Thermal Conductance in Cross-linked Polymers: Effects of Non

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Thermal Conductance in Crosslinked Polymers: Effects of Non-Bonding Interactions Vahid Rashidi, Eleanor J. Coyle, Katherine Sebeck, John Kieffer, and Kevin P. Pipe J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.7b01377 • Publication Date (Web): 31 Mar 2017 Downloaded from http://pubs.acs.org on April 3, 2017

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The Journal of Physical Chemistry

Thermal Conductance in Crosslinked Polymers: Effects of Non-Bonding Interactions Vahid Rashidi,† Eleanor J. Coyle,‡ Katherine Sebeck,‡ John Kieffer,‡ and Kevin P. Pipe∗,†,¶ Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI, 48109, Department of Materials Science and Engineering, University of Michigan, Ann Arbor, MI, 48109, and Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI, 48109 E-mail: [email protected]

Abstract Weak inter-chain interactions have been considered to be a bottleneck for heat transfer in polymers, while covalent bonds are believed to give a high thermal conductivity to polymer chains. For this reason, crosslinkers have been explored as a means to enhance polymer thermal conductivity; however, results have been inconsistent. Some studies show an enhancement in the thermal conductivity for polymers upon crosslinking, while others show the opposite trend. In this work we study the mechanisms of heat transfer in crosslinked polymers in order to understand the reasons for these discrepancies, in particular examining the relative contributions of covalent (referred to here as “bonding”) and non-bonding (e.g., van der Waals and electrostatic) interactions. ∗

To whom correspondence should be addressed Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI, 48109 ‡ Department of Materials Science and Engineering, University of Michigan, Ann Arbor, MI, 48109 ¶ Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI, 48109 †

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Our results indicate crosslinkers enhance thermal conductivity primarily when they are short in length and thereby bring polymer chains closer to each other; this increases inter-chain heat transfer by enhancing non-bonding interactions between the chains (non-bonding interactions being highly dependent on inter-chain distance). This suggests that enhanced non-bonding interactions (rather than thermal pathways through crosslinker covalent bonds) are the primary transport mechanism by which thermal conductivity is increased. We further illustrate this by showing that energy from THz acoustic waves travels significantly faster in polymers when non-bonding interactions are included versus when only covalent interactions are present. These results help to explain prior studies that measure differing trends in thermal conductivity for polymers upon crosslinking with various species.

1. Introduction Polymers have been extensively studied in recent decades for many applications, such as organic field effect transistors and other electronic devices, 1,2 thermoelectric devices, 3–7 photovoltaic devices, 8 thermal devices, 9 lithography and patterning, 10 drug delivery, 11 photonic devices, 12 and plastics products. 13,14 In many of these applications, the rate of thermal transport through the polymer plays a significant role in device reliability and performance. 15,16 For example, enhancing the thermal conductivity of polymers (which is on the order of 0.1 W/mK for most amorphous polymers 17,18 ) could improve their applications in solar cells 16 and the plastics industry, 19 while for a better thermoelectric figure of merit a polymer with a lower thermal conductivity is desired. 20 Previous computational 17,19,21–24 and experimental 25–34 studies have largely shown that the thermal conductivity of aligned linear polymer chains in the direction of chain alignment is significantly higher than that of bulk amorphous polymers. This has been attributed to a relatively large thermal conductivity for covalent bonds. 17,29,35 Meanwhile, the low thermal conductivities of amorphous bulk polymers are generally believed to be due to a bottleneck associated with weak inter-chain forces such as 2


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van der Waals (vdw) interactions, 18 which have been calculated to be an order of magnitude less efficient at transferring heat than covalent bonds. 36 Mechanisms to enhance thermal transport in polymers have traditionally been based on the inclusion of thermally conductive fillers 37–39 and more recently on enhanced inter-chain interactions. 18,40–43 Crosslinking of polymer chains through covalent bonds is a common synthesis approach for polymers and would seem to be a very compelling technique to enhance their thermal conductivity. 18,41,44–47 However, computational and experimental studies of thermal conductivity in crosslinked polymers have shown significant discrepancies. For example, experimental studies have shown 50% enhancement for polyethylene samples 41 at high crosslinking densities and approximately 30% enhancement for polystyrene at 20% crosslinking density, 46 while computational studies have predicted a threefold enhancement in the bulk thermal conductivity of polyethylene upon crosslinking 18 and almost no change in the thermal conductivity of polystyrene at 20% crosslinking density. Other studies have measured or predicted only a very small enhancement 18 or even a reduction in thermal conductivity upon crosslinking. 48–50 For example, experimental results by Yu et al. 49 show a nearly 30% reduction in the thermal conductivity of polyethylene upon crosslinking. Numerical simulations by Ni et al. 48 also suggest that crosslinking 10% of the carbon atoms in polyethylene chains results in a 44.2% reduction in bulk thermal conductivity. While both Yu et al. 49 and Ni et al. 48 predict a monotonic decrease in thermal conductivity with increasing crosslinking density for polyethylene, Kikugawa et al. 18 predict an opposite trend. Given the many applications of crosslinked polymers 11,12,51–60 and the importance of thermal conductivity to many of these applications, it is crucially important to build a comprehensive understanding of the effects of crosslinking on thermal transport. Thermal conduction in electrically insulating crosslinked polymers occurs through four mechanisms, as shown in Figure 1. These mechanisms may be closely coupled. For example, the length of the crosslinking agent is expected to strongly influence the spacing between linked polymer chains. This spacing in turn greatly influences heat flow via non-bonding in-

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fixed in order to control the chain endpoints. The system was then evolved as an NVT ensemble for 1.5 ns at 300 K using the Langevin 81 thermostat followed by a 1.5 ns relaxation at 300K using the Nose-Hoover 82,83 thermostat to reach thermal equilibrium. Sample temperature and potential energy profiles during thermal equilibrium are shown in the Supporting Information. A temperature gradient was then established in the system under NVE conditions by pumping "heat" into one chain and removing it from the other chain at the same rate. This heat flux was generated by rescaling the velocities of atoms in each chain to stimulate or suppress thermal motion. The system was allowed to evolve for 3 ns to reach equilibrium. Thermal data was then collected from the system. For example, thermal conductance (TC) between two chains was calculated using Fourier's law of conduction: 63,64,67,84,85

TC =

q ∆T

(1)

where q is the thermal energy transferred between the two chains and ∆T is the difference between the average temperatures of the chains. This technique has been used for intertube thermal transport in one dimensional systems such as CNTs. 84 Sample temperature profiles are shown in the Supporting Information. All relaxation steps used free, non-periodic boundary conditions. After relaxation, boundaries were fixed for thermal data collection and trajectory analysis. The OPLS force field was used to model the interactions of PMMA chains. 86–88 The parameters are reported in the Supporting Information. This force field has been successfully used in other studies for heat transfer modeling. 89–94 The force fields used for PVA and PE are described in their relevant sections, and their parameters are reported in the Supporting Information. We have used a variety of force fields to ensure the results are not force field dependent. For non-bonding interactions, a cutoff distance longer than the size of the simulation box was used in order to account for all interactions between the atoms in the system and to avoid any drift in the total energy of the system. This inclusion of all interactions in our simulations is crucial for drawing clear conclusions regarding the 6


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effects of these interactions. Inter-chain distances were measured by calculating the average distance between the centers of mass in each chain using the coordinates of the atoms in each chain.

3. Results and Discussion 3.1. Effects of Crosslinking Species and Non-bonding Interactions on Inter-chain Thermal Transport To study the effects of crosslinking on thermal transport in polymers and to understand the differing results reported in the literature, we simulated systems of two parallel PMMA polymer chains consisting of 20 monomers in each chain (an icosamer), terminated with hydrogens (similar to Figure 2). We first varied the crosslinker species in PMMA chains with 100% crosslinking to assess the maximum impact of the crosslinking agent on interchain thermal conductance. Although 100% crosslinking density for PMMA may not be practically achievable, we include it in our study as it gives us an understanding of the theoretical maximum by which crosslinkers may enhance heat transfer. In the following sections we also study lower crosslinking densities that are achievable in practice. Note that such high crosslinking densities are often found in epoxies and resins. 95 The species used (shown in Figure 3) span a range of shapes, masses, bond strengths, lengths, and vibrational characteristics, all of which potentially play a role in thermal transport. Although most of the crosslinkers are real chemical structures, some such as "light benzene-1,4-diyl" (a benzene-1,4-diyl crosslinker in which the carbon masses are reduced to half of the true carbon mass) do not have a real-world chemical representation. In order to isolate the effect of crosslinker mass, we performed a parametric study in which all of the other parameters (chemical structure, charge, force field, etc.) were kept constant and only the mass of the crosslinker was varied. We performed a similar simulation to elucidate the effect of bond strength by representing a carbon atom that has only two (rather than four) covalent bonds 7




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for systems in which non-bonding interactions are excluded (Figure 4(d)). Although the polymer chains get closer in this case, because there are no non-bonding interactions in the system, inter-chain thermal transport does not benefit from this reduction in the interchain distance. We note that the reduction in inter-chain distance due to the absence of the repulsive portion of the vdW interactions is more significant for systems crosslinked with benzene-1,4-diyl, since these repulsive forces are strong in benzene-1,4-diyl and the additional degrees of freedom associated with its ring structure enable a more significant collapse upon their removal. For systems in which all non-bonding interactions are included (squares, Figure 4(b)), chains crosslinked with CH2 (carbene) have the highest enhancement for inter-chain thermal conductance. This is because the crosslinkers bring the chains close to each other and hence enhance the non-bonding inter-chain interactions between the chains. The interchain distance for this crosslinker is smaller than the radius of effectiveness for non-bonding interactions (∼8-10Å or 2.5 times the Lennard Jones σ parameter; widely used in molecular simulations as a distance beyond which vdW interactions are negligible 61 ) and hence these interactions significantly enhance heat transfer between the chains. Moreover, there is an overlap between the vibrational density of states of the crosslinker and that of the main polymer chain, which further contributes to inter-chain heat transfer. This is discussed in more detail in the Supporting Information. When all non-bonding interactions are removed from the system, the only path for heat transfer between the chains is through the crosslinker covalent bonds. In this case we observe a large drop in inter-chain thermal conductance (triangles in Figure 4(d)). For example, a factor of ∼5 reduction in thermal conductance is observed for the system crosslinked with CH2 . This drastic drop in thermal conductance suggests that the majority of inter-chain thermal transport in crosslinked polymers occurs through enhanced inter-chain non-bonding interactions rather than thermal conduction along crosslinkers. Quantifying the number of bonding and non-bonding interactions offers another perspective on this effect, and is

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discussed in Section 3.4. Above we have shown that crosslinking species that bring the chains closer have a higher impact on thermal conductance between the chains. It is interesting to consider whether inter-chain distance also has an influence on the portion of thermal transport between the chains that occurs through covalent bonds. To study whether the presence or absence of non-bonding interactions influences heat transfer through crosslinker covalent bonds, we calculated the thermal conductance between polymer chains crosslinked with carbene as a function of inter-chain distance in the absence of non-bonding interactions. Here, changing the inter-chain distance is meant to approximate the physical effect that non-bonding interactions may have on the crosslinked chains. These results are shown in Figure 5. For these simulations we manually moved the chains with respect to each other in order to change the inter-chain distance. This will result in some residual stress in the system. The range of inter-chain distances in Figure 5 is chosen to cover the range of inter-chain distances in Figure 4(b). In the absence of non-bonding interactions, we did not see any clear relation or enhancement in thermal conductance between the chains for different inter-chain distances. Thus, based on Figures 4(d) and 5, we conclude that the thermal conductance between the chains in the absence of non-bonding interactions is not a function of inter-chain distance. This indicates that heat conduction through crosslinker covalent bonds is not greatly perturbed by the presence or absence of non-bonding interactions, and thus can be considered approximately to be independent. Therefore, the large drop in thermal conductance observed in Figure 4(a) when non-bonding interactions are removed is expected to be due primarily to removal of heat transfer through non-bonding interactions rather than an alteration of heat conduction through crosslinker covalent bonds. Together, the results of Figures 4 and 5 suggest that the primary thermal function of a crosslinker is to bring chains closer together so that non-bonding interactions are increased. Eiermann 36 calculated that thermal conductance through vdW interactions is 10 times weaker than thermal transport through covalent bonds. While this applies for a single

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vdW bond in comparison with a single covalent bond, our results indicate that inter-chain heat transfer through vdW interactions in larger systems is greater than that through covalent interactions due to the much larger number of vdW interactions. In addition, short crosslinker length enhances non-bonding interactions and hence increases their contribution to heat transfer. For example, for each PMMA monomer that forms a crosslink with another PMMA monomer, there is only one covalent bond between them to conduct heat. However, since each PMMA monomer has 15 atoms, there are 225 (15 × 15) heat transfer pathways through non-bonding interactions between the two crosslinked monomers. Although each non-bonding interaction transfers less heat than a covalent bond, the significantly larger number of them has a greater effect.

Figure 5: Change in thermal conductance (TC) between two PMMA polymer chains crosslinked with carbene crosslinkers in the absence of non-bonding interactions, with respect to the distance between the two chains. The importance of non-bonding interactions revealed here could explain results such as that of Kikugawa et al. 18 in which the thermal conductivity of simulated polystyrene was not calculated to increase even for high crosslinking densities. In this case the benzene side group in polystyrene may prevent the inter-chain distance from decreasing even in the presence of a large density of crosslinkers. Likewise, Yu et al. 49 reported a reduction in thermal conductivity for polyethylene sam12


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ples crosslinked by dicumyl peroxide. The length of this crosslinking agent may lead to increased inter-chain distance and hence a reduction in the contribution of non-bonding interactions to thermal transport. This is supported by the reduced density they measure.

3.2. Acoustic Wave Propagation Velocity Analysis In Figure 4(b) we showed that the thermal conductance between two polymer chains crosslinked with CH2 (carbene) crosslinkers is approximately 5 times greater when non-bonding interactions are included. The long-range nature of such non-bonding interactions (versus the short-range nature of covalent bonds) has important implications for thermal transport. In this section, we show that acoustic waves (which carry heat) travel faster when non-bonding interactions are included versus when only covalent bonds are present. Figure 6 shows a crosslinked polymer system for simple acoustic wave propagation analysis, with input wave location and "probe" locations at which we detect the energy of the wave after it propagates from its initial point. We first reduce the temperature of the structure to 0K to remove all molecular motion. We then create a 1 THz oscillation input in the transverse direction with respect to the polymer backbone. The 1 THz oscillation is chosen to represent a wave that carries heat in the materials at representative temperatures. 63,96 The method used to detect the wave at the probe locations is discussed in detail in the Supplementary Information. In Table 1 we show the normalized wave propagation velocities for a 1 THz wave. The results show that a wave at 1 THz frequency travels approximately 9 times faster from one chain to another chain when non-bonding interactions are included. This wave propagation analysis agrees with our previous results in which we showed that thermal conductance between two chains is 5 times higher when non-bonding interactions are included versus when only covalent bonds conduct heat between the chains. This analysis

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Above 65% crosslinking density, the system with all interactions included shows a nonlinear increase in thermal conductance. This is due to the fact that the PMMA polymer chains simulated here have an intrinsically syndiotactic conformation, and when they are 100% crosslinked, the monomer side groups in the two chains all face each other and inter-chain distance is reduced drastically. As further discussed in the Supporting Information, we also find that inter-chain distance for the results shown in Figure 7 are inversely proportional with the inter-chain thermal conductance. This means that reducing crosslinking density results in a larger inter-chain distance which in turn weakens the non-bonding interactions between the chains.

3.4. Inter-chain Thermal Transport through Non-bonding Interactions In the previous section we have shown that non-bonding interactions have a more significant effect on thermal transport in crosslinked polymers than covalent bonds when the crosslinking agent reduces the inter-chain distance. Additionally, in Figure 7 we show that the thermal conductance between polymer chains scales approximately linearly with crosslinking density whether the non-bonding interactions are included or not. This means that the difference between the red and blue curves in Figure 7 also scales linearly with crosslinking density. The values obtained by subtracting the two curves in Figure 7 are proportional to the portion of thermal transport through non-bonding interactions. Therefore, even in polymers without crosslinkers, there should be a linear relationship between the number of interactions between polymer chains (which we quantify below) and the inter-chain thermal conductance. Here we simulate relatively long (120 monomers in each chain, terminated with hydrogens) pairs of different polymer chains (PMMA, PVA, and PE) to study the effect of inter-chain distance and entanglement on inter-chain thermal transport for polymers without crosslinkers. We used nanoHUB Polymer Modeler 78 to generate the polymer chains geometry and the force field provided to study PE chains. The force-field developed by 16


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Muller 97 was used to model the PVA chains; however, since the lengths of the bonds in their model were fixed, we used bond stiffness values from the OPLS force field 86 to allow the bonds to vibrate. The average bond length using the OPLS bond stiffness values is the same as the original bond length set in Muller’s force field 97 (post-simulation bond length distributions are given in the Supplementary Information). The diversity of force fields 78,88,97 that we have used here reinforces the generality of the results. Polymers were relaxed, and the thermal transport between the chains was measured similar to previous simulations using NEMD. A time step of 0.5 fs was used for all polymer simulations in this section. The cutoff values for the non-bonding interactions were set to be larger than the simulation box in order to account for all of them. After each simulation we analyzed the atomic trajectories to find the degree of inter-chain interaction between the monomers in the polymer chains. We define this degree of interaction as the number of monomers from one chain that are within a specific distance from a monomer in another chain. This distance is approximately equal to 2.5 times the average value of σ parameter in Lennard Jones potential for carbon atoms (∼8Å) in each polymer. This 2.5 σ factor is widely used in molecular simulations as the distance beyond which vdW interactions are negligible. 61 Figure 8 shows the method we have used to determine the number of interactions per monomer in our simulations. We then calculate the average number of interactions per monomer for all monomers in a chain and plot the inter-chain thermal conductance with respect to this average. When calculating the number of interactions for a given monomer, we measure the distance between the center of mass of the monomer and the centers of mass of monomers in the other chain. If the distance is smaller than the specified distance cut-off then we consider the monomers as interacting. To control the level of entanglement of polymer chains, we let them relax for different periods of time ranging from a few femtoseconds to several thousands of femtoseconds, before we fix the terminating hydrogen atoms. The results for inter-chain conductance as a function of average number of interacting monomers are shown in Figure 9. The inter-chain thermal conductance exhibits a linear relation with the average number

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of monomers that are interacting. Each pair of polymer chains has a unique slope that could be used to inform a new theoretical model for the thermal conductivity of bulk samples or to improve existing models. 47,98 The relation observed here is in agreement with a previous study on the effect of chain length on thermal transport in polymers. 99 For different polymers, thermal transport between the chains scales linearly with the average number of interacting monomers and hence scales with inter-chain distance. This is also in agreement with the linear regime observed in Figure 7. Also, the slopes of the lines in Figure 9 are strongly related to the strength and number of non-bonding interactions between the chains. For example, the PMMA monomer has 15 atoms, and hence if two PMMA monomers interact with each other there will be 225 (15 × 15) interactions that can transfer heat. This number for two interacting PE monomers is only 9 (3 × 3) and 49 (7 × 7) for two PVA monomers.

3.5. Effects of Different Inter-chain Interactions on Thermal Transport along Polymer Chains High thermal conductivity along polymer chains is usually associated with robust thermal transport through covalent bonds in the polymer. 17,29,35 In the previous section we showed the significant contribution of non-bonding interactions on inter-chain thermal transport in crosslinked polymers. Single polymer chains could also be considered as several monomers that are sequentially linked together through covalent bonds. Thus, one may expect a significant contribution from non-bonding interactions on thermal transport along the chain. To investigate this possibility, we ran two sets of simulations using the NEMD method detailed in Section 2. In one of them we removed the non-bonding interactions from the PMMA chain and then measured the thermal conductance along the chain. In the other we kept all of the non-bonding interactions included and measured the thermal conductance. These results are reported in Table 2. We found a factor of ten increase in thermal transport along the PMMA polymer chain when non-bonding interactions are included versus when they are not. Similar to the previous section, we observed that non-bonding interactions 19



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significantly enhance the intra-chain thermal transport. These results suggest that thermal transport along a single polymer chain could be enhanced by improving the non-bonding interactions along the chain through introduction of polar side groups or hydrogen bonding within the chain. Table 2: Effects of non-bonding interactions on intra-chain thermal conductance in polymers. With non-bonding interactions 0.288 (nW/K) Without non-bonding interactions 0.029 (nW/K)

4. Conclusions Our investigation of heat transfer in polymers through non-bonding and bonding interactions suggests that non-bonding interactions are very important to overall thermal transport. Despite previous beliefs that strong covalent bonds increase heat transfer in polymers, studies have shown both enhancement 18 and reduction 49 in the thermal conductivities of polymers upon crosslinking. We showed that conduction along covalent crosslinkers between polymer chains is not the primary mechanism for heat transfer enhancements in crosslinked polymers. If the crosslinker is short enough to bring the chains close to each other, the large number of non-bonding interactions between the chains is enhanced, leading to a large contribution to heat transfer. This is reasonable since the number of crosslinkers is small compared to the number of non-bonding interactions. Thus, enhancing non-bonding interactions could result in a large improvement in thermal conductivity. For example, recent measurements of polymers with inter-chain hydrogen bonds have shown an order of magnitude enhancement in thermal conductivity. 40 We further illustrated the importance of non-bonding interactions by showing that acoustic waves that carry heat travel significantly faster when non-bonding interactions are included compared to when only covalent bonds are included, because of the long-range nature of non-bonding interactions. The effects of crosslinking species chemical structure on inter20


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chain thermal transport were also studied. Our results suggest that in order to synthesize crosslinked polymers with high thermal conductivity, one should choose a short crosslinker and a polymer that is capable of forming strong non-bonding interactions between its chains.

Supporting Information 1) Sample temperature and potential energy profiles 2) Sample force fields used 3) Bond length distribution 4) Inter-chain distance for various crosslinking densities 5) Bulk density and thermal conductivity predictions for force field validation 6) Effects of crosslinker species vibrational density of states of crosslinker agent on interchain thermal transport 7) Method of acoustic wave detection at the probe location

Acknowledgement This work used the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National Science Foundation grant number OCI-1053575.

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Thermal Conductance in Cross-linked Polymers: Effects of Non

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