Molecular Dynamics Study on the Thermal Conductivity of the End

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Molecular Dynamics Study on the Thermal Conductivity of the Endgrafted Carbon Nanotubes Filled Polyamide-6.6 Nanocomposites Yangyang Gao, and Florian Müller-Plathe J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.7b11310 • Publication Date (Web): 26 Dec 2017 Downloaded from http://pubs.acs.org on December 26, 2017

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The Journal of Physical Chemistry C is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

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The Journal of Physical Chemistry

Molecular Dynamics Study on the Thermal Conductivity of the End-grafted Carbon Nanotubes Filled Polyamide-6.6 Nanocomposites Yangyang Gao1,2*, Florian Müller-Plathe2 1

Key Laboratory of Beijing City on Preparation and Processing of Novel Polymer Materials, Beijing University of Chemical Technology, People’s Republic of China 2 Eduard-Zintl-Institut für Anorganische und Physikalische Chemie and Profile Area Thermofluids & Interfaces, Technische Universität Darmstadt, Alarich-Weiss-Str.8, 64287 Darmstadt, Germany ABSTRACT It is very important to improve the thermal conductivity of polymer nanocomposites to widen their application. In this work, the effect of grafted chains and mechanical deformation on the thermal conductivity of end-grafted carbon nanotubes (CNTs) filled polyamide-6.6 nanocomposites has been investigated by molecular dynamics simulation. The results show that the thermal conductivity increases with the grafting density, while it first increases and then saturates with the length of the grafted chains. The dependence of the thermal conductivity on the density and the length of the grafted chains is described by an empirical equation. Moreover, it is further improved if all CNTs are linked by chains or CNTs align along one direction, especially the latter. By fitting the present simulation results with an effective medium approximation model, the interfacial thermal resistance is obtained, which indicates that a stronger enhancement of the thermal conductivity is realized when chains are grafted at the end atoms of CNTs. Under deformation, the orientation of both the chains and the CNTs improves the thermal conductivity parallel to the tensile direction, but reduces the thermal conductivity perpendicular to it. Lastly, the *Corresponding author: [email protected] 1

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The Journal of Physical Chemistry 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

contribution of the polymer alignment and the CNT alignment to the anisotropy of thermal conductivity is quantified.

2


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1. Introduction A high thermal conductivity is very important for many applications of polymer nanocomposites. For example, when used as heat sinks in electric or electronic systems, nanocomposites with a thermal conductivity approximately from 1 to 30 W/m·K are required.1 In general, amorphous polymers have a very low thermal conductivity of 0.1-1 W/m·K at room temperature due to phonon scattering from numerous defects.2 On the other hand, carbon nanotubes (CNT) possess a thermal conductivity 104 times as large as that of polymers. It reaches about 6600 W/ m·K.3, 4 Recently, addition of CNTs into the polymer matrix has been regarded as an effective approach to improve the thermal conductivity of materials.5, 6 However, obtaining nanocomposites with high thermal conductivities is still very challenging. On the one hand, the high interfacial thermal resistance hinders the heat flow across the CNT-polymer interface.7 An effective method to increase the interfacial conductance is to graft the chains on the surface of CNTs.8, 9 On the other hand, the CNT network formed at high filler concentration is not effective to transfer the heat, which results in a lack of thermal percolation in composite materials.6 Currently, the effective medium theory (EMA)10 has been developed to evaluate the macroscopic thermal conductivity of nanocomposites. It is based on multiple-scattering theory11 for arbitrary filler composites with interfacial thermal resistance. Through analysis with the EMA model, the interfacial thermal resistance is estimated in experiments.12, 13 In simulations,14 the interfacial thermal resistance is inserted into the EMA model to obtain the thermal conductivity of nanocomposites. 3



The second way to improve the thermal conductivity of nanocomposites is to stretch the materials to align the CNTs and polymer chains along the tensile direction. For CNT filled epoxy nanocomposites,15 the thermal conductivity has been enlarged by ~50% and ~90% in this direction; Perpendicular, however, it was reduced by ~30% and ~53% at strain=25% and 40%. Therefore, the anisotropy of the thermal conductivity is about 2.0 and 4.0 at strain=25% and 40%, respectively. For CNT-filled polyethylene nanocomposites, Haggenmueller et al.5 attributed the increase of the thermal conductivity along the alignment direction to the alignment of the polyethylene chains, not the CNTs. The CNT-CNT contacts in the polymer nanocomposites become increasingly important for the effective heat transport across the systems for aligned CNTs.16 Thus, grafting and orientation can cooperate to improve the thermal conductivity of nanocomposites. On the basis of the aforementioned descriptions, although a large amount of research works have been devoted to enhance the thermal conductivity of nanocomposites, systematic research is still needed to study the fundamental mechanisms on how the grafted CNT and mechanical deformation is used to tune the thermal conductivity. In this work, we explored the influence of grafted chains, the orientation, and volume fraction ( f ) of CNTs on the thermal conductivity of CNT-filled polyamide-6,6 (PA) nanocomposites. By fitting our simulation results with the EMA model,10 first the interfacial thermal resistance is estimated for different systems. Then, by inserting the microscopic parameters into EMA model, the macroscopic thermal conductivities are obtained. At last, we investigate the influence 4


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of mechanical deformation on the thermal conductivity of nanocomposites. A realistic united-atom model for CNT and polyamide-6,6 (PA) has been employed. Meanwhile, the other two fundamental issues are also discussed in this work: (i) Do the grafting positions (at the ends of CNTs or at the lateral surfaces of CNTs) influence the thermal conductivity of nanocomposites? (ii) How do the polymer alignment and CNT alignment contribute to the anisotropy of thermal conductivity? 2. Details of the model. Similar to our previous work,17 the simulation box contains 160 PA chains of four repeat units. The length of the chosen (3,3) CNT is 2.5nm, which has been synthesized in experiment18 and adopted by others19. The volume fraction of CNT (between 0% and 34%) is defined as the ratio of volume occupied by CNT to the total volume. The volume occupied by CNT is the ratio of the total mass of CNT to the density of CNT (2.17 g/cm3)20. Depending on the CNT volume fraction, the number of CNTs in the systems varies from 0 to 125 and the box length varies from 8.3 nm to 8.9 nm. The grafting density (gd) is defined as the ratio of the number of grafted chains to the number of CNT atoms (120). The grafting length (L) is the number of backbone atoms of the grafted chains. The force-field parameters of the systems are the same as in our previous work.17 Similar to our previous work,17 first, we put the ungrafted or grafted CNTs and free PA chains into a large box. Then, we adopt the isothermal-isobaric (NPT) ensemble to compress the system for 20 ns, where the temperature and pressure are fixed at T = 350 K and P = 101.3 kPa respectively by using the Berendsen’s thermostat and barostat21 with coupling times of 1 ps and 10 ps, respectively. During 5



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this process, we used a harmonic potential to fix the center of mass of the initial dispersed CNTs until the CNTs are surrounded by the polymer chains. This procedure ensures that the CNTs are uniformly dispersed in the PA matrix. Further equilibration under isothermal-isovolumetric (NVT) conditions is performed for 10 ns, where CNTs can move freely. The temperature T=350 K has been chosen because it makes certain that the polymer is above its glass transition temperature ( Tg ≈ 320K ).22 Periodic boundary conditions are employed in all three directions of the simulation box. The time step for the leapfrog integration scheme is set to be1.0 fs. The uniaxial deformation is carried out by stretching the simulation box in one direction at a constant rate:

k(t) = k0 + αt

(1)

where k0 is the initial length of the box in this direction, t is the time and α is the deformation rate. The strain is defined as

k − k0 ×100% . The two perpendicular box k0

dimensions are coupled to the bath pressure (1 bar) using a Berendsen barostat with pressure coupling time 1.0 ps. The temperature of the system is kept constant during the deformation by using a Berendsen thermostat with temperature coupling time 1.0 ps. The results have been obtained with a constant deformation rate of α =10 nm/ns. A maximum strain of 100% is reached, i.e. K = 2K0 . It takes about 0.8 ns for the strain equal to 100%. It takes about 3ns to calculate thermal conductivity. Final results have been averaged over three deformation directions. We have carried out additional tests with strain rates α between 5 and 50 nm/ns. For systems of gd=6.67%, L=20, and α =5, 10 and 50 nm/ns, the calculated parallel thermal conductivities are 0.313, 6


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0.312 and 0.279 W/m·K, respectively. For these calculation, the degree of orientation (defined below) for the polymer is 0.19, 0.19, 0.15 and for CNTs is 0.34, 0.34 and 0.32, respectively. We decided to proceed with α =10 nm/ns. With this setting of α , there is also no formation of voids or crazes at the chosen temperature of 350 K. The density of the system after stretching stays within 2% of the unstretched state. We adopted reverse non-equilibrium molecular dynamics23 to calculate the thermal conductivity of the composites. The calculation details of the thermal conductivity and the standard deviation can be found in our previous work.17 Our in-house package YASP is adopted to run the molecular simulation.24 When we calculated the thermal conductivity

κ of

nanocomposites, the simulation box is

divided into 16 slabs along the heat flux direction z. The velocity of the coldest atom in the 1st slab is exchanged with that of the hottest atom in the 9th slab as a primary perturbation.

The

conductivity

is

calculated

by

using

Fourier’s

law23

j z = κ dT / dz ，where jz is the heat flux; dT / dz is the temperature gradient in the intervening region. Detailed information can be found in our previous work.25 3. Results and discussion 3.1 Effect of grafting density and grafting length The length of the chosen (3,3) CNT is 2.5nm, which contains 120 atoms. The number of grafted PA chains per CNT is chosen to be 0, 2, 4, 8, and 12, which corresponds to gd=0.00%, 1.67%, 3.33%, 6.67%, and 10.0%. The grafting length (L) is chosen as 0 (ungrafted), 6, 14, 20 and 28 backbone atoms. The PA chains are grafted at the end carbon atoms of CNT as shown in Fig. S1, unless mentioned 7



otherwise. Figure S1 shows the PA grafted CNTs with different grafting densities (gd). Here, f stands for the volume fraction of CNT. First, we investigated the influence of grafted chains on the thermal conductivity

κ

of nanocomposites. Detailed information on the studied systems is shown in Table

S1 in Supporting Information (SI) for different grafting densities and Table S2 in SI for different grafting lengths. The normalized thermal conductivity κ / κ0 in Fig. 1(a) increases linearly with the grafting density, which is well fitted by a linear relation.

κ0 stands for the thermal conductivity of the ungrafted system. The linear increase of the κ / κ0 with gd indicates that the grafted chains play a very important role in transferring the heat across the nanocomposites. As displayed in Fig. 1(b), κ / κ0 increases first quickly and then plateaus as a function of the grafting length. This indicates that (1) κ / κ0 can increase at most by ~20%, as one goes from no grafted chains to infinitely long chains; (2) κ / κ0 increases quickly with the length between no grafted chains and short grafted chains. Then, extending the chains to long chain 2

× 5.0 ≈ 10 atoms will have only a small effect. These results prove that the grafted chains at the CNT end atoms can support the heat transfer between CNT and free PA chains. However, the limited increase of κ / κ0 is attributed to the short CNTs adopted in the simulation. In our previous work on graphene,17 the maximum thermal conductivity appears at a medium grafting density because grafting decreases the intrinsic in-plane thermal conductivity of graphene. However, the thermal conductivity always increases with the grafting density in this work. On one hand, the effect of grafting in decreasing the longitudinal thermal conductivity is limited in 8


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short CNT.26 On the other hand, the PA chains are grafted only at the CNT end atoms. Thus, they nearly do not influence the longitudinal thermal conductivity of CNT. As a result, the thermal conductivity does not decrease at high grafting density. 1.4

1.4

(a) ƒ=6.13 %: κ/κ0=0.026⋅gd(%)+1.0 ƒ=10.2 %: κ/κ0=0.033⋅gd(%)+1.0 ƒ=14.5 %: κ/κ0=0.035⋅gd(%)+1.0

1.3 1.2

(b)

ƒ=6.13 %: κ/κ0=1+0.19(1-exp(-(L/atoms)/4.81)) ƒ=10.2 %: κ/κ0=1+0.24(1-exp(-(L/atoms)/4.43)) ƒ=14.5 %: κ/κ0=1+0.26(1-exp(-(L/atoms)/5.19))

1.3

1.1

Simulation (ƒ=6.13 %) Simulation (ƒ=10.2 %) Simulation (ƒ=14.5 %) Fit (ƒ=6.13 %) Fit (ƒ=10.2 %) Fit (ƒ=14.5 %)

1.0 0.9 0

2

4

gd (%)

6

8

Fig. 1 The normalized thermal conductivity

κ/κ0

1.2

κ/κ0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Simulation (ƒ=6.13 %) Simulation (ƒ=10.2 %) Simulation (ƒ=14.5 %) Fit (ƒ=6.13 %) Fit (ƒ=10.2 %) Fit (ƒ=14.5 %)

1.1

1.0

0

10

κ/κ0

5

10

15

L/atoms

20

25

of nanocomposites as a function of (a) grafting

density (gd) (L(grafting length)=20) and (b) grafting length (L) (gd(grafting density)=6.67%) at three carbon nanotube (CNT) volume fractions

f

.

κ0

is the thermal conductivity of the

ungrafted system. (T=350K)

3.2 Effect of polyamide-linked carbon nanotubes Let us come to the normalized thermal conductivity κ / κm for ungrafted CNTs, PA-grafted CNTs as well as PA-linked CNTs. κm stands for the thermal conductivity of pure PA without CNTs. Our simulations calculate κm to be 0.22 W/m.K, in line with experimental values and previous simulations.27-29 Figure S2 presents the details of the structure of PA-linked CNTs where all CNTs are linked by PA chains. The free PA chains are around the PA-linked CNTs (not shown). First, during the NPT simulation, the center of mass of each CNT is fixed until the free PA chains enter into the interstice between CNTs. There are two kinds of PA chains which connect the CNTs together (Fig. S3): (1) one PA chain which links two CNTs, (2) one PA chain which connects one CNT, which is same as a grafted chain. The details of the systems 9



for this section are shown in Table S3 in SI. The calculated normalized thermal conductivity κ / κm is shown in Fig. 2. The total number of atoms in grafted PA chains is similar for PA-grafted CNT and PA-linked CNTs at the same CNT volume fraction. For ungrafted CNTs, κ / κm is less than 1.0. This indicates that the thermal conductivity of nanocomposites is lower than that of pure PA, in contrast to the experimental results30, 31. This contradiction is due to the short CNTs used in the simulation, which has a low intrinsic thermal conductivity32. The largest enhancement of the thermal conductivity appears for PA-linked CNTs, which can help effectively transfer heat across the nanocomposites. In addition, experimental efforts33, 34 have been published already to synthesize interconnected CNTs of the end-to-side as well as end-to-end types. 1.3

(a) Ungrafted CNT (b) PA-grafted CNT (c) PA-linked CNT

1.2 1.1

κ/κm

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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1.0 0.9 0.8 0.7 0

3


6

f (%)

κ / κm

9

12

15

of nanocomposites for ungrafted carbon

nanotube (CNT), polyamide (PA)-grafted CNT (gd(grafting density)=6.67%, L(grafting length)=20), and PA-linked CNT as a function of CNT volume fraction ( conductivity of pure PA. (T=350K, random orientation of CNTs)

3.3 Orientation of carbon nanotubes 10


f

).

κm is the thermal

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The effect of the random, perpendicular and parallel orientation of CNTs with respect to the temperature gradient on the thermal conductivity is investigated. The systems 1, 4 in Table S2 are adopted here. Figure S4 shows the systems with perpendicular and parallel CNT orientations. To prepare them, for perpendicular CNTs, free PA chains are distributed around the CNTs along the x direction (not shown). First, during the NPT simulation, 12 atoms of each CNT are randomly chosen to be fixed at their initial position by harmonic potential (force constant=99999 kJ/mol.nm2). Meanwhile, the lengths of y (7.23 nm) and z (9.71 nm) directions are fixed. The x direction is gradually compressed until the density does not change any more, which is about 1.15 g/cm3. In the Berendsen barostat, the isothermal compressibility is 10-6 kPa-1 for the y direction, while it is set to 0.0 kPa-1 for the other two directions. The systems continue to run under NVT conditions for 10ns. For parallel CNTs, the method is similar except that the lengths of x and z directions are set to be 9nm and 12.9nm, respectively while the y direction is gradually compressed until the density approaches about 1.15 g/cm3. It is noted that z axis is the direction of the heat flux. The normalized thermal conductivity κ / κm is shown in Fig. 3 for different orientations of CNTs. It shows that κ || / κ m is largest, next is κrandom / κm and κ⊥ / κm is smallest. This is consistent with experiment in graphene filled epoxy nanocomposites.35 This is because the thermal conductivity of CNT in longitudinal direction is larger than that in perpendicular directions.4, 36 Especially, κ || / κ m are 1.17 and 1.43 for ungrafted and grafted CNTs. But κrandom / κm is about 1.1 for the PA-linked CNTs in Fig. 2 at same CNT volume fraction. Thus, CNT orientation is 11



more effective to improve the nanocomposite’s thermal conductivity than chemically linking CNTs. This is because if CNTs align along one direction, it can provide direct thermal conductivity pathways across the nanocomposite, which leads to the improvement of thermal conductivity.37, 38 It is noted that this effect should be much larger for typical commercial µ m -sized CNTs. 1.5

Ungrafted CNT Grafted CNT

1.0

κ/κm

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

0.5

0.0 Random

Perpendicular


κ / κm

Parallel

of nanocomposites for ungrafted carbon

nanotube (CNT) and grafted CNT (gd(grafting density)=6.67%, L(grafting length)=20) with random, perpendicular and parallel orientations. polyamide. (T=350K,

κm

is the thermal conductivity of pure

f (CNT volume fraction)=6.13%)

3.4 Combination with effective medium approximation In the above simulations, the short CNT is adopted because of limited computer resources. To compare with experiments, we first fit the simulation results with the effective medium theory (EMA)10 to obtain the interfacial thermal resistance. This is because the interfacial thermal resistance between the end-grafted CNT and polymer is difficult to be obtained from the simulation. Lastly we insert the parameters (thermal conductivity, aspect ratio) of the µ m -sized CNT into the EMA model to estimate the macroscopic thermal conductivity. According to the EMA model, the 12


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thermal conductivity

κ of a nanocomposite with randomly dispersed CNTs can be

derived as10

κ 3 + f [βx + βz ] = κm 3 − f βx

(2)

2(κ11 −κm) , κ11 +κm

(3)

with

βx =

βz =κ33 /κm −1

where f is the volume fraction of CNTs; κm is the thermal conductivity of the pure PA matrix. κ11 and

κ33 can be expressed as κ11 =

κc , 2αk κc 1+ d κm

κ33 =

κc 2α κ 1+ k c l κm

where d and l are the diameter and length of the CNTs;

(4)

κ c is the longitudinal

thermal conductivity of the CNT; and α is a so-called Kapitza radius defined by

α = κ m Ri (5) where R i is the interfacial thermal resistance across CNT-PA interfaces. Equation (2) contains the influence of the CNT diameter, aspect ratio, volume fraction, the interface thermal resistance, and thermal conductivity ratio

κc / κm

on the thermal

conductivity of composites. The simulated systems are described in Table S4 in SI for this section. For the calculations, we took κc (l = 2.5 nm) =24 W/m.K32 and κ m =0.22 W/m.K17 The thermal conductivity of CNTs

κc

depends on its length.32 In the simulation,

CNT with a length of 2.5 nm has a thermal conductivity 24 W/m.K, which is 13



consistent with that in ref 32. Because of the short CNT adopted here, we neglect the effect of grafting on the longitudinal thermal conductivity κc of CNT.26 The length

l and diameter d of CNT are 2.5 nm and 0.407 nm, respectively. The influence of the positions of the grafted chains (at the terminal carbon atoms or at the lateral surface (randomly grafted atoms) of CNT) on the thermal conductivity is investigated. For comparison, the thermal conductivity for ungrafted CNT is also calculated. The normalized thermal conductivity

κ / κm

from simulation and the fitted results from

the EMA model as a function of CNT volume fraction is shown in Fig. 4(a). We fit the simulation results with the EMA model by tuning the interfacial thermal resistance

R i . For ungrafted CNTs, the fitted interfacial thermal resistance R i is 5.26.10-8 m2.K/W. For end grafted CNTs, it is about 0.25.10-8 m2.K/W. And it is about 0.625.10-8 m2.K/W for lateral grafted CNTs. It has been reported that the interfacial thermal resistance between ungrafted CNTs and polymer is 2.5-5.10-8 m2.K/W from simulation9, 14, 39 and 1.2-8.10-8 m2.K/W from experiments40-42. It is about 0.4-1.10-8 m2.K/W between lateral grafted CNTs (gd=6.67%) and polymer.9, 39 These results are consistent with our results. It indicates the low interfacial thermal resistance when the grafted chains are at the end carbon atoms of CNTs. For end grafted CNT, the sp2 structure of CNTs is not broken, which does not influence the thermal conductivity of CNT itself regardless of the CNT length. Thus, this should also be true for experimental µ m -sized CNT. In total, our results indicate that the thermal conductivity of nanocomposites can be enhanced more when the PA chains are grafted at the end atoms of CNT. In order to prove that our results can be transferred to 14


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systems with µ m -sized CNTs, we compared our simulation with the experiments for ungrafted CNTs. To obtain the thermal conductivity of infinitely long CNTs, an inverse fitting method is employed. Figure S5 presents the variation of the inverse thermal conductivity of (5, 5) CNT ( 1 / κ c ) as a function of the inverse CNT length ( 1/ l ), which can be fitted by an approximate linear relationship. The effective thermal conductivity of CNT with infinite length can be extrapolated when the inverse CNT length ( 1/ l ) is equal to 0, which yields a thermal conductivity of ~600 W/m.K. From simulation, it is reported to be in the range of 320 W/m.K at l =100nm43, 190 W/m.K at l =40nm44, 150 W/m.K at l =20nm45, 215 W/m.K at l =50nm, 230 W/m.K at l =50nm32 and 350 W/m.K at l =100nm46. In experiments, the thermal conductivity of µ m -sized CNT is reported to be 350 W/m.K from the four-point three-ω method47, 600 W/m.K from the electrical breakdown phenomenon48, 1100 W/m.K from direct transmission electron microscopy characterization49, and 1750-5800 W/m.K from thermocouples50. Thus, our results fall into the range of simulated and experimental results. Then, we insert the interfacial thermal resistance

R i =5.26.10-8m2.K/W, the thermal conductivity of the CNT

κ c =600

W/m.K, the

CNT length l , the diameter d=1.356 nm, and the CNT volume fraction f into the EMA model. We show the comparison between experiments30, 31, 51-53 and model10 in Fig. 4(b). In total, the results from the model are larger than those from experiments, but they are comparable. This is because the thermal conductivity of nanocomposites also depends on other factors, such as the purity and dispersion of CNT53, processing conditions54 and so on. 15



(a)

1.0 0.9

κ/κm

0.8 0.7

Ungrafted CNT (simulation) Ungrafted CNT (EMA model) End grafted CNT (simulation) End grafted CNT (EMA model) Lateral grafted CNT (simulation) Lateral grafted CNT (EMA model)

0.6 0.5 0 20

10

10

15

20

f (%) 15

25

30

35

(c)

Ungrafted CNT (EMA model) Lateral grafted CNT (EMA model) End grafted CNT (EMA model) Exp[30] Exp[56] Exp[57] Exp[55]

12

κ/κm

15

5

(b)

Model (1µm) Model (3µm) Model (10µm) Model (15µm) Exp[51] (40µm) Exp[31] (3µm) Exp[53] (10µm) Exp[30] (15µm) Exp[52] (10µm)

κ/κm

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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9

6

5 3

0.00

0.02

0.04

0.06

f (%)

0.08

0.10

Fig. 4 (a) The normalized thermal conductivity

0

κ /κm

1

2

3

4

f (%)

5

6

of nanocomposites from the simulation

and EMA model as a function of carbon nanotube (CNT) volume fraction (

f

) for ungrafted

CNT and end grafted CNT (gd(grafting density)=6.67%, L(length of grafted chains)=20) as well as lateral grafted CNT. (b) Comparison of thermal conductivity κ / κm of nanocomposites for random orientation of ungrafted CNT between experiments30, 31, 51-53 and model10. (c) Comparison of thermal conductivity κ / κm of nanocomposites for ungrafted CNT and end grafted CNT as well as lateral grafted CNT from EMA model10 and for lateral grafted CNT from experiments30, 55-57.

κm is the thermal conductivity of pure polyamide. (T=350K) At last, the macroscopic thermal conductivity is estimated by the EMA model for ungrafted CNT, lateral grafted CNT and end grafted CNT with a length of 10 µ m . In experiments, the grafted chains are generally at the lateral surface of the CNTs. Figure 4(c) presents that the thermal conductivity from the model10 is larger than that from the experiments30,

55-57

for lateral grafted CNT. In experiments,58, 16


59

the chemical

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functionalization damages the structure of CNT side walls, which is not considered in our model. It is noted experimentally that it is very difficult to graft the chains only at the end carbon atoms of CNTs.58 We inferred that for end grafted CNT, the heat flows from the matrix into the end carbon atoms of CNT. Then, it flows along the longitudinal axis of the CNT and leaves from the other end atoms of the CNT into the matrix. It will not flow into matrix from the lateral surface of CNT because of high interfacial thermal resistance between polymer and lateral surface. The heat flux can take full advantage of the high thermal conductivity of CNT. For the lateral grafted CNT, the heat flows from the matrix into the lateral surface of CNT. Then it will flow along the longitudinal axis of CNT and leaves from the lateral surface of CNT into the matrix. Then it flows into the lateral surface of another CNT and so on. The path where the heat flows within the CNTs for the lateral grafted CNT is shorter than that for the end grafted CNT. In addition, the sp2 structure of the lateral grafted CNT is interrupted by grafting points. To better understand it, the schematic diagrams of the heat transfer process for the end grafted CNTs and the lateral grafted CNTs are presented in Fig. S6. Thus, the end grafted CNT is more efficient in improving the thermal conductivity of nanocomposites compared with the lateral grafted CNT, which is consistent with theoretical predictions.60 From the EMA model, the thermal conductivity for end grafted CNTs is about 40% larger than that for lateral grafted CNTs at a CNT volume fraction=10%. 3.5 Effect of stretching In general, thermal energy transports more efficiently along the polymer chain 17



(the strong covalent bonds) than perpendicular to the polymer chain (Van der Waals force). Thus, thermal conductivity can be enhanced along the chain alignment direction.45,61 The systems 1, 11-15 in Table S1 in SI are adopted here. The thermal conductivities κ

||

parallel to the tensile direction and κ ⊥ perpendicular to it are

shown in Fig. 5 for pure PA and nanocomposites. κ

||

continuously increases while

κ ⊥ continuously decreases with increasing strain. As the simulation box is deformed, the chains and CNTs are stretched and tend to align along the tensile direction. The anisotropy of the thermal conductivity κ || / κ ⊥ increases approximately linearly with the strain in Fig. 5(c); however, it falls into a common line for all systems. 0.35

pure polyamide gd=1.67% gd=6.67%

(a)

gd=0.00% gd=3.33% gd=10.0%

κ|| (W/m⋅K)

0.30

0.25

0.20

0.15 0 0.24

(b)

20


0.21

40

60

Strain (%)

80

100

2.4

gd=0.00% gd=3.33% gd=10.0%

κ⊥ (W/m⋅K)

pure polyamide gd=1.67% gd=6.67% Linear fit

2.1

(c)

gd=0.00% gd=3.33% gd=10.0%

1.8

0.18

κ|| /κ⊥

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 18 of 30

0.15

1.5

0.12

1.2

0.09

0.9 0

20

40

60

Strain (%)

80

100

0

20

40

60

Strain (%)

80

100

κ

Fig. 5 Change in the thermal conductivities (a) κ|| (W/m.K) parallel to tensile direction, (b ) ⊥ (W/m.K) perpendicular to tensile direction, and (c) anisotropy of thermal conductivity

κ|| /κ⊥

of

pure polymer and nanocomposites during the deformation for different grafting densities (gd). (T=350K,

f(carbon nanotube volume fraction)=14.5%, L(grafting length)=20) 18


Page 19 of 30 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


We define the orientation of free and grafted chains and the CNTs by the second-order Legendre polynomials < P2 >, defined by

=(3−1)/2 (6)

θ

denotes the angle between a given vector joining bead i and bead i+4 of the

backbone atoms of chain or the longitudinal direction of CNT and the tensile direction.62 The parameter < P2 > is 0.0 if the chains or CNTs are randomly oriented, 1.0 if the chains or CNTs are perfectly parallel to the tensile direction, and -0.5 if the chains or CNTs are perpendicular to it. Figure 6 shows the orientation of the chains and the CNTs as a function of the strain for different grafting densities at f =14.5%. The parameter < P2 > is zero at strain=0.0 which indicates a random orientation of the chains and the CNTs. Both the orientations < P2 > of PA chains and CNTs gradually increase with the strain. However, these < P2 > of the PA chains is different for different systems. It decreases with increasing grafting density. This maybe because when more chains are grafted at the end atoms of CNTs, their conformational freedom decreases. Such packing inhibits the reorientation of the chains. However, the behavior of thermal conductivities is similar for different grafting densities. Thus, we infer that the heat

19



0.25

0.4

(a)

0.20

(b)

0.3

0.15

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 20 of 30

0.2

0.10 0.1

0.05


0.00 0

20

40

60

Strain (%)

Fig. 6 Orientation parameter

gd=0.00% gd=3.33% gd=10.0%

80

< P2 >

gd=0.00% gd=3.33% gd=10.0%

0.0 0

100

20

40

60

Strain (%)

gd=1.67% gd=6.67% 80

100

of (a) free and grafted chains and (b) carbon nanotubes

(CNTs) as a function of strain for different grafting densities (gd). (T=350K, f (CNT volume fraction)=14.1%, L(grafting length)=20)

transfer is not completely borne by the PA chains. Next, we analyze the relationship of

κ || / κ ⊥ with the orientation of PA chains or CNTs, which is shown in Fig. 7. Figure 7(a) shows that, even though κ || / κ ⊥ shows a linear dependence on the orientation < P2 > of PA chains for every system, the slope is different and varies from 3.57 to 6.24 for different grafting densities. In addition, from Fig. 7(b) we find that the dependence of the κ || / κ ⊥ on the orientation < P2 > of CNTs falls into a common line, whose slope is about 3.1. In Figs. 7(a) and 7(b), the slopes are comparable, which may indicate that the heat transfers via both PA chains and CNTs. Recently, Liu et. al.61 found that the dependence of thermal conductivity of pure polyethylene on the chain orientation can be described by an exponential equation. In previous work63 in our group, a linear relationship between the anisotropy of the thermal conductivity

κ || / κ ⊥ and the average direction cosine of the C-C backbone bonds has been found for amorphous atactic polystyrene. For amorphous polyamide-6,6,64 κ || increases by

20


Page 21 of 30 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


37%, while κ ⊥ decreases by 12% at strain=0.3 and T=300K, which leads to κ || / κ ⊥ ~1.5. For the polystyrene,65 κ || increases by 8%, while κ ⊥ decreases by 11% at strain=0.3 and T=400K, which leads to κ || / κ ⊥ ~1.2. With the increase of the intersurface width from 1 nm to 6 nm, the anisotropy in thermal conductivity κ || / κ ⊥ of polyamide-6,6 nanoconfined between graphene surfaces varies from 1.91 to 1.05.66, 67

For our systems, κ || increases by 18%, κ ⊥ decreases by 12%, and κ || / κ ⊥ is

~1.32 at strain=0.3 and T=350K. These results for different polymers appear to be comparable. In experiments,15 for CNT filled epoxy nanocomposites κ || / κ ⊥ is reported to be ~2.0 and ~4.0, respectively at strain=25% and 40%. The anisotropies are larger than ours, which can be attributed to the large aspect ratio of the CNTs in the real experiments. There are few reports about the contribution of polymer alignment, CNT alignment and grafting density to κ || / κ ⊥ , which is discussed below. We propose an empirical formulas to describe the relationship among the anisotropy of the thermal conductivity, the orientation of polymer and CNTs, and the grafting density:

κ || / κ ⊥ = A ∗ < P2 ( poly ) > + B ∗ < P2 ( C N T ) > (1 + C ∗ gd ) + 1

(7)

where < P2 ( poly ) > and < P2 ( CNT ) > are the orientation of polymer and CNT, respectively. The gd is the grafting density. A, B and C are fitting parameters, which here are 3.02, 1.20 and 0.60, respectively. The average error between simulation and this formula (Eq. (7)) is about 3.82%, which is acceptable. The contribution of the polymer

alignment

and

A ∗ < P2 ( poly ) > /(κ || / κ ⊥ − 1)

CNT and

alignment

is

calculated

by

B ∗ ( < P2 ( CN T ) > + C ∗ gd ) /(κ || / κ ⊥ − 1) 21


the ,


respectively. All the results are shown in Table I. The contribution percentage of CNT gradually increases with the strain and the grafting density. It is less than 50% in all systems, which indicates that the main contribution to κ || / κ ⊥ is the polymer alignment. In summary, both PA chains and CNTs play an important role in transferring heat, especially at high strain and high grafting density. 2.4

2.4

(a)

pure polyamide gd=0.00% gd=1.67% gd=3.33% gd=6.67% gd=10.0% Linear fit (pure polyamide) Linear fit (gd=10.0%)

2.1

κ|| /κ⊥

2.1

κ|| /κ⊥

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 22 of 30

1.8 1.5

gd=0.00% gd=3.33% gd=10.0%

(b)

gd=1.67% gd=6.67% Linear fit

1.8 1.5 1.2

1.2 0.9 0.00

0.05

0.10 0.15

0.20

0.9 0.0

0.25

0.1

Fig. 7 The relationship of the anisotropy of thermal conductivity

0.2

κ|| /κ⊥

0.3

0.4

with the orientation

< P2 > of (a) polyamide chains and (b) carbon nanotubes (CNTs) for different grafting densities (gd). (T=350K,

f

(CNT volume fraction)=14.5%, L(grafting length)=20)

5. Conclusions We investigated the effect of grafted chains and mechanical deformation on the thermal conductivity of the carbon-nanotube-(CNT)-filled polyamide-6.6 (PA) nanocomposites. First, the thermal conductivity of nanocomposites gradually increases with the grafting density and the length of the grafted chains, which is described by an empirical equation. It can be further improved when all CNTs are chemically connected by bridging PA strands. By fitting our simulation results with an effective medium approximation (EMA) model, the interfacial thermal conductivity is derived. With the EMA model, the macroscopic thermal conductivities for the systems 22


Page 23 of 30 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


with 10 µ m -sized CNT are estimated. The results from EMA model indicate that the thermal conductivity for end-grafted CNTs is 40% higher than that for lateral surface grafted CNT at a volume fraction=10%. Following stretching, the orientation of both the chains and the CNTs contribute to improve the thermal conductivity parallel to the tensile direction, but reduce the thermal conductivity perpendicular to it. The contributions of the polymer alignment and CNT alignment to the anisotropy of thermal conductivity are quantified. The comparison indicates that both play an important role in transferring heat, especially at high strain and high grafting density.

Supporting Information Supporting information referenced in the text can be found online. It includes some snapshots of the typical systems and the tables for the details of all the simulated systems.

Acknowledgements The authors acknowledge financial supports from the National Natural Science Foundation of China (21704003), the start-up funding of Beijing University of Chemical Technology (BUCT) for excellent introduced talents (buctrc201710) and the Fundamental

Research

Funds

for

the

Central

Universities

(JD1711).

CHEMCLOUDCOMPUTING of Beijing University of Chemical Technology and Technische Universität Darmstadt is greatly appreciated. 23



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Table I. Comparison between the simulation results and Eq.(7) for the anisotropy of thermal 27



conductivity

Page 28 of 30

κ|| /κ⊥. Contribution

of

terms Strain (%)

gd(%) Simulation Formula

Error Polymer CNT (%) (%) (%)

0

0

1.00

1.00

0.00

0.00

0.00

10

0.032

1.07

1.10

2.51

100

0.00

20

0.062

1.19

1.19

-0.51

100

0.00

30

0.094

1.34

1.28

-4.19

100

0.00

40

0.124

1.40

1.38

-1.43

100

0.00

50

0.150

1.54

1.45

-5.50

100

0.00

70

0.206

1.71

1.62

-4.98

100

0.00

85

0.233

1.80

1.70

-5.16

100

0.00

100

0.265

2.03

180

-11.3

100

0.00

0

0

0

0

1.00

1.00

0.00

0.00

0.00

10

0.042

0.041

0

1.09

1.18

7.91

72.1

27.9

20

0.078

0.092

0

1.21

1.35

11.3

68.2

31.8

30

0.109

0.130

0

1.31

1.49

13.8

67.9

32.1

40

0.124

0.174

0

1.44

1.59

9.94

64.2

35.8

50

0.159

0.237

0

1.56

1.77

13.0

62.9

37.1

70

0.200

0.325

0

1.89

1.99

5.43

60.7

39.3

85

0.222

0.362

0

2.04

2.11

3.50

60.7

39.3

100

0.251

0.407

0

2.27

2.25

-1.06

60.8

39.2

0

0

0

1.67

1.00

1.00

0.00

0.00

0.00

10

0.036

0.049

1.67

1.14

1.17

2.65

65.1

34.9

20

0.062

0.074

1.67

1.29

1.28

-1.31

67.4

32.6

30

0.091

0.114

1.67

1.38

1.41

2.39

66.5

33.5

40

0.107

0.175

1.67

1.47

1.54

4.69

60.3

39.7

50

0.132

0.188

1.67

1.57

1.63

3.73

63.6

36.4

70

0.167

0.263

1.67

1.76

1.82

3.34

61.2

38.8

85

0.185

0.322

1.67

1.96

1.95

-0.71

58.9

41.1

100

0.218

0.364

1.67

2.24

2.10

-6.22

59.9

40.1

Pure polymer

0

0

0

3.33

1

1.00

0.00

0.00

0.00

10

0.030

0.046

3.33

1.10

1.15

4.62

61.9

38.1

20

0.058

0.091

3.33

1.20

1.29

7.70

61.3

38.7

30

0.083

0.118

3.33

1.37

1.40

2.07

63.1

36.9

40

0.103

0.171

3.33

1.47

1.52

3.34

59.7

40.3

50

0.117

0.194

3.33

1.61

1.59

-1.22

59.8

40.2

70

0.164

0.283

3.33

1.85

1.84

-0.51

58.8

41.2

85

0.183

0.333

3.33

1.94

1.96

1.26

57.5

42.5

100

0.200

0.370

3.33

2.13

2.06

-3.17

57.1

42.9

0

0

0

6.67

1.00

1.00

0.00

0.00

0.00

28


Page 29 of 30 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


10

0.029

0.032

6.67

1.08

1.13

4.68

69.2

30.8

20

0.050

0.064

6.67

1.17

1.23

5.55

65.5

34.5

30

0.075

0.116

6.67

1.31

1.37

4.57

61.0

39

40

0.097

0.135

6.67

1.48

1.46

-1.57

63.4

36.6

50

0.120

0.191

6.67

1.64

1.60

-2.18

60.4

39.6

70

0.152

0.260

6.67

1.78

1.78

-0.03

58.5

41.5

85

0.168

0.288

6.67

1.96

1.87

-4.78

58.5

41.5

100

0.186

0.337

6.67

2.16

1.98

-8.29

57.2

42.8

0

0

0

10.0

1.00

1.00

0.00

0.00

0.00

10

0.021

0.035

10.0

1.07

1.11

3.73

58.6

41.4

20

0.037

0.064

10.0

1.18

1.19

0.56

57.7

42.3

30

0.061

0.101

10.0

1.30

1.31

0.98

58.7

41.3

40

0.072

0.138

10.0

1.41

1.39

-1.43

55.4

44.6

50

0.088

0.179

10.0

1.58

1.49

-5.20

53.9

46.1

70

0.112

0.243

10.0

1.72

1.65

-4.23

52.2

47.8

85

0.129

0.298

10.0

1.83

1.77

-3.31

50.7

49.3

100

0.157

0.336

10.0

2.00

1.90

-4.83

52.5

47.5

The Polymer and CNT terms denote the contribution of the polymer alignment and CNT alignment to the anisotropy of thermal conductivity

κ|| /κ⊥, respectively.

29



Table of Content Graphics

30


Page 30 of 30

Molecular Dynamics Study on the Thermal Conductivity of the End

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