High thermal conductivity of bulk epoxy resin by bottom-up

parallel-linking and strain: a molecular dynamics study

Shouhang Li1,2,#, Xiaoxiang Yu2,3,#, Hua Bao1,* , Nuo Yang2,3,*

1 University of Michigan-Shanghai Jiao Tong University Joint Institute, Shanghai Jiao Tong University, Shanghai 200240, P. R. China 2 State Key Laboratory of Coal Combustion, Huazhong University of Science and Technology, Wuhan 430074, P. R. China 3 Nano Interface Center for Energy (NICE), School of Energy and Power Engineering, Huazhong University of Science and Technology, Wuhan 430074, P. R. China

Abstract

The ultra-low thermal conductivity (~0.3 Wm-1K-1) of amorphous epoxy resins

significantly limits their applications in electronics. Conventional top-down methods e.g.

electrospinning usually result in aligned structure for linear polymers thus satisfactory

enhancement on thermal conductivity, but they are deficient for epoxy resin polymerized

by monomers and curing agent due to completely different cross-linked network structure.

Here, we proposed a bottom-up strategy, namely parallel-linking method, to increase the

intrinsic thermal conductivity of bulk epoxy resin. Through equilibrium molecular

dynamics simulations, we reported on a high thermal conductivity value of parallel-linked

epoxy resin (PLER) as 0.80 Wm-1K-1, more than twofold higher than that of amorphous

structure. Furthermore, by applying uniaxial tensile strains along the intra-chain direction,

a further enhancement in thermal conductivity was obtained, reaching 6.45 Wm-1K-1.

Interestingly, we also observed that the inter-chain thermal conductivities decrease with

increasing strain. The single chain of epoxy resin was also investigated and, surprisingly,

its thermal conductivity was boosted by 30 times through tensile strain, as high as 33.8

Wm-1K-1. Our study may provide a new insight on the design and fabrication of epoxy

resins with high thermal conductivity.

# S.L. and X.Y. contributed equal to this work. * To whom correspondence should be addressed. Email: hua.bao@sjtu.edu.cn (HB); nuo@hust.edu.cn (NY)

Introduction

Epoxy resins are classical thermoset polymers widely used in coatings, adhesives and

electronic packaging for their excellent thermal mechanical properties and high electrical resistance

[1]. However, most of epoxy resins have very low thermal conductivity on the order of 0.1 Wm-1K-

1 [2] owing to the existence of entanglement, voids etc. [3]. In some special situations, for example,

aerospace and flexible electronics fields, heat dissipation is becoming a crucial issue and the epoxy

resin with higher thermal conductivity is desirable [4]. The general strategy to increase the thermal

conductivity of polymers is doping materials with high thermal conductivities, such as ceramics [5-

8], metals [9], carbon nanotubes [10] and graphene [11]. Adding fillers increases the cost. More

importantly, the fillers will degrade the original electrical and mechanical properties of polymers

and affect the performance of the relevant products. Thus, it is necessary to enhance the intrinsic

thermal conductivity of epoxy resin.

The heat transport properties of cross-linked epoxy resin (CLER) are well revealed

theoretically and experimentally. Varshney et al. [12] calculated the thermal conductivity of CLER

employing both equilibrium molecular dynamics method (EMD) and non-equilibrium molecular

dynamics method (NEMD) and the thermal conductivity value is determined to be in the range of

0.30-0.31 Wm-1K-1 at 300 K. By using EMD method, Kumar et al. [2] found the positive

temperature dependence of thermal conductivity. Kline et al. [13] measured the thermal

conductivity of epoxy resin and found the thermal conductivity value goes from 0.23 Wm-1K-1 to 0.27

Wm-1K-1 when the temperature raises from 275 K to 375 K. The heat dissipation ability of epoxy resin

is similar to other polymers and it cannot meet the stringent requirement in relevant fields.

In the past decades, researchers found that polymers with highly ordered structures have

appreciable thermal conductivities. Chen and Henry [14, 15] found that the thermal conductivity of

polyethylene single chain can be even divergent which is several orders of magnitude larger than

that of amorphous bulk PE. The amazing discovery reveals that the intrinsic thermal conductivities

of polymers have not been fully excavated. Luo [16-19] and his cooperators found that polymer

nanofibers with intrinsically ordered backbones, strong backbone bonds, and strong dihedral angles

could be the high thermal conductivity candidates. The aligned polymers have very high thermal

conductivities is also verified by experiments. Shen et al. [3] found the thermal conductivity of

ultra-drawn polyethylene nanofibers can be as high as ~104 Wm-1K-1 at 300K. Xu et al. [20] found

polyethylene film consists of nanofibers with crystalline and amorphous regions owns high thermal

conductivity of 62 Wm-1K-1. Cao et al. [21] investigated on the thermal conductivity of polyethylene

nanowire arrays and they found the value of the nanowire array with the diameter of 100 nm can

reaches 21.1 Wm-1K-1 at 80 ℃. Singh et al. [22] measured the thermal conductivity of chain-

oriented amorphous polythiophene and the value can be as high as ~4.4 Wm-1K-1 at room

temperature.

To attain ordered structure in epoxy resin, some arisen technology [23]and materials [24] based

on top-down methods were adapted to enhance thermal conductivity. Akatsuka et al. [25] measured

the thermal conductivity of liquid-crystalline epoxy resin with macroscopic isotropic structure at

30°C and the value can be as high as 0.3 - 0.96 Wm-1K-1. The exact value depends on the type of

monomer and curing agent. Correspondingly, Koda et al. [26] calculated the thermal conductivity

of liquid crystalline epoxy resins at 300K using molecular dynamics and they got the value of 0.37

- 0.96 Wm-1K-1, which is consistent with the experimental results. Zeng et al. [27] measured the

thermal conductivity of the epoxy resin fibers made by electrostatic spinning technology and they

found the value can be as high as 0.8 Wm-1K-1. In their following work [28], they found the thermal

conductivity of epoxy resin fibers increases on decreasing the fiber diameter.

However, it should be noted that the conventional epoxy resin is amorphous cross-linked

network prepared by the polymerization of monomers and curing agents, which is different from

those amorphous linear polymers consisting of many chains. The top-down methods mentioned

above like electrospinning could be applied to straighten many unconnected chains and achieve the

long-range alignment. Nevertheless, for epoxy resin, top-down methods could only construct short-

range ordered structure, and also fail to avoid crosslinking and long-range disorder [28], thus form

network structure giving rise to strong intra-molecule phonon scatterings in heat transport

contributed by covalent bonding. This is the dominant reason why top-down methods based on

cross-linking does not work well in enhancement of thermal transport in epoxy resin. Therefore,

how to carefully design the linking structure with long-range alignment and better understanding of

heat transport mechanism in epoxy resin remains to be explored.

In this work, to further enhance the intrinsic thermal conductivity, we propose a bottom-up

linking strategy to construct parallel-linked epoxy resin (PLER). The procedure is similar to the

bottom-up manufacture technology, molecular layer deposition (MLD), which can be used to make

high quality polymer films [29, 30]. At the same time, the effects of morphology on the thermal

conductivity of epoxy resin are investigated. We first studied the relationship between the degree of

crosslinking and thermal conductivity of amorphous cross-linked network. Then we calculated the

thermal conductivities of bulk PLER and the strain effects are discussed. In addition, the intrinsic

thermal conductivity of PLER single chain is presented.

Model and Simulation details

The commonly used monomer and curing agent of epoxy resin are EPON-862 (di-glycidyl

ether of bisphenol F) and DETDA (diethylene toluene diamine), respectively. The corresponding

chemical structures are shown in Fig. 1(a) and (b). The two end carbon atoms in EPON-862 and the

two nitrogen atoms in DETDA are reaction sites and they can form a new covalent bond as shown

in Fig. 1(c). Several methods have been proposed to construct crosslinking atomic model of epoxy

resin network, for which the properties can be consistent with experiments [31-33]. In this work, we

employed the methods described in the Ref. [33], and the relaxed cross-linked network is shown as

the Fig. 1(d). In order to build ordered epoxy resin structure, we also changed the conventional

crosslinking method and proposed a new parallel-linking method, which is similar to the principle

of MLD. MLD is widely used to manufacture polymer films with high quality [29, 30]. The polymer

films are fabricated by stacking molecules on substrates layer by layer. The schematic of MLD is

shown in Fig. 1(e). For parallel-linking model, the monomer and curing agent form different

molecular layers. Then one monomer reacts with one curing agent and then we could get one cross-

linked segment, as shown in Fig. 1(c). Several molecular layers replicate along the chain direction

(z axis) and then we could get the stacked semi-crystal structure, as shown in Fig. 1(f). In this study,

there are 4 chains on x and y axes for bulk PLER, respectively.

The molecular dynamics simulations are performed at the temperature of 300K using the

Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) package[34]. According

to the former simulations [2, 12], the consistent valence force field (CVFF) [35, 36] is accurate

enough to characterize the thermal properties of epoxy resins and therefore it was employed to

describe the interatomic interactions in this work. As Kumar et al. [2] pointed out, the simulation

results can be consistent with the experiments when the long-range interactions are included. Thus,

both van der Waals (vdW) force and electrostatic force are included in our simulation. The exact

formula of CVFF force field is:

2 2

0 0

12 6

1 1, 0

1 cos

14

2

total bb

N Ni j

i j i j j iij ij ij

E K r r K K d n

q q

r r r

. (1)

The terms on right hand side of Eq. (1) stand for bond force, angle force, dihedral force, vdW

force and Coulomb interaction, respectively. The Lennard-Jones (LJ) potential parameters across

different types of atoms were calculated by using the Lorentz-Berthlot mixing rules (i.e., ��� =

����������, ��� = ��� + ���/2). The cutoff distance of the 12-6 LJ potential for all structures was

set to 9 Å . All potential parameters are available in Ref. [35] and [36].

The calculation of heat flux is a critical step in EMD method. The heat flux is defined as:

1 1

2i i ij i j ij

i i j

J e v f v v xV

, (2)

where V is the volume of the simulation cell, i and j are the indexes of atoms, ei is the total energy

of atom i, iv and jv are atom velocities of atom i and j, ijf is the force between atom i and j,

ijx

is

the relative position of atom i and j.

We can obtain the thermal conductivity by directly integrating heat current autocorrelation

function (HCACF), namely

0

2 00

3 B

VJ J d

k T

, (3)

where T is the system temperature, Bk is the Boltzmann constant,

0 is the integral upper limit of

HCACF, which is also called correlation time. The angular bracket denotes an ensemble average.

The velocity Verlet algorithm [37] is employed to integrate equation of motion, and the time

step is set as 0.25 fs. The initial structures are first minimized by standard conjugate-gradient energy-

minimization methods in LAMMPS. After that, the system runs in the isothermal-isobaric ensemble

(NPT) for 250 ps to relax the whole system at the given temperature and pressure. Then it runs

another 250 ps in the canonical ensemble (NVT) at the given temperature 300 K. Later on, it runs

in the microcanonical ensemble (NVE) for 250 ps for further relaxation. Finally, it runs at least

another 2.5 ns in NVE, during which the heat current is recorded at an interval of 2.5 fs. To achieve

better ensemble average, we obtained the final results on the average of at least 4 independent

simulations with different initial conditions. All the aforementioned simulation parameters were

carefully checked to ensure the results were converged [38].

Note that the thermal conductivities were averaged over three orthogonal directions for CLER

to get better statistics as they were considered to be isotropic in nature for cross-linked network

structure. For parallel-linked bulk and single chain structures, the thermal conductivity on the z axis

is different from that on the other two axes due to anisotropy. For these anisotropic cases, we

calculate the thermal conductivities on the three axes separately.

Fig. 1. (a-b) Chemical structures of monomer EPON-862 and curing agent DETDA. (c) One cross-

linked segment of epoxy resin, consisting of carbon (blue), hydrogen (green), oxygen (red) and nitrogen

(purple). (d) The relaxed cross-linked epoxy resin. (e) An illustration of MLD method. (f) An illustration

of PLER structure.

Results and Discussions

We first calculated the thermal conductivity of CLER with 50% degree of crosslinking and the

results are depicted in Fig. 2(a) as an illustration of the EMD method. HCACF fluctuates

dramatically for 1 ps, which means the strong reflection of heat current, and then decays to zero

rapidly within 3ps for the sake of strong phonon scatterings in CLER, thus giving rise to the quick

convergence of integral, i.e., thermal conductivity value. In practical usage of CLER, degree of

crosslinking raises with the increase of curing reaction time, so CLER could have different degrees

of crosslinking [39]. Therefore, we further investigated the thermal conductivity of amorphous

CLER with respect to degree of crosslinking. As Fig. 2(b) shows, the value is in the range of 0.32 -

0.35 Wm-1K-1. It is comparable to the previous simulation [12] and experiment results [40], which

validates our calculations results about epoxy resins. We can see that degree of crosslinking has little

impacts on thermal conductivity of CLER. It is quite different from polyethylene [41] whose thermal

conductivity increases with degree of crosslinking. This is attributed to the different contributions

of heat transport in these two kinds of cross-linked polymers. Cross-linking bridges the separated

molecules in polymers and thus paves the way for heat transport by covalent bonding interactions

while it almost has no influences on heat transport by nonbonding interactions. For linear polymer

chains, bonding interactions are main contributors [41], so crosslinking further improves the heat

transport. However for epoxy resin, it has been verified that nonbonding interactions are the most

predominant among different interactions contributions to thermal conductivity [12]. Accordingly,

crosslinking has little effects on heat transport in epoxy resin.

Fig. 2. (a) Normalized heat current autocorrelation function (black line) and thermal conductivity (red

line) of cross-linked epoxy resin with 50% degree of crosslinking. (b) Thermal conductivity of cross-

linked network epoxy resin as a function of degree of crosslinking.

To show the enhancement on thermal conductivity by bottom-up parallel linking, we performed

further simulation on bulk PLER. Due to the well-known simulation size effects of EMD, we

performed convergence test with different simulation domain sizes. The thermal conductivity values

of systems with 1, 2, 4, 8 segments along chain are calculated and the results are shown in Fig. 3

(a). It can be seen that the simulation domain size effect is not significant for bulk PLER. We use 8

segments as the length of simulation system in the subsequent simulations for bulk PLER. The

converged thermal conductivity along intra-chain (z axis) direction (κz) at 300K is 0.80 Wm-1K-1,

which is twofold higher than that of CLER. Our results are comparable with the thermal conductivity

of highly ordered liquid-crystalline epoxy resins as Koda et al. [26] reported. Unlike common liquid

epoxy resin, the relaxed PLER owns long-range ordered structure, as shown in the inset of Fig. 3(a).

Besides, there is no bonding interactions between chains, which can further decrease the intra-chain

scatterings. It is noteworthy that the thermal conductivity value can be even further enhanced if

monomer and curing agent with stronger backbone are employed in PLER. The thermal

conductivities along inter-chain (x and y axes) directions (κx and κy) are ~ 0.25 Wm-1K-1, which are

close to that of CLER. This is to be expected because the nonbonding interactions are the main

contributors for heat transport along x and y directions, while the bonding interactions dominate

heat transport along z direction. It can be seen that PLER owns strong anisotropy due to semi-crystal

structure (the inset of Fig. 3 (a)), which is much more ordered than amorphous bulk CLER in Fig.

1 (d). However, the thermal conductivity value of PLER is still much smaller than that of bulk

aligned PE [18, 42-44] . Previous studies suggested that the thermal conductivities of nanowires and

nanoribbons can be decreased with the increasing number of kinks and folds [45-47]. There is also

buckling in chains of the relaxed bulk PLER, which indicates that semi-crystalline structure still

owns low degree of crystallinity and it can induce intra-chain phonon scatterings, which is also

observed in other bulk aligned buckled polymers [48].

Fig. 3. (a) Thermal conductivity of PLER as a function of number of segments in the chain

direction. The inset is a side-view of the relaxed bulk PLER (16 chains are allocated with different

colors). (b) The thermal conductivities of stretched PLER with respect to different strains. (c)

XRD patterns of different epoxy resin structures. (d) The radial distribution function g(r) as a

function of radius r of PLER with different strains. The dash line and dot dash line are the radiuses

where peaks in g(r) of PLER without strain and with 0.15 strain locate, respectively.

For further improvement of thermal conductivity, we studied the thermal conductivities of bulk

PLER with uniaxial tensile strains [49]. Figure 3(b) shows the strain dependence of thermal

conductivity along different directions. It can be seen that κz increases dramatically with tensile

strain. When the strain is smaller than 0.15, κz increases slowly with strain. Then it increases

significantly for larger strain. The value of κz reaches 6.45 Wm-1K-1 with 0.50 strain, which is around

20 times larger than that of the bulk CLER. However, κx and κy decrease from 0.25 Wm-1K-1 to 0.12

Wm-1K-1. Similarly, Zeng et al. [27] found the thermal conductivity of fibrous epoxy on the through-

plane decreases with the strain rising. We did not compute the cases with even larger strain for the

confinement of computational ability for the well-known size effects of EMD method.

In order to quantitatively describe the morphologies of CLER, pristine PLER and stretched

PLER, we compared the X-ray diffraction (XRD) patterns of three different structures, as it can

reflect the crystallinity of structure. The XRD intensity is calculated based on a mesh of reciprocal

lattice nodes defined by entire simulation domain using a simulated radiation of wavelength lambda

[34, 50]. As presented in Fig. 3(c), the XRD patterns show significant differences among these three

structures. As we can see, stretched PLER shows several discrete peaks, and PLER without strain

has fewer peaks. CLER shows many broad continuous peaks. The phenomenon indicates that the

stretched PLER has a higher crystallinity of structure compared to the structure without strain, while

the latter owns more ordered structure compared to the amorphous cross-linked network. There are

less phonon scatterings in more ordered structures. That is why the stretched PLER has the highest

thermal conductivity among the three structures. Besides, the XRD pattern of PLER with 0.15 strain

is similar to that with 0.50 strain. It elucidates that the crystallinity of PLER with strain larger than

0.15 is higher than the ones with smaller strains which results in the saltation of κz at the strain 0.15.

We analyzed the inter-chain radial distribution function (RDF) of stretched PLER to further

understand the negative strain dependence of inter-chain thermal conductivity. The heat transfer

along inter-chain directions is dominated by nonbonding interactions, which are related to the atom

distribution around the reference atom. We choose all the atoms in an isolate chain as the reference

atoms of RDF. The distance from a reference atom to the atom in other chains is defined as � =

�(� − ��)� + (� − ��)

� + (� − ��)�. The RDF is recorded as �(�) =

��

��[(����)����]

, where r is

the distance between reference atoms and other atoms, and � is the number of atoms with a

distance of �(� < � < � + ��) to the reference atoms [18], �� is set to be 0.2 Å. The results are

shown in Fig. 3 (d). It can be seen �(�) decreases with the increase of strain. This demonstrates

that the reduction of density around the reference atoms and then the decrease of atomic interactions

on x and y axes because nonbonding interactions are the leading contributors of heat propagation

on these two axes. Hence, the interactions on x and y axes would be smaller for stretched PLER

with larger tensile strains and then κx and κy would be lower. The differences of �(�) are more

significant when the strain is larger than 0.3. Corresponding, κx and κy are decrease dramatically.

To gain insights into how the inter-chain nonbonding interactions impact on the thermal

conductivity of PLER, the single chain structure (Fig. 4 (a)) was investigated which can avoid such

inter-chain phonon scatterings, and help us obtain the upper limit thermal conductivity of epoxy

resin. In this study, a sufficiently large vacuum space was introduced artificially to ensure that the

single chain is isolate and there are no other chains interacting with it. The relaxed chain length

obtained from the bulk PLER is used for isolated single chain at the same temperature. We deformed

the foregoing relaxed single chain and bulk structure with corresponding strains and we could get

the stretched structures. The thermal conductivity of unstretched PLER single chain structure is

shown in Fig. 4 (b). The converged value is ~ 1.13 Wm-1K-1 on z axis, which is about threefold

higher than that of amorphous cross-linked network and larger than that of bulk PLER. The

converged length of single chain structure is much larger than the bulk PLER, which can also verify

that the absence of inter-chain interactions could reduce the phonon scatterings and facilitate the

propagation of the long-wavelength phonons. We also calculated the thermal conductivity of

stretched PLER single chain. The tensile strain increases from 0 to 0.80 and it can have strong effects

on the morphology of the single chain and greatly affect κz of PLER single chain, as shown in Fig.

4 (a). The values increase by 30 times, from ~ 1.13 Wm-1K-1 to ~ 33.82 Wm-1K-1. The variation

trend of thermal conductivity with strain is analogous to that of stretched bulk PLER: The thermal

conductivity increases slowly with small strain and then increases linearly when the strain is higher

than 0.15. As shown in Fig. 4(a), the morphology of single chain with 0.15 strain is straighter than

that of unstretched structure and this phenomenon is more significant in the structure with 0.30

strain. The thermal conductivity still keeps the increasing trend with strain raising. This implies the

thermal conductivity of bulk PLER could be further enhanced through decreasing both intra-chain

and inter-chain phonon scatterings.

Fig. 4. (a) Epoxy resin single chains with 0.15 and 0.30 strain. (b) The thermal conductivities of

single chain with respect to different number of segments. (c) The thermal conductivities of

stretched single chain as a function of strains.

Conclusions

In summary, we performed equilibrium molecular dynamics simulations to investigate the

thermal conductivities of cross-linked and parallel-linked epoxy resins as well as epoxy resin single

chain. Our results show that the thermal conductivity of cross-linked epoxy resin is in the range of

0.32 - 0.35 Wm-1K-1 and the degree of crosslinking has little effects because the nonbonding

interactions are the main contributors in heat transport. We proposed a bottom-up parallel-linking

method, which is shown to be efficient and the along-chain thermal conductivity of bulk parallel-

linked epoxy resin structure can be enhanced more than twofold compared to the cross-linked

network structure due to the much more ordered morphology. The thermal conductivity can be

boosted with the increasing strain, and reaches a value as high as 6.45 Wm-1K-1 with 0.50 strain.

The thermal conductivity firstly increases slightly and then increases significantly with the strain

raising. X-ray diffraction pattern shows that deformed parallel-linked epoxy resin owns perfect

crystalline structure, which explains the surprising high thermal conductivity. The thermal

conductivity on x and y axes would decrease with the increase of strain for degraded nonbonding

interactions. For parallel-linked single chain, the thermal conductivity is about threefold higher than

that of cross-linked network and the value can be enhanced greatly by 30 times, reaching as high as

~ 33.82 Wm-1K-1 when strain reaches 0.80. The stretched structures own higher thermal

conductivities than that of unstreched one due to more ordered morphology and the suppression of

both inter-chain and intra-chain phonon scatterings. The results may provide a meritorious guide on

the design and fabrication of epoxy resin with high thermal conductivity.

Acknowledgements

This work was supported by the National Natural Science Foundation of China No. 51676121

(HB), 51576076 (NY) and 51711540031 (NY), and the Hubei Provincial Natural Science

Foundation of China No. 2017CFA046 (NY). Simulations were performed with computing

resources granted by HPC () from Shanghai Jiao Tong University and the National

Supercomputing Center in Tianjin (NSCC-TJ).

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