# Anomalous Temperature-Dependent Thermal Transport in Crystalline Polyethylene Driven by Strong Anharmonicity

Shouhang Li, Philippe Dollfus, Jérôme Saint-Martin, Davide Romanin

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Shouhang Li, Philippe Dollfus, Jérôme Saint-Martin, Davide Romanin. Anomalous Temperature-Dependent Thermal Transport in Crystalline Polyethylene Driven by Strong Anharmonicity. Journal of Physical Chemistry Letters, 2025, 16 (39), pp.10210-10215. 10.1021/acs.jpclett.5c02084. hal-05284461

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<figure><img src="image_1.png" /></figure>
Anomalous Temperature-Dependent Thermal
Transport in Crystalline Polyethylene Driven by
Strong Anharmonicity

Shouhang Li,\*,† Philippe Dollfus,† Jérôme Saint-Martin,†,‡ and Davide Romanin†

†Centre de Nanosciences et de Nanotechnologies, CNRS, Université Paris-Saclay, 10

Boulevard Thomas Gobert, Palaiseau 91120, France

‡ SATIE, CNRS, ENS Paris-Saclay, Université Paris-Saclay, 91190 Gif-sur-Yvette, France

E-mail: shouhang.li@universite-paris-saclay.fr
Abstract

Thermal conductivity typically decreases with increasing temperature along the three principal crystalline directions, primarily due to enhanced phonon anharmonicity. In this work, we conducted a comprehensive first-principles investigation of thermal transport in crystalline polyethylene by solving the Wigner transport equation, assisted with the stochastic self-consistent harmonic approximation. It is found that the thermal conductivity of crystalline polyethylene decreases along the chain direction, but increases nearly linearly in the out-of-chain directions. This anomalous contrasting behavior stems from the dominance of particle-like transport along the chain and wave-like transport in the out-of-chain directions. The strong anharmonicity facilitates phonon tunneling between high- and low-frequency modes in the out-of-plane directions. Therefore, further enhancement of thermal conductivity in those directions could benefit from increased anharmonicity and the introduction of additional disorder. These findings provide fundamental insights into the thermal transport mechanisms of anisotropic crystalline polymers, offering valuable guidance for rationally engineering their thermal properties.

**KEYWORDS:** thermal conductivity, particle-like transport, wave-like transport, quantum anharmonic effects, polyethylene
Bulk polymers are typically regarded as thermal insulators, exhibiting room-temperature thermal conductivities ($\kappa$) on the order of ~0.2 W/mK.¹ In contrast, many inorganic materials possess much higher thermal conductivities, often exceeding 100 W/mK.² However, although thermally conductive inorganic fillers are commonly incorporated to enhance the $\kappa$ of amorphous polymers,³ they suffer from drawbacks such as high cost and degradation of the mechanical and dielectric properties. This highlights the urgent need for polymers with intrinsically high thermal conductivity. As a matter of fact, it has been demonstrated that the intrinsic $\kappa$ of polyethylene (PE) can be enhanced by one to three orders of magnitude through the formation of highly oriented structures using gel spinning⁴ and ultradrawing techniques.⁵,⁶ These advances open new opportunities for the integration of polymers in thermal management applications.

Above 100 K, $\kappa$ of amorphous polymers either increases with temperature or remains nearly constant.⁷ Historically, this behavior has been attributed to the temperature-dependent nature of heat capacity,⁸ and more recently to contributions from non-propagating vibrational modes.⁹,¹⁰ In contrast, the $\kappa$ along the chain direction of oriented PE exhibits a $\kappa \sim 1/T$ dependence, similar to that observed in thermally conductive inorganic crystals, indicating phonon-dominated transport.¹¹ This behavior has been further supported by molecular dynamics (MD) simulations¹² and density functional theory (DFT) calculations.¹³,¹⁴ However, thermal transport in the out-of-chain directions of crystalline polyethylene remains underexplored due to measurement challenges. While DFT calculations have predicted a $\kappa \sim 1/T$ scaling in the out-of-plane directions as well,¹¹,¹³ such conclusions are questionable in light of recent experimental and theoretical results showing glass-like behavior of $\kappa$ in materials with weak interatomic interactions.¹⁵⁻¹⁸

While DFT has proven highly effective for predicting the lattice thermal conductivity of both dielectrics¹⁹,²⁰ and metals,²¹,²² its application to crystalline polymers remains limited. The large unit cell, often containing more than ten atoms, makes DFT calculations computationally demanding. Moreover, the harmonic approximation used to extract interatomic force
constants is inadequate for polymers, which typically exhibit much stronger anharmonicity than conventional crystalline solids. The abundance of light atoms (i.e. hydrogen and carbon) further amplifies quantum nuclear effects, which must be properly treated.²³ These factors likely contribute to the significant overestimation of thermal conductivity in DFT-based predictions¹⁴ compared to results from the time-domain thermoreflectance (TDTR) experiment,¹¹ the steady-state measurement,⁶ and MD simulations.¹²,²⁴ Although MD simulations offer a practical alternative, they are inherently classical and thus fail to capture quantum nuclear effects. Additionally, the reliance on empirical force fields introduces further uncertainty, limiting their predictive accuracy for crystalline polymers.

In this Letter, we employed the stochastic self-consistent harmonic approximation (SSCHA)²⁵ combined with the Wigner transport equation (WTE)¹⁷,²⁶ to investigate lattice vibrations and thermal transport in crystalline polyethylene (PE). The SSCHA fully incorporates thermal, quantum, and anharmonicity effects in the evaluation of interatomic force constants, while the WTE unifies the particle-like and the wave-like thermal transport in complex crystals. As a result, our predicted thermal conductivity along the chain direction quantitatively agrees with the steady-state experimental value of 62 W/mK obtained from high-quality samples.⁶ Moreover, we find that thermal transport in PE exhibits a hybrid character, sharing features of both crystalline and amorphous materials. Specifically, the chain-direction thermal conductivity ($\kappa^{zz}$) is dominated by particle-like transport, whereas the thermal conductivities in out-of-chain directions ($\kappa^{xx}$ and $\kappa^{yy}$) are governed by wave-like mechanisms. While $\kappa^{zz}$ decreases rapidly with increasing temperature, both $\kappa^{xx}$ and $\kappa^{yy}$ increase almost linearly over the entire temperature range. This anomalous temperature dependence is attributed to the distinct impact of anharmonicity on heat transport along different crystallographic directions.

*First principles theory.*-Given that lightweight hydrogen atoms constitute the majority of atoms in polyethylene (PE), quantum nuclear effects are significant, as emphasized in Ref. ¹³,²³ To accurately capture these effects, we extracted the second-order (2nd-FCs) and
third-order (3rd-FCs) force constants using the stochastic self-consistent harmonic approximation (SSCHA).<sup>25</sup> The 2nd-FCs were obtained through free energy minimization within the SSCHA, while the 3rd-FCs are extracted via stochastic sampling of atomic displacements. DFT calculations were performed with the Perdew-Burke-Ernzerhof (PBE) functional<sup>27</sup> and norm-conserving pseudopotentials from the PseudoDojo repository.<sup>28</sup> We employed a plane-wave energy cutoff of 80 Ry and a 6 × 8 × 12 Monkhorst-Pack k-point mesh.

In the SSCHA calculations, the initial dynamical matrices were obtained from density functional perturbation theory (DFPT). For each T considered, a set of random ionic configurations were generated in the supercell of 2 × 3 × 5 according to the Gaussian ionic probability distribution, with zero-point motion incorporated into the atomic displacements. The temperature-dependent lattice structures and dynamical matrices were obtained by minimizing the free energy within SSCHA. The thermal effects have marginal effects on lattice parameters, and they were not optimized in the SSCHA calculations. To mitigate the computational burden in DFT calculations, we adopted the neural equivariant interatomic potential (NequIP)<sup>29</sup> to evaluate the energies and forces of the atomic configurations. This approach significantly accelerates SSCHA and Hessian calculations while retaining accuracy comparable to DFT, as demonstrated in Section II of the Supplemental Material. To ensure convergence, We employed 8×10<sup>3</sup> configurations for each SSCHA population, and 6×10<sup>4</sup> configurations for calculating the Hessian of the free energy and the third-order force constants. Convergence details are provided in Section III of the Supplemental Material.

In the WTE framework, the thermal conductivity $\kappa$ consists of a particle-like term ($\kappa_P$) and a wave-like term ($\kappa_C$, also referred to as the coherence term), such that $\kappa_{tot} = \kappa_P + \kappa_C$.<sup>17</sup> When only the diagonal elements of the velocity operator are considered, the WTE reduces to the Boltzmann transport equation (BTE). In this limit, the particle-like contribution $\kappa_P^{\alpha\beta}$ is given by

$$ \kappa_P^{\alpha\beta} = \frac{1}{\Omega N_q} \sum_{\lambda} C_{\lambda} v_{\lambda}^{\alpha} v_{\lambda}^{\beta} \tau_{\lambda}, \quad (1) $$
where $\alpha$ and $\beta$ are Cartesian indices, $\Omega$ is the volume of the primitive cell, and $N_q$ is the number of q-points used in Brillouin zone sampling. The index $\lambda \equiv (q, \nu)$ denotes a phonon mode with wave vector $q$ and polarization $\nu$. Here, $C_\lambda$, $v_\lambda$, and $\tau_\lambda$ are the phonon mode's heat capacity, group velocity, and relaxation time, respectively.30,31

The wave-like contribution $\kappa_C$ arises from off-diagonal elements ($\nu \neq \nu'$) of the velocity operator and is described in Ref.,17 as

$$ \kappa_C^{\alpha\beta} = \frac{\hbar^2}{k_B T^2 \Omega N_q} \sum_{q} \sum_{\nu \neq \nu'} \frac{\omega_\lambda + \omega_{\lambda'}}{2} V_{q,\nu,\nu'}^\alpha V_{q,\nu,\nu'}^\beta \\ \times \frac{\omega_\lambda n_\lambda(n_\lambda + 1) + \omega_{\lambda'} n_{\lambda'}(n_{\lambda'} + 1)}{4(\omega_\lambda - \omega_{\lambda'})^2 + (\Gamma_\lambda + \Gamma_{\lambda'})^2} \times (\Gamma_\lambda + \Gamma_{\lambda'}) \quad (2) $$

where $\hbar$ is the reduced Planck constant and $k_B$ is the Boltzmann constant. $\omega_\lambda$ and $n_\lambda$ denote the phonon frequency and the Bose-Einstein distribution, respectively. $V_{q,\nu,\nu'}$ is the velocity operator which captures the coupling strength between phonon modes $q\nu$ and $q\nu'$.32 The phonon linewidth $\Gamma_\lambda$ is the inverse of the relaxation time, i.e., $\Gamma_\lambda = 1/\tau_\lambda$.

We adopted the direct solution approach to calculate $\kappa_P$,33 which is particularly important along the chain direction, where normal phonon-phonon scattering processes dominate, as shown in Section IV of the Supplemental Material. Both $\kappa_P$ and $\kappa_C$ were evaluated using the phono3py package.34 Isotope disorder was included in all calculations. Brillouin zone integrations were carried out using the tetrahedron method. Convergence with respect to the q-point mesh is discussed in Section III of the Supplemental Material.

*The anisotropic lattice structure and lattice vibration.*- The lattice structure of the crystalline PE is depicted in Figure 1. The primitive cell contains two PE chains and 12 atoms. The optimized lattice constants from DFT are $a = 6.9302\AA$, $b = 4.8273\AA$, and $c = 2.5564\AA$, in good agreement with previous reports.14 Along the chain direction, interatomic interactions are governed by strong covalent Carbon–Carbon bonds. In contrast, there are no chemical bonds in the out-of-chain directions, where the interactions are dominated by weak van der Waals (vdW) forces. This stark anisotropy in bonding strength is further illustrated by the
electron localization function (ELF), shown in Figure 1(b). The ELF exhibits delocalized behavior along the chain direction, while it nearly vanishes between polymer chains along out-of-chain directions.

<figure><img src="image_2.png" /><figcaption>Figure 1: (a) Crystal structure of polyethylene (PE), with four carbon atoms and eight hydrogen atoms per unit cell. (b) Electron localization function (ELF) projected along the chain direction [001] and the transverse direction [110]. (c) Phonon dispersion relations obtained from density functional perturbation theory (harmonic) and from SSCHA incorporating the QA effects at 300 K in the bubble approximation.²⁵ For clarity, only phonon branches below 20 THz are shown. The inset highlights the region near the Γ-point indicated by the green box. The square dots and circle dots are Experimental data taken from Ref. ³⁵ and Ref. ³⁶ respectively.</figcaption></figure>

The strong anisotropy in atomic interactions gives rise to pronounced anisotropy in lattice dynamics, as shown in Figure 1. There are 36 phonon branches in total, and we only present the lowest 8 branches here, which have significant contributions to thermal transport. The full phonon dispersion can be found in Section V of the Supplemental Material. The longitudinal acoustic (LA) mode along the chain direction (z-axis) reaches a maximum frequency of ~15THz, whereas the corresponding mode in the out-of-chain direction (x-axis)
remains below 3THz. Along the z-axis, the three acoustic branches and several optical branches exhibit significant dispersion, indicative of strong interatomic bonding. In contrast, all phonon modes along the x-axis are nearly flat and clustered, reflecting the weak van der Waals interactions. Overall, the dispersive branches calculated from DFPT (harmonic approximation) are in good agreement with those obtained by incorporating quantum anharmonic (QA) effects, and both match well with experimental measurements. However, for the flat modes, DFPT and QA results diverge significantly, as highlighted in the inset of Figure 1(c), suggesting that QA effects are particularly pronounced for these low-dispersive phonon branches.

*Anomalous thermal transport.*-The temperature-dependent total thermal conductivity is shown in Figure 2. The total thermal conductivity along the chain direction, $\kappa_{\text{tot}}^{zz}$, exhibits a marked decrease with temperature, consistent with the TDTR experiment.¹¹ At 300 K, $\kappa_{\text{tot}}^{zz}$ is 66.7 W/mK, in good agreement with values reported for samples with excellent crystallite orientation and crystallinity.⁶ Our results are also consistent with those from MD simulations,¹²,²⁴ but are notably lower than those predicted by conventional DFT calculations¹⁴ that neglect quantum nuclear effects. This phenomenon is also emphasized in Ref. ¹³ Above 300 K, $\kappa_{\text{tot}}^{zz}$ follows a temperature dependence of $\kappa_{\text{tot}}^{zz} \sim T^{-1.23}$, slightly deviating from the $\kappa_{\text{tot}}^{zz} \sim T^{-1.0}$ behavior observed in Ref. ¹¹ This weaker temperature dependence in Ref. ¹¹ can be attributed to the presence of transition and amorphous segments in commercial samples⁶ which introduce static phonon scattering.

The total thermal conductivities in the out-of-chain directions ($\kappa_{\text{tot}}^{xx}$ and $\kappa_{\text{tot}}^{yy}$) exhibit a behavior distinct from that of $\kappa_{\text{tot}}^{zz}$. As a matter of fact, both $\kappa_{\text{tot}}^{xx}$ and $\kappa_{\text{tot}}^{yy}$ increase almost linearly with temperature. The relatively smaller increase at low temperatures is attributed to the particle-like term which contributes significantly to $\kappa_{\text{tot}}$ at low temperatures but diminishes as the temperature increases. This behavior of $\kappa_{\text{tot}}^{xx}$ and $\kappa_{\text{tot}}^{yy}$ contrasts with the reports in Refs. ¹⁴ and, ¹³ where only the particle-like term was considered by solving the phonon BTE. The anomalous temperature dependence observed in $\kappa_{\text{tot}}^{xx}$ and $\kappa_{\text{tot}}^{yy}$ is due to
the wave-like term. As shown in Figure 2(b), the wave-like thermal conductivities in the out-of-chain directions ($\kappa_C^{xx}$ and $\kappa_C^{yy}$) increase linearly with temperature, as the phonon lifetime decreases with temperature, promoting phonon tunneling processes, as shown in Section VI of the Supplemental Material. The wave-like thermal conductivity along the chain direction ($\kappa_C^{zz}$) initially increases at low temperatures and then decreases at higher temperatures. This behavior is primarily due to the increasing anharmonicity of low-frequency phonons at elevated temperatures, which limits $\kappa_C^{zz}$. Although $\kappa_C^{zz}$ is significantly larger than $\kappa_C^{xx}$ and $\kappa_C^{yy}$, it remains much smaller than $\kappa_P^{zz}$ and has little impact on the temperature dependence of $\kappa_{tot}^{zz}$, as shown in Figure 2(a).

<figure><img src="image_3.png" /><figcaption>Figure 2: (a) Total thermal conductivity along the chain direction. The inset shows the total thermal conductivities in the out-of-chain directions. The triangles ($\bigtriangledown$) and diamonds ($\diamond$) are experimental data reported by Wang *et al.* for commercial Dyneema and Zylon, respectively.¹¹ The pentagram ($\star$) is the experimental data for the sample with excellent crystallite orientation and crystallinity reported by Xu *et al.*.⁶ (b) Wave-like thermal conductivity in the three crystalline orientations.</figcaption></figure>

The contrasting temperature dependence of $\kappa_{tot}$ along the chain direction and in the out-of-chain directions can be attributed to the strongly anisotropic phonon band structure of crystalline PE. As shown in Figure 3, the largest phonon group velocity in the longitudinal
acoustic (LA) branch along the chain direction reaches approximately 17.3 km/s. Additionally, there are significant interbranch spacings between the LA branch and other phonon branches in the frequency range of 10 to 16 THz, which reduces phonon-phonon scattering and results in small phonon linewidths. Consequently, thermal transport is primarily dominated by the particle-like mechanism, as indicated by the large red bubbles along the ΓZ path in Figure 3(b). In contrast, the phonon polarizations along the XΓ path are very flat, suggesting a small group velocity, as shown in Figure 3(a). Furthermore, the phonon linewidths for phonon modes along the XΓ path are extremely broad. These two factors contribute to the very small $\kappa_P^{xx}$, which is only 0.048 W/mK at 300 K. However, the flat phonon polarizations along the XΓ path are closely packed. The significant broadening of these phonon polarizations further eliminates small interbranch spacings. As a result, the wave-like term dominates thermal transport in the out-of-chain directions, as indicated by the green and blue bubbles in Figure 3(b). Traditionally, flat optical phonon branches are considered to contribute little to thermal conductivity. However, even high-frequency flat optical phonon branches can make substantial contributions to $\kappa_C^{xx}$, as shown by the blue bubbles along the XΓ path. The low-frequency phonon modes close to the Γ-point along the ΓZ path exhibit small interbranch spacing and large broadenings, which makes the phonon tunneling processes available along the chain direction and resulting in the relatively large $\kappa_C^{zz}$.
<figure><img src="image_4.png" /><figcaption>Figure 3: The SSCHA phonon band structure at 300 K, mapped with (a) the group velocity represented by the color and (b) the half of the phonon linewidth $\Gamma_{\lambda}/2$ ($\Gamma_{\lambda}$ is the inverse of the phonon lifetime.) indicated by the gray shaded areas. The mode contributions to $\kappa_{tot}$ are represented by the diameters of the colored bubbles. To clarify, the scale factor of the bubbles along the XΓ path is 20 times larger than that along the ΓZ path. The color of the bubbles corresponds to the dominant thermal conduction terms, as shown by the color bar at the top of the panel.</figcaption></figure>

The quantitative analysis of the contributions from different phonon modes to $\kappa_C$ is shown in Figure 4. As seen in Figure 4(a), both low-frequency and high-frequency phonons contribute significantly to $\kappa_C^{xx}$. Phonon tunneling from one mode to another occurs between two phonon modes with similar frequencies for low-frequency phonons, but it can also occur between phonons with quite different frequencies due to the large phonon linewidths shown in Figure 3(b). In contrast, only low-frequency phonons significantly contribute to $\kappa_C^{zz}$, and phonon tunneling can only occur between phonons with similar frequencies, as illustrated in Figure 4(c).

To further clarify the effects of anharmonicity on $\kappa_C$, we artificially decrease the phonon linewidths by a factor of 10. As expected, $\kappa_P$ in all crystalline orientations increases by a factor of 10. However, $\kappa_C^{xx}$ is decreased by ~60%. As shown in Figure 4(b), phonon tunneling between phonon modes with significantly different frequencies is severely suppressed, while tunneling between low-frequency phonon modes with similar frequencies is slightly enhanced.
In contrast, $\kappa_C^{zz}$ is increased by ~30% due to the increase in contributions from low-frequency phonons, as shown in Figure 4 (d). The different trends in the responses of $\kappa_C^{xx}$ and $\kappa_C^{zz}$ to the reduction in anharmonicity highlights the need for caution when modulating the thermal conductivity of anisotropic materials. For thermal transport in crystalline orientations dominated by the wave-like term, increasing anharmonicity and introducing more disorder is necessary to further enhance $\kappa$.

<figure><img src="image_5.png" /><figcaption>Figure 4: (a) $\kappa_C^{xx}$ associated with various pairs of phonon frequencies ($\omega^s$ and $\omega^{s'}$) at 300 K. (b) The same content as (a), but the phonon-phonon scattering rate ($\Gamma$) is artificially reduced to $0.1 \times \Gamma$. (c) $\kappa_C^{zz}$ associated with various pairs of $\omega^s$ and $\omega^{s'}$ at 300 K. (d) The same content as (c), but $\Gamma$ is artificially reduced to $0.1 \times \Gamma$.</figcaption></figure>

In summary, we employed the stochastic self-consistent harmonic approximation combined with the Wigner transport equation to predict thermal transport in crystalline polyethylene. Our theoretical results along the chain direction are in excellent agreement with recent experimental data, and the thermal conductivity is lower than previously anticipated. We find that the thermal transport along the chain direction and in the out-of-chain directions
exhibits opposite temperature dependencies. In the chain direction, the particle-like term dominates the thermal conductivity, whereas in the out-of-chain directions, the wave-like term prevails. Additionally, the anharmonicity in the out-of-chain directions is significant, facilitating phonon tunneling between modes with considerably different frequencies.

## Acknowledgement

We would like to thank Dr. Xiaoxiang Yu, Dr. Zhen Tong, Mr. Andrea Corradini, and Prof. Matteo Calandra for fruitful discussions. We acknowledge support from the HPC resources of Ruche, as well as those from IDRIS, CINES, and TGCC under Allocation No. 2024-A0160914101 made by GENCI. We also acknowledge funding from the projects PLACHO (ANR-21-CE50-0008) and TyLDE (Grant No. ANR-23-CE50-0001-01) founded by the French National Research Agency.

## Supporting Information Available

Detailed information regarding convergence tests on the phonon dispersion, validation of the machine learning potential, convergences tests on the thermal conductivity, the phonon-phonon scattering rate, the quantum anharmonicity effects on the lattice vibration, and the temperature effects on the mode-level wave-like thermal conductivity.

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