Introduction

Reasonable regulation of heat energy is one of the crucial elements that enable our societies to develop and advance in a sustainable way, e.g., professional heat management helps alleviate the impact of temperature on the performance of equipment [1, 2]. Over the years, noteworthy efforts have been made in developing cutting-edge technologies to solve these issues [3, 4]. Among these techniques, nanotechnologies are one of the effective and promising technical roadmaps to tune thermal properties because they often exhibit surprising behavior [5]. With the continuous advancement of nanotechnology, it is found that the reasonable design of the microstructure of polymers can significantly improve the thermal conductivity of polymers [6]. Choy et al. [7] have testified that the in-plane thermal conductivity of ultrahigh molecular weight polyethylene could exceed 40 \({\text{ W/mK}}\) by increasing the crystallinity and rationally aligning the polymer chains. Even the thermal conductivities of commercially available microfibers are measured to be as high as 23 \({\text{W}}/{\text{mK}}\) with optimized microstructure [8]. In comparison, general commercial polyethylene (PE) products are commonly regarded as thermal insulators with a thermal conductivity value of about 0.4 \({\text{W}}/{\text{mK}}\) [9, 10].

During the last decade, one-dimensional or quasi-one-dimensional materials have attracted the attention of many researchers because of their excellent physical properties. For a single carbon nanotube (CNT), the thermal conductivity along the backbone \(\kappa_{\parallel }\) of CNT exceed 103 \({\text{W}}/{\text{mK}}\) [2]. However, the arbitrary-designed and unordered CNT-based forests, bundles, sheets, and networks show a terrible reduction of thermal conductivity, for example, \(\kappa_{\parallel }\) of CNT forests [11] is reduced to 0.5–1.2 \({\text{W}}/{\text{mK}}\), \(\kappa_{\parallel }\) is around 100 \({\text{W}}/{\text{mK}}\) for CNT bundles [12], \(\kappa_{\parallel }\) is about 43 \({\text{W}}/{\text{mK}}\) for CNT sheets [13]. This sharp decrease in thermal conductivity is not a specific phenomenon for CNT and can also be found in silicon nanowire (NW) arrays [14], PE fibers [8], crystalline assemblies of PE [15], and poly-3,4-ethylene dioxythiophene (PEDOT) [16]. At present, people have realized the impact of the unreasonable design of microstructures on physical properties; besides, rational design of microstructures of existing materials has overwhelming advantages over synthesis of new materials because the former is more environmental-friendly and resource-saving.

Nowadays, interfacial nanotechnologies with surface-initiated polymer brushes have seen rising interest both in industry and academia due to their potential to improve interfacial performance [17]. The so-called surface-initiated polymer brushes usually consist of one-dimensional or quasi-one-dimensional materials, which are robust functional coating materials with unique end-anchoring structures and extraordinary physical characteristics. Most of these functionalities are achieved by chemically grafting one end of prefabricated one-dimensional or quasi-one-dimensional materials, especially polymer chains, at a specific density to a substrate. Specifically, the synthesis method [17] of surface-initiated polymer brushes can be roughly divided into two types: “grafting to” and “grafting from”, which are consistent with their synthetic approach. In the former approach, the polymer chains with functional groups that can react or interact with the target substrate are first prepared, and then the prefabricated polymers are tethered to the substrate through chemical reaction subsequently. In contrast, in the “grafting from” method, the polymer brushes gradually grow from the tethered initiators on the substrate. The different types of polymer chains tethered on the target surface will provide the substate with desirable properties [18] including controllable wettability, corrosion resistance, adhesive performance, lubrication and friction properties. Besides, as the rational structural design of materials has the potential to improve the efficiency of interfacial heat exchange, research interest in the optimal design of surface-initiated polymer brushes has grown rapidly in heat management. Cola et al. [19] have experimentally testified that the improvement of alignment in polymer brushes further enhances the thermal conductivity. Mukherji et al. [20] numerically prove that the thermal conductivity of polymer forests is an intriguing balance between the interaction of nonbonded polymer chains and the bending of polymer chains. However, the thermal transportation for surface-initiated polymer brushes consisting of a wide range of one-dimensional or quasi-dimensional materials is still not clear.

In this study, we report simulation results of thermal transportation in surface-initiated polymer brushes on a two-dimensional flat plane using a multi-scale molecular dynamics method. We map the physical properties of polymer chains into a CGMD model by a simple strategy. Detailed equilibrium molecular dynamics simulations using the Green–Kubo method are employed to obtain the thermal conductivity of different surface-initiated polymer brushes. Furthermore, structure analysis and phonon density of states (PDOS) are used to illustrate the change of thermal conductivity in different polymer brushes. Our purpose is to use a generic model to unveil the thermal characteristics of an extensive range of materials with similar structural features.

Simulation details and methods

Simulation details

To mimic the configuration of real surface-initiated polymer brushes, we use a simple mapping scheme to transform the nanoscale physical features into a coarse-grained (CG) model. Because one-dimensional and quasi-one-dimensional materials are the main components of polymer brushes, surface-initiated polymer brushes inherit their excellent flexibility. For these materials, their intrinsic flexibility is governed by persistence length \(l_{{\text{p}}}\). The polymer chain with a high value of \(l_{{\text{p}}}\) usually possesses high bending stiffness, which is easier to keep the shape of a linear rod. Notably, the polymer chain with a considerably small value of \(l_{{\text{p}}}\) is soft and easy to collapse into a pile. To model this feature into our coarse-grained model, we summarize the relationship between persistence length and diameter of several one-dimensional and quasi-one-dimensional materials in Fig. 1. This perspective provides essential information about the length scale of a single polymer chain in our CG models.

Figure 1
figure 1

Persistence length versus diameter of different single chains [21,22,23,24,25,26,27,28,29]

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In our CG model, we consider a spherical bead represents several monomers, and the diameter of the bead is set as \(\sigma\). Thus, the range of persistence length is circumscribed from \(0.65\sigma\) to \(5 \times 10^{5} \sigma\). As the persistence length (\(l_{{\text{p}}} = \frac{k}{{k_{{\text{B}}} T}}\)) is positively dependent on the bending stiffness, it is convenient to change the persistence length by changing the bending stiffness (\(k\)). The bending energy of a polymer chain is defined as: \(E_{{\text{B}}} = \frac{k}{2}\int_{0}^{{\text{L}}} {\left( {\frac{\partial T}{{\partial s}}} \right)}^{2} {\text{d}}s\) [30], where L is the length when the polymer chain is straightened and not extensible, T is the position vector of a CG bead, \(k\) is the bending stiffness reflecting the flexibility of a polymer chain, and \(\frac{\partial T}{{\partial s}}\) represents the degree of local bending. As the bending energy of this model is given as:

$$\begin{array}{*{20}c} {E_{{\text{B}}} = \frac{k}{2}\mathop \smallint \limits_{0}^{{\text{L}}} \left( {\frac{\partial T}{{\partial s}}} \right)^{2} {\text{d}}s } \\ \end{array}$$
(1)

the discretized version of Eq. (1) can be illustrated as follows:

$$\begin{aligned} & \approx \frac{k}{{2r_{0} }}\mathop \sum \limits_{i} \left( {T_{i + 1}^{2} + T_{i}^{2} - 2\cos \alpha } \right) \\ & = \frac{k}{{r_{0} }}\mathop \sum \limits_{i} \left( {1 - \cos \alpha } \right) \\ & = \frac{k}{{r_{0} }}\mathop \sum \limits_{i} \left( {1 - \cos \left( {\pi - \theta } \right)} \right) \\ & = \frac{{l_{{\text{p}}} k_{{\text{B}}} T}}{{r_{0} }}\mathop \sum \limits_{i} \left( {1 - \cos \left( {\pi - \theta } \right)} \right) \\ \end{aligned}$$
(2)

where \(\theta\) is the bond angle between the bond vectors (among three successive CG beads), \(\alpha\) is the complementary angle, and \(r_{0}\) is the balanced bond length.

The Kremer–Grest model [31] is employed to describe the CG polymer chain in this work, which has been testified to be able to describe the thermal transportation of polymers [6, 20]. Within our CG model, consecutively bonded spherical beads with the mass of m form a polymer chain. The nonbonded interactions, which are mainly the van der Waals (vdW) interactions, within and between polymer chains are expressed by a Lennard–Jones (LJ) potential:

$$\begin{array}{*{20}c} {U_{{{\text{LJ}}}} \left( {r_{ij} } \right) = 4\varepsilon \left[ {\left( {\frac{\sigma }{{r_{ij} }}} \right)^{12} - \left( {\frac{\sigma }{{r_{ij} }}} \right)^{6} } \right], r_{ij} < r_{{\text{c}}} } \\ \end{array}$$
(3)

All other physical parameters are deduced and based on these three fundamental reduced Lennard–Jones units (\(\varepsilon\), \(\sigma\), and m are the reduced units of energy, length, and mass, respectively.) with Boltzmann constant set as \(k_{{\text{B}}} =\) 1.The time unit can then be deduced to \(\tau = \sigma \left( {{m \mathord{\left/ {\vphantom {m \varepsilon }} \right. \kern-\nulldelimiterspace} \varepsilon }} \right)^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}}\). The cutoff radius is set as \(r_{{\text{c}}} = 1.12246\sigma\) in all types of interactions for enhancing the calculation efficiency. Within the Lennard–Jones potential, \(\sigma\) is the point where the LJ potential crosses the zero energy line, and the parameter \(\varepsilon\) equals the strength of interaction.

For bonded beads, the Kremer-Grest model uses finite extensible nonlinear elastic (FENE) bonds to connect each bead. The FENE bond potential can be expressed as:

$$\begin{array}{*{20}c} { U_{{{\text{bond}}}} \left( {r_{ij} } \right) = - 0.5k_{{{\text{fene}}}} R_{0}^{2} \ln \left[ {1 - \left( {\frac{{r_{ij} }}{{R_{0} }}} \right)} \right]} \\ \end{array}$$
(4)

which can effectively confine the maximum of bond length to \(R_{0} = 1.5 \sigma\). The bond strength is set to \(k_{{{\text{fene}}}} = 30.0 \varepsilon \cdot\sigma^{ - 2}\), and the balanced bond length is \(r_{{{\text{balance}}}} = 0.97\sigma\). The following equation defines the total energy of a polymer brush system:

$$\begin{array}{*{20}c} { U_{{{\text{total}}}} = U_{{{\text{bond}}}} + U_{{{\text{LJ}}}} + U_{{{\text{angle}}}} + U_{{{\text{kinetic}}}} } \\ \end{array}$$
(5)

Each polymer chain in all brush systems consists of 501 beads. The contour length of a polymer chain is \(N_{{\text{l}}} \cong 500\sigma ,\) because the balanced bond length is \(r_{{{\text{balance}}}} = 0.97\sigma\). Furthermore, the first beads are randomly fixed onto a square plane with lateral dimensions \(L_{x} = L_{y} \cong 35\sigma\), and the chains are placed perpendicular to the square plane along the z direction in the initial configuration. The surface grafted density \(\rho_{{{\text{grafted}}}}\) is varied from 0.2 \(m\cdot\sigma^{ - 2}\) to 0.65 \(m\cdot\sigma^{ - 2}\). \(\rho_{{{\text{grafted}}}}\) is obtained by dividing the number of fixed CG beads on the surface by the total surface area (\(L_{x} \,L_{y}\)). Periodic boundary conditions are applied in the x and y directions, and fixed boundary condition is applied in the z direction.

Simulations are performed in two steps: the initial relaxation of polymer brushes, and the calculation of physical properties including thermal conductivity, density, virial stress, and phonon analysis. The initial relaxation is carried out under the canonical ensemble (NVT ensemble) with a timestep of \(\Delta t = 0.001\tau\), and the relaxation is run for \(10^{6} \tau\). The NVT ensemble is realized by coupling the Langevin thermostat and microcanonical ensemble (NVE ensemble). The Langevin thermostat possesses a damping constant \(\gamma = 1\tau^{ - 1}\) and \(T = 1\varepsilon \,k_{{\text{B}}}^{ - 1}\) in the initial equilibration, and the NVE ensemble is used for updating the position and velocity of CG beads. It should be emphasized that only thermal conductivities are calculated in the NVE ensemble, and the Langevin thermostat is decoupled in this process. The equations of motion are integrated using the velocity Verlet algorithm, and all the simulations are performed using the LAMMPS package [32].

Besides, one should notice that a simple CGMD model with artificially selected features is not suitable for elucidating all the physical properties of polymer brushes. Our purpose is trying to explore the effect of the morphological characteristics of surface-initiated polymer brushes on thermal transportation through numerical calculations. Also, we try to give rudimentary explanations of the reduction of thermal conductivity observed in experiments [13, 33] using a simple CGMD model.

Calculation of thermal conductivity and virial stress

After the preparation and equilibrium of initial configurations, the components of \(\kappa\) are calculated using the Green–Kubo method in the microcanonical ensemble (NVE ensemble). The equations of motion are integrated in the NVE ensemble with a timestep of \(10^{ - 3} \tau\), and the estimation of the heat flux auto correlation (\({\text{HACF}}\left( t \right) = \left\langle {J\left( t \right)\,J\left( {0} \right)} \right\rangle\)) is calculated by sampling the heat flux vector \(J\left( t \right)\). Here, the correlation function is acquired in a time length of \(2 \times 10^{3} \tau\) during a total simulation time of \(2 \times 10^{5} \tau\); then, ten different simulation results will be obtained. Finally, \(\kappa\) is calculated by averaging these different plateau values of Green–Kubo integral for the component along the chain orientation direction, i.e., z direction,

$$\begin{array}{*{20}c} {\kappa_{\parallel } = \frac{V}{{k_{{\text{B}}} T^{2} }}\mathop \smallint \limits_{0}^{\infty } \left\langle {J_{z} \left( t \right)\,J_{z} \left( 0 \right)} \right\rangle {\text{d}}t} \\ \end{array}$$
(6)

and in the lateral directions, i.e., x and y directions,

$$\begin{array}{*{20}c} {\kappa_{ \bot } = \frac{V}{{2k_{{\text{B}}} T^{2} }}\mathop \smallint \limits_{0}^{\infty } \left[ {\left\langle {J_{x} \left( t \right)\,J_{x} \left( 0 \right)} \right\rangle + \left\langle {J_{y} \left( t \right)\,J_{y} \left( 0 \right)} \right\rangle } \right]{\text{d}}t} \\ \end{array}$$
(7)

It should be noted that every surface-initiated polymer brush has a certain number of fixed beads. However, these beads are not treated as rigid bodies. In order to make the simulations conformable to the Geen–Kubo method, several virtual springs are attached to the fixed beads. Therefore, the fixed beads can still oscillate at their initial position and keep the shape of surface-initiated polymer brushes.

Figure 2 is a numerical example for calculating \(\kappa_{\parallel \left( 0 \right)}\) of a fully extended chain with two fixed ends with \(l_{{\text{p}}} = 10\). It should be noted that Fan et al. prove the many-body heat flux is not correct in LAMMPS [34]. As we introduce the angle potential in this work, a many-body heat flux corrected LAMMPS version is used for the computation of thermal conductivity [35]. The virial stress \(\sigma_{\parallel }\) is calculated using the standard subroutine in LAMMPS and following the procedure in the previous study [36].

Figure 2
figure 2

a Heat current autocorrelation function (HCACF) versus time, b \(\kappa_{\parallel \left( 0 \right)}\) versus time

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Calculation of phonon density of states

The calculation of phonon density of states (PDOS) of materials is an effective way to describe phonon activities [4], which also provides explanations for the change of thermal conductivity. The PDOS is computed by integrating the fourier transform [37] of the velocity autocorrelation function (VACF) of a selected group of atoms as follows:

$$\begin{array}{*{20}c} {{\text{PDOS}}\left( \omega \right) = \mathop \smallint \limits_{ - \infty }^{\infty } e^{i\omega t} {\text{VACF}}\left( t \right){\text{d}}t} \\ \end{array}$$
(8)

where \({\text{PDOS}}\left( \omega \right)\) is the entire PDOS at the vibrational frequency \(\omega\) and \({\text{VACF}}\left( t \right)\) is described as:

$$\begin{array}{*{20}c} {{\text{VACF}}\left( t \right) = \frac{1}{N}\mathop \sum \limits_{i}^{N} \left\langle {v_{i} \left( 0 \right)v_{i} \left( t \right)} \right\rangle } \\ \end{array}$$
(9)

\(v_{i} \left( 0 \right)\) is the velocity vector for \(i{\text{th}}\) particle at time origin, \(v_{i} \left( t \right)\) is the velocity vector for \(i{\text{th}}\) particle at time origin \(t\), \(N\) is the number of the selected atoms. The ensemble average (denoted by \(\left\langle \cdots \right\rangle\)) in VACF is realized by time-average in molecular dynamics simulations over a period of \(3\tau\), with the velocities extracted from the simulation under NVT ensemble every \(0.001\tau\). For decomposed PDOS [38], the correlation function is calculated as \(v_{ix} \left( 0 \right)v_{ix} \left( t \right) + v_{iy} \left( 0 \right)v_{iy} \left( t \right)\) and \(v_{iz} \left( 0 \right)v_{iz} \left( t \right)\) for the directions vertical to the polymer chain and along the polymer chain, respectively. The peak of \({\text{PDOS}}\left( \omega \right)\) predicates more states are occupied by the phonons with frequency \(\omega\). Note that when no phonon with the frequency of \(\omega\) exists in polymer brushes, \({\text{PDOS}}\left( \omega \right)\) is equal to 0.

Calculation of phonon participate rate

The calculation of phonon participation ratio (PPR) is another efficient technique providing perspectives into the phonon activities, especially for quantitatively describing the phonon mode localization, e.g., PPR equals 1 if all atoms participate in the heat conduction; otherwise, PPR equals to \(1/N\) if only one atom participates in the heat conduction. That is, if the PPR increases, the localization of phonon modes will be mitigated. Therefore, the thermal conductivity will be higher. The PPR at any temperature can be calculated conveniently from the molecular dynamics simulations without the calculation of lattice dynamic [4], and is defined as [39, 40]:

$$\begin{array}{*{20}c} {{\text{PPR}}\left( \omega \right) = \frac{1}{N}\frac{{\left( {\mathop \sum \nolimits_{i} {\text{PDOS}}_{i} \left( w \right)^{2} } \right)^{2} }}{{\mathop \sum \nolimits_{i} {\text{PDOS}}_{i} \left( w \right)^{4} }}} \\ \end{array}$$
(10)

Results and discussions

Thermal conductivity of a single polymer chain

As mentioned in the introduction, a single polymer chain can exhibit a high thermal conductivity. Related numerical efforts have predicted that the thermal conductivity of a single polyethylene chain to be as high as 350 \({\text{W}}/{\text{mK}}\) using all-atom molecular dynamics simulations [41]. To explore the influence on the thermal conductivity of a single chain, we firstly consider only one chain in a crowded environment. Then we use a single chain with no interactions induced by other chains in a three-dimensional fixed boundary box for comparison. The grafted polymer brush to a two-dimensional rigid substrate is depicted in Fig. 3a, with the first bead of each polymer chain fixed to the substrate. All polymer brushes are configured in a rectangular box of dimensions \(35\sigma \times 35\sigma \times 502\sigma\) (x, y, z, respectively). Periodic boundary conditions are applied in x and y directions, and fixed boundary conditions is used in z direction. Figure 3b is a configuration when a single polymer chain is fully extended in a three-dimensional fixed boundary condition. For the single chain with two fixed ends, the dimensions of the simulation box are \(15\sigma \times 15\sigma \times 502\sigma\) (x, y, z, respectively) in order to save memory and reduce computation load in simulation.

Figure 3
figure 3

a Schematic diagram of surface-initiated polymer brushes represented by bead spring chains, along with the interactions involved in the brush and the boundary conditions of the simulation box. Yellow-colored beads are fixed to the rigid substrate. b A single polymer chain with two fixed ends (yellow-colored beads) to form a fully extended configuration in a simulation box

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The line chart of Fig. 4a depicts how the longitudinal thermal conductivity of a single polymer chain with two fixed ends \(\kappa_{\parallel \left( 0 \right)}\) depends on the persistence length. We first focus on the red region of the line chart in Fig. 4a, where the longitudinal thermal conductivity is positively correlated with persistence length. However, the thermal conductivity decreases as the persistence length increases in the blue region, and then the thermal conductivity saturates in the green region. One should notice that for a polymer chain with unconstrained ends, its thermal conductivity is positively dependent on the persistence length. The decrease in the thermal conductivity in Fig. 4a is mainly due to the overall rotation of the chain relative to the axis of its initial position when \(l_{{\text{p}}}\) reaches a high value. In Fig. 4b, for the calculation of \(\kappa_{\parallel \left( 0 \right)}\) in a brush, we randomly choose one chain in each polymer brush for computation of \(\kappa_{\parallel \left( 0 \right)}\) in the crowded environment. It can be seen that, within the entire range of persistence length, \(\kappa_{\parallel \left( 0 \right)}\) in a brush is reminiscently smaller than \(\kappa_{\parallel \left( 0 \right)}\) of a single chain with two fixed ends. This sharp decrease of \(\kappa_{\parallel }\) can also be seen in CNT forests and sheets [11, 13, 33]. Moreover, we find that with the increase in the grafted density, \(\kappa_{\parallel \left( 0 \right)}\) in a brush is improved considerably. Furthermore, the collision between chains, the twinning of chains, and the rotation of chains are all inevitable in reality and simulations, which may be the reasons for the nonmonotonic increase of \(\kappa_{\parallel \left( 0 \right)}\) when increasing the persistence length shown in Fig. 4b. We believe if these factors can be reduced or controlled, the thermal conductivity of a single chain in a crowded environment will be further improved.

Figure 4
figure 4

a Thermal conductivity of a fully extended polymer chain varying with the persistence length. Note that the distance between these two fixed ends is 501 \(r_{{{\text{balance}}}}\), and the normalization volume in the Green–Kubo formula is taken as the volume of one chain, i.e., \(v = 500\;v_{{{\text{bead}}}}\) with \(v_{{{\text{bead}}}}\) being the volume of one bead. b Thermal conductivity of a single polymer chain in a crowded environment with different grafted densities. The dashed line is a polynomial fit to the data

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A single polymer chain in a polymer brush configuration has two different modes of heat transfer: the heat transfer along the backbone of a chain is usually dominated by repeated covalent bonds; the heat transfer between chains is governed by the vdW interactions. Previous perspectives believe the longitudinal thermal conductivity of polymers is dominated by the bonded interaction because the strength of a covalent bond is about 100 times larger than that of vdW interactions. However, we find \(\kappa_{\parallel \left( 0 \right)}\) of a single polymer chain in a brush is a delicate balance of its persistence length and the grafted density, and the persistence length and the grafted density are found to largely influence the morphology of a polymer system [30]. Therefore, we speculate that the maintenance of a rational and ordered micro-morphology of a polymer chain has a crucial impact on improving thermal conductivity.

Density distribution, stress, and thermal conductivity

Then we start by calculating the number density distribution along the longitudinal direction for each polymer brush. It is expected that the orientation of every bond of a fully extended polymer chain should be aligned in one direction. Thus, the height of the polymer chain should be equal to the contour length (500 \(\sigma\)), and the beads should be distributed equally along the longitudinal backbone direction, i.e., the z direction. The green lines in Fig. 5a–c indicate the number density distribution along the z direction for polymer brushes when all the chains are fully extended. However, we find that polymer brushes with different grafted densities have a similar density concentration phenomenon. The density concentration indicates the disorder of bond orientation; therefore, the height of a polymer chain in a polymer brush is always smaller than its contour length. With the increase in the persistence length, the degree of density concentration will be alleviated. Furthermore, for polymer chains with a smaller persistence length, e.g., \(l_{{\text{p}}} = 10\sigma\), it can be seen that \(\rho_{\parallel }\) sketches show ridges at the top interface. The increment of the grafted density can effectively lower down the ridge.

Figure 5
figure 5

a–c The number density of different surface-initiated polymer brushes as a function of the height along the z direction. d–f The virial stress of different surface-initiated polymer brushes as a function of the height along the z direction. Data are shown for different grafted densities, and the black arrows indicate the increase in the persistence length

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The density concentration also gives a stress concentration, as shown in Fig. 5d–f. The stress concentration is caused by bending in chains and entangling between chains, which also means the aggregation of kinks [42]. Since the kinks can be considered as a phonon scattering center, reducing the number of kinks is beneficial for phonon propagation. In essence, the uniform distribution of polymer beads in three-dimensional space, i.e., no bending or curling for polymer chains will give a fairly high thermal conductivity, however, it is hard to achieve. Increasing the persistence length and grafted density can ameliorate the irregular distribution of polymer beads in three-dimensional space as shown in Fig. 6. Therefore, a single chain in a crowded environment could maintain the rod-like shape and is not easy to entangle with other chains, and the number of kinks decreases correspondingly. Thus the thermal conductivity of a single chain increases accordingly as shown in Fig. 4b,

Figure 6
figure 6

The morphology of different surface-initiated polymer brush systems captured from CGMD simulations

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Figure 7 shows the overall thermal conductivity of different polymer brushes, and the thermal conductivities are typically anisotropic. \(\kappa_{ \bot }\) is predominantly influenced by the grafted density, as the weak van der Waals interactions between chains are the principal contributor to \(\kappa_{ \bot }\). Also, \(\kappa_{\parallel }\) is significantly increased by increasing the grafted density, which is due to the alleviation of the bending of a polymer chain in a crowded environment. To explain the influence of persistence length on thermal conductivity, we further make phonon analysis in the next part.

Figure 7
figure 7

Thermal conductivity for surface-initiated polymer brushes with different grafted densities varying with the persistence length

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Phonon analysis

As phonons are the primary carrier of heat transfer in polymers, we analyze the phonon density of states (PDOS) to explain the change of \(\kappa_{\parallel }\) in Fig. 8. The restraint in thermal conductivity of polymer brushes could be further attributed to the enhanced phonon scatterings, as evidenced by the phonon density of states. The PDOSs along the backbone direction are calculated separately for polymer brushes of different persistence lengths with the same grafted density of 0.65 \(m\cdot\sigma^{ - 2}\). Besides, as the timestep is \(\Delta t = 0.001\tau\), its inverse is almost the maximum attainable frequency. However, the PDOS curves decay to zero at about \(50\tau^{ - 1}\). The results are shown in Fig. 8.

Figure 8
figure 8

a, b PDOSs for surface-initiated polymer brushes with a grafted density of 0.65 \(m\cdot\sigma^{ - 2}\) along the z direction

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The PDOSs all concentrate on the low-frequency regime which is similar to the previous work [43], and it can be found that the high-frequency region of PDOSs is not sensitive to the change of the persistence length. Thus, we mainly focus on the change of the low-frequency region of PDOSs. As shown in Fig. 8a, with the increase in the persistence length, it can be seen that the peak of the low-frequency region is enhanced when \(l_{{\text{p}}} < 800\), which means the increase in phonon group velocities and has a favorable effect on heat conduction. It also indicates that the increase of persistence length leads to the attenuation in phonon scattering and the increase in the thermal conductivity. However, when \(l_{{\text{p}}} \ge 800\) the PDOS peaks in their low-frequency region are significantly damped, which induces the decrease in phonon group velocities and has an adverse effect on heat conduction. Although the overall PDOSs of polymer brushes with \(l_{{\text{p}}} \ge 800\) show attenuation in low frequency, the peak of PDOSs with \(l_{{\text{p}}} \ge 800\) is still higher than PDOSs with lower persistence length (e.g., \(l_{{\text{p}}} = 10\sigma\)) in the peak of the low-frequency region. Specially, as the persistence length keeps increasing, the curves of PDOS roughly overlap with each other, which indicates that they have almost the same phonon modes. Therefore, the thermal conductivity of polymer brushes with high persistence length is almost consistent shown in Fig. 7. The PDOS analysis may help explain the non-monotonic variation of longitudinal thermal conductivity \(\kappa_{\parallel }\) shown in Fig. 7.

The PPRs of polymer brushes with the grafted density of 0.65 \(m\cdot\sigma^{ - 2}\) varying with persistence length along the z direction are summarized in Fig. 9. From Fig. 9a, the increase in persistence length (\(l_{{\text{p}}} \le 500\sigma\)) leads to the increase in PPR over the entire range of frequency. Thus, phonon modes in polymer brushes tend to be delocalized, and more atoms participate in the motion. The delocalized phonons are capable of transporting thermal energy in polymer brushes, which results in an increment in thermal conductivity. As the persistence length of polymer brushes keeps increasing from Fig. 9b, the PPRs are almost close over the entire range of frequency which also proves the validity of PDOS analysis. However, the PPRs of polymer brushes with high persistence length are still higher than the PPRs of polymer brushes with low persistence length, e.g., the polymer brush with \(l_{{\text{p}}} = 10\sigma\).

Figure 9
figure 9

Phonon participation ratios for surface-initiated polymer brushes with the grafted density of 0.65 \(m\cdot\sigma^{ - 2}\) along the z direction

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Conclusions

In summary, in-depth investigations of heat conduction in polymer brushes are performed using the CGMD simulations. A simple and useful mapping strategy is used to model one-dimensional and quasi-one-dimensional materials into CGMD models. We provide a feasible explanation for the attenuation of thermal conductivity in surface-initiated polymer brushes, i.e., the sharp reduction of \(\kappa_{\parallel }\) in polymer brushes while a single polymer chain can have a high thermal conductivity. Although the reduction of thermal conductivity in polymer brushes is inevitable, we find that the \(\kappa_{\parallel }\) can be significantly improved by tuning the grafted density and the persistence length of polymer chains, and the mechanism is explained using the structure and phonon analysis. We demonstrate that the polymer brushes consisting of one-dimensional materials or quasi-one-dimensional materials are found to have controllable thermal conductivity, thus indicating their potential applications in heat dissipation coatings. We also show a numerical verification that a reasonable selection and design of structure can effectively broaden the application of existing materials. Therefore, upon experimental validation, these results may help design advanced coating materials with controllable thermal properties.