Thermal conductivity of polymers: A simple matter where complexity matters

Debashish Mukherji [email protected] Quantum Matter Institute, University of British Columbia, Vancouver V6T 1Z4, Canada

Abstract

Thermal conductivity coefficient $\kappa$ measures the ability of a material to conduct a heat current. In particular, $\kappa$ is an important property that often dictates the usefulness of a material over a wide range of environmental conditions. For example, while a low $\kappa$ is desirable for the thermoelectric applications, a large $\kappa$ is needed when a material is used under the high temperature conditions. These materials range from common crystals to commodity amorphous polymers. The latter is of particular importance because of their use in designing light weight high performance functional materials. In this context, however, one of the major limitations of the amorphous polymers is their low $\kappa$ , reaching a maximum value of about 0.4 W/Km that is 2–3 orders of magnitude smaller than the standard crystals. Moreover, when energy is predominantly transferred through the bonded connections, $\kappa\geq 100$ W/Km. Recently, extensive efforts have been devoted to attain a tunability in $\kappa$ via macromolecular engineering. In this work, an overview of the recent results on the $\kappa$ behavior in polymers and polymeric solids is presented. In particular, computational and theoretical results are discussed within the context of complimentary experiments. Future directions are also highlighted.

I Introductory remarks

When a material is subjected to a non–equilibrium condition of temperature $T$ , such that one end of the material is kept at an elevated temperature $T_{\rm high}$ and a lower temperature $T_{\rm low}$ is maintained at the other end, a heat current flows from the hot to the cold region. This is a direct consequence of the second law of thermodynamics and is quantified in terms of the heat flux vector $\vec{j}$ . Here, the Fourier’s law of heat diffusion states that ${j}\propto\left(T_{\rm high}-T_{\rm low}\right)/\ell$ across a sample of length $\ell$ . The proportionality constant is thermal transport coefficient $\kappa$ , which is a key material property that commonly dictates the usefulness of a material for a particular application [1, 2, 3, 4, 5, 6, 7, 8].

Traditionally, heat flow in crystalline materials and in nano–structures have been of primary interest [9, 10, 11, 12, 13, 14]. Due to the long range order in crystals, the phonon mean free path $\Lambda$ are rather large and thus $\kappa\geq 100$ W/Km [15, 16, 17, 7]. In the carbon–based materials, $\kappa$ can even exceed 1000 K/Wm [18, 19, 20, 21]. A complete opposite class to the crystals is the amorphous solids, where $\Lambda$ is small, i.e, within the direct atom–to–atom contact. Here, $\kappa\leq 2$ W/Km and heat propagates via localized vibrations [22, 23, 24].

Within the class of amorphous solids, polymers are of particular importance because they usually provide a flexible platform for the design of advanced functional materials [25, 26, 27, 28]. Some examples include, but are not limited to, organic solar cells [12, 29, 30], electronic packaging and/or heat sinking materials [31, 32, 29, 33], thermal switches [34, 35, 36, 37, 38] and thermoelectric applications [39, 40]. However, the typical $\kappa$ values of the amorphous polymeric solids are further 5–10 times smaller [31, 32, 6, 41] than the standard amorphous materials (such as amorphous silicon). This often hinders the usefulness of polymers under the high $T$ conditions.

Most commonly known (non–conducting) commodity polymers can be categorized into the systems where non–bonded monomer–monomer interactions are either dictated by the van der Waals (vdW) forces or by the hydrogen bonds (H–bonds) [42]. Here, the interaction strength of vdW is about $k_{\rm B}T$ at a temperature $T=300$ K and the Boltzmann constant $k_{\rm B}$ , while the strength of a H–bond is between 4–8 $k_{\rm B}T$ depending on the dielectric constant of the medium [27, 43]. A few examples of the commodity polymers is shown in Fig. 1. Note that these particular polymers are chosen because their experimental and simulation data is readily available.

Refer to caption — Figure 1: Schematics of a few examples of different commodity polymeric structures. Top panel shows systems with van der Waals (vdW) interactions, i.e., poly(lactic acid) (PLA), polystyrene (PS), poly(methyl methacrylate)(PMMA), and poly(N-acryloyl piperidine) (PAP). Bottom panel shows hydrogen bonded (H–bond) systems, i.e., poly(vinyl alcohol) (PVA), poly(acrylic acid) (PAA), poly(acrylamide) (PAM), and poly(N-isopropyl acrylamide) (PNIPAM).

Experiments have reported that $\kappa\simeq 0.1-0.2$ W/Km for the vdW systems [6, 31, 44, 45, 46, 24] and for the H–bonded polymers $\kappa\to 0.4$ W/Km [31, 32, 47].

Table 1: Thermal transport coefficient

\kappa

for different commodity polymers and their corresponding glass transition

T_{\rm g}

. The data is compiled from the experimental literature, except for PNIPAM which is taken from simulation.

Interaction	Polymer	$\kappa$ [W/Km]	$T_{\rm g}$ [K]
vdW	PLA	0.064–0.090 [45]	335 [42]
	PS	0.175 [23]	373 [42]
	PMMA	0.200 [32]	378 [42]
	PAP	0.160 [32] & 0.200 [31]	380 [31]
H–bond	PVA	0.310 [32]	348 [42]
	PAA	0.370 [32]	385 [31]
	PAM	0.380 [32]	430 [42]
	PNIPAM	$0.316^{\rm sim}$ [47]	413 [47]

Table 1 lists $\kappa$ and the corresponding glass transition temperatures $T_{\rm g}$ for the polymers in Fig. 1. It can be seen in Table 1 that PMMA (a vdW–based polymer) has $\kappa\simeq 0.20$ W/Km and its $T_{\rm g}\simeq 378$ K, while PVA (a H–bonded system) has $\kappa\simeq 0.310$ W/Km and a lower $T_{\rm g}\simeq 348$ K [42]. These values indicate that $T_{\rm g}$ and $\kappa$ are not correlated, which is also visible across many polymeric systems [42]. Furthermore, a closer look suggests that– within a simple approximation, $T_{\rm g}$ is directly related to the trans–to–gauche free energy barrier $\Delta E_{\rm t-g}$ (i.e., local fluctuations), which is dictated by a delicate combination of the bonded, the angular and the dihedral interactions along a chain backbone. The higher the $\Delta E_{\rm t-g}$ , the larger the $T_{\rm g}$ . Here, PMMA with a larger side group has a higher $\Delta E_{\rm t-g}$ than a PVA, hence the observed trend in $T_{\rm g}$ . Within this simple discussion it becomes reasonably apparent that the exact $T_{\rm g}$ is a completely irrelevant quantity within the context of $\kappa$ in polymers.

Amorphous polymeric solids are a special case because even when their macroscopic $\kappa$ values are very small, at the monomer level they have different rates of energy transfer. For example, energy can be transferred between the bonded monomers and that between the neighboring non–bonded monomers. In this context, a closer investigation of the polymer structures reveal that the carbon–carbon (C–C) covalent bond constitute the most common backbone of commodity polymers, see Fig. 1. Here, it is known that the stiffness of a C–C contact is $E\geq 250$ GPa [48], while $E$ of a vdW or a H–bond system vary between 2–5 GPa [32]. Given that $\kappa\propto E$ [49, 14], the energy transfer between the two non–bonded monomers (soft contacts) is significantly smaller than along a (stiff) bond [50]. A simple schematic of this scheme is shown in Fig. 2. The thermal behavior in polymers are predominantly dictated by the non–bonded contacts, while the energy transfer along a bonded contact plays a lesser important role. This is particularly because a chain in a frozen configuration follows the random walk statistics [51, 52, 53], i.e., when it is quenched to $T\ll T_{\rm g}$ from a melt configuration. Within this picture, when heat flows along a chain contour, it experiences scattering due to the bends and the kinks along the path [54, 55]. Energy also occasionally hops off to a non–bonded neighboring monomer. A combination of these two effects facilitates a knocking down of $\kappa$ in the amorphous polymers [23, 24, 56]. Note also that the exact monomer structure, i.e., the side groups connected to a backbone play an additional (delicate) role [57, 58, 59], which will be discussed at a later stage within this short overview.

Over the last 2–3 decades, extensive efforts have been devoted to study the heat flow in polymers using the experimental, theoretical, and computational approaches [60, 31, 32, 29, 61, 35, 62, 36, 63, 64, 65, 66]. In particular, even when the polymers are a class of simple matter with their great potential in designing flexible materials, establishing a microscopic understanding in polymers is rather complex. Here, one of the grand challenges in this field is to attain a predictive tunability in $\kappa$ (almost at will) using macromolecular engineering. This requires a protocol that can properly account for a delicate balance between the bonded to the non–bonded interactions, the chain conformations, and their morphology. Motivated by the above, this manuscript aims to highlight the latest developments in the field of polymer thermal conductivity. For this purpose, comparative experimental and simulation results will be discussed to put forward the key concepts.

II Effect of blending on the thermal conductivity of polymers

In a system consisting of one (linear) polymer component, $\kappa$ is rather limited because of the restrictive monomer–level interactions [32, 31, 47]. To circumvent this problem, studies have suggested that $\kappa$ of a polymeric solid may be enhanced by blending a second component with a relatively higher $\kappa$ . Here, an obvious choice is the carbon–based materials, such as the carbon nanotube (CNT) and polymer composites [29, 67, 68]. In such a composite, a significant increase in $\kappa$ requires concentration of CNT $\phi_{\rm CNT}$ exceeding their typical percolation threshold. While a CNT–polymer composite certainly show a significantly higher $\kappa$ than the bare polymers [29, 68], it also has two major drawbacks: (1) it looses the underlying flexibility (typical of polymers) because of a large $\phi_{\rm CNT}$ and their physical properties then get dominated by the CNTs present in the background polymers. (2) Polymers are rather cost effective, having their typical prices of about 2–3 orders of magnitude lower than the CNTs and thus a CNT–polymer composite inherently becomes significantly costlier.

A more plausible alternative is the polymer blends, where the non–bonded interactions can be altered by changing $\phi_{\rm second}$ of the second polymer component. Here, however, a prerequisite is that the two components remain fairly miscible over the full range of $\phi_{\rm second}$ [32, 31, 69, 70, 71]. On the contrary, when the two components in a blend phase separate, they create zones within a sample consisting of the individual components. These separate zones usually have very weak interfacial interaction and thus induce resistance for the heat flow, akin of the Kapitza resistance [72].

In a symmetric polymer blends (i.e., when both polymer components have a comparable degree of polymerization $N_{\ell}$ ), $\kappa$ varies monotonically between the two pure components [32, 71]. A recent experimental study, however, reported that an asymmetric blend (consisting of the longer PAA and shorter PAP chains) shows a larger enhancement in $\kappa\simeq 1.5$ W/Km around a PAP concentration of $\phi_{\rm PAP}\simeq 30\%$ [31]. It was argued that the PAP chains act as the H–bonded cross–linkers between the neighboring PAA chains, forming a 3–dimensional H–bonded stiff network. On the contrary, another set of experiments did not attain the same enhancement, instead found that PAA–PAP phase separate around $\phi_{\rm PAP}\simeq 30\%$ [32].

Motivated by the above contradicting experimental results, a simulation study suggested that the miscibility can be enhanced when PAP is replaced with PAM, i.e., a PAA–PAM system consisting of a long PAA and a short PAM [62]. PAA–PAM showed a weak non–monotonic variation in $\kappa$ with $\phi_{\rm PAM}$ , attaining a maximum $\kappa\simeq 0.4$ W/Km around $\phi_{\rm PAM}\simeq 30\%$ . Note also that the size of PAM molecules in this simulation study was chosen to be of the order of persistence length $\ell_{\rm p}\simeq 0.75$ nm (or 3 monomers), i.e., a PAM as a stiff linker that fits perfectly between two PAA chains. When the cross–linker length $N_{\ell}\gg\ell_{\rm p}$ , they form flexible cross–linking. This on one hand makes a network soft [73], on the other they also induce effective free volume (weak spots) within a network [64]. Collectively, these two effects reduce $\kappa$ . Something that speak in this favor is that the experimental results of cross–linked PAA reported $\kappa\simeq 0.27$ W/Km, which is about 25% smaller than $\kappa\simeq 0.37$ W/Km measured in a linear PAA [32].

The above discussions suggest that there is a need to look beyond the simple amorphous polymers. Therefore, in the following section some analytical approaches are first presented that may provide a guiding tool for the remaining discussions presented herein.

III Analytical models

In an isotropic material, $\kappa$ is directly related to the volumetric heat capacity $c$ , the group velocity $v_{{\rm g},i}(\nu)$ , and the phonon mean–free path $\Lambda(\nu)=\tau(\nu)v_{{\rm g},i}(\nu)$ . Here, $\tau(\nu)$ is the phonon life time and $\nu$ is the vibrational frequency. Starting from the above description, $\kappa$ can be written as,

\kappa(\nu)=\frac{1}{3}\sum_{i}c(\nu)v_{{\rm g},i}^{2}(\nu)\tau(\nu).

(1)

For an non–conducting amorphous material a well known theoretical description is the minimum thermal conductivity model (MTCM) [49] that is discussed in the following.

III.1 The minimum thermal conductivity model

Following Eq. 1, the general expression of $\kappa$ for a 3–dimensional isotropic system reads [5],

\kappa=\left(\frac{\rho_{\rm N}h^{2}}{3k_{\rm B}T^{2}}\right)\sum_{i}\int\tau(\nu)v_{{\rm g},i}^{2}(\nu)\frac{\nu^{2}e^{{h\nu}/{k_{\rm B}T}}}{\left(e^{{h\nu}/{k_{\rm B}T}}-1\right)^{2}}g(\nu){\rm d}\nu,

(2)

where $\rho_{\rm N}$ , $g(\nu)$ , $h$ are the total particle number density, the vibrational density of states, and the Planck constant, respectively. Within this description, MTCM uses the Debye model of lattice vibrations in Eq. 2 and proposes that a sample can be divided into regions of size $\Lambda(\nu)/2$ , whose frequencies are given by the low $\nu$ sound wave velocities $v_{i}$ , and thus approximates $\tau=1/2\nu$ [49] and $v_{{\rm g},i}=v_{i}$ . This gives,

\kappa=\left(\frac{\rho_{\rm N}h^{2}}{6k_{\rm B}T^{2}}\right)\left(v_{\ell}^{2}+2v_{t}^{2}\right)\int{\mathcal{I}}(\nu)g(\nu){\rm d}\nu,

(3)

with

{\mathcal{I}}(\nu)=\frac{\nu e^{{h\nu}/{k_{\rm B}T}}}{\left(e^{{h\nu}/{k_{\rm B}T}}-1\right)^{2}}.

(4)

$v_{\ell}=\sqrt{C_{\rm 11}/\rho_{\rm m}}$ and $v_{t}=\sqrt{C_{\rm 44}/\rho_{\rm m}}$ are the longitudinal and the transverse sound wave velocities, respectively. Here, $C_{\rm 11}=K+4C_{\rm 44}/3$ , $K$ is the bulk modulus, $C_{\rm 44}$ is the shear modulus, and $\rho_{\rm m}$ is the mass density.

One key quantity in Eq. 3 is $g(\nu)$ , which can be calculated by the Fourier transform of the mass–weighted velocity auto–correlation function $c_{\rm vv}(t)=\sum_{i}m_{i}\langle{\overrightarrow{v}}_{i}(t)\cdot{\overrightarrow{v}}_{i}(0)\rangle$ [74, 75] obtained from the classical simulations,

g(\nu)=\frac{1}{C}\int_{0}^{\infty}\cos(2\pi\nu t)\frac{c_{\rm vv}(t)}{c_{\rm vv}(0)}{\rm d}t.

(5)

The prefactor $C$ ensures that $\int g(\nu){\rm d}\nu=1$ . The representative $g(\nu)$ for four different commodity polymers are shown in Fig. 3. It can be appreciated that the polymers have many high $\nu$ quantum degrees–of–freedom that do not contribute to $\kappa$ at $T=300$ K. For example, a C–H bond vibration frequency in a polymer is $\nu\simeq 90$ THz, while the representative $\nu_{\rm room}\simeq 6.2$ THz at $T=300$ K. Such a mode, together with many other stiff modes (for $\nu>\nu_{\rm room}$ ), remain quantum–mechanically frozen at $T=300$ K [75]. If the contributions of these individual modes are not properly incorporated in Eq. 3 via the Bose–Einstein weighted function in Eq. 4, one can easily overestimate $\kappa$ within the classical simulations in comparison to the experimental data [47, 76, 75, 77, 56].

Standard analytical descriptions typically use the Debye estimate of parabolic vibrational density of states $g_{\rm D}(\nu)=3\nu^{2}/\nu_{\rm D}^{3}$ in Eq. 3. Here, $\nu_{\rm D}$ is the Debye frequency and is written as [5, 74],

\nu_{\rm D}=\left(\frac{9\rho_{\rm N}}{4\pi}\right)^{1/3}\left(\frac{1}{v_{\ell}^{3}}+\frac{2}{v_{t}^{3}}\right)^{-1/3}.

(6)

While $g_{\rm D}(\nu)$ is certainly a good approximation for the standard amorphous solids when $T\ll\Theta_{\Theta}$ , with $\Theta_{\rm D}=h\nu_{\rm D}/k_{\rm B}$ being the Debye temperature. Typical examples are amorphous silica and/or silicon, where $\Theta_{\rm D}\geq 480$ K [74, 78, 79]. Moreover, in the case $g(\nu)$ is complex, such as in the polymers (see Fig. 3), simplistic $g_{\rm D}(\nu)$ may lead to a wrong estimate of the low $\nu$ vibrational modes and thus leads to the default artifacts in computed $\kappa$ . This is particularly because $\Theta_{\rm D}\simeq 180-220$ K for the commodity polymers listed in Table 1, i.e., 20–40% smaller than $T=300$ K [32, 56] where typical experiments and simulations are performed.

When exact $g(\nu)$ from Fig. 3 are used in Eq. 3, $\kappa$ values can be reasonably reproduced within 5–20% error, see Fig. 4.

Note that this data is obtained by taking $v_{\ell}$ and $v_{t}$ from the experiments [32], while $\rho_{\rm N}$ and $g(\nu)$ are taken from simulations [56].

III.1.1 High temperature approximation with correction of the stiff modes

Within the high $T$ classical limit, i.e., when all modes are considered in Eq. 3 [80], the original MTCM for amorphous polymers can be written as,

\displaystyle\kappa_{\rm MTCM}

\displaystyle=\left(\frac{\pi}{48}\right)^{1/3}k_{\rm B}{N}^{2/3}\left(v_{\ell}+2v_{t}\right).

(7)

The corrections for the stiff modes in a polymer can then be incorporated in Eq. 7 by considering an effective number of atoms ${\overline{N}}=2(N-N_{\rm H})/3$ and $c=3{\overline{N}}k_{\rm B}$ . Here, ${\overline{N}}$ eliminates the stiff modes associated with the number of hydrogen atoms $N_{\rm H}$ and other stiff backbone modes. Following this, Eq 7 can be simply written as,

\displaystyle\kappa_{\rm MTCM}=\left(\frac{\pi}{432}\right)^{1/3}k_{\rm B}^{1/3}{c}^{2/3}\left(v_{\ell}+2v_{t}\right).

(8)

Eq. 8 gives estimates that are 30% larger than the corresponding experimental data [32, 47, 73].

III.1.2 Accurate computation of $\kappa$ via specific heat correction

An accurate computation of $\kappa$ within the standard classical molecular simulation setups is a daunting task, because polymers have quantum degrees–of–freedom whose exact contribution to the heat balance is rather non–trivial. Furthermore, $T>\Theta_{\rm D}$ for the commodity polymers (see Section III.1) and thus the polymer thermal properties are dominated by the low $\nu$ classical modes that are dominated by the non–bonded interactions (or the localized vibrations). On the contrary, the stiff modes (i.e., for $\nu>\nu_{\rm room}$ ) remain quantum–mechanically frozen and do not contribute to $\kappa$ . In this context, one of the key quantities that dictates $\kappa$ behavior in Eq. 1 for polymer is its $c$ .

In classical simulations, every mode in a polymer contribute equally to $c$ , i.e., given by the Dulong–Petit classical estimate. Here, it is well documented that the classically computed $c$ are always overestimated in comparison to the corresponding experimental data [74, 81, 75, 82] and thus also leads to an overestimation of $\kappa$ [76, 77, 56].

Given the above discussion, if $c$ is estimated accurately, it will automatically lead to an accurate computation of $\kappa$ . Recently a method has been proposed to compute the quantum corrected $c$ . This method uses the Binder approach to estimate the contributions of the stiff harmonic modes [74], which is then used to get the difference between the classical and the quantum descriptions [75],

\frac{\Delta c_{\textrm{rel}}(T)}{k_{\rm B}}=\int_{0}^{\infty}\left\{1-\left(\frac{h\nu}{k_{\textrm{B}}T}\right)^{2}\frac{e^{{h\nu}/{k_{\rm B}T}}}{\left(e^{{h\nu}/{k_{\rm B}T}}-1\right)^{2}}\right\}g(\nu){\rm d}{\nu}.

(9)

Finally the quantum corrected estimate of $c(T)$ is can be given by,

c(T)=c_{\rm cl}(T)-\Delta c_{\textrm{rel}}(T).

(10)

Here, the classical heat capacity is calculated using $c_{\rm cl}=[H(T-\Delta T)-H(T-\Delta T)]/2\Delta T$ and $H(T)$ is enthalpy. The main advantage of using Eq. 10 is that the stiff harmonic modes are corrected, while the contributions from the anharmonic (low $\nu$ ) modes remain unaffected.

Using Eq. 10, one can then calculated the quantum corrected $\kappa(T)$ ,

\kappa(T)=c(T)\frac{\kappa_{\rm cl}(T)}{c_{\rm cl}(T)}.

(11)

Here, $\kappa_{\rm cl}(T)$ is classically computed thermal transport coefficient using the standard equilibrium [83] and/or non–equilibrium [84, 85] methods. Fig. 5 show the computed $\kappa(T)$ for PMMA using $c(T)$ [56]. It can be seen that the quantum corrected $\kappa(T)$ compares reasonably with the corresponding experimental data, while the classical estimate is about a factor of three too high. The method proposed in Ref. [75] also highlighted different strategies to estimate $c(T)$ accounting for the missing degrees–of–freedom (DOF) within the united–atom and/or coarse–grained models. A direct implication is that a certain percentage error in $\kappa$ computed in the united–atom models comes from the missing DOFs [86].

It is also important to highlight that the simple scaling in Eq. 11 works reasonably for polymers because only the low $\nu$ modes dominate their thermal properties. When dealing with the crystalline materials special attention is need. For example, in a crystal, not only $c$ that has quantum effects, rather $v_{\rm g}$ (i.e., stiffness) [87] and $\tau$ also has quantum contributions at low $T\ll\Theta_{\rm D}$ .

III.2 Single chain energy transfer

In the introduction, it is discussed that a C–C bond is significantly stiffer [48] than the typical non–bonded contacts. A direct consequence of this microscopic interaction contrast is that the energy transfer between two bonded monomers is over 100 times faster than the energy transfer between two non–bonded monomers [88, 50]. Taking motivation from such distinct microscopic interactions, experimental and computational/theoretical studies have reported a large enhancement in $\kappa$ in the systems where bonded interactions dominate, such as in the single extended chains [54, 89, 90, 59], polymer fibers [60, 91], and/or molecular forests [55, 92]. However, until recently there existed no direct theoretical framework that could quantitatively decouple the effects of these two separate microscopic interactions in dictating the macroscopic heat flow in polymers [88, 50]. Therefore, in this section, the key ingredients of this simple chain energy transfer model (CETM) will be discussed.

Starting from a homogeneous sample consisting of linear polymers, CETM considers the diffusion of energy along a chain contour, i.e., between the bonded monomers. This involves multiple hops along a chain before infrequent energy transfers to the neighboring non–bonded monomer belonging to another chain. Note that there may also be non–bonded contacts between the two monomers belonging to the same chain, but topologically far from one another. However, this will require loop–like conformations of a chain in a dense polymeric system. Furthermore, the free energy difference to form such a loop of segment length $\mathcal{N}$ is given by ${\mathcal{F}}\left(\mathcal{N}\right)=mk_{\rm B}T\ln\left(\mathcal{N}\right)$ with a critical exponent $m=1.95$ [53]. Within this picture, a loop can only form when it overcomes a free energy barrier of several $k_{\rm B}T$ , which has a very low probability in a dense system. Note also that the CETM method does not distinguish between the intra– and intre–molecular non–bonded hopping.

Considering the first and the second neighboring bonded monomer transfers along a chain contour, The rate of change in the internal energy $\mathcal{E}$ for any monomer $i$ can then be simply written as [88, 50],

$\displaystyle\frac{{\rm d}{\mathcal{E}}_{i}}{{\rm d}t}=c_{\rm m}\frac{{\rm d}T_{i}}{{\rm d}t}$	$\displaystyle=G_{\mathrm{b}}(T_{i+1}-2T_{i}+T_{i-1})$	(12)
	$\displaystyle\quad+\tilde{G}_{\mathrm{b}}(T_{i+2}-4T_{i+1}+6T_{i}-4T_{i-1}+T_{i-2})$
	$\displaystyle\quad+nG_{\mathrm{nb}}(T_{\mathrm{bulk}}-T_{i})\,,$

with $G_{\rm b}/c_{\rm m}$ , ${\tilde{G}}_{\rm b}/c_{\rm m}$ , and $G_{\rm nb}/c_{\rm m}$ are the bonded, next nearest bonded, and non–bonded energy transfer rates, respectively. Here, individually $G$ values are thermal conductances, $c_{\rm m}$ is the specific heat of one monomer, $n$ is the number of non–bonded neighbors, $T_{i}$ is the temperature of the $i^{\rm th}$ monomer, and $T_{\rm bulk}=300$ K. Following the treatment presented in Ref. [50], diagonalizing Eq. 12 along the chain contour will lead to an exponential relaxation of the eigen–modes,

{\hat{T}}_{p}\left(t\right)\propto{e}^{-\alpha_{p}t},

(13)

with,

{\hat{T}}_{p}(t)=\sum_{i=0}^{N-1}\{T_{i}(t)-T_{\rm bulk}\}\cos\left[\frac{p\pi}{N}\left(i+\frac{1}{2}\right)\right],

(14)

and

\alpha_{p}=4\frac{G_{\rm b}}{c_{\rm m}}\sin^{2}\left(\frac{p\pi}{2N}\right)-16\frac{{\tilde{G}}_{\rm b}}{c_{\rm m}}\sin^{4}\left(\frac{p\pi}{2N}\right)+n\frac{G_{\rm nb}}{c_{\rm m}}.

(15)

In a nutshell, $p$ gives the effective length scale in a system, i.e., a particular $p$ mode corresponds to a length scale of $N_{\ell}/p$ . Fig. 6 shows the variation in $\alpha_{p}$ for PMMA. Fitting the simulation data (symbols) with Eq. 15 gives $G_{\rm b}/G_{\rm nb}\simeq 63$ . Note also that $G_{\rm b}/G_{\rm nb}\simeq 155$ for a polyethylene (PE) chain [50]. This difference is because of the rather bulky side group in PMMA that act as an additional scattering center for the energy transfer. This aspect will be discussed at a later stage.

It can also be appreciated in Fig. 6 that $\alpha_{p}$ almost plateaus for $4\sin^{2}\left(\pi p/2N_{\ell}\right)\geq 0.9$ (or $p\geq 11$ ). This length scale is comparable to $\ell_{p}\simeq N_{\ell}/p=2.7$ monomers (or 0.7 nm for PMMA), i.e., a length scale where ballistic energy transfer may dominate that is not considered within the formalism of CETM.

The energy transfer rates obtained using Eq. 15 can also be used to get a theoretical estimate of thermal transport coefficient within the Heuristic Random–Walk model [50],

\kappa_{\rm HRW}=\frac{\rho_{\rm N}}{6}\left[nG_{\mathrm{nb}}r_{\mathrm{nb}}^{2}+\left(G_{\mathrm{b}}-4\tilde{G}_{\mathrm{b}}\right)r_{\mathrm{b}}^{2}+\tilde{G}_{\mathrm{b}}\tilde{r}_{\mathrm{b}}^{2}\right].

(16)

Here, $r_{\rm nb}$ , $r_{\rm b}$ , and ${\tilde{r}}_{\rm b}$ are the average distances between a monomer and its first bonded, second bonded and non-bonded first shell neighboring monomers, respectively. It should, however, be noted that $\kappa_{\rm HRW}$ is underestimated for all investigated commodity polymers [50, 86].

One of the central assumptions in Eq. 12 is that the monomers surrounding the reference chain is kept at a constant $T_{\rm bulk}=300$ K [50]. This is certainly a good approximation for the common amorphous polymers, where the heat leakage between the non–bonded monomers is rather weak and mostly restricted upto the first non–bonded neighbor. Moreover, in the polymers and lubricants under high pressure [80, 77, 93], in the confined hydrocarbons [94, 95], and/or in the systems where $\pi-\pi$ stacking is dominant [30, 29, 39], heat leakage between the non–bonded monomers can be significantly enhanced. In these cases, the formalism within CETM may not be directly applicable without properly accounting for $T_{\rm bulk}$ that will have a gradient as a function of the radial distance from the central chain.

IV Thermal conductivity of cross–linked polymer networks

A predictive tuning of $\kappa$ purely based on the non–bonded interactions is certainly a non–trivial task, if not impossible. Therefore, a more plausible protocol might be to make use of the distinct microscopic interactions (i.e., bonded vs non–bonded), chain conformation, and possibly also their morphology to understand their effects on $\kappa$ . In this context, one of the most common classes of polymeric materials where the bonded interactions dominate their properties is the epoxies, commonly also referred to as the highly cross–linked polymer (HCP) networks.

In a typical HCP, an individual monomer can form more than two bonds (unlike in a linear chain) and thus forms a 3–dimensional bonded network. The HCPs are usually light weight high performance materials with extraordinary mechanical response [97, 98, 99, 100, 101, 102], attaining $E$ values that can be 2–3 orders of magnitude larger than the common amorphous polymers, consisting of linear chains, and may provide a suitable materials platform toward the enhancement of $\kappa$ [73, 64, 96, 66]. Therefore, recent interest has been devoted in investigating the $\kappa$ behavior in HCPs.

IV.1 Amine cured epoxies

One common example of amorphous HCP is the amine cured epoxy networks [96, 101], where monomers are cross–linked with different amine hardeners with varying stiffness and cross–linker bond length $\ell_{\rm cb}$ , see the insets in Fig. 7. It can be seen from the experimental data in Fig. 7 that just by changing the hardener, $\kappa$ can be tuned by about a factor of two [96]. Moreover, even in the best case (shown by the red circles in Fig. 7), $\kappa$ only increases by a factor of 1.35 in comparison to a linear PMMA, i.e., $\kappa\simeq 0.20$ W/Km [32, 24, 44]. The specific enhancement in $\kappa$ is rather small considering that $\kappa$ in epoxies is expected to be dominated by the bonded interactions. What causes such a small variation in $\kappa$ for HCPs? To answer this question, direct information about the network micro–structure is needed where several competing effects control their physical properties. In this context, obtaining any reasonable information regarding such microscopic details is a rather difficult task within the commonly employed experimental techniques. Therefore, simulations may be of particular interest in studying the $\kappa$ behavior in epoxy networks, where a direct access to the microscopic network details are reasonable available [98, 100, 103].

Give the chemical specificity of epoxies, one may expect that the all–atom simulations might be the best possible choice. Moreover, creating a network structure at the all–atom level is difficult and also is computationally expensive, especially when dealing with a broad range of system parameters. Complexities get even more elevated because of the large system sizes coupled with spacial and temporal heterogeneity [103]. Therefore, an alternative (and possibly a better choice) is a bead–spring type generic simulation technique [98, 64]. Broadly, generic simulations address the common polymer properties that are independent of any specific chemical details and thus a large number of systems can be explained within one physical framework [104]. Additionally, tuning the system parameters is rather straightforward within a generic setup, such as the relative bond lengths, their stiffness, and/or bond orientations that are usually inspired by the underlying chemical specific systems [96, 105, 106].

Fig. 8 shows $\kappa$ as a function of $\ell_{\rm cb}$ for a set of model HCPs with different bond stiffness [64]. It can be appreciated that the (relatively) soft bonds (representing the amine hardeners) give reasonably consistent values as in the experiments [96], i.e., $\kappa/\kappa_{\rm amorphous}\simeq 1.25-1.50$ and it is about 1.35 in Ref. [96] (represented by an arrow in Fig. 8).

For the stiff bonds, a significant increase in $\kappa$ is observed (represented by the diamond symbols in Fig. 8). This large enhancement can be understood by looking into the networks micro–structures. From the simulation snapshots in Fig. 9, it can be appreciated that there are large voids (or free volume) in all the cured samples. Such voids exist when the neighboring monomers form all their bonds pointing out of each other [100, 64] and the monomers along the periphery of a void only interact via vdW forces. These are usually the weak spots within a network, hence resist the heat flow.

The observed void sizes are larger for the tri–functional moderately cross–linked polymers (MCP), while the tetra–functional HCP have relatively smaller free volume. A direct consequence is that the MCPs (open symbols) usually have lower $\kappa$ than the HCPs (solid symbols) in Fig. 8. In summary, it is not only that the increasing bonded contacts can by default increase $\kappa$ . Instead the cross–linked bond stiffness, $\ell_{\rm cb}$ , and their effects on the network micro–structures control $\kappa$ .

IV.2 Ethylene cured epoxies

Another class is the ethylene cured epoxy networks [73], where the free volume can be tuned by changing the length of the ethylene linkers, see the schematic in Fig. 10(a). Here, it is important to mention that $\ell_{\rm p}\simeq 0.65$ nm for of a PE chain (or equivalent of one ethylene monomer) [50]. When an ethylene linker is longer than $N_{\ell}\geq 4$ , it is soft because of its small flexural stiffness. The longer the $N_{\ell}$ , more flexible is the linker and thus there are also larger free volume in a sample. A direct consequence of such a linker is that– together the free volume and soft linker– they significantly reduce stiffness of a materials and as a result $\kappa$ decreases with increasing $N_{\ell}$ , see Fig. 10(b).

V Thermal conductivity of chain oriented systems

V.1 Extended chain configurations

A typical representing system where bonded interactions dominate $\kappa$ is the polymer fibers [60, 91], where individual chain are extended along the direction of heat flow and thus $\kappa$ is dominated by the energy transfer between the bonded monomers [60, 91, 107, 54, 89, 90, 59]. In this context, it has been experimentally reported that a PE fiber can attain $\kappa>100$ W/Km [60], which is significantly higher than $\kappa\simeq 0.3$ W/Km for an amorphous PE [42].

A closer look at an extended chain configuration reveals that it can be viewed as a quasi one–dimensional (Q1–D) crystalline material [108]. This is a direct consequence of the periodic arrangement of monomers along a chain backbone, see the top schematic in the inset of Fig. 11. In such a system, phonons carry a heat current and the coupling strength between the lattice sites is dictated by the bonded interactions. Usually a pure (pristine) sample has a large $\Lambda$ and thus also a high $\kappa\simeq 160$ W/Km [54], see the main panel in Fig. 11. However, whenever there appears a kink or a bend along a chain contour (see the bottom schematic in the inset of Fig. 11), it scatters the phonons. The larger the number of kinks along a chain, the larger the resistance for heat flow and thus a lower $\kappa$ . This picture is well supported by the simulation results of an extended PE chain with varying number of kinks, see Fig. 11.

To investigate the effects of kinks on $\kappa$ , different PE configurations were specifically engineered [54]. However, a natural system where the number of kinks and the backbone stiffness can be controlled almost at will is the bottle–brush polymers (BBP) [109, 110]. A polymer is referred to as a BBP when a linear polymer of length $N_{\ell}$ is grafted with the side chains with varying length $N_{\rm s}$ and grafting density $\rho_{\rm g}$ . Here, $\rho_{\rm g}$ is defined as the number of side chains grafted per backbone monomer. For example, if every backbone monomer is grafted with one side chain, then $\rho_{\rm g}=1$ . BBPs are of interest because of their potential in designing one–dimensional organic nano–crystals [108].

In a BBP, the backbone flexural stiffness (controlling the number of kinks along a chain) is dictated by $N_{\rm s}$ and $\rho_{\rm g}$ [109, 110]. The heat management in the BBPs is of particular interest because their $\kappa$ is dictated by two competing effects [57, 59]: (1) The presence of side chains act as the pathways for heat leakage that effectively reduce $\kappa$ . (2) The side chains increase the backbone flexural stiffness and thus there exists less kinks (or defects) along the backbone, which effectively increases $\kappa$ . To investigate the extent by which these two effects control $\kappa$ , recent simulations have been performed using a generic model. The representative data is shown in Fig. 12(a).

It can be appreciated in Fig. 12(a) that $\kappa$ shows a non–monotonic variation with $\rho_{\rm g}$ , where two regimes are clearly visible: For $\rho_{\rm g}\leq 1$ , scattering because of the side chains reduces $\kappa$ , while the backbone stiffening via side chains increases $\kappa$ for $\rho_{\rm g}>1$ [59]. This backbone stiffening scenario is also supported by $g(\nu)$ . It can be seen in Fig. 12(b) that there are almost indistinguishable changes in $g(\nu)$ for $\rho_{\rm g}\leq 1$ . Moreover, when $\rho_{\rm g}>1$ a peak becomes more prominent around $\nu\simeq 9-10t_{\circ}^{-1}$ . This is associated with the flexural stiffness that increases with increasing $\rho_{\rm g}$ , as revealed by the shift in this peak towards the higher $\nu$ values. The full–width–of–half–maxima $\nu_{\rm FWHM}$ also decreases with increasing $\rho_{\rm g}$ and thus increases the phonon life time $\tau\propto 1/\nu_{\rm FWHM}$ (or $\kappa$ ).

V.2 Molecular forests

The knock down in $\kappa$ via kinks is a concept that can also be helpful in dictating the heat flow in more complex molecular assemblies. One example is molecular forest, where Q1–D are grafted perpendicularly on a surface forming a two dimensional assembly, such as the forests of CNT [111, 112, 113], silicon nanowires [114, 115], and/or polymers [60, 90, 55]. These forests often exhibit intriguing and counter–intuitive physical behavior. In this context, it had been experimentally reported that– while a single CNT has $\kappa\geq 10^{3}$ W/Km [18, 19, 20, 21], the same CNT in a forest shows a drastic reduction in $\kappa$ [111, 112, 113]. This phenomenon is commonly referred to as the heat trap effect (HTE) in the carbon nanotube (CNT) forests [112].

Even when the counter–intuitive HTE phenomenon was known for over a decade, there existed no clear understanding of this behavior. Here, the simple concepts known from soft matter physics turned out to be reasonably useful in understanding certain aspects of a hard matter problem of complex molecular assemblies. For this purpose, generic simulations were performed [55]. The key assumption in this model is that a Q1–D is considered as a single extended polymer chain and thus represents a molecular forest as a polymer brush. The only input parameter in such a modelling approach is $\ell_{\rm p}$ . In this context, it was readily observed that a CNT can be characterized by their bending stiffness, as measured in terms of $\ell_{\rm p}$ , that increases with CNT diameter $d$ [116]. For example, a CNT of $d=1$ nm has $\ell_{\rm p}=50-60~{}\mu$ m. Within this picture, a CNT forest of $2$ mm height can have about $30-40\ell_{\rm p}$ or as many number of kinks. Note that in this simple argument we do not discuss the effect of grafting density $\Gamma$ . The representative data and the corresponding simulation snapshot is shown in Fig. 13. As expected, a chain in a forest shows a significant reduction in $\kappa$ [55], see the left panel in Fig. 13. As discussed above, this knock down is direct consequence of the kinks that act as the scattering centers for the heat flow. The kinks are also evident from the simulation snapshot in the right panel of Fig. 13.

VI Thermal conductivity of crystalline polymers

A somewhat different class to the amorphous polymers is the polymers with certain degree of crystalline order, where the long range order facilitates phonon propagation that carry a heat current and thus results in an enhanced rate of energy transfer. The typical examples include liquid–crystalline materials [117, 37, 38], poly–peptide sequences [61], and/or semi–crystalline polymers [30, 118, 91].

VI.1 Liquid crystalline polymers

Liquid crystals usually have $\kappa\simeq 0.3$ W/Km [37, 38], i.e., similar to the commodity polymers. However, one of the advantages of a liquid crystalline material, such as the azobenzene–based liquid crystals, is that an azobenzene undergoes a re–entrant trans–to–cis transition when they are exposed to near–ultraviolet light [119, 120]. Such a transition also alters the molecular order in a liquid crystal and hence $\kappa$ switches between 0.1 W/Km (in cis state) and 0.3 W/Km (in trans state) [37]. This can simply be viewed as a light responsive thermal switch.

When a liquid crystalline polymer is cross–linked with the ethylene linkers, they give very interesting and counter–intuitive trends in $\kappa$ [121, 122]. For example, earlier experimental studies have reported that $\kappa$ shows a zig–zag variation with increasing $N_{\ell}$ , varying between 1.0–0.2 W/Km [122]. For an even number of carbon atoms in a linker, $\kappa$ always has a higher value than the next system with a linker with an odd number of carbon atoms. This behavior is commonly known as the odd–even effect in $\kappa$ , which was initially reported in a simulation study [121]. Moreover, a more detailed investigation is recently reported [122]. Such an odd–even effect is also well–known in various other properties of the liquid crystalline polymers [123, 124]. While these studies gave very nice insight into the $\kappa$ behavior of these complex systems, an exact molecular level understanding of such a non–trivial odd–even effect is still somewhat lacking.

VI.2 Conjugated polymers

Another polymeric system, where crystalline ordering is probably most important, is the conjugated polymer because they are often used under the high temperature conditions. The crystalline order in such a system is because of the $\pi-\pi$ stacking of their backbone consisting of the aromatic structures [28, 30, 29, 39]. In this context, a significantly large value of $\kappa\to 2.0$ W/Km was reported in poly(3-hexylthiophene) (P3HT) [30]. This $\kappa$ can also be further increased by blending with the multi–wall CNTs [29]. The latter study reported a non–monotonic variation with $\phi_{\rm CNT}$ , reaching a maximum value of about 5.0 W/Km around $\phi_{\rm CNT}\simeq 30\%$ .

VI.3 Poly–peptide sequences

A natural soft matter that shows structural order is a poly–peptide sequence. Previous experimental results have shown that– by controlling the specific amino acid residues along a poly–peptide sequence, one can significantly alter its degree of secondary structure $d_{\rm s}$ . In such a system, an enhancement of up to $\kappa\simeq 1.5$ W/Km is observed in a hydrated poly–peptide [61]. Note also that $\kappa\simeq 0.6$ W/Km for water and hence the observed increase is a direct consequence of $d_{\rm s}$ . Here, however, it is important to mention that the controlled synthesis (aka precision polymerization) of a specific poly–peptide sequence is a grand challenge and they are commercially expensive [26].

VI.4 Semi–crystalline polymers

A more plausible alternative to poly–peptides might be the semi–crystalline synthetic commodity polymers, such as the PVA, PLA, and PE systems [125, 126]. While $\kappa$ of a semi–crystalline sample can be rather large because of the long range order, it may be inferred that if they are cross–linked, the combination of these two effects might lead to an even greater increase in $\kappa$ than the bare amorphous polymers. Motivated by this, a recent experimental study has reported that a cross–linked semi–crystalline network can only achieve an increase of up to a factor of 2.5 times than the pure PMMA sample, i.e., $\kappa\simeq 0.5$ W/Km [65]. This rather surprising behavior was also investigated in a simulation study, where a similar increase in $\kappa$ was observed for a critical $\ell_{\rm cb}$ , see Fig. 14(a).

A maxima in $\kappa$ is only observed when $\ell_{\rm cb}$ is comparable to the lattice constant $\ell_{\rm a}$ , such that the degree of crystallinity $d_{\rm c}$ increases with $\ell_{\rm cb}$ , as revealed by the peak heights in the scattering function in Fig. 14(b) [66]. When $\ell_{\rm cb}$ increases beyond a certain threshold (i.e., $\ell_{\rm cb}/\ell_{\rm a}\geq 0.9$ )– in one hand $d_{\rm c}$ increases only slightly, on the other hand there is a large decrease in the bond density $\rho_{\rm cb}$ , see Fig. 14(c). To summarize the data in Fig. 14(a), the observed initial increase in $\kappa$ for $\ell_{\rm cb}/\ell_{\rm a}\leq 0.9$ is due to the increased $d_{\rm c}$ and the decrease in $\kappa$ for $\ell_{\rm cb}/\ell_{\rm a}\geq 0.9$ is due to the reduced $\rho_{\rm cb}$ . This readily suggests that a delicate combination of $d_{\rm crystal}$ and $\rho_{\rm cb}$ controls $\kappa$ [65, 66].

VII Thermal conductivity of polyelectrolytes

In the preceding sections, a short overview of the $\kappa$ behavior in neutral polymers are presented. Possible ideas are also discussed that can be used to tune $\kappa$ in amorphous systems by macromolecular engineering. However, there are systems where electrostatic interaction also plays an important role, examples include but are not limited to, organic (soft) electronics [29, 30, 39], bio–inspired materials [61], and flexible chips [127, 128, 129]. In these systems it is always desirable to attain a large $\kappa$ that can act as a heat sink and thus improves device performance/durability. Because of this need, extensive efforts have been devoted in studying the $\kappa$ behavior in electrostatically modified polymers [33, 130].

One of the classical examples of polyelectrolytes is the modified PAA with varying degree of ionization. Here, a recent experimental study has investigated the effect of pH on ionized PAA, which reported $\kappa\to 1.2$ W/Km [33]. This is an enhancement of about 3–4 times than the neutral amorphous PAA, where $\kappa\simeq 0.20-0.37$ W/Km [32, 31, 33]. This enhancement was also coupled with a significant increase in the materials stiffness $E$ , i.e., consistent with the predictions of the MTCM that $\kappa\propto E$ [49, 14]. The increased $E$ was predominantly because electrostatic interaction stretches a PAA and thus makes bonded interaction more dominant than in the case of an uncharged PAA system. This characteristic extension of an ionized chain is also consistent with the earlier studies investigating the effective stretching and $\ell_{\rm p}$ of polyelectrolytes [131, 132, 133].

The $\kappa$ behavior in the polyelectrolytes suggest that the influence of the electrostatic interaction is rather indirect, i.e., they help stretch a chain and thus the bonded interactions become more dominant, which increases $\kappa$ . Something may speak in this favor that the electrostatics alone do not influence $\kappa$ , as in the case of the ionic liquid consisting of small molecules [134], where $\kappa\simeq 0.2$ W/Km [135].

VIII Thermal transport in smart responsive polymers

The backbone structure of the commodity polymers are commonly dominated by the C–C covalent bonds, see Fig. 1. Such a bond is extremely strong with its strength of about 80 $k_{\rm B}T$ and thus these bonds live forever under the unperturbed environmental conditions. This creates severe ecological problems, which get even worse when dealing with water insoluble polymers, as shown by a few example in the top panel of Fig. 1. This is one of the main reasons why the recent interest has been diverted to water soluble (H–bonded) polymers, referred to as the “smart” polymers [136, 27], as shown by a few examples in the bottom panel of Fig. 1. Additionally, it is also preferred if a polymer can be bio–degradable and/or pH responsive, such as the acetal–linked copolymers [137].

A polymer is referred to as a “smart” responsive when a small change in the external stimuli can significantly alter their structure, function, and stability. These stimuli can be temperature [138, 136, 139, 27], pressure [140, 141, 142], pH [137], light [143, 144, 145], and/or cosolvent [146, 147, 148, 149]. One common example of smart polymer is PNIPAM that shows a coil–to–globule transition in water around $T_{\ell}\simeq 305$ K (or $32^{\circ}$ C) [138, 136]. This is a typical lower critical solution (LCST) behavior [136, 137, 106] driven by the solvent entropy [52, 51, 53].

The fast conformational switching of PNIPAM in water may be extremely useful in the thermal applications. Thermal switching is one such application that controls heat flow in various systems, including, but are not limited to, thermoelectric conversion, energy storage, space technology, and sensing [150, 34, 35, 36]. In this context, the conventional thermal switches often suffer from their slow transition rates and thus also have poor performance.

Recently there has been considerable interest in studying $\kappa$ in the smart polymer with a goal to attain a fast switching in $\kappa$ [34, 35, 36]. In particular, experimental studies in the aqueous solutions of PNIPAM [35] and PNIPAM–based hydrogels [36] have shown that their $\kappa$ behavior follow the same trend as the LCST transition around $T_{\ell}\simeq 305$ K (or $32^{\circ}$ C) [138, 136], see Fig. 15.

It is important to note that $\kappa$ increases with $T$ in the liquids and in the amorphous materials [49, 86, 56, 45, 151, 44, 24, 23] because of an increased vibrations. This behavior is also visible in pure water, where a weak increase in $\kappa$ is observed with $T$ , see the black data set in Fig. 15. Moreover, the sudden drop in $\kappa$ around $T\geq 303$ K (or $30^{\circ}$ C) for the PNIPAM concentration above $5\times 10^{-3}$ g/mL is predominantly due to the coil–to–globule transition of PNIPAM. This drop is likely due to the loss in the number of hydrogen bonds needed to stabilize a PNIPAM configuration and the resultant breakage of the water caging around PNIPAM [35, 36]. These broken water–PNIPAM H-Bonds effectively creates weak interfaces that act as resistance for the heat flow. Contrary to these results, another experimental study has reported an opposite trend for the concentrated PNIPAM solutions, i.e., $\kappa$ increases above $T_{\rm cloud}$ [34]. This is simply because a chain under a high concentration does not collapse into a globule, instead it remains rather expanded surrounded by the other neighboring chains and hence the $\kappa$ behavior become dominated by the energy transfer between the bonded monomers. Furthermore, these distinct results highlight that the polymer concentration, chain size at a given concentration, relative interaction/coordination (monomer–monomer, monomer–solvent, and solvent–solvent), water tetrahedrality around a PNIPAM and a delicate balance between these effects play key role in dictating $\kappa$ behavior in the polymer solutions. A detailed understanding of such effects on the $\kappa$ behavior is a rather open discussion.

Lastly, it might also be important to highlight another (possible) system for the thermal switching application. In this context, elastin–like poly–peptides (ELP) are a modern class of biomimetic polymers that also shows LCST transition [152, 153]. One important aspect of the ELPs is their proline isomerization (ProI) that can have either a cis or a trans conformation, which can dictate their relative conformations [154, 155, 156]. The free energy barrier of such a cis–to–trans transition is about 30 $k_{\rm B}T$ , which the free energy difference between these two states in only about 2 $k_{\rm B}T$ and thus can be switched via light. Given the above discussion, ELPs with ProI may also be alternatively used as a thermal switch, similar to that in the liquid crystalline materials [37, 38].

IX Concluding remarks

Ever since the seminal publication of Hermann Staudinger [157], the field of polymer science has traversed a long journey with many new interesting developments for the future design of advanced functional materials. In the constant quest to find new polymeric materials with improved performance, significant attention has been devoted within the field thermal transport of polymers over the last 2–3 decades. Especially because the polymeric plastics usually have very low thermal conductivity coefficient $\kappa$ , which is typically a few orders of magnitude smaller than the common crystals. Here, one of the grand challenges is to attain a predictive tuning of $\kappa$ via macromolecular engineering. In the context, experiments have investigated a plethora of systems that include– linear polymers, symmetric and asymmetric polymer blends, polymer composites, cross–linked networks, polymer fibers, crystalline polymers and electrostatically modified polymers, to name a few. Motivated by these studies, computational studies have also been conducted to establish a structure–property relationship in polymers within the context of their $\kappa$ behavior.

While it is certainly rather difficult to address all aspects of a huge field of research within one short overview, in this work an attempt has been made to highlight some of the latest developments in the field of heat conductivity in polymers and polymeric materials. In particular, computational results are discussed within the context of the complementary experiments with a goal to establish a detailed microscopic understanding that dictates macroscopic polymer properties. Available theoretical models are also discussed that may pave the way to guide the future experimental and/or simulation studies.

Some discussions are also presented that showed that the simple concepts know from the basic polymer (soft matter) science [55] can be used to understand a complex problem from an opposite class of hard matter physics [112, 111]. This further highlights why polymer science is such a vibrant and active field of research which is not only restricted within the soft matter community. Rather, it reaches across a wide range of interdisciplinary fields.

Acknowledgement: The contents presented in this review have greatly benefited from the discussions with many colleagues. In particular, the development of two key concepts presented here would not have been possible without very fruitful collaborations with Martin Müser and Marcus Müller, whom I take this opportunity to gratefully acknowledge.

This draft is a contribution towards a special issue to celebrate the 40th anniversary of Max Planck Institute for Polymer Research. I take this opportunity to gratefully acknowledge very fruitful continual collaborations with many MPIP colleagues, especially Kurt Kremer for numerous stimulating discussions that led to the foundation of my works in MPIP and also after.

I further thank Kyle Monkman for useful comments on this draft.

Conflict of interest: The author declares no conflicting financial interest.

References

[1] C. Choy, “Thermal conductivity of polymers,” Polymer, vol. 18, no. 10, pp. 984–1004, 1977.
[2] J. Taler and P. Duda, “Solving direct and inverse heat conduction problems,” Springer Link, 2006.
[3] A. Henry, “Thermal transport in polymers,” Annual Review of Heat Transfer, vol. 17, pp. 485–520, 2014.
[4] O. A. Ezekoye, “Chapter: Conduction of heat in solids,” Springer Link, pp. 25–52, 2016.
[5] Z. M. Zhang, Nano/Microscale Heat Transfer, end edition. Switzerland: Springer Nature Switzerland, 2020.
[6] P. Keblinski, “Modeling of heat transport in polymers and their nanocomposites,” Handbook of Materials Modeling, pp. 975–997, 2020.
[7] X. Qian, J. Zhou, and G. Chen, “Phonon-engineered extreme thermal conductivity materials,” Nature Materials, vol. 20, pp. 1188–1202, Sep 2021.
[8] J. H. Lienhard, V and J. H. Lienhard, IV, A Heat Transfer Textbook. Cambridge, MA: Phlogiston Press, 6th ed., Apr. 2024. Version 6.00.
[9] J. R. Olson, K. A. Topp, and R. O. Pohl, “Specific heat and thermal conductivity of solid fullerenes,” Science, vol. 259, pp. 1145–1148, 1993.
[10] D. G. Cahill, W. K. Ford, K. E. Goodson, G. D. Mahan, A. Majumdar, H. J. Maris, R. Merlin, and S. R. Phillpot, “Nanoscale thermal transport,” Journal of Applied Physics, vol. 93, pp. 793–818, 2003.
[11] A. I. Boukai, Y. Bunimovich, J. Tahir-Kheli, J.-K. Yu, W. A. Goddard, III, and J. R. Heath, “Silicon nanowires as efficient thermoelectric materials,” Nature, vol. 451, pp. 168–171, Jan. 2008.
[12] S. Ren, M. Bernardi, R. R. Lunt, V. Bulovic, J. C. Grossman, and S. Gradecak, “Toward efficient carbon nanotube/p3ht solar cells: Active layer morphology, electrical, and optical properties,” Nano Letters, vol. 11, pp. 5316–5321, 2011.
[13] T. Mikolajick, A. Heinzig, J. Trommer, S. Pregl, M. Grube, G. Cuniberti, and W. M. Weber, “Silicon nanowires – a versatile technology platform,” physica status solidi (RRL) – Rapid Research Letters, vol. 7, no. 10, pp. 793–799, 2013.
[14] J. L. Braun, C. M. Rost, M. Lim, A. Giri, D. H. Olson, G. N. Kotsonis, G. Stan, D. W. Brenner, J. P. Maria, and P. E. Hopkins, “Charge-induced disorder controls the thermal conductivity of entropy-stabilized oxides,” Advanced Materials, vol. 30, pp. 1805004–8, 2018.
[15] M. G. Holland, “Phonon scattering in semiconductors from thermal conductivity studies,” Phys. Rev., vol. 134, pp. A471–A480, Apr 1964.
[16] G. Slack, “Nonmetallic crystals with high thermal conductivity,” Journal of Physics and Chemistry of Solids, vol. 34, no. 2, pp. 321–335, 1973.
[17] G. Fugallo and L. Colombo, “Calculating lattice thermal conductivity: A synopsis,” Physica Scripta, vol. 93, pp. 043002–23, 2018.
[18] S. Berber, Y.-K. Kwon, and D. Tománek, “Unusually high thermal conductivity of carbon nanotubes,” Phys. Rev. Lett., vol. 84, pp. 4613–4616, May 2000.
[19] E. Pop, D. Mann, Q. Wang, K. Goodson, and H. Dai, “Thermal conductance of an individual single-wall carbon nanotube above room temperature,” Nano Letters, vol. 6, no. 1, pp. 96–100, 2006. PMID: 16402794.
[20] V. Lee, C.-H. Wu, Z.-X. Lou, W.-L. Lee, and C.-W. Chang, “Divergent and ultrahigh thermal conductivity in millimeter-long nanotubes,” Phys. Rev. Lett., vol. 118, p. 135901, Mar 2017.
[21] Q.-Y. Li, K. Takahashi, and X. Zhang, “Comment on “divergent and ultrahigh thermal conductivity in millimeter-long nanotubes”,” Phys. Rev. Lett., vol. 119, p. 179601, Oct 2017.
[22] H. W. H. Wada and T. K. T. Kamijoh, “Thermal conductivity of amorphous silicon,” Japanese Journal of Applied Physics, vol. 35, p. L648, may 1996.
[23] S. Shenogin, A. Bodapati, P. Keblinski, and A. J. H. McGaughey, “Predicting the thermal conductivity of inorganic and polymeric glasses: The role of anharmonicity,” Journal of Applied Physics, vol. 105, 02 2009. 034906.
[24] B. Li, F. DeAngelis, G. Chen, and A. Henry, “The importance of localized modes spectral contribution to thermal conductivity in amorphous polymers,” Communications Physics, vol. 5, p. 323, Dec 2022.
[25] M. Müller, “Process-directed self-assembly of copolymers: Results of and challenges for simulation studies,” Progress in Polymer Science, vol. 101, p. 101198, 2020.
[26] C. Chen, D. Y. W. Ng, and T. Weil, “Polymer bioconjugates: Modern design concepts toward precision hybrid materials,” Progress in Polymer Science, vol. 105, p. 101241, 2020.
[27] D. Mukherji, C. M. Marques, and K. Kremer, “Smart responsive polymers: Fundamentals and design principles,” Annual Review of Condensed Matter Physics, vol. 11, pp. 271–299, 2020.
[28] N. C. Forero-Martinez, K.-H. Lin, K. Kremer, and D. Andrienko, “Virtual screening for organic solar cells and light emitting diodes,” Advanced Science, vol. 9, no. 19, p. 2200825, 2022.
[29] M. K. Smith, V. Singh, K. Kalaitzidou, and B. A. Cola, “High thermal and electrical conductivity of template fabricated p3ht/mwcnt composite nanofibers,” ACS Applied Materials Interfaces, vol. 8, pp. 14788–14794, 2016.
[30] Y. Xu, X. Wang, J. Zhou, B. Song, Z. Jiang, E. M. Y. Lee, S. Huberman, K. K. Gleason, and G. Chen, “Molecular engineered conjugated polymer with high thermal conductivity,” Science Advances, vol. 4, p. eaar3031, 2018.
[31] G. Kim, D. Lee, A. Shanker, L. Shao, M. S. Kwon, Gidley, J. Kim, and K. P. Pipe, “High thermal conductivity in amorphous polymer blends by engineered interchain interactions,” Nature Materials, vol. 14, pp. 295–300, 2015.
[32] X. Xie, D. Li, T. Tsai, J. Liu, P. V. Braun, and D. G. Cahill, “Thermal conductivity, heat capacity, and elastic constants of water-soluble polymers and polymer blends,” Macromolecules, vol. 49, pp. 972–978, 2016.
[33] A. Shanker, C. Li, G.-H. Kim, D. Gidley, K. P. Pipe, and J. Kim, “High thermal conductivity in electrostatically engineered amorphous polymers,” Science Advances, vol. 3, p. e1700342, 2017.
[34] E. Pallecchi, Z. Chen, G. E. Fernandes, Y. Wan, J. H. Kim, and J. Xu, “A thermal diode and novel implementation in a phase-change material,” Mater. Horiz., vol. 2, pp. 125–129, 2015.
[35] C. Li, Y. Ma, and Z. Tian, “Thermal switching of thermoresponsive polymer aqueous solutions,” ACS Macro Letters, vol. 7, no. 1, pp. 53–58, 2018.
[36] H. Feng, N. Tang, M. An, R. Guo, D. Ma, X. Yu, J. Zang, and N. Yang, “Thermally-responsive hydrogels poly(n-isopropylacrylamide) as the thermal switch,” The Journal of Physical Chemistry C, vol. 123, no. 51, pp. 31003–31010, 2019.
[37] J. Shin, J. Sung, M. Kang, X. Xie, B. Lee, K. M. Lee, T. J. White, C. Leal, N. R. Sottos, P. V. Braun, and D. G. Cahill, “Light-triggered thermal conductivity switching in azobenzene polymers,” Proceedings of the National Academy of Sciences, vol. 116, no. 13, pp. 5973–5978, 2019.
[38] T. Ishibe, T. Kaneko, Y. Uematsu, H. Sato-Akaba, M. Komura, T. Iyoda, and Y. Nakamura, “Tunable thermal switch via order–order transition in liquid crystalline block copolymer,” Nano Letters, vol. 22, no. 15, pp. 6105–6111, 2022. PMID: 35883274.
[39] W. Shi, Z. Shuai, and D. Wang, “Tuning thermal transport in chain-oriented conducting polymers for enhanced thermoelectric efficiency: A computational study,” Advanced Functional Materials, vol. 27, pp. 1702847–8, 2017.
[40] M. Goel and M. Thelakkat, “Polymer thermoelectrics: Opportunities and challenges,” Macromolecules, vol. 53, no. 10, pp. 3632–3642, 2020.
[41] D. Mukherji and K. Kremer, “Smart polymers for soft materials: from solution processing to organic solids,” Polymers, vol. 15, no. 15, p. 3229, 2023.
[42] J. Brandrup, E. H. Immergut, and E. A. Grulke, “Polymer handbook, 2 volumes set, 4th edition,” 2003.
[43] G. R. Desiraju, “Hydrogen bridges in crystal engineering: Interactions without borders,” Accounts of Chemical Research, vol. 35, no. 7, pp. 565–573, 2002.
[44] S. Agarwal, N. S. Saxena, and V. Kumar, “Temperature dependence thermal conductivity of zns/pmma nanocomposite,” pp. 737–739, 2014.
[45] M. S. Barkhad, B. Abu-Jdayil, A. H. I. Mourad, and M. Z. Iqbal, “Thermal insulation and mechanical properties of polylactic acid (pla) at different processing conditions,” Polymers, vol. 12, no. 9, 2020.
[46] B. Salameh, S. Yasin, D. A. Fara, and A. M. Zihlif, “Dependence of the thermal conductivity of pmma, ps and pe on temperature and crystallinity,” Polymer(Korea), vol. 45, no. 2, pp. 281–285, 2021.
[47] C. Ruscher, J. Rottler, C. E. Boott, M. J. MacLachlan, and D. Mukherji, “Elasticity and thermal transport of commodity plastics,” Physical Review Materials, vol. 3, p. 125604, 2019.
[48] B. Crist and P. G. Hereña, “Molecular orbital studies of polyethylene deformation,” Journal of Polymer Science Part B: Polymer Physics, vol. 34, no. 3, pp. 449–457, 1996.
[49] D. G. Cahill, S. K. Watson, and R. O. Pohl, “Lower limit to the thermal conductivity of disordered crystals,” Physical Review B, vol. 46, pp. 6131–6140, 1990.
[50] L. Pigard, D. Mukherji, J. Rottler, and M. Müller, “Microscopic model to quantify the difference of energy-transfer rates between bonded and nonbonded monomers in polymers,” Macromolecules, vol. 54, no. 23, pp. 10969–10983, 2021.
[51] P.-G. de Gennes, Scaling Concepts in Polymer Physics. Cornell University Press, 1979.
[52] M. Doi and S. F. Edwards, The Theory of Polymer Dynamics. UK: Oxford Science Publications, 1986.
[53] J. D. Cloizeaux and G. Jannink, Polymers in Solution: Their Modelling and Structure. Clarendon Press, 1990.
[54] X. Duan, Z. Li, J. Liu, G. Chen, and X. Li, “Roles of kink on the thermal transport in single polyethylene chains,” Journal of Applied Physics, vol. 125, 04 2019. 164303.
[55] A. Bhardwaj, A. S. Phani, A. Nojeh, and D. Mukherji, “Thermal transport in molecular forests,” ACS nano, vol. 15, no. 1, pp. 1826–1832, 2021.
[56] D. Mukherji, “Computing the thermal transport coefficient of neutral amorphous polymers using exact vibrational density of states: Comparison with experiments,” Phys. Rev. Mater., vol. 8, p. 085601, Aug 2024.
[57] H. Ma and Z. Tian, “Effects of polymer topology and morphology on thermal transport: A molecular dynamics study of bottlebrush polymers,” Appl. Phys. Lett., vol. 110, no. 9, p. 091903, 2017.
[58] C. Huang, X. Qian, and R. Yang, “Thermal conductivity of polymers and polymer nanocomposites,” Mater. Sci. Eng. R Rep., vol. 132, pp. 1–22, 2018.
[59] M. K. Maurya, T. Laschuetza, M. K. Singh, and D. Mukherji, “Thermal conductivity of bottle–brush polymers,” Langmuir, vol. 40, no. 8, pp. 4392–4400, 2024.
[60] S. Shen, A. Henry, J. Tong, R. Zheng, and G. Chen, “Polyethylene nanofibres with very high thermal conductivities,” Nature Nanotechnology, vol. 5, no. 4, pp. 251–255, 2010.
[61] J. A. Tomko, A. Pena-Francesch, H. Jung, M. Tyagi, B. D. Allen, M. C. Demirel, and P. E. Hopkins, “Tunable thermal transport and reversible thermal conductivity switching in topologically networked bio-inspired materials,” Nature Nanotechnology, vol. 13, no. 10, pp. 959–964, 2018.
[62] D. Bruns, T. E. de Oliveira, J. Rottler, and D. Mukherji, “Tuning morphology and thermal transport of asymmetric smart polymer blends by macromolecular engineering,” Macromolecules, vol. 52, no. 15, pp. 5510–5517, 2019.
[63] S. Xu, J. Liu, and X. Wang, “Thermal conductivity enhancement of polymers via structure tailoring,” Journal of Enhanced Heat Transfer, vol. 27, pp. 463–489, 2020.
[64] D. Mukherji and M. K. Singh, “Tuning thermal transport in highly cross-linked polymers by bond-induced void engineering,” Phys. Rev. Materials, vol. 5, p. 025602, Feb 2021.
[65] G. Lv, E. Jensen, C. M. Evans, and D. G. Cahill, “High thermal conductivity semicrystalline epoxy resins with anthraquinone-based hardeners,” ACS Appl. Polym. Mater., vol. 3, p. 4430–4435, Dec 2021.
[66] M. K. Maurya, J. Wu, M. K. Singh, and D. Mukherji, “Thermal conductivity of semicrystalline polymer networks: Crystallinity or cross-linking?,” ACS Macro Letters, vol. 11, no. 7, pp. 925–929, 2022.
[67] M. Moniruzzaman and K. I. Winey, “Polymer nanocomposites containing carbon nanotubes,” Macromolecules, vol. 39, no. 16, pp. 5194–5205, 2006.
[68] Z. Han and A. Fina, “Thermal conductivity of carbon nanotubes and their polymer nanocomposites: A review,” Progress in Polymer Science, vol. 36, no. 7, pp. 914–944, 2011. Special Issue on Composites.
[69] X. Hou, S. Chen, J. J. Koh, J. Kong, Y.-W. Zhang, J. C. C. Yeo, H. Chen, and C. He, “Entropy-driven ultratough blends from brittle polymers,” ACS Macro Letters, vol. 10, no. 4, pp. 406–411, 2021.
[70] A. Kulshreshtha, R. C. Hayward, and A. Jayaraman, “Impact of composition and placement of hydrogen-bonding groups along polymer chains on blend phase behavior: Coarse-grained molecular dynamics simulation study,” Macromolecules, vol. 55, no. 7, pp. 2675–2690, 2022.
[71] D. Mukherji, T. E. de Oliveira, C. Ruscher, and J. Rottler, “Thermodynamics, morphology, mechanics, and thermal transport of pmma-pla blends,” Phys. Rev. Materials, vol. 6, p. 025606, Feb 2022.
[72] G. L. Pollack, “Kapitza resistance,” Rev. Mod. Phys., vol. 41, pp. 48–81, Jan 1969.
[73] G. Lv, B. Soman, N. Shan, C. M. Evans, and D. G. Cahill, “Effect of linker length and temperature on the thermal conductivity of ethylene dynamic networks,” ACS Macro Letters, vol. 10, no. 9, pp. 1088–1093, 2021.
[74] J. Horbach, W. Kob, and K. Binder, “Specific heat of amorphous silica within the harmonic approximation,” The Journal of Physical Chemistry B, vol. 103, pp. 4104–4108, May 1999.
[75] H. Gao, T. P. W. Menzel, M. H. Müser, and D. Mukherji, “Comparing simulated specific heat of liquid polymers and oligomers to experiments,” Phys. Rev. Mater., vol. 5, p. 065605, Jun 2021.
[76] M. Lim, Z. Rak, J. L. Braun, C. M. Rost, G. N. Kotsonis, P. E. Hopkins, J.-P. Maria, and D. W. Brenner, “Influence of mass and charge disorder on the phonon thermal conductivity of entropy stabilized oxides determined by molecular dynamics simulations,” Journal of Applied Physics, vol. 125, p. 055105, 2019.
[77] J. Ahmed, Q. J. Wang, O. Balogun, N. Ren, R. England, and F. Lockwood, “Molecular dynamics modeling of thermal conductivity of several hydrocarbon base oils,” Tribology Letters, vol. 71, p. 70, May 2023.
[78] B. L. Zink, R. Pietri, and F. Hellman, “Thermal conductivity and specific heat of thin-film amorphous silicon,” Phys. Rev. Lett., vol. 96, p. 055902, Feb 2006.
[79] D. Dahal, R. Atta-Fynn, S. R. Elliott, and P. Biswas, “Ab initio studies of the impact of the debye-waller factor on the structural and dynamical properties of amorphous semiconductors: The case of $a$ -si,” Phys. Rev. B, vol. 108, p. 094206, Sep 2023.
[80] W.-P. Hsieh, M. D. Losego, P. V. Braun, S. Shenogin, P. Keblinski, and D. G. Cahill, “Testing the minimum thermal conductivity model for amorphous polymers using high pressure,” Physical Review B, vol. 83, p. 174205, 2011.
[81] R. Bhowmik, S. Sihn, V. Varshney, A. K. Roy, and J. P. Vernon, “Calculation of specific heat of polymers using molecular dynamics simulations,” Polymer, vol. 167, pp. 176–181, 2019.
[82] F. Demydiuk, M. Solar, H. Meyer, O. Benzerara, W. Paul, and J. Baschnagel, “Role of torsional potential in chain conformation, thermodynamics, and glass formation of simulated polybutadiene melts,” The Journal of Chemical Physics, vol. 156, 06 2022. 234902.
[83] R. Zwanzig, “Time-correlation functions and transport coefficients in statistical mechanics,” Annual Review of Physical Chemistry, vol. 16, no. Volume 16,, pp. 67–102, 1965.
[84] F. Müller-Plathe, “A simple nonequilibrium molecular dynamics method for calculating the thermal conductivity,” The Journal of Chemical Physics, vol. 106, pp. 6082–6085, 04 1997.
[85] E. Lampin, P. L. Palla, P.-A. Francioso, and F. Cleri, “Thermal conductivity from approach-to-equilibrium molecular dynamics,” Journal of Applied Physics, vol. 114, no. 3, p. 033525, 2013.
[86] J. Wu and D. Mukherji, “Comparison of all atom and united atom models for thermal transport calculations of amorphous polyethylene,” Computational Materials Science, vol. 211, p. 111539, 2022.
[87] P. Schöffel and M. H. Müser, “Elastic constants of quantum solids by path integral simulations,” Phys. Rev. B, vol. 63, p. 224108, May 2001.
[88] S. Gottlieb, L. Pigard, Y. K. Ryu, M. Lorenzoni, L. Evangelio, M. Fernández-Regúlez, C. D. Rawlings, M. Spieser, F. Perez-Murano, M. Müller, and A. W. Knoll, “Thermal imaging of block copolymers with sub-10 nm resolution,” ACS Nano, vol. 15, no. 5, pp. 9005–9016, 2021.
[89] J. Liu and R. Yang, “Length-dependent thermal conductivity of single extended polymer chains,” Phys. Rev. B, vol. 86, p. 104307, Sep 2012.
[90] A. Crnjar, C. Melis, and L. Colombo, “Assessing the anomalous superdiffusive heat transport in a single one-dimensional pedot chain,” Phys. Rev. Mater., vol. 2, p. 015603, Jan 2018.
[91] T. Zhang and T. Luo, “High-contrast, reversible thermal conductivity regulation utilizing the phase transition of polyethylene nanofibers,” ACS Nano, vol. 7, no. 9, pp. 7592–7600, 2013. PMID: 23944835.
[92] Z. Wang, J. A. Carter, A. Lagutchev, Y. K. Koh, N.-H. Seong, D. G. Cahill, and D. D. Dlott, “Ultrafast flash thermal conductance of molecular chains,” Science, vol. 317, no. 5839, pp. 787–790, 2007.
[93] N. J. Mosey, M. H. Müser, and T. K. Woo, “Molecular mechanisms for the functionality of lubricant additives,” Science, vol. 307, no. 5715, pp. 1612–1615, 2005.
[94] I. M. Sivebaek, V. N. Samoilov, and B. N. J. Persson, “Effective viscosity of confined hydrocarbons,” Phys. Rev. Lett., vol. 108, p. 036102, Jan 2012.
[95] H. Gao and M. H. Müser, “Why liquids can appear to solidify during squeeze-out – even when they don’t,” Journal of Colloid and Interface Science, vol. 562, pp. 273–278, 2020.
[96] G. Lv, E. Jensen, C. Shen, K. Yang, C. M. Evans, and D. G. Cahill, “Effect of amine hardener molecular structure on the thermal conductivity of epoxy resins,” ACS Applied Polymer Materials, vol. 3, no. 1, pp. 259–267, 2021.
[97] S. R. White, N. R. Sottos, P. H. Geubelle, J. S. Moore, M. R. Kessler, S. R. Sriram, E. N. Brown, and S. Viswanathan, “Autonomic healing of polymer composites,” Nature, vol. 409, pp. 794–797, Feb 2001.
[98] M. J. Stevens, “Interfacial fracture between highly cross-linked polymer networks and a solid surface: Effect of interfacial bond density,” Macromolecules, vol. 34, no. 8, pp. 2710–2718, 2001.
[99] X. Chen, M. A. Dam, K. Ono, A. Mal, H. Shen, S. R. Nutt, K. Sheran, and F. Wudl, “A thermally re-mendable cross-linked polymeric material,” Science, vol. 295, no. 5560, pp. 1698–1702, 2002.
[100] D. Mukherji and C. F. Abrams Phys. Rev. E, vol. 79, p. 061802, Jun 2009.
[101] M. Sharifi, C. W. Jang, C. F. Abrams, and G. R. Palmese Journal of Materials Chemistry A, vol. 2, pp. 16071–16082, 2014.
[102] R. M. Elder, D. B. Knorr, J. W. Andzelm, J. L. Lenhart, and T. W. Sirk, “Nanovoid formation and mechanics: a comparison of poly(dicyclopentadiene) and epoxy networks from molecular dynamics simulations,” Soft Matter, vol. 12, pp. 4418–4434, 2016.
[103] M. Huang and C. Abrams Macromolecular Theory and Simulations, vol. 28, p. 1900030, aug 2019.
[104] K. Kremer and G. S. Grest The Journal of Chemical Physics, vol. 92, no. 8, pp. 5057–5086, 1990.
[105] C. F. Abrams and K. Kremer, “The effect of bond length on the structure of dense bead–spring polymer melts,” The Journal of Chemical Physics, vol. 115, pp. 2776–2785, 08 2001.
[106] C. C. De Silva, P. Leophairatana, T. Ohkuma, J. T. Koberstein, K. Kremer, and D. Mukherji, “Sequence transferable coarse-grained model of amphiphilic copolymers,” The Journal of Chemical Physics, vol. 147, no. 6, p. 064904, 2017.
[107] V. Singh, T. L. Bougher, A. Weathers, Y. Cai, K. Bi, M. T. Pettes, S. A. McMenamin, W. Lv, D. P. Resler, T. R. Gattuso, D. H. Altman, K. H. Sandhage, L. Shi, A. Henry, and B. A. Cola, “High thermal conductivity of chain-oriented amorphous polythiophene,” Nature Nanotechnology, vol. 9, pp. 384–390, May 2014.
[108] X. Pang, Y. He, J. Jung, and Z. Lin, “1d nanocrystals with precisely controlled dimensions, compositions, and architectures,” Science, vol. 353, no. 6305, pp. 1268–1272, 2016.
[109] C. M. Marques, Les polymères aux interfaces. PhD thesis, Éditeur inconnu, 1989.
[110] P. E. Theodorakis, H.-P. Hsu, W. Paul, and K. Binder, “Computer simulation of bottle-brush polymers with flexible backbone: Good solvent versus theta solvent conditions,” J. Chem. Phys., vol. 135, p. 164903, 10 2011.
[111] M. B. Jakubinek, M. A. White, G. Li, C. Jayasinghe, W. Cho, M. J. Schulz, and V. Shanov, “Thermal and electrical conductivity of tall, vertically aligned carbon nanotube arrays,” Carbon, vol. 48, no. 13, pp. 3947–3952, 2010.
[112] P. Yaghoobi, M. V. Moghaddam, and A. Nojeh, ““heat trap”: Light-induced localized heating and thermionic electron emission from carbon nanotube arrays,” Solid State Communications, vol. 151, no. 17, pp. 1105–1108, 2011.
[113] S. Yamaguchi, I. Tsunekawa, N. Komatsu, W. Gao, T. Shiga, T. Kodama, J. Kono, and J. Shiomi, “One-directional thermal transport in densely aligned single-wall carbon nanotube films,” Applied Physics Letters, vol. 115, p. 223104, 11 2019.
[114] J. P. Feser, J. S. Sadhu, B. P. Azeredo, K. H. Hsu, J. Ma, J. Kim, M. Seong, N. X. Fang, X. Li, P. M. Ferreira, S. Sinha, and D. G. Cahill, “Thermal conductivity of silicon nanowire arrays with controlled roughness,” Journal of Applied Physics, vol. 112, p. 114306, 12 2012.
[115] G. Pennelli, S. Elyamny, and E. Dimaggio, “Thermal conductivity of silicon nanowire forests,” Nanotechnology, vol. 29, p. 505402, oct 2018.
[116] N. Fakhri, D. A. Tsyboulski, L. Cognet, R. B. Weisman, and M. Pasquali, “Diameter-dependent bending dynamics of single-walled carbon nanotubes in liquids,” Proceedings of the National Academy of Sciences, vol. 106, no. 34, pp. 14219–14223, 2009.
[117] S. M. Ha, H. L. Lee, S.-G. Lee, B. G. Kim, Y. S. Kim, J. C. Won, W. J. Choi, D. C. Lee, J. Kim, and Y. Yoo, “Thermal conductivity of graphite filled liquid crystal polymer composites and theoretical predictions,” Composites Science and Technology, vol. 88, pp. 113–119, 2013.
[118] H. Ma, Y. Ma, and Z. Tian, “Simple theoretical model for thermal conductivity of crystalline polymers,” ACS Applied Polymer Materials, vol. 1, no. 10, pp. 2566–2570, 2019.
[119] A. Cembran, F. Bernardi, M. Garavelli, L. Gagliardi, and G. Orlandi, “On the mechanism of the cis-trans isomerization in the lowest electronic states of azobenzene: S0, s1, and t1,” Journal of the American Chemical Society, vol. 126, no. 10, pp. 3234–3243, 2004.
[120] F. Aleotti, L. Soprani, A. Nenov, R. Berardi, A. Arcioni, C. Zannoni, and M. Garavelli, “Multidimensional potential energy surfaces resolved at the raspt2 level for accurate photoinduced isomerization dynamics of azobenzene,” Journal of Chemical Theory and Computation, vol. 15, no. 12, pp. 6813–6823, 2019. PMID: 31647648.
[121] T. Koda, T. Toyoshima, T. Komatsu, Y. Takezawa, A. Nishioka, and K. Miyata, “Ordering simulation of high thermal conductivity epoxy resins,” Polymer Journal, vol. 45, pp. 444–448, Apr 2013.
[122] G. Lv, C. Shen, N. Shan, E. Jensen, X. Li, C. M. Evans, and D. G. Cahill, “Odd–even effect on the thermal conductivity of liquidcrystalline epoxy resins,” Proceedings of the National Academy of Sciences of the United States of America, vol. 119, p. e2211151119, 2022.
[123] D. A. Pink, “The even–odd effect in liquid crystals. A simple model,” The Journal of Chemical Physics, vol. 63, pp. 2533–2539, 09 1975.
[124] R. Saha, G. Babakhanova, Z. Parsouzi, M. Rajabi, P. Gyawali, C. Welch, G. H. Mehl, J. Gleeson, O. D. Lavrentovich, S. Sprunt, and A. Jákli, “Oligomeric odd–even effect in liquid crystals,” Mater. Horiz., vol. 6, pp. 1905–1912, 2019.
[125] Q. Fang and M. A. Hanna, “Rheological properties of amorphous and semicrystalline polylactic acid polymers,” Industrial Crops and Products, vol. 10, no. 1, pp. 47–53, 1999.
[126] H. Quan, Z.-M. Li, M.-B. Yang, and R. Huang, “On transcrystallinity in semi-crystalline polymer composites,” Composites Science and Technology, vol. 65, no. 7, pp. 999–1021, 2005.
[127] A. Sarı and A. Biçer, “Thermal energy storage properties and thermal reliability of some fatty acid esters/building material composites as novel form-stable pcms,” Solar Energy Materials and Solar Cells, vol. 101, pp. 114–122, 2012.
[128] F. Wu, X. Li, yingao Jiao, C. Pan, G. Fan, Y. Long, and H. Yang, “Multifunctional flexible composite membrane based on nanocellulose-modified expanded graphite/electrostatically spun fiber network structure for solar thermal energy conversion,” Energy Reports, vol. 11, pp. 4564–4571, 2024.
[129] B. Yang, X. Zhang, J. Ji, W. Hua, and M. Jiang, “Multifunctional phase change film with high recyclability, adjustable thermal responsiveness, and intrinsic self-healing ability for thermal energy storage,” Journal of Energy Chemistry, vol. 97, pp. 216–227, 2024.
[130] X. Wei and T. Luo, “Role of ionization in thermal transport of solid polyelectrolytes,” J Physical Chemistry C, vol. 123, pp. 12659–12665, 2019.
[131] S. W. Cranford and M. J. Buehler, “Variation of weak polyelectrolyte persistence length through an electrostatic contour length,” Macromolecules, vol. 45, no. 19, pp. 8067–8082, 2012.
[132] A. A. Gavrilov, A. V. Chertovich, and E. Y. Kramarenko, “Conformational behavior of a single polyelectrolyte chain with bulky counterions,” Macromolecules, vol. 49, no. 3, pp. 1103–1110, 2016.
[133] M. Muthukumar, “50th anniversary perspective: A perspective on polyelectrolyte solutions,” Macromolecules, vol. 50, no. 24, pp. 9528–9560, 2017. PMID: 29296029.
[134] N. C. Forero-Martinez, R. Cortes-Huerto, A. Benedetto, and P. Ballone, “Thermoresponsive ionic liquid/water mixtures: From nanostructuring to phase separation,” Molecules, vol. 27, no. 5, 2022.
[135] D. Tomida, Thermal Conductivity of Ionic Liquids. Rijeka: IntechOpen, 2018.
[136] A. Halperin, M. Kröger, and F. M. Winnik, “Poly(n-isopropylacrylamide) phase diagrams: Fifty years of research,” Angewandte Chemie International Edition, vol. 54, no. 51, pp. 15342–15367, 2015.
[137] S. Samanta, D. R. Bogdanowicz, H. H. Lu, and J. T. Koberstein, “Polyacetals: Water-soluble, ph-degradable polymers with extraordinary temperature response,” Macromolecules, vol. 49, no. 5, pp. 1858–1864, 2016.
[138] C. Wu and X. Wang, “Globule-to-coil transition of a single homopolymer chain in solution,” Phys. Rev. Lett., vol. 80, pp. 4092–4094, May 1998.
[139] T. E. de Oliveira, C. M. Marques, and P. A. Netz, “Molecular dynamics study of the lcst transition in aqueous poly(n-n-propylacrylamide),” Phys. Chem. Chem. Phys., vol. 20, pp. 10100–10107, 2018.
[140] S. Grobelny, C. H. Hofmann, M. Erlkamp, F. A. Plamper, W. Richtering, and R. Winter, “Conformational changes upon high pressure induced hydration of poly(n-isopropylacrylamide) microgels,” Soft Matter, vol. 9, pp. 5862–5866, 2013.
[141] T. E. de Oliveira, P. A. Netz, D. Mukherji, and K. Kremer, “Why does high pressure destroy co-non-solvency of pnipam in aqueous methanol?,” Soft Matter, vol. 11, pp. 8599–8604, 2015.
[142] C. M. Papadakis, B.-J. Niebuur, and A. Schulte, “Thermoresponsive polymers under pressure with a focus on poly(n-isopropylacrylamide) (pnipam),” Langmuir, vol. 40, no. 1, pp. 1–20, 2024. PMID: 38149782.
[143] F. D. Jochum and P. Theato, “Temperature- and light-responsive polyacrylamides prepared by a double polymer analogous reaction of activated ester polymers,” Macromolecules, vol. 42, no. 16, pp. 5941–5945, 2009.
[144] K. Y. Lam, C. S. Lee, M. R. Pichika, S. F. Cheng, and R. Y. Hang Tan, “Light-responsive polyurethanes: classification of light-responsive moieties, light-responsive reactions, and their applications,” RSC Adv., vol. 12, pp. 15261–15283, 2022.
[145] J. Volarić, J. Buter, A. M. Schulte, K.-O. van den Berg, E. Santamaría-Aranda, W. Szymanski, and B. L. Feringa, “Design and synthesis of visible-light-responsive azobenzene building blocks for chemical biology,” The Journal of Organic Chemistry, vol. 87, no. 21, pp. 14319–14333, 2022. PMID: 36285612.
[146] H. G. Schild, M. Muthukumar, and D. A. Tirrell, “Cononsolvency in mixed aqueous solutions of poly(n-isopropylacrylamide),” Macromolecules, vol. 24, no. 4, pp. 948–952, 1991.
[147] F. M. Winnik, H. Ringsdorf, and J. Venzmer, “Methanol-water as a co-nonsolvent system for poly(n-isopropylacrylamide),” Macromolecules, vol. 23, no. 8, pp. 2415–2416, 1990.
[148] G. Zhang and C. Wu, “Reentrant coil-to-globule-to-coil transition of a single linear homopolymer chain in a water-methanol mixture,” Phys. Rev. Lett., vol. 86, pp. 822–825, Jan 2001.
[149] D. Mukherji, C. M. Marques, , and K. Kremer, “Polymer collapse in miscible good solvents is a generic phenomenon driven by preferential adsorption,” Nature Communications, vol. 5, p. 4882, 2014.
[150] R. Zheng, J. Gao, J. Wang, and G. Chen, “Reversible temperature regulation of electrical and thermal conductivity using liquid–solid phase transitions,” Nature Communications, vol. 2, p. 289, 2011.
[151] K.-P. Le, R. Lehman, J. Remmert, K. Vanness, P. M. L. Ward, and J. D. Idol Journal of Biomaterials Science, Polymer Edition, vol. 17, no. 1-2, pp. 121–137, 2006.
[152] D. E. Meyer and A. Chilkoti, “Purification of recombinant proteins by fusion with thermally-responsive polypeptides,” Nature Biotechnology, vol. 17, pp. 1112–1115, 1999.
[153] B. Zhao, N. K. Li, Y. G. Yingling, and C. K. Hall, “Lcst behavior is manifested in a single molecule: Elastin-like polypeptide (vpgvg)n,” Biomacromolecules, vol. 17, no. 1, pp. 111–118, 2016.
[154] D. Kern, M. Schutkowski, and T. Drakenberg, “Rotational barriers of cis/trans isomerization of proline analogues and their catalysis by cyclophilin,” Journal of the American Chemical Society, vol. 119, no. 36, pp. 8403–8408, 1997.
[155] A. Valiaev, D. W. Lim, T. G. Oas, A. Chilkoti, and S. Zauscher, “Force-induced prolyl cis-trans isomerization in elastin-like polypeptides,” Journal of the American Chemical Society, vol. 129, no. 20, pp. 6491–6497, 2007.
[156] Y. Zhao, R. Cortes–Huerto, and D. Mukherji, “A simple generic model of elastin–like polypeptides with proline isomerization,” Macromolecular Rapid Communications, vol. n/a, no. n/a, p. 2400304.
[157] H. Staudinger, “über polymerisation,” European Journal of Inorganic Chemistry, vol. 53, no. 6, pp. 1073–1085, 1920.