# Numerical and Experimental Evaluation of Thermal Conductivity: An Application to Al-Sn Alloys

^{*}

## Abstract

**:**

## 1. Introduction

_{eff}), can become a little tricky. Equilibrium properties of composites, like density and heat capacity, can be determined as a weighted average of constituents’ properties with respect to volume or mass. On the other hand, morphology must be taken into account as well for transport properties like thermal conductivity [3]. In more detail, this means that distribution, shape, and orientation of phases must be considered in addition to thermal conductivity of each phase and its volume fraction.

_{eff}) of a two-dimension or three-dimension multiphase material, considering the random walk of energy particles in a lattice model describing its phase volume fraction, distribution, and size. The starting point for the application of this method can be a micrograph (2D) or a tomography (3D) [4,11]. Each node of the lattice derived from images, arranged in a simple cubic structure, has the properties of the local material. In this way, any geometry of interest can be represented such as starting from a micrograph [12]. In LMC methods applied to the estimation of a multiphase material, a fundamental role is played by the jump probability [9]. The jump probability of a particle to one of the surrounding nodes is a function of thermal conductivity of the phases. The effective thermal conductivity is related to the probability that N jump attempts are successful or unsuccessful in a time interval t. This probability depends on thermal conductivities of original and destination nodes and it is described by the particle displacement vector

**R**. This relation is described by Equation (1).

_{eff}= <

**R**

^{2}> * λ

_{max}/N

_{max}is the maximum thermal conductivity of all phases, <

**R**

^{2}> is the mean square displacement of all energy particles, and N is the number of jump attempts. Therefore, λ

_{eff}is the average effective thermal conductivity. In addition, λ

_{eff}can be estimated along directional components of

**R**. Further details about the equations used in the LMC method are presented in a dedicated paper by Li and Gariboldi [13].

## 2. Materials and Methods

#### 2.1. Numerical Method: Lattice Monte-Carlo

_{AB}) is calculated using Equation (2), according to Reference [13].

_{Al}and λ

_{Sn}are the thermal conductivities of Al and Sn, respectively. The number of particles Num is usually chosen to be equal to the total number of lattice nodes. The number of jump attempts N is suggested be equal to 10

^{5}when L < 100 and to 10

^{6}for larger L [13]. The simulation is repeated for a set number of cycles, at least 3, to have statistically representative results. A study on the effect of the different parameters was presented in Reference [13].

#### 2.2. Validation for 2-Phase Materials

^{3}for Sn and Al, respectively. Thermal conductivity was measured using a radial heat flow apparatus as a function of temperature [20].

_{eff}was repeated through a direct simulation (DS) with finite elements on the same micrographs at the three reference temperatures. In this steady analysis, λ

_{eff}can be derived using Equation (3).

#### 2.3. Validation for Three-Phase Material

_{p}) and density (ρ). Specific heat capacity for Al-10vol%Sn was determined with thermodynamic calculations as the derivative of enthalpy with respect to temperature at a constant pressure. These calculations were done using Thermo-Calc Software (Version 2020b with TCAL5.1 Al-Alloys Database, Thermo-Calc Software, Stockholm, Sweden) [24]. Density was either calculated with Thermo-Calc Software and measured experimentally using Archimedes’ method (Analytical Balance ME204 with Density Kit Standard and Advanced, Mettler Toledo, Greifensee, Switzerland). The experimental error for thermal conductivity (ε

_{λ}) is calculated from the relative errors on density (ρ) and thermal diffusivity (α) (Equation (4), [25]).

_{ρ}and δ

_{α}are the standard deviations of density and thermal diffusivity obtained experimentally, while ${\stackrel{\u203e}{\lambda}}_{exp}$, $\stackrel{\u203e}{\rho},$ and $\stackrel{\u203e}{\alpha}$ are the average values of experimental thermal conductivity, density, and thermal diffusivity, respectively.

## 3. Results

#### 3.1. Application of Two-Phases LMC Method

_{eff}plotted vs. temperature in Figure 2b show a decreasing trend with temperature. At low temperature, the agreement between experimental and calculated data for the same nominal composition are good. The data calculated at a higher temperature are not able to describe the downward curvature displayed by the thermal conductivity of Al-Sn. The downward curvature of high-temperature data from Reference [20] is not displayed by pure Al and Sn.

#### 3.2. Application of Three-Phase LMC Method

#### 3.2.1. Experimental Characterization of Al-10Sn

^{3}. Experimental density, measured with Archimedes’ method, is 2.90 g/cm

^{3}, corresponding to 91.84% of the theoretical value. This result is consistent with porosity measured with metallographic analysis. The average thermal diffusivity measured with LFA is 0.357 cm

^{2}/s with a standard deviation of 0.002 cm

^{2}/s, i.e., 0.4%. The resulting thermal conductivity is 76.863 ± 0.352 W/(m·K).

#### 3.2.2. LMC Calculation

_{eff}obtained with two different numbers of particles (Num) are compared plotting the series of values with Num = 22,201 on the x-axis and the one with Num = 10,000 on the y-axis in Figure 5. Datapoints are on the bisector of the first quadrant (y = x), which means that the two different parameters for the number of particles result in a close value. The difference is in the standard deviation for the same area after a simulation repetition, which is higher for Num = 10,000 (average value of 0.96 W/(K·m)) than for Num = 22,201 (average value of 0.52 W/(K·m)). In both cases, the standard deviation is quite small, at about 1% or lower. Due to its low value, standard deviation is not reported in any plot, since it would not be possible to appreciate it.

_{eff}, 125 W/(K·m), is observed for the lowest porosity content (2.49% volume) and for 40% Sn. It is difficult to identify a clear trend to correlate λ

_{eff}values with Sn or porosity values.

_{eff}calculated over all the areas is 95.65 ± 14.52 W/(K·m) for Num = 22,201 and 95.47 ± 14.82 W/(K·m) for Num = 10,000.

_{eff}along horizontal and vertical directions of the selected regions of micrographs, values can be higher in the direction or, in the other direction, suggesting isotropic thermal conductivity, since the ratio between the difference of thermal conductivity in horizontal and vertical directions and the vertical one is less than 1%. Furthermore, since the orthotropic material microstructure was observed for produced samples, thermal conductivity in a perpendicular metallographic section would lead to values corresponding to those in a horizontal (x) direction.

## 4. Discussion

_{eff}, it reduces as Sn volume content increases as well as with temperature. The predicted thermal conductivities have a good agreement with experimental data at relatively low temperatures. At 460 K, the equilibrium Al-Sn phase diagram [29] suggests that the mutual solubility of the elements increases. It is known that the formation of solid solutions reduces the thermal conductivity with respect to those of metals. This could be the case for Al-Sn alloys, for which the experimental data presented by Reference [20] rapidly decrease above a certain temperature. In these conditions, the use of the thermal conductivity of Al and Sn cannot be considered anymore, since the two phases identified as black and white are not pure Al and pure Sn. The available data did not allow us to verify this explanation. The results show also that the material is likely to have an isotropic response in planes perpendicularly to the solidification axis of the sample from which micrographs have been taken by Reference [20].

_{eff}, with a predominant effect of porosity, as it can be easily expected. Nevertheless, even a slight variation of volume content of phases can give very different results. For example, this can be observed for about 12% of porosity and 20% of Sn in volume (Figure 7). Despite a very similar composition, these areas have a different phase distribution, which can explain differences up to 18% in the λ

_{eff}value. Further studies are required to obtain a deep understanding on the role of the three phase distributions.

_{eff}for Al-10vol%Sn is 76.863 ± 0.352 W/(m·K), while the average result of the LMC simulation is 95.65 ± 14.52 W/(K·m) (Num = 22,201). The standard deviation of the LMC simulations is very large. Thus, the experimental and simulation values can be considered to be similar. Moreover, since the sample was sintered at 500 °C, it is reasonable to expect the presence of mutual solute atoms, which reduce the actual thermal conductivity of the material with respect to the one calculated considering the thermal conductivity of pure Al and Sn, as mentioned above. The presence of impurities consisting of other elements could affect thermal conductivity of the phases as well. However, according to nominal composition of raw materials, impurity content should be very low and chemical analysis did not show significant quantities.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Burger, N.; Laachachi, A.; Ferriol, M.; Lutz, M.; Toniazzo, V.; Ruch, D. Review of thermal conductivity in composites: Mechanisms, parameters and theory. Prog. Polym. Sci.
**2016**, 61, 1–28. [Google Scholar] [CrossRef] - Meydaneri, F.; Saatçi, B.; Özdemir, M. Thermal conductivities of solid and liquid phases for pure Al, pure Sn and their binary alloys. Fluid Phase Equilib.
**2010**, 298, 97–105. [Google Scholar] [CrossRef] - Rawson, A.J.; Kisi, E.; Wensrich, C. Microstructural efficiency: Structured morphologies. Int. J. Heat Mass Transf.
**2015**, 81, 820–828. [Google Scholar] [CrossRef] - Rawson, A.; Kisi, E.; Sugo, H.; Fiedler, T. Effective conductivity of Cu-Fe and Sn-Al miscibility gap alloys. Int. J. Heat Mass Transf.
**2014**, 77, 395–405. [Google Scholar] [CrossRef] [Green Version] - Progelhof, R.C.; Throne, J.L.; Ruetsch, R.R. Methods for predicting the thermal conductivity of composite systems: A review. Polym. Eng. Sci.
**1976**, 16, 615–625. [Google Scholar] [CrossRef] - Veyhl, C.; Fiedler, T.; Andersen, O.; Meinert, J.; Bernthaler, T.; Belova, I.V.; Murch, G.E. On the thermal conductivity of sintered metallic fibre structures. Int. J. Heat Mass Transf.
**2012**, 55, 2440–2448. [Google Scholar] [CrossRef] - Fiedler, T.; Löffler, R.; Bernthaler, T.; Winkler, R.; Belova, I.V.; Murch, G.E.; Öchsner, A. Numerical analyses of the thermal conductivity of random hollow sphere structures. Mater. Lett.
**2009**, 63, 1125–1127. [Google Scholar] [CrossRef] - Karkri, M.; Ibos, L.; Garnier, B. Comparison of experimental and simulated effective thermal conductivity of polymer matrix filled with metallic spheres: Thermal contact resistance and particle size effect. J. Compos. Mater.
**2015**, 49, 3017–3030. [Google Scholar] [CrossRef] - Fiedler, T.; Belova, I.V.; Rawson, A.; Murch, G.E. Optimized Lattice Monte Carlo for thermal analysis of composites. Comput. Mater. Sci.
**2014**, 95, 207–212. [Google Scholar] [CrossRef] - Ye, H.; Ni, Q.; Ma, M. A Lattice Monte Carlo analysis of the effective thermal conductivity of closed-cell aluminum foams and an experimental verification. Int. J. Heat Mass Transf.
**2015**, 86, 853–860. [Google Scholar] [CrossRef] - Fiedler, T.; Rawson, A.J.; Sugo, H.; Kisi, E. Thermal capacitors made from Miscibility Gap Alloys (MGAs). In WIT Transactions on Ecology and the Environment, Proceedings of the 5th International Conference on Energy and Sustainability, Putrajaya, Malaysia, 16–18 December 2014; WIT Press: Southampton, UK, 2014; Volume 186, pp. 479–486. [Google Scholar] [CrossRef]
- Belova, I.V.; Murch, G.E. Bridging Different Length and Time Scales in Diffusion Problems Using a Lattice Monte Carlo Method. Solid State Phenom.
**2007**, 129, 1–10. [Google Scholar] [CrossRef] - Li, Z.; Gariboldi, E. Reliable estimation of effective thermal properties of a 2-phase material by its optimized modelling in view of Lattice Monte-Carlo simulation. Comput. Mater. Sci.
**2019**, 169, 109–125. [Google Scholar] [CrossRef] - Gariboldi, E.; Colombo, L.P.M.; Fagiani, D.; Li, Z. Methods to Characterize Effective Thermal Conductivity, Diffusivity and Thermal Response in Different Classes of Composite Phase Change Materials. Materials
**2019**, 12, 2552. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Shinzato, K.; Baba, T. A Laser Flash Apparatus for Thermal Diffusivity and Specific Heat Capacity Measurements. J. Therm. Anal. Calorim.
**2001**, 64, 413–422. [Google Scholar] [CrossRef] - ASTM International. E1461-13 Standard Test Method Thermal Diffusivity by the Flash Method; ASTM International: West Conshohocken, PA, USA, 2013. [Google Scholar] [CrossRef]
- Liu, X.; Zeng, M.Q.; Ma, Y.; Zhu, M. Promoting the high load-carrying capability of Al–20wt%Sn bearing alloys through creating nanocomposite structure by mechanical alloying. Wear
**2012**, 294–295, 387–394. [Google Scholar] [CrossRef] - Sugo, H.; Cuskelly, D.; Rawson, A.; Erich, K. High conductivity PCM for thermal energy storage—Miscibility Gap Alloys. In Proceedings of the Solar2014: The 52nd Annual Conference, Australian Solar Energy Society (Australian Solar Council), Melbourne, Australia, 9 May 2014; pp. 201–210. [Google Scholar]
- Confalonieri, C.; Bassani, P.; Gariboldi, E. Microstructural and thermal response evolution of metallic form-stable phase change materials produced from ball-milled powders. J. Therm. Anal. Calorim.
**2020**, 142, 85–96. [Google Scholar] [CrossRef] - Meydaneri Tezel, F.; Saatçi, B.; Arı, M.; Durmuş Acer, S.; Altuner, E. Structural and thermo-electrical properties of Sn–Al alloys. Appl. Phys. A
**2016**, 122, 906. [Google Scholar] [CrossRef] - Rasband, W.S. ImageJ, Version 2.0.0-rc-69/1.52i, Distribution Fiji. 2018. Available online: https://imagej.net/Welcome (accessed on 29 March 2021).
- Li, Z.; Gariboldi, E. Review on the temperature-dependent thermophysical properties of liquid paraffins and composite phase change materials with metallic porous structures. Mater. Today Energy
**2021**, 20, 100642. [Google Scholar] [CrossRef] - Gariboldi, E.; Perrin, M. Metallic Composites as Form-Stable Phase Change Alloys. Mater. Sci. Forum
**2018**, 941, 1966–1971. [Google Scholar] [CrossRef] - Thermo-Calc Software. Version 2020b with TCAL5.1 Al-Alloys Database; Thermo-Calc Software: Stockholm, Sweden, 2020. [Google Scholar]
- Taylor, J.R. Introduzione All’Analisi Degli Errori, 1st ed.; Zanichelli: Bologna, Italy, 1986. [Google Scholar]
- Yamasue, E.; Susa, M.; Fukuyama, H.; Nagata, K. Deviation from Wiedemann–Franz Law for the Thermal Conductivity of Liquid Tin and Lead at Elevated Temperature. Int. J. Thermophys.
**2003**, 24, 713–730. [Google Scholar] [CrossRef] - Bakhtiyarov, S.I.; Overfelt, R.A.; Teodorescu, S.G. Electrical and thermal conductivity of A319 and A356 aluminum alloys. J. Mater. Sci.
**2001**, 36, 4643–4648. [Google Scholar] [CrossRef] - Baehr, H.D.; Stephan, K. Heat and Mass Transfer, 2nd ed.; Springer: Berlin/Heidelberg, Germany, 2006; ISBN 978-3-540-29526-6. [Google Scholar]
- Pilote, L.; Gheribi, A.E.; Chartrand, P. Study of the solubility of Pb, Bi and Sn in aluminum by mixed CALPHAD/DFT methods: Applicability to aluminum machining alloys. Calphad
**2018**, 61, 275–287. [Google Scholar] [CrossRef]

**Figure 1.**Area of Al-10vol%Sn microstructure prepared for the Lattice Monte-Carlo (LMC) method: (

**a**) Original cropped image and (

**b**) three-zone image after application of a multi-threshold. The side of the micrograph is 80.54 µm in length.

**Figure 2.**(

**a**) Effective thermal conductivity of Al-Sn alloys vs. Sn volume fraction experimentally measured by Meydaneri Tezel et al. [20] at 340 K and estimated by the Lattice Monte-Carlo (LMC) method and direct simulation (DS) method at the same temperature. In addition to average values of effective thermal conductivity, those obtained along the horizontal (x) and vertical, (y) direction in their micrographs are given. (

**b**) Experimental and calculated effective thermal conductivity as a function of temperature data for pure Al, pure tin, and for different Al-Sn alloys.

**Figure 3.**SEM micrograph of Al-10vol%Sn sample, sectioned on a plane parallel to the compression direction.

**Figure 4.**SEM micrograph for LMC simulations of Al-10vol%Sn sample, sectioned on a plane parallel to the compression direction.

**Figure 5.**Comparison between results obtained with a different number of particles: 22,201 on the x-axis and 10,000 on the y-axis. The line is the bisector of the first quadrant of the plot (y = x).

**Figure 6.**Average effective thermal conductivity (values next to datapoints). The color should help to order values from a minimum to a maximum. (

**a**) Results with the number of particles (Num) equal to 22,201. (

**b**) Results with the number of particles (Num) equal to 10,000.

**Table 1.**Thermal conductivity values for Al and Sn at different temperatures inferred from the plot of experimental data provided in Reference [20].

Phase | Thermal Conductivity [W/(m·K)] | ||
---|---|---|---|

at 340 K | at 400 K | at 460 K | |

Al | 306.56 | 296.66 | 286.76 |

Sn | 69.18 | 65.28 | 61.38 |

**Table 2.**Calculated (C

_{p}) or experimental (density and thermal diffusivity) values to determine thermal conductivity.

Value | Density [g/cm ^{3}] | C_{p}[J/(g·K)] | Thermal Diffusivity [cm ^{2}/s] | Thermal Conductivity [W/(m·K)] |
---|---|---|---|---|

Average | 2.901 | 0.743 | 0.357 | 76.863 |

Error (%) | 0.164 ^{1} | - | 0.428 ^{1} | 0.458 ^{2} |

^{1}standard deviation%

^{2}experimental error (Equation (1)).

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Li, Z.; Confalonieri, C.; Gariboldi, E.
Numerical and Experimental Evaluation of Thermal Conductivity: An Application to Al-Sn Alloys. *Metals* **2021**, *11*, 650.
https://doi.org/10.3390/met11040650

**AMA Style**

Li Z, Confalonieri C, Gariboldi E.
Numerical and Experimental Evaluation of Thermal Conductivity: An Application to Al-Sn Alloys. *Metals*. 2021; 11(4):650.
https://doi.org/10.3390/met11040650

**Chicago/Turabian Style**

Li, Ziwei, Chiara Confalonieri, and Elisabetta Gariboldi.
2021. "Numerical and Experimental Evaluation of Thermal Conductivity: An Application to Al-Sn Alloys" *Metals* 11, no. 4: 650.
https://doi.org/10.3390/met11040650