⇦Chinese Physics Letters, 2020, Vol. 37, No. 10, Article code 104401Express Letter A Ubiquitous Thermal Conductivity Formula for Liquids, Polymer Glass, and Amorphous Solids Qing Xi (席晴)1, Jinxin Zhong (钟金鑫)1, Jixiong He (何集雄)2, Xiangfan Xu (徐象繁)1, Tsuneyoshi Nakayama1,3, Yuanyuan Wang (王元元)4, Jun Liu (刘君)2*, Jun Zhou (周俊)1†*, and Baowen Li (李保文)5* Affiliations 1Center for Phononics and Thermal Energy Science, China-EU Joint Lab for Nanophononics, School of Physics Science and Engineering, Tongji University, Shanghai 200092, China 2Department of Mechanical and Aerospace Engineering, North Carolina State University, Raleigh, NC 27695, USA 3Hokkaido University, Sapporo, Hokkaido 060-0826, Japan 4School of Environmental and Materials Engineering, Shanghai Polytechnic University, Shanghai 201209, China 5Paul M Rady Department of Mechanical Engineering, Department of Physics, University of Colorado, Boulder, CO 80305-0427, USA Received 11 August 2020; accepted 15 September 2020; published online 23 September 2020 This work is supported by the National Key R&D Program of China (Grant No. 2017YFB0406004), the National Natural Science Foundation of China (Grant No. 11890703). JH and JL are supported by the National Science Foundation of USA (Award No. CBET-1943813) and the Faculty Research and Professional Development Fund at North Carolina State University.
†Current address: School of Physics and Technology, Nanjing Normal University, Nanjing 210046, China
*Corresponding author. Email: jliu38@ncsu.edu; zhoujunzhou@tongji.edu.cn; Baowen.Li@Colorado.edu Citation Text: Xi Q, Zhong J X, He J X, Xu X F and Nakayama T et al. 2020 Chin. Phys. Lett. 37 104401    Abstract The microscopic mechanism of thermal transport in liquids and amorphous solids has been an outstanding problem for a long time. There have been several approaches to explain the thermal conductivities in these systems, for example, Bridgman's formula for simple liquids, the concept of the minimum thermal conductivity for amorphous solids, and the thermal resistance network model for amorphous polymers. Here, we present a ubiquitous formula to calculate the thermal conductivities of liquids and amorphous solids in a unified way, and compare it with previous ones. The calculated thermal conductivities using this formula without fitting parameters are in excellent agreement with the experimental data. Our formula not only provides a detailed microscopic mechanism of heat transfer in these systems, but also resolves the discrepancies between existing formulae and experimental data. DOI:10.1088/0256-307X/37/10/104401 PACS:44.10.+i, 66.25.+g, 66.70.-f © 2020 Chinese Physics Society Article Text Thermal conductivity is a fundamental physical quantity describing a material's capability of heat conduction. It is a macroscopic quantity reflecting the statistical properties of heat carriers such as phonons, electrons, or molecules at microscopic level. It has been a challenge to describe this macroscopic quantity in terms of only fundamental parameters specifying material characteristics. Although many works have been devoted to this problem, a unified formula for calculating the thermal conductivities of all these systems is still lacking, as the heat carriers are different in different states of matters. In this Letter, we first discuss the existing (different) formulae of thermal conductivity for various states at high temperatures, exemplified by gases, liquids, crystals, amorphous solids, and polymers. Then we present a unified formula for thermal conductivity of these material forms using only fundamental structural components. The thermal conductivity formulae proposed for different forms of materials are listed in Table 1. The $\kappa_{\rm G}$ is derived from the kinetic theory of gases (KTG),[1] where $\epsilon$ is a pure number, $\eta$ is the viscosity, and $C_{v}$ is the heat capacity at constant volume. Bridgman[2] proposed in 1923 an empirical formula $\kappa_{\rm L}$ to calculate the thermal conductivity of liquids, where $n_{\rm mole}=\rho/m_{\rm mole}$ is the number density of molecules, $\rho$ the mass density, and $m_{\rm mole} = M_{\rm mole}/ N_{\rm A}$ the mass of molecules. $k_{\rm B}$ is the Boltzmann constant, $v_{\rm s}$ the sound velocity, $N_{\rm A}$ the Avogadro constant, and $3k_{\rm B}$ the heat capacity per molecule. For $\kappa_{\rm L}$, Bridgman[2] assumed that the neighboring molecules form a simple cubic lattice. This model works well for liquids (such as methanol, carbon tetrachloride, chloroform, and water) and breaks down for liquids with complicated polyatomic molecules (such as liquid alkane). Later on, another formula of thermal conductivity of liquid was proposed by Eyring et al.[3] by taking into account the parts of gas-like and solid-like thermal conduction in liquids,[4] namely, combining $\kappa_{\rm G}$ and $\kappa_{\rm C}$ together. In non-metallic crystalline solids, heat carriers are phonons — the collective excitation of lattice vibrations.[5] By treating phonons as gas-like, the thermal conductivity is given by the kinematic formula $\kappa_{\rm C}$ in Table 1,[5–7] where $C$ is the phonon heat capacity, $\Lambda$ the mean free path (MFP), and $\bar{v}_{\rm s}=\frac{1}{3}(v_{\rm sl}+2v_{\rm st})$ the average sound velocity where $v_{\rm sl}$ and $v_{\rm st}$ are longitudinal and transverse sound velocities, respectively. The derivation of thermal conductivities in amorphous solids is much less rigorous because of the ill-defined phonons due to the lack of lattice periodicity. Kittel[8] employed $\kappa_{\rm C}$ to calculate the thermal conductivity of amorphous solids by assuming a nearly constant MFP ($\Lambda_{0}$) which is close to the size of unit cell. However, Kittel's attempt was not successful in amorphous polymers, because $\Lambda_{0}$ in polymers should be less than 1 Å to recover experimental data.[9] This value is much shorter than the bond length of molecules. Cahill and Pohl[10–12] extended this idea and proposed the concept of the minimum thermal conductivity (MTC) based on Einstein's theory.[13] Einstein[13] postulated the heat transfer as a random walk of energy elements between neighboring localized oscillators with 26 atomic neighbors. Cahill and Pohl[10–12] modified Einstein's theory by assuming a cluster of atoms as a vibrational “entity” where the size of cluster is one half of the phonon wavelength. At high temperatures, the MTC is reduced into a simple form of $\kappa_{\rm A}$, where $n_{\rm atom}=\rho/m_{\rm atom}$ is the number density of atoms, and $m_{\rm atom} = M_{\rm atom}/ N_{\rm A}$ is the average mass of atoms. The $\kappa_{\rm A}$ works well for many inorganic amorphous solids such as vitreous silica (v-SiO$_2$), whose atomic structure can be described by the continuous random network (CRN) model.[14] However, it significantly overestimates thermal conductivities of amorphous solids such as amorphous selenium (a-Se) composed of crown ring Se$_8$ molecules and polymeric chains of Se. Such molecular solids are not described by the CRN model. Very recently, a thermal resistance network (TRN) model was proposed by us to calculate the thermal conductivity of polymers whose structures can be described by the random coil model,[15] see $\kappa_{\rm P}$ in Table 1, where $R_{\rm inter}$ is the thermal resistance across contact points between molecular chains, i.e., the inter-chain resistance. $L$ is the length of molecular chains, $n_{\rm mole}L=\rho L/m_{\rm mole}=\rho l_{\rm unit}/m_{\rm unit}$. Note that $l_{\rm unit}$ and $m_{\rm unit}$ are length and mass of repeating unit, respectively.
Table 1. The heat carriers and the formulae for calculating thermal conductivity in different states of materials.
Heat carrier Thermal conductivity Ref.
Gas Molecule $\kappa_{\rm G}=\epsilon\eta C_{v}$  [1]
Liquid Vibration of molecule $\kappa_{\rm L}=3k_{\rm B}v_{\rm s}n_{\rm mole}^{{2}/{3}}$  [2,4]
Crystal Phonon $\kappa_{\rm C}=\frac{1}{3}C\bar{v}_{\rm s}\Lambda$  [5–7]
Amorphous solid Vibration of atom/molecule $\kappa_{\rm A}=1.2k_{\rm B}\bar{v}_{\rm s}n_{\rm atom}^{{2}/{3}}$  [10–13]
Polymer Vibration of molecule $\kappa_{\rm P}={\left(n_{\rm mole}L \right)}^{{1}/{2}} \frac{1}{R_{\rm inter}}$  [15]
It is worth pointing out that $\kappa_{\rm L}$ of liquids and $\kappa_{\rm A}$ of amorphous solids are analogous because they are proportional to $n_{\rm mole}^{2/3}$ and $n_{\rm atom}^{2/3}$, respectively, and include the sound velocity. One may ask: is there any common physical mechanism between them? Indeed, $\kappa_{\rm L}$ and $\kappa_{\rm A}$ can be rewritten as $\kappa_{\rm L} \propto C v_{\rm s}n_{\rm mole}^{-{1}/{3}}$ and $\kappa_{\rm A}\propto C \bar{v}_{\rm s}n_{\rm atom}^{-{1}/{3}}$, respectively, where $C \sim k_{\rm B}n_{\rm mole}$, and $\sim k_{\rm B}n_{\rm atom}$ at high temperatures. These two forms are basically the same as $\kappa_{\rm C}$ except a dimensionless prefactor 3 in $\kappa_{\rm L}$ and 1.2 in $\kappa_{\rm A}$, when choosing the MFP $\Lambda$ to be on the same order of magnitude of $n_{\rm mole}^{-{1}/{3}}$ and $n_{\rm atom}^{-{1}/{3}}$. Further comparison of $\kappa_{\rm P}$ with $\kappa_{\rm L}$ and $\kappa_{\rm A}$ makes it clear that the number density dependence of thermal conductivity in polymers $n_{\rm mole}^{1/2}$ is different from that in $\kappa_{\rm L}$ and $\kappa_{\rm A}$ and the sound velocity dependence vanishes. We note that the actual value of $\frac{(Ln_{\rm mole}^{-1/3})^{1/2}}{R_{\rm inter}}$ is of the same order of magnitude of $k_{\rm B}v_{\rm s}$ because $L$ and $n_{\rm mole}^{-1/3}$ are typically a few Angstroms and the thermal resistance $R_{\rm inter}$ is between $0.65$–$1.6 \times 10^{10}$ K/W from fitting results.[15] Therefore, $\frac{(Ln_{\rm mole}^{-1/3})^{1/2}}{R_{\rm inter}}\approx 10^{-20}$ Wm/K is very close to the value of $k_{\rm B}v_{\rm s}$. Based on the above analysis, we conjecture that $\kappa_{\rm L}$, $\kappa_{\rm A}$, and $\kappa_{\rm P}$ might be expressed as a single formula. The difference for different materials lies only in a prefactor. In the following, we shall derive such a formula from the fundamental heat carriers in these materials. We adopt Einstein's idea[13] that thermal transport is a random walk of heat through the coupling between vibrational modes.[16] Einstein[13] considered that the random walk occurs between an atom and its surrounding neighbors. The difference from our theory is: we postulate that molecules or clusters of atoms work as individual fundamental units in which the atomic arrangements are ordered or nearly ordered that maintaining short-range order. Then the random walk of heat through the network formed by these fundamental units plays a key role for the overall thermal conductivity. Our hypothesis is based on the fact that both liquids and amorphous solids possess short-range order only.[17] The size of fundamental units usually ranges from several Angstroms to several nanometers as demonstrated by x-ray diffraction measurements.[17] It should be emphasized that such a length scale of fundamental units is too small to define a local temperature distribution. It is natural that the thermal equilibrium in each fundamental unit is achieved very fast compared with overall one, not affecting the overall thermal resistance above room temperature. Thus, the thermal transport between neighboring fundamental units are governed by the interaction at the contact, for example, via weak van der Waals (VDW) interaction, H-bond, chemical bond, etc, where these units are regarded as phase uncorrelated oscillators. In complex network systems, heat conduction generally depends on the degree of networks (the average coordination number surrounding a constituent element).[18,19] We can regard fundamental units as nodes and the contact between fundamental units as links among nodes in analogy to the nodes-links-blobs model for complex network systems.[20] In this regard, the number of links per unit length along the direction of heat current is the cubic root of the number density of links. Then we can write the universal formula of thermal conductivity which is valid for a wide range of complex disordered systems, at high temperatures as $$ \kappa={\left(\frac{Z}{3\varXi}\tilde{n}\right)}^{{1}/{3}}h,~~ \tag {1} $$ where $Z$ is the degree of network (the average coordination numbers), 3 is the dimension, $\varXi$ is the number of nodes (fundamental units) sharing one link, and $\tilde{n}$ is the number density of nodes (fundamental units). $h$ is the heat conductance between blobs, which means the average energy flow across links per unit time per temperature drop. Its unit is J$\cdot$K$^{-1}$$\cdot$s$^{-1}$. The heat conductance can be written as $$ h=\frac{1}{\gamma}\frac{C_{\rm per} v_{\rm s}}{\delta},~~ \tag {2} $$ where $\gamma$ is the proportion of inter-molecular heat transfer, $C_{\rm per}$ is the heat capacity per particle whose high-temperature limit is $(\frac{3}{2}+\frac{D_{\rm v}}{2})k_{\rm B}$ with 3 in the numerator corresponding to the translational degree of freedom and $D_{\rm v}$ the average vibrational degree of freedom of particle. The value of $D_{\rm v}$ should be from 0 to 3. $\delta$ is the length of link, namely, the distance between neighboring atoms or functional groups (such as -CH$_{3}$ in paraffin or polymers) at contact points. Here we apply Eqs. (1) and (2) to calculate thermal conductivity of liquids and amorphous solids with different fundamental units in different network structures. Substances are categorized into three types according to the characteristics of fundamental units, instead of the matter state. The illustrations in Fig. 1 show different types of networks with different fundamental units. In liquids and amorphous solids made of simple molecules, such as alkane and paraffin, the fundamental units are individual molecules. In inorganic amorphous solids with covalent bonded atoms such as v-SiO$_2$, the fundamental unit is a dense cluster of atoms as marked by blue circles in Fig. 1(b). In polymers made of macromolecules, fundamental units are drawn by purple ellipse in Fig. 1(c), namely, the chain segments between adjacent contact points of different chains. The average segment length is defined by $\xi$. The fast thermal equilibrium achieved in individual small fundamental unit is indubitable for all cases above room temperature. Thus, the thermal transport can be described as a random walk from one fundamental unit to its touching neighbor units. At low temperatures, these two processes should be considered simultaneously. In this Letter, we focus only on high-temperature regime, where the equipartition theorem is valid and quantum effects are negligible. Low-temperature cases will be considered elsewhere.
cpl-37-10-104401-fig1.png
Fig. 1. The schematic illustration of different types of fundamental units transferring heat in liquids and amorphous solids and the prefactor $\alpha$ in Eq. (3). (a) The fundamental unit for liquid or molecular amorphous solid is individual molecule. (b) The fundamental unit for covalent amorphous solid is dense cluster (shaded areas in blue). $V_{\rm cl}$ is the volume of atomic cluster. (c) The fundamental unit for polymer is a chain segment (shaded areas in purple).
Table 2. Selections of fundamental units and corresponding parameters for different types of materials.
Material type Fundamental unit $\tilde{n}$ Typical size $\varXi $ $\gamma$ $D_{\rm v}$ $\delta$
of fundamental unit
Molecular liquids and solids Nearly-spherical molecules $=n_{\rm mole}$ $a\times a\times a$ $2 $ 1 1–3 $= a\approx n_{\rm mole}^{-1/3}$
Molecular liquids and solids Capsule-like molecules $=n_{\rm mole}$ $L\times a\times a$ $2 $ 1 1 $\approx a < n_{\rm mole}^{-1/3}$
Molecular liquids and solids Disk-like molecules $=n_{\rm mole}$ $L\times L\times a$ $2 $ 1 1 $\approx a < n_{\rm mole}^{-1/3}$
Covalently bonded amorphous solids Atomic clusters $\ll n_{\rm atom}$ $\zeta\times\zeta\times\zeta$ $2 $ 1 2 $\approx b < n_{\rm atom}^{-1/3}$
Polymer liquids and solids Chain segments $\gg n_{\rm mole}$ $\xi\times a\times a$ $4 $ 1/2 1 $=a^{\prime} \ll n_{\rm mole}^{-1/3}$
By properly taking the parameters for different fundamental units, Eq. (1) reduces to $$ \kappa=\alpha\frac{1}{\delta}\left(\frac{3}{2}+\frac{D_{\rm v}}{2}\right) k_{\rm B}v_{\rm s},~~ \tag {3} $$ where the prefactor $\alpha$ differs from state to state as presented in Fig. 1. Its physical meaning is given in the Supplementary Information. The fundamental units for different types of liquids and amorphous solids are summarized in Table 2. The parameters $\varXi$, $\gamma$, $D_{\rm v}$, and $\delta$ are also presented (see details in Supplementary Information).
cpl-37-10-104401-fig2.png
Fig. 2. Comparison of calculated thermal conductivities from Eq. (3) with experimental ones. The diagonal (dotted) line is given to guide for eyes. Symbol lies on this line means calculated thermal conductivity has the same value with experimental one. (a) Calculated thermal conductivities (black symbols) of thirteen different liquids at 300 K versus the experimental values. $D_{\rm v}$ is chosen to be 1–3 for nearly spherical molecules (methanol, carbon tetrachloride, benzene, chloroform, and water) and $D_{\rm v}=1$ for non-spherical molecules. The experimental data of thermal conductivities are taken from Refs. [2,21]. Thermal conductivities calculated from the Bridgman formula (red symbols) are also given for comparison. (b) Calculated thermal conductivities (black symbols) of ten different alkane liquids and paraffin at 300 K versus the observed values when $D_{\rm v}=1$. The calculated values of $Z$ are listed in Supplementary Information. The experimental thermal conductivities are taken from Ref. [21]. Thermal conductivity calculated according to the Bridgman formula (red symbols) are also given for comparison. (c) Calculated thermal conductivities (black symbols) for six different inorganic amorphous solids at 300 K versus the observed values when $D_{\rm v}=2$ except for a-As$_{2}$ S$_3$ and a-Se, in which case we take As$_{2}$ S$_3$ and Se$_8$ molecules as fundamental units with $5 < Z < 7$ and $D_{\rm v}=1$. The experimental thermal conductivities are taken from Ref. [11]. Thermal conductivity calculated using the MTC formula (blue symbols) are also given for comparison.[11] (d) Calculated thermal conductivities (black symbols) for six polymers and a-Se composed of 100$\%$ polymeric chains at 300 K versus the experimental values when $D_{\rm v}=1$. The experimental data are taken from Ref. [22].
We shall point out that the number density of fundamental units is the key parameter in determining the thermal conductivity, as can be seen from Eq. (1). When the fundamental unit is the entire small molecule, $\tilde{n}$ can be replace by $n_{\rm mole}$. When the molecules are large as in polymers, $\tilde{n}$ should be much larger than $n_{\rm mole}$ by choosing chain segments as fundamental units which contain tens of atoms. Similarly, $\tilde{n}$ should be the number density of atomic clusters which contains tens of atoms in covalently bonded amorphous solids. It needs to be cautious when applying Eq. (3) to polymer glasses consisting of extremely long chains, as it is assumed that fundamental units (polymer segments) reach to the thermal equilibrium within the time scale $\tau_{\rm C}$, faster than the time-scale $\tau_{\rm V}$ of the heat conductance between those units. Namely, the situation $\tau_{\rm C}/ \tau_{\rm V} < 1$ gives the criterion for the applicability of Eq. (3) to polymer glass. Since the polymer segments execute coherent vibrations at finite temperatures, its characteristic time scale is considered to be connected with the strength of covalent bonding in polymer chain, while the time scale $\tau_{\rm V}$ is governed by the strength of VDW force. The ratio can be estimated from the strengths of covalent bonding in polymer chains and VDW coupling between chain segments, which is around 100. This means that Eq. (3) is applicable for polymer glasses consisting of polymer segments within around 100 times of periodic unit of polymer chains. Our formula is in excellent agreement with the experimental data when $1\le D_{\rm v}\le 3$ for all substances. We further find that $D_{\rm v}$ is reduced to 1 when the shape of molecules is capsule-like or disk-like. This is due to the shape effect on the liquid relaxation process. In covalently bonded amorphous solids, the value of $D_{\rm v}$ is found to be 2. Possible explanation is the strongly distorted atomic structure at the edge of fundamental unit where the internal stress shifts from negative value to positive one.[23] Another major difference of our model from the Bridgman's formula or the MTC one is the intermolecular/interatomic separation $\delta$. The Bridgman's formula uses $n^{-1/3}_{\rm mole}$ as the intermolecular separation. This is valid only for simple liquids with spherical-like molecules because $a\approx n_{\rm mole}^{-1/3}$, where $a$ is approximately equal to the length scale of the occupying space of molecules/atoms. When the molecule is non-spherical, for example, Se$_8$ is disk-like and acetone is capsule-like, the intermolecular separation is not the distance between the centers of adjacent molecules. The distance should be the separation between contact atoms or functional groups. Therefore, the value of $\delta$ is approximately $a$ which is obviously much smaller than $n^{-1/3}_{\rm mole}$. In covalently bonded amorphous solids, it is convenient to chose $\delta$ to be the bond length, $b$, which is smaller than $n^{-1/3}_{\rm atom}$. When the molecule is extremely long in polymer, $\delta$ is the average value of the size of atoms or functional groups at the contact points, $a^{\prime}$. In Fig. 2, we compare the calculated thermal conductivities from our formula with experimental values for various liquids, molecular solids, inorganic amorphous solids, and polymers. The symbols given by black are calculated results obtained from Eq. (3), and the diagonal (dotted) lines are used to guide eyes to judge the agreement between calculated and experimental thermal conductivities. It shows that our formula agrees very well with the experimental data. For more comprehensive analysis, please refer to the Supplementary Information. Equation (3) holds in the temperature regime that fundamental units execute incoherent/independent vibrations respectively. This situation is achieved at high temperatures where the measured thermal conductivities are saturated or show weak temperature dependence. This temperature regime has been universally observed for a wide range of complex disordered materials. We note that Eq. (3) recovers the weak temperature dependence of $\kappa$ via the temperature dependences of macroscopic parameters included in Eq. (3) such as mass density, distance between neighboring fundamental units, and sound velocity. We shall emphasize here that thermal conductivities are highly dependent on the detailed atomic/molecular structure of fundamental unit. We further point out that the mechanism of thermal conductivity of liquids and non-crystalline solids has the same origin. For heat transport, liquids should behave as amorphous solids rather than as fluids, when the oscillation period $1/\omega$ of atoms in liquids is much shorter than their structure relaxation time $\tau$, namely, $\omega\tau>1$.[24–26] In summary, by introducing the concept of the fundamental units characterizing the types of complex disordered systems such as liquids, amorphous solids, and amorphous polymers, we have proposed a ubiquitous formula for thermal conductivity that applicable for both liquids and amorphous solids. With properly defined fundamental units, we have demonstrated that the thermal conductivity is dominated by the thermal resistance between neighboring fundamental units. This is because the fundamental units reach thermal equilibrium much faster than the heat transfer between units. Our formula of Eq. (1) not only provides a deeper physical understanding compared with the existing well-accepted empirical/theoretical ones, but also resolves the discrepancies between these formulae and experiment measurements. References The Thermal Conductivity of Liquids under PressureThermal Conductivity of LiquidsDiffusion, Thermal Conductivity, and Viscous Flow of LiquidsZur kinetischen Theorie der Wärmeleitung in KristallenInterpretation of the Thermal Conductivity of GlassesHeat flow and lattice vibrations in glassesLower limit to the thermal conductivity of disordered crystalsHigh and low thermal conductivity of amorphous macromoleculesElementare Betrachtungen über die thermische Molekularbewegung in festen KörpernThermal resistance network model for heat conduction of amorphous polymersUnified theory of thermal transport in crystals and glassesHeat flux distribution and rectification of complex networksThermal-siphon phenomenon and thermal/electric conduction in complex networksDynamical properties of fractal networks: Scaling, numerical simulations, and physical realizationsAmorphous solids: their structure, lattice dynamics and elasticityDuality of liquidsColloquium : The glass transition and elastic models of glass-forming liquids
[1]Loeb L B 1927 Kinetic Theory of Gases (New York: McGraw-Hill Book Co., Inc.)
[2] Bridgman P W 1923 Proc. Am. Acad. Arts Sci. 59 141
[3] Lin S H, Eyring H and Davis W J 1964 J. Phys. Chem. 68 3017
[4] Powell R E, Roseveare W E and Eyring H 1941 Ind. Eng. Chem. 33 430
[5]Kittel C 2005 Introduction to Solid State Physics 8th edn (New York: John Wiley & Sons)
[6]Debye P J W 1914 Vorträgeüber die Kinetische Theorie der Materie und der Elektrizität (Leipzig: Teubner B G)
[7] Peierls R 1929 Ann. Phys. 395 1055
[8] Kittel C 1949 Phys. Rev. 75 972
[9]Van Krevelen D W and Te Nijenhuis K 2009 Properties of Polymers 4th edn (Amsterdam: Elsevier)
[10] Cahill D G and Pohl R O 1989 Solid State Commun. 70 927
[11] Cahill D G, Watson S K and Pohl R O 1992 Phys. Rev. B 46 6131
[12] Xie X, Yang K, Li D, Tsai T H, Shin J, Braun P V and Cahill D G 2017 Phys. Rev. B 95 035406
[13] Einstein A 1911 Ann. Phys. 340 679
[14]Zallen R 1998 The Physics of Amorphous Solids (New York: John Wiley & Sons) pp 107, 133
[15] Zhou J, Xi Q, He J, Nakayama T, Wang Y Y and Liu J 2020 Phys. Rev. Mater. 4 015601
[16] Simoncelli M, Marzari N and Mauri F 2019 Nat. Phys. 15 809
[17]Stachurski Z H 2015 Fundamentals of Amorphous Solids: Structure, Properties (Beijing: Higher Education Press) pp 23, 116
[18] Liu Z, Wu X, Yang H, Gupte N and Li B 2010 New J. Phys. 12 023016
[19] Xiong K, Liu Z, Zeng C and Li B 2020 Natl. Sci. Rev. 7 270
[20] Nakayama T, Yakubo K and Orbach R L 1994 Rev. Mod. Phys. 66 381
[21]Yaws C L 1995 Handbook of Thermal Conductivity (Houston: Gulf Publishing Company) vol 1–3
[22]Mark J E 2009 Polymer Data Handbook (Oxford: Oxford University Press)
[23] Alexander S 1998 Phys. Rep. 296 65
[24] Trachenko K and Brazhkin V V 2013 Sci. Rep. 3 2188
[25]Frenkel J 1947 Kinetic Theory of Liquids (Oxford: Oxford University Press)
[26] Dyre J C 2006 Rev. Mod. Phys. 78 953