The *trans-gauche* state transformation commonly exists in polymers. However, the fundamental understanding of the roles of kink (*gauche* state) on the thermal energy transport in polymer chains is rather limited. From atomic simulations, we show that kinks greatly scatter phonons in single polyethylene chains, and even a single kink can reflect more than half of the phonons. Further studies show that kinks not only add extra thermal resistance to the chain but also break the whole chain into small segments and each with reduced thermal conductivity. A simple series thermal resistance model is proposed to estimate the effective thermal conductivity of single polymer chains with multiple kinks.

## I. INTRODUCTION

Polymers are usually flexible, light-weight, and electrically and thermally insulating materials, with low-cost and easy fabrication. For thermal energy applications, foams made from polymers are widely used as thermal insulation materials for energy saving due to their extremely low thermal conductivity. High thermal conductivity for polymers and their composites, however, is still in urgent need for the development of polymer-based heat exchangers, underfills in electronic packaging, optical lenses of light emitting diodes (LEDs), separators for Li-ion batteries, and wearable and flexible electronic devices. Over the last decade, studies have shown that polymers can have high thermal conductivities with good chain alignment.^{1–10} The measured thermal conductivity is up to ∼100 W m^{−1 }K^{−1} for polyethylene (PE) nanofibers,^{5} ∼50 W m^{−1 }K^{−1} for PE microfibers,^{9} and ∼60 W m^{−1 }K^{−1} for PE films.^{8,10} These recent experimental advances greatly rely on the progress of fundamental understandings of thermal transport in polymers.^{11–15} Yet, the limit of high thermal conductivity for polymers has not been reached.

Realistic polymers are made from long, often entangled and folded molecular chains. In the synthesis of highly thermal conductive polymers, one key technique is to align the nonoriented polymer chains via mechanical stretching.^{5,16,17} However, even in ultradrawn polymer fibers and films, polymer chains are still not perfectly aligned and crystallized. They are still folded or curved in either the lamellar region or the amorphous region because the rotational degree of freedom of the polymer chains gives flexibility to the chains and minimizes the total energy. For example, in PE, the backbone can rotate around the sp3-hybridized carbon–carbon bonds, forming torsion angles (φ) between −180° and 180°.^{18} Three stable conformations can be found in a PE chain: the *trans* state with *φ *= 0° and the *gauche* plus and minus states with *φ *= ±120°, which correspond to the global minimal and local minimal of the rotational energy, respectively.^{18} The *trans* state can convert to the *gauche* state when the PE chains are curved. The *gauche* states are usually called “kinks,” which bring disorder to the polymer structure. Kinks are common structures in realistic polymers, which largely exist at the folded ends of polymer chains, such as those in lamellas.

The effects of molecular-level structural disorders (e.g., curvature, kinks, orientational disorder) on thermal conductivity of polymer chains have been explored in the literature using simulations of molecular dynamics (MD).^{17,19–21} Sasikumar and Keblinski^{19} studied the thermal transport in curved polymer chains at 300 K and found that for the curved chains, the chain resistance greatly depends on the number of kinks, which were considered as centers for defect scattering. A similar conclusion has been made in the studies of strained single polymer chains.^{21} Kinks change the orientation of the polymer chains, which can be characterized with the orientational order parameter.^{22} The correlation between the thermal conductivity and the orientational order parameter also suggests that the thermal conductivity of polymer chains decreases with increasing disorder.^{17,20} However, all the previous studies present the collective effects of kinks on the thermal conductivity of curved polymer chains. The fundamental understanding of the roles of kink on the thermal energy transport in polymer chains is rather limited.

## II. METHODOLOGY

We carried out MD simulations to study the roles of kink on the thermal energy transport in single PE chains using LAMMPS.^{23} The atomistic Green's function approach^{24} was further applied to investigate the spectral phonon energy transport. Polymer Consistent Force Field (PCFF)^{25} was applied for the covalent interaction and Van der Waals potential was applied for the nonbonded interaction with a cutoff distance of 10 Å. Due to the classical feature of MD, we applied the quantum correction to match the total vibrational energy of the MD system to the quantum phonon system without zero-point energy,

where *T*_{MD} and *T* are temperature for the MD and the phonon systems, respectively; *N* is the number of atoms; *k*_{B} is the Boltzmann constant; $\u210f$ is the Planck constant; ω is phonon frequency; *Dos* is the phonon density of state; and *f* is the Bose–Einstein distribution. We performed MD simulations at about *T*_{MD} = 60 K, which corresponds to a quantum phonon system with T ≈ 320K.

Figures 1(a) and 1(b) show both the structures of straight PE chain and the PE chain with a single kink in the middle. The thermal conductivity of PE chains was calculated using the nonequilibrium molecular dynamics (NEMD) simulations. As shown in Figs. 1(c) and 1(d), hot and cold reservoirs (marked with red and blue colors) using the Langevin method with the phonon quantum temperature at T = 370 K and T = 260 K (average temperature of T = 320 K) were applied at the two ends of the polymer chains, respectively. The corresponding MD temperature (*T*_{MD}) is 75 K and 40 K, respectively. The heat flowing in (*q*_{in}) and out (*q*_{out}) of the two reservoirs, respectively, was recorded. The steady-state condition is considered to be reached when the magnitudes of the accumulated heat flow in the two reservoirs (*q*_{in} and *q*_{out}) are within 5%. A 0.5 ns NVE (constant total number of atoms, *N*; constant volume, *V*; constant energy, *E*) run is adequate to reach such condition. Another 0.5 ns NVE run is performed to obtain the average temperature along the polymer chains. Figure 1(c) shows the temperature profile along straight PE chains under the steady-state condition. Near the hot or cold reservoir, there is an apparent nonlinear temperature distribution, which is commonly observed in NEMD simulations^{26} and is due to the mismatch of the phonon spectra between the phonons in the internal region and the phonons launched from the reservoir.^{27} In the center region, the scattered temperature data align well within a straight line, which agrees with the one-dimensional steady-state heat transfer model. Figure 1(d) shows the temperature profile along a PE chain with a single kink in the center. Between the reservoirs and the kink, the temperature is linear. Across the kink, there is a significant temperature drop. The above results qualitatively suggest that kinks greatly scatter phonons in single polymer chains.

The thermal conductivity and thermal boundary conductance obtained in this study are phenomenally defined by the Fourier's law. The thermal conductivity of PE chains can be calculated by *k* = *qL*_{0}/(*A*Δ*T*_{L}), where *q* is the average heat flow, *A* is the cross-sectional area of the polymer chain, *L*_{0} is the chain length between the two reservoirs, and Δ*T*_{L} is the corresponding temperature difference. The typical value of 0.18 nm^{2} is used for *A*.^{13} The thermal boundary conductance (TBC, *G*) across the kink can be obtained by *G* = *q*/(*A*Δ*T*), where Δ*T* is the temperature jump across the kink. The spectral phonon transmission is calculated using the atomistic Green's function approach, and the details can be found in Ref. 28.

## III. RESULTS AND DISCUSSION

### A. The phonon dispersion curve and the thermal conductivity of straight PE chains

In MD simulations, interatomic potential plays a key role to accurately model the phonon transport. For validation, the phonon dispersion of a straight PE chain with PCFF is shown in Fig. 2, which agrees well with the one from a first-principles calculation for a PE chain in all frequency ranges.^{15}

Previous studies have shown that the thermal conductivity of straight single polymer chains strongly depends on the length of these chains.^{20} In order to further validate our NEMD simulation results, the thermal conductivity of straight PE chains at 320 K is calculated as a function of the chain length, which is shown in Fig. 3. The thermal conductivity $(kL0)$ does not converge with increasing chain length (*L*_{0}) up to 431 nm. The straight line from the log–log plot indicates an exponential relation, which agrees with the literature.^{20} The numerical fitting suggests that

with *C* = 19.58, in the unit of $W/(nm\alpha mK)$; *α* = 0.382 and *L*_{0} in the unit of nanometers. The value of coefficients *C* and *α* depends on the polymer type, temperature, etc. Henry and Chen^{11} suggest that the divergent thermal conductivity of PE chains is likely due to the nonattenuating modes, which is similar to the FPU (Fermi, Pasta, and Ulam) problem.^{29} Such an exponential relation has been observed in other low dimensional systems, for example, carbon nanotubes.^{30} It is worth mentioning that density-functional perturbation theory based simulations do not include the possibility of wave effects and divergence that can be included in molecular dynamics simulation.^{15}

### B. Phonon transport across single and multiple kinks in PE chains

For the PE chains with a single kink in the middle, a *gauche* state was created in the initial input structure by rotating the torsion angle from 0° to 120° (or −120°). Within the temperature range we applied, the kink does not convert back to the *trans* state during the entire MD simulation. The thermal boundary conductance across the kink is shown in Fig. 4(a) with different total lengths. The results show that when the chain length varies, the TBC does not change. The average TBC is about 2200 MW m^{−2 }K^{−1} or the average thermal resistance (TBR), defined as *R* = 1/*G*, is 4.55 × 10^{−10} m^{2}K W^{−1}. For a single PE chain, the upper limit of the thermal conductance is 5200 MW m^{−2} K^{−1} calculated by assuming that all the phonon modes are transmitted along the chain without any reflection. Thus, the effective phonon transmission coefficient across a single kink is 0.42.

Figure 4(b) shows the spectral phonon transmission per mode across a single kink in the PE chain calculated from the atomistic Green's function approach^{28} (only low and middle frequency data are shown). At low frequency (< 0.15 × 10^{14} Hz), about half of the phonons transmit across the kink. At middle frequency, phonons transmit effectively in the range from 0.27 − 0.38 × 10^{14} Hz and near the 0.43 × 10^{14} Hz peak, while phonons are greatly scattered at other frequencies, especially from 0.14 − 0.19 × 10^{14} Hz and near 0.4 × 10^{14} Hz where the phonon stop bands form. Due to the directional structural feature of the kinks, there would be strong mode conversion when the vibration of atoms transfers across the kinks. For the acoustic phonons (< 0.15 × 10^{14} Hz), the average phonon transmission is around 0.5. For the optical phonons, the average phonon transmission is about 0.36 in the range 0.19 − 0.29 × 10^{14} Hz and is about 0.74 in the range 0.29 − 0.38 × 10^{14} Hz. As shown in Fig. 2, there are more branches overlapping with each other in the frequency range 0.29 − 0.38 × 10^{14} Hz; therefore, we speculate that this could be the reason why it has the largest average phonon transmission by effective phonon mode conversion. In contrast, in the frequency range 0.19 − 0.29 × 10^{14} Hz, the two branches do not overlap with other branches, which results in the lowest average phonon transmission with the absence of mode conversion. For the acoustic phonons, the phonon transmission has a middle average phonon transmission that corresponds to a middle level of phonon dispersions overlapping. The thermal boundary conductance calculated from the phonon transmission is ∼1970 MW m^{−2 }K^{−1} (with effective phonon transmission ∼0.38), which agrees reasonably with the NEMD results. From the studies of both methods, the results show that more than half of the phonons are reflected even by a single kink. Although the TBC of a single kink is much larger than the typical TBC of interfaces (1–700 MW m^{−2 }K^{−1}),^{31} it greatly affects the effective thermal conductivity of polymer chains due to its large amount in realistic polymer chains. We need to emphasize that due to the complexity of PCFF potential, it is challenging to obtain the force constant analytically, and in our studies, the force constant was obtained by differential methods. Due to the numerical errors, the phonon dispersion is not accurate at very low frequency and small wave vectors when comparing with Ref. 15 (longitudinal branch with frequency <0.027 × 10^{14} Hz; transverse branches with frequency <0.01 × 10^{14} Hz; the twisting acoustic branch agrees with the literature). This could be the reason why phonon transmission does not approach 1 when frequency approaches zero. Since this frequency range is very small, it would not affect the main conclusions in this paper.

The method of generating multiple kinks is similar to the one for single kink, and all the kinks are stable during the MD simulations. The typical temperature profile of PE chains with multiple kinks is shown in Fig. 5(a). The chain length is 242 nm and there are 6 kinks evenly distributed in the PE chain. The profile shows that there is an apparent temperature drop across each kink, which indicates strong phonon scattering across every kink. In order to evaluate the effects of phonon scattering from multiple kinks, a total thermal resistance (*R*) is defined as

where *n* is the number of kinks and Δ*T*_{i} is the temperature drop across the *i*th kink. Here, the effects from the straight part of the PE chain are not included. Figure 5(b) shows the total thermal resistance as a function of *n* for the 242 nm PE chain. The linear fitting shows that *R* increases linearly with *n*, though a detailed calculation shows that the thermal resistance of each kink varies from each other. Such a linear relation indicates that the overall effect from multiple kinks in the PE chain is dominant by diffusive-like phonon transport, and the slope from Fig. 5(b) suggests a value of $\u223c3.7\xd710\u221210m2KW\u22121$ for the thermal resistance per kink. In contrast, the intercept from the linear fitting leads to a value ∼1 × 10^{−10} m^{2 }K W^{−1}, i.e., a conductance of 10^{10} W m^{−2 }K^{−1}. From the extrapolation, the uncertainty of the intercept is about 12% and the uncertainty of the slope is about 0.75%. The uncertainty from the data fitting in MD is about 6%. We should emphasize that this extrapolation does not include the effects of temperature jumps at the two ends and the effects from the straight part of the chain. For the nonzero intercept, we speculate that this nonzero intercept is possibly due to the relatively weak wave effect—some long wavelength phonons that can propagate through all kinks and the chain segments in between the kinks, as in the case of superlattices^{32} in which long wavelength phonons extend through multiple interfaces.

### C. The effects of kinks and a theoretical model of thermal energy transport in PE chains

Kinks affect the thermal conductivity of single PE chains not only by adding extra thermal resistance but also by breaking a single chain into smaller segments that have lower thermal conductivity due to their reduced length. For example, for a single chain with a length of 141 nm, if no kinks are present, the thermal conductivity is 134 W m^{−1 }K^{−1}, as shown in Fig. 3; however, if a single kink is present, simulation results show that the thermal conductivity of the left half (or the right half) of the chain (about 70 nm) is only around 109 W m^{−1} K^{−1}, which is close to the thermal conductivity of a 70 nm single straight chain (∼100 W m^{−1} K^{−1}), as shown in Fig. 3. Furthermore, from the above discussions, we found that phonon scattering across kinks in PE is dominant by a diffusivelike scattering and the wave effect is relatively weak. Based on these observations, we propose a simple model to estimate the effective thermal conductivity of a PE chain with a length *L*_{0} and *n* kinks by assuming that (1) phonon scattering at each kink is fully diffusive and (2) the kinks are evenly distributed along the chain, as shown in Fig. 6(a). For simplicity, the effective thermal conductivity *k*_{L0} defined by the chain length is used for the discussions. The effective thermal conductivity of the chain can be derived from the series thermal resistor model in Fig. 6(a). From correlation Eq. (2), the thermal resistance of each segment with length *l* = *L*_{0}/(*n* + 1) is

The total thermal resistance of the chain from Fig. 6(a) is

where *R*_{2} = 3.7 × 10^{−10} m^{2} K W^{−1} from Fig. 5(b).

Finally, the effective thermal conductivity along the chain is *k*_{eff} = *L*_{0}/*R*_{total}.

In order to validate this model, NEMD simulations were performed on two PE chains with different lengths and multiple kinks. Figure 6(b) shows the simulated results from MD and the modeled results, which agree with each other. For the 242 nm chain, the discrepancy increases with an increasing number of kinks. For example, with n = 8, the effective thermal conductivity from the above model is 11% smaller than the one from MD simulation, which could be due to the wave effect that is not included in the model. Overall, the results from Fig. 6 indicate that when kinks are added in PE chains, the effective thermal conductivity of PE chains can be reasonably modeled by the superposition of a series of thermal resistors, and more accurate modeling is needed to include the wave effects.

## IV. CONCLUSIONS

In conclusion, we have studied the roles of kink(s) on the thermal energy transport in single polyethylene chains. It is found that kinks are strong phonon scattering centers. Even a single kink can reflect more than half of the phonons in a single PE chain. Phonon scattering across kinks are dominated by diffusivelike scattering. There are two major roles of kinks on the thermal transport in single PE chains: (1) kinks add extra thermal resistance and (2) kinks break the whole chain into segments and each segment has a reduced thermal conductivity. With both effects, the thermal conductivity of a single PE chain decreases rapidly with an increasing number of kinks. A simple series thermal resistor model is proposed and verified with NEMD simulations on PE chains with multiple kinks. The new findings from this study could enlighten new ideas of making polymers with high thermal conductivity for energy applications.

## ACKNOWLEDGMENTS

X. Li acknowledges the support of the National Natural Science Foundation of China (NSFC Grant Nos. 51506062 and 51776080) and G. Chen acknowledges the support of DOE BES (No. DE-FG02-02ER45977). The authors would like to thank B. L. Huang for providing a full phonon dispersion curve of a single PE chain through private communications.