Received: 09 May 2019
Revised: 26 Jun 2019
Accepted: 28 Jul 2019
Published online: 29 Jul 2019

 Modulating Thermal Transport in Polymers and Interfaces: Theories, Simulations, and Experiments

 Bo Zhang1, Peng Mao1, Yunmin Liang1, Yan He2, Wei Liu1* and Zhichun Liu1*


1 School of Energy and Power Engineering, Huazhong University of Science and Technology (HUST), Wuhan 430074, China

2 College of Electromechanical Engineering, Qingdao University of Science & Technology, Qingdao 266061, China

*Corresponding Author: E-mail: (Zhichun Liu), (Wei Liu)



Bulk polymers are often regarded as thermal insulators due to low thermal conductivity, which extremely limits their applications in the field of heat transfer. Over the past decades, thermal transport in polymers, and polymer nanocomposites has been intensively studied on both theoretical and experimental levels. In addition, novel thermal transport phenomenon involving divergent thermal conductivity in individual polymer chains, giant thermal rectification, has been observed. In this review, the mechanism behind thermal transport in materials, interfacial thermal transport, thermal rectification in polymers, and enhancing thermal transport of polymers are firstly addressed. Secondly, the computational methods for investigating the thermal property of materials mainly focused on molecular dynamics (MD) simulation are summarized and compared. The advanced spectral decomposition methods in non-equilibrium molecular dynamics (NEMD) simulation are highlighted. Thirdly, experimental advances relevant to thermal transport of polymers are briefly reviewed. Finally, the challenges and outlook about modulating thermal transport in polymers and polymer nanocomposites are pinpointed.

Table of Content

The review systematically summarizes the progress about thermal transport of polymers from theoretical, computational and experimental level.






Keywords: Thermal transport; Thermal rectification; Molecular dynamics simulation; Spectral decomposition; Polymers

1. Introduction 

Polymers have been ubiquitously used in industry and daily life owing to various advantages, including chemical inertness, light density, easy of processing, and low cost. As shown in Fig. 1, the thermal conductivity of bulk polymers is often in the range of 0.1~0.5 Wm-1K-1, which hinders their applications in thermal management.1-3 In contrast, Henry and Chen observed divergent thermal conductivity of individual polymer chains in molecular dynamics simulations.4,5 Experimentally, polymer fibers with excellent thermal and mechanical properties have been fabricated.6-8


Fig 1 Thermal conductivities of bulk polymers at room temperature.


As shown in Fig. 2(a), heat can spontaneously flows from high temperature zone to low temperature zone due to temperature difference. At the macroscale, heat conduction can be described by Fourier’s law

                                                                   J=-κ∙∇T                                                (1.1)

where J is heat flux, κ is thermal conductivity, and ∇T is temperature gradient. Fourier’s law can also be written as

                                                                   Q=-G∙∆T                                                (1.2)

where Q is heat current, G is thermal conductance, and ∇T is temperature difference. The relationship between thermal conductivity and thermal conductance of the homogeneous material can be given by

                                                                   G=κ∙A/L                                                 (1.3)

where A is the cross-sectional area and L is the length of the system. In solid materials, electrons and phonons both contribute to thermal conduction.9 The total thermal conductivity of non-magnetic materials can be given by

                                                                  κ=κe+κph                                               (1.4)

where κe represents the electronic thermal conductivity, and κph represents the phononic thermal conductivity. In dielectric material like polymers, the contribution from electrons is negligible and lattice vibration contributes most to the thermal conductivity. Hence phonons, the quanta of lattice vibration modes, are the main heat carriers in dielectric system. Based on kinetic theory, the phononic thermal conductivity can be roughly estimated by Debye equation10,11

                                                                 κph=13CvvgΛ                                               (1.5)

where Cv is the volumetric specific heat, vg  is the average phonon group velocity, and Λ is the phonon mean free path.

    As shown in Fig. 2(b), the structure disorder and various defects in bulk polymers extremely reduce the thermal conductivity. Align the orientation of polymer chains can effectively enhance the intrinsic thermal conductivity of polymers.12 However, in order to tune the chains’ orientation, the giant drawing ratio is often necessary.6 Incorporating high thermal conductive fillers like graphene,13 carbon nanotube,14 boron nitride15 in polymers is another way to elevate the thermal conductivity of polymers. Whereas the thermal conductivity of polymer composites is not as high as expected due to the large interfacial thermal resistance (ITR) between matrix and fillers. Hence declining ITR is critical to enhancing the thermal conductivity of polymer composites. Understanding the thermal transport mechanism in polymers, fillers, across the interface will be helpful to design high thermal conductive materials. Apart from dissipating heat, actively manipulating heat flow is also desired in thermal management.16 In the past decades, thermal rectification has been extensively studied on the theoretical level while only a few papers report the thermal rectification about polymers.17 This paper will recount the origin of thermal rectification and summarize recent developments.

Fig 2. Schematic illustration of heat conduction and a polymer. (a) Heat conduction in a homogeneous cylinder. (b) The morphology and structure of a polymer.18

     In this article, we will first recount the theory about thermal transport, thermal rectification, followed by a brief discussion about the underlying mechanism in modulating thermal transport of polymers. In Sec. 3, state-of-the-art computational methods and spectral decomposition methodology based on NEMD are summarized. In Sec. 4, advanced experimental methods about fabricating high thermal conductive polymers are reviewed. In Sec. 5, the remarks and outlook are given to pinpoint the limitations and directions in theories, simulations, and experiments.

2. Thermal transport theorie

2.1 Thermal transport in crystalline materials

In dielectric medium, phonons play a dominant role in heat conduction, whose occupation obeys Bose-Einstein statistics. The thermal conductivity is related to the phonon energy ℏω(q,j), phonon group velocity νg (q,j), phonon occupation function fBE(q,j), phonon relaxation time19-21 τ(q,j)


where V is the volume of the system, q and j denote the phonon wave vector and phonon polarization, respectively. The volumetric phonon specific heat and phonon mean free path can be written as



Substituting Eq. (2.2), (2.3) into Eq.(2.1), we can find that


Hence thermal conductivity can be determined by volumetric phonon specific heat, phonon group velocity, and phonon mean free path. Phonon group velocity is defined by


Hence phonon dispersion relation is necessary to get phonon group velocity, which is determined by the following secular equation


where α, β = 1, 2, 3 represent x, y, z, respectively. D(q) is the 3n × 3n dynamical matrix (n is the number of atoms per unit cell.). The elements of the dynamical matrix are


where s and s denote an atom in the l -th unit cell, l = 0 refers to the reference unit cell, M and r denote the mass and position of the atom. In order to accurately determine phonon dispersion, accurate empirical potential function or first principle calculation is essential. In addition, the size of the system cannot be too large due to the time consuming calculation. Fig. 3 displays some common atomic structures and the corresponding phonon dispersion relation. It can be seen that all the phonon frequency of the system is positive. More phonon branches will emerge with the increasing number of atoms in the primitive cell. Compared with low frequency phonons, the group velocity of high frequency phonons is much smaller. This confirms that low frequency phonons often play an important role in heat conduction.

    Based on Eq. (2.1)~(2.4), another crucial factor affecting thermal conductivity is phonon relaxation time, which is intensively related to phonon interaction in the material. In pure dielectric crystal, phonon-phonon scattering induced by anharmonic interaction becomes the main source of scattering. The extremely high thermal conductivity of carbon nanotube22 (CNT) and graphene23-25 is attributed to the strong atomic bonding and low anharmonicity. For low frequency phonons, the phonon mean free path in CNT and graphene can reach the order of microns. Compared with CNT and graphene, the high thermal conductivity of boron arsenide (BAs) is mainly due to the large phonon band gap and relatively weak phonon-isotope scattering.26-29 The large band gap and conservative conditions can effectively increase the phonon relaxation time.


Fig. 3 Schematic of atomistic structure and corresponding phonon dispersion. (a) Single PE chain. (b) Graphene sheet. (c) C-honeycomb structure.30 (d) Silicon. (e) Phonon dispersion of the single PE chain. (f) Phonon dispersion of the graphene sheet.31 (g) Phonon dispersion of the C-honeycomb.30 (h) Phonon dispersion of the silicon.32


As shown in Fig. 4, in realistic dielectric materials, more phonon scattering channels will emerge owing to various defects like vacancy, impurity, electron, dislocation, grain boundary, external perturbation, et al.33,34 According to Matthiessen’s rule, the total phonon scattering rates are equal to the summation of phonon scattering rates in individual process,


where ,  and denote the total phonon scatterings, phonon-impurity scatterings, phonon-electron scatterings, phonon-boundary scatterings, and phonon-phonon scatterings, respectively. However, Matthiessen’s rule may overestimate the thermal conductivity owing to the neglect of the coupling among different scattering process.35 Meanwhile, accurately calculating phonon scattering rates in individual process like electron-phonon scattering is not trivial and a better understanding of phonon scatterings will be useful to modulate the thermal conductivity of materials.


Fig. 4 Schematic illustration of various phonon scatterings. (a) Phonon-impurity scattering. (b) Phonon-electron scattering. (c) Phonon-boundary scattering. (d) Phonon-Phonon scattering.


2.2 Thermal transport in amorphous materials

In contrast with crystal, lacking translational symmetry is the most distinguishable feature of amorphous dielectric material. Hence phonons are only used to denote the energy quanta of lattice vibration for amorphous materials. Phonon wave vector and group velocity are not well defined for the majority of states. For amorphous polymers, thermal transport more likely depends on the phonon hopping process.36  Feldman and Allen et al propose to use “propagons, diffusons, and locons” to replace the phonons.37-40 The total thermal conductivity can be written as


where κpr, κdif, and κlo are the contribution from propagons, diffusons, and locons, respectively. As shown in Fig. 5, propagons and diffusons are extended modes, while locons are localized modes.


Fig. 5 Phonon density of states (DOS) of amorphous silicon39.

Note: the x-axis unit here is the energy unit (ℏω ).


For amorphous material, propagons only belong to very low frequency modes, whose behavior is like phonons. The contribution to thermal conductivity from propagons can be written as


where ωIR is the cutoff frequency of the propagons. In contrast, diffusons and locons don’t have well-defined group velocity, whose contribution to thermal conductivity can be described by



where ωmax is the maximum frequency of density of states, Di is the mode diffusivity, Sij = < i|S|j > is the intermode matrix element of the heat current operator, and δ is the Dirac delta function. The heat current operator denotes the interaction of vibrational modes, which is related to the spatial overlap and spring constants.41 Propagons and diffusons can be distinguished by specifying Ioffe-Regellimit.42 Larkin and McGaugyhey et al propose to use structure factors to distinguish propagons and diffusons. The structure factor is defined as


where the summation is over the gamma modes, EL and ET refer to the longitudinal and transverse polarization, respectively.43 However, this method requests that the mode character must change abruptly in the vicinity of Ioffe-Regel crossover. In order to distinguish the propagons and diffusons, Lv and Henry et al recommend measuring the eigenvector periodicity (EP) for each mode.44,45 The normalized EP can be given by


where e, e’, refer to the eigenvector of the real mode and fictitious periodic mode, n is the mode index, q’, φ’ refer to the wave vector and phase of the fictitious mode, respectively. The large value of γn denotes the mode is propagating, while the small value of γn denotes the mode is non-propagating.

Locons are localized modes whose eigenvector decays dramatically with distance from the center. The participation ratio46 (PR) can be used to distinguish extended modes (propagons, diffusons) and localized modes, which is given by


The PR characterizes the proportion of atoms participating in an eigenvibration, whose numerical range is 0~1. With respect to harmonic system, all the atoms participate in eigenvibration and thus PR is 1. For real system with impurities, boundaries and interfaces, only a part of atoms participate in eigenvibration and thus PR will decrease. Accurate eigenvector may be not accessible for large size super cell due to limited computational resource. Without considering phonon polarization, the mode PR47,48 can be given by


where N is the number of atoms in the system, DOSS(ω) is the local DOS of s-th atom calculated from Fourier transformation of normalized velocity autocorrelation function (VACF)49


where υ is the atomic vibrational velocity. Compared with extended modes, the localized modes have smaller PR. Extended modes and localized modes play different role in heat conduction with respect to different materials. Fig. 6 displays the structure image, corresponding PR spectra and DOS of silicon phononic crystals (Si-PnCs), amorphous silicon (a-Si), amorphous silica (a-SiO2), and atactic PS. As shown in Fig. 6(a) and (e), phonons are localized in the vicinity of spherical hole and have low participation ratio.50 The DOS and PR spectra in Fig. 6 (f)-(h) indicate that high frequency phonons are more likely to be localized compared with low frequency phonons. For Si-PnCs and a-Si, localized modes make negligible contribution to the thermal conductivity. Harmonic theory can be successfully applied to these systems with one type of atom and identical interaction potential. However, for a-SiO2 and atactic PS, localized modes make a moderate contribution to the thermal conductivity through anharmonic coupling. Harmonic theory fails to predict thermal conductivity of these systems with complex composition.51 In contrast, MD simulation with full anharmonic potentials is suitable for these complex systems.

Fig. 6 Structure of four materials with the corresponding participation ratio spectra and DOS. (a) Structure image and normalized energy distribution of the Si PnCs. (b) Structure of a-Si. (c) Structure of a-SiO2. (d) Structure of atactic PS. (e) Participation ratio spectrum of Si-PnCs. Localization ratio refers to localization ratio. (f) DOS and PR spectra of a-Si. (g) DOS and PR spectra of a-SiO2. (h) DOS and PR spectra of atactic PS. (Ref. [50,51])

2.3 T​​​​​hermal transport across interface

Interfacial thermal transport is crucial to thermal management of microelectronics52,53 and thermal transport of polymer nanocomposites54-56 due to high interface density in these systems. As shown in Fig. 7(a), the temperature difference ΔT will emerge at the interface when heat flows through different medias. The interfacial thermal conductance (ITC) G is given by57-59


where J is the heat flux across the interface. The ITR is the reciprocal of ITC. The acoustic mismatch model60 (AMM) and diffuse mismatch model61 (DMM) are two simple models in estimating ITC. However, AMM assumes phonons undergo specular reflection and transmission, which is only reasonable at low temperature. DMM assumes that phonons at the interface undergo diffusive scattering, which suits for the wavelength of phonons much shorter than the surface roughness. The transmissivity of DMM can be given by


      It can be seen from Eq. (2.20) that DMM only considers elastic scattering. Hence DMM cannot apply in high temperature condition, where anharmonic effect plays an important role in interfacial thermal transport. Hida and Shiomi discover that ITC of CNT/PE composites increases with the increasing temperature due to the enhanced inelastic thermal transport.62 Wu and Luo discover that the anharmonicity inside the material can facilitate phonon mode conversion and thus contribute to interfacial thermal transport.63 Feng et al find that the inelastic transport can contribute more than 50% to the ITC of silicon/germanium (Si/Ge) heterostructure.64 The anharmonic interaction enables optical phonons to contribute to interfacial thermal transport.64-66 Different from transitional lattice dynamics method, spectral decomposition method in MD simulation can fully capture the anharmonic effect and deliver deep insight about interfacial thermal transport from frequency level. With respect to thermal transport in polymers, spectral decomposition based on MD has less limits, which will be described in Sec 3.3.

     Fig. 7(b) shows the h-BN and graphene heterostructure with different interface. Defects like isotopes,67-69 vacancies,70,71 and dislocations72,73 are often detrimental to thermal transport. Contrary to conventional belief, interfacial defects unexpectedly facilitate thermal transport across the interface. Liu et al find that h-BN/graphene interface with topological defects has a higher ITC than the pristine interface due to the local stress field near the pores.74 Giri and Hopkins et al discover that atomic mass defects at the interface can enhance ITC of amorphous SiOC:H/SiC:H interface due to the emergence of interfacial modes.75 The coupling between interfacial modes and bulk modes facilitates the thermal transport across the interface.

    Fig. 7(c) and (d) show that ITC can be enhanced by employing the intermediate layer (IL) to bridge vibrational power spectra (VPS). The VPS is similar to DOS and can be given by63




where ms is the mass of s-th atom, vs  is the vibrational velocity of s-th atom, ω is the phonon angular frequency. The spectral temperature Tsp(ω) can be defined as


where Teq is the temperature of equilibrium state, VPSneq(ω) and VPSeq(ω) represent the VPS of non-equilibrium state and equilibrium state, respectively.76 The overlap of VPS can demonstrate the elastic thermal transport across the interface, which can be given by77


Clearly, the larger value of S indicates the better match of VPS and more efficient interfacial thermal transport. Luo et al discover that ITC between gold (Au) and PE can be increased by 7 folds by employing SAM as IL owing to the better match of VPS between SAM and PE.78


Fig. 7 Schematic of interfacial thermal transport and simulation set up in MD. (a) Schematic illustration of interfacial thermal transport. (b) Heterostructure of h-BN and graphene with coherent interface and incoherent interface with topological defects.74 (c) Schematic of SiC/GaN and SiC/IL/GaN structure.66 (d) Schematic of Au-SAM/hexylamine structure.79


Apart from matching degree of VPS, the alternative indexes like interfacial binding energy, the effective contact area, the atomic number density near the interface can be used to evaluate the interfacial thermal transport. The higher interfacial binding energy79,80 and larger effective contact area81 can be realized by choosing proper SAM as IL, which can effectively enhance the ITC.

2.4 Thermal rectification of polymers 

Thermal rectification (TR) can be used to control the heat current, which has great potential in thermal management82 and phononic information technology.83-85 The TR ratio η can be defined as


where J+, J-, κ+, κ- denote forward heat flux, backward heat flux, forward thermal conductivity, and backward thermal conductivity, respectively. The nonlinear interaction is the key in TR, which ensures the phonon mode conversion among different frequencies.86-88 As shown in Fig. 8(a), the TR can be induced by the mismatch of VPS for dissimilar anharmonic systems, which results from the weak coupling at the interface.89-92 The asymmetry of geometry,93-95 mass distribution,96,97 and defects distribution98-100 can also induce mismatch of VPS to get high TR ratio in conjunction with nonlinear interactions. However, the TR ratio of CNT deposited C9H16Pt at room temperature is only 2%.96 Previous simulations have shown that the considerable TR ratio can be reached in asymmetrical nanostructures like asymmetrical graphene nanoribbons94,95,101 and junctions.77,102 Nevertheless, the giant temperature difference in a small system is undesired in applications. The temperature itself has a nonlinear effect on phonons. Phonons are more likely to be nondiffusive transport in small size graphene and CNT, which is difficult to implement in industry due to large size of the material. Recently, a TR factor of 26% has been achieved in a large-area monolayer graphene with nanopores on one side.100 However, the expensive cost and elaborate manufacturing of graphene with nanopores hinder the application. Compared with graphene, the cost of polymers is so low that they are hopeful for large-scale industrial applications. 

    Fig. 8(b) shows that the TR can emerge at the dissimilar bimaterial junction, which is related to the phonon localization.101,103,104 The degree of phonon localization determines the available phonon transport channels, which is intimately related to the thermal transport. Based on phase dependent thermal conductivity, the thermal diode can be constructed by PE with different morphology (Fig. 8(c)). The linear PE will become disordered at high temperature, but the cross-link PE (PEX) still keeps the pristine phase. The reverse temperature bias can change the phase of linear PE and thus result in TR.105 Moreover, Tian et al discover that the phonon transport mechanism will change when the direction of heat flow reverses.106 The underlying mechanism is related to the structural transition and phonon transmission between side chains and the backbone.


Fig. 8 (a) Schematic illustration of thermal rectification.92 (b) Schematic of PA/Si junction.103 (c) Schematic of PE/PEX junction.105 (d) Schematic of bottlebrush polymer and the XRD pattern.106


2.5 Enhancing intrinsic thermal transport and interfacial thermal transport

    Enhancing thermal conductivity of polymers can greatly expand their application in heat transfer. For crystalline polymers, thermal transport can be well described with the help of phonon picture. For noncrystalline polymers, thermal transport mechanism still needs to be investigated. As shown in Fig. 9(a), high crystallinity,8 good chain orientation,6,14,107 ordered morphology,108 less side chains,109 stiff and extended polymer backbone,110,111 confined angular bending,112 efficient thermal conductive pathways113,114 are beneficial to thermal transport of polymers. The underlying mechanism is that they can increase phonon group velocity or phonon relaxation time.

    Interfacial thermal transport is essential for improving the thermal conductivity of polymer nanocomposites due to large specific surface area. With respect to thermal transport across the interface, not only thermal transport inside polymers and fillers but also interfacial atomic interaction is worthwhile to consider. As shown in Fig. 9(b), similar atomic vibration near the interface,79 high interfacial binding energy,115 large contact area,81 strong filler-matrix coupling116 can lead to efficient interfacial thermal transport. The reason is that they can provide more channels for phonons to transmit through the interface. Hence in order to enhance the thermal conductivity of polymers, not only high thermal conductive fillers but also proper polymer matrix is necessary to select.


Fig. 9 Crucial factors affecting the thermal conductivity of polymers and polymer nanocomposites. (a) Crucial factors affecting the intrinsic thermal conductivity of polymers. (b) Crucial factors affecting the thermal conductivity of polymer nanocomposites.


3. Computational methods and their applications in polymers

3.1 Comparison of micro/nanoscale computational methods

Detailed depiction of simulation methods at micro/nanoscale has been reviewed from different perspectives.48,117,118 Table 1 summarizes the features about the popular simulation methods involving Boltzmann transport equation (BTE), MD simulation, and atomistic Green’s function (AGF). Compared with BTE and AGF, MD is more suitable to investigate thermal transport of polymers due to many atoms in the super cell.

Table 1 Comparison of popular simulation methods at micro/nanoscale.


Simulation methods



Application range

Boltzmann transport equation48

(1) Phonons obey quantum distribution  

(2) Couple with first principle calculation

(1) Ignore wave effect        (2) Successive phonon scatterings are independent5

crystalline material with weak anharmonicity

Molecular dynamics simulation118

(1) Permit simulating large super cell 

(2) Involve all order  anharmonicity of lattice vibration

(1) Ignore quantum effect      (2) Depend on the accuracy  of potential energy function

System at moderately high temperature

Atomistic Green's function119

(1) Involve phonon wave effect             (2) Phonons obey quantum distribution

Ignore phonon anharmonicity

System at low temperature


3.2 Fundamentals about molecular dynamics simulation

    Although MD is a powerful tool to investigate thermal transport of polymers at the molecular level, the assumptions and limitations are worthwhile to be noted. One assumption is the Born-Oppenheimer approximation, which separates the motion of atomic nuclei and electrons. Theoretically, the atomic motion needs to be described by Schrodinger equation. However, solving Schrodinger equation is time consuming and computational expensive. One critical approximation is to assume the atomic motion obeys classical Newton’s second law120


    where t is the time, mi, ri, Fi, Ui are the mass, position, force, and potential energy of atom i, respectively. The phonons in MD are considered to obey Maxwell Boltzmann distribution, which is invalid at low temperature. Hence MD isn’t suitable for system with high Debye temperature. In order to overcome the size effect, the periodic boundary condition is likely to be used. When the system reaches the equilibrium state, the macroscopic property can be acquired according to statistical physics.

     The microscopic connection is provided via the notion of an ensemble, which is an imaginary collection of systems described by the same Hamiltonian with each system in a unique microscopic state at any given instant in time. One of the basic ensembles is the microcanonical ensemble, whose Hamilton’s equation conserves the total Hamiltonian


The microcanonical ensemble consists of all microscopic states on the constant energy hypersurface determined by eqn. (3.2). The ensemble average of an observable A  by a phase space function α(r) is given by


Based on ergodic hypothesis, the microcanocial phase space averages can be equivalent to time averages over the trajectory according to


One thing needs to note is that a single dynamical trajectory conveys little information because a slight change in initial conditions can change the trajectory dramatically. Observables require averaging over an ensemble of trajectories each with different initial conditions. Other equilibrium ensemble like canonical ensemble can be derived from microcanonical ensemble through Legendre transformation. All statistical ensembles are equivalent in the thermodynamic limit.121

    Thermostat can be used in MD simulation to maintain the temperature of the system. The Nose -Hoover thermostat and Langevin thermostat are two commonly used thermostats. The former is a global thermal bath while the latter is a local thermal bath. With respect to Nose -Hoover thermostat, the Nose -Hoover chain algorithms are implemented to tune the temperature of the system and can generate a correct canonical distribution. The equation of atomic motion can be written as122




Where ζ is the deterministic damping term and τ is the relaxation time. The Langevin thermostat is a stochastic thermal bath whose temperature-control equation can be written as


where ξ is the random force and γ is the dissipation rate. Compared with the Nose -Hoover thermostat, the Langevin thermostat is more suitable to generate a linear temperature profile with small temperature jump.48


3.3 Spectral decomposition in non-equilibrium molecular dynamics simulation

   Phonons are the main heat carriers in dielectric polymers and polymer nanocomposites, which are essentially wave packets.9 Hence understanding thermal transport from the frequency perspective can deliver deep insights. Spectral decomposition methods based on lattice dynamics usually need eigenvectors and eigenvalues as inputs, which strongly limit their application in polymers. However, spectral decomposition methods based on NEMD don’t have these limitations, which can be used as a powerful tool to unravel the underlying mechanisms about thermal transport in polymers.

    The core of spectral decomposition is to acquire frequency-dependent heat current Q(ω). Hence the frequency-dependent ITC G(ω) can be defined as


Similarly, the frequency-dependent thermal conductivity κ(ω) can be written as


For two-body potential, the heat current between atoms s and s'  (belong to chunk A and chunk B, respectively) can be written as123-125


where Fss’ is the interatomic force between atoms s and s’; υs and υs’ are the velocity of atoms s and υs’, respectively. The auxiliary correlation function for interparticle heat current can be defined as126


The total heat current between chunk S and chunk S’ can be written as


The auxiliary correlation function for total heat current between chunk A and chunk B can be written as


For many-body potential, the heat current between atoms s and s’ depends on the atoms’ neighbor and cannot be simply described by Eqs. (3.4). Fortunately, a well-defined many-body heat current formula has been derived by Fan et al 127


where Us and Us’ are the site potential of atoms s and s’, respectively. rss’ is the relative position and can be written as


With respect to many-body potential, the auxiliary correlation function for interparticle heat current can be given by


The total heat current between chunk A and chunk B can be given by


The auxiliary correlation function for total heat current between chunk A and chunk B can be given by


The Fourier transformation pairs about the auxiliary correlation function can be defined as127-129



Since KAB(t) is real, the spectrally decomposed heat current between chunk A and chunk B can be written as                                                         

                                                        Q                                   (3.15)

The phonon transmission function T (ω) between chunk A and chunk B can be further defined as130

In order to distinguish elastic and inelastic spectral thermal conductance, Saa skilahti et al128 and Zhou et al131 propose to use second-order force constants and third-order force constants when calculating spectral heat current. Fig. 10 displays the typical spectrally decomposed heat current, thermal conductance, thermal conductivity and phonon transmission function. It can be seen that the spectral decomposition methodology has been successfully applied in crystalline material such as graphene and CNT. However, applying spectral decomposition method to thermal transport in polymers has been little reported. It will be hopeful to reveal the complex mechanism about thermal transport of polymers with the help of spectral decomposition method.



Fig. 10 Typical spectrally decomposed physical quantity. (a) Spectrally decomposed heat current of graphene.127 (b) Spectrally decomposed thermal conductance of polycrystalline graphene.132 (c) Spectrally decomposed thermal conductivity of graphene.133 (d) Phonon transmission spectra of CNT.130


3.4 Molecular dynamics simulation in heat transfer

According to fluctuation-dissipation theorem134 and linear response theory,135-137 thermal conductivity can be calculated by Green-Kubo formula in equilibrium molecular dynamics (EMD) simulation


For two body potential, heat current J can be defined as


For many body potential, heat current can be defined as138,139


Apart from EMD, thermal conductivity can also be calculated by NEMD simulation in conjunction with Fourier’s law. Dong and Fan et al have demonstrated that EMD and NEMD are essentially equivalent in terms of computing thermal conductivity.140 Fig. 11 displays the useful methods to enhance thermal conductivity of polymers in MD simulation, including mechanical stretching, molecular layer deposition, increasing the stiffness of backbone and forming hydrogen bond.

Fig. 11 Enhancing thermal conductivity of polymers by different methods. (a) Mechanical strain.141 (b) Construct parallel-linked epoxy resin by molecular layer deposition.108 (c) Increase the stiffness of polymer backbone.111 (d) Confine structural disorder by forming hydrogen bond.142


   Meng and Yang et al discover that thermal conductivity of polymers is strongly associated with the morphology.143 The aforementioned methods can align the polymer chain, increase the length of thermal conductive path, and reduce the structural disorder and thus facilitate thermal transport in polymers. The thermal conductivity of typical sample and computing methodology are listed in Table 2.


Table 2 Thermal conductivity of typical sample and simulation details.


κ (W/m-1K-1)

Computing method

Potential function


Single PE chains





Bulk PE crystals





DPG-1 zigzag





DPG-1 armchair





Aligned CNT-PE



Morse + cosine + LJ + AIREBO


Crystalline PEO





Twisted PE chains





Bulk single PE crystal





Single PT chains





Nylon 10





Single PVA chains





Single PDMS chain





Single PP chains





Aligned single PNb chains





PAAm hydrogels





Parallel-linked epoxy resin





Bulk PNb crystal





Amorphous PEO





Amorphous paraffin wax





Amorphous paraffin wax





Amorphous PT





Crystalline PE

0.26 (radial)




Amorphous PS film





Amorphous PE



NERD united potential


Amorphous PE





Amorphous PDMS





Amorphous PS





Amorphous PS





Amorphous PE

0.14 (600K)




Amorphous PP

0.07 (600K)





It can be seen that the thermal conductivity of polymers is strongly related to the potential function. The all-atom potential like PCFF and COMPASS is usually more accurate than united-atom potential like OPLS-UA. The accurate potential often means time-consuming computation. Researchers need to consider the accuracy and computational cost. Secondly, the thermal conductivity of single polymer chains is always higher than bulk polymers due to less phonon scattering. Compared with amorphous polymers, thermal conductivity of crystalline polymers is always higher due to well-organized structure. Thermal energy can transport more effectively along polymer backbones. Excessive branching can cause more phonon scattering, thus is unfavorable for heat transport. Therefore, polymers with less side chains, aligned molecular chains, ordered structure are desired thermal conductive material.

   Apart from tuning the intrinsic thermal conductivity of polymers, doping high thermal conductive fillers is another popular way to increase the thermal conductivity of polymers. As previously mentioned, the major barrier to fabricate thermal conductive polymer nanocomposites is the ITR between fillers and matrix. The ITC between typical fillers and matrix is shown in Table 3.


Table 3 Interfacial thermal conductance between typical fillers and polymer matrix.





Computing method

Potential function








Graphene with Stone-Wales defect











































Hexanoic acid





Gold + CH3 SAM






Gold + hetero-length SAMs






Gold + hetero-length SAMs






Gold + hetero-CH3 SAMs






Gold + pseudo-hetero-

length SAMs






Gold + mixed short SAMs






Gold + COOH SAMs






Gold + CH3 SAMs






Gold + COOH SAMs







As can be seen from Table 3, the ITC also depends on the computational method and potential function. Therefore, it is necessary to check the potential function before the simulation. In addition, the ITC is related to the contact area, binding energy and molecular polarity. Generally speaking, covalent connections can be more efficient to enhance the ITC than nonconvalent connections. Nevertheless, covalent bonding can damage the intrinsic structure of the fillers. SAMs with the similar backbone to the matrix can protect the fillers and be used as the phonon bridge to increase the vibrational coupling, which can effectively enhance the thermal transport across the interface.


4. Experimental advances in fabricating high thermal conductive polymers

Fig. 12 Fabricate high thermal conductive polymers in experiments. (a) Schematic of stretched PE microfiber.6 (b) Fabricating PE nanofibers from PE microfibers by local heating and drawing.8

(c) Schematic of electrospinning equipment.157 (d) Aligned PE chains in the electrospinning process.158

(e) Schematic of polymers with rigid backbone and strong inter-chain interaction.114


   High thermal conductive polymers have been realized not only in simulations but also in experiments. Fig. 12 displays the typical preparation methods including mechanical stretching, electrospinning and molecular engineering. The common mechanism behind these methods is that they can increase the alignment of polymer chains. In addition, when applying large strain to the sample, mechanical stretching can elevate the crystallinity of sample. Electrospinning is a popular method to prepare polymer fibers. Molecular engineering can modulate thermal transport property of polymers from the molecular level, which can be used in conjunction with electrospinning technology to fabricate high thermal conductive polymers. The typical preparation methods, samples and corresponding thermal conductivity are listed in Table 4.


Table 4 Typical samples, thermal conductivity and preparation methods.



κ  (W/m-1K-1)

Preparation method


Ultra-drawn PE nanofibers


Two-stage heating and drawing


Crystalline PE nanofibers

90 (150K)

Local heating and drawing


PE films


Flow extrusion and drawing


UHMW PE microfibers


Heat-stretching method


Liquid crystalline PBO fibers


Embedding fibers and microdissection


PEO naofibers








HDPE nanowire arrays


Nanoporous template wetting


Electrospun PE nanofibers




HDPE nanofibers


Melt infiltration of AAO templates


Cross-linked LCER


Surface treatment


P3HT/MWCNT nanofibers


Template method


PMDA/ODA nanofibers




Epoxy resin/3D CNT


Ice-templating SAM and infiltration


Amorphous PT nanofibers


Nanoscale templates




Melt extrusion


P3HT film


Oxidative chemical vapor deposition


Nylon 11 nanofibers


Electrospinning and post-stretching


Electrospun PVA nanofibers




PAP:PAA blend fims


Mixing method






Amorphous PAA films


Systematic ionization






Diamond-ABS filament


3D printing


PVA membrane




PVA/Fe3O4 films


Film casting and magnetic alignment


PAAm hydrogels


Free radicals copolymerization method


Ordered LCER


Linear polymerization of the epoxy



As shown in Fig. 12(a), Shen et al fabricated ultra-drawn PE nanofibers with extremely high thermal conductivity.6 Kim et al fabricated high thermal conductive polyacrylic acid (PAA) film with the help of coulombic force172. Singh et al fabricated high thermal conductive polythiophene nanofibers by nanoscale templates.12 Xu and Chen et al fabricated high thermal conductive PE film by employing flow extrusion and mechanical strain.159 The common point here is to increase the polymer chain orientation.

    Fig. 12(b) shows that the process of preparing PE nanofibers with both high strength and thermal conductivity by local heating method.8 Cahill7 et al measured thermal conductivity of different fibers like PE, PVA, PAA etc. They discovered that the polymers with higher strength have higher thermal conductivity.7,179 Zhu et al fabricated high thermal conductive PVA/Fe3O4 composites in conjunction with magnetic field. The common underlying mechanism is that the higher mechanical strength leads to higher sound speed, which can facilitate thermal transport in polymers.

     Fig. 12(c) and (d) display the vertical set up of electrospinning and oriented PE chains in the electrospinning process, respectively. Ma and Li et al discover that the thermal conductivity of electrospun polymer nanofiber is associated with the electric field.158 The electric filed can influence the arrangement of polymer chains, which may lead to scattered thermal conductivity of electrospun nanaofibers.

    Polymer blends with high thermal conductivity can be prepared by forming strong interchain bonds.113 Zhu et al fabricated high thermal conductive films by employing small organic linkers.180 The underlying mechanism is that strong interchain interaction promotes phonon transport.181 Apart from interchain interaction, intrachain interaction also plays an important role in thermal transport. As shown in Fig. 12(e), Xu and Chen et al fabricated high thermal conductive conjugated polymers by simultaneously tuning intrachian interaction and interchain interaction.114 The high thermal conductivity results from efficient thermal conductive network in conjugated polymers.

    Compared with the simulation, measuring the ITC in experiment is arduous due to the such small sample, expensive equipment and scrupulous operation. The electrothermal 3ω  method, pump/probe thermoreflectance techniques like time-domain thermoreflectance (TDTR) and frequency-domain thermoreflectance (FDTR) are the predominant methods to measure the ITC.182 The typical experimental results are listed in Table 5.


Table 5 Typical interfacial thermal conductance and measuring methods.




Measuring method




Thermal measurements


Cooper/SAM-CH3/ epoxy


Thermal measurements


Cooper/SAM-NH2/ epoxy


Thermal measurements


CNT/hydrocarbon liquid


Laser measurement




Laser flash method


Magnetic oriented BN/epoxy


Laser flash method


Au/paraffin wax


Transient thermoreflectance




Transient thermoreflectance


Au/ alkanethiol

SAMs/ paraffin wax


Transient thermoreflectance


Au/ alkanethiol

SAMs/ hexadecane


Transient thermoreflectance














Sapphire/silane SAM/PS
























As can be seen from table 5, the typical ITC between polymers and fillers is in the range of 5~170 MWm-2K-1. The SAMs with different end-group molecules can have different effects on the interfacial thermal transport. SAMs that form strong interaction with the matrix often facilitate interfacial thermal transport, whereas SAMs that still keep weak interaction with the matrix can impede interfacial thermal transport. The simulations and experimental results confirm that the SAMs can be a good candidate to enhance interfacial thermal transport.

5.  Summary and outlook

For thermal transport in crystalline materials, lattice dynamic theory can help us to understand the underlying mechanisms very well. However, thermal transport in amorphous material is still not well understood. Although A-F theory provides a useful methodology to specify important carriers to thermal transport, distinguishing propagons and diffusons is insurmountable for large size systems. Some theoretical models have been proposed to understand the interfacial thermal transport and calculate the interfacial thermal conductance. Nevertheless, these models often need to input the eigenvector, eigenfrequency, and phonon group velocity. For polymers with many atoms in super cell, the lattice dynamic calculation is often intractable due to huge amount of calculation. The emergence of thermal rectification provides hope for manipulating the heat current while the low thermal rectification ratio and expensive cost hinder the applications. Powerful computational methods can accelerate the research on thermal properties of materials. The thermal conductivity can be obtained according to Green-Kubo relations in EMD simulation or Fourier’s law in NEMD simulation. However, accurate potentials are essential to get the accurate thermal conductivity, which will consume large amount of computing resource due to the complicated potentials and vast neighbor atoms. Furthermore, multiple independent simulations are necessary to get the thermal conductivity with small standard error. Advanced spectral decomposition methods are introduced to assist us to real the phonon transport mechanism inside the material and at the interface. High thermal conductive polymers and polymer nanocomposites have been fabricated in experiments, which promotes the practical applications of thermal conductive polymers. Some consensuses have been reached on enhancing thermal conductivity of polymers, like aligning the polymer chain, increasing the stiffness of polymer backbone, restricting the angular bending freedom, and elevating the crystallinity of polymers,

      However, there still remain some open questions. First and foremost, the experimentally measured thermal conductivity and interfacial thermal conductance results are scarce, especially for low dimensional polymers like one-dimensional polymer chain and two-dimensional polymer films. Only a few papers report the thermal conductivity of low dimensional polymers at room temperature. Preparing low dimensional polymers and measuring their thermal conductivities still remain challenges. The scattering simulation results sometimes contain numerical tricks and can’t provide effective guidance to design high thermal conductive polymers. The accurate experimental results can efficiently complement the shortages of the simulation and speed up the industrialization of the thermal conductive polymers. Therefore, developing feasible method to overcome the experimental challenges will be a research focus. Secondly, the reliable and pragmatic theoretical model that can accurately predict the thermal conductivity of polymers and polymer nanocomposites is still lacking. The results based on the empirical relations can reflect a general trend and fail to produce the precise thermal conductivity value. The NEMD simulation for small system is often accompanied by large temperature gradient on the order of 109 K/m, which is extremely rare in the realistic situation. In addition, the simulation results don’t always agree with the experimental results due to the complicated simulation details like suitable initial structure, proper boundary condition, accurate potential function, rational time step and enough simulation time. Phonons are considered to obey Maxwell-Boltzmann distribution in MD simulation whereas phonons obey Bose-Einstein distribution in reality. Hence the theory of MD simulation needs to be further developed to have more physical meaning and to agree with the simulation result. Thirdly, large interfacial thermal resistance at the material interface is still a bottleneck in enhancing thermal conductivity of polymer nanocomposites. Novel strategies to reduce interfacial thermal resistance are still worthwhile to develop. Although polymer fibers can have high thermal conductivity along the chain, the thermal transport vertical to the chain is still unsatisfactory. Isotropic polymers with high thermal conductivity needs to be prepared. Finally, in order to apply thermal rectification in thermal management, more investment are needed to study polymer thermal diode in experiment.


The authors are grateful to Quanwen Liao, Xiaoxiang Yu, Runchun Tu, Zheyong Fan, Dengke Ma, Ji Li, Shan Gao, Meng An and Nuo Yang for valuable discussions. This work was supported by the National Natural Science Foundation of China (Grant No. 51776079) and the National Key Research and Development Program of China (No.2017YFB0603501-3).

Conflict of interest

The authors declare no financial interest.


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electronic thermal conductivity


dissipation rate


phononic thermal conductivity


acrylonitrile butadiene styrene


propagons’ thermal conductivity


liquid crystalline epoxy resin


diffusons’ thermal conductivity




locons’ thermal conductivity




forward thermal conductivity




backward thermal conductivity


polyacrylic acid 


total thermal conductivity




forward heat flux


poly (acrylamide)


backward heat flux


poly (n-acryloyl piperidine)


heat flux




heat current




thermal conductance


cross-linked polyethylene


cross-sectional area


low-density polyethylene


temperature gradient


high-density polyethylene


temperature difference




volumetric specific heat




phonon group velocity




average phonon group velocity




phonon relaxation time


poly (3-hexylthiophene)


phonon mean free path



Reduced Planck’s constant




Phonon wave vector




phonon polarization


polyvinyl chloride


Phonon frequency


polyvinylidene fluoride




boron arsenide


Bose-Einstein distribution


boron nitride


force constant matrix


boron nitride nano-sheets


dynamical matrix


anodic aluminum oxide


mode diffusivity








thermal rectification ratio


carbon nanotube


potential energy function


multiwalled carbon nanotube


total energy


dubbed porous graphene




silicon carbide


atomic velocity


gallium nitride


observable property




phase space function


Self-assembled monolayer


Dirac δ -function


phononic crystals


damping term




random force


intermediate layer










Boltzmann transport equation




atomistic Green’s function




anharmonic lattice dynamics




molecular dynamics




equilibrium molecular dynamics




non-equilibrium molecular dynamics


density of states


participation ratio


eigenvector periodicity


localization ratio


interfacial thermal conductance


interfacial thermal resistance


thermal rectification


Time-domain thermoreflectance


Frequency-domain thermoreflectance


velocity autocorrelation function


vibrational power spectra




Adaptive intermolecular reactive empirical bond order potential


Condensed-phase optimized molecular potentials for atomistic simulation studies


Consistent valence forcefield


Optimized potential for liquid simulations-all atom model


Optimized potential for liquid simulations-united atom model


Polymer consistent force field


Reactive force-field