a filled epoxy polymer were prepared and investigated. We studied the effects of filler size, filler material, filler volume fraction, and surface structure, focusing on monomodally filled polymers with filler amounts between 30 and 60v%. The obtained results and the uncertainties of the new method are discussed within this paper.
Keywords: thermal contact resistance, filled polymers, TIM junctions, micro thermography, microscale TCR * Author to whom any correspondence should be addressed.
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Introduction
Thermally conductive filled polymers are used in a wide range of applications. As thermal interface materials (TIMs), they improve the thermal transition between solid surfaces, as a potting they protect electronic components and conduct excessive heat to the environment and as a case for electronic devices they help with minimizing operating temperatures. Various theoretical and experimental studies have investigated the effective thermal conductivity of the filled polymer, and the effects of filler loading [1][2][3][4], filler materials [1,[4][5][6][7][8], particle shapes [1,9], and sizes [10,11]. Thermal conductivity measurements were performed using laser flash analysis (LFA) (calculated from thermal diffusivity, heat capacity and density) [1,3,10], transient hot-bridge, -wire or -disc methods [4,6,7,9], or the steady-state cylinder method [2,8,11]. The latter can be referred to as the industrial standard for TIMs, and is described in ASTM D5470-17. All methods used are suitable for specific fields of application, depending on the temperature range, ambient conditions, and the sample consistency. In general, the investigated samples must be sufficiently large to avoid the heterogeneous microstructure of the filled polymer affecting the results. In applications where the filled polymers are only used in thin layers (<2 mm) and touch solid surfaces, the effective thermal conductivity as well as thermal contact resistance (TCR) between the filled polymer layer and the contacting solid must be considered. Figure 1 illustrates the typical use of a TIM between an electronic chip and a heat sink.
The filled polymer is used to displace air between the microscopic rough surfaces of the chip and heat sink and to improve thermal transfer. The lower the thermal resistance R th of the transition between the chip and heat sink, the lower the operating temperature of the chip, when considering a constant power loss. This thermal resistance can be described as a serial connection of three single resistances: the thermal resistance of the TIMs bulk R th,TIM and the two contact resistances R th,C to the solid surfaces. The TCRs often significantly increase the total thermal resistance of a TIM-filled gap.
In addition to the absolute thermal resistances R th , it is common, to specify and analyze the thermal resistance of a unit area R th × A, called thermal insulance. Without area dependency, comparisons between different geometries with different heat transfer areas are much easier. In the heat transfer literature, thermal insulance is also called the thermal impedance or specific thermal resistance.
However, the physical principles of thermal transport when considering this type of contact have not yet been investigated in detail, and only a few experimental results are available. In 2018, Xian et al [12] published an extensive summary of transient and steady-state measurement methods for TCR. However, these techniques are used for solid-solid contacts. Typically, solid-solid contacts are described with a thermal contact conductance h, with the resulting contact resistance R th,C = 1 h·A for a contacting area A or the thermal contact insulance (R th × A) C . The TCR are projected on the transition surface and considered to be a surface phenomenon. When using the steady-state cylinder method according to ASTM D5470-17, information about the TCR can be obtained in two different ways. In the steady-state cylinder method, samples are typically placed between two metallic cylinders. One cylinder is heated and the other is cooled. An approximately one-dimensional heat flow is generated from the heated cylinder through the sample into the cooled cylinder. Based on the temperature gradient in the cylinders, the heat fluxq through the measuring section is determined. From this heat flux and the measured temperature drop across the sample ∆T, the thermal insulance can be calculated as (R th × A) = ∆T/q. With the model assumption that the thermal insulance (R th × A)increases linearly with thickness for a homogeneous sample, the TCR between the sample surface and the contacting metallic surfaces can be determined with a zero gap A direct measurement is possible when no sample is inserted between the metallic cylinders. In this case, the result is directly the TCR between the two surfaces of the metallic cylinders. Two exemplary studies using this technique for various solid-solid contacts were published by Rao et al [13] and Mo and Segawa [14]. Rao et al measured the TCR between oxygen free copper samples at different surface pressure levels (direct method). Mo and Segawa started with direct measurements and extended the method to measure the contact resistances between thin solid layers.
In 2007, Teertstra [15] published experimental studies on the thermal resistances of bonded joints with thermally conductive filled adhesives using the steady-state cylinder method. They extracted TCR by zero-gap extrapolation (indirect method) and found that the TCR are of a similar magnitude to the bulk resistances. Schacht et al [16] and Prasher and Matayabas [17] also reported zero-gap extrapolations to predict the TCRs between filled polymers and substrate surfaces.
However, as shown in figure 1, contact resistance can be affected by the overall transition region, when considering microscopic heterogeneous materials such as filled polymers. With the experimental methods described in [12], they are not considered to be a volumetric phenomenon, affected by a layer of a certain thickness. In particular, zero-gap extrapolation using the steady-state cylinder method neglects the fact that TCR can be affected by several microscopic effects in the boundary region. To analyze the effects of the surface structure, filler amount, filler size, and filler material, the thermal resistances in the boundary region need to be investigated on the microscale and resolved spatially. In general, local resistances can be calculated when heat flux and local temperature difference are known. The finer the local resolution of temperature, the finer the local resolution of the thermal resistance. Contacting temperature measurements always disturb the temperature field in the sample and the local resolution depends on the dimensions of the sensor used.
In the present study, we used micro thermography to achieve high spatial resolution of temperature measurements without disturbing the temperature field of the sample. From the single-pixel information, we were able to resolve thermal resistances at the pixel level and thus study the phenomenon of TCR between filled polymers and adjacent solid surfaces, including local inhomogeneities, on the microscale.
In 2015, Burghold et al [18] measured the TCR between two steel samples using infrared (IR) thermography. Two cylindrical samples with direct face-to-face contact were simultaneously heated to two different temperatures during the measurement. An IR camera was used to record the axial temperature profile of both samples during heating. The timedependent temperature field in the samples was used to evaluate the contact resistance between the specimens by solving the corresponding inverse heat conduction problem.
Ishizaki et al [19] used lock-in thermography to measure the TCR between bonded graphite layers on the microscale. The bonded graphite layers were periodically heated with a laser. The temperature response of a cross-section through the two-layer system was resolved spatially, using a lock-in thermography device with an IR microscope. The measured temperature field across the interface was used to calculate local TCR.
Warzoha and Donovan [20] already used micro thermography to measure thermal resistances of TIM junctions. The setup was similar to the steady-state cylinder method. However, the temperature gradient in the two metallic cylinders was not determined with individual temperature sensors but was finely resolved with a microscopic IR camera.
Since only the temperatures in the metallic cylinders were measured, a separation of bulk and contact resistances could only be made with the zero gap extrapolation already described. Warzoha and Donovan reported a high sensitivity of TCR evaluation for the gap width determination during measurement. A local resolution in the contacting zones was not achieved. In 2016, Smith et al [21] reported a similar technique, but without the extraction of contact resistances.
We extended the micro thermography method to obtain spatially resolved thermal information, also within the sample, and analyze the effects of the surface structure, filler amount, filler material, and particle size on the TCRs.

Sample preparation
For our investigations, we prepared multi-layer samples consisting of two aluminum substrates (EN AW-5754) and a filled polymer layer in between. We selected the two-component epoxy polymer SikaBiresin ® TD150 + TD165 and several fillers from different materials with different particle sizes and shapes. Table 1 shows the basic properties of the epoxy system used. Further details can be found in the product data sheet [22]. Table 2 lists the filler properties. All the fillers showed a monomodal size distribution. The median particle sizes (volumetric) range from 7.86 µm to 76.29 µm and were measured using a laser particle sizer Fritsch Analysette 22 NanoTec. The Fraunhofer diffraction theory was applied for evaluation (see e.g. [23].). For the spherical fillers, the median particle size corresponds to the median particle diameter. In the case of irregularly shaped aluminum hydroxide, the size refers to the laser diffraction equivalent diameter. The filler volume fraction ϕ = V filler /V total was varied between 0.3 and 0.6. After the liquid polymer and the granular fillers were mixed, the open sample containers with a maximum filling of 70 ml were degassed for 10 min in a vacuum chamber at room temperature and at ≈0.8 Pa. Aluminum substrates were prepared with different surface structures by sandblasting with different grain sizes (blasting abrasive: glass spheres) and cleaning with isopropanol. The filled polymer was placed and cured between the two aluminum substrates, which were kept at a constant distance using spacers of 0.9-1.2 mm. No surface pressure was applied and the samples were cured for 200 h. The resulting multi-layer samples had dimensions of 100 × 100 mm 2 . For the micro thermography investigations, we cut smaller specimens out of the middle using a waterjet cutter. The cut surfaces were ground and polished. As a result, we obtained precise multi-layer specimens of 20 × 20 × 5 mm 3 with a filled epoxy layer in the middle. Finally, the front surfaces of the samples were coated with an acrylic resin-based graphite spray to obtain a defined and uniform emissivity for the thermography measurements. We initially determined the emissivity ε achieved with this method using an aluminum sample with high thermal conductivity. We coated it with graphite spray, placed it in the measuring section, and inserted a thermocouple directly behind the coated surface. The upper and lower aluminum bars were controlled to the same absolute temperature and the emissivity was calibrated using the measured temperature of the aluminum sample. The result was ε = 0.98 ± 0.01. Since all samples were coated with the same graphite spray, this value can be expected to be constant. Uncertainties in the determination of the emissivity have only a minor influence on our measurement result, since the influence of the emissivity is compensated for by the calibration of the measurement setup, see section 4.

Micro thermography setup
For the micro thermography measurements of the prepared samples, a thermography device was designed and built (figures 2 and 3). The samples are placed between an upper and lower aluminum bar with evenly milled contacting surfaces and the same cross-sectional size as the samples. By heating the upper bar and cooling the lower bar, a onedimensional heat flow through the sample is achieved. Using a side-mounted IR camera, the temperature profile through the multilayer sample is captured. Based on the measured temperature profile and the heat flow through the sample, the local resistance profile of the sample can be calculated.
The front surfaces of the aluminum bars were maintained matte, to avoid excessive reflection. The upper bar is mechanically fixed with a constant clamping force. To avoid high TCRs between the aluminum bars and aluminum substrates of the samples, a water-glycol mixture is used as contact agent. The lower bar is mounted on a cold plate with cooling channels, connected to the coolant circuit of a lab thermostat and kept on T cold = const.. The upper bar is electrically heated to T hot = const.. How T cold and T hot must be set to avoid high environmental losses and with this reduce uncertainties is discussed in section 6.1. For the micro thermography measurements, steady-state temperatures must be reached. The temperature field on the front surface of the samples is captured using the IR camera VarioCAM ® HD head 980 from Infratec with an additional close-up lens 0.5×.
The thermal images have a pixel size of 14.836 µm, when the internal optomechanical MicroScan mode is activated.  Using the MicroScan mode, the spatial resolution is increased by the superimposition of single thermal images with a slight local offset. Therefore, the heat radiation is deflected by a mechanically oscillating optical unit between the lens and detector. The resulting temperature field, T (x, z) is saved and evaluated in the second step. Figure 4 shows an example of the captured thermal image of a sample with a filled polymer layer. The graph below shows the extracted average temperature increase along the z-axis.
To obtain information about the thermal insulance (R th × A) of the different layers and the transition insulances in between, not only temperature data but also information about the heat flowQ through the sample is necessary. Therefore, two thermocouples (type K, calibrated to an absolute accuracy of ±0.02 K in the desired temperature range by using a certified PT100 reference sensor) were placed in the lower aluminum bar and used for the heat flow evaluation with The thermal reference resistance R th,ref of the aluminum between the two thermocouples is determined during a calibration procedure performed in advance, see section 4.

Calibration procedure
Temperature measurements are surrounded by high uncertainty, particularly when using an IR camera. In our setup, the local assignment of data using the pixel information of the thermal image causes further high uncertainty.
We decided to perform relative measurements based on a reference sample as opposed to absolute measurements.
We have defined the thermal reference resistance R th,ref of the lower aluminum bar as the calibration variable. Thus, the heat flowQ is no longer an absolute value, but only a relative measurement value for comparisons between a reference sample and the actual sample. Therefore, absolute temperature measurements are no longer necessary.
Prior to each series of measurements, the complete setup is calibrated with a multi-point calibration using a reference sample, as shown in figure 5. Both, calibration, and measurement were performed under controlled environmental conditions with ambient temperature changes of less than 1 K. However, recalibrations were performed at least every 3 h. To ensure that the thermal conditions in the measuring section are the same for the calibration measurement and the actual measurement, the thermal resistance of the reference sample must be similar to that of our multi-layer samples. To attain the overall thermal resistance of the reference sample of the same magnitude as that of the typical samples to be investigated, a stainless steel 304 was selected. Stainless steel 304 was proposed as a reference material by the National Physical Laboratory, UK [24], and its thermal properties have been widely investigated (e.g. [25][26][27][28].). However, the thermal conductivities of the reference materials are often reported at 300 K or higher. As there was no suitable and certified material available for our use, we purchased non-certified stainless steel 304 and determined its thermal conductivity as a function of temperature by LFA (ASTM E1461-13). The thermal diffusivity a, and specific heat capacity c were measured separately. The materials density ρ was measured using the buoyancy method. The thermal conductivity was calculated with λ = aρc. As result we got for our relevant temperature range of T = 15 • C-30 • C. Even though the uncertainty of our LFA is typically ≈ 8%, we set ∆λ ref = 4.8%, following [24] after also comparing our results with the previously published values in [25][26][27][28] and finding a very good agreement in the overlapping temperature ranges.
The reference sample was produced in the same dimension A ref = A as the multi-layer samples and a flat groove of width ∆z M = 3.0 mm and a depth of 0.2 mm was milled into the outer surfaces, see figure 5. The ground of the groove was coated with a graphite spray, while the remaining rib surfaces above and below were polished. To calibrate the local resolution of the IR camera measurements simultaneously, the evaluation window of the thermography software (IRBIS ® 3) is adjusted to the clearly visible edges of the groove in the reference sample surface. The settings are saved and used for all subsequent measurements.
During the calibration procedure, the cooling temperature T cold = 12 • C is kept constant, and the heating temperature T hot is increased stepwise from 30 • C to 70 • C. At each temperature step, steady-state conditions are reached, the temperature field on the reference samples surface is measured, and the temperature differences ∆T TC = (T TC,1 − T TC,2 ) in the lower aluminum bar is recorded. Steady-state conditions were usually reached in five to six minutes, so that changes in ambient conditions during a measurement can be neglected. Using the captured temperature field T (x, z), we determine the mean temperature gradient (dT/dz) calib and with it, the heat flowQ calib during the calibration measurement, as shown in figure 5. Finally, the thermal reference resistance R th,ref is calculated usingQ calib and the respective temperature differences ∆T TC .
In the calibration procedure, the inaccuracies of the temperature measurement and local assignment of the IR camera affect the calculated temperature gradient and thus the reference resistance, R th,ref . However, as our actual measurements are performed with an identical setup and in the same temperature range, the inaccuracies are compensated for when evaluating the thermal resistance of the sample. If we assume an equal sample area A and reference sample area A ref and a temperature-independent reference resistance R th,ref , the increase in thermal insulance of a sample layer to be measured at the height z can be described as If the calibration and measurements are performed at very similar absolute temperature, systematic uncertainties in the measurement of the temperature gradients (dT/dz) and in the temperature difference measurement ∆T TC are compensated as they are just included as a ratio in the evaluation equation, shown above. Additional uncertainties of the thermography due to an inaccurate emissivity or the emitted and reflected radiation of the adjacent components of the measurement setup are included and compensated as well. The only requirement is that these occur equally in calibration and measurement. This is achieved by the unmodified setup and the same absolute temperature level. With this procedure, we avoid the high uncertainties of the IR camera system affecting our results and calibrate the temperature measurements and local resolution simultaneously. A detailed description of the evaluation strategy considering the temperature dependencies  can be found in section 5. In addition to systematic uncertainties in temperature difference and local temperature gradient measurement, random variation, occurring between calibration and measurement needs to be considered. Their consideration in the uncertainty calculation can be found in section 6.2. Figure 6 shows the results of an exemplary calibration run.
The calculated thermal reference resistance R th,ref of the lower aluminum bar between the two thermocouples is shown as a function of the mean temperature T TC,mean = 1/2 (T TC,1 + T TC,2 ). The solid error bars show the associated uncertainty of the linear regression of the measured temperature field to determine the temperature gradient. The dashed error bars additionally contain the uncertainty components coming from the reference sample. It is good to see that more than 80% of the total uncertainty is caused by the reference sample (thermal conductivity λ ref and cross-sectional area A ref ). If an even better-known reference sample were available, the micro thermography method could be implemented with even higher accuracy. Only the uncertainty in the temperature gradient is method-related and cannot be reduced. Further details regarding the error propagation calculation can be found in section 6.2. The resulting correlation of the calibration is and can be used for the evaluations of all subsequent measurements.

Evaluation strategy
For each micro thermography measurement, we capture the temperature field on the samples surface T (x, z) and record the temperature difference at the thermocouple ∆T TC = (T TC,1 − T TC,2 ). The heat flowQ through the sample during the measurement is calculated viȧ .
As thermal images are always blurry, we numerically sharpen the captured temperature fields T (x, z) using a reverse Gaussian low-pass filter, considering a point spread function with a standard deviation of 2.3 pixels and taking into account the T 4 -dependency of the radiative heat detector signal. The standard deviation of the respective point spread function was determined by capturing the temperature field on a specimen with a local temperature step function set using different local emissivities. To avoid amplification of the highfrequency noise of the temperature fields, we use a standard Wiener filter to remove the noise in advance. As the same geometrical and optical setup is used for all specimens, no individual adjustment of the sharpening parameters is necessary. After sharpening, the temperature field T (x, z) is first averaged along the x-axis in total to T 0 (z) and section-wise to T i (z), see figure 4. The section-wise evaluation allows us to analyze local deviations and to quantify the effects of random filler and surface structures, see section 7.1. The thermal insulance (R th × A) across the sample is calculated pixel-wise with a constant heat flowQ, the cross-sectional area A of the sample, and the pixel-to-pixel temperature gradient (dT/dz) (z) Finally, the thermal insulance is accumulated across the sample height with a start of (R th × A) (z = 0) = 0. Figure 7 shows the typical course of the cumulative thermal insulance across a multi-layer sample with two aluminum substrates and a filled epoxy composite in between.
The increase in thermal insulance in the aluminum substrate zones is negligible. However, for further evaluations, we fit a linear function to the measured data with a slope of Al . The thermal conductivity of the aluminum substrates is λ Al = 135 W m −1 K −1 (EN AW-5754). To separate the bulk and contacting zones of the filled epoxy, the z-positions of the transitions z C,1 and z C,2 between the filled polymer and substrates is measured externally using a digital light microscope VHX with a dual-light high-magnification zoom lens VH-Z250R and an xy-measurement system VH-M100E from Keyence. The surface roughness and particle arrangement close to the surface cause the thermal transition between the filled polymer and substrate to be smooth and not abrupt. The smooth transition, typically a zone of 20-400 µm, can clearly be seen in the course of thermal insulance R th × A along the z-axis and is marked with gray bars in figure 7. The determined z-levels of the transitions are used to perform zero-gap extrapolations and to calculate macroscopically meaningful thermal contact insulances by surface projection. For the z-level measurements, we aligned the rough substrate surfaces using the mean roughness value. Thus, the measured substrate thicknesses are slightly lower than the material thicknesses measured macroscopically using a caliper gauge or micrometer gauge. However, this method provides the more representative results, in comparison to evaluations projected on minimum or maximum roughness values.
From a macroscopic perspective, one would expect a linear increase in thermal insulance with layer thickness for a homogeneous material. The investigated filled polymers have heterogeneous microstructures. However, as the layer thickness is typically at least ten times the maximum particle size, it seems reasonable to assume a linear increase and fit a linear function (R th × A) (z) to the measured data. The thermal conductivity of the bulk zone can be evaluated using as the reciprocal of the slope of the fitted linear function (R th × A) (z). By extrapolating the linear function to the measured z-levels z C,1 and z C,2 , the share of thermal insulance (R th × A) C solely caused by the contact zones can be determined. The thermal contact conductance can be calculated as the reciprocal of the thermal insulance h = (R th × A) −1 C . In addition to this surface-projected evaluation of contact resistances, the spatially resolved micro thermography results allow us to evaluate the overall boundary region and summarize all the effects on thermal contact insulance (R th × A) * C . We identify the z-level, where the course of cumulated thermal insulance diverges from the linear fit, as the starting point of the boundary effect. In addition to the extrapolated thermal contact insulances, we evaluate the thickness of these boundary layers and the proportion of thermal insulance caused by those layers. Figure 8 shows an exemplary evaluation of a specimen with an unfilled polymer layer, and a microscopic image of the investigated cross section before coating with graphite spray.
For the unfilled polymer, a smooth and small transition to the substrate surfaces was observed, and extrapolation to the zero-gap width seems reasonable for the evaluation of contact insulances. For the specimen shown in figure 8, we measured 2 × (R th × A) C = 157.95 mm 2 K W −1 , for both contacts in total. We did not perform separate evaluations because both transitions were produced equally, and the total evaluation compensates for uncertainties in the z-level-alignment. The obtained value corresponds to a pure polymer layer of approximately 36 µm, which is of the same magnitude as the two roughness depths of the substrates (14 µm per side) and is therefore very sensitive to inaccuracies in z-level alignment. Even though the zero-gap-extrapolation when using the steady-state cylinder method is much easier to perform, this example clearly shows the possible issues when spatially resolved data are not available. The smooth transition between the polymer and substrate could be explained by the rough surface structure, resulting in a thin transition layer, where heat is partially conducted in the substrate and polymer. In this transition layer, the heat flow concentrates on the more conductive substrates' roughness peaks, causing an unequal distribution of heat flux. As a result, we observe a constriction resistance in the first substrate layers, similar to the electrical constriction resistance when considering the electrical contact resistances. However, the smooth transitions at the substrate surfaces are not necessarily a purely thermal phenomenon. The blurriness of the thermal image remaining after image sharpening may also contribute, at least in part, to this smooth transition. In this case, the local resolution limit of the thermography system is reached, and a more detailed evaluation of the transition is not possible. The limit of the local resolution is given on the one hand by the point spread function of the optical system and on the other hand by the physical resolution limit in the used spectral range of 7.5-14 µm (Abbe limit). However, the upcoming evaluations with filled polymers show significantly larger transition areas that are far from this resolution limit. Figure 8 also shows the results of evaluating the overall transition region. We observed boundary layers of ∆z C,1 = 127 µm and ∆z C,2 = 57 µm which cause thermal contact insulances of 2 × (R th × A) * C = 347.81 mm 2 K W −1 in total. Figure 9 shows an exemplary evaluation of a specimen with a filled polymer layer, and a microscopic image of the investigated cross section before coating with graphite spray.
The differences in the courses of the thermal insulance in the boundary region between figures 8 and 9 can be clearly seen. We observed thicker boundary layers of ∆z C,1 = 212 µm and ∆z C,2 = 184 µm, mainly caused by the filled polymer, which did not exhibit typical bulk behavior. The boundary layer thickness is the same magnitude as three to four times the median particle size of the filler used (Alox-07: D 50 = 64.78 µm). The boundary layers are again marked gray and cause thermal insulances of 2 × (R th × A) * C = 411.23 mm 2 K W −1 . For comparison: extrapolation of the linear function (R th × A) (z) of the bulk zone to the measured z-levels z C,1 and z C,2 results in contact insulances of 2 × (R th × A) C = 232.12 mm 2 K W −1 . Our evaluation results in much higher contact insulances because we consider the overall boundary region to be part of the transition. It is obvious, that with such an evaluation the layer thickness of the polymer must be reduced for the calculation of the bulk and substrate resistance. The total thermal resistance of the multilayer-samples remains the same. However, a separate description of the transition region leads to a completely new understanding of the term contact resistance at transitions between filled polymers and solid surfaces. Only through this proposed evaluation is it possible to include the entire transition area and to examine all microscopic effects acting there. The significance of this differentiation is shown in section 7.6, where we present the results for thin-layer samples.

Discussion of uncertainties
To quantify the measurement uncertainty of micro thermography studies, we investigated the effects of environmental losses using a numerical simulation model and performed an error propagation calculation with maximum error estimations (type B uncertainties according to GUM terminology [29]), see section 6.2. Additionally, studies on reproducibility and random variations have been carried out to determine type A uncertainties, see section 6.3. For quantitative validation, we performed additional measurements using the steady-state cylinder method and LFA and compared the results with those of the new micro thermography method.

Dealing with environmental heat losses
First, the effects of the environmental heat losses must be discussed. They can distort the results in two ways: • Environmental heat losses can lead to significant differences in the heat flow through the sample and the reference temperature measurement position. • Environmental heat losses cause the heat conduction within the measurement setup to differ from the assumed one dimensionality. The measured temperature field on the samples surface may not be representative for the inner temperature conditions of the sample.
Madhusudana [30] published an extensive investigation on the effects of environmental losses on thermal contact conductance measurements in 2000. He analyzed radiative and convective losses, and showed their effects at different measurement temperatures and surface pressures for solid-solid contacts. The effects on our micro thermography measurements were investigated using a thermal simulation model. We found that the expected deviations of the measured surface temperature and mean temperature of the respective crosssectional areas are <0.2 K for our samples and temperature conditions, and therefore neglectable. Smith et al [21] reported similar calculations for their comparable setup and also found that environmental heat losses did not significantly affect the measurement results when heating and cooling temperatures were maintained near ambient temperature. Using a quasione-dimensional thermal RC-model with analytical heat radiation and convection modeling, we additionally quantified the environmental losses on the lateral surface of the measuring section and checked the deviations in heat flow determination using the two thermocouples in the lower aluminum bar. The charts in figure 10 show the progression of the environmental loss affected heat flow in the z-direction for different heating and cooling temperature combinations. The sample is placed at z = 0 and illustrated with gray bars in the background of the charts in figure 10. The thermocouple positions are indicated by the dashed lines. Environmental heat losses cause the heat flow through the measuring section to change significantly along the z-axis. Depending on the temperature combination set, the course of the heat flow curve in the lower aluminum bar (negative z-positions) varies. For the temperature combination of T cold = 12 • C and T hot = 50 • C, the difference between the heat flow in the middle of the sample and the heat flow between the two thermocouples for heat flow measurement is the lowest. However, for all relevant temperature combinations (30 • C − 12 • C and 70 • C − 12 • C) also shown in figure 10, the absolute deviation is less than 1% and therefore neglected in the further course. The remaining deviations and effects of further error sources not considered, such as e.g., reflected ambient radiation are compensated for during the calibration procedure. The deviations would be more important if the heat flow measurement would be performed absolute and not be calibrated. The results depend on the ambient temperature T amb , which was set to 20 • C in this simulation, and on the total thermal resistance of the investigated sample. The optimum is reached, when the temperature course along the z-axis crosses the ambient temperature between the thermocouple positions, as shown in figure 10 (vertical black line).
For our measurements, we performed the presented simulations for the recent ambient temperature and actual thermal conductivity of the samples, and adjusted the heating temperature respectively. It varied between T hot = 45 • C and T hot = 55 • C.

Estimation of systematic measurement uncertainty
To estimate systematic measurement errors, we considered all steps and components, beginning with the reference sample and the described calibration procedure. Within the calibration procedure, the thermal reference resistance R th,ref of the lower aluminum bar between the two mounted thermocouples is determined as a function of the mean temperature T TC,mean .
and includes the potentially incorrect determination of the reference sample area ∆A ref , the uncertainty of the linear regression for temperature gradient calculation ∆ (dT/dz), the uncertainty of the reference materials' thermal conductivity ∆λ ref and for the sake of completeness, the expected uncertainty in the temperature difference measurement ∆ (T TC,1 − T TC,2 ) . To determine ∆A ref we considered an uncertainty of 0.01 mm for the manual measurements of sample length and width using a caliper gauge. The uncertainty of linear regression is calculated using for each single measurement based on the pixel data of the thermal images with a height of n pixels and the corresponding z-coordinates z i and Temperatures T i . The uncertainty of thermal conductivity of the reference sample was constantly set to ∆λ ref = 4.8%, as described previously. We set the uncertainty in the temperature difference measurement, ∆ (T TC,1 − T TC,2 ) to zero. The calibration procedure makes absolute temperature difference measurements unnecessary. An uncertainty must be considered only when it occurs while performing the actual measurement and setting the temperature differences into a relation. If the inaccuracy remains constant and allows for comparable measurements, it will not affect the final results. Figure 11 shows the calculated uncertainties and their components corresponding to the calibrated reference resistance, as shown in figure 6.
It can clearly be seen that almost 70% of the total uncertainty is caused by the inaccuracy of thermal conductivity of the reference sample. The value does not change with the temperature level, as well as the ≈ 1 % uncertainty caused by ∆A ref . Only the uncertainty of linear regression for temperature gradient calculation ∆ (dT/dz) changes slightly with the temperature and counts a little bit more than 1 %. To For the actual sample measurement, we estimated the total uncertainty of thermal insulance with The uncertainty of the temperature difference measurement with respect to the previously performed measurement with the same magnitude and without any changes in the measurement chain was estimated as ∆ (T TC,1 − T TC,2 ) = 0.05 K. While all other uncertainty components were determined as type B uncertainty, we consider a statistically determined type A uncertainty for the temperature difference measurement.
Several test measurements at constant conditions have shown that the temperature difference measurement shows stochastic fluctuations of maximum 0.05 K. This value is considered as the maximum deviation to be assumed between calibration and measurement. An additional uncertainty in the IR temperature measurements was not considered since only the differences in comparison to the calibration procedure were evaluated.
The total uncertainty of the thermal insulance is divided proportionally between the bulk and contact insulances. Figure 12 shows the calculated systematic uncertainties and their components for the representative selection of eight samples.
The total uncertainty of the measured thermal insulance ∆(R th × A) increases significantly with the thermal insulance of the sample. The uncertainty components of ∆R th,ref and ∆A ref remain constant with increasing thermal insulance. However, the expected uncertainty in temperature difference measurement ∆ (T TC,1 − T TC,2 ) becomes more important for highly insulating samples, as they cause the temperature difference (T TC,1 − T TC,2 ) to decrease and therefore the considered uncertainty ∆ (T TC,1 − T TC,2 ) = 0.05 K to become more decisive. Over the entire range of investigated samples with total thermal insulances between (R th × A) tot = 500 mm 2 K W −1 and (R th × A) tot = 4500 mm 2 K W −1 , the uncertainty is estimated between 10% and 17%.
For the separation of bulk and contact insulances, also the uncertainty in the linear regression in the bulk zone must be considered. Depending on the quality of the thermal image and the remaining high frequency noise after filtering, this additional uncertainty was evaluated between 1% and 2% of total contact insulances.

Random measurement uncertainty
In addition to the previously described systematic error propagation calculations, studies on reproducibility and random variations have been carried out (type A uncertainties according to GUM terminology [29]). We analyzed a representative selection of specimens, performed several micro thermography measurements on each single lateral surface, and compared the results on each surface and between the surfaces. Within all the samples, a good reproducibility with deviations lower than the variations within one surface was observed. However, we regularly observed significant random variations within one of the four lateral surfaces of the specimen and between the lateral surfaces of the square shaped specimen. We attribute this to the high sensitivity of the TCR to small random variations at the microscopic scale. The local contact resistances are affected by several statistical phenomena, such as local particle-substrate contacts, local particle agglomerations, and different polymer layer thicknesses between the substrate surface and the first particle layer. As a result, the contact resistances of similarly prepared test samples exhibited statistical fluctuations in the range of 25%. Details of the local variations in TCR are discussed in section 7.1.

Method comparison
To validate the obtained results, we performed additional measurements using the LFA (ASTM E1461-13) and the steady-state cylinder method (ASTM D5470-17), and compared the results with our values. As there is no suitable method available that can be used to reproduce the spatially resolved results of TCR, we focused on the thermal conductivity of the filled polymers bulk zone. Systematic errors of the micro thermography method affect the thermal conductivities in the same way as they affect the measurements of the TCR. For this validation study, we prepared additional single-layer samples without aluminum substrates and sequentially used the same specimens for all measurement methods. All specimens have a thickness between 2.0 mm and 2.5 mm. For the measurements according to ASTM D5470-17, discs with a diameter of 30 mm were used. For micro thermography, squares with 20 × 20 mm 2 and for LFA discs with a diameter of 12.7 mm were prepared. Figure 13 shows the measured thermal conductivities of 12 different filled epoxy samples.
Aluminum (Sample no. 1-6) and alumina (Sample no. [8][9][10][11][12] and different filler contents were used. Sample no. 7 is an unfilled epoxy sample. The study showed an overall good agreement among the three different methods. The measurements agreed within a range of 12.9 % in average. The error bars indicate the respective systematic measurement uncertainties. Based on the described calibration procedure and the uncertainties of the reference sample, the expected systematic uncertainties of micro thermography (15.8 % in average) were mainly higher than those of the other two methods (11.8 % in average for ASTM D5470-17 and 8 % for ASTM E1461-13). This comparison has shown that the calibration procedure described in section 4 has been successful and has eliminated all sources of error that cannot be influenced or quantified in any other way.

Results
To identify and study the most crucial effects on TCR, several samples with different combinations of substrate surface structure, filler material, filler size, and filler content were prepared and measured using the introduced micro thermography method.
This section describes the acquired results and their interpretations, divided into several single studies with individual objectives. We present the majority of our results based on the zero-gap extrapolation results, as these surface projected values are more meaningful from a macroscopic perspective. However, as shown in section 5, the overall contact resistances and thicknesses of the transition zones must be considered when analyzing the phenomena from a microscopic perspective.

Section-wise evaluation and variations
As addressed in section 5, we not only evaluated the mean contact resistances on the surfaces of the specimens, but also used the spatially resolved data to investigate local variations along the x-axis. Using the section-wise averaged temperature profile along the z-axis T i (z), the local course of the thermal insulance (R th × A) i (z) was determined using the same algorithm as described in section 5, see figure 14.
Typically, 20 individual sections (intervals) were defined to be considered separately. Figure 14 shows the parallel course of the sections and the proportions of thermal contact insulance for the lower and upper transitions. Visually, only slight differences can be observed. For detailed investigation of local variations, we calculated the standard deviation of the thermal contact insulance and thermal conductivity of the filled polymers bulk and plotted the results for the overall sample width, as shown in figure 15.
We observed significant variations in the local thermal contact insulance along the x-axis. However, no significant variations were observed in the thermal conductivity of the filled polymers bulk. Both, thermal conductivity, and contact conductance can be affected by random filler structures and therefore show local variations. The fact that these variations were not detected for thermal conductivity indicates that the random surface structures of the aluminum substrates, the close-to-surface particle arrangement, and local particlesurface contacts play an important role. For the example shown in figure 15, we determined a standard deviation of 3.2% for  the bulk thermal conductivity λ bulk and 9.1% for thermal contact insulance 2 × (R th × A) C . By performing an equal evaluation for all measured samples (95), we found the following dependencies: • The lower the filler volume fraction, the higher the standard deviation of the evaluated intervals. With fewer particles in the transition zone, direct particle-surface contacts become more unlikely, and thus, the local variations increase when evaluating section-wise with a section-width of approx. 0.8 mm. • The smaller the filler particles, the higher the evaluated standard deviation of the thermal contact insulance along the x-axis. Larger particles are expected to form uniform particle layers close to the surface more likely than the smaller particles. For smaller particles, we expect more irregularities and agglomerations in the filler packing, causing higher local variations. • The lower the thermal conductivity of the filler, the lower are the local variations. Local particle-surface contacts become less effective as the established heat path is less conductive.

Effects of filler content
In the first study on filler properties, we prepared samples with different filler loadings of three spherical alumina fillers of different sizes. We combined with two different surface structures (R0 and R1) and measured the thermal contact insulances, as shown in figure 16. There were no clear observable differences between the two different substrate surfaces. However, it can be clearly seen that the different-sized fillers show significantly different effects depending on the filler volume fraction ϕ . For the smallest filler, an overall increase in the contact insulance with increasing filler volume fraction by a factor of ≈ 10 was observed. The medium-sized filler Alox-05 shows a similar behavior for filler volume fractions of ϕ = 0.4 and higher. Only with the lowest filler volume fraction ϕ = 0.3 were significantly higher contact insulances of up to (R th × A) C = 121 mm 2 K W −1 measured. The contact insulance level with the largest filler, Alox-07, is higher over the entire range of filler volume fraction and measures (R th × A) C = 132 mm 2 K W −1 on average. The curves do not show a clear trend.
We expect three different phenomena to affect the thermal contact insulance, superimposing on each other and causing the shown results and dependencies: • Particles close to the surface tend to form uniform layers, which are only superimposed by random structures at a certain distance from the surface. The closer the first particle layer is to the surface, the thinner the remaining polymer layer between the particles and surface, and the lower the contact resistance. Larger particles cause a larger distance between the substrate surface and the center of the first particle layer, and thus, higher contact insulances. Additionally, the number of local particlesurface contacts per area decreases with increasing particle size. • For very low filler volume fractions, direct particle-surface contacts become more unlikely, and thus, the contact insulance tends to increase. • At very high filler volume fractions, the thermal conductivity of the bulk zone increases significantly. The slope of the (R th × A) (z) course decreases, and the zero-gap extrapolation across the first particle layers of the transition zone leads to higher evaluated contact resistances than those for lower filler volume fractions and higher slopes. When evaluating the contact insulance (R th × A) * C of the overall transition zone, no increasing values for higher filler volume fractions were observed.

Effects of filler size
To study the effects of the filler size in detail, we carried out further evaluations using several samples with varying sizes of spherical alumina fillers with different filler loadings combined with different substrate surface structures.
The measured thermal contact insulances clearly increase with filler size. For the smallest filler particles (Alox-03), we measured (R th × A) C = 14 mm 2 K W −1 in average. For the medium-sized filler (Alox-05), we obtained (R th × A) C = 53 mm 2 K W −1 and for the largest filler particles (Alox-07), we obtained (R th × A) C = 139 mm 2 K W −1 . Almost an increase by a factor of 10. The course for the lowest filler volume fraction ϕ = 0.3 differs from those at higher filler volume fractions, as shown in figure 17.
However, these results support our theory as described in the previous section. Larger particles cause a larger distance between the substrate surface and the center of the first particle layer. The mean polymer layer thickness increases and with it the thermal contact insulance. Additionally, the number of local particle-surface contacts per area is decreasing with increasing particle size.

Effects of filler material
To study the effects of the filler material, focused on the fillers thermal conductivity, we selected three fillers with significantly different thermal conductivities. In addition to the alumina Alox-07 already used in the previous studies with an expected thermal conductivity of ≈ 35 Wm −1 K −1 , we selected an aluminum hydroxide (ATH) with a lower thermal conductivity of ≈ 10 Wm −1 K −1 and an aluminum (Al) with a higher thermal conductivity of ≈ 150 Wm −1 K −1 . The exact thermal conductivities of the granular filler materials are unknown and cannot be easily measured. Therefore, the results do not refer to a specific thermal conductivity, but show the qualitative effects of different filler materials, see figure 18.
All fillers had a similar median particle size. However, it should be considered, that only the alumina and the aluminum fillers have spherical particle shapes, whereas the ATH filler has irregularly shaped particles. However, a clear trend was observed. The use of fillers with higher thermal conductivity leads to lower thermal contact insulances. For the most conductive filler particles (Al), we measured (R th × A) C = 80 mm 2 K W −1 in average. For the lowest conductive filler particles (ATH), we obtained (R th × A) C = 150 mm 2 K W −1 and with Alox-07 we obtained (R th × A) C = 122 mm 2 K W −1 with the filler volume fraction applied.
All samples investigated had a filler volume fraction of ϕ = 0.5. The filled polymers were combined with different surface structures (R0, R1, and R2), but no significant differences were observed.

Effects of surface roughness
Variation in surface roughness was included in the studies, as presented in sections 7.2-7.4. No clear trends were observed. However, the standard deviation of the local contact resistances decreases with increasing surface roughness. The higher the roughness peaks, the more likely it is that local particle surface contacts are formed. In the measurements shown in figure 18, the standard deviation of local contact resistances decreased from 11% for surface roughness R0 to 8% for surface roughness R2.

Thin layer phenomena
It is obvious that the effects of the thermal contact insulance become more important with a decreasing thickness of the filled polymer layer between the substrates. TIMs are often used in gaps that are as thin as possible. Our experimental studies showed that the thermal transition zone between the substrate surface and the filled polymer extends over several particle diameters. In the presented studies it was always possible to separate the transition zones from the bulk zone of the filled polymer. Additional samples were produced without using spacers between the substrates and thus allowing the gap to become as thin as possible. Again, no pressure was applied. The designation 'thin' must always be related to the size of the filler particles. To gain a good spatial resolution, we selected the largest spherical alumina Alox-07 with D 50 = 64.78 µm for this study. Figure 19 shows an exemplary evaluation for a specimen with a filled polymer layer of 596 µm. On the left side of figure 19, a microscopic image of the investigated cross section before coating with graphite spray is shown.
The behavior of the filled polymer layer is clearly distinct from the typical bulk behavior. We were able to measure the total thermal insulance of the specimen (R th × A) tot = 499.32 mm K W −1 , but we were not able to separate the contributions of thermal bulk and contact insulances, as there is a gradual transition from one material to the other.
For comparison, a specimen with the same filled polymer composition but a layer thickness of 1.08 mm was investigated. We determined transition zones of approx. 180 µm, a thermal conductivity of 1.48 W m −1 K −1 and by zerogap extrapolation 2 × (R th × A) C = 233.92 mm 2 K W −1 .
Considering these values, one would expect a total thermal insulance (R th × A) calc tot = 2 × (R th × A) C + (R th × A) bulk = 636.93 mm 2 K W −1 for the total layer thickness of the filled polymer of 596 µm. This value is 28 % higher than that obtained in the thin layer measurement. Considering the transition zones of approximately 180 µm in the thicker sample, it appears comprehensible that there was not enough space for the filler packing to develop a microstructure independent of the structure of the adjacent surfaces, as shown in figure 19. Because of the size of the particles compared to the separation of the two boundary surfaces, only a few particle layers in the middle of the sample relaxed to a random orientation and thus exhibited distinctive bulk behavior. This example clearly demonstrates the importance of spatially resolving the transition zone. Macroscopic measurements of different layered samples with zero-gap extrapolation would yield misleading results. However, the spatially resolved micro thermography data contains all the relevant data to analyze the thermal transport phenomena in this multi-layer application.

Conclusion and outlook
In this work, we used micro thermography to investigate the TCRs between filled polymer composites and solid surfaces. This new method resolves the thermal insulance spatially across the investigated transitions and provides new insights into microscale effects at the particle level. We designed and set up a micro thermography device, based on an Infratec VarioCAM ® HD head 980 with a close-up lens 0.5x, and performed measurements with a local resolution of 14.836 µm. Thermal measurements were performed in steady state and were based on a calibrated heat flow measurement, using temperature difference measurements. We extracted the thermal contact insulances using zero-gap extrapolation and proposed a new method to extract and evaluate the thermal insulances caused by the overall boundary regions. The contact insulances were measured with a potential systematic uncertainty between 10% and 17%, depending on the total thermal resistance of the sample.
For the first study, we prepared multi-layer samples with aluminum substrates and a filled epoxy layer in between. Using this simplified reference system, we were able to study the basic effects on TCR.
We found that TCR varies much more with particle arrangement and the materials microstructure than the thermal conductivity of the filled polymer and shows high random variations. We conclude that the TCR can be reduced significantly when direct particle-surface contacts are achieved. Within a random mixture and arrangement of particles, single surface contacts have a decisive effect on the contact resistance, while the randomness affects the thermal conductivity of the filled polymers bulk to a smaller extent. The properties of the filler and the surface both modify the thermal contact insulances. We were able to isolate several of these effects, such as particle size, filler volume fraction, and filler thermal conductivity and to investigate their impact on contact insulances. In general, small particles with high intrinsic thermal conductivity and medium filler volume fractions cause the lowest TCRs. High local variability is expected for smooth substrate surfaces, small particles, and low filler volume fractions.
Numerical simulations have shown that environmental heat losses during measurement are negligible if the setup is kept at ambient temperature. In future studies, we will investigate the effects of multi-modal particle size distributions and filler mixtures. Furthermore, we plan to use this method for commercially relevant filled polymer composites. Adjustments will be made to be able to measure on specimens with elastic or even viscoelastic filled polymer layers. For such samples we also plan to control the surface pressure during the measurement and study the effects of different surface pressures.
Additionally, we will set up a microscale simulation model to obtain further insight into microscale heat transport phenomena at the investigated transitions between the filled polymers and solid surfaces. Experimental approaches are always limited to a certain scale. Using a numerical simulation model, we can overcome these restrictions and investigate the effects of smaller particles and thinner layers. Additionally, numerical studies can help to support, extend, or disprove our interpretation of the different effects presented.

Data availability statement
The data that support the findings of this study are available upon reasonable request from the authors.