# Microstructural features governing the effective thermal conductivity of Cu-25Cr sintered composites

Lucas Varoto, Elodie Courtois, Raymond Kwesi Nutor, Sophie Roure, Anthony Papillon, Mélissa Chosson, Baptiste Gault, Pierre Lhuissier, Jean-Jacques Blandin, Guilhem Martin

► To cite this version:

Lucas Varoto, Elodie Courtois, Raymond Kwesi Nutor, Sophie Roure, Anthony Papillon, et al. Mi-crostructural features governing the effective thermal conductivity of Cu-25Cr sintered composites. Acta Materialia, 2025, 301, pp.121601. 10.1016/j.actamat.2025.121601. hal-05319082

HAL Id: hal-05319082

https://hal.univ-brest.fr/hal-05319082v1

Submitted on 17 Oct 2025

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# Microstructural features governing the effective thermal conductivity of Cu-25Cr sintered composites

<figure><img src="image_2.png" /></figure>

Lucas Varoto<sup>a,b,d,*</sup> <sup>●</sup>, Elodie Courtois<sup>c</sup> <sup>●</sup>, Raymond Kwesi Nutor<sup>d</sup> <sup>●</sup>, Sophie Roure<sup>a</sup> <sup>●</sup>, Anthony Papillon<sup>a</sup>, Mélissa Chosson<sup>a</sup>, Baptiste Gault<sup>d,e</sup>, Pierre Lhuissier<sup>b</sup> <sup>●</sup>, Jean-Jacques Blandin<sup>b</sup>, Guilhem Martin<sup>b,**</sup> <sup>●</sup>

## ARTICLE INFO

Keywords:

Cu alloys

Thermal conductivity

Composites

X-ray computed tomography

Solid solution

Interfacial resistance

## ABSTRACT

Medium voltage vacuum interrupters are high-performance current interruption devices in the contemporary energy transition and electrification. The current interruption performance is directly correlated to the properties of their electrical contacts made of Cu-Cr alloys. Therefore, enhancing their performance to respond to the continuously increasing demand for higher current, higher power, and high voltage applications inherently requires the optimization of the microstructure of the Cu-Cr alloys to increase their electrical and thermal conductivity. Herein, we unveil the microstructural features governing the effective thermal conductivity of Cu-25Cr sintered composites, a subject of much less scientific attention than their electrical conductivity due to the complex microstructure-thermal conduction relationships. We coupled advanced 3D characterization techniques, namely X-ray computed tomography and atom probe tomography, with experimental and full-field numerical investigation of the effective thermal conductivity for three Cu-25Cr sintered composites having different final relative density (94, 96, and 98%). We demonstrate the synergistic effect of solid solution, interfacial thermal resistance, phase distribution on the effective thermal conductivity. Interfacial pores hinder the thermal conduction across phases. The effective thermal conductivity also decreases due to elements in solid-solution that diffused in the Cu matrix during sintering, along with interfacial thermal resistance across Cu/Cr phases. Using a full-field numerical approach, we unravel a microstructure heterogeneity-induced heat flux anisotropy contributing to the anisotropy in effective thermal conductivity.

## 1. Introduction

Green and reliable electrical devices throughout the energy distribution chain are essential to meet the new standards and specifications of the contemporary energy production transition and electrification. This inherently involves the improvement of various components required to ensure the safe and efficient distribution of electricity. Among these components, medium voltage vacuum interrupters (MV-VI) are an important link in this chain because they enable the safe distribution between high and low voltage applications [1]. In addition, the previous generation of MV-interrupters often relied on SF₆, a

notorious greenhouse gas, as current interruption media. In MV-VIs, vacuum is used instead of SF₆ as the interruption media. Thus, MV-VIs have been increasingly replacing them as a sustainable alternative for current interruption [1]. MV-VIs interrupt currents during failure events, such as an overvoltage or short-circuit in the distribution line. The current interruption performance of MV-VIs is directly correlated to the properties of their electrical contacts, and thus to their microstructure [2,3]. This is due to the requirement that electrical contacts remain engaged to maintain continuous current flow during normal operation. However, in the case of a failure event, the electrical contacts separate to interrupt the current and protect critical electrical circuits upstream.

<footnote><sup>a</sup> Schneider Electric Industries, F-38000 Grenoble, France</footnote>

<footnote><sup>b</sup> Univ. Grenoble Alpes, CNRS, Grenoble INP, SIMaP, F-38000 Grenoble, France</footnote>

<footnote><sup>c</sup> Univ. Bretagne Sud, CNRS 6027, IRDL, Lorient F-56100, France</footnote>

<footnote><sup>d</sup> Max-Planck-Institute for Sustainable Materials, Max-Planck-Str. 1, Düsseldorf 40237, Germany</footnote>

<footnote><sup>e</sup> Faculty of Engineering, Department of Materials, Imperial College London, London SW7 2AZ, United Kingdom</footnote>

<footnote>* Corresponding author at: Max-Planck-Institute for Sustainable Materials, Max-Planck-Str. 1, Düsseldorf 40237, Germany.</footnote>

<footnote>** Corresponding author at: Univ. Grenoble Alpes, CNRS, Grenoble INP, SIMaP, F-38000 Grenoble, France.</footnote>

<footnote>E-mail addresses: l.varoto@mpi-susmat.de (L. Varoto), guilhem.martin@simap.grenoble-inp.fr (G. Martin).</footnote>
Therefore, improving the performance of vacuum interrupters directly requires optimization of the properties of alloys used as electrical contacts [3–7].

Cu-Cr based alloys with Cr content varying between 20 and 50 wt % have been extensively used as electrical contacts in vacuum interrupters because of their desirable combination of electrical and thermal conductivities and good arc erosion resistance [3,4,8–12]. These properties result from the good electrical and thermal conductivity of the Cu matrix and the semi-refractory and hard Cr phase. Solid-state sintering (SSS) is one of the most widely used routes to produce Cu-Cr electrical contacts for medium voltage applications [2,3]. To ensure high MV-VIs performance, Cu-Cr composites must show high electrical conductivity for the current flow when the vacuum interrupter is closed (electrical contacts in touch) [3,4,6,13,14]. Once a short-circuit or an electrical overvoltage takes place, Cu-Cr contacts will separate from one another and an electrical arc will be created between them [15–19]. The latter causes localized heating and subsequent melting of their microstructure [15–19]. Heat dissipation is thus essential to minimize the electrical arc lifecycle and contact erosion during this extreme thermal condition event. As a result, electrical contacts must show high thermal conductivity since it contributes to the efficiency of the current interruption while increasing the contacts' lifetime.

Establishing the microstructure-properties relationships of Cu-Cr composites used as electrical contacts has been of interest to optimize their current interruption performance and increase their life-cycle [5–7,12,13,16,20–24]. On the one hand, their mechanical properties depend significantly on the size, morphology and distribution of the hard Cr phase in the microstructure [23,24]. The main strengthening mechanisms have been recently investigated in [24]. Regarding the electrical conductivity, because Cr conductivity is nearly 10-times lower than that of Cu ($\sigma_{Cr}/\sigma_{Cu} \approx 0.10$), the effective conductivity of the composites depends primarily on the Cu matrix conductivity, the Cr content and its spatial distribution [4–7,12,13,16,23].

In contrast to the extensive scientific interest in the mechanical and electrical properties of the Cu-Cr composites, their effective thermal conduction and their governing mechanisms have received much less attention. This is in part due to the inherent complexity of the relationships between the microstructure and the thermal conduction, i.e., the synergy between different microstructural features and the effective thermal conduction across the Cu and Cr phases. The discussion on the effective thermal conductivity of Cu-Cr composites is often restricted to a simple and direct correlation with their electrical conductivity. Such correlation can often neglect differences in conduction mechanisms between electrical and thermal conductivity, deviating from the direct Wiedemann-Franz law. Very limited studies have been dedicated to investigating the effective thermal conductivity of Cu-Cr composites [20,25].

Herein, we investigate the microstructural features governing thermal conduction in solid-state-sintered Cu-25Cr composites and hence the performance of MV-VIs. The microstructure of solid-state sintered Cu-25Cr with different final relative densities (94, 96, and 98 %) is fully characterized in 3D using X-ray computed microtomography (XCT). The effective thermal conductivity is also measured in these samples. The composition of the matrix is measured by atom probe tomography (APT) as elements retained in the solid solution are expected to decrease the thermal conductivity of the Cu matrix. Then, we use a full-field modeling approach to investigate, discuss, and assess the effects of different microstructural features such as pores, composition of the Cu matrix, and interfacial thermal resistance, on the effective thermal conductivity of sintered Cu-25Cr composites. Interfacial pores, trace elements in solid-solution in Cu matrix, interfacial thermal resistance and phases heterogeneous distributions governs the effective thermal conductivity of Cu-25Cr sintered composites. This provides guidelines to optimize the thermal conductivity of the future generation of Cu-Cr electrical contacts.

# 2. Materials and methods

## 2.1. Materials

Cu-25Cr samples with three different final relative densities were fabricated by solid-state sintering under a secondary vacuum. Pure dendritic copper powder and pure irregular chromium powder are mixed in a 3:1 mass ratio. The Cu powder has an average equivalent diameter of $D_{50} = 40 \pm 5$ µm and the Cr powder has a particle size distribution from 30 to 80 µm in equivalent diameter. The Cr powder was provided by Ampere Alloys and was processed by mechanical milling. To obtain three different final densities, three samples with different green-body densities were fabricated by varying the uniaxial compression conditions. Green densities were estimated geometrically by weighing green samples and measuring their dimensions using a digital micrometer. The green densities of these samples were approximately $85 \pm 1\%$, $89 \pm 1\%$, and $91 \pm 1\%$, respectively. To maximize solid-state diffusion, the sintering heat treatment was done in a secondary vacuum near copper's melting point at 1050 °C as in [26,27]. All samples were sintered under the same conditions and using the same Cu and Cr powder batches. More details regarding the sintering parameters of Cu-25Cr materials can be found in references [26,27]. The final relative densities were measured by Archimedes' method in a Mettler Toledo Static Detect model XPR204 balance in triplicate for each sample. After solid-state sintering, the final relative densities reached $94 \pm 0.5\%$, $96 \pm 0.5\%$ and $98 \pm 0.5\%$, respectively.

## 2.2. Chemical composition

The bulk chemical composition of Cu-25Cr SSS (at 94, 96, and 98 %) composites was measured to verify the Cu and Cr contents, but also to detect the overall presence of other trace elements, especially those known to be detrimental to the thermal conductivity of Cu (Ti, O, Fe, and S). An average was calculated to represent the overall chemical composition of Cu-25Cr SSS composites. Glow Discharge Mass Spectrometry (GDMS) was used to measure the traces in Al, Si, P, S, Ti, Fe, Co, Ni, Zn, and Sn. Such a technique is based on a glow discharge atomization/ionization source combined with a sector-field mass analyzer. Then, for elements such as C, N, and O, quantification was based on Instrumental Gas Analysis (IGA). The analytical method is based on the heating of the solid sample above 2000 °C. Any gas released during such a process is measured by infrared detectors. The same was done for the raw Cu and Cr powders used to fabricate the sintered Cu-25Cr SSS composites. Measurements were performed by Eurofins EAG Laboratories.

The average bulk chemical composition of all the Cu-25Cr SSS composites is shown in Table 1. The chemical composition of the Cu and Cr powder used to fabricate the SSS composites is also shown for comparison. Cu-25Cr SSS composites are relatively free of trace elements. One can see that the Cr powder used to manufacture Cu-25Cr SSS composites contains a relatively high Fe content (non-desirable element here). We believe that this is the result of contamination from the mechanical milling process during Cr powder manufacturing.

## 2.3. Microstructure characterization

### 2.3.1. X-ray computed tomography

XCT was performed on cylindrical samples (7 mm in height and 1 mm in diameter). Samples were prepared by electrical discharge machining out of the three Cu-25Cr samples having final relative density of 94 %, 96 %, and 98 % respectively. An EasyTom XL laboratory tomograph from RX Solutions was used at 100 kV, equipped with a $LaB_6$ filament, with a 0.75 mm aluminum filter and 1.5 s acquisition per image with a total of 1792 projections. A volume of 1750 × 1125 × 1350 voxels (corresponding to 1575 × 1012 × 1215 µm$^3$) was reconstructed by filtered back projection using XAct software from RX Solutions.
<caption>**Table 1**</caption>

Bulk chemical composition (ppm wt %) of impurity elements in the Cu-25Cr SSS composites and its average, as well as in the pure Cr and Cr powders used for fabricating the SSS composites.* indicates quantities from GDMS measurements and # indicates from IGA analysis.

<table><thead><tr><th rowspan="2">Alloy</th><th colspan="5">wt %</th><th colspan="8">Chemical composition (ppm wt %)</th></tr><tr><th>Cu</th><th>Cr*</th><th>Al*</th><th>Si*</th><th>P*</th><th>S*</th><th>Ti*</th><th>Fe*</th><th>Co*</th><th>Ni*</th><th>Zn*</th><th>Sn*</th><th>O#</th></tr></thead><tbody><tr><td>Cu-25Cr SSS 94 %</td><td>Bal.</td><td>25.0</td><td>&lt;20</td><td>80</td><td>&lt;10</td><td>&lt;20</td><td>&lt;5</td><td>650</td><td>&lt;5</td><td>&lt;20</td><td>&lt;5</td><td>&lt;30</td><td>225</td></tr><tr><td>Cu-25Cr SSS 96 %</td><td>Bal.</td><td>24.3</td><td>&lt;20</td><td>65</td><td>&lt;10</td><td>&lt;20</td><td>&lt;5</td><td>530</td><td>&lt;5</td><td>&lt;20</td><td>&lt;5</td><td>&lt;30</td><td>250</td></tr><tr><td>Cu-25Cr SSS 98 %</td><td>Bal.</td><td>24.5</td><td>&lt;20</td><td>70</td><td>&lt;10</td><td>&lt;20</td><td>&lt;5</td><td>580</td><td>&lt;5</td><td>&lt;20</td><td>&lt;5</td><td>&lt;30</td><td>230</td></tr><tr><td>Cu-25Cr SSS</td><td>Bal.</td><td>24.6</td><td>&lt;20</td><td>&lt;100</td><td>&lt;10</td><td>&lt;20</td><td>&lt;5</td><td>&lt;600</td><td>&lt;5</td><td>&lt;20</td><td>&lt;5</td><td>&lt;30</td><td>&lt;250</td></tr><tr><td>Cu powder</td><td>Bal.</td><td>0.04</td><td>&lt;1</td><td>&lt;1</td><td>&lt;1</td><td>&lt;20</td><td>&lt;5</td><td>&lt;50</td><td>&lt;5</td><td>&lt;20</td><td>&lt;5</td><td>&lt;50</td><td>660</td></tr><tr><td>Cr powder</td><td>0.01</td><td>Bal.</td><td>140</td><td>180</td><td>26</td><td>8.4</td><td>&lt;5</td><td>&lt;2000</td><td>1.2</td><td>10</td><td>6.2</td><td>5.1</td><td>490</td></tr></tbody></table>

Under these conditions, each experimental setup resulted in a voxel size 0.90 µm for all three samples. The spatial resolution enables the detectability of pores as small as 3 µm in diameter and an accurate morphological description for pores as small as 5 µm in diameter. All objects smaller than 3 µm in diameter were excluded from the 3D analysis.

### 2.3.2. Image segmentation and 3D quantitative analysis

The 3D images were then segmented and analyzed using a workflow based on a machine learning-assisted segmentation and an in-house Python algorithm for image analyses, see our previous study for more details [28]. The developed workflow allows the quantification of the volume fraction of the different phases (pores, Cu and Cr phases), their characteristic size, as well as the description of their respective morphology and interconnection within the microstructure.

The concept of granulometry is used here to characterize the average 'characteristic size' of the segmented Cr particles. To evaluate the local orientation and spatial distribution of the Cr phase with respect to the compression direction used during the fabrication of the green bodies, the characteristic cord length in specific directions of the segmented microstructure is calculated. This is done using the concept of mean intercept length in 3D For example, a distribution of the mean intercept length for the Cr phase will be presented with respect to the compression direction, namely the parallel (CD) and transverse (TD) direction. Then, we investigate the fraction of each phase (Cu, Cr and total porosity) in each segmented slice from the reconstructions. This enables the direct correlation to potential restrictions of the heat flux in the Cu phase imposed by the spatial distribution of the Cr phase as a function of sample height or width. This is done first following the CD direction. Then, the same is done in the other two directions, i.e. parallel to the compression direction.

### 2.3.3. Atom probe tomography

APT specimens were prepared using site-specific lift-outs [29] from the Cu matrix and Cr particles in a FEI Helios Nanolab 600i Dual Beam scanning-electron microscope focused ion beam (SEM-FIB) equipped with a Ga-source, from the Cu-25Cr SSS composite at 96 % final relative density. The Cu tips were extracted far, approximately 20–30 µm, from the Cu/Cr interfaces. The APT experiments were performed in a local electrode atom probe (LEAP 5000XR, Cameca) in laser pulsing mode at a base temperature of 60 K, a pulsing frequency of 125 kHz, a detection rate of 0.7 % and laser pulse energy of 60 pJ. APT data were reconstructed using AP Suite 6.3, and elemental compositions were extracted using its peak decomposition function.

## 2.4. Measurement of the effective thermal conductivity

The thermal conductivity $k$ (W·m⁻¹·K⁻¹) of the Cu-25Cr SSS composites with different final relative densities was calculated from the experimental measurements of the thermal diffusivity ($\alpha$ - m²·s⁻¹), specific heat capacity ($c_p$ - J·kg⁻¹·K⁻¹), and density ($\rho$ - kg·m⁻³) at T = 200 °C, following Eq. (1):

$$k = \alpha \cdot \rho \cdot c_p \quad (1)$$

The thermal diffusivity measurements were achieved using the flash laser method with a NETZSCH LFA 457 apparatus under an argon atmosphere at T = 200 °C. Measurements at lower temperatures were found to show significant variability and uncertainty in the thermal diffusivity. Disk-shaped samples of 25 mm in diameter and 3 mm in thickness for the Cu-Cr composites were machined out of the initial Cu-25Cr SSS samples along their thickness. Therefore, the thermal diffusivity as a function of temperature is measured parallel to the compression direction (CD) for all the Cu-Cr SSS composites. For the Cu-25Cr SSS composite with a final relative density of 96 %, the thermal diffusivity perpendicular to the compression direction was additionally characterized to investigate the potential effect of anisotropy. The specific heat capacity was measured by differential scanning calorimetry (DSC). A Labsys evo SETARAM DSC was used to measure the heat flow from the Cu-25Cr SSS samples under Ar. A heating rate of 2 K/min was used until reaching T = 200 °C, and then the temperature was kept constant for 12 min. The $c_p$ was then determined by integrating the heat flow signal from the Cu-25Cr SSS sample and a reference sample made of silver (Ag), corrected by a blank measurement. The dimensional variation was used to obtain the coefficient of thermal expansion (CTE) of the different samples. A LINSEIS L76/10 dilatometer was used to measure the dimensional variation during heating and cooling from RT to 200 °C under an argon flow. An average linear CTE was calculated using linear regression from three samples and thus used to calculate the density evolution at T = 200 °C, $\rho(T = 200 °C)$. The uncertainties associated with the thermal conductivity calculation are estimated to be approximately 9 %.

# 3. Experimental results

## 3.1. 3D microstructure characterization

### 3.1.1. Pore volume fraction and evolution with final relative density

The typical Cu-25Cr SSS microstructures with different final relative densities are shown in Fig. 1a-c through a single cross-section of the reconstructed XCT volume. The effect of the final relative density is clear when comparing Fig. 1a-c. Here, Cr particles are dispersed in the Cu matrix together with two different pore populations. Pores are observed at the Cu/Cr interfaces, regarded as interfacial pores, and relatively smaller pores are found in the Cu matrix. The total porosity in the Cu-25Cr SSS microstructure is thus the sum of both populations. From Fig. 1a-c, the evolution of Cu matrix porosity with increasing density is evident, especially from 94 % to 96 %. Although it is less evident to comment on the evolution of interfacial porosity based on such 2D cross-sections, it seems that the interfacial porosity decreases when increasing the relative density. This will be investigated more quantitatively based on the 3D images.

The quantitative results of volume fraction for the overall porosity from XCT analysis are given in Table 2. They are in good agreement compared to those found by the Archimedes method. On one hand, the volume fraction of Cu matrix porosity decreases significantly from Cu-25Cr SSS-94 % to SSS-98 % composites, as qualitatively illustrated in Fig. 1a-c. In the Cu-25Cr SSS-94 % sample, the volume fraction of matrix
<figure><img src="image_3.png" /><figcaption>**Fig. 1.** 2D cross-section extracted from the XCT after 3D reconstruction for Cu-25Cr SSS at a final relative dentistry of a) 94 %, b) 96 %, and c) 98 %.</figcaption></figure>

porosity is 1.6 ± 0.4 % (26 % of the total porosity), whereas in the Cu-25Cr SSS-98 % sample, there is only 0.1 % ± 0.05 % of matrix porosity (5 % of the total porosity). Thus, one can consider the Cu matrix nearly fully dense from a final relative density of Cu-25Cr SSS-96 % (i.e., only 0.4 ± 0.1 % of residual matrix porosity). On the other hand, interfacial

porosity also decreases when increasing the relative density, but decreases to a greater extent from 96 to 98 %, see Table 2. Regardless of the final relative density after sintering, the volume fraction of interfacial porosity is larger than that of matrix porosity. In the 94 % and 98 % dense samples, interfacial porosity represents 74 % (4.5 % absolute) and 95 % (1.8 % absolute) of the total porosity, respectively.

The Cu matrix and interfacial pores volume fraction evolution as final relative density increases is summarized in Fig. 2a. In addition, Fig. 2b and c illustrate from a small region of interest (ROI) extracted from the 96 % dense sample, the Cu matrix and interfacial pores typical size, spatial distribution, and morphology. From Fig. 2b, Cu matrix pores are mostly spherical, rather small, typically < 10 µm, and randomly distributed in the matrix. Interfacial pores are relatively large (20–300 µm) and have complex morphology, interconnecting around adjacent Cr particles resulting in a pore envelope. This isolates the Cr phase from the Cu matrix, as shown in Fig. 2c.

On the one hand, the size and morphology of Cu matrix do not evolve significantly with increasing final relative density. On the other hand, interfacial pores evolve with increasing final density, decreasing in size and losing partly their complex morphology. Fig. 3a-c shows a larger ROI from the Cu-25Cr SSS (94, 96, 98 %) segmented microstructure where the size and morphology evolution of Cu matrix and interfacial pores can be followed. Such ROI corresponds to the vicinity of the largest interfacial pore from each Cu-25Cr SSS segmented microstructure. From 94 % to 96 % relative density, there is a significant decrease in Cu matrix pores and their distribution in the microstructure. From 96 % relative density to 98 %, the number and distribution of Cu matrix pores evolve significantly, and very few isolated pores are left in the microstructure. No significant changes in Cu matrix pore size is observed from 94 % to 98 %, remaining < 10 µm. Contrary to Cu matrix pores, interfacial pores present a significant size and morphology evolution from 94 % to 98 % relative density. At 94 % relative density, interfacial pores can reach up to 300 µm in the microstructure, see Fig. 3a, and are interconnected around adjacent Cr particles through large distances. With the increase of relative density from 94 % to 98 %, interfacial pores decrease in size, reaching a maximum of 200 µm, but keep their complex morphology, still isolating Cr particles as smaller envelopes, see e.g. Fig. 3a-c. Note that in the 94 % and 96 % dense samples, such networks are highly anisotropic, expanding preferentially in the transverse direction to the CD. Interfacial pores gain in isotropy as final relative density increases while decreasing in size. This is qualitatively illustrated by the 3D rendering in Fig. 3b-d.

### 3.1.2. Cr phase: size, morphology, and spatial distribution

The Cr volume fraction measured based on 3D image analysis also yields close values when compared to the nominal value of 25wt %Cr (29vol %Cr), see Table 2. The differences in the Cr phase quantification can result from the segmentation as well as local microstructure heterogeneity. The error on the measurement of the volume fraction of Cr was estimated to be of the order of 2–3 % in [28]. The variations in volume fraction of Cr measured in the different samples are thus considered acceptable. More details regarding the segmentation procedure and the corresponding error can be found in [28]. Overall, the Cu-25Cr SSS composites microstructure characterization using XCT analysis and segmentation yields representative volume fraction, size, and morphology of the main phases (pores and Cr).

The characteristic size of Cr particles (granulometry) as well as their characteristic cord length with respect to the CD are summarized in Fig. 4a-b. First, no significant differences in Cr particle size from a final relative density of 94 % to 98 % are found. Cr particles exhibit the same size as in the initial powder, suggesting that they are not affected during the fabrication (compression of the green body followed by solid-state sintering). However, the characteristic cord length of Cr particles in the parallel and transverse directions with respect to compression direction (CD) changes significantly, see Fig. 4b This suggests that Cr particles take a preferential orientation with respect to the CD after the
<caption>**Table 2**</caption>

Summary of the quantitative analysis of the Cr phase volume fraction and porosity in the three samples with different relative densities from 3D images. The total porosity volume fraction measured by Archimedes is also given for comparison. The error bars correspond to the results of the triplicate measurements in the Archimedes case and result from the segmentation procedure in the case of JACK images. The segmentation procedure is detailed in ref [28].

<table><thead><tr><th>Cu-25Cr SSS</th><th>Total Porosity Fraction (absolute value in %) from Archimedes</th><th>Total Porosity Fraction (absolute value in %) from JACK</th><th>Fraction of Interfacial Porosity (absolute value in %)</th><th>Contribution of interfacial pores to total porosity in %</th><th>Fraction of Cu matrix Porosity (absolute value in %)</th><th>Contribution of Cu matrix pores to total porosity in %</th><th>Cr phase Fraction (absolute value in %) from JACK</th></tr></thead><tbody><tr><td>94 %</td><td>5.7 ± 0.2</td><td>6.1 ± 1.0</td><td>4.5 ± 0.6</td><td>74 %</td><td>1.6 ± 0.4</td><td>26 %</td><td>26 ± 2</td></tr><tr><td>96 %</td><td>4.2 ± 0.3</td><td>4.3 ± 0.5</td><td>3.8 ± 0.4</td><td>88 %</td><td>0.4 ± 0.1</td><td>12 %</td><td>25 ± 3</td></tr><tr><td>98 %</td><td>1.9 ± 0.2</td><td>1.9 ± 0.2</td><td>1.8 ± 0.15</td><td>95 %</td><td>0.1 ± 0.05</td><td>5 %</td><td>29 ± 2</td></tr></tbody></table>

<figure><img src="image_4.png" /><figcaption>**Fig. 2.** a) Evolution of the fraction of Cu matrix and interfacial pores. b-c) 3D rendering of the segmented microstructure from a selected volume of interest (VOI) extracted from the 96 % dense Cu-25Cr SSS composite, showing the Cu matrix and interfacial pore characteristic size and morphology, respectively.</figcaption></figure>

compression and solid-state sintering. The latter is characterized by the preferred orientation of the longer axis of Cr particles, which can be morphologically considered as ellipsoids (Fig. 4c), perpendicular to the CD.

Fig. 5 shows the effect of the preferential Cr particle orientation and heterogeneous distribution along by the fraction of each phase with increasing sample height. This is done by plotting the corresponding fraction of each phase (Cu, Cr and total porosity) in each slice of the JACK reconstructed images. From Fig. 5, one can note that Cu and Cr phase fraction profile is heterogeneous regardless of the final density. Moreover, once considering the sum of Cr and porosity phase fraction, it shifts the corresponding heterogeneity to higher values. Such heterogeneity in Cr phase fraction, amplified by porosity (especially interfacial porosity) translates to heterogeneous Cu phase distribution. The latter refers to the potential restrictions of the heat flux in the conductive Cu phase imposed by the spatial distribution of the Cr phase. Regions with depleted fraction in Cu (i.e., enriched in Cr) locally restrains the heat flux across the microstructure along the CD direction. This is systematically observed for all the Cu-25Cr SSS composites microstructure. Such heterogeneity is much less evident in the other two transverse direction to CD, see Figure S1. As a result, the Cr phase spatial distribution can be considered anisotropic and interfacial pores amplify this anisotropy. Such Cr distribution heterogeneity does not break the Cu matrix percolation within the microstructure. Therefore, in all the samples investigated, the Cu matrix percolates in the microstructure.

## 3.2. Chemical compositions of the studied samples

A summary of the Cu and Cr powders and the bulk chemical

compositions of the sintered materials is shown in Table 1. Here, Si, Fe, and Al are found in the Cu-25Cr SSS composites and are believed to originate from Cr powder due to its chemical composition, see Table 1. The content and distribution of the secondary elements in the Cu matrix critically impact the thermal conductivity by impeding electron or phonon mobility [30]. APT specimens were extracted from the Cu matrix from Cu-25Cr SSS composite at 96 % relative density to determine the impurities in solid solution. Fig. 6 displays the tomographic reconstruction obtained from APT analysis. The most commonly found solute in the Cu matrix is Cr, with an average concentration of 0.040 at % based on peak decomposition analyses (at least 3 APT samples). This corresponds to approximately 300 ppm Cr in wt %. First nearest-neighbor distribution (1NND) analysis (Figure S2) reveals a homogenous distribution of Cr in the Cu matrix, evidenced by a perfect fit between experimental data and random distribution curves [31]. This presence of Cr in Cu is likely due to Cr diffusion in Cu during sintering. Indeed, the diffusion length of Cr in Cu at the sintering temperature (1050 °C) (reaching about 50 µm in around 10 h). This corresponds to the characteristic cord length in the Cu matrix. Thus, Cr diffuses into Cu through large lengths at the sintering temperature. Interestingly, no Fe, Al, and Si were measured above the level of background in the Cu matrix.

The composition of the Cr phase was also measured by APT. Here, an average amount of 0.03 at % Si was measured in the Cr phase. This corresponds well with the measured Si content through GDMS analysis for the Cr powder composition shown in Table 1. Thus, we can assume all Si content is in solid solution in the Cr phase. An example of an APT analysis from a Cr particle is shown in Figure S3. Cu is also uniformly distributed in the Cr particle according to the 1NND analysis in Figure S2. Similarly, no Fe or Al was found in the Cr phase. The absence
<figure><img src="image_5.png" /><figcaption>Fig. 3. 3D rendering from a larger ROI from the segmented microstructure showing the size, distribution, and morphology evolution of Cu matrix and interfacial pores for a) 94 %, b) 96 %, and c) 98 % dense samples. Cr phase is shown in transparent grey and Cu matrix in total transparency. All three ROI corresponds to the vicinity of the largest interfacial pore found in each segmented microstructure.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Fig. 4. a) Characteristic Size of Cr particles calculated by granulometry analysis from the segmented XCT data, b) characteristic cord-length of the Cr phase in the microstructure of Cu-25Cr SSS composites, and c) SEM secondary electrons image of the as-received Cr powder from the provider, before uniaxial compression and sintering) showing its typical morphology.</figcaption></figure>

of these impurity elements in either of the Cu matrix or Cr particles, especially Al and Fe coming from the Cr powder suggest they might be segregated at Cr grains boundaries or at Cu/Cr interfaces. Table 3 summarizes the APT analysis of the elemental distribution in the Cu and

Cr phases.

Here, we suggest that the detrimental effect of solutes retained in the Cu solid solution on the Cu matrix conductivity is essentially from Cr atoms in the solid solution.
<figure><img src="image_7.png" /><figcaption>Fig. 5. Phase fraction as a function of sample height for each slice from the segmented 3D image along the CD direction.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 6. Example of a reconstructed APT tip extracted from the Cu matrix. Note that for the Cu inset image the size of the spheres representing the atoms is smaller than in the Cr inset for visualization purpose.</figcaption></figure>

### Table 3

Summary of the composition of the Cu and Cr phases in the Cu-25Cr SSS 96 % composite. *Reliable quantification of oxygen content in Cu was not possible due to high incertitude from experimental conditions.

<table><thead><tr><th>Element in solid solution</th><th>Cu<br>wt %</th><th>Cr<br>wt %</th></tr></thead><tbody><tr><td>Cr</td><td>0.030</td><td>Bal.</td></tr><tr><td>Cu</td><td>Bal.</td><td>0.004</td></tr><tr><td>Si</td><td>-</td><td>0.018</td></tr><tr><td>O</td><td>*</td><td>0.006</td></tr></tbody></table>

### 3.3. Effective thermal conductivity measurements

The effective thermal conductivity measured on the Cu-25Cr SSS composites with different densities is given in Table 4. Note that such values correspond to the effective conductivity measured parallel to CD. One additional measurement along TD is presented for the 96 % dense sample. There is a significant difference in the effective thermal conductivity with increasing final relative density. The effective thermal conductivity increases by 25 % as the relative density increases from 94 % to 96 %. A subsequent increase of about 18 % is observed when the relative density increases from 96 % to 98 %. Interestingly, the effective

### Table 4

Average effective thermal conductivity at 200 °C, k (W.m⁻¹.K⁻¹), calculated from experimental measurements following Eq. (1).

<table><thead><tr><th rowspan="2">Cu-25Cr SSS</th><th colspan="2">k (W.m<sup>-1</sup>.K<sup>-1</sup>) at 200 °C</th></tr><tr><th>// to CD</th><th>⊥ to CD</th></tr></thead><tbody><tr><td>94 %</td><td>135 ± 12</td><td>—</td></tr><tr><td>96 %</td><td>170 ± 16</td><td>200 ± 18</td></tr><tr><td>98 %</td><td>200 ± 18</td><td>—</td></tr></tbody></table>

thermal conductivity measured along the CD direction is lower compared to the effective thermal conductivity in the transverse direction, suggesting an anisotropy in thermal conduction. We discuss the different microstructural features governing thermal conduction in Cu-25Cr SSS composites in the next section based on full-field simulations.

## 4. Full field modeling of the effective thermal conductivity

### 4.1. The full-field modeling approach

To investigate the microstructural features governing the effective thermal conductivity shown in Table 4, we employ a full-field modeling approach to simulate the effective property of the samples with different final relative densities (94, 96, and 98 %). The use of tomography images to faithfully describe microstructures coupled with FFT or FE modeling to simulate mechanical [32–35], functional [36,37] and fluid dynamic [38–40] properties has been recently used as a full-field approach to investigate microstructure-property relationships of composites materials. In our case, the segmented Cu-25Cr SSS microstructures from XCT reconstructions are used as voxelized meshes to simulate effective thermal conduction. This approach enables the investigation of the effect of the main microstructural features of the Cu-Cr composites on their effective thermal conductivity: the 3D spatial distribution of the Cu and Cr phase, the presence of porosity, and its spatial distribution. For this, GeoDict® software from Math2Market is used (GeoDict 2022, Math2Market GmbH, Germany). The conductivity equations are solved using a Fast-Fourier-Transformation (FFT) solver and the introduction of jump variables (from an Explicit Jump solver) at the boundary conditions, as well as at material interfaces. Therefore, Fourier’s law for the heat transfer as well as Poisson’s equation (steady state equilibrium of temperature) are solved:

$$ \dot{h} = -K\nabla T \qquad (2) $$
$$ \nabla \cdot (k \nabla T) = 0 \qquad (3) $$

where, $\dot{h}$ is the heat flux (W/m²), K is the effective thermal conductivity (W/m.K) here expressed as the thermal conductivity tensor, T is the temperature (K) and k is the local thermal conductivity.

Isotropic thermal conductivity is attributed to the constitutive phases, pure Cu and Cr, to simulate the Cu-25Cr SSS composite's effective thermal conductivity. The values of pure Cu ($k = 383 \text{ W.m}^{-1}.\text{K}^{-1}$ at 200 °C) and Cr ($k = 89 \text{ W.m}^{-1}.\text{K}^{-1}$ at 200 °C) thermal conductivity are used [41–43]. Pores are considered a non-conductive phase. Symmetric (Dirichlet) boundary condition in the heat flux direction is imposed, i.e. the temperature is constant along the border planes in the flux direction. Boundary conditions in tangential directions are set to be symmetrical. As a result, the voxelized mesh structure is flipped (mirrored) and repeated at the mesh end of each tangential direction. Lastly, a temperature difference of 600 °C, i.e., an absolute temperature of 800 °C in the in-flux plane and 200 °C in the out-flux plane, is set as the last boundary condition. The simulated effective thermal conductivity results will be systematically compared to the experimental effective thermal conductivity results at 200 °C. Simulations of the effective thermal conductivity in the three main sample directions, along the CD direction and both transverse directions (TD) are performed for all the three Cu-25Cr SSS composites having different relative density (94 %, 96 %, and 98 %).

## 4.2. Strategy to unveil the microstructural features

We will make use of the full-field numerical approach to simulate the effective thermal conductivity of Cu-25Cr SSS composites along the CD direction while investigating the effect of three main microstructural features: (i) porosity, (ii) trace elements in the Cu solid solution, and (iii) interfacial thermal resistance (ITR). The simulation of the effective thermal conductivity in the transverse direction and its comparison with that along CD will be discussed in a dedicated section. All three microstructural features that have been identified to play a significant role in the loss of thermal conductivity will be investigated based on simulations following the approaches described hereafter. We systematically compare the simulated with the experimental effective thermal conductivity.

(i) The effect of both interfacial and matrix pores will be investigated to be able to estimate their respective impact on the effective conductivity. First, the effective thermal conductivity is simulated considering no pores in the microstructure (fully dense samples). Then, only the Cu matrix pores (interfacial pores filled) are considered, and finally only the interfacial porosities are considered (Cu matrix pores filled). Such a strategy requires filling in the different pore populations with Cu. This has been done artificially using an image analysis routine.

(ii) The effect of trace elements retained in the Cu solid solution on the effective thermal conductivity is investigated by artificially deteriorating the Cu thermal conductivity. The decrease in the Cu matrix thermal conductivity due to the elements in solid solution can be estimated considering the APT data while taking into account all the pores.

(iii) The effect of an interfacial thermal resistance ($m^2.K/W$) is also evaluated. The interfacial resistance is numerically introduced at the Cu/Cr interfaces. The effects from (i) and (ii) are considered, and the ITR is then estimated by comparison with the experimental measurements of the effective thermal conductivity.

The idea is to weigh each of the above-mentioned microstructural feature contributions to the effective Cu-25Cr SSS thermal conductivity.

# 5. Discussion

## 5.1. Effect of pores: Cu matrix pores and interfacial pores

Fig. 7 shows the full-field simulation results considering no pores in the microstructure, then only Cu matrix pores (green symbol), and finally only the interfacial pores (magenta symbol). The simulated effective thermal conductivity considering all the pores and the experimental results are shown by the light-blue bars and the star symbols, respectively. Note that for this prediction, the thermal conductivity of the Cu matrix is taken as that of pure Cu and the interfacial thermal resistance is ignored.

The effective thermal conductivity predictions of fully dense samples (i.e. considering 100 % final relative density) results in significantly higher effective conductivity compared to the experimental results. From the simulation results, one can estimate that the Cu matrix porosity reduces the effective property by only 4 % for the 94 % dense sample (1.6 ± 0.4 % of pores in the Cu matrix, see Table 2). At 96 % final relative density, the loss is reduced to approximately 1 %, and at 98 % in final relative density, the Cu matrix pores do not decrease the effective thermal conductivity (the green symbol overlaps with the horizontal line). This result agrees well with previous studies found in the literature [44–46] where it was reported that the thermal conductivity of copper starts to decrease when the volume fraction of porosity exceeds 1–2 vol %. Therefore, Cu matrix porosity impacts significantly the effective thermal conductivity of Cu-25Cr sintered composites only when the relative density is below 96 %. Meanwhile, Cu/Cr interfacial porosities appear to have a greater impact. From the simulation results, one can see that at 94, 96, and 98 % of final relative density, the interfacial porosity decreases the effective thermal conductivity by 15, 10, and 5 %, respectively. For the 96 % and 98 % dense samples, interfacial porosity accounts for the total decrease in effective thermal conductivity, see magenta symbols in Fig. 7. These results suggest that the interfacial

<figure><img src="image_9.png" /><figcaption>Fig. 7. Simulated effective thermal conductivity of Cu-25Cr SSS composites considering no pores (black symbol), then with only Cu matrix pores (green symbol), and with only interfacial pores (magenta symbol). The simulated effective thermal conductivity considering both families of pores and the experimental values are represented by the light-blue bars and star symbols, respectively.</figcaption></figure>
pores forming envelopes around the Cr particles account for the major thermal conductivity loss. The presence of interfacial pores hinders the Cr contribution to the effective property. This is of great importance in composite materials where both constituent phases contribute to the effective thermal condition [22,47–52]. From XCT analysis, in the 94 % dense sample, 35 % of the total Cr particles' surface area in the volume analyzed is covered by interfacial porosity. While in the 96 and 98 % dense samples, 30 and 16 % of the total Cr phase surface is covered, respectively.

Considering both populations of pores in the microstructure, one can note that the predicted effective thermal conductivity reaches lower values compared to the fully dense samples. Here, the effect of porosity (matrix and interfacial pores) on the reduction of the effective thermal conductivity in the 94 %, 96 %, and 98 % dense samples is estimated to be 20 %, 11 % and 5 %, respectively. However, the simulations still overestimate the effective thermal conductivity measured experimentally. This suggests that additional microstructural features would be worth considering.

## 5.2. Effect of the composition of the Cu matrix

APT results suggest that Cr is trapped in the Cu matrix. Approximately 300 ppm (%wt) of Cr is found in the Cu matrix through APT measurements. Under thermodynamic equilibrium, the maximum Cr solubility in Cu can reach up to 0.7 wt %Cr (7000 ppm) at the eutectic temperature (1076 °C) and at low temperatures (< 400 °C) such solubility is very limited, being inferior to 0.04 wt %Cr (400 ppm) [43]. During high temperature solid-state sintering, Cr diffuses into Cu thus increasing the Cr content in the solid solution. The slow cooling following the sintering heat treatment due to the secondary vacuum environment allows Cr precipitation within the Cu matrix. These can be easily found homogeneously distributed in the microstructure, see **Figure S4a**. Therefore, the measured Cr content in the Cu solid solution seems in reasonable agreement with the solubility of Cr in Cu at low temperatures. For such Cr amount in the Cu matrix, the detrimental effect expressed as the reduction of the thermal conductivity of pure Cu is estimated to be approximately 10 %, see **Figure S5** in supplementary materials [43].

One can also account for the effect of O on the thermal conductivity of the Cu matrix. IGA results indicate that Cu-25Cr SSS composites have a relatively low O content, around 250 ppm. It has been shown that this residual amount of O is associated with small nodules of Cr oxides ($Cr_2O_3$) due to the fragmentation of the Cr powder oxide layer during sintering at the Cu/Cr interface, as shown in **Figure S4b** [26,27]. Table 1 also indicates that the Cu powder has a higher O content compared to the bulk sintered Cu-25Cr SSS composites. The solid solubility of O in Cu is very low. Under thermodynamic equilibrium, the maximum solid solubility of O in pure Cu can reach approximately 100 ppm (0.01 wt % or 0.03 at %) at 1050 °C and only around 10 ppm (0.001 wt % or 0.004 at %) at 700 °C [43]. Thus, we believe that the O content found through IGA is mostly associated with the $Cr_2O_3$ oxides nodules present at the Cu/Cr interface and that a limited amount of O is trapped in solid solution in the Cu matrix. High temperature stable Cu-oxides formed during vacuum sintering can also contribute to this O content found [26, 27]. Thus, the detrimental effect of O on the Cu matrix thermal conductivity is thought to be limited. We account the additional 5 % reduction in Cu thermal conductivity caused by oxygen in the Cu matrix, based on the maximum solid solubility of oxygen in Cu (100 ppm). Considering a cumulative effect of all potential elements in solid-solid solution based on the APT and IGA results, being mostly associated to Cr content in solid solution, we take the overall detrimental effect of elements on the thermal conductivity of the Cu matrix of 15 %.

Introducing a detrimental effect of 15 % on Cu's thermal conductivity into the full-field model decreases the predicted effective thermal conductivity for all Cu-25Cr SSS composites (94 %, 96 %, and 98 %, respectively), as shown in **Fig. 8**. Here, the predicted values for effective

<figure><img src="image_10.png" /><figcaption>**Fig. 8.** Simulated effective thermal conductivity of Cu-25Cr SSS composites considering both families of pores and the detrimental effect of elements in solid-solution in the Cu matrix expressed as –15 % reduction on pure Cu's thermal conductivity (black symbols).</figcaption></figure>

thermal conductivity get closer to the experimental measurements. However, full-field simulations considering the effect of pores and a 15 % decrease in the thermal conductivity of the Cu phase due to solid-solution impurities still fail to predict the experimentally effective thermal conductivity of Cu-25Cr SSS composites. This suggests that we are probably missing another important feature. Thus, we considered an additional microstructural feature: the interfacial thermal resistance (ITR).

## 5.3. Effect of interfacial thermal resistance across Cu/Cr interfaces

The effect of interfacial thermal resistance (ITR) on the effective thermal conductivity of Cu-25Cr SSS composites is shown in **Fig. 9**. The effective property is normalized with respect to the effective thermal conductivity values found in the full-field simulation, considering both pore populations (matrix and interfacial pores) as well as the 15 % decrease in matrix conductivity due to the elements in solid-solution (**Fig. 8**). **Fig. 9a** shows the evolution of the normalized effective thermal conductivity with increasing ITR. Here, in the case of Cu-25Cr SSS composites, from $1 \cdot 10^{-8}$ to $1 \cdot 10^{-5}$ m²·K/W, there is a maximum decrease in the effective thermal conductivity of approximately 20 %, 23 %, and 27 %, respectively, for 98 %, 96 %, and 94 % final relative densities. Interestingly, the latter results show that the effect of ITR on the normalized effective thermal conductivity is slightly different for Cu-25Cr SSS composites with different volume fractions of pores, thus different pore sizes and morphologies, especially interfacial pores. This suggests a synergistic effect of ITR and other microstructural features, such as the interfacial pores. This will be discussed further in the next section. Beyond $1 \cdot 10^{-5}$ m²·K/W, no significant decrease in the effective thermal conductivity is observed.

In essence, interfacial thermal resistance is the inverse of interfacial thermal conductance, leading to a finite temperature discontinuity at the interface, reducing the conduction properties [48,50,52–55]. Such a discontinuity is the result of scattering across the interface due to
<figure><img src="image_11.png" /><figcaption>**Fig. 9.** a) Simulated effective thermal conductivity with increasing interfacial thermal resistance for all Cu-25Cr SSS composites (94 %, 96 % and 98 % final relative density). The latter was simulated considering all pores and the detrimental effect of solid solution impurities in Cu matrix conductivity (–15 %). Simulated effective thermal conductivity of Cu-25Cr SSS composites considering all the microstructural features (pores, solid-solution impurities, and ITR represented by black symbols).</figcaption></figure>

different electron structures between the two metals as well as different conduction mechanisms, e.g. when the phonon-mediated thermal conduction becomes non-negligible in one of the two phases. Studies have shown that in composite materials, the nature of the interface, whether metal/ceramic or metal/metal interfaces, plays an important role in the effective properties [48,56–59]. Additional phases, such as oxides and/or intermetallics, also contribute significantly to increasing ITR. Metal/ceramic interfaces are usually characterized by high interfacial thermal resistance ($> 10^{-7}$ m²·K/W) due to the differences in phonon density of states which are low for metals and high for ceramics. For metal/metal interfaces, one can assume that electrons contribute significantly and synergistically with phonons to the heat transport across interfaces. Interfacial thermal resistance of metal/metal interface is characterized by one to two orders of magnitude lower than metal/-ceramics ($10^{-7} < \text{ITR} < 10^{-10}$ m²·K/W). Indeed, the full-field modeling results considering an ITR $> 1·10^{-6}$ m²·K/W practically isolate the Cr phase from the conduction properties, drastically reducing the Cr contribution to the effective thermal conductivity.

From the full-field predictions shown in Fig. 9a, the introduction of an ITR of $5·10^{-7}$ m²·K/W best fits the experimental results for all the Cu-25Cr SSS composites, see Fig. 9a and b. A relative difference is found for the Cu-25Cr SSS at 94 % final density, where the experimental effective thermal conductivity is lower than the predicted values. We believe that this is the result of the significant isolation of Cr particles by the large and interconnected interfacial pores in the case of Cu-25Cr SSS 94 %, extending to large distances, which significantly amplifies the anisotropy, as shown in the next section. Such difference found in the 94 % case can also be associated with different interfaces properties due to low densification. This would result into higher interfacial resistance, hindering significantly the effective thermal conductivity.

It is worth discussing the value of ITR of $5·10^{-7}$ m²·K/W found from our FFT-modeling by making comparisons with values of ITR estimated based on the first-order models proposed in the literature such as acoustic-mismatch model (AMM) and diffuse-mismatch model (DMM) [57,60–63]. For both AMM and DMM, the interfacial thermal resistance can be estimated based on the Debye-model for the phonon density. Here we consider a simple two-segment system case of Cu/Cr interface where only electron-electron and phonon-phonon coupling represents the thermal resistance from one metal to the other. The first is the significant difference in electronic structure of both metals, contributing to

electron-scattering, thus, to the ITR. The latter indicates that the phonon thermal conduction contribution of one of the phases effectively participates to the heat transport, causing acoustic mismatch and increasing the ITR [64]. In transition metals such as Ti, Zr, Co, Fe, and Cr, the contribution of phonons to thermal conductivity becomes more important than in highly conductive metals [42,65,66]. From first principles calculations, phonon thermal conductivity contributes approximately 5 % to the total thermal conductivity for pure copper, whereas in Ti and Cr such contribution is significantly higher, surpassing 20 % contribution [66]. In composite materials where the dispersed phase contributes effectively, the inherent conduction mechanisms in each phase play an important role to the final properties. This is because with different conduction mechanisms, additional scattering effects are present across interfaces. This corresponds to acoustic mismatch from one phase to another, reducing the efficiency of the heat transport [58,67,68]. This has been largely shown for Cu-diamond composites (Cu mostly electronic and Diamond purely by phonons). The conduction from the matrix to the dispersed phase suffers from significant scattering (electron-phonon) at the interface as an interfacial thermal resistance [58,67–70]. The corresponding equations and constants for the DMM model can be found in the Supplementary Materials.

From DMM, one can estimate the metal-metal phonon interfacial thermal resistance, $R_p$, through equation S1 ($R_p = 1.5·10^{-9}$ m²·K/W); and the metal-metal electronic interfacial thermal resistance $R_e$ ($R_e = 2.9·10^{-10}$ m²·K/W) can be estimated by equation S2 given in the Supplementary Materials, respectively. The $R_e$ estimated using DMM is relatively low compared to our FFT simulations and is characteristic of metal-metal interfaces in which the electrons-mediated thermal conduction is dominant. The $R_p$ corresponds to a relatively high interfacial thermal resistance for a metal-metal interface when compared to other metal-metal coupling (Cu-Al coupling, $R_p = 7.3·10^{-10}$ m²·K/W). This result indicates that the phonon thermal conduction contribution from Cr-phase effectively participates to the heat transport, increasing the ITR. However, such value of $R_p$ is still two orders of magnitude lower than the one found from our FFT simulations.

We suggest that a relatively high ITR, typically $5·10^{-7}$ m²·K/W, characterizes the Cu/Cr interfacial conduction properties. Besides the previous two contributions to the ITR accounted for in the DMM, we believe that the high value of ITR $5·10^{-7}$ m²·K/W found through FFT simulations results from the nature of the Cu/Cr interfaces in our
sintered composites which are not as perfect as assumed in the DMM. It has been shown that during sintering under vacuum, oxygen can get trapped within the interfacial pores [26,27]. This can create conditions to form a stable oxide layer at the interface. In addition, the presence of Cr-oxides at the Cu/Cr interface as shown in **Figure S4** as well as the poor contact between both phases during sintering prevent the formation of solid/solid interface, and are believed to contribute to substantially increasing the ITR.

In the case of Cu-25Cr SSS composites, coupling such a numerical full-field modelling approach with microstructure characterization and thermal conductivity measurements, different microstructural features synergistically govern the effective thermal conduction. In addition to the detrimental effects of pores, especially interfacial pores and elements in solid solution, high ITR is also present at the Cu/Cr interfaces. This ITR results from the different electronic structure in Cu and Cr, the difference in conduction mechanisms between Cu and Cr (contributions of phonon), and more importantly from the presence of oxide particles or an oxide layer at the Cu/Cr interfaces which negatively affect the thermal conduction across the Cu/Cr interfaces.

### 5.4. Effect of Cr phase distribution: anisotropy of the thermal conduction

To investigate the effect of the Cr phase distribution on the effective

thermal conductivity, we analyze the heat flux ($\hat{h}$) across the three composites having different relative densities. Here, localized heat flux within the microstructure will be given attention. For such analysis, the simulation results considering all the microstructural features previously discussed (pores, -15 % on Cu's thermal conductivity and ITR equal to $5.10^{-7}$ m²·K/W) are considered. Fig. 10 shows a 2D cross-section of the segmented Cu-25Cr SSS at different final relative densities (94, 96, 98 %) with the corresponding phases' color labels. From this cross-section, the difference in porosity from 94 % to 98 % Cu-25Cr SSS composites is seen. The corresponding heat flux map of the same cross-section in the compression direction (CD) is shown in Fig. 10b. One can easily see that the heat flux is less intense through the Cr phase than in the Cu matrix, as well as the presence of non-conductive regions (dark blue) such as pores. Additionally, attention must be given to the regions carrying a large heat flux (displayed in red). It is clear from Fig. 10b that Cu-25Cr SSS composites present regions that have an intense heat flux. In other words, within the microstructure, restrained heat flux zones can be found in narrow Cu matrix sections. Moreover, large bands of such restrained heat flux, transverse to the CD, are easily identified. Such bands are formed by adjacent Cr particles and their surrounding porosity, especially interfacial porosity.

This is further illustrated by the 3D rendering of the heat flux shown in Fig. 11. Here, a threshold in the heat flux scale was applied to better

<figure><img src="image_12.png" /><figcaption>**Fig. 10.** a) Single cross-section from segmented microstructures used in the simulation of the effective thermal conductivity. b) The corresponding heat flux in the same cross-section as in a).</figcaption></figure>
<figure><img src="image_13.png" /><figcaption>**Fig. 11.** 3D rendering of the heat flux along the CD direction for all the Cu-25Cr SSS and VAR composites. Note that a threshold is applied on the heat flux scale to better visualize the less conducting (in blue) zones and restrained heat flux (in red) zones.</figcaption></figure>

reveal the less conducting regions (in blue) as well as the high heat flux zones (in red). According to the 3D heat flux observations, such bands of restrained flux and less-conducting regions expand within the microstructure. The effect of porosity on the size of these bands is also clear. In the 94 % dense sample, large and thick bands are found in the microstructure. The 3D renderings allow visualization of their expansion from one side to another. Increasing the final relative density to 96 % and further to 98 %, bands get thinner but are still present in the microstructure. In addition, the restrained heat flux zones become less intense and smaller in the transverse direction, see **Figure S6**. These results suggest that the anisotropy in the Cr phase distribution previously characterized reflects on the effective thermal conductivity of Cu-25Cr SSS composites. This is confirmed by the additional experimental measurement of the effective thermal conductivity in the transverse direction, see Table 4. Full-field predictions in all directions following the microstructural features discussed in the previous sections and the experimental points are summarized in **Fig. 12**. The effective thermal conductivity of Cu-25Cr SSS composites is, thus, anisotropic. As previously discussed, we suggest that this strong anisotropy promoted by the Cr spatial distribution, amplified by the large and interconnected interfacial pores, explains the significantly hindered measured effective

thermal conductivity along the CD direction for the 94 % [71,72].

## 6. Conclusion

The microstructural features governing the effective thermal conductivity of Cu-25Cr solid-state sintered composites with different final relative density have been thoroughly investigated. The synergistic effect of interfacial porosity, trace elements in the Cu solid solution, and interfacial thermal resistance on the effective thermal conductivity of Cu-25Cr composites shows the complex microstructure-thermal conduction property relationships. The advanced microstructural characterization enabled precise 3D description of the microstructure and the chemical composition of Cu and Cr phases. Through the full-field modeling approach, we unraveled effects of different complex microstructural features, such as interfacial resistance and phases spatial distributions on the effective thermal conductivity. Coupling both experimental and numerical investigations allowed one to decouple such effects and can be summarized as follows.

* The presence of large and interconnected interfacial pores around the Cr phase contributes to their isolation from participating in
<figure><img src="image_14.png" /><figcaption>Fig. 12. Simulated effective thermal conductivity of Cu-25Cr SSS composites in all three directions (along the CD and both transverse directions) considering both families of pores (matrix and interfacial pores), the detrimental effect of elements in solid-solution in the Cu matrix expressed as –15 % reduction on pure Cu’s thermal conductivity and an interfacial thermal resistance across Cu/Cr interface of 5·10⁻⁷ m²·K/W.</figcaption></figure>

thermal conduction. Interfacial pores also contribute to the anisotropy in Cr spatial distribution in the microstructure, especially in the parallel direction to the compression direction, hindering the thermal conduction for low density Cu-25Cr composites (<96 % final relative density). With increasing final relative density, interfacial pores decrease in size, gaining isotropy, allowing Cr participation in thermal conduction and decreasing anisotropy in thermal flow. Cu matrix pores have an effect on the effective thermal conductivity only at low densities (<96 % final relative density). From 96 % relative density, Cu matrix can be considered fully dense with randomly distributed small and spherical residual Cu matrix pores, with a limited effect on the effective thermal conductivity.

* Cr is the main trace element found in the Cu matrix and is held responsible for the reduction in thermal conductivity of the Cu phase. Cr is found in the Cu matrix because it has diffused in the Cu matrix upon sintering. The Cr content in the Cu solid solution is close to its maximum solid solubility in Cu at low temperatures.

* Cu/Cr interfaces are characterized by a relatively high interfacial thermal resistance. This is mostly due to the different electronic structures from the two dissimilar metals and higher phonon-mediated thermal conduction in the Cr phase compared to Cu, as well as the presence oxides nodules and/or oxide layer at the Cu/Cr interface, which all contribute to electron-phonon scattering leading to an increase of the ITR.

* The effective thermal conductivity of Cu-25Cr solid-state sintered composites shows anisotropy. There is a difference between the measured property in the parallel direction and the compression direction. The ellipsoidal Cr particles take a preferential orientation with their longer axis perpendicular to the compression direction during compression and sintering. This results in a heterogeneous distribution of the Cr particles in the microstructure. Such an effect is amplified by the large and interconnected interfacial pores, resulting in lower effective thermal conductivity in the compression direction

compared to the transverse direction. Such anisotropy is reduced, however not completely eliminated, with increasing final relative density.

All the conclusions drawn from this work could be used in the future as guidelines for the optimization of the future generation of Cu-Cr electrical contacts.

### CRediT authorship contribution statement

**Lucas Varoto:** Writing – original draft, Visualization, Validation, Methodology, Investigation, Formal analysis, Data curation, Conceptualization. **Elodie Courtois:** Writing – review & editing, Validation, Methodology, Formal analysis, Data curation. **Raymond Kwesi Nutor:** Writing – review & editing, Investigation, Data curation. **Sophie Roure:** Writing – review & editing, Validation, Project administration, Methodology, Investigation, Funding acquisition, Conceptualization. **Anthony Papillon:** Validation, Project administration, Methodology, Investigation, Conceptualization. **Mélissa Chosson:** Validation, Methodology, Data curation. **Baptiste Gault:** Writing – review & editing, Validation, Methodology, Investigation, Data curation, Conceptualization. **Pierre Lhuissier:** Writing – review & editing, Validation, Methodology, Investigation, Formal analysis, Data curation, Conceptualization. **Jean-Jacques Blandin:** Writing – review & editing, Validation, Supervision, Methodology, Investigation, Funding acquisition, Conceptualization. **Guilhem Martin:** Writing – review & editing, Visualization, Validation, Supervision, Project administration, Methodology, Investigation, Funding acquisition, Data curation, Conceptualization.

### Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

### Acknowledgements

The authors are grateful to Schneider Electric Industries and the Association Nationale de la Recherche Technologique (ANRT) for the financial support of this work in the framework of the Ph.D. of Lucas Varoto (CIFRE 2021/1619). This work has benefited from the characterization equipment of the Grenoble INP CMTC platform supported by the Centre of Excellence of Multifunctional Architectured Materials "CEMAM" n°ANR-10-LABX-44-01 funded by the Investments for the Future program. The authors are also thankful to all Max-Planck-Institute for Sustainable Materials technical team members and characterization platform. Baptiste Gault and Raymond Kwesi Nutor are grateful for Uwe Tezins and Andreas Sturm for their support of the FIB and APT facilities at the institute.

### Supplementary materials

Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.actamat.2025.121601.

### References

[1] P.G. Slade, 1. High-Voltage Vacuum Interrupter Design, Taylor & Francis Group (Ed.). The Vacuum Interrupter: Theory, Design, and Application, CRC Press, 2008, pp. 9–111.

[2] M. Braunovic, V. V Konchits, N.K. Myshkin, S. Edition, Fundamentals of Electrical Contacts, 138, Taylor & Francis Group, 2006, p. 660.

[3] P.G. Slade, 3. Materials, Desgin, and Manufacture of Vacuum Interrupter, Taylor & Francis Group (Ed.). The Vacuum Interrupter: Theory, Design, and Application, CRC Press, 2008, pp. 177–249.
[4] J. ru Han, Z. he Dou, T. an Zhang, W. An, Review of the recent Chinese research on the electrical properties of CuCr contacts for vacuum interrupters, J. Mater. Res. Technol. 25 (2023) 1585-1598, https://doi.org/10.1016/j.jmrt.2023.06.060.

[5] B. Ding, Z. Yang, X. Wang, Influence of microstructure on dielectric strength of CuCr contact materials in a vacuum, IEEE Trans. Compon. Packag. Manuf. Technol. A 19 (1996) 76-81, https://doi.org/10.1109/95.486565.

[6] K.V. Klinski-Wetzel, C. Kowanda, M. Heilmaier, F.E.H. Müller, The influence of microstructural features on the electrical conductivity of solid phase sintered CuCr composites, J. Alloys Compd. 631 (2015) 237-247, https://doi.org/10.1016/j.jallcomm.2014.12.249.

[7] W.F. Rieder, M. Schussek, W. Glätzle, E. Kny, The influence of composition and Cr Particle Size of Cu/Cr contacts on chopping current, contact resistance, and breakdown voltage in vacuum interrupters, IEEE Trans. Compon. Hybrids Manuf. Technol. 12 (1989) 273-283, https://doi.org/10.1109/33.31434.

[8] R. Muller, Arc-Melted CuCr alloys as contact materials for vacuum interrupters, Siemens Forsch. -Entwicklungsberichte 17 (1988) 105-111.

[9] C. Zhang, Y. Wang, Z. Yang, Y. Guo, D. Bingjun, Microstructure and properties of vacuum induction melted CuCr25 alloys, J. Alloys Compd. 366 (2004) 289-292, https://doi.org/10.1016/j.jallcomm.2003.07.001.

[10] S.X. Xiu, R. Yang, J. Xue, J.X. Wang, J.Y. Wang, Microstructure and properties of CuCr contact materials with different Cr content, Trans. Nonferrous Met. Soc. China (Engl. Ed.) 21 (2011), https://doi.org/10.1016/S1003-6326(11)61612-9.

[11] S. Spaić, M. Komac, A. Fetahagić, Microstructure and properties of sintered Cu-25Cr alloy, Mater. Sci. Technol. (U. K.) 5 (1989) 1069-1073, https://doi.org/10.1179/mst.1989.5.11.1069.

[12] W. Li, R.L. Thomas, R.Kirkland Smith, Effects of Cr content on the interruption ability of CuCr contact materials, IEEE Trans. Plasma Sci. 29 (2001) 744-748, https://doi.org/10.1109/27.964467.

[13] K. Von Klinski-Wetzel, C. Kowanda, M. Böning, M. Heilmaier, F.E.H. Müller, Parameters influencing the electrical conductivity of CuCr alloys, in: Proceedings - International Symposium on Discharges and Electrical Insulation in Vacuum, ISDEIV, 2012, pp. 392-395, https://doi.org/10.1109/DEIV.2012.6412536.

[14] W. Huang, H. Yu, L. Wang, X. Wu, C. Ouyang, Y. Zhang, J. He, State of the art and prospects in sliver- and copper-matrix composite electrical contact materials, Mater. Today Commun. 37 (2023) 107256, https://doi.org/10.1016/j.mtcomm.2023.107256.

[15] P.G. Slade, 2. Vacuum Arc, Taylor & Francis Group (Ed.). The Vacuum Interrupter: Theory, Design, and Application, CRC Press, 2008, pp. 121-176.

[16] U. Hauf, A. Kauffmann, S. Kauffmann-Weiss, A. Feilbach, M. Boening, F.E. H. Mueller, V. Hinrichsen, M. Heilmaier, Microstructure formation and resistivity change in CuCr during rapid solidification, Metals 7 (2017), https://doi.org/10.3390/met7110478.

[17] X. Wei, J. Wang, Z. Yang, Z. Sun, D. Yu, X. Song, B. Ding, S. Yang, Liquid phase separation of CuCr alloys during the vacuum breakdown, J. Alloys Compd. 509 (2011) 7116-7120, https://doi.org/10.1016/j.jallcom.2011.04.017.

[18] L. Varoto, M. Chosson, J.-J. Blandin, A. Papillon, S. Roure, G. Martin, Microstructural evolutions induced by an electrical breakdown in a binary Cu-25Cr alloy, IOP. Conf. Ser. Mater. Sci. Eng. 1249 (2022) 012023, https://doi.org/10.1088/1757-899X/292/1/012024.

[19] W. Sun, H.L. Liu, Y.G. Cai, Y.L. Li, H.Y. Zhou, Y. Zhou, Simulation of the temperature field distribution in medium-voltage vacuum interrupter and experimental verification. IOP Conf Ser Mater Sci Eng, Institute of Physics Publishing, 2018, https://doi.org/10.1088/1757-899X/292/1/012024.

[20] X. Zhang, X. Guo, K. Song, X. Wang, J. Feng, S. Li, H. Lin, Simulation and verification of thermal conductivity of CuCr30 contact material based on morphological changes of Cr particles, Mater. Today Commun. 26 (2021), https://doi.org/10.1016/j.mtcomm.2021.102153.

[21] J.J. Shea, X. Zhou, Contact material and arc current effect on post-current zero contact surface temperature, IEEE Trans. Compon. Packag. Technol. 29 (2006) 286-293, https://doi.org/10.1109/TCAPT.2006.875881.

[22] Baihe Miao, Hui Guo, Yan Zhang, Guoxun Liu, Effects of interface characteristics of Cu/Cr phases on the contact performance of Cu-25Cr alloy contact material, in: 2008 23rd International Symposium on Discharges and Electrical Insulation in Vacuum, 2008, https://doi.org/10.1109/DEIV.2008.4676750.

[23] S.X. Xiu, R. Yang, J. Xue, J.X. Wang, J.Y. Wang, Microstructure and properties of CuCr contact materials with different Cr content, Trans. Nonferrous Met. Soc. China (Engl. Ed.) 21 (2011), https://doi.org/10.1016/S1003-6326(11)61612-9.

[24] L. Varoto, P. Lhuissier, J.J. Blandin, S. Roure, A. Papillon, M. Chosson, G. Martin, Influence of the processing route on the mechanical properties of Cu-35Cr metal matrix composites, Mater. Sci. Eng.: A 908 (2024), https://doi.org/10.1016/j.msea.2024.146953.

[25] Z. Chen, F. Wang, W. Yu, Y. Wang, Thermal conductivity prediction of metal matrix particulate composites: theoretical methodology and application, Int. J. Thermophysics 44 (2023) 94, https://doi.org/10.1007/s10765-023-03204-3.

[26] A. Papillon, J.M. Missiaen, J.M. Chaix, S. Roure, H. Schellekens, Sintering mechanisms of Cu-Cr metallic composites, Int. J. Refract Met. Hard Mater. 65 (2017) 9-13, https://doi.org/10.1016/j.jirrmhm.2016.11.010.

[27] A. Papillon, S. Roure, H. Schellekens, J.-M. Missiaen, J.-M. Chaix, E. Rigal, Investigation on the chemical reactions affecting the sinterability and oxide content of Cu-Cr composites during the solid state sintering process, Mater. Des. 113 (2017) 353-360, https://doi.org/10.1016/j.matdes.2016.09.038.

[28] L. Varoto, J.J. Blandin, P. Lhuissier, S. Roure, A. Papillon, M. Chosson, G. Martin, 3D microstructure characterization of Cu-25Cr solid state sintered alloy using X-ray computed tomography and machine learning assisted segmentation, Mater. Charact. 203 (2023), https://doi.org/10.1016/j.matchar.2023.113107.

[29] K. Thompson, D. Lawrence, D.J. Larson, J.D. Olson, T.F. Kelly, B. Gorman, In situ site-specific specimen preparation for atom probe tomography, Ultramicroscopy 107 (2007) 131-139, https://doi.org/10.1016/j.ultramicro.2006.06.008.

[30] L. Weber, R. Tavangar, On the influence of active element content on the thermal conductivity and thermal expansion of Cu-X (X = Cr, B) diamond composites, Scr. Mater. 57 (2007) 988-991, https://doi.org/10.1016/j scriptamat.2007.08.007.

[31] L.T. Stephenson, M.P. Moody, P.V. Liddicoat, S.P. Ringer, New techniques for the analysis of fine-scaled clustering phenomena within Atom Probe Tomography (APT) data, Microsc. Microanal. (2007) 448-463, https://doi.org/10.1017/S1431927607070900.

[32] J. Lienhard, T. Barisin, H. Grimm-Strele, M. Kabel, K. Schladitz, T. Schweiger, Microstructural characterisation of 3D printed and injection-moulded glass fibre-reinforced polypropylene by image analysis, simulation and experimental methods, Strain 60 (2024), https://doi.org/10.1111/str.12463.

[33] E. Gregorová, P. Simonová, W. Pabst, T. Uhlířová, V. Nečina, Temperature dependence of Young's modulus and the occurrence of an elastic anomaly in porous alumina-mullite composites prepared by starch consolidation casting, Ceram. Int. 50 (2024) 6309-6323, https://doi.org/10.1016/j.ceramint.2023.11.359.

[34] R. Umer, M.A. Ali, R.A. Alia, K.A. Khan, W.J. Cantwell, Creep compaction and µCT based permeability measurement of aerospace-grade out-of-life prepregs, Mater. Today Commun. 25 (2020), https://doi.org/10.1016/j.mtcomm.2020.10149.

[35] J. Genée, S. Berbenni, N. Gey, R.A. Lebensohn, F. Bonnet, Particle interspacing effects on the mechanical behavior of a Fe-TiB2 metal matrix composite using FFT-based mesoscopic field dislocation mechanics, Adv. Model. Simul. Eng. Sci. 7 (2020), https://doi.org/10.1186/s40323-020-0141-z.

[36] Z. Zheng, X.X. Lu, L. Xu, Y. Liu, J. Zeng, H. Zhu, J. Wen, P. Zhu, R. Sun, W. Wu, Constructing phonon transport bridges via low-temperature sintering in Diamond@Ag/EP composite to achieve efficient 3D networks structure, Chem. Eng. J. 495 (2024), https://doi.org/10.1016/j.cej.2024.153499.

[37] J. Schöne, W. Beckert, B. Matthey, M. Herrmann, Modelling of the microstructure and thermal conductivity of SiC-bonded diamond materials, Open Ceram. 18 (2024), https://doi.org/10.1016/j.oceram.2024.100594.

[38] M. Azimian, M. Naderi, P. Soltani, L. Cheng, K. Naderi, S. Linden, A. Wiegmann, Experimental and CFD analysis of fluid flow through nanofiber filter media, Sci. Rep. 14 (2024), https://doi.org/10.1038/s41598-024-67066-x.

[39] M.A. Ali, T. Khan, K.A. Khan, R. Umer, Micro Computed Tomography based stochastic design and flow analysis of dry fiber prefabricated manufactured by automated fiber placement, J. Compos. Mater. 57 (2023) 2075-2090, https://doi.org/10.1177/00219983231168791.

[40] C. Kim, S. Seo, J. Kim, Y. Park, S. Yoon, P. Lee, D. You, Effects of wettability and pore size on the transport dynamics of multiphase flow in porous media, Phys. Fluids 37 (2025), https://doi.org/10.1063/5.0279033.

[41] C.Y. Ho, R.W. Powell, P.E. Liley, Thermal Conductivity of the Elements, J. Chem. Ref. Data 1 (1972) 279-421, https://doi.org/10.1063/1.3253100.

[42] P.G. Klemens, R.K. Williams, Thermal conductivity of metals and alloys, Int. Met. Rev. 31 (1986).

[43] ASM Specialty Handbook, Copper and Copper Alloys, 2003.

[44] J. Ott, A. Burghardt, D. Britz, F. Mücklich, Influence of porosity and impurities on the thermal conductivity of pressure-less sintered Cu powder green bodies, Powder Metall. 64 (2021) 85-96, https://doi.org/10.1080/00325899.2021.1871806.

[45] S.J. Raab, R. Guschlbauer, M.A. Lodes, C. Körner, Electrical and Electrical Conductivity of 99.9% Pure Copper Processed via Selective Electron Beam Melting, Adv. Eng. Mater. 18 (2016) 1661-1666, https://doi.org/10.1002/adem.201600078.

[46] C. Vincent, J.F. Silvain, J.M. Heintz, N. Chandra, Effect of porosity on the thermal conductivity of copper processed by powder metallurgy, J. Phys. Chem. Solids 73 (2012) 499-504, https://doi.org/10.1016/j.jpcs.2011.11.033.

[47] X. Chen, H. Zhou, T. Zhang, L. Bi, W. Tian, S. Fu, W. Li, X. Liu, F. Ma, K. Zhang, H. Sun, P. Liu, Mechanism of interaction between the Cu/Cr interface and its chemical mixing on tensile strength and electrical conductivity of a Cu-Cr-Zr alloy, Mater. Des. 180 (2019), https://doi.org/10.1016/j.matdes.2019.1079796.

[48] B.C. Gundrum, D.G. Cahill, R.S. Averback, Thermal conductance of metal-metal interfaces, Phys. Rev. B Condens. Matter. Mater. Phys. 72 (2005), https://doi.org/10.1103/PhysRevB.72.245426.

[49] A.R. Moustafa, R.B. Dinwiddie, A.E. Pawlowski, D.A. Splitter, A. Shyam, Z. C. Cordero, Mesostructure and porosity effects on the thermal conductivity of additively manufactured interpenetrating phase composites, Addit. Manuf. 22 (2018) 223-229, https://doi.org/10.1016/j.addma.2018.05.018.

[50] D.P.H. Hasselman, L.F. Johnson, Effective Thermal Conductivity of Composites with Interfacial Thermal Barrier Resistance, J. Compos. Mater. 21 (1987).

[51] K. Wang, Y. Lu, Y. Rao, N. Wei, J. Ban, Y. Peng, S. Yao, S. Ahzi, New insights into the synergistic influence of voids and interphase characteristics on effective properties of unidirectional composites, Compos. Struct. 255 (2021) 112862, https://doi.org/10.1016/j.compstruct.2020.112862.

[52] S. Torquato, M.D. Rintoul, Effect of the Interface on the Properties of Composite Media, Phys. Rev. Lett. 75 (1995).

[53] N. Chawla, K.K. Chawla, Metal Matrix Composites, Springer New York, New York, 2013, https://doi.org/10.1007/978-1-4614-9548-2, 2nd ed.

[54] T.William Clyne, Thermal and electrical conduction in metal matrix composites. Comprehensive Composite Materials II, Elsevier, 2017, pp. 188-212, https://doi.org/10.1016/B978-0-12-803581-8.09970-7.

[55] G.L. Pollack, S. Properties, O. Metals, P. Dependence, C. Mazo, The, Iv. Conclusions, A.R. eferences, Kapitza Resistance, Rev. Mod. Phys. 4 (1969).
[56] L. Tian, I. Anderson, T. Riedemann, A. Russell, Modeling the electrical resistivity of deformation processed metal-metal composites, Acta Mater. 77 (2014) 151–161, https://doi.org/10.1016/j.actamat.2014.06.013.

[57] M. Kida, L. Weber, C. Monachon, A. Mortensen, Thermal conductivity and interfacial conductance of AlN particle reinforced metal matrix composites, J. Appl. Phys. 109 (2011), https://doi.org/10.1063/1.3553870.

[58] M. Battabyal, O. Beffort, S. Kleiner, S. Vaucher, L. Rohr, Heat transport across the metal-diamond interface, Diam. Relat. Mater. 17 (2008) 1438–1442, https://doi.org/10.1016/j.diamond.2008.01.023.

[59] L. Weber, Non-conducting inclusions in a conducting matrix: influence of inclusion size on electrical conductivity, Acta Mater. 53 (2005) 1945–1953, https://doi.org/10.1016/j.actamat.2005.01.004.

[60] X. Liu, F. Sun, L. Wang, Z. Wu, X. Wang, J. Wang, M.J. Kim, H. Zhang, The role of Cr interlayer in determining interfacial thermal conductance between Cu and diamond, Appl. Surf. Sci. 515 (2020), https://doi.org/10.1016/j.apsusc.2020.146046.

[61] X. Liu, F. Sun, W. Wang, J. Zhao, L. Wang, Z. Che, G. Bai, X. Wang, J. Wang, M. J. Kim, H. Zhang, Effect of chromium interlayer thickness on interfacial thermal conductance across copper/diamond interface, Int. J. Miner. Metall. Mater. 29 (2022) 2020–2031, https://doi.org/10.1007/s12613-021-2336-9.

[62] Z. Tan, Z. Li, D.B. Xiong, G. Fan, G. Ji, D. Zhang, A predictive model for interfacial thermal conductance in surface metallized diamond aluminum matrix composites, Mater. Des. 55 (2014) 257–262, https://doi.org/10.1016/j.matdes.2013.09.060.

[63] G. Chang, F. Sun, J. Duan, Z. Che, X. Wang, J. Wang, M.J. Kim, H. Zhang, Effect of Ti interlayer on interfacial thermal conductance between Cu and diamond, Acta Mater. 160 (2018) 235–246, https://doi.org/10.1016/j.actamat.2018.09.004.

[64] Salvatore Torquato, Random Heterogeneous Materials, Springer New York, NY, 2013, https://doi.org/10.1007/978-1-4757-6355-3, 1st ed.

[65] S. Sæther, M.F. Erichsen, S. Xiao, Z. Zhang, A. Lervik, J. He, Phonon thermal transport in copper: the effect of size, crystal orientation, and grain boundaries, AIP Adv. 12 (2022), https://doi.org/10.1063/5.0094170.

[66] Z. Tong, S. Li, X. Ruan, H. Bao, Comprehensive first-principles analysis of phonon thermal conductivity and electron-phonon coupling in different metals, Phys. Rev. B 100 (2019), https://doi.org/10.1103/PhysRevB.100.144306.

[67] B.C. Gundrum, D.G. Cahill, R.S. Averback, Thermal conductance of metal-metal interfaces, Phys. Rev. B Condens. Matter. Mater. Phys. 72 (2005), https://doi.org/10.1103/PhysRevB.72.245426.

[68] L. Chen, S. Chen, Y. Hou, Understanding the thermal conductivity of Diamond/ Copper composites by first-principles calculations, Carbon N. Y. 148 (2019) 249–257, https://doi.org/10.1016/j.carbon.2019.03.051.

[69] S.Q. Jia, L. Bolzoni, T. Li, F. Yang, Unveiling the interface characteristics and their influence on the heat transfer behavior of hot-forged Cu-Cr/Diamond composites, Carbon N. Y. 172 (2021) 390–401, https://doi.org/10.1016/j.carbon.2020.10.036.

[70] K. Chu, C. Jia, H. Guo, W. Li, On the thermal conductivity of Cu-Zr/diamond composites, Mater. Des. 45 (2013) 36–42, https://doi.org/10.1016/j.matdes.2012.09.006.

[71] I.Y. Forero-Sandoval, A.P. Franco-Bacca, F. Cervantes-Álvarez, C.L. Gómez-Heredia, J.A. Ramírez-Rincón, J. Ordonez-Miranda, J.J. Alvarado-Gil, Electrical and thermal percolation in two-phase materials: a perspective, J. Appl. Phys. 131 (2022) 230901, https://doi.org/10.1063/5.0091291.

[72] J.K. Carson, S.J. Lovatt, D.J. Tanner, A.C. Cleland, Thermal conductivity bounds for isotropic, porous materials, Int. J. Heat Mass Transf. 48 (2005) 2150–2158, https://doi.org/10.1016/j.ijheatmasstransfer.2004.12.032.