Determination of thermal conductivity of eutectic Al–Cu compounds utilizing experiments, molecular dynamics simulations and machine learning

In this study, the thermal conductivity ( κ ) of Al–Cu eutectics were investigated by experimental and computational methods to shed light on the role of these compounds in thermal properties of Al–Cu connections in compound casting. Specifically, the nonequilibrium molecular dynamics (MD) method was utilized to simulate the lattice thermal conductivity ( κl ) of six compositions of Al–Cu with 5–30 at.% Cu. To extend the results of the MD simulations to bulk materials, instead of using conventional linear extrapolation methods, a machine learning approach was developed for the dataset acquired from the MD simulations. The bootstrapping approach was utilized to find the most suitable method among the support vector machine (SVM) with polynomial and radial basis function (RBF) kernels and the random forest method. The results showed that the SVM model with RBF kernel performed the best, and thus was used to predict the bulk κl . Subsequently, the chosen compositions were produced by induction casting and their electrical conductivities were measured via eddy current method for calculating the electronic contribution of κ using the Wiedemann–Franz law. Finally, the actual κ of the alloys were measured using the xenon flash method and the results were compared with the computational values. It was shown that the MD method is capable of successfully simulating the thermal conductivity of this system. In addition, the experimental results demonstrated that the κ of Al–Cu eutectics decreases almost linearly with formation of the Al2Cu phase due to increase in the Cu content. Overall, the current findings can be used to enhance the κ of cooling devices made via Al–Cu compound casting.

In this study, the thermal conductivity (κ) of Al-Cu eutectics were investigated by experimental and computational methods to shed light on the role of these compounds in thermal properties of Al-Cu connections in compound casting. Specifically, the nonequilibrium molecular dynamics (MD) method was utilized to simulate the lattice thermal conductivity (κ l ) of six compositions of Al-Cu with 5-30 at.% Cu. To extend the results of the MD simulations to bulk materials, instead of using conventional linear extrapolation methods, a machine learning approach was developed for the dataset acquired from the MD simulations. The bootstrapping approach was utilized to find the most suitable method among the support vector machine (SVM) with polynomial and radial basis function (RBF) kernels and the random forest method. The results showed that the SVM model with RBF kernel performed the best, and thus was used to predict the bulk κ l . Subsequently, the chosen compositions were produced by induction casting and their electrical conductivities were measured via eddy current method for calculating the electronic contribution of κ using the Wiedemann-Franz law. Finally, the actual κ of the alloys were * Author to whom any correspondence should be addressed.
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Introduction
Cooling efficiency is of utmost importance for the design of electronic components with superior performance [1]. One of the essential parameters affecting the cooling performance is the thermal conductivity (κ), and copper is the material of choice for this purpose due to its high κ; however, the high density of copper (8.96 g cm −3 ) and material costs restrict its utility for monolithic components. As a result, cooling components are made by coupling copper with aluminum that has a lower density (2.7 g cm −3 ) at the expense of lowering the overall κ. Previous studies showed that poor connection at the Al and Cu component surfaces is one of the main issues that decrease the overall cooling capability of the components [2]. Currently, various thermal interface materials (TIMs) such as polymer-based composites containing copper or alumina are used between the components in the form of pads, gels or adhesives to reduce contact resistance [1,3,4]. However, such materials have substantially lower κ than the bulk Al and Cu [5], and thus significantly decrease the efficiency of the cooling component. A solution to this problem is to replace the TIMs with compound casting of Al on Cu components. With this approach, the molten Al is cast on the copper component, forming a metallic bond, eliminating the need for TIMs, and lowering the contact resistance. Although this process is deemed simple, the natural oxides on the surface of the components suppress wetting and work as a barrier, resulting in a poor connection, and thereby leading to undesirable mechanical properties and limited thermal conductivity [2,6].
In order to eliminate the adverse effect of natural oxides on parts manufactured by compound casting, a new approach has been proposed, where the casting process is performed under extremely high vacuum (XHV) adequate atmosphere using silane gas to remove the residual oxygen present in the environment [7][8][9]. Since the partial pressure of the oxygen is extremely low in such atmospheres, the formation of natural oxides on the cleaned surfaces of copper is inhibited, enhancing the metallic bond formation [7,8]. As expected, during solidification and formation of such bonds, diffusion of atoms would create a compositional gradient at the connection interface. Previous studies showed that under XHV atmosphere, the interface, where Al-Cu connection takes place, is mainly composed of the Al-rich eutectic range of the Al-Cu phase diagram and the θ-phase (Al 2 Cu), followed by thin layers of intermetallic compounds such as AlCu (η) and Al 4 Cu 9 (γ) [8]. Considering that the compositional gradient of these layers could significantly affect the overall κ of the cooling component, it is crucial to optimize the casting parameters so that the relative fraction of each phase can be controlled to obtain the best thermal conductivity. However, this goal is not achievable without a proper understanding of κ in this region.
The thermal conductivity of metals and alloys is made of two components κ = κ e + κ l , where κ e is the electronic and the κ l is the lattice contribution to κ. The component κ e is commonly calculated via the Wiedemann-Franz law (WFL) equation that converts the electrical conductivity (σ) to κ e : where T is the absolute temperature, and L is a constant known as the Lorenz number. The substantial contribution of electrons to κ is known for materials with high electrical conductivity, such as pure Cu, Al, and Au [10]. Therefore, the WFL is a common method to estimate the κ since the σ is readily available for many materials. Also, if σ is not available, its experimental measurement is simple and less complicated than the direct measurement of κ. Although some studies approved this estimation for some alloys and intermetallics, such as Sn-37 Pb, Cu 3 Sn and Cu 6 S 5 [11], several other studies showed that this equation does not hold for other alloy systems and intermetallic compounds. The shortcoming of the WFL method is a consequence of the change in the electrons' mean free path, which is affected by crystal defects and scattering of electrons. In addition, the higher contribution of κ l in such materials is discarded by the WFL, which can result in some deviation from the actual value of κ [12,13]. Therefore, it can be deduced that by complementing the κ e calculated using the WFL by κ l , the overall κ can be calculated with enhanced accuracy. However, the calculation of κ l demands a good understanding of the thermal conduction through the crystal lattice.
The κ l is generated from the transfer of vibrational energy of atoms by progressive waves, also known as phonons. These phonons propagate through materials and have a mean free path that is governed by interactions with the other phonons, free electrons, and defects [14]. Such scatterings, also known as Umklapp scatterings, are more pronounced in impure materials and it is necessary to consider them when calculating the κ l [15]. Some studies have tried the calculation of the κ l via analytical methods [16]; however, due to the complex behaviors of such scatterings, an accurate calculation of the κ l is only possible through computational methods that consider the vibration of atoms.
One of the methods to calculate the κ l is atomistic simulations, and for this purpose, one of the strongest candidates in terms of accuracy is density functional theory (DFT). However, the computational cost of employing DFT simulations in the solution of problems involving numerous compositions is significant. A more efficient solution for calculating κ l is molecular dynamics (MD), where κ l can be simulated via equilibrium or nonequilibrium approaches. In an equilibrium approach, heat flux is applied to the system, and the resulting thermal gradient until equilibration is used for calculating the κ l . As an example, the Green-Kobo method uses an equilibrium approach that has proved to be successful in predicting κ l of various materials [17,18]. However, in some materials, this method has the disadvantage of slow convergence, as it is necessary to simulate periods several times longer than the longest phonon relaxation time to achieve accurate results [19]. Another approach is to use nonequilibrium methods such as the Müller-Plathe method, also known as reverse nonequilibrium molecular dynamics (NEMD). As opposed to the Green-Kobo method, this method applies the thermal gradient in the system and measures the resulting heat flux to calculate the κ l . As a result, it provides several advantages, such as substantially lower computational cost, faster convergence, and better compatibility with periodic boundary conditions. It should be noted that the relative efficiency of the equilibrium and nonequilibrium methods are dependent on the properties of the material, and since the Al-Cu system studied in this work is isotropic, the NEMD method can be a useful and efficient tool to simulate κ l for this system [20,21].
With this motivation, the Müller-Plathe method was employed to determine the κ l of six Al-Cu compositions with 5,10,15,20,25, and 30 at.% Cu in the current study investigating the thermal conductivity of Al-Cu eutectic compounds. In order to adapt the computed values to bulk materials, a machine learning (ML) approach was utilized to predict the bulk κ l as several recent studies suggest the use of ML to predict the thermal transport properties of materials with enhanced accuracy [22][23][24]. Subsequently, to validate the results, samples with the aforementioned compositions were cast utilizing vacuum induction melting, heat treated for recrystallization and growth, and their electrical conductivity was measured utilizing an eddy current method, such that it could be incorporated into the WFL equation to calculate their κ e . Finally, their thermal conductivities were directly measured using the xenon flash method and were compared with the sum of κ e and κ l . The results suggested that the use of ML in conjunction with MD simulations for predicting κ l for bulk materials is a promising method that can eliminate the problems associated with linear extrapolation. Moreover, the addition of κ l to the values acquired from the WFL equation can enhance the accuracy of the calculated κ, providing a replacement for the direct measurement of κ. Finally, both the experimental and computational results suggest that decreasing the fraction of the θ phase in the eutectic compounds increases the overall κ of the Al-Cu connection, which can be exploited further for enhancing the overall thermal conductivity of cooling devices.

Numerical techniques
In order to simulate the κ l of the Al-Cu system, a 10 × 10 × 10 face centered (fcc) single crystal Al cube (40.4 Å width) was constructed via Atomsk [25]. Next, Al atoms were randomly substituted with Cu atoms based on the desired atomic percentage of 5, 10, 15, 20, 25 and 30. The interatomic potential used in this study was obtained by a second nearest-neighbor modified embedded atoms method (2NN MEAM) that already showed its ability to predict different phases of the Al-Cu system and their properties with acceptable accuracy [26]. In addition, this type of interatomic potential has shown better accuracy for κ l prediction [27]. To find out the preferred position of atoms, the systems were initially relaxed at 300 K for 20 ps via a Canonical (NVT) ensemble, then an isothermal-isobaric (NPT) ensemble for 20 ps. Subsequently, to imitate the effect of heat treatment at the temperature that the eutectic compounds form, a Monte Carlo algorithm was run for 200 ps. This algorithm switched the position of atoms to minimize the system's potential energy while the system evolved at 773 K (500 • C) under the NPT ensemble [28]. This method has already proven successful in predicting thermodynamically stable phases at desired temperatures for other materials [29]. Subsequently, the systems were cooled down to 300 K in 100 ps and were reproduced in the [001] direction to 2, 3, 4, 5, 6, 7, 8, 9, 10, 15, 20, and 25 times the original length of the system to incorporate the effect of system size on the thermal conductivity. The acquired systems were again relaxed via NVT and NPT ensembles before measuring the thermal conductivity. Notably, all MD simulations were performed utilizing the large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS, 23 June 2022 version) [30].
A reverse NEMD method proposed by Müller-Plathe and implemented in LAMMPS [20,31] was utilized to calculate the thermal conductivity of the systems. In this method, the systems were equilibrated in a microcanonical (NVE) ensemble for 100-250 ps and were divided into ten bins in the z direction. Every 50 fs, the kinetic energy of the five atoms with the lowest kinetic energy at the center bin was swapped with the five atoms possessing the highest kinetic energy in the bin at the end of the system. To increase the accuracy of the temperature sampling, overlapping bins half the size of previous bins were utilized. After the equilibration, the temperature gradient and the energy accumulation were measured for 100 ps, then converted to thermal conductivity.
Previous studies have shown a significant size effect on the predicted κ l , which is due to the sample length being smaller than the bulk mean free path of phonons. More specifically, if a phonon is emitted from the hot reservoir of the sample (here the central bin) and the length it travels is smaller than its mean free path, it would pass through without scattering, which is also referred to as ballistic transportation. However, as the sample size is increased, the chance of phonon scattering will increase, thus transforming the ballistic to diffusive transport. This transformation is essential as the phonons of ballistic transport contribute less to the κ l , which leads to an underestimation as compared to the bulk material (κ l, bulk ) [32,33]. The primary constraint in simulating the full mean free path of phonons is their large size, which demands extensive MD simulations. To solve this issue, Sellan et al [32] proposed an analytical solution establishing a relationship between the κ l and the sample length L: where c ph is the volumetric specific heat of each mode, K is the wave vector, ν is the dispersion branch, and υ g (K, ν) is the group velocity vector and τ is the relaxation time. It is possible to describe the above relationship via 1 κ l = χ × 1 L in which χ is the unknown function describing the relationship. Therefore, it can be argued that as 1 L → 0, L → ∞ and κ l → κ ∞ , which is the bulk value κ l, bulk . Thus, using a Taylor-series expansion, this relationship can be approximated: Based on this approach, a method has been proposed to predict κ l via truncating the terms after the first order, i.e. using a linear extrapolation to predict bulk thermal conductivity. A noticeable number of works have already used this method to simulate the κ l [19,[34][35][36][37][38]. However, several studies pointed out possible issues with this method. One problem is that, as this method truncates some of the parameters of the relationship, it loses accuracy if the relationship between the κ l and the sample length deviates from linearity [32,39]. In addition, if the κ l values collected for the linear extrapolation do not represent a sufficient length, it might misrepresent a non-linear behavior as linear. To solve this issue, ML methods were implemented in the current study as a replacement for linear extrapolation. Specifically, support vector machine with radial basis function (SVM-RBF) kernel, SVM with polynomial (SVM-POLY) kernel, and the random forest (RF) algorithm were trained utilizing the data acquired from the MD simulations. Subsequently, the accuracy of the models was evaluated via a test dataset 3 using the root mean squared error (RMSE). To increase the prediction power of the models, their hyperparameters were optimized utilizing the grid search method and bootstrapping.  Finally, the best model was chosen to predict κ bulk for each composition. All the ML methods were implemented in Python programming language using sci-kit learn libraries [40].

Experimental methods
In order to measure the thermal diffusivity of the eutectic Al-Cu samples, pellets of high-purity aluminum (Hydro Aluminium High Purity GmbH, Grevenbroich, Germany) with a purity of >99.87% and copper pieces of 99.65% purity (Hans-Erich Gemmel & Co. GmbH, Berlin, Germany) were weighted via a laboratory scale for each composition as shown in table 1. A vacuum induction melting device (MC50, Indutherm GmbH, Walzbachtal, Germany) was utilized to melt and mix the elements via inductive stirring for 5 min before casting them into a cylindrical cavity of a steel die. To increase the homogeneity of the ingots in terms of chemical composition and grain size of the samples, the ingots were homogenized inside a furnace at 500 • C for 48 h in air atmosphere. To incorporate the effect of gravity on the composition of the ingots, four cylindrical samples were sectioned from the top, middle, and bottom parts of the ingots as shown in figure 1. Subsequently, these samples were machined down to 10 mm diameter and 3 mm thickness. Their average densities were measured at room temperature (table 1) using the Archimedes principle via a density measurement scale (2200, MK Industrievertretungen, Germany).
To measure the thermal diffusivity of each composition, the sample surfaces were ground employing a 200-grit emery paper to remove the oxide layers, cleaned via ethanol, and spraycoated with a thin layer of graphite. This black graphite layer was applied to increase the energy absorption by the samples ensuring a homogeneous distribution of the energy in the samples. Subsequently, the xenon flash pulse method was used (Netzsch, LFA 447 NanoFlash), where an instantaneous pulse of energy is radiated to the sample, causing a temperature gradient detected via an infrared camera to measure the thermal diffusivity [41,42]. The thermal diffusivity value can be converted to thermal conductivity utilizing equation (4): where α is the thermal diffusivity, C p is the specific heat and ρ is the density. For each sample, five shots of the xenon flash were used to calibrate the device at room temperature, and then three shots were applied at higher temperatures of 50 • C, 100 • C, and 150 • C. The average diffusivities of these measurements are reported for each composition. For measuring the electrical conductivity of the samples, their surfaces were cleaned with ethanol and ten measurements were carried out on each sample by a portable eddy current device (SIGMATEST 2.069, Institut Dr Foerster GmbH & Co, Germany) at room temperature.
To investigate the chemical composition and surface morphology of the samples, following metallographic preparation of the samples, scanning electron microscopy (SEM, Zeiss Supra 55 VP) and energy dispersive x-ray (EDX) analysis (Bruker AXS GmbH, Karlsruhe, Germany) were carried out. For detecting grain boundaries and evaluating the grain size, electron backscatter diffraction (EBSD, Zeiss Auriga with Oxford detector) was utilized. Figure 2(a) presents results of the NEMD simulations of the κ l for eutectic compositions of the Al-Cu compounds with various sample lengths. Accordingly, the κ l of the compounds decreases notably with increasing Cu content, with a more pronounced change from 5 to 15 at.%. However, for the samples with a Cu content higher than 15 at.%, the decrease in κ l is not substantial. When the change of the simulated 1/κ l with 1/L is compared for the six Al-Cu compounds (figure 2(b)), two regions with different slopes are prominent: one for the shorter and one for the longer samples. Implementing the conventional linear extrapolation method to predict the κ l, bulk would result in an inaccurate prediction due to high fitting error, and thus a different approach should be taken. However before performing any fitting or prediction, it was necessary to ensure that the change in the slope was not an artifact of the simulations.

Results and discussion
There are three possible reasons for the observed changes induced by varying the Cu content of the compounds (figure 2), and the first one might be associated with the divergent term of the κ l as proposed by Hu et al [39]. In particular, for one-dimensional samples with small sizes, the divergent term of the κ l is negligible. However, as the length of the samples increase, κ l approaches infinite thermal conductivity, such that the intercept of the extrapolation on the 1/κ l would be negative. In contrast to one-dimensional samples, for nanowires, the long wavelength phonons result in higher scattering that prevents divergence in infinite length limit. Therefore, the values of κ l converge to finite values [39], which is the case for all the present compositions. Thus, the change in the slope cannot be due to the divergent term of the κ l .
The second possible reason could be the dependence of κ l on temperature along the sample length, which was proposed by Talaat et al [38]. Specifically, in NEMD methods, materials with κ l highly dependent on temperature, would possess different κ l values along the length of a sample. This difference would increase as the length of the sample increases, resulting in a non-linear temperature gradient along the sample length. This non-linearity would cause longer samples to have higher κ l as a simulation artifact [38]. However, in the present samples, the magnitude of the temperature gradient was linear even for the longest Al30Cu samples ( figure 3), which corresponds to the composition with the highest change in the slope of the  2(b)). Thus, the effect of temperature dependence of κ l can be ruled out as a potential reason for the observed changes in the current study.
The third possible reason is the incorporation of the contribution of phonons with a long mean free path to the κ l value of the samples, which has been previously suggested as a reason for the increase in the κ l values of the longer samples [19,32,37,39]. As a result, one can deduce that the difference in the slope was not an artifact of simulation and the values of κ l calculated via the NEMD method can be used for accurate prediction of the bulk κ l in the Al-Cu eutectic system.
To extrapolate the simulated κ l and predict the κ l, bulk , three different ML methods, namely the SVM-RBF, SVM-POLY, and RF, were implemented on the dataset constructed with the MD simulation results. This dataset included the Cu at.% and 1/L as the input features, and the 1/κ l as the output. In the first step, a grid search method was used to find the optimal hyperparameters for each of these methods 4 . In order to compare the performances of the models, a bootstrapping approach was used. This approach was utilized to establish the effect of the  training dataset on the performance of the models. Thus, the accuracy of the models can be assessed over different portions of the dataset, minimizing the bias caused by data sampling [43]. The three models were bootstrapped 100 times, and their RMSEs 5 were determined utilizing different test datasets. As a result, the SVM-RBF exhibited the lowest average RMSE of 0.056 with the least standard deviation of 0.018 (figure 4), and therefore, it was chosen for the prediction of κ l .
Subsequently, the data were randomly separated into training and test datasets with a 20%-80% ratio to assess the ability of the SVM-RBF model to predict the unseen data. After training the model with the training dataset, both training and test datasets were predicted (cf. figure 5), and the SVM-RBF model could predict the unseen data of the test dataset with high accuracy, such that the RMSE was only 0.0121. This, indeed, demonstrates the ability of the SVM-RBF model to extrapolate and interpolate the values of 1/κ l . Therefore, the SVM-RBF model was run on a synthetic dataset that incorporated 1/L values of 0 (for bulk samples) to 0.01 (10 Å), where 1/κ l was predicted for different compositions. The results presented in figure 6 demonstrate that the model could capture the behavior of the 1/κ l for both interpolation and extrapolation, which permits the use of the model to predict the κ l, bulk from the intercept of  [10]. However, the present results also indicate the effect of Cu content on the κ l , which follows the expected behavior of substitutional elements on the thermal conductivity [44,45].
The actual chemical composition of the samples taken from the middle part of each ingot was measured utilizing EDX, and the measured values that are presented in table 3 are close to the intended compositions. Subsequently, the microstructures of the samples were investigated further utilizing backscattered electron microscopy ( figure 7), revealing the change in the microstructure of samples with the increase in the Cu content. To be more specific, figure 7 shows that the point EDX measurements from the dark regions correspond to a composition of 98.5 at.% Al-1.5 at.% Cu, which is attributed to α-aluminum, while the bright areas represent a chemical composition of about 66.5 at.% Al-33.5 at.% Cu, which is attributed to the θ phase (Al 2 Cu). Therefore, the Al30Cu sample contains mostly the θ phase, in addition to a small volume of α-Al, while the other compositions exhibit a rather mixed microstructure containing both α-Al and θ phases of different ratios. As a result of the difference in the amount of θ  phase, it is expected that the Al30Cu sample would have low thermal and electrical conductivity due to the fact that the θ phase has a higher amount of substitutional defects, promoting different scattering of phonons and electrons. In addition to the chemical composition, grain boundaries also have a substantial effect on the thermal conductivity of materials [46]. Therefore, before cutting the cast ingots, they were heat treated in the furnace for 48 h at 500 • C for recrystallization and grain growth. To ensure  the success of minimizing the grain boundaries, EBSD analysis was conducted on the samples prepared for SEM. Analyzing the EBSD images with the ImageJ software, it was deduced that each grain had an average area of 1 mm 2 . Exemplary EBSD images of the surfaces of Al25Cu and Al30Cu are presented in figures 8 (a) and (b) 6 . The apparent grain sizes of the samples are larger than the ASTM G number of 0 [47], and thus, the grain boundaries are few enough to have a minimal effect on thermal conductivity. The results of thermal diffusivity (α) measurements conducted with the xenon flash method at 25 • C and 100 • C are summarized in figure 9. The error bars shown in the figure include both the sampling position effect and α for several xenon shots, implying that varying the sample position causes statistically insignificant differences in thermal diffusivity.
A sharp decrease in α is noted when the Cu content increases from 0 at.% (i.e. pure Al) to 5 at.% (Al-5Cu), which is attributed to the introduction of both phonon and electron scattering   of Cu atoms as substitutional defects [48,49]. For the remaining samples, α decreases almost linearly concomitant with increasing the Cu content. In addition, there is a minor temperature effect on α (roughly a 1% increase from 25 • C to 100 • C); however, such an increase can be ignored within the temperature range intended for cooling devices: although not shown herein for the sake of brevity, similar measurements at 50 • C and 150 • C support this argument. Using the experimentally determined thermal diffusivity values (calibrated by pure aluminum thermal diffusivity), the ρ from table 1, and the extrapolation of specific heat data acquired from the literature [50,51], the experimental values of κ were calculated, and are presented in figure 11.
In order to evaluate the accuracy of the WFL equation for Al-Cu eutectics, it is necessary to measure the electrical conductivity of samples using a method that is not affected by contact resistance. Therefore, the σ of the samples were measured by the eddy current method and the results are presented in figure 10. Accordingly, the Cu addition significantly decreased the σ of the samples, reducing the σ to roughly half in the inspected range. Therefore, it is concluded that the increase in the θ phase, which results from increased Cu content as shown in figure 7, significantly decreases the σ. In addition, unlike the effect of increased Cu content on α, the values of σ do not decrease linearly, exhibiting a higher rate of decrease as the Cu content increases. Thus, the formation of the θ phase has a more pronounced effect on the σ as compared to α. This can be interpreted as κ e being more affected by the θ phase formation than the overall κ. To calculate the κ e , the σ values shown in figure 10 and the Lorenz value from the literature [52] were used in the WFL equation and the results are presented in figure 11.
A comparison of the κ l calculated with MD simulations, the κ e calculated using the WFL and the κ acquired via the xenon flash method (figure 11) revealed that the addition of κ l to the κ e enhances the accuracy of the WFL method with errors less than 10% for all compositions. If a measurement of the thermal conductivity via experimental methods is not possible, calculation of κ l via MD simulations and calculation of κ e via WFL equation can be a useful replacement. In addition, although the contribution of κ e is the dominating factor of κ in Al-Cu system, in other systems with lower electronic and higher lattice contributions, κ l calculated via MD simulations is indispensable for accurate prediction. The acquired results are also helpful in optimizing the parameters of the compound casting to enhance the overall κ of Figure 11. Comparison of the results acquired using the xenon flash method (κexp) and those obtained by the MD simulations (κ l ) and upon conversion of electrical conductivity utilizing the WFL (κe). a cooling device. For instance, a previous study by Fromm et al showed that the connection made by compound casting under an XHV atmosphere mainly consists of a eutectic composition with about 17 at.% Cu and θ phase, followed by a small zone of Cu-rich intermetallics [8]. Based on both the experimental and the simulated results of this study, for an enhanced κ of an Al-Cu connection, increasing the portion of the Al-rich eutectic region would be beneficial. This is due to the fact that regions with compositions close to the θ phase (such as the Al30Cu sample) have roughly 50% less κ than the Al-rich ones. However, since in compound casting of Al-Cu, the connection relies on diffusion of elements, a short diffusion duration, resulting in a lower amount of Cu content, would cause a connection with insufficient mechanical properties. Considering the outcome of this study that suggests the minimization of the θ phase, and the fact that the elimination of the θ phase will deteriorate the mechanical properties of the Al-Cu bonds, superior Al-Cu connections could be achieved through the optimization of compound casting parameters to control the diffusion of Cu during the casting process.

Conclusions
In this study, the thermal conductivity (κ) of Al-Cu eutectic compositions was determined utilizing a combined computational and experimental approach with the aim of enhancing the overall κ of parts made with compound casting. In particular, using reverse NEMD, the lattice thermal conductivity (κ l ) of six Al-Cu eutectic compounds with 5-30 at.% Cu was simulated for various sample lengths. Subsequently, three ML models (SVM-RBF kernel, SVM-POLY kernel, and RF), were developed to predict the bulk κ l . The performances of these models were compared using the bootstrap method, and the SVM-RBF kernel was found to have the best predicting capabilities. Employing this model, the bulk κ l was predicted for the samples, with results showing that the addition of Cu has a significant effect on the κ l of the Al-Cu compounds. In addition, these results remained in good agreement with those of the previous studies. To validate the computational approach and enhance the knowledge of the κ in Al-Cu eutectics, the samples were produced by the induction melting method and heat-treated for 48 h at 500 • C for recrystallization and growth. Three samples were machined from the heattreated ingots' top, middle, and bottom parts to measure the thermal diffusivity for increased accuracy and carry out microstructural characterization. The experimental results showed a similar decrease in the κ due to an increased amount of Cu. In addition, it was demonstrated that utilizing the MD simulations to calculate κ l can enhance the estimations made by WFL, presenting a new approach in cases where the measurement of κ via experimental methods is not possible. Moreover, the present results suggest the importance of elemental diffusion on κ, as the compounds with higher Cu content and θ phase can have less than 50% κ compared to the Al-rich eutectics. Overall, in addition to providing a novel approach to determine κ, the results of this study also warrant further investigation of the compound casting parameters influencing the diffusion in the Al-Cu connections.

Data availability statement
The data cannot be made publicly available upon publication because they are not available in a format that is sufficiently accessible or reusable by other researchers. The data that support the findings of this study are available upon reasonable request from the authors.