A generalized thermal conductivity model for nanoparticle packed bed considering particle deformation

International Journal of Heat and Mass Transfer 129 (2019) 28–36

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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt

A generalized thermal conductivity model for nanoparticle packed bed considering particle deformation Lin Zizhen a,b, Huang Congliang a,⇑, Li Yinshi b,⇑ a b

School of Electrical and Power Engineering, China University of Mining and Technology, Xuzhou 221116, China Key Laboratory of Thermo-Fluid Science and Engineering of MOE, School of Energy and Power Engineering, Xi’an Jiaotong University, Xi’an 710049, China

a r t i c l e

i n f o

Article history: Received 10 July 2018 Received in revised form 3 September 2018 Accepted 14 September 2018

Keywords: Nanoparticle packed bed Thermal conductivity model Thermal resistance Thermal insulation material

a b s t r a c t Theoretically understanding the thermal conductivity of the nanoparticle packed beds (NPBs) is critical for designing high-performance thermal insulation materials. Currently, the classical effective medium assumption (EMA) model, Nan model, just show their good prediction at a high porosity of NPBs (
0.75). Herein, we propose a generalized model of the thermal conductivity that almost covers the whole porosity range by considering the effect of the nanoparticle deformations on the thermal contact resistance (R). It has been demonstrated that our model matches the experimental results great well. It is also found that at high porosity R is dominated by the phonon diffusive scatterings (Rcd ), while it is determined by the phonon ballistic scatterings (Rcb ) at a low porosity. More interestingly, R can determine the porosity at which the lowest thermal conductivity of NPBs appears. This work opens a new way to design the desired thermal insulation materials. Ó 2018 Elsevier Ltd. All rights reserved.

1. Introduction The silica nanoporous materials, such as silica aerogel [1,2], MCM-41 [3] and SBA-15 [4], have drawn a wide interest because of their quite low thermal conductivities (k) for potential applications as thermal insulation materials. For such kinds of materials, even without considering the air and the thermal radiation in pores, the thermal conductivity could still be higher than the Einstein limit (kE ) [5] corresponding to the amorphous bulk. These thermal insulation materials are still somewhat not competent for their applications in several areas, such as high temperature energy storage tanks [6–9] and space applications [10]. To probe a nanoporous material with a quite low thermal conductivity is still desirable. The nanoparticle packed bed (NPB) as one typical powdermoulding material is focused on in this work. Compared to the traditional nanoporous material, the NPB has an advantage of high density of inter-nanoparticle contact interface, which could provide an effective scattering mechanism for mid-and longwavelength phonons that contribute heavily to the thermal conductivity [11–13]. Because of large amounts of interfaces, the NPB could possess a quite low thermal conductivity [14–17]. To ⇑ Corresponding authors. E-mail addresses: [email protected] (H. Congliang), [email protected] (L. Yinshi). https://doi.org/10.1016/j.ijheatmasstransfer.2018.09.067 0017-9310/Ó 2018 Elsevier Ltd. All rights reserved.

understand the effect of the interface on the thermal conductivity of NPBs, an effective medium approach (EMA) phonon thermal conductivity model is proposed in this work including the interface effect. Before introducing our model, the feasibility of using the phonon concept in describing the thermal transport properties of NPBs is firstly discussed. Phonon is usually introduced to describe the heat transfer in crystalline materials by the lattice vibration approach. In general, the definition of phonons is applicable when the character size of materials is larger than the lattice constant which is less than 1 nm for most materials. Therefore, if the pore is not less or not much less than 1 nm in a nano-porous material, the phonons can be applied to describe the heat transfer. On the contrary, the phonon framework fails to describe the heat transfer when the materials is amorphous or the character size of materials is less than the lattice constant, such as single atom or chemical bond, where an ab-initio calculations should be used to calculate the heat transport properties. Considering the sizes and the crystalline structure of the silica nanoparticles in this work, the phonon framework should be applicable. Several phonon models have already been proposed to theoretically understand the effect of interfaces on the thermal conductivity of composites. The classical acoustic mismatch model (AMM) [18–20] and the diffuse mismatch model (DMM) [21] are originally proposed to predict the effective thermal conductivity (ke ) of composites by taking phonons as mechanical waves in the processes of

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29

Nomenclature ke ks ka kr R Ek

sk hD P T r lm a S

xm x

effective thermal conductivity thermal conductivity of solid phase thermal conductivity of air radiative thermal conductivity thermal contact resistance spectral extinction coefficient spectral transmittance Debye temperature pressure temperature radius of nanoparticles phonon mean free path radius of contact surface cross-sectional area of nanoparticles maximum phonon frequency phonon frequency

their colliding with interfaces, where the interface is treated to be specular or diffusive respectively. While normally the interface is not fully specular or diffusive, these two models (AMM and DMM) only give the upper and lower limits of thermal contact resistances (R). Importantly, the size effect of interface on R is not included in these two models, when different sizes of interface at nanoscale could give different R. Taking the size effect of interface into account but assuming interface to be diffusive, the ke model has been widely established based on the effective medium approximation (EMA) approach, including the size effects of interface by shortening the mean free path of phonons for the restriction of the small interface, such as the Bahadur model [22] and the widely-used Nan model [23]. By carefully taking the ballistic and diffusive transport of phonons into account, Prasher [24,25] have successfully taken the size effect of interface into account in modelling R between two nanoparticles. Although the size effect of interface has already been carefully considered, the nanoparticle deformation which could importantly affect the interface and will show up at a low porosity, has not been considered before. In this work, by modeling the nanoparticle-deformation influenced R, we extend the theoretical understanding of the thermal conductivity of NPBs from a high porosity to a low porosity. This paper is organized as follows: In Section 2, the preparation of silica NPB and the measurement of thermal conductivity are represented. In Section 3, the ke -prediction model with nanoparticle deformation considered is established. For Section 4, we further separate this part into four parts. In Section 4.1, the ke -prediction model is testified with the experimental approach. The ke predicted by the classical EMA model also added for comparison. In Section 4.2, the thermal conductivity of radiation (kr ), confined air (ka ) and solid phase (ks ) are experimentally studied in this part, and the experimental ks is also compared with that predicted by the classical EMA model to reveal the effect of nanoparticle deformation on NPB-thermal conductivity. Finally, the influence of nanoparticle deformation on R is discussed to reveal the influence of nanoparticle deformation on ke . 2. Sample preparation and characterizations 2.1. NPB preparation and structure characterization Commercial silica nanoparticles (Beijing DK Nano Technology Co., Ltd) with diameters of 10 nm and 50 nm are adopted in this work. The macrostructure of 50-nm silica nanoparticle powders are shown in Fig. 1(a), and the microstructure is shown in Fig. 1

kE

u q1 q2 r ER k Ik P dpore

c d K Ac

v

m

thermal conductivity of amorphous bulk porosity density of NPB density of bulk Stefan–Boltzmann constant Rosseland extinction coefficient wavelength of the radiation black body spectral radiative intensity reduced pressure diameter of nanopores Gruneisen constant pore characteristic length elastic constants NPB cross-sectional area phonon group velocity effective electron mass

(b), observed with the field-emission scanning electron microscopy (SEM) on a QuantaTM-250 instrument (FEI Co., USA) with an accelerating voltage of 30 kV. The diameter of nanoparticles keeps almost same, and some clusters are formed by nanoparticle aggregations in Fig. 1(b). To eliminate the aggregation, the nanoparticles are firstly treated with a physical ultrasonic dispersion for 60 min, and then the ball-milling method is utilized to uniformly blend nanoparticles (200 rpm for 60 min, then 100 rpm for 60 min). The microstructures of the physical-dispersion silica nanoparticles are shown in Fig. 1(c), observed by the transmission electron microscope (TEM) method with an accelerating voltage of 80 kV performing on Tecnai G2 F20 instrument (FEI Co., USA). It shows that the nanoparticles distribute uniformly and the aggregations of nanoparticles are eliminated. Finally, the widely-used coldpressing method [13,26,27] is applied to prepare the NPBs, the pressing process is illustrated in Fig. 1(d). Sixteen different stamping pressures (1, 2, 3, 4, 8, 14, 20, 24, 28, 32, 34, 36, 38, 40, 42 and 44 MPa) are applied to press the nanoparticles into differentporosity NPBs. The microstructure of 50-nm NPB with a tableting pressure of 42 MPa is shown in Fig. 1(e), and the details of interface in Fig. 1(e) is further depicted in Fig. 1(f), where the deformation of nanoparticles can be obviously seen. The length scale of the internanoparticle contact area is 10–30 nm, as illustrated by the red line in the inset of Fig. 1(f). The high resolution transmission electron microscope (HRTEM) is further utilized to character the microstructure of silica nanoparticles in our work as shown in Fig. 1(g). An ordered crystal structure is observed indicating that the silica nanoparticles used in this work is crystalline. Additionally, the inset diffraction patterns also confirm the crystalline nature of the silica nanoparticles. 2.2. Thermal conductivity measurement The ke measurement is performed on a commercial device (Model TC3000, Xian XIATECH Technology Co.), according to the hot-wire method [28,29]. More details about the measurement can be found in our previous works [15,17]. With every sample independently measured for five times, a mean ke is obtained with a deviation less than 4.5%. The air thermal conductivity (ka ) in a NPB is estimated by subtracting ke measured at indoor atmospheric pressure with that measured in a vacuum environment (kv ) (detail refer to Appendix A). The radiative thermal conductivity (kr ) is calculated by the diffusion approximation model based on the measurement of spectral transmittance (sk ), which is observed with a Fourier transform infrared spectrometer

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Fig. 1. NPB preparation and characterizations: (a) Optical image of 50-nm silica nanoparticles; (b) microstructures of 50-nm silica nanoparticles observed by the TEM before the physical dispersion process; (c) microstructures of 50-nm silica nanoparticles observed by the TEM after the physical dispersion process; (d) schematic of NPB preparation process; (e) microstructure of 50-nm NPB observed by SEM; (f) a larger view of (e), where the red line indicates the inter-nanoparticle contact surface; (g) HRTEM image of silica nanoparticles with the diffraction pattern shown in the inset. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

(VERTEX 80v Germany Bruker). The calculation of kr are added in Appendix B.

for silica, S is the sectional area of a nanoparticle. For a NPB, the loaded pressure is a function of u, can be expressed as PP0 ¼ f ðuÞ,

3. Theoretical model

where P0 is the maximum pressure used in the experiment, and the f ðuÞ function can be obtained by experimental methods. Here, a simple equation is applied to fit the experimental results,

Firstly, the influence of the nanoparticle deformation is included to get the inter-nanoparticle contact area model. Then the contact area model is further used to derive the R model. Finally, by considering both of the size effect of nanoparticles and the R, the ks model for solid phase is derived. Under the assumption that the NPBs is optical thick which has been proved to be reasonable, [30] the ke of a NPB can be calculated by ke ¼ ks þ kr þ ka .

pffiffiffiffiffiffiffiffiffiffiffiffiffi P ¼ f ðuÞ ¼ x 1  u; P0

ð2Þ

where x is the fitting parameter. Substituting Eq. (2) in Eq. (1), the radius of the contact surface can derived as,

a¼

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 3 3p KrP0 ð1  uÞx ; 4

ð3Þ

Therefore, the contact surface area can be calculated by, 3.1. Inter-nanoparticle contact surface area For simplicity, we assume all nanoparticles in a NPB distribute uniformly according to a simple cubic lattice as illustrated in Fig. 2(a), and the unit cell used for model derivation is shown in Fig. 2(b). The radius ðaÞ of the contact surface can be described by the simplified Johnson-Kendall-Roberts theory [31,32],

a3 ¼

3p KrP; 4

ð1Þ

where P is the loaded pressure in the preparation of NPBs, r and K is respectively the radius and elastic constants of nanoparticles, K ¼ ES=2r, where E is the elasticity modulus equaling 270 Nm2

 2 pffiffiffiffiffiffiffiffiffiffiffiffiffi 3 2 3p Au ¼ p KrP 0 x 1  u ¼ A0 ð1  uÞ3x ; 4

ð4Þ

2

where A0 ¼ pð34p KrP0 Þ3 . 3.2. Thermal contact resistance (R) The R in NPB can be divided into two components: the one caused by the mesoscopic contact area defect (Ra ) and the one caused by the interfacial phonon scatterings (Rc ). Under the classical assumption of elastic scattering of phonons at interface, given the radius of the inter-nanoparticle contact surface is comparable

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Fig. 2. Schematic structures of a NPB and the contact surface: (a) NPB structure; (b) heat transfer in a unit cell.

to the wavelength of the dominant phonons in heat transfer in this work, the Rc comes from the contribution of both the phonon ballistic scattering (Rcb ) and phonon diffusive scattering (Rcd ). (a) Contact-defect thermal contact resistance ðRa Þ The Ra can be calculated by Ra ¼ R1 þ R2 , where R1 is the thermal resistance aroused by the inadequate nanoparticle contact, R2 is the thermal resistance aroused by the lattice defect in the contact areas between nanoparticles. The Ra can therefore be written as [33],

Ra ¼ R1 þ R2 ¼

w þ k0 eAu

"

2 4 kB Au 60 3

p


h

#1

v 2 T 3

;

ð5Þ

where k0 is the thermal conductivity of nanoparticles which can be predicted by the Callaway model [34], the calculation detail is added in Appendix C. w is the root mean square roughness, w ¼ 1 nm is usually applied for nanoparticles. e is the compactness of nanoparticles in NPBs, e ¼ 0:5 for spherical nanoparticles. Au is the contact surface area, is the reduced Planck constant. (b) Thermal contact resistance caused by phonon scatterings ðRc Þ Given the mean free path (l) of phonons in silica is about 1 nm [35–37], which has the same order of magnitude of character length of the contact area between nanoparticles, the interface thermal resistance caused by the diffusive and ballistic phonon scatterings (Rc ) should be both concerned. In this work, the Rc is calculated by a classical model given by Prasher [38],

Rcb Rc ¼ Rcd þ 2 ; pa

ð6Þ

1 : 2k0 a

Rcb ¼

T1  T2 4 ¼ ; Au Cv qAu

ð9Þ

The value of C  v can be estimated by the kinetic theory, 3k=l. So, Rcb is further written as,

Rcb ¼

4l : 3Au k0

ð10Þ

Including Ra , Rcd and Rcb , the total thermal contact resistance can be calculated by,

" #1 w p2 k4B 2 3 1 4l R¼ þ Au v T þ ; þ 3 k0 eAu 2k 3A 60 h a 0 u k0


ð11Þ

3.3. Solid-phase thermal conductivity Under the one-dimensional heat transfer hypothesis, the heat transfer through the solid part in Fig. 2(b) can be calculated by,

Au DT Ac DT Q ¼ 2L ¼ 2L ; þR k0 ks

ð12Þ

where Ac is the sectional area of the unit cell. k0 is the thermal conductivity of silica nanoparticles; ks is the thermal conductivity in solid phase of silica NPB. The ks of NPB can be therefore derived as,

ks ¼

Au k 0 : Ac 1 þ k 0 R 2L

ð13Þ

Substituting expression of Au and L ¼ is rewritten as,

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 2  a2 in Eq. (13), the ks

2

where a is the radius of contact surface, and the Rcd is given by the Maxwell constriction resistance model derived from solving the heat conduction equation based on Fourier’s law,

Rcd ¼

where C is the volume heat capacity. The Rcb can be therefore expressed as,

ð7Þ

ks ¼

A0 ð1  uÞ3x k0 ; k0 R Ac 1 þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2r

ð14Þ

A 1 02 ð1uÞ3x pr

3.4. Confined-air thermal conductivity (ka)

By viewing phonon flow (heat transfer) as the gas flow, the Rcb can be calculated by considering the flow rate of gas molecules through an orifice in the free molecular flow regime [25]. Considering the same materials on both sides of the interface in this work, the heat flux (q) under the circular constriction can be expressed as,

ka ¼

Cv q¼ ðT1  T2 Þ; 4

the gas, kg is the thermal conductivity of the free air. The mean free path of confined air molecules is calculated by [39],

The Kaganer model [39] is used to estimate the ka ,

1þ

0 kg ; 2c 1 2a 2L cþ1 Pr a la

where a is the accommodation coefficient, c is the adiabatic ratio of 0

ð8Þ

ð15Þ

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L. Zizhen et al. / International Journal of Heat and Mass Transfer 129 (2019) 28–36

1 kB T la ¼ pffiffiffi 2 ¼ pffiffiffi 2 ; 2pdg ng 2pdg p

ð16Þ

where ng is the amount of substance of air molecules, dg is the character dimension of pores in NPB, which can be estimated by dg ¼ r  a. Substituting the expression of radius of contact surface into Eq. (16), la can be expressed as,

1 kB T la ¼ pffiffiffi 2 ¼ h i2 ; 1 3 2pdg ng pffiffiffi 3p 2p 4 KrP0 ð1  uÞx p

with an uncertainty of 0.0001 g, V is the volume of NPBs calculated by the sectional-area multiplied by the thickness, here the sectional diameter and the thickness of NPBs are measured by a spiral micrometer with an uncertainty of 0.001 mm. q2 for the bulk silica is 2.2  103 kg/m3. The porosity uncertainty is calculated to be less than 0.002, small enough to be ignored. The experimental ke of 50-nm and 10-nm silica NPBs are used to testify our theoretical model as shown in Fig. 3(a). Before calculating

ð17Þ

3.5. Radiative thermal conductivity (kr) Due to the optical thickness of NPB is much larger than 1, the Rosseland diffusion approximation model can be used to calculate kr as follows, [40]

kr ¼

16rT 3 ; 3ER

ð18Þ

where r is the Stefan–Boltzmann constant, ER is the Rosseland extinction coefficient which can be calculated by,

E1 R ¼

Z

1

0

Z 1 1 Z 1 1 dIk ðT Þ dIk ðT Þ 1 dIk ðT Þ dk dk dk; ¼ tr Ek dT dT dIðT Þ E 0 0 k ð19Þ

Etr k

where ¼ Ek ð1  xk g k Þ is the spectral transport extinction coefficient [41]. Ek , xk and g k are the spectral extinction coefficient, albedo and asymmetry factor of scattering respectively. Ik is the spectral radiative intensity of a black body and k is the wavelength of the radiation. When a spectral radiative intensity Ik is incident on a volume element of NPB with a thickness of dl, considering the absorption and scattering, the reduced Ik can be expressed as,

dIk ¼ Ek Ik dl:

ð20Þ

According to the Lambert-Bell law [42], the relationship between the spectral extinction coefficient and the radiative intensity can be written as,

Z

Ik ðsÞ

I k ð0 Þ

dIk ¼ Ik

Z

S

Ek ðSÞdl:

ð21Þ

0

Eq. (21) can be simplified to

ln ðsk Þ Ek ¼  ; H

ð22Þ

Fig. 3. Comparisons between our model result and experimental data. The experimental ke of Silica NPB is obtained in this work, and the experimental ke of copper and nickel NPB were reported in Refs. [15,17,43].

where H is the thickness of the sample and sk is measured with a Fourier transform infrared spectrometer (VERTEX 80v Germany Bruker). The experimental results of sk are shown in Appendix B. 4. Results and discussions Firstly, the theoretical model is verified by the experimental measurements. And then, the kr , ka and ks is studied respectively to unveil the effect of nanoparticle deformation on ke . Finally, the relationship between the radius of inter-nanoparticle contact surface and R is discussed to reveal the influence of nanoparticle deformation on R. 4.1. Model verification The porosity of a NPB is calculated by u ¼ 1  q1 =q2 , where q1 is the density of the NPB, and q2 is the density of the corresponding bulk material. The density of the NPB is calculated by q1 ¼ m=V, where m is the mass of NPBs measured with an electronic balance

Fig. 4. Comparison between our model and classical EMA model.

L. Zizhen et al. / International Journal of Heat and Mass Transfer 129 (2019) 28–36

33

Fig. 5. ka and kr of NPB at the room temperature: (a) ka , with the inset showing the pore size distribution of 10 nm NPB and 50 nm NPB at u ¼ 0:8; (b) kr .

ke , the relationship between the u and P  is firstly obtained by experimental methods as shown in Fig. 3(b), where P is the reduced pressure calculated by P ¼ PP0 and P 0 is the maximum pressure used in the experiment. By fitting Eq. (2) to the experimental results, the fitting parameter x is obtained with a correlation coefficient larger than 0.93, which suggests that Eq. (2) could capture the experimental results well. After obtaining the fitting parameter x, the ke is theoretically predicted and shown in Fig. 3(a). Our model result could match the experimental data well with a related coefficient larger than 0.95, also confirming the reliability of our model. The experimental data in our previous works [15,17,43] carried out for copper and nickel NPBs are also added in Fig. 3(a) for comparison. Our model also exhibits a good agreement with the experimental data with a related coefficient larger than 0.95. The parameters used in our model prediction of thermal conductivity of copper and nickel NPBs refer to Ref. [44].

ment at u > 75% between our model and the classical model can be easily understood by that the nanoparticle deformation is negligible at a large porosity. To further reveal the mechanism of nanoparticle deformation on ke , the kr , ka and ks are investigated respectively in the next section. 4.3. Effect of nanoparticle deformation on kr, ka and ks The ka and kr of 50-nm and 10-nm NPB are shown in Fig. 5(a) and (b) respectively. The ka of 50-nm and 10-nm NPB are both smaller than that of free air, which is about 0.026 W m1 K1 [46], because of the smaller size of pores in NPB than the l of free air molecules (about 70 nm). The effect of nanoparticle deformation on ka can be

4.2. Comparison between our model and the classical EMA model The ke prediction of our model is compared with the ones predicted by classical EMA model [22,45] where R is estimated under an assumption that the phonon scatterings are fully diffusive. The comparison results are shown in Fig. 4. Both of our model and the classical EMA model capture the experiments well at u > 75%, while only our model could give a good prediction at low porosities when the classical EMA model overestimates the ke . The ke overestimation by classical EMA model should arrive from omitting the Rcb introduced by the nanoparticle deformation. The good agree-

Fig. 6. ks in 10-nm NPB and 50-nm NPB, the ks predicted by our model and the classical EMA model also added for comparison.

Fig. 7. Interface area per unit volume (Au ), radius of contact surface (a) and the thermal contact resistance (R): (a) Au and a versus u; (b) R versus a for 50-nm NPB; (c) R versus a for 10-nm NPB.

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L. Zizhen et al. / International Journal of Heat and Mass Transfer 129 (2019) 28–36

ignored when the ka only contributes about 3% of ke . The Rosseland diffusion model which is applied to calculate the kr has not a direct relation with the nanoparticle deformation. Thus, we infer that the nanoparticle-deformation only takes an important role on the ke of NPB through affecting the ks . The influence of the nanoparticle deformation on ks is therefore discussed in next paragraph. The ks is experimentally obtained by the ks ¼ ke  kr  ka as shown in Fig. 6. The ks theoretically predicted by the classical EMA model and our model is also added in Fig. 6 for comparison. Our model results coincide with experimental data reasonably well for all u, while the classical EMA model only gives a good prediction at a very high porosity (u P 0:75). The classical EMA model overestimates the ks at 0:3 6 u 6 0:75, due to the underestimated R with respect to the ones predicted by our model. This underestimated R results from the underestimated interface area because of omitting the plastic deformation of nanoparticles. In the following section, the influence of the nanoparticle deformation on the R is carefully discussed. The interface area per unit volume (Au ) and the a are respectively calculated and shown in Fig. 7(a). The a decreases with the increase of the u. Further depicting the dependence of the R on a in Fig. 7(b) and (c), it shows that the Rcd dominates R at a small a (high u) while Rcb dominates at a large a (low u). The underestimated R in the classical EMA model arrives from the ignored Rcb which will dominate R at a low u, thus the classical EMA model could give a good prediction at a high u but not at a low u. Additionally, Ra , Rcd and Rcb all decrease with the increasing a or degree of deformations in nanoparticles.

Appendix A. ke of NPB measured in vacuum See Fig. A1. Appendix B. Spectral transmittance () in NPB See Fig. A2. Appendix C. Calculation of k0 The k0 is calculated by the Callaway model

 3 Z hD T kB T sc g4 eg dg; 2 2p v
h ðeg  1Þ2 0 kB

k0 ¼

ðA1Þ

here kB is the Boltzmann constant,
h is the reduced Planck constant, hD is the Debye temperature of the bulk silica, v is the phonon group velocity, g ¼ ðx=kD T Þ, x is the frequency of phonons, an sc is the relaxation time. The sc can be predicted by the Matthiessen law, [47]

1

sc

¼

1

sU

where

þ

1

sM

þ

1

sphe

þ

1

sB

;

ðA2Þ

1 1 1 s1 U , sM , sphe , sB are corresponding to the relaxation time

caused by phonon Umklapp scattering, the mass difference, the phonon-electron scattering and the boundary scattering. In the room temperature, s1 can be obtained by following U equation, [48]

1

¼ 2c2

k B T x2 ; lV 0 xD

ðA3Þ

5. Conclusion

sU

In this work, the ke model including the effect of nanoparticle deformations is firstly established for NPBs. Secondly, the coldpressing method is applied to prepare NPB, then the ke of NPBs is measured for verification of our model. Thirdly, the ke predicted by our model is further compared with the one predicted by the classical EMA model for highlighting the effect of the nanoparticle deformation on ke . Finally, the influence mechanism of nanoparticle deformations on ke is revealed by analyzing the effect of nanoparticle deformations on ka , kr and ks , especially on R. Results show that our model could give a good prediction of ke at almost all u, while the classical EMA model just match the experimental data well at a high porosity (u P 0:75). The classical EMA model overestimate the ke at a low porosity (0:3 6 u 6 0:75) because of without considering the influence of nanoparticle deformation. The nanoparticle deformation affects the ke predominantly through influencing ks , more exactly R. By calculating the Ra , Rcd and Rcb at different degree of deformation of nanoparticles, results show that the Rcd dominates R at a small a (high u) while Rcb dominates at a large a (low u). The overestimation of ke by the classical EMA model arrives from the ignored Rcb . This study extend the theoretical understanding of the ke of NPBs from a high porosity to a low porosity, while previous model only could be used to predict the ke at a high porosity because of neglecting the nanoparticle deformations. This work is also expected to provide some physical insights into the R between nanoparticles.

where c is Gruneisen constant, xD is the Debye frequency, V 0 is the molecular volume, l is the shear modulus which can be calculated by l ¼ v 2t q. s1 M can be predicted by, [49]

1

sM

¼

V 0 x4 ; 4pv 3

ðA4Þ

where v is the phonon group velocity, and C ¼ 1 for silica materials. [50] s1 phe can be calculated by, [51]

1

sphe

ne f2 x ¼ qv 2 kB T

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   pme v 2 mv 2 ; exp  2kB T 2kB T

Conflict of interest The authors declare that there is no conflict of interest. Acknowledgements This work has been supported by the Fundamental Research Funds for the Central Universities (2018XKQYMS17).

Fig. A1. ke of NPB measured in vacuum (kv ).

ðA5Þ

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35

Fig. A2. Spectral transmittance in NPBs at u = 35% and u = 90%: (a) 50-nm NPB, u = 35%; (b) 50-nm NPB, u = 90%; (c) 10-nm NPB, u = 35%; (d) 10-nm NPB, u = 90%.

Table A1 Parameters used in the calculation of k0 . Parameters

Value

Literature

Density, q (g cm3) Gruneisen constant, c Phonon group velocity, v , m s1 Longitudinal phonon group velocity, v L , m s1 Transverse phonon group velocity, v T , m s1 Debye temperature, hD , K Electron mass, m , m0 Molecular volume, V0, m3

2.28 0.45 4.47  103 5.90  103 3.75  103 511 0.26 3.76  1029

[53] [54] [53] [55] [55] [5] [54] [56]

where ne is the electron concentration, f is the silica deformation potential. For silica materials, ne f2 = 8.75  1012 J m3. [52] me is the effective electron mass. s1 B can be predicted by,

1

sB

¼

  1p ; 2r 1 þ p

v

ðA6Þ

where p is the surface reflection coefficient. With a diffusive surface, p ¼ 0 (see Table A1). References [1] H. Zhang, W.Z. Fang, Z.Y. Li, W.Q. Tao, The influence of gaseous heat conduction to the effective thermal conductivity of nano-porous materials, Int. J. Heat Mass Transf. 68 (2015) 158–161. [2] T. Xie, Y.L. He, Heat transfer characteristics of silica aerogel composite materials: structure reconstruction and numerical modeling, Int. J. Heat Mass Transf. 95 (2015) 621–635. [3] A.P. Cote, A.I. Benin, N.W. Ockwig, M. O’ Keeffe, A.J. Matzger, O.M. Yaghi, Porous, crystalline, covalent organic frameworks, Science 310 (2005) 1166– 1170. [4] C. Bi, G.H. Tang, Effective thermal conductivity of the solid backbone of aerogel, Int. J. Heat Mass Transf. 64 (2013) 452–456.

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