A Ubiquitous Thermal Conductivity Formula for Liquids, Polymer Glass, and Amorphous Solids^*

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 κ_G is derived from the kinetic theory of gases (KTG),^[1] where is a pure number, η is the viscosity, and C_v is the heat capacity at constant volume. Bridgman^[2] proposed in 1923 an empirical formula κ_L to calculate the thermal conductivity of liquids, where n_mole = ρ/m_mole is the number density of molecules, ρ the mass density, and m_mole = M_mole/N_A the mass of molecules. k_B is the Boltzmann constant, v_s the sound velocity, N_A the Avogadro constant, and 3k_B the heat capacity per molecule. For κ_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 κ_G and κ_C together.

Table 1. The heat carriers and the formulae for calculating thermal conductivity in different states of materials.

	Heat carrier	Thermal conductivity	Ref.
Gas	Molecule	κG = ηCv	[1]
Liquid	Vibration of molecule	${\kappa }_{{\rm{L}}}=3{k}_{{\rm{B}}}{v}_{{\rm{s}}}{n}_{{\rm{mole}}}^{2/3}$	[2,4]
Crystal	Phonon	${\kappa }_{{\rm{C}}}=\displaystyle \frac{1}{3}C{\bar{v}}_{{\rm{s}}}\Lambda$	[5–7]
Amorphous solid	Vibration of atom/molecule	${\kappa }_{{\rm{A}}}=1.2{k}_{{\rm{B}}}{\bar{v}}_{{\rm{s}}}{n}_{{\rm{atom}}}^{2/3}$	[10–13]
Polymer	Vibration of molecule	${\kappa }_{{\rm{P}}}={({n}_{{\rm{mole}}}L)}^{1/2}\displaystyle \frac{1}{{R}_{{\rm{inter}}}}$	[15]

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 κ_C in Table 1,^[5–7] where C is the phonon heat capacity, Λ the mean free path (MFP), and ${\bar{v}}_{{\rm{s}}}=\displaystyle \frac{1}{3}({v}_{{\rm{sl}}}+2{v}_{{\rm{st}}})$ the average sound velocity where v_sl and v_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 κ_C to calculate the thermal conductivity of amorphous solids by assuming a nearly constant MFP (Λ₀) which is close to the size of unit cell. However, Kittel's attempt was not successful in amorphous polymers, because Λ₀ 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 κ_A, where n_atom = ρ/m_atom is the number density of atoms, and m_atom = M_atom/N_A is the average mass of atoms. The κ_A works well for many inorganic amorphous solids such as vitreous silica (v-SiO₂), 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 Se₈ 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 κ_P in Table 1, where R_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_moleL = ρ L/m_mole = ρ l_unit/m_unit. Note that l_unit and m_unit are length and mass of repeating unit, respectively.

It is worth pointing out that κ_L of liquids and κ_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, κ_L and κ_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 ∼ k_Bn_mole, and ∼ k_Bn_atom at high temperatures. These two forms are basically the same as κ_C except a dimensionless prefactor 3 in κ_L and 1.2 in κ_A, when choosing the MFP Λ to be on the same order of magnitude of ${n}_{{\rm{mole}}}^{-1/3}$ and ${n}_{{\rm{atom}}}^{-1/3}$.

Further comparison of κ_P with κ_L and κ_A makes it clear that the number density dependence of thermal conductivity in polymers ${n}_{{\rm{mole}}}^{1/2}$ is different from that in κ_L and κ_A and the sound velocity dependence vanishes. We note that the actual value of $\displaystyle \frac{{(L{n}_{{\rm{mole}}}^{-1/3})}^{1/2}}{{R}_{{\rm{inter}}}}$ is of the same order of magnitude of k_Bv_s because L and ${n}_{{\rm{mole}}}^{-1/3}$ are typically a few Angstroms and the thermal resistance R_inter is between 0.65 – 1.6 × 10¹⁰ K/W from fitting results.^[15] Therefore, $\displaystyle \frac{{(L{n}_{{\rm{mole}}}^{-1/3})}^{1/2}}{{R}_{{\rm{inter}}}}\approx {10}^{-20}$ Wm/K is very close to the value of k_Bv_s.

Based on the above analysis, we conjecture that κ_L, κ_A, and κ_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

$$\begin{eqnarray}&&\kappa ={\left(\displaystyle \frac{Z}{3\varXi }\tilde{n}\right)}^{1/3}h,\end{eqnarray} \tag{ 1 }$$

where Z is the degree of network (the average coordination numbers), 3 is the dimension, Ξ 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 ⋅ K⁻¹ ⋅ s⁻¹. The heat conductance can be written as

$$\begin{eqnarray}&&h=\displaystyle \frac{1}{\gamma }\displaystyle \frac{{C}_{{\rm{per}}}{v}_{{\rm{s}}}}{\delta },\end{eqnarray} \tag{ 2 }$$

where γ is the proportion of inter-molecular heat transfer, C_per is the heat capacity per particle whose high-temperature limit is $(\displaystyle \frac{3}{2}+\displaystyle \frac{{D}_{{\rm{v}}}}{2}){k}_{{\rm{B}}}$ with 3 in the numerator corresponding to the translational degree of freedom and D_v the average vibrational degree of freedom of particle. The value of D_v should be from 0 to 3. δ is the length of link, namely, the distance between neighboring atoms or functional groups (such as -CH₃ 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-SiO₂, 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 ξ. 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.

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By properly taking the parameters for different fundamental units, Eq. (1) reduces to

$$\begin{eqnarray}&&\kappa =\alpha \displaystyle \frac{1}{\delta }\left(\displaystyle \frac{3}{2}+\displaystyle \frac{{D}_{{\rm{v}}}}{2}\right){k}_{{\rm{B}}}{v}_{{\rm{s}}},\end{eqnarray} \tag{ 3 }$$

where the prefactor α 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 Ξ, γ, D_v, and δ are also presented (see details in Supplementary Information).

Table 2. Selections of fundamental units and corresponding parameters for different types of materials.

Material type	Fundamental unit	$\tilde{n}$	Typical size of fundamental unit	Ξ	γ	Dv	δ
Molecular liquids and solids	Nearly-spherical molecules	=nmole	a × a × a	2	1	1–3	$=a\approx {n}_{{\rm{mole}}}^{-1/3}$
Molecular liquids and solids	Capsule-like molecules	=nmole	L × a × a	2	1	1	$\approx a\lt {n}_{{\rm{mole}}}^{-1/3}$
Molecular liquids and solids	Disk-like molecules	=nmole	L × L × a	2	1	1	$\approx a\lt {n}_{{\rm{mole}}}^{-1/3}$
Covalently bonded amorphous solids	Atomic clusters	≪natom	ζ × ζ × ζ	2	1	2	$\approx b\lt {n}_{{\rm{atom}}}^{-1/3}$
Polymer liquids and solids	Chain segments	≫nmole	ξ × a × a	4	1/2	1	$={a}^{^{\prime} }\ll {n}_{{\rm{mole}}}^{-1/3}$

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_mole. When the molecules are large as in polymers, $\tilde{n}$ should be much larger than n_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 τ_C, faster than the time-scale τ_V of the heat conductance between those units. Namely, the situation τ_C/τ_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 τ_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 ≤ D_v ≤ 3 for all substances. We further find that D_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_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 δ. The Bridgman's formula uses ${n}_{{\rm{mole}}}^{-1/3}$ 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, Se₈ 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 δ is approximately a which is obviously much smaller than ${n}_{{\rm{mole}}}^{-1/3}$. In covalently bonded amorphous solids, it is convenient to chose δ to be the bond length, b, which is smaller than ${n}_{{\rm{atom}}}^{-1/3}$. When the molecule is extremely long in polymer, δ is the average value of the size of atoms or functional groups at the contact points, a'.

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.

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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 κ 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/ω of atoms in liquids is much shorter than their structure relaxation time τ, namely, ω τ > 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.