77 Characterization and Application of Nanomaterials (2021) Volume 4 Issue 1 doi:10.24294/can.v4i1.1292 Original Research Article Mesoscale computational prediction of lightweight, thermally conduc- tive polymer nanocomposites containing graphene-wrapped hollow particle fillers Jianjun Wang1,a,*, Zhonghui Shen2,a, Wenying Zhou3*, Yang Shen2, Cewen Nan2, Qing Wang1, Longqing Chen1* 1 Department of Materials Science and Engineering, The Pennsylvania State University, University Park, Pennsylvania 16802, United States. E-mail: wjj8384@gmail.com (J Wang); lqc3@psu.edu (L Chen) 2 School of Materials Science and Engineering, State Key Lab of New Ceramics and Fine Processing, Tsinghua Univer- sity, Beijing 100084, China 3 College of Chemistry and Chemical Engineering, Xi'an University of Science & Technology, Xi’an 710054, China. E-mail: wyzhou2004@163.com(W Zhou) ABSTRACT Heat removal has become an increasingly crucial issue for microelectronic chips due to increasingly high speed and high performance. One solution is to increase the thermal conductivity of the corresponding dielectrics. However, traditional approach to adding solid heat conductive nanoparticles to polymer dielectrics led to a significant weight in- crease. Here we propose a dielectric polymer filled with heat conductive hollow nanoparticles to mitigate the weight gain. Our mesoscale simulation of heat conduction through this dielectric polymer composite microstructure using the phase-field spectral iterative perturbation method demonstrates the simultaneous achievement of enhanced effective thermal conductivity and the low density. It is shown that additional heat conductivity enhancement can be achieved by wrapping the hollow nanoparticles with graphene layers. The underlying mesoscale mechanism of such a microstructure design and the quantitative effect of interfacial thermal resistance will be discussed. This work is expected to stimulate future efforts to develop light-weight thermal conductive polymer nanocomposites. Keywords: Thermal Conductivity; Polymer Nanocomposites; Materials Design; Graphene-Wrapped Hollow Nanoparti- cles ARTICLE INFO Received: 2 Febuary 2021 Accepted: 17 March 2021 Available online: 25 March 2021 COPYRIGHT Copyright © 2021 Jianjun Wang, et al. EnPress Publisher LLC. This work is licensed under the Creative Commons Attribution- NonCommercial 4.0 International License (CC BY-NC 4.0). https://creativecommons.org/licenses/by- nc/4.0/ 1. Introduction The effective thermal management in applications, such as LED (light emitting diode) lighting, batteries, automobile cooling systems, and high-power density microelectronic devices, where heat accumu- lation can have deleterious effects, is critically important to ensure the device performance and reliability, and therefore to enhance the lifetime and accuracy of the system[1–3]. With further miniaturization, integration and functionalization of microelectronics and the emerg- ing applications, such as electronic assembly and packaging, and solar the thermal dissipation has become a challenge[4–6]. Addressing this challenge requires the development of novel materials with enhanced thermal conductivity as well as light weight, low cost, good process- ability, and corrosion resistance. Polymers have many of these charac- a These authors contributed equally to this work. 78 teristics, but they generally have very low thermal conductivity (0.1–0.4 Wm−1∙K−1)[7,8]. Therefore, heat conductive fillers, such as carbon nanotube[9–13] (> 2,000 Wm−1∙K−1), graphene[14–18] (–5,000 Wm−1∙K−1), aluminum oxide[19–21] (> 20 Wm−1∙K−1), boron ni- tride[22–26] (–350 Wm−1∙K−1), and metal particles[27–31] (> 100 Wm−1∙K−1), etc., are traditionally added into polymers to enhance their thermal conductivity while preserving the above-mentioned advantages of polymers. The influences of the filler type, size, shape, alignment, and loading level on the effective thermal conductivity of the resulted polymer composites have been extensively investigated, see e.g., the recent reviews[5,12,32]. It was generally accepted that a high filler loading level (≥ 30% in volume) is nec- essary in order to achieve the appropriate level (≥ 1 Wm−1∙K−1) of thermal conductivity in a polymer nanocomposite. For example, heat sinks in micro- electronic systems require polymer nanocomposites with a thermal conductivity approximately from 1 to 30 Wm−1∙K−1, which normally needs a filler loading level higher than 30% in volume[7,33]. The high load- ing level of the filler, particularly for metallic fillers, usually significantly increases the mass density and costs, and weakens the mechanical performances, such as tensile strength and flexibility, and processi- bility, which prevents the polymer composites from being used commercially, in particular in aerospace where a lightweight is extremely desired[32,34]. There- fore, it is imperative to seek for alternative approach- es to developing novel material microstructures with enhanced thermal conductivity but low density and costs. To effectively reduce the weight and improve the specific thermal conductivity of filled polymers, in this work, we propose to fill the polymer matrix with hollow nanoparticles to increase the thermal conductivity while preserving a low mass density of the nanocomposite. In particular, we computed the effective thermal conductivity (κeff) and the effective mass density (ρeff) of the polyethene (PE) polymer nanocomposites filled with various hollow nanoparti- cles. It is predicted that by wrapping a thin graphene layer onto the hollow nanoparticles, the effective thermal conductivity can be further significantly en- hanced. Materials Thermal Conductivity (Wm−1∙K−1) Mass Density (kg/m3) Literature Values (Wm−1∙K−1) Refs. Ag 417 10,490 427 31 Al 237 2,700 247 28 Fe 40 7,900 67 38 Cu 397 8,900 398 28 Al2O3 33 3,700 30–36 39 AlN 300 3,260 100–300 5 BN 57 2,290 185–300 5,40 Graphene 4,000 2,250 2,000–6,000 41,42 PE polymer 0.24 1,000 0.3–0.45 5 Air 0.024 1.225 0.024 5 Table 1. Thermal conductivities and mass density of the filler materials used in the simulation. For BN nanoparticles, the thermal conductivity used is smaller than the literature values which were reported for the in-plane thermal conductivity in BN nanosheets. For PE polymer, the thermal conductivity depends on the density Figure 1. Computationally-generated microstructures for poly- mer nanocomposites filled with (a) 20 vol.% solid nanoparticles, (b) 40 vol.% solid nanoparticles, (c) 20 vol.% hollow nanopar- ticles, and (d) 40 vol.% hollow nanoparticles, (e) the effective thermal conductivity and (f) mass density as function of filler volume fraction for polymer nanocomposites filled with various solid and hollow nanoparticles. 79 2. Methods The heat conduction equation in the polymer nanocomposite can be written as: ( ) ( ) ( ) ( ) ( ) ( ) ij p i j T T k q c x x t ρ  ∂ ∂∂ + =  ∂ ∂ ∂  x x x x x x where kij(x), T(x), ρ(x), and cp(x) represent the spa- tial-dependent thermal conductivity tensor, tempera- ture, the mass density, and the specific heat capacity, respectively. Those spatial-dependent material prop- erties such as kij(x), ρ(x), and cp(x) are determined by the microstructure of the polymer nanocomposite specified by a phase-field variable. The internal heat source of the material is represented by q(x). Eq. (1) can be solved using the spectral iterative perturbation method which was developed in previous work[35] or using the finite element method via the COMSOL software. When incorporating the interfacial thermal resistance, slit boundary conditions are applied at the heterointerfaces between phase A and phase B, i.e., ( ) k A B A A A T TT R − − ⋅ − ∇ =n k , ( ) k B A B B B T TT R − − ⋅ − ∇ =n k (2) where nA and nB represent the normal directions of the interface pointing to phase A and phase B, re- spectively. The variable TA and TB represent the tem- perature at the two boundaries of the heterointerfac- es, and kA and kB represent the thermal conductivity of phase A and phase B, respectively. Once the tem- perature distribution is solved, the heat flux density that flows through a unit area per unit time can be determined from the Fourier’s law, i.e., ( ) ( ) /i ij jJ k T x= − ∂ ∂x x (3) The effective thermal conductivity tensor eff ijk of the polymer nanocomposite can then be determined from Eq. (3) by solving ( )eff /i ij jJ k T x= − ∂ ∂x (4) where represents the average property per vol- ume. 2. Results and discussions Figures 1a-d show the microstructures of PE nanocomposites filled with 20 vol.% and 40 vol.% solid nanoparticles, 20 vol.% and 40 vol.% hollow nanoparticles, respectively, computationally gen- erated assuming random distributions of the filler nanoparticles. For hollow nanoparticles, the thick- ness of the filler layer is about 4% to 7% of the radius of the nanoparticles. The effective thermal conductivity for the polymer nanocomposite is cal- culated by solving the steady-state heat conduction equation using the phase-field spectral iterative perturbation method[35–37]. The intrinsic thermal con- ductivities and mass densities of the filler materials, polyethylene (PE) polymer, and air used in the com- putation are listed in Table 1. Figure 1e shows the effective thermal conduc- tivity as function of the filler volume fraction for PE nanocomposites filled with various nanoparticles. For all listed nanocomposites, the effective thermal conductivity increases with the volume fraction of the fillers for both the solid and hollow nanoparticles. For a filler material at a given Vf, the solid nanoparti- cles are more effective than the hollow nanoparticles in enhancing the thermal conductivity. For example, at a Vf of ∼50%, the polymer nanocomposites filled with solid Ag, solid Al, solid Fe, solid Cu, solid Al2O3, solid AlN, and solid BN nanoparticles have a κeff of ∼28 Wm−1∙K−1, ∼14 Wm−1∙K−1, ∼3.6 Wm−1∙K−1, ∼20 Wm−1∙K−1, ∼4.1 Wm−1∙K−1, ∼22 Wm−1∙K−1, and ∼5.0 Wm−1∙K−1, while their counterparts filled with corresponding hollow nanoparticles have a κeff of ∼1.9 Wm−1∙K−1, ∼1.2 Wm−1∙K−1, ∼0.5 Wm−1∙K−1, ∼1.7 Wm−1∙K−1, ∼0.5 Wm−1∙K−1, ∼1.4 Wm−1∙K−1, and ∼0.6 Wm−1∙K−1, respectively (see Table 2). However, the effective mass density of the PE nanocomposite is significantly increased by the solid nanoparticles compared to the hollow nanoparticles. As shown in Figure 1f, at a Vf of ∼50%, the nano- composites filled with solid Ag, solid Cu, and solid Fe nanoparticles respectively have a ρeff of ∼5,646 kg/m3, ∼4,971 kg/m3, and ∼4,374 kg/m3, while their counterparts filled with hollow nanoparticles have a much lower ρeff of ∼1,246 kg/m3, ∼1,135 kg/m3, and ∼1,109 kg/m3, respectively (see Table 2). More in- terestingly, the effective mass density of the polymer nanocomposites filled with hollow Al, hollow Al2O3, hollow AlN, and hollow BN even decreases with (1) 80 the volume fraction of the nanoparticles. More spe- cifically, the effective mass density of the polymer nanocomposites can be reduced from ∼1,000 kg/m3 to ∼681 kg/m3, ∼756 kg/m3, ∼731 kg/m3, and ∼663 kg/m3 when it is respectively filled with hollow Al, hollow Al2O3, hollow AlN, and hollow BN nanopar- ticles at a Vf of 50 percent (see Table 2). Nanoparticle Vf = 50% Ag Al Fe Cu Al2O3 AlN BN Solid κeff(Wm−1∙K−1) 28 14 3.6 20 4.1 22 5.0 ρeff (kg/m3) 5,646 1,849 4,374 4,971 2,380 2,097 1,663 Hollow κeff(Wm−1∙K−1) 1.9 1.2 0.5 1.7 0.5 1.4 0.6 ρeff (kg/m3) 1,246 681 1,109 1,135 756 731 663 Graphene-wrapped κeff(Wm−1∙K−1) ~11 ~11 ~5 ~11 ~5 ~11 ~5 ρeff (kg/m3) 1,290 817 1,153 1,196 877 847 789 Table 2. Effect thermal conductivity and mass density for polymer nanocomposites filled with various solid, hollow, and graphene- wrapped hollow nanoparticles at a volume fraction of 50%. Figure 2a shows the mapping result of the κeff as function of the thermal conductivity of the filler ma- terial (κfiller) and Vf for the polymer nanocomposites filled with hollow nanoparticles. The calculations in Figure 1e are included in this more comprehensive mapping result. For example, for nanocomposites filled with 10 vol.% hollow AlN, 20 vol.% hollow Cu, 30 vol.% hollow Al, and 40 vol.% hollow Ag nanoparticles, which are marked on Figure 2e, they have a κeff of ∼0.31 Wm−1∙K−1, ∼0.41 Wm−1∙K−1, ∼0.54 Wm−1∙K−1, ∼0.91 Wm−1∙K−1, respectively, as in- dicated by the color bar. The κeff increases with both κfiller and Vf. At κfiller = 2000 Wm−1∙K−1 and Vf = 50%, the effective thermal conductivity can be increased to ∼4.8 Wm−1K−1. In contrast, Figure 2b shows the mapping result for the nanocomposite filled with sol- id nanoparticles. The solid nanoparticles are indeed more effective in enhancing the κeff than their hollow counterparts. For example, at κfiller = 2000 Wm−1∙K−1 and Vf = 50%, the κeff can be enhanced to ∼103 Wm−1∙K−1 by the solid nanoparticles. However, in order to achieve a specific thermal conductivity by filling different types of nanoparti- cles into the PE polymer, the nanocomposites filled with hollow nanoparticles are shown to require much less filler materials and hence show much lower mass density. For example, as shown in Figure 3a, for a PE nanocomposite with an effective thermal conductivity of ∼1 Wm−1∙K−1, the mass density is ∼3,566 kg/m3, ∼1,506 kg/m3, ∼3,206 kg/m3, and ∼1,614 kg/m3 when the nanocomposite is filled with solid Ag, solid Al, solid Cu, and solid AlN nanopar- ticles, respectively. However, their counterparts only show a mass density of ∼1,271 kg/m3, ∼774 kg/m3, ∼1,156 kg/m3, and ∼805 kg/m3 when filled with the corresponding hollow nanoparticles. Figure 2. The effective thermal conductivity as function of the volume fraction and the filler thermal conductivity for polymer nano- composites filled with (a) hollow nanoparticles and (b) solid nanoparticles. 81 Furthermore, the materials cost in the nanocom- posites filled with the hollow nanoparticles will also be much less. As shown in Figure 3b, the unit costs of the polymer nanocomposites filled with hollow Ag, hollow Al, hollow Cu, and hollow AlN nanopar- ticles are ∼0.839 $/cm3, ∼0.001 $/cm3, ∼0.0069 $/ cm3, and ∼0.161 $/cm3, while the unit costs of their counterparts filled with solid nanoparticles are ∼2.35 $/cm3, ∼0.0019 $/cm3, ∼0.019 $/cm3, and ∼0.322 $/ cm3, respectively. Therefore, the usage of hollow nanoparticles reduces the weight of the nanocom- posite and the materials cost of the nanocomposite. However, the hollow nanoparticles might not yield sufficient enhancement of the effective thermal con- ductivity. For instance, by filling hollow nanoparti- cles such as Fe, Al2O3, and BN into the PE polymer, a target κeff of 1 Wm−1∙K−1 may not be achieved un- less denser PE polymer with higher thermal conduc- tivity is used as the matrix. Figure 3. For a PE nanocomposite with a targeted effective ther- mal conductivity of 1 WK1m1, (a) the mass density and (b) the unit cost of materials it has when it is filled with various solid and hollow nanoparticles. The inset Table of (b) lists the rough costs of different filler materials, which might change depending on the market. In the light of the mapping result for the κeff as function of the κfiller and the Vf shown in Fig- ure 2a, we propose a hierarchical architecture for the hollow nanoparticles. As shown in Figure 4, we suggest wrapping one graphene layer onto the shell of the hollow nanoparticle. This design is ra- tionalized by the super-high thermal conductivity (∼4,000 Wm−1∙K−1) of the graphene[41,42], which can be employed to possibly wrap the shell of a hol- low nanoparticle[43–45]. The technique of wrapping a graphene layer onto the shell of a nanoparticle has been used to improve the performances of batter- ies[43–45], and here we predict that it can be used to improve the effective thermal conductivity of the polymer nanocomposites. It can be seen from Figure 5a that the effective thermal conductivity of the nanocomposite filled with graphene-wrapped hollow nanoparticles in- creases much faster with the volume fraction of the filler nanoparticles. At a Vf of 50%, the effective thermal conductivity of the nanocomposite can be enhanced to ∼11 Wm−1∙K−1, which is about 10 times of their counterparts filled with hollow nanoparticles without a graphene layer. Meanwhile, this hierarchi- cal architecture does not increase the effective mass density much. As shown in Figure 5b, the effective mass density of the nanocomposite at a Vf of 50% is ∼1,290 kg/m3, ∼817 kg/m3, ∼1,153 kg/m3, ∼1,196 kg/ m3, ∼877 kg/m3, ∼847 kg/m3, and ∼789 kg/m3 when filled with graphene-wrapped hollow Ag, Al, Fe, Cu, Al2O3, AlN, and BN nanoparticles, respectively (see Table 2). These values are about ∼3.5%, ∼20.0%, ∼3.9%, ∼5.3%, ∼16.0%, ∼15.8%, and ∼19.0% higher than their counterparts filled with corresponding hol- low nanoparticles without wrapping graphene. Figure 6a shows the comparison of the effective mass density between the nanocomposites filled with solid nanoparticles and graphene-wrapped hollow nanoparticles. It can be seen that a target κeff of 1 Wm−1∙K−1 now can be achieved by all listed filler materials. The effective mass densities of the nano- composites filled with graphene-wrapped hollow nanoparticles are much lower than their counterparts filled with solid nanoparticles. For nanocomposites filled with heavy fillers such as Ag, Fe, and Cu, the effective mass density can be reduced by ∼70% by using graphene-wrapped hollow nanoparticles rather than solid nanoparticles, while still preserving the 82 same κeff of 1 Wm−1∙K−1. This is not only beneficial to the reduction of the weight and materials costs, but also beneficial to the preservation of the flexibility performances of the polymers which can easily be damaged by a high loading level[46–50]. As shown in Figure 6b, the Vf of the nanocomposites filled with graphene-wrapped hollow nanoparticles is univer- sally reduced, compared with the counterpart in the nanocomposites filled with solid nanoparticles. Now turn to the underlying mechanisms of the advantages of using hollow nanoparticles and graphene-wrapped nanoparticles over the solid nanoparticles. We consider three nanocompos- ites, which are filled with solid nanoparticles, hol- low nanoparticles, and graphene-wrapped hollow nanoparticles, respectively. The filler materials are same, e.g., Cu metal, and the sizes of the filler nanoparticles are assumed to be similar. The volume fraction of Cu metal in the three nanocomposites are set to be at the same value of 6%. While in the nanocomposite filled with solid nanoparticles the thermally conductive Cu metal concentrates at each solid particle, the Cu metal in the nanocomposite filled with hollow nanoparticles distributes on the surface of each hollow particle. Since the surface layer volume of the hollow particle is much lower than the whole volume of the solid particle, there must be more hollow particles in the same polymer. As a result, the probability of forming thermally con- ductive channels through surfaces connection of the hollow particles is increased, leading to the enhance- ment of the effective thermal conductivity. This can be understood from the comparison of the thermal energy flux distributions shown in Figures 7a-b. By wrapping a more thermally conductive graphene layer on the surfaces of the hollow nanoparticles, the formation probability of heat conductive channels and hence the effective thermal conductivity will be 83 further increased, as revealed in Figure 7c. Figure 6. For a PE nanocomposite with a targeted effective ther- mal conductivity of 1 WK1m1, (a) the mass density and (b) the filler volume fraction it has when it is filled with various solid and graphene-wrapped hollow nanoparticles. Figure 8. Effect of the interfacial thermal resistance on the effective thermal conductivity for PE nanocomposites filled with 25 vol.% solid Cu nanoparticles, hollow Cu nanoparticles, and graphene-wrapped hollow Cu nanoparticles. In above simulations, the strategy of adding hol- low and graphene-wrapped hollow nanoparticles into the polymer to enhance the thermal conductivity and reduce the mass density is illustrated without consid- ering the interfacial thermal resistance (Rk). Figure 8a shows the parameterized study of Rk effects on the effective thermal conductivity for polymer nano- Figure 7. Thermal energy flux distributions for polymer nanocomposites filled with (a) solid Cu nanoparticles, (b) hollow Cu nanoparticles, and (c) graphene-wrapped hollow Cu nanoparticles. The volume fractions of Cu metal for these three polymer nano- composites are at the same value of 6%. Due to the introduction of the hollow structure, the volume fractions of the hollow nanopar- ticles are 30%. composites filled with solid, hollow, and graphene- wrapped Cu hollow nanoparticles. For polymer nanocomposites filled with solid Cu and graphene- wrapped hollow Cu nanoparticles, Rk is important when it is great than 10−10 m2∙K/W, whereas it is important when Rk > 10−6 m2∙K/W for the polymer nanocomposite filled with hollow Cu nanoparticles. Specifically, the effective thermal conductivity can be decreased by the interfacial thermal resistance from 1.0 Wm−1∙K−1 to 0.19 Wm−1∙K−1, from 0.71 Wm−1∙K−1 to 0.16 Wm−1∙K−1, and from 0.48 Wm−1∙K−1 to 0.14 Wm−1∙K−1 for polymer nanocomposite filled with 25 vol.% graphene-wrapped hollow Cu nanoparticles, solid Cu nanoparticles, and hollow Cu nanoparti- cles, respectively. Therefore, the effective thermal conductivity predicted in this work should be lower 84 when the interfacial thermal resistance is considered. In order to accurately predict the effective thermal conductivity as function of the microstructure, the knowledge of the interfacial thermal resistance is necessary. While it is challenging to measure the in- terfacial thermal resistance experimentally, it may be obtained via molecular dynamic simulations[51–53]. 3. Conclusions The effective thermal conductivity and effective mass density of the polymer nanocomposites filled with solid nanoparticles and hollow nanoparticles are computed. It is predicted that the usage of hol- low nanoparticles rather than the solid nanoparticles as fillers can enhance the thermal conductivity but preserve the low mass density of the polymer nano- composites. By wrapping a graphene layer onto the surface of the hollow nanoparticles, the effective thermal conductivity can be further significantly en- hanced while still preserving a low mass density of the polymer nanocomposite. The underlying mech- anism of this microstructure design and the quanti- tative effect of the interfacial thermal resistance are presented. The present work is expected to stimuli future experimental and theoretical efforts to design light-weight thermally conductive polymer nano- composites. Conflict of interest No conflict of interest was reported by the au- thors. Acknowledgements J. J. Wang and L. Q. Chen are partially support- ed by the US Air Force Office of Scientific Research through tasks (FA9550-17-1-0318) and by the Hamer Professorship. W Y Zhou gratefully acknowledge the financial supports from the National Natural Science Foundation of China (No.51577154), the Key Labo- ratory of Engineering Dielectrics and Its Application, Ministry of Education, Harbin University of Science and Technology (No. JZK201301, KF20151111). References 1. Gurrum SP, Suman SK, Joshi YK, et al. Thermal issues in next-generation integrated circuits. IEEE Transactions on Device and Materials Reliability 2004; 4 (4): 709–714. 2. Ghosh S, Calizo I, Teweldebrhan D, et al. 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