Role of Chain Morphology and Stiffness in Thermal Conductivity of

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The Role of Chain Morphology and Stiffness in Thermal Conductivity of Amorphous Polymers Teng Zhang, and Tengfei Luo J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.5b09955 • Publication Date (Web): 11 Jan 2016 Downloaded from http://pubs.acs.org on January 21, 2016

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The Journal of Physical Chemistry

The Role of Chain Morphology and Stiffness in Thermal Conductivity of Amorphous Polymers

Teng Zhang† and Tengfei Luo*,†,‡

†

Aerospace and Mechanical Engineering and ‡Center for Sustainable Energy at Notre Dame,

University of Notre Dame, Notre Dame, Indiana 46556, United States

Abstract:

Designing thermally conductive polymer is of scientific interest and practical importance for applications like thermal interface materials, electronics packing and plastic heat exchangers. In this work, we study the fundamental relationship between the molecular morphology and thermal conductivity in bulk amorphous polymers. We use polyethylene as a model system and performed systematic parametric study in molecular dynamics simulations. We find that the thermal conductivity is a strong function of the radius of gyration of the molecular chains, which is further correlated to persistence length, an intrinsic property of the molecule that characterizes molecular stiffness. Larger persistence length can lead to more extended chain morphology and thus higher thermal conductivity. Further thermal conductivity decomposition analysis shows that thermal transport through covalent bonds dominates the effective thermal conductivity over other contributions from non-bonded interactions (van der Waals) and translation of molecules disregarding the morphology. As a result, the more 1

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extended chains due to larger persistence length provide longer spatial paths for heat to transfer efficiently and thus lead to higher thermal conductivity. In addition, rigid rod-like polymer with very large persistence length tends to spontaneously crystalize and forms orientated chains, leading to a thermal conductivity increase by more than one order of magnitude. Our results will provide important insights to the design of thermally conductive amorphous polymers.

Keywords: Thermal Transport, Molecular Dynamics, Amorphous Polyethylene, Backbone Stiffness, and Chain Morphology.

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Introduction

Achieving high thermal conductivity in polymers is desirable for contemporary applications such as electronics packaging, thermal interface materials and polymeric heat exchangers. These applications have triggered a renewed interest in exploiting thermally conductive polymers using a variety of fabrication methods, such as mechanical stretching,1-3 electrospining,4 nanoscale templating,5 compositing,6 and polymer blending.7 By drawing amorphous polyethylene into highly aligned fibers, chain entanglements and voids, which act as stress concentration points and phonon scattering centers in the amorphous form, are significantly reduced, leading to a thermal conductivity increase by two to three orders of magnitude.1, 8-10 By compositing amorphous polymers with thermally conductive particles, thermal conductivity can be improved to as much as ~10 W/mK,11-15 and such improvement can be potentially enhanced by strong particle-matrix interfacial adhesion (e.g., covalent bond,16, 17 π- π stacking,18 and hydrogen bond19, 20) and better vibrational spectra coupling.21 However, the details of thermal transport in pure amorphous polymers are still not thoroughly understood, leaving limited guidance on designing polymers with intrinsically high thermal conductivities.

Precise prediction of the thermal transport in amorphous polymer has not been easy, since many factors can influence the thermal performance, such as strength of bonds, local and global morphologies, processing conditions, temperature, and so on. Dashora and Gupta attempted to explain thermal conductivity behavior of non-conducting linear amorphous 3



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polymer around the glass transition temperature (Tg).21 Below Tg, an increasing trend of the thermal conductivity was reported and explained by straightened chain morphology and longer phonon mean free path.21 Above Tg, thermal conductivity shows three different behaviors (decreasing trend, increasing trend, and plateau), since temperature has two competing effects on polymer structure: creating more vacancy sites due to thermal expansion, which serve as phonon scattering sites, and improving chain local order due to larger chain mobility, which increases phonon mean free path.21 Zhong et al. developed a group contribution model to establish the correlation between thermal conductivity and chemical structure in glassy and liquid amorphous polymers.22 However, the prediction was far from accurate, since the prediction model did not consider any other effects except polymer chemical formula and temperature.22 Although there are more refined theories for thermal conductivity of amorphous polymer, none of them are able to provide detailed relationship between molecule details and macroscopic thermal conductivity, impairing their predictive power.22-25 An investigation of such a relation will be beneficial for understanding the thermal transport mechanism in amorphous polymers and help guide the design of high thermal conductivity polymers.

Some recent studies have pointed the major factor of polymer thermal conductivity to morphology. Our previous studies as well as others have shown that thermal transport along polymer chains is a strong function of the chain segmental rotation when they are aligned.8, 26, 27

However, how the chain morphology influences the thermal conductivity of bulk

amorphous polymers remains unknown. Recently, Zhao et al. studied the thermal conductivity 4


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dependence on chain length in amorphous polyethylene using Molecular Dynamics (MD) simulations. There is a clear correlation between morphology and thermal conductivity, and the major difference maker is the phase of the polymer (gas, gas-liquid, and liquid) due to different chain lengths.28 Using MD, Ma and Tian studied the chain morphology due to confinement and found lower thermal conductivity of polymer films when the chains are strongly confined and less entangled.29 However, their discussion of correlation between structure and thermal conductivity is limited to spatial confinement in ultra-thin films, which might not be directly transferrable to bulk amorphous polymers. Through both MD and experimental measurements, anisotropic thermal conductivity in stretched amorphous is reported, and explained by the anisotropy of polymer chain orientation due to applied strain.30-32 However, the relationship between morphology and thermal conductivity in isotropic amorphous polymers is still absent.

In this paper, using polyethylene as a model system, we employ MD simulations to study the correlation between morphology and thermal conductivity in bulk amorphous polymers. The importance of chain morphology on thermal conductivity in the amorphous state is investigated by comparing chains with different radius of gyration due to different processing conditions and backbone stiffness. Persistence length, an intrinsic property of polymer chains, is used to characterize the stiffness. It is found that large persistence length leads to more extended chain morphology and increases thermal conductivity. For rigid rod-like polymer with very large persistence length, spontaneously chain alignment and crystal domain formation are found, and they can increase thermal conductivity significantly by more than 5



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one order of magnitude. This study can provide important guidance for the design of amorphous polymers with higher thermal conductivities.

Method

Force Field and Non-equilibrium MD (NEMD) simulation. In our MD simulations, the united-atom (UA) version of OPLS (optimized potentials for liquid simulations) potential distributed by TINKER is used to simulate methylene group (-CH2-) and methyl group (-CH3) in polyethylene (Table S1 in Supporting Information (SI)).33,

34

Thermal conductivity

calculations using NEMD are performed on well equilibrated structures, which are achieved after 4 - 20 ns NPT (constant number of atom, temperature and pressure) relaxation at 1 atmosphere. All the simulations are carried out using the large-scale atomic/molecular massively parallel simulator (LAMMPS)35 with a time step of 1 fs and data files prepared using Moltemplate.36

Non-equilibrium MD (NEMD) simulation is used to calculate the thermal conductivity of all polymer structures. A Langevin thermostat50 with a temperature 15 K higher than the average temperature is applied at the middle of simulation box as the heat source, and another Langevin thermostat of a temperature 15K lower than the average temperature is applied at the ends of simulation box as the heat sink to establish a temperature gradient and impose heat flux across the polymer structure (see SI section 1). Expect the thermostated regions, the rest of the system is simulated using NVE ensemble (constant number of atoms, constant 6


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volume, and constant energy). This setup with mirror symmetry allows us to calculate two thermal conductivity data in one simulation (see Figure S1 in SI). At steady state, the temperature gradient (/) is obtained by fitting the linear portion of the temperature profile, and heat flux () is calculated using = / / /2, where / is the average of the energy input and output rates in the thermostated regions, and S is the cross sectional area. The thermal conductivity ( ) is calculated by Fourier's law, = −/(/). For each simulation, which ranges from 1.5 to 4 ns long depending on the convergence, eight thermal conductivity values are obtained from four different time blocks at steady state, and the final value is the average of all calculated values with the error bars being the standard deviation (see SI section 1 for details). Different temperature gradients are tested, but no dependence of thermal conductivity on the temperature gradient is observed (see SI section 2). Structures consisting of different numbers of polymer chains are studied, and a chain number of 1000 is chosen so that it does not influence the result (see SI section 3). The chain length effect on thermal conductivity is also observed.28 Polymer chains with 100 carbon atoms per chain are chosen due to the balance between efficiency and accuracy of simulations. This choice is reasonable since the correlation between thermal conductivity and morphology is the same for longer chains (see SI sections 4).

Amorphous Structure Construction. To construct amorphous polyethylene, we use the following procedure (Figure 1): (1) a single extended polyethylene chain containing 100 carbon atoms is simulated and equilibrated at 300 K for 1 ns to form a compacted particle. (2) 1000 of these particles are randomly packed into a super-cell using packmol.37, 38 (3) After 7



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minimization, an NPT run is used to increase the system temperature to 600 K by a rate of 50 K/ns, and then a 12 ns NPT run at 600 K is used to generate amorphous polyethylene melt with fully relaxed and amorphous structure. To ensure that the morphologies obtained from such a procedure is well equilibrated, we monitored the energy and radius of gyration for each system, and plateaus are reached (Figure S6 in SI sections 5). The obtained structures are then quenched to different temperatures, and NPT ensemble runs for 4 ns at each temperature are used to further equilibrate the structure at the quenched temperatures (SI sections 5). This will, to the largest extend, ensure that we are simulating realistic morphologies at different temperatures. After the stable structures are obtained, thermal conductivity is calculated using NEMD.

Figure 1. Schematics of initial structure preparation.

Results and Discussion

Thermal Conductivity of Amorphous Polyethylene at Different Temperatures. The calculated thermal conductivities from 200 K to 600 K show a maximum value around 350 K 8


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(red symbols, Figure 2). Radius of gyration, Rg, is calculated for each chain to characterize chain morphology, and the averaged values also show a peak, but at 400 K (black symbols, Figure 2). The increase of radius of gyration from 200 K to 400 K may come from thermal expansion, which will lead to more free space for the originally confined chains to expand, while the decrease of radius of gyration above 400 K may be due to the increase in gauche conformation population and weaker inter-chain interaction at higher temperature, which will enable the chains to curl more and thus pose smaller radius of gyration (Figure S9 in SI section 6 and Figure 3b). The similar trend in thermal conductivity and radius of gyration suggests that more extended chain morphology enhances the thermal transport in amorphous polymer. Besides the chain morphology, Figure 2 also shows that temperature significantly influences the thermal conductivity: for the structures with similar radius of gyration (e.g., 300 K & 550 K, Figure 2), the thermal conductivity at higher temperature is much lower than that at lower temperature (illustrated by arrows in Figure 2). The observed negative effect on thermal conductivity from temperature may also explain why the peak in thermal conductivity appears at lower temperature compared to that in the radius of gyration (Figure 2): if there is only morphology effect, the thermal conductivity should peak at the same location as the radius of gyration. However, since higher temperature can make the thermal conductivity lower (as discussed in the next section), the thermal conductivity peak will shift to lower temperature compared to that of the radius of gyration. We have illustrated such combined effect in SI section 7 of using a simplified picture.

9



17.5 0.22

larger κ at lower temperature

17.0 0.20 16.5 0.18 16.0

similar Rg

0.16 15.5 0.14 smaller κ at higher temperature

0.12

200

300

400

500

15.0

Radius of Gyration (Å)

Thermal Conductivity (W/mK)

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600

Temperature (K)

Figure 2. Thermal conductivity and radius of gyration of amorphous polyethylene. The structures are quenched from 600 K.

To further study the impact of temperature on thermal conductivity, polymer density as a function of temperature is plotted in Figure 3a (red line). Figure 3a shows that polymer density decreases in the whole temperature range, which is due to thermal expansion. This leads to larger molecular spacing and thus weaker van der Waals (vdW) interactions. The red line in Figure 3b shows that the vdW interaction, calculated as the total vdW energy of the system divided by mole of chains, becomes much weaker as the temperature increases from 300 K to 600 K. Thus, the decreasing trend of thermal conductivity above 400 K can be partially attributed to the lower density and thus weaker vdW interaction, besides the radius of gyration effect. The decreasing trend of density and thermal conductivity as a function of temperature are also widely observed experimentally in hydrocarbons melts.39 However, the monotonic decreasing trend in density can only explain the thermal conductivity decrease above 400 K, and the increasing trend of polymer thermal conductivity below 400 K has to be explained by chain morphology (i.e., larger radius of gyration).

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(b) 0.90

From tangled chains From compact chains

0.85 0.80 0.75 0.70 0.65

200

300

400

500

600

van der Waals Energy(kcal/mol)

(a)

Density (g/cm-3)

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Temperature (K)

-100 -120

From tangled chains From compact chains

-140 -160 -180 -200 -220 -240 100

200

300

400

500

600

700

Temperature Difference (K)

Figure 3. (a) Density and (b) vdW energy per mole of molecule of amorphous polyethylene prepared differently. “Tangled chains” are from the heating and quenching process and “compact chains” are prepared without such a step.

To better understand the morphology dependency of amorphous polymer thermal conductivity, polymers constructed from randomly packed compacted chains are used as initial structures directly without heating up to 600 K and quenching down (i.e., step 3 of the procedure in Figure 1 is skipped). 4 ns of NPT relaxations are performed at each temperature and 1 atmosphere to release large stress in the system. This procedure leads to more compacted chains than those achieved from the heating and quenching methods. Without the heating and quenching treatment, the structures below 400 K still have some memory of the initial compacted chain morphology, and the radius of gyration is much smaller (blue line, Figure 4a). Although compacted chains lead to lower density due to more inter-chain gaps and voids at very low temperatures (blue line at 200 K, Figure 3a), the impact of initial structure on density is not significant above 250 K (Figure 3a), where chains are more mobile and move to fill the gaps and voids. Below 400 K, compacted chains will also lead to weaker vdW 11



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interaction. In other words, well tangled chains can help strengthen the inter-chain interaction due to large inter-chain contacting area.

The calculated thermal conductivities are plotted in Figure 4b (blue line) together with those of the structured prepared with the heating and quenching process (red line). The thermal conductivities of the structures using compacted chains as initial structures are much lower below 400 K. When temperature is above 400K, radius of gyration (Figure 4a), density (Figure 3a) and vdW interaction (Figure 3b) are found to be similar for the two sets of polymers prepared differently, indicating that the memory of initial structure is eliminated (i.e., melting happened). Thus the two cases are found to have similar thermal conductivities above 400K (Figure 4b).

Figure 4. Comparison of polyethylene prepared using different methods in (a) morphology -radius of gyration, and (b) thermal conductivity.

In the temperature range from 300 K to 400 K, where the densities of the two cases are not much different (Figure 3a), the large differences in thermal conductivity are believed to be 12


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directly related to the morphology, i.e., more extended chains with larger radius of gyration lead to higher thermal conductivity. There are two possible mechanisms for the thermal conductivity difference: (1) since thermal transport via covalent bond is highly efficient, larger radius of gyration means that the efficient energy path via covalent bonds within each individual molecule are more extended; (2) well tangled chains often have larger inter-chain vdW interaction due to larger inter-chain contacting area, and thus inter-chain thermal transport is more efficient. To identify the dominant mechanism, it is desirable that the total thermal conductivity can be decomposed into contributions from different types of interactions, i.e., = + + , where , ,

and are the contributions from bonded interaction, non-bonded interaction, and translation of atoms, respectively. It is known that the total heat flux ( ) of a molecular

system can be calculated as Equation 1:40, 41

=

1 1 2

*#

!"# ∙ %& + &# '( )# +

+ & , (1)

where - and . are the indices of atoms, V is the volume, "# is interatomic force, & and

&# are the atom velocities, )# is the relative position vector, + is the total energy per-atom,

including potential and kinetic energies. The first term in Equation 1 refers to the heat flux contributed by interatomic interactions (i.e., conduction), and the second term is due to translation of atoms (i.e., convection). Since the interatomic force "# can be decomposed

into contributions from bonded and non-bonded interactions ("# = " + " ), # #

the total heat flux can be decomposed into three parts ( = + +

).

13



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1 ∙ %& + &# '( )# !" 2 = # 2 0 *# 0 1 = ∙ %& + &# '( )# (2) !" # 2 1 *# 0 1 0 = + & /

With the decomposition of the heat flux, the thermal conductivity decomposition can be evaluated as 34 =

56789 5

56:6;? ) of the

dihedral angle energy (Torsion, Table S1 in SI) is systematically changed (Figure 6a). As >?

increases, the energy curve with respect to the dihedral angle shows shallower energy wells around the Gauche conformation and deeper energy wells around the trans conformation (Figure 6a). For each of these “polyethylene”, we construct the amorphous phases and fully relaxed them for 16 ns at 600 K (Figure S8a in SI). To characterize the effect of the dihedral energy change, the dihedral angle distribution is calculated. It shows more trans conformations for larger >? , and the peaks of Gauche conformation almost disappear when

>? increases to −0.40 CDEF/GHF

(Figure 6b), confirming that more rigid chains are

created by increasing >? . Figure 6c shows that chains with more rigid backbone have larger 16


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radius of gyration.

Figure 6. (a) Dihedral angle energy curve, (b) dihedral angle distribution, (c) radius of gyration of chains, and (d) persistence length calculated using different methods for chains with different dihedral angle energy constants (>? ).

To quantitatively characterize the rigidity, the persistence length is also calculated. In MD simulations, there are mainly three different methods for calculating the persistence length, and we used all of them. Formally, the persistence length is defined as the length over which correlations in the direction of the tangent are lost. Considering atoms connected by freely rotating bonds with length F, the correlation of chain directions can be calculated 17



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using < DHJK# >, where K# is the angle between bond - and bond ., and < > means ensemble average over all the chains. For bonds away from the chain ends, an exponential decay of < DHJK# > is often observed, and persistence length (M4 ) can be calculated by fitting the decay using < DHJK# >= exp [−ERJ(. − -)F/M4 ] (blue line, Figure 6d).42 To exclude the end effects, the bond correlation is calculated using the middle bonds in each chain. Although the exponential fit of < DHJK# > yields a good prediction of the persistence length of the original polyethylene ( M4 = 6.9 Å from our simulation, and 6.5 Å from

experimental measurements43), very rigid chains do not exhibit a perfect exponential decay in < DHJK# > (e.g., >? ≥ −0.9 >DEF/GHF).

In another method, the persistence length of a simple long linear polymer chain is 44 defined as M4 = ∑\ #] < Z# ∙ Z /|Z | >, which is the summation (from bond . to bond ∞)

of the bond vector (Z# ) projected on bond - (Z ). Here, the middle bond of each chain is

chosen as bond -, and . increases from - to the ends of each chain. With this definition, a similar increasing trend of persistence length is observed, and the sharp increase of persistence length at >? = −0.40 CDEF/GHF is due to the formation of straight chains (red line, Figure 6d). For semi-flexible polymer chains, persistence length can also be calculated by fitting the end-to-end distance < ℎ` > using the worm-like chain model45 (black line, Figure 6d). The persistence length calculated from < ℎ` > agrees well with the other two methods for >? < −0.9 CDEF/GHF. However, it predicts a constant Lp for >? ≥ −0.9 CDEF/

GHF. This is due to the failure of the worm-like chain model,45 which also implies the formation of rigid rod-like chains when >? ≥ −0.9 CDEF/GHF. Despite the differences in the 18


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three calculation methods, the trend and the magnitude in persistence length are very similar, except in the regions where some of the calculation models are not applicable.

As >? increases from -3.90 kcal/mol to -1.40 kcal/mol, chains become more rigid, and thermal conductivities in all three dimensions increases (Figure 7a). This can be linked to the morphology character – radius of gyration (Figure 6c), and more fundamentally, to the intrinsic molecular property – persistence length (Figure 6d). When >? is larger than -1.40 kcal/mol, the originally isotropic thermal conductivity becomes anisotropic, and the thermal conductivity in one certain direction suddenly increases by one order of magnitude (red line, Figure 7a). This corresponds well to the sudden jump in radius of gyration (Figure 6c) and in the persistence length (Figure 6d). For >? = −0.4 CDEF/GHF, only the thermal conductivities in the X- and Y-directions will be discussed here, because the temperature profile in the Z-direction does not show linear region due to some domain interface structures, which prevent a linear fit to obtained the temperature gradient (see SI section 8). In addition, it is found that the formation of crystalline structure in stiffer polymers (>? = −0.4 CDEF/GHF) is a combined effect from both backbone stiffness and temperature (Figure S14 in SI section 9). For example, isotropic amorphous structures are observed for >? from −3.9 CDEF/GHF to

−0.4 CDEF/GHF at 1000 K (Figure S14 in SI section 9). Though the formation of crystalline structure is closely related to the chosen temperature, at either 600 K or 1000 K, stiffer chains always have larger radius of gyration and larger trans conformation population, showing the importance of chain stiffness on chain morphology (Figure 6 and Figure S14 in SI section 9). To understand the origin of the anisotropic thermal conductivity, we characterized the 19



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chain orientation. Herman’s orientation factor (b) is used to characterize the segment alignment along different directions: b = 1.5 < DHJ ` K > −0.5, where K can be the angle between a carbon-carbon bond and the X-axis, Y-axis, or Z-axis.26, 46 A value of f = -0.5 means that the orientation is perpendicular to the selected direction, and f = 1 means that the orientation is parallel to the selected direction. A value of 0 indicates a completely random orientation. Figure 7b shows that when >? c −1.40 kcal/mol , the chains Herman’s orientation factors in all three dimensions are 0, indicating random orientations. When >? = −0.40 kcal/mol, a positive Herman’s orientation factor along the Y-axis indicates that chains are more aligned along the Y-axis, and a negative factor along the X-axis indicates that chains are perpendicular to the X-axis (Figure 7b). These explain the sudden increase in thermal conductivity in the Y-direction and the decrease in the X-direction. Such observation is consistent with previous findings on the relation between alignment and thermal conductivity.5, 9, 46-48

Figure 7. (a) Thermal conductivity of polymer chains with different stiffness, and (b) characterization of chain orientation using Herman’s orientation factor to explain the thermal 20


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conductivity anisotropy.

To further characterize the structure, radial distribution function is calculated as k(l) = m(l)/( 4n l ` opl), where m(l) is the number of atoms in a shell of width pl at

distance l, and o is the average atom density (Figure 8a). To name different pairs of atoms

more conveniently, the reference atom is designated as atom 1, the one directly bonded to the reference atom is atom 2, and then atoms 3, 4, 5… are labeled along the polymer chain (right inset, Figure 8a). As expected, the most notable two peaks are found at 1.5 Å (l?`) and 2.5 Å (l?q), corresponding to the distances separated by one and two carbon-carbon bonds,

respectively (left insets, Figure 8a). Figure 8b provides a zoom-in view of the radial distribution function for distance from 3 Å to 6 Å (green boxed, Figure 8a). The distance between atoms 1, 4 in gauche conformation is around 3.3 Å, and Figure 8b shows that the

peak around 3.3 Å disappears as >? becomes larger. Figure 8b also show that two peaks at

3.9 Å and 5.0 Å, which correspond to atom pair 1, 4 in trans conformation and atom pair 1, 5 in straight chains, become higher as >? increases (l?r & l?s , Figure 8b). Thus, larger

>? can reduce gauche conformation and increase trans conformation, leading to more straight chains. It is known that straighter chains are highly beneficial to thermal transport.1, 8, 26, 27, 49

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Figure 8. Radial distribution function of carbon atom for chains with different stiffness, >? : (a)over all view, (b) zoom-in view to highlight intra-chain order, and

(c)zoom-in view to

identify inter-chain order.

Besides conformation information along chains, radial distribution function can also help identify the formation of crystalline structure. For chains with >? c −1.40 CDEF/GHF, the radial distribution function above 8 Å remains flat (Figure 8c), meaning the lack of long range order. However, for >? = −0.40 CDEF/GHF, the radial distribution function above 8 Å shows several peaks, indicating long range inter-chain orders. The inter-chain lattice structures corresponding to 9.3 Å and 13.1 Å are illustrated in Figure 8c (cross sectional

view of crystalline chains). Thus, from these analyses, we can see that larger >? (i.e., stiffer chains) can lead to more extended chains, increasing the thermal conductivity of amorphous polymer. In addition, a large enough >? will lead to straight and aligned chains, which lead to crystalline domains with highly anisotropic thermal conductivities. The formation of crystalline domains will significantly influence the phonon density of state, which should be relevant to the sudden change in thermal conductivity, although more detailed analyses of phonon modes and their mean free paths are needed to quantify this relation (see SI section 10 22


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for details).

Thermal conductivity decomposition is also performed on the polymers with different backbone stiffness (Figure 9). For >? c −0.90 CDEF/GHF, since the thermal conductivity in all three dimensions are almost the same (Figure 7a), the decomposition of thermal conductivity in the X-direction is plotted as a representative case (left legends, Figure 9). For >? c −0.90 CDEF/GHF, the thermal conductivity due to bonded interaction counts more than 80% of the total thermal conductivity, and the increase in total thermal conductivity with respect to >? mainly comes from the increase in bonded contribution (blue & green lines,

Figure 9). For >? c −0.90 CDEF/GHF , the thermal conductivity due to non-bonded

interaction also increases slightly as >? increases (red line, Figure 9), which can be explained by larger density and larger vdW interaction in stiffer polymers (Figure 10). However, the vdW contribution is always much smaller than the non-bonded contribution for >? c −0.90 CDEF/GHF, confirming that the more extended chains in stiffer polymers improve the total thermal conductivity by mainly increasing the bonded contribution. When >? = −0.4 CDEF/GHF, the thermal transport in polymer becomes anisotropic, as discussed above. Since chains become more aligned with the Y-axis (Figure 7b), the thermal transport in the X-direction should rely more on the inter-chain vdW interaction. As expected, for the thermal conductivity in the X-direction, the non-bonded contribution increases and bonded contribution decreases significantly (red & green lines, Figure 9) as >? increases from -0.90 kcal/mol to -0.40 kcal/mol (the formation of orientated structures). On the other hand, the thermal conductivity in the Y-direction significantly increases due to larger bonded 23



contribution (yellow right triangle, Figure 9).

Thermal Conductivity (W/mK)

4.2 Total (Y) Bonded (Y) Non-bonded (Y) Translation (Y)

4.0 3.8 3.6 Total (X) Bonded (X) Non-bonded (X) Translation (X)

0.3 0.2 0.1 0.0 -4

-3

-2

-1

Energy Constant, Κ1 (kcal/mol)

Figure 9. Bonded, non-bonded and translation contributions to thermal conductivity of polymers with different backbone stiffness.

(a)

(b) van Waals Energy (kcal/mol)

0.90

0.85

Density (g/cm-3)

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0.80

0.75

0.70

0.65

-4

-3

-2

-1


-100

-150

-200

-250

-300 -4

-3

-2

-1


Figure 10. (a) Density and (b) vdW energy of polymers with different stiffness.

Summary and Conclusion

In summary, we report the morphology-dependent thermal conductivity in amorphous 24


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polymers. Polymer structures with different morphologies are generated by changing the preparation process, temperature and backbone stiffness. Larger density and more extended chain morphology can lead to larger thermal conductivity. To find out the fundamental mechanism behind the morphology-dependent thermal conductivity, thermal conductivity decomposition according to different types of molecular interactions is performed. The increase of thermal conductivity is found to mainly come from the increase in bonded contribution. vdW contribution, on the other hand, has a much smaller impact on thermal conductivity. Larger persistence length, an intrinsic property that characterizes the polymer chain stiffness, is found to help polymer chain adopt extended chain morphology, and thus can significantly increase the thermal conductivity by enabling longer thermal transport path along the covalently bonded chain backbone. In addition, very stiff backbone can induce crystallization and lead to one order of magnitude thermal conductivity increase in the chain aligned direction. Our study relates the molecular level information to macroscopic thermal conductivity, and provides insight to thermal transport mechanism in amorphous polymers. It will also provide useful guidance for fabricating (e.g., heat treatment) and choosing molecules (e.g., larger persistence length) for thermally conductive amorphous polymers.

Author Information Corresponding Authors *E-mail: [email protected]. Tel: + 15746319683 (T.L.). Notes The authors declare no competing financial interest. 25



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Acknowledgements This research was supported in part by the Notre Dame Center for Research Computing and NSF through XSEDE resources provided by SDSC Comet, TACC Stampede and NICS Darter under grant number TG-CTS100078. The authors acknowledge the financial support from American Chemical Society (PRF# 54129-DNI10).

Supporting Information Details for NEMD setup, temperature gradient effect in NEMD, isotropic thermal conductivity in amorphous PE, thermal conductivity dependence on chain number, thermal conductivity dependence on chain length, structure relaxation, dihedral angle distribution analysis, brief illustration of combined effects from morphology and temperature-induced density change on thermal conductivity, temperature profile and structure visualization for very rigid chains, morphology characterization at 1000 K for polymers of different stiffness, phonon density of state of polymers with different stiffness, and force field parameters. This information is available free of charge via the Internet at http://pubs.acs.org

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for Table of Contents use only

31


Role of Chain Morphology and Stiffness in Thermal Conductivity of

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