
==== Front
iScience
iScience
iScience
2589-0042
Elsevier

S2589-0042(24)02040-6
10.1016/j.isci.2024.110815
110815
Article
Heat flux concentrator based on nanophononic metamaterials
Zhang Jian 12
Zhang Haochun hczhang@hit.edu.cn
14∗
Zhang Gang zhanggang@bitjx.edu.cn
3∗∗
1 School of Energy Science and Engineering, Harbin Institute of Technology, Harbin 150001, China
2 Institute of High Performance Computing, Agency for Science, Technology and Research (A∗STAR), Singapore 138632, Singapore
3 Yangtze Delta Region Academy of Beijing Institute of Technology (Jiaxing), Jiaxing 314019, China
∗ Corresponding author hczhang@hit.edu.cn
∗∗ Corresponding author zhanggang@bitjx.edu.cn
4 Lead contact

26 8 2024
20 9 2024
26 8 2024
27 9 1108156 3 2024
4 6 2024
22 8 2024
© 2024 The Author(s)
2024
https://creativecommons.org/licenses/by-nc-nd/4.0/ This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Summary

Heat flux concentrators have important potential applications in thermoelectric generators. In this study, we demonstrated the heat flux concentration characteristic in nanophononic metamaterials using molecular dynamics simulations. The ratio of heat flux (RHF) is used to evaluate the concentration performance, the RHF can reach 1.62. The performance is optimized by varying the height of the nanopillar and the atomic mass of the atoms in the nanopillar. Increasing the atomic mass of the atoms in the nanopillars yields better performance. We found that the main mechanism for concentration is phonon localization in the nanopillar region. Furthermore, when the distance from the surface increases, the low-frequency peak of phonon density of states (PDOS) decreases, the high-frequency peak increases, and the mode participation rate (MPR) transforms from localized to delocalized. This work provides a new design of heat flux concentrator based on nanophononic metamaterials for regulating thermal conduction.

Graphical abstract

Highlights

• Nanophononic metamaterials can produce heat flux concentration

• The mechanism for concentration is phonon localization in the nanopillar region

• The localized resonance effects occur in the surface layer of the base film

Heat transfer; Thermal engineering; Metamaterials; Modeling in materials science

Subject areas

Heat transfer
Thermal engineering
Metamaterials
Modeling in materials science
Published: August 26, 2024
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pmcIntroduction

In recent years, regulating heat flux has attracted extensive research attention.1,2,3,4,5,6 At the nanoscale, with the emergence of nanophononic metamaterials,7 numerous studies have shown that the strong resonance hybridization between the nanopillars and the host film, combined with Bragg diffraction,8,9,10,11,12,13,14 will reduce the thermal conductivity of the host film. In addition, researchers have conducted extensive studies on the design of various phononic metamaterials15,16,17,18,19 and explained the underlying mechanism.20,21,22,23,24 The increase in local heat flux is also highly significant. For example, thermoelectric generators, which are applied in areas with concentrated heat flow, convert the low-grade waste heat generated into electricity.25 Researchers have designed various thermal concentrators based on the transformation thermotics theory, with a primary focus on temperature concentration.26,27,28,29,30,31 For example, Huang et al.26 designed a single-layer concentrator with isotropic thermal conductivity and the temperature concentration efficiency is about 12. Li et al.28 first presented the spatial distribution of thermal conductivity in arbitrarily shaped thermal concentrators. Subsequently, they constructed cylindrical thermal concentrators using alternating composite conventional materials. Recently, with the advancement of topology optimization theory, it has also emerged as a powerful tool for designing temperature concentrators.29,30 Compared with these studies on temperature concentrators, little attention has been given to heat flux concentration. In the microscope, Yu et al.31 developed a thermal concentrator that enables heat flux concentration. Zhang et al.32 designed two heat flux concentrators using patterned amorphous and nanomesh structures, respectively. They found that the heat flux in the central regions is much higher than that in the adjacent regions, reaching a concentration ratio of 6–9 times. Furthermore, ray phononics33,34,35,36 are used to achieve heat flux concentration.

Nanophononic metamaterials reduce the thermal conductivity of the base film through the local resonance effect of nanopillars and nanofilms. This effect ensures the integrity of the base film and significantly reduces the interfacial thermal resistance. Currently, there are several experimental preparation methods available, such as vapor-liquid-solid phase epitaxy,37 dry etching,38 and wet etching.39 However, the preparation methods of nanophononic crystals and nanophononic metamaterials are significantly different. For example, Graczykowski et al.40 patterned silicon insulator wafers using the electron-beam lithography (EBL) method and obtained nanophononic crystals by cutting the pristine films and arranging the periodically aligned nanoholes using the focused ion beam (FIB) method. Nanophononic metamaterials are created by arranging nanopillars in a periodic pattern within the pristine film using the evaporation method. In addition, compared with nanophononic crystals, nanophononic metamaterials can alter the atomic mass and height of the nanopillars. Consequently, the thermal conductivity can be regulated over a wider range.

Therefore, in this paper, we designed a heat flux concentrator using nanophononic metamaterials, analyzed its heat flux concentration properties, and investigated the effect of the height of nanopillar and atomic mass of the atoms in the nanopillars on its performance. In addition, the potential mechanism of heat flux concentration is analyzed using phonon localization theory. This work provides insight into the significant applications of nanophononic metamaterials for regulating heat flux.

Nanophononic metamaterials heat flux concentrator model

In this paper, we cleave bulk Si supercells along the [001] direction and use a Si unit cell (UC) as the basic unit to construct pristine nanofilms by replicating 100, 80, and 4 UCs in the x, y, and z directions, respectively. We selected nanopillars with a side width of 4 × 4 UCs, a height of 25 UCs, and a spacing of 2 UCs to build the heat flux concentrator. The atomic model is illustrated in Figure 1. For comparison, we also selected the pristine Si film. The length of the fixed region is 2 UCs, while the thermostat regions are 8 UCs. Therefore, the length of the calculated region is 80 UCs in both the x and y directions.Figure 1 The model of the heat flux concentrator

Molecular dynamics simulation of heat flux concentrators

We utilize the large-scale atomic/molecular massively parallel simulator (LAMMPS) packages41 to calculate the heat flux through non-equilibrium molecular dynamics (MD) simulations. The Stillinger-Weber potential42 is employed to describe the interactions among the Si atoms. Firstly, the system is fully optimized and then all atoms are given an initial velocity corresponding to 300 K, with the velocities follow the Gaussian distribution. The system is relaxed in a canonical ensemble (NVT) for 100 ps. Next, the atoms in the end regions are fixed. We use the Nose-Hoover thermostat43 to establish the hot bath and the cold bath at 320 K and 280 K, respectively. Finally, all atoms except those in the fixed and thermostat regions are placed in a microcanonical ensemble (NVE) for 10 ns. During the first 5 ns, the system arrives to a non-equilibrium steady state, and the temperature and heat flux distributions are calculated in the later 5 ns. To visualize the phenomenon of heat flux concentration, we divided the system into 20 × 20 small blocks to calculate the temperature and heat flux.

Results and discussion

Performance of heat flux concentrator

We plot the temperature and heat flux profiles to illustrate the phenomenon of heat flux concentration. Figure 2A shows the heat flux profile of the concentrator with the height of nanopillar is 25 UC, and the temperature profile is shown in Figure S2A. The results of the pristine film are displayed in Figure S1. In the pristine Si film, the temperature decreases from the hot to the cold bath and the heat flux is the same. This demonstrates the system is in a non-equilibrium steady state. However, the temperature profile in the concentrator is different due to the presence of nanopillars, leading to an extremely large heat flux in the center. To quantitatively evaluate the performance, the ratio of heat flux (RHF) is defined as RHF = J/Jnanopillars, where J represents the local heat flux while Jnanopillars denotes the average heat flux in the nanopillars regions, all the nanopillar regions are selected as a reference. Figure 2B shows the RHF distribution of the concentrator. The highest RHF value is observed in the center, reaching up to 1.62 times that of the nanopillar region. This is primarily due to the local resonance of the nanopillars with the nanofilm, resulting in a reduction of the thermal conductivity in the nanopillar region.8,9,10,11,12,13,14 This ultimately leads to the heat flux toward the central region. In addition, Hossein et al.44 discovered that the nanopillar’s stronger coupling to the base nanofilm resulted in more pronounced curve flattening and a greater reduction in group velocity. And the phonon lifetime is also decreased, further reducing the thermal conductivity. In addition, Liu et al.45 found that nanopillars with non-uniform heights induce stronger phonon resonance effect and can further reduce thermal conductivity compared to models with nanopillars of uniform heights. Therefore, the cloaking efficiency will increase when using nanopillars with non-uniform heights.Figure 2 The thermal properties of the concentrator with the height of nanopillar is 25 UCs

(A) Heat flux.

(B) RHF.

Optimization of heat flux concentrator

To optimize the performance of the heat flux concentrator, two strategies were employed: one involved increasing the height of the nanopillars to 50 UC while keeping the nanopillar positions the same as in Figure 1, and the other involved increasing the atomic mass of the atoms in the nanopillars to 112. We plot the heat flux profiles of the two optimized concentrators in Figures 3A and 3C. The temperature profiles are depicted in Figures S2B, and S2C. The temperature profiles are consistent with the non-optimized concentrator. The heat flux in the center is higher than that in the nanopillar region. We further calculated the RHF, and the RHF distributions are shown in Figures 3B and 3D. The RHF in the center is the highest and both are higher than the unoptimized concentrator. This indicates that the performance can be enhanced by implementing both strategies. This is because increasing the height of nanopillar and atomic mass of the atoms in the nanopillars enhances the local resonant hybridization of the nanopillar with the base nanofilm. Not only does the phonon band structure enrich for a higher vibrator ratio but the level of agreement between phonon and vibrator DOS distributions is significantly enhanced.44 The resonant hybridization results in a decrease in the phonon mean free path, ultimately leading to a reduction in thermal conductivity. In addition, as shown in Figure 3, increasing the atomic mass of the atoms in the nanopillars leads to better performance compared to increasing the height of the nanopillars. This suggests that, based on all the calculations in this paper, increasing the atomic mass in the nanopillar produces stronger phonon localization compared to increasing the nanopillar height.Figure 3 The thermal properties of the concentrator with the height of nanopillar is 50 UCs and the atomic mass of the atoms in the nanopillars is 112

(A) The heat flux profile of the concentrator with the height of nanopillar is 50 UCs.

(B) The temperature profile of the concentrator with the height of nanopillar is 50 UCs.

(C) The heat flux profile of the concentrator with the atomic mass of the atoms in the nanopillars is 112.

(D) The temperature profile of the concentrator with the atomic mass of the atoms in the nanopillars is 112.

Underlying mechanisms of heat flux concentrator

In the previous section, we designed a heat flux concentrator using nanophononic metamaterials and optimized it using two strategies. Previous studies have shown that the resonant hybridization of the nanopillar with the base nanofilm reduces the group velocity and results in a decrease in thermal conductivity.45,46 Using phonon localization theory, we further investigate the underlying mechanisms. The same region (excluding the fixed and thermostat regions) of the three concentrators is selected for velocity autocorrelation, with an autocorrelation time of 30 ps. The pristine film is selected as a reference. We calculate the phonon density of states (PDOS) using the following equation,47(Equation 1) PDOS(ω)=1N2π∫e−iωt〈∑j=1Nvj(0)vj(t)〉dt

where ω is phonon frequency, v is velocity vector, and N is the number of atoms.

The mode participation rate (MPR) can identify both delocalized and localized phonon modes and can be calculated as follows,48(Equation 2) MPR(ω)=1N[∑iPDOSi(ω)2]2∑iPDOSi(ω)4

where PDOSi (ω) is the PDOS at a specific atomic i location. Besides, the intensity off locaalised phonon modes (Λ∈MPR< 0.4) is calculated.49,50(Equation 3) ϕiα,Λ=∫ΛPDOSiαdω1N∑1≤i≤N∫ΛPDOSiαdω

where α = x, y, and z. In this paper, the distribution in the xy-plane of the localized phonon modes is plotted by averaging the ϕiα values over all the same x/y. The larger the ϕiα, the stronger the phonon mode localization at the ith atom. The calculated results of the three concentrators are displayed in Figure S3. The localized mode is distributed in the nanopillar region, which impedes the heat flux transfer and leads to the concentration of heat flux toward the central region.

Next, we plot the spatial distribution of heat flux and localized mode intensity in the P region. Figure 4 shows the calculated results of the concentrator with the atomic mass of the atoms in the nanopillars is 112, and the other results are presented in Figures S4–S6. In pristine Si films, the heat flux along the y axis is the same. The heat flux in the central region along the y axis of the concentrator with the atomic mass of the atoms in the nanopillars is 112 is much higher than that in the nanopillar region as shown in Figure 4A. In Figure 4B, along the y axis the localized phonon modes generate oscillations, exhibiting strong localization in the nanopillar region and weak localization in the nanopillar spacing. This ultimately results in an oscillatory distribution of heat flux. This suggests the heat flux concentration is due to the phonon modes localization in the nanopillars region.Figure 4 The structural diagrams and mechanistic analyses of the concentrator with the atomic mass of the atoms in the nanopillars is 112

(A) Heat flux.

(B) Localized phonon mode.

Furthermore, to investigate the localization effect of the nanopillars on the base nanofilm, we divided the base film into 12 layers along the z axis. We specifically focused on layers L1–L4, as illustrated in Figure 1. We calculated the PDOS and MPR for the L1-L4 in the same region of the three concentrators, using the pristine film as a reference. Figure 5 shows the PDOS and MPR in L1-L4 of the concentrator with the atomic mass of the atoms in the nanopillars is 112, while the remaining results are presented in Figures S7–S9. The PDOS and MPR of the pristine film in L1-L4 are the same, and all MPRs are above 0.4. The PDOS and MPR of the three concentrators exhibit the same trend. For the PDOS, from L1 to L4, the low-frequency peak decreases while the high-frequency peak increases. However, the MPR gradually increases. In the L1, most of the MPRs are less than 0.4, producing the strongest localization. In the L4, the MPR is mostly greater than 0.4, indicating the weakest localization. This suggests that phonon localization weakens with increasing distance from the nanopillar. Zhang et al.51 also observed the phenomenon of surface layer localization by designing periodic nanophononic metamaterials. In addition, Xiong et al.52 calculated the phonon dispersion curve of the nanophononic metamaterials. They found that local resonance may reduce the low-frequency phonon group velocities, and these resonance modes are mostly confined to the first few layers near the interface. This provides further physical insight into our results.Figure 5 The PDOS and MPR in different layers of the concentrator with the atomic mass of the atoms in the nanopillars is 112

Previous research has demonstrated that phonon localization results in a decrease in thermal conductivity, which in turn affects the transport of heat flux.53,54 Therefore, we further investigate the localization effect on heat flux. We calculated the spatial distribution of heat fluxes in the P region of the L1-L4. To quantitatively evaluate the localization effect on heat flux, we defined the inhomogeneous rate of heat flux (α) as α = Jmid/Jedge, where Jmid represents the average heat flux at the central region (along the y axis, ˗10 to 10 UC) and Jedge represents the average heat flux at the edges (along the y axis, ˗40 to ˗15 UC and 15 to 40 UC; see Figure 4 for details). The inhomogeneous rate of heat flux with two optimized concentrators is depicted in Figure 6. From L1 to L4, the inhomogeneous rate first decreases and then remains constant. This indicates that the nanopillars have a limited effect on the underlying nanofilm, reaching only a certain depth. Furthermore, this demonstrates that the stronger the localization, the lower the thermal conductivity and the smaller the heat flux, which is consistent with previous studies.55,56 For example, Chen et al.55 found that surface localization occurred in both silicon nanowires and silicon nanotubes within the same cross-sectional region. However, the increased surface area-to-volume ratio in silicon nanotubes decreased the proportion of delocalized modes. The mode participation rate of both low-frequency and high-frequency phonons in silicon nanotubes is lower compared to that in silicon nanowires. This results in the thermal conductivity of silicon nanotubes being only about 33% of that of silicon nanowires at room temperature. Giri et al.56 found that the thermal conductivity of the a-Si/a-Ge superlattice structure can be reduced by about 30% compared to the thermal conductivity of a-Si0.5Ge0.5 alloys. This reduction is attributed to the stronger localized modes of the a-Si/a-Ge superlattice structure at 10–13 THz, at an interfacial density of 1.28 nm−1. In addition, increasing the atomic mass of the atoms in the nanopillars produces a stronger heat flux concentration effect compared to increasing the height of the nanopillars.Figure 6 The rate of heat flux inhomogeneity with different layers in the P region

In summary, we constructed a heat flux concentrator using nanophononic metamaterials. The phenomenon of heat flux concentration was demonstrated by calculating the ratio of heat flux between the center and the edges. The performance of heat flux concentration is up to 1.62. Furthermore, we optimized the heat flux concentrator by increasing the height of nanopillar and atomic mass of the atoms in the nanopillars. Compared to increasing the height of the nanopillars, the optimization of the atomic mass of the atoms in the nanopillars has a greater effect. We employed phonon localization theory to investigate the mechanism. Results indicate that phonon localization in the nanopillar region is the primary cause of heat flux concentration. In addition, we investigated the localization effect of the nanopillar on the base film and calculated the phonon density of states and mode participation rate for different layers. The low-frequency peak of PDOS decreases and the high-frequency peak increases with increasing distance from the surface. However, the mode participation rate shifts from localized to non-localized, which further suggests that the localization effect only exists for a certain thickness of the base film. Finally, we investigated the effect of localization on the heat flux and calculated the inhomogeneous rate of heat flux along the y axis for different layers. As the localization effect diminishes, the inhomogeneous rate of heat flux decreases. This work provides a new design of heat flux concentrator based on nanophononic metamaterials for regulating thermal conduction.

Limitations of the study

This paper has two limitations. The first is the limited computational resources, which resulted in fewer structures being selected for performance optimization. The second is the absence of quantitative analysis regarding the relationship between the intensity of localized modes and thermal conductivity.

Resource availability

Lead contact

Further information and requests should be directed to and will be fulfilled by the Lead Contact, Haochun Zhang (hczhang@hit.edu.cn).

Materials availability

The study did not generate any new materials.

Data and code availability

• All data will be shared upon request to the lead contact.

• This paper does not report original code.

• Any additional information required to reanalyze the data reported in this work paper is available from the lead contact upon request.

Acknowledgments

This work is supported by the A∗STAR Computational Resource Centre through the use of its high performance computing facilities. J. Z. gratefully acknowledges the financial support from the 10.13039/501100004543 China Scholarship Council (no. 202206120136 ).

Author contributions

Conceptualization, J.Z., and G.Z.; methodology, J.Z., H.C.Z., and G.Z.; software and formal analysis, J.Z.; data curation, J.Z.; writing – original draft, J.Z., H.C.Z., and G.Z.; writing – review and editing, J.Z., H.C.Z., and G.Z.; funding acquisition: J.Z.; supervision, H.C.Z. and G.Z.

Declaration of interests

The authors declare no competing interests.

STAR★Methods

Key resources table

REAGENT or RESOURCE	SOURCE	IDENTIFIER	
Software and algorithms	
	
LAMMPS 3 Mar 2020	LAMMPS	www.lammps.org	

Method details

We modeled a nanophononic metamaterial heat flux concentrator using LAMMPS 3 Mar 2020 and performed molecular dynamics simulations to calculate its performance and optimize it using LAMMPS 3 Mar 2020. Several simulations were performed by varying the initial velocity, all of which produced heat flux concentration. The Stillinger-Weber potential is employed to describe the interactions among the Si atoms. Firstly, the system is fully optimized, then all atoms are given an initial velocity corresponding to 300 K, with the velocities follow the Gaussian distribution. The system is relaxed in a canonical ensemble (NVT) for 100 ps. Next, the atoms in the end regions are fixed. We use the Nose-Hoover thermostat to establish the hot bath and the cold bath at 320 K and 280 K, respectively. Finally, all atoms except those in the fixed and thermostat regions are placed in a microcanonical ensemble (NVE) for 10 ns. During the first 5 ns, the system arrives to a non-equilibrium steady state.

Quantification and statistical analysis

No statistical analysis is used.

Supplemental information

Document S1. Figures S1–S9

Supplemental information can be found online at https://doi.org/10.1016/j.isci.2024.110815.
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