Received: 09 May 2019
Revised: 26 Jun 2019
Accepted: 28 Jul 2019
Published online: 29 Jul 2019
Bo Zhang^{1}, Peng Mao^{1}, Yunmin Liang^{1}, Yan He^{2}, Wei Liu^{1*} and Zhichun Liu^{1*}
^{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: Email: zcliu@hust.edu.cn (Zhichun Liu), w_liu@hust.edu.cn (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 nonequilibrium 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.
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
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^{1}K^{1}, which hinders their applications in thermal management.^{13}^{ }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.^{68}
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 crosssectional 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 nonmagnetic materials can be given by
κ=κ_{e}+κ_{ph } (1.4)
where κ_{e} represents the electronic thermal conductivity, and κ_{ph} 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 equation^{10,11}
κph=13C_{v}v_{g}Λ (1.5)
where C_{v} 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 nitride^{15} 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, stateoftheart 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.
2.1 Thermal transport in crystalline materials
In dielectric medium, phonons play a dominant role in heat conduction, whose occupation obeys BoseEinstein statistics. The thermal conductivity is related to the phonon energy ℏω(q,j), phonon group velocity νg (q,j), phonon occupation function f_{BE}(q,j), phonon relaxation time^{1921} τ(q,j)
(2.1)
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
(2.2)
(2.3)
Substituting Eq. (2.2), (2.3) into Eq.(2.1), we can find that
(2.4)
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
(2.5)
Hence phonon dispersion relation is necessary to get phonon group velocity, which is determined by the following secular equation
(2.6)
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
(2.7)
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, phononphonon scattering induced by anharmonic interaction becomes the main source of scattering. The extremely high thermal conductivity of carbon nanotube^{22 }(CNT) and graphene^{2325} 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 phononisotope scattering.^{2629} 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) Choneycomb 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 Choneycomb.^{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,
(2.9)
where , , and denote the total phonon scatterings, phononimpurity scatterings, phononelectron scatterings, phononboundary scatterings, and phononphonon 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 electronphonon 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) Phononimpurity scattering. (b) Phononelectron scattering. (c) Phononboundary scattering. (d) PhononPhonon 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.^{3740} The total thermal conductivity can be written as
(2.10)
where κ_{pr}, κ_{dif}, and κ_{lo} 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 silicon^{39}.
Note: the xaxis 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
(2.11)
where ω_{IR} is the cutoff frequency of the propagons. In contrast, diffusons and locons don’t have welldefined group velocity, whose contribution to thermal conductivity can be described by
(2.12)
(2.13)
where ω_{max }is the maximum frequency of density of states, D_{i} is the mode diffusivity, S_{ij }= < iSj > 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 IoffeRegellimit.^{42} Larkin and McGaugyhey et al propose to use structure factors to distinguish propagons and diffusons. The structure factor is defined as
(2.14)
where the summation is over the gamma modes, E^{L} and E^{T} refer to the longitudinal and transverse polarization, respectively.^{43} However, this method requests that the mode character must change abruptly in the vicinity of IoffeRegel 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
(2.15)
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 γ_{n} denotes the mode is propagating, while the small value of γ_{n} denotes the mode is nonpropagating.
Locons are localized modes whose eigenvector decays dramatically with distance from the center. The participation ratio^{46} (PR) can be used to distinguish extended modes (propagons, diffusons) and localized modes, which is given by
(2.16)
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 PR^{47,48} can be given by
(2.17)
where N is the number of atoms in the system, DOS_{S}(ω) is the local DOS of sth atom calculated from Fourier transformation of normalized velocity autocorrelation function (VACF)^{49}
(2.18)
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 (SiPnCs), amorphous silicon (aSi), amorphous silica (aSiO_{2}), 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 SiPnCs and aSi, 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 aSiO_{2} 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 aSi. (c) Structure of aSiO_{2}. (d) Structure of atactic PS. (e) Participation ratio spectrum of SiPnCs. Localization ratio refers to localization ratio. (f) DOS and PR spectra of aSi. (g) DOS and PR spectra of aSiO_{2}. (h) DOS and PR spectra of atactic PS. (Ref. [50,51])
2.3 Thermal transport across interface
Interfacial thermal transport is crucial to thermal management of microelectronics^{52,53 }and thermal transport of polymer nanocomposites^{5456 }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 by^{5759}
(2.19)
where J is the heat flux across the interface. The ITR is the reciprocal of ITC. The acoustic mismatch model^{60} (AMM) and diffuse mismatch model^{61} (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
(2.20)
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.^{6466} 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 hBN and graphene heterostructure with different interface. Defects like isotopes,^{6769} vacancies,^{70,71} and dislocations^{72,73 }are often detrimental to thermal transport. Contrary to conventional belief, interfacial defects unexpectedly facilitate thermal transport across the interface. Liu et al find that hBN/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 by^{63}
(2.21)
or^{66}
(2.22)
where m_{s} is the mass of sth atom, vs is the vibrational velocity of sth atom, ω is the phonon angular frequency. The spectral temperature T_{sp}(ω) can be defined as
(2.23)
where T^{eq} is the temperature of equilibrium state, VPS^{neq}(ω) and VPS^{eq}(ω) represent the VPS of nonequilibrium state and equilibrium state, respectively.^{76} The overlap of VPS can demonstrate the elastic thermal transport across the interface, which can be given by^{77}
(2.26)
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 hBN 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 AuSAM/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 energy^{79,80} and larger effective contact area^{81} 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 management^{82} and phononic information technology.^{8385} The TR ratio η can be defined as
(2.27)
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.^{8688} 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.^{8992} The asymmetry of geometry,^{9395} mass distribution,^{96,97} and defects distribution^{98100} can also induce mismatch of VPS to get high TR ratio in conjunction with nonlinear interactions. However, the TR ratio of CNT deposited C_{9}H_{16}Pt 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 nanoribbons^{94,95,101} 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 largearea 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 largescale 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 crosslink 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 pathways^{113,114 }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 fillermatrix coupling^{116 }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.
3.1 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 
Features 
Shortcomings 
Application range 
Boltzmann transport equation^{48} 
(1) Phonons obey quantum distribution (2) Couple with first principle calculation 
(1) Ignore wave effect (2) Successive phonon scatterings are independent^{5} 
crystalline material with weak anharmonicity 
Molecular dynamics simulation^{118} 
(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 function^{119} 
(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 BornOppenheimer 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 law^{120}
(3.1)
where t is the time, m_{i}, r_{i}, F_{i}, U_{i} 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
(3.2)
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
(3.3)
Based on ergodic hypothesis, the microcanocial phase space averages can be equivalent to time averages over the trajectory according to
(3.4)
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 as^{122}
(3.5)
(3.6)
(3.7)
Where ζ is the deterministic damping term and τ is the relaxation time. The Langevin thermostat is a stochastic thermal bath whose temperaturecontrol equation can be written as
(3.8)
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 nonequilibrium 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 frequencydependent heat current Q(ω). Hence the frequencydependent ITC G(ω) can be defined as
(3.2)
Similarly, the frequencydependent thermal conductivity κ(ω) can be written as
(3.3)
For twobody potential, the heat current between atoms s and s' (belong to chunk A and chunk B, respectively) can be written as^{123125}
(3.4)
where F_{ss’} is the interatomic force between atoms s and s’; υ_{s} and υ_{s’} are the velocity of atoms s and υ_{s’}, respectively. The auxiliary correlation function for interparticle heat current can be defined as^{126}
(3.5)
The total heat current between chunk S and chunk S’ can be written as
(3.6)
The auxiliary correlation function for total heat current between chunk A and chunk B can be written as
(3.7)
For manybody 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 welldefined manybody heat current formula has been derived by Fan et al^{ 127}
(3.8)
where U_{s} and U_{s’} are the site potential of atoms s and s’, respectively. r_{ss’} is the relative position and can be written as
(3.9)
With respect to manybody potential, the auxiliary correlation function for interparticle heat current can be given by
(3.10)
The total heat current between chunk A and chunk B can be given by
(3.11)
The auxiliary correlation function for total heat current between chunk A and chunk B can be given by
(3.12)
The Fourier transformation pairs about the auxiliary correlation function can be defined as^{127129}
(3.13)
(3.14)
Since K_{A}_{→}_{B}(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 as^{130}
In order to distinguish elastic and inelastic spectral thermal conductance, Saa skilahti et al^{128} and Zhou et al^{131} propose to use secondorder force constants and thirdorder 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 fluctuationdissipation theorem^{134} and linear response theory,^{135137} thermal conductivity can be calculated by GreenKubo formula in equilibrium molecular dynamics (EMD) simulation
(3.16)
For two body potential, heat current J can be defined as
(3.17)
For many body potential, heat current can be defined as^{138,139}
(3.18)
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 parallellinked 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.
Sample 
κ (W/m^{1}K^{1}) 
Computing method 
Potential function 
Refs 
Single PE chains 
350 
EMD 
AIREBO 
[4] 
Bulk PE crystals 
237 
ALD 

[144] 
DPG1 zigzag 
84.4 
EMD 
PCFF 
[145] 
DPG1 armchair 
110.8 
EMD 
PCFF 
[145] 
Aligned CNTPE 
99.5 
NEMD 
Morse + cosine + LJ + AIREBO 
[14] 
Crystalline PEO 
60 
EMD 
PCFF 
[143] 
Twisted PE chains 
60 
EMD 
AIREBO 
[146] 
Bulk single PE crystal 
50 
EMD 
AIREBO 
[147] 
Single PT chains 
43.3 
EMD 
ReaxFF 
[12] 
Nylon 10 
7.83 
NEMD 
OPLSAA 
[142] 
Single PVA chains 
7.01 
EMD 
COMPASS 
[68] 
Single PDMS chain 
6 
NEMD 
COMPASS 
[148] 
Single PP chains 
5.83 
EMD 
COMPASS 
[68] 
Aligned single PNb chains 
2.54 
EMD 
PCFF 
[109] 
PAAm hydrogels 
0.98 
EMD 
OPLS + TIP4P 
[149] 
Parallellinked epoxy resin 
0.8 
EMD 
CVFF 
[108] 
Bulk PNb crystal 
0.72 
EMD 
PCFF 
[150] 
Amorphous PEO 
0.37 
EMD 
PCFF 
[143] 
Amorphous paraffin wax 
0.332 
NEMD 
AIREBO 
[13] 
Amorphous paraffin wax 
0.327 
NEMD 
COMPASS 
[13] 
Amorphous PT 
0.3 
EMD 
ReaxFF 
[5] 
Crystalline PE 
0.26 (radial) 
NEMD 
PCFF 
[151] 
Amorphous PS film 
0.249 
EMD 
CVFF 
[152] 
Amorphous PE 
0.27 
NEMD 
NERD united potential 
[153] 
Amorphous PE 
0.22 
NEMD 
COMPASS 
[141] 
Amorphous PDMS 
0.2 
NEMD 
COMPASS 
[148] 
Amorphous PS 
0.17 
EMD 
PCFF 
[109] 
Amorphous PS 
0.16 
NEMD 
PCFF 
[153] 
Amorphous PE 
0.14 (600K) 
NEMD 
OPLSUA 
[154] 
Amorphous PP 
0.07 (600K) 
NEMD 
OPLSUA 
[154] 
It can be seen that the thermal conductivity of polymers is strongly related to the potential function. The allatom potential like PCFF and COMPASS is usually more accurate than unitedatom potential like OPLSUA. The accurate potential often means timeconsuming 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 wellorganized 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.
Fillers 
Matrix 
G(MWm2K1) 
Computing method 
Potential function 
Refs 
Graphene 
Epoxy 
135.53 
NEMD 
PCFF 
[71] 
Graphene with StoneWales defect 
Epoxy 
162.642 
NEMD 
PCFF 
[71] 
Graphene 
PE 
136.2 
NEMD 
ReaxFF 
[155] 
Graphene 
Paraffin 
71 
NEMD 
AIREBO+COMPASS 
[13] 
Graphene 
PE 
56 
NEMD 
PCFF 
[156] 
BN 
Hexane 
90.47 
NEMD 
Tersoff+PCFF 
[57] 
BN 
Hexanamine 
113.38 
NEMD 
Tersoff+PCFF 
[57] 
BN 
Hexanol 
136.16 
NEMD 
Tersoff+PCFF 
[57] 
BN 
Hexanoic acid 
155.17 
NEMD 
Tersoff+PCFF+UFF 
[57] 
Gold + CH_{3} SAM 
Hexylamine 
55.33 
EMD 
Morse+PCFF+UFF 
[81] 
Gold + heterolength SAMs 
Hexylamine 
92.85 
EMD 
Tersoff+PCFF+UFF 
[81] 
Gold + heterolength SAMs 
Propylamine 
102.11 
EMD 
Morse+PCFF+UFF 
[81] 
Gold + heteroCH_{3} SAMs 
Hexylamine 
109.60 
EMD 
Morse+PCFF+UFF 
[81] 
Gold + pseudohetero length SAMs 
Hexylamine 
120.13 
EMD 
Tersoff+PCFF+UFF 
[81] 
Gold + mixed short SAMs 
Hexylamine 
168.75 
EMD 
Morse+PCFF+UFF 
[81] 
Gold + COOH SAMs 
Hexylamine 
208.78 
EMD 
Morse+PCFF+UFF 
[81] 
Gold + CH_{3} SAMs 
Hexane 
55.18 
EMD 
Tersoff+PCFF+UFF 
[81] 
Gold + COOH SAMs 
Hexane 
73.76 
EMD 
Morse+PCFF+UFF 
[81] 
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.
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 interchain 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.
Sample 
κ (W/m^{1}K^{1}) 
Preparation method 
Refs 
Ultradrawn PE nanofibers 
104 
Twostage heating and drawing 
[6] 
Crystalline PE nanofibers 
90 (150K) 
Local heating and drawing 
[8] 
PE films 
62 
Flow extrusion and drawing 
[159] 
UHMW PE microfibers 
51 
Heatstretching method 
[160] 
Liquid crystalline PBO fibers 
20 
Embedding fibers and microdissection 
[7] 
PEO naofibers 
13~29 
Electrospinning 
[161] 
PVDF/BNNS film 
16.3 
Electrospinning 
[162] 
HDPE nanowire arrays 
10 
Nanoporous template wetting 
[107] 
Electrospun PE nanofibers 
9.3 
Electrospinning 
[158] 
HDPE nanofibers 
9 
Melt infiltration of AAO templates 
[163] 
Crosslinked LCER 
5.8 
Surface treatment 
[164] 
P3HT/MWCNT nanofibers 
4.7 
Template method 
[165] 
PMDA/ODA nanofibers 
4.6 
Electrospinning 
[166] 
Epoxy resin/3D CNT 
4.42 
Icetemplating SAM and infiltration 
[167] 
Amorphous PT nanofibers 
4.4 
Nanoscale templates 
[12] 
HDPE/BN 
3.57 
Melt extrusion 
[168] 
P3HT film 
2.2 
Oxidative chemical vapor deposition 
[114] 
Nylon 11 nanofibers 
1.6 
Electrospinning and poststretching 
[169] 
Electrospun PVA nanofibers 
1.5 
Electrospinning 
[170] 
PAP:PAA blend fims 
1.5 
Mixing method 
[113] 
PMMABNNS/Ag 
1.48 
Solutionblending 
[171] 
Amorphous PAA films 
1.2 
Systematic ionization 
[172] 
PAI/3D BN 
1.17 
crosslinking−freezedrying−infiltration 
[173] 
DiamondABS filament 
0.94 
3D printing 
[174] 
PVA membrane 
0.7 
Electrospinning 
[175] 
PVA/Fe_{3}O_{4} films 
0.63 
Film casting and magnetic alignment 
[176] 
PAAm hydrogels 
0.51 
Free radicals copolymerization method 
[177] 
Ordered LCER 
0.48 
Linear polymerization of the epoxy 
[178] 
As shown in Fig. 12(a), Shen et al fabricated ultradrawn PE nanofibers with extremely high thermal conductivity.^{6} Kim et al fabricated high thermal conductive polyacrylic acid (PAA) film with the help of coulombic force^{172}. 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} Cahill^{7} 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/Fe_{3}O_{4} 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 timedomain thermoreflectance (TDTR) and frequencydomain 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.
Sample 
G(MWm2K1) 
Measuring method 
Refs 
Cooper/epoxy 
12.5 
Thermal measurements 
[183] 
Cooper/SAMCH_{3}/ epoxy 
7.1 
Thermal measurements 
[183] 
Cooper/SAMNH_{2}/ epoxy 
142.9 
Thermal measurements 
[183] 
CNT/hydrocarbon liquid 
12 
Laser measurement 
[184] 
BN/epoxy 
13.2 
Laser flash method 
[185] 
Magnetic oriented BN/epoxy 
120.5 
Laser flash method 
[185] 
Au/paraffin wax 
25 
Transient thermoreflectance 
[78] 
Au/hexadecane 
28 
Transient thermoreflectance 
[78] 
Au/ alkanethiol SAMs/ paraffin wax 
165 
Transient thermoreflectance 
[78] 
Au/ alkanethiol SAMs/ hexadecane 
169 
Transient thermoreflectance 
[78] 
Au/ethanol 
17.7 
TDTR 
[186] 
Au/toluene 
13.8 
TDTR 
[186] 
Sapphire/PS 
7.6 
TDTR 
[187] 
Sapphire/silane SAM/PS 
58 
TDTR 
[187] 
Al/PS 
19.35 
TDTR 
[188] 
Si/PS 
16.28 
TDTR 
[188] 
Al/PMMA 
30 
3ω 
[189] 
Sapphire/HDPE 
5 
TDTR 
[190] 
Sapphire/PS 
7.0~21.0 
TDTR 
[191] 
As can be seen from table 5, the typical ITC between polymers and fillers is in the range of 5~170 MWm^{2}K^{1}. The SAMs with different endgroup 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.
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 AF 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 GreenKubo 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 onedimensional polymer chain and twodimensional 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 10^{9} 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 MaxwellBoltzmann distribution in MD simulation whereas phonons obey BoseEinstein 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.2017YFB06035013).
The authors declare no financial interest.
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Nomenclature 


κe 
electronic thermal conductivity 
γ 
dissipation rate 
κph 
phononic thermal conductivity 
ABS 
acrylonitrile butadiene styrene 
κpr 
propagons’ thermal conductivity 
LCER 
liquid crystalline epoxy resin 
κdif 
diffusons’ thermal conductivity 
PA 
polymide 
κlo 
locons’ thermal conductivity 
PA6 
nylon6 
κ+ 
forward thermal conductivity 
PA66 
nylon6.6 
κ 
backward thermal conductivity 
PAA 
polyacrylic acid 
κ 
total thermal conductivity 
PAI 
Polyamideimide 
J+ 
forward heat flux 
PAAm 
poly (acrylamide) 
J 
backward heat flux 
PAP 
poly (nacryloyl piperidine) 
J 
heat flux 
PBO 
polybenzobisoxazole 
Q 
heat current 
PE 
polyethylene 
G 
thermal conductance 
PEX 
crosslinked polyethylene 
A 
crosssectional area 
LDPE 
lowdensity polyethylene 
∇T 
temperature gradient 
HDPE 
highdensity polyethylene 
∆T 
temperature difference 
PDMS 
poly(dimethylsiloxane) 
Cv 
volumetric specific heat 
PI 
polyimide 
vg 
phonon group velocity 
PMMA 
poly(methylmethacrylate) 
vg 
average phonon group velocity 
PNb 
polynorbornene 
τ 
phonon relaxation time 
P3HT 
poly (3hexylthiophene) 
Λ 
phonon mean free path 
PP 
polypropylene 
ℏ 
Reduced Planck’s constant 
PS 
polystyrene 
q 
Phonon wave vector 
PT 
polythiophene 
j 
phonon polarization 
PVC 
polyvinyl chloride 
ω 
Phonon frequency 
PVDF 
polyvinylidene fluoride 
e 
eigenvector 
BAs 
boron arsenide 
fBE 
BoseEinstein distribution 
BN 
boron nitride 
Φ 
force constant matrix 
BNNS 
boron nitride nanosheets 
D(q) 
dynamical matrix 
AAO 
anodic aluminum oxide 
Di 
mode diffusivity 
Ag 
silver 
T 
transmissivity 
Au 
gold 
η 
thermal rectification ratio 
CNT 
carbon nanotube 
U 
potential energy function 
MWCNT 
multiwalled carbon nanotube 
E 
total energy 
DPG 
dubbed porous graphene 
H 
Hamiltonian 
SiC 
silicon carbide 
υ 
atomic velocity 
GaN 
gallium nitride 
A 
observable property 
Ge 
germanium 
a 
phase space function 
SAM 
Selfassembled monolayer 
δ 
Dirac δ function 
PnCs 
phononic crystals 
ζ 
damping term 
BB 
bottlebrush 
ξ 
random force 
IL 
intermediate layer 




LJ 
LennardJones 


BTE 
Boltzmann transport equation 


AGF 
atomistic Green’s function 


ALD 
anharmonic lattice dynamics 


MD 
molecular dynamics 


EMD 
equilibrium molecular dynamics 


NEMD 
nonequilibrium molecular dynamics 

DOS 
density of states 

PR 
participation ratio 

EP 
eigenvector periodicity 

LR 
localization ratio 

ITC 
interfacial thermal conductance 

ITR 
interfacial thermal resistance 

TR 
thermal rectification 

TDTR 
Timedomain thermoreflectance 

FDTR 
Frequencydomain thermoreflectance 

VACF 
velocity autocorrelation function 

VPS 
vibrational power spectra 

UHMW 
ultrahighmolecularweight 

AIREBO 
Adaptive intermolecular reactive empirical bond order potential 

COMPASS 
Condensedphase optimized molecular potentials for atomistic simulation studies 

CVFF 
Consistent valence forcefield 

OPLSAA 
Optimized potential for liquid simulationsall atom model 

OPLSUA 
Optimized potential for liquid simulationsunited atom model 

PCFF 
Polymer consistent force field 

ReaxFF 
Reactive forcefield 