Normal Mode Analysis of The Thermal conductivity in Amorphous Polymers: The Importance of Localized Modes

: Polymers are a unique class of materials from the perspective of normal mode 12 analysis. Polymers consist of individual chains with repeating units and strong intra-chain 13 covalent bonds, and amorphous arrangements among chains with weak inter-chain van 14 der Waals and for some polymers also electrostatic interactions. Intuitively, this strong 15 heterogeneity in bond strength can give rise to interesting features in the constituent 16 phonons, but such effects have not been studied deeply before. Here, we use lattice 17 dynamics and molecular dynamics to perform modal analysis of the thermal conductivity 18 in amorphous polymers for the first time. We find an abnormally large population of 19 localized modes in amorphous polymers, which is dramatically different from amorphous 20 inorganic materials. Contrary to the common picture of thermal transport, localized modes 21 in amorphous polymers are found to be the dominant contributors to thermal conductivity. 22 We find that a significant portion of the localization happens within individual chains, but 23 heat is dominantly conducted when localized modes involve two chains. These results 24 suggest that even though each polymer is different, localized modes play a key role. The 25 results provide new perspective on why polymer thermal conductivity is generally quite 26 low and gives insight into how to potentially change it. 27

1 associated with their disordered structure, amorphous polymer TC is still an order of 2 magnitude lower than that of amorphous inorganic materials, and the reason why is not 3 entirely clear from a theoretical perspective. For crystalline materials, the TC is usually 4 well explained by the phonon gas modal (PGM). [15][16][17] However, there is growing evidence 5 to suggest that the PGM is insufficient for understanding the thermal behavior of 6 disordered materials 18-39 , such as an amorphous polymer. The structural or compositional 7 disorder within disordered materials causes a drastic change in the character of 8 vibrational modes, which are no longer spatially periodic, as they are in crystalline 9 materials. 40 As a result, the PGM can become ill-suited for describing the behaviors and 10 contributions of phonons/normal modes with non-periodic mode shapes, which we will 11 henceforth refer to as "non-propagating". 12 13 When a significant fraction of the modes in a material cease to have propagating 14 character, the PGM can fail at describing the TC in ways that cannot be corrected with 15 simple modifications of terms that treat it as a perturbation. 41 This is because there is an 16 intrinsic inapplicability of the PGM physical picture to modes that are non-propagating, 17 since a well-reasoned group velocity or wave vector cannot be assigned to non-18 propagating modes. 21 For amorphous materials, Allen-Feldman (AF) proposed a model 19 based on supercell lattice dynamics calculations that naturally classifies modes into three 20 distinct categories, namely propagons, diffusons, and locons. 19,42,43 Propagons are modes that most closely resemble the traditional picture of a phonon, 23 which arises from the periodicity in crystalline materials. Propagons have a well-defined 24 wave vector and a periodic mode shape that can produce a propagating wave packet, 44 25 and we therefore refer to them herein as propagating. They tend to be low-frequency 26 modes that are delocalized throughout the entire system, and their propagating nature is 27 what allows them to be accurately described by the PGM. Diffusons, while spatially 28 delocalized, exhibit no apparent periodicity in the atomic vibrations/mode shape. Instead, 29 these vibrations appear random, and it is not clear how one could define a wave-vector 30 and/or a group velocity for them. Thus, diffusons are not likely to be well described by the 31 PGM. Finally, locons are spatially localized modes that exist in a portion of the supercell 32 that have their energy highly concentrated onto a subset of atoms. In all previous studies, 33 locons have comprised the smallest group of modes 21,35,41,45 and they have been 34 assumed to be too localized to contribute to TC in a significant way, 19 We performed LD calculations on well-relaxed amorphous polymers to determine the 29 vibrational normal modes. Mode resolved TC was calculated using GKMA with 30 equilibrium MD simulations and the details associated with the relaxation procedure, 31 interatomic potential, and LD/MD are provided in the supplementary information. Three 32 amorphous thermoplastics are studied: amorphous poly(methyl methacrylate) (a-PMMA), 33 amorphous polystyrene (a-PS), and amorphous polyvinyl chloride (a-PVC). All three 34 polymers contain carbon (C) and hydrogen (H), while PMMA contains oxygen (O) atoms, 35 and PVC contains chlorine (Cl). The molecular structure of each monomer is shown below 36 in Fig. 1. 37 38 1 Figure 1| Chemical structures of thermoplastics in study: a-PMMA (left), a-PS (center), 2 and a-PVC (right).

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We chose to study these three thermoplastics for two primary reasons: 1) Due to the 5 nature of thermoplastics, they can be annealed and/or melted using MD to relax the 6 structure without concern about significant changes to the chemical bonding, as opposed 7 to thermosets, which are irreversibly cured. 2) Since there is a practical limit to the number 8 of polymers that can be studied herein, given finite computational resources, we 9 attempted to study thermoplastics with some significant heterogeneity in monomer 10 structure/chemistry. By focusing on these thermoplastics, there will perhaps be some 11 observations that may apply more broadly to thermoplastics in general. Due to the 12 significant differences between the three thermoplastics selected, the findings that hold 13 across all three polymers may have implications for a large number of other 14 thermoplastics, as opposed to the alternative of studying three very chemically similar 15 thermoplastics.

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To characterize the normal modes in amorphous polymers, we studied the normal modes  Figure  19 2a shows the PR versus mode frequency. The PR measures how many atoms are 20 actively involved in a given mode, and therefore is a measure of the degree of localization. 21 The definition of PR can be found in supporting information. Surprisingly, as compared to 22 the modes in amorphous silicon, 19,44,80 which are similar to the modes in amorphous 23 carbon 81 and amorphous germanium, the modes in amorphous polymers are much more 24 localized than any of the aforementioned amorphous semiconductors. The modes in the 25 polymers studied here have extremely low PR across a broad range of frequencies as 26 shown in Fig. 2a. Based on the PR, the vast majority of the modes are considered locons 27 using the criterion PR < 0.1, which is more conservative than the widely used delineation 28 criterion of 0.2 in other studies, 41,43,44,84 to classify modes as locons. Interestingly, if one 29 were to adopt this less conservative criterion, almost every mode could be considered a 30 locon with PR < 0.2 (i.e., 82%, 95%, and 98% of the modes in a-PMMA, a-PS, and a-31 PVC, respectively). Such a high degree of localization, especially at low frequencies has not been observed 10 in any other bulk material to date. However, when visualizing the eigenvectors, which 11 describe the square root mass weighted displacement of each atom, as it participates in 12 a given mode, many of the locons seem to have their eigen vectors spread over a The width of the mode in direction is calculated as as ( 1 + 2 )/2. The MSE of one mode 4 is the average of the fitted width in three directions. A Gaussian was chosen based on 5 inspection of the modes, but different trial functions other than a two-term Gaussian were 6 also used and the resulting MSE was not sensitive to the choice of trial function. MSE in 7 essence attempts to quantify how large a mode is, by measuring the size of the region to 8 the participating atoms occupy. Note that the calculation of MSE involves fitting a two-9 term Gaussian, which might yield values greater than the simulation domain. This is 10 reasonable as in the ideal case the MSE of a periodic mode is infinite. The details of MSE 11 calculation can be found in SI.

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In on the order of the atomic spacing or lower (~1 Å), the mode can be considered highly  19 spatially localized, while a MSE of 10 Å is closer to the order of the dimensions of the 20 supercell in this work. In Fig. S2, several modes are shown that have low PR (~0.01) yet 21 large MSE (~100 Å) in a-PMMA. Despite the way the eigenvectors appear upon 22 inspection, the cumulative DOS vs. MSE in Fig. 2c confirms that fact that the majority of 23 modes are in fact spatially localized, with a MSE < 10 Å. For a-PMMA, a-PS, and a-PVC, 24 the factions of modes with MSE < 10 Å are 75%, 89%, and 89%, respectively. 25 26 In Fig. 3 reasonable to postulate that similar results might also be observed for other 44 thermoplastics, and maybe even thermosets. Nonetheless, more materials need to be 45 studied to confirm or deny this hypothesis. Regardless, the difference between the two 1 descriptors is an indication that perhaps "locons" (specifically, modes with PR < 0.1 and 2 high MSE > 10 Å) may be able to interact with other modes more than previously thought, 3 therebycontrary to the findings of many previous studies, 47-50 playing an important role 4 in thermal transport. The large degree of localization observed in amorphous polymers prompted the 11 hypothesis that it may be possible that modes localize onto individual chains where there 12 is more homogeneity in bond strength. We therefore evaluated a polymer chain 1 participation ratio (PCPR) as a measure of the involvement of each individual chain in 2 each mode. The definition of PCPR can be found in SI, as it contains a selective 3 summation for PR, that is limited to the atoms on a given polymer chain. Thus, for each 4 polymer chain in a simulation domain, there will be a PCPR value measuring its 5 involvement in a given normal mode. Consequently, we assign M different PCPR values, 6 PCPR1, PCPR2, … PCPRM, for each individual mode, where M is the number of polymer 7 chains in the supercell. The subscripts, however, do not denote which specific chain a 8 mode is localized onto. Instead, since all of the chains are the same i.e., chemically 9 indistinguishable, the values of the subscripts are sorted by decreasing value. In this way, 10 the PCPR1 value for a given mode will always correspond to the chain that has the largest 11 PCPR for that given mode. The PCPR2 is therefore by definition always smaller than the 12 PCPR1, and it represents the extent of participation the mode has on whichever chain 13 has the 2 nd highest participation. The PCPR values then continue in decreasing order of 14 participation, with PCPRM being be the smallest. The PCPR spectrums of three polymers 15 are provided in the SI (see Fig. S3).

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With this definition for PCPR, the ratio 2 = PCPR2/PCPR1 becomes an important quantity, 18 because it describes the extent to which a mode is localized onto more than one chain.

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For example, when 2 is less than 0.1, it indicates that at least 53% of the energy of the 20 mode is confined to one polymer chain. Here, it should be noted that there is a somewhat 21 complex relationship between the value of 2 and the amount of energy confined to a 22 single chain, as explained in the SI. However, most modes with 2 < 0.1 have more than 23 90% of their energy localized onto a single chain. Similarly, the ratio will tell whether 24 there are k chains involved in this mode. The 2 spectrums of three polymers are shown 25 in Fig. S3d in SI, which suggests a large number of modes have a small 2 . To better 26 understand this, we show the cumulative DOS with respect to 2 in Fig. 2d. If we take 27 2 < 0.1, to mean a mode is primarily localized onto a single chain, then for a-PMMA, 52% 28 of the localized modes are localized onto a single chain, and this fraction is 75% for a-PS 29 and 74% for a-PVC. This is a somewhat surprising result, as it shows a strong tendency 30 for modes to localize onto individual chains, which is a special feature of polymers. The 31 strength of covalent bonds are usually three orders of magnitude higher than that of van 32 der Waals interactions. The disparity in bond strength for inter vs. intra chain interactions 33 is rather unique to polymers and possibly some 2D materials (e.g., graphite, MoS2, W2Se, 34 Hexagonal BN etc. together/collectively at the same high frequencies as those associated with the covalently 4 bonded atoms on a different chain. This observation provides intuition on how heat 5 conduction through amorphous polymers occurs. One might expect that these two groups 6 of modes, i.e., modes confined to a single chain, versus modes that are able to spread 7 across multiple chains, might exhibit different transport characteristics, as the latter 8 category are more likely to couple/share energy between among different chains. This 9 point will become clearer when we discuss thermal conductivity. 10 11 We emphasize again that a-PMMA, a-PS, and a-PVC have significantly different 12 structures i.e., the monomers are comprised of 15, 16, and 6 atoms respectively. PS is 13 the only polymer with an aromatic ring. And the polymers contain different functional 14 groups. However, the observations about the high degree of localization in these 15 polymers are consistent amongst all three and are independent of supercell size (see SI 16 - Fig. S1). Thus, we postulate that it may be also the case that most of the modes in many 17 other amorphous polymers are locons. However, it is not clear to what extent these modes 18 contribute to TC, and if they do, which classifications of modes contribute most (i.e., 19 modes localized onto individual chains or modes delocalized over multiple chains)? 20 Conventional wisdom, based on the PGM, would suggest that localized modes in general 21 do not contribute much to thermal transport, but if they're the predominant mode type in 22 amorphous polymers, it stands to reason that they may be the dominant contributors to 23 TC, simply because they're the dominant mode type.

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Contributions to Thermal conductivity 26 27 In Fig. 4 we show the normalized TC accumulation with respect to PR, MSE, and 2 , 28 based on GKMA. Fig.4a shows the locons, defined by PR < 0.1, contribute 82%, 95%, 29 and 98% to TC for a-PMMA, a-PS, and a-PVC respectively. Therefore, they are the 30 dominant contributors to thermal transport. This is very different than prior studies, some 31 of which have suggested that locons do not contribute significantly, 19 Figure 4c plots thermal conductivity as a function of 2 that 46 represents localized phonons spreading over two chains, and it clearly shows heat is 1 mainly conducted by localized modes able to across chains. chain. For locons defined as modes with PR below 0.1, they contribute 82%, 95%, and 8 98% of the total TC in a-PMMA, a-PS, and a-PVC, respectively. (Colors online) 9 10 With this mode-resolved TC, we calculated the temperature dependent TC with quantum 11 correction. In Fig. 5, the quantum corrected temperature dependent TC for the three The GKMA results show the correct qualitative trends, although there are significant 5 quantitative discrepancies at different temperatures. The calculations underestimate the 6 TC for a-PMMA and a-PVC, while slightly overestimate that of a-PS. We believe the 7 reason can be attributed to a combination of the difference of polymer molecular weights 8 (MW) and size effects. It is known that for polymers, the TC has a strong positive 9 dependence on its MW. 92 Due to the limitations of computational power, the MW of a-10 PMMA, a-PVC, and a-PS are 5,000, 1,696, and 7,813 g/mol in our simulations (every 11 chain contains 200 repeating units), which are orders of magnitudes smaller than the MW 12 of the common products of these polymers. Nevertheless, obtaining good quantitative 13 agreement in temperature dependence of TC is not the main objective here, but the TC 14 results seem to suggest that the mode level observations made in our simulation could 15 still be meaningful and provide important insights for understanding the thermal transport 16 in amorphous polymers.

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Although delocalized modes contribute more to TC than locons on a per mode basis, 19 interestingly, the most active modes with the highest TC contributions are strongly 20 localized. As shown in Fig. 3, the modes with highest modal TC are more likely to be 21 localized. It is clear that for all three polymers in study, the modes with highest modal TC 22 appear in the regime with low PR and low MSE. Similar observations are found in the 23 modal TC spectrums shown in SI Fig. S4, where the TC outliers are strongly localized.

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This finding suggests that there is a group of locons participating in thermal transport 5 more actively than the others. Note that this observation is not in conflict with our previous 6 statement that delocalized modes contribute more to thermal transport on a per mode 7 basis. To explain what is special about these active locons, we studied how the locons 8 are localized in polymer structures. In a-PMMA for example, we notice there are a few 9 modes involving two chains with large (> 10 -5 ) PCPR2's and low (< 10 -10 ) PCPR3's around 10 50 THz, where most modes in the same frequency range are confined to a single chain, 11 and no modes are spread across three or more chains. There are a total of 62 out of 9024 12 of such modes that are delocalized onto a second chain but not a third chain. Interestingly, 13 the average magnitude of these 62 modes' TC is 18.4 x 10 -5 Wm -1 K -1 . This value is 60% 14 higher than the average magnitude of TC for SLM with both a low PR and MSE and nearly 15 twice the average magnitude of all mode in a-PMMA. Generally, we have found most of 16 the locons are confined onto a single chain, as suggested by the more-than-half 17 cumulative contribution to DOS by modes with 2 < 0.1 in Fig. 2d. As shown in the 18 cumulative TC contribution with respect to 2 in Fig. 4c, however, modes confined onto a 19 single chain only comprise less than half of the total TC, despite their populations being 20 more than half (Fig. 2d). Even if we exclude the contributions from the minority of 21 delocalized modes and focus on the locons, the contribution to TC from locons localized 22 onto multiple chains (i.e., for modes with PR < 0.1 yet 2 > 0.1), is still higher than the 23 modes localized onto a single chain, as summarized in Tab Therefore, we conclude that the modes with very large TC values are more likely to be 32 strongly localized modes and as a result, the average TC value of the strongly localized 33 modes (SLM) is slightly higher than other modes. And the locons that spread onto multiple 34 chains contribute more to thermal transport than the ones confined onto single chains, 35 even though the later type of locons are the majority in amorphous polymers. These 1 findings differ from the behavior of locons in other disordered materials.

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Conclusions 4 5 In this work, we studied the normal modes and TC of three amorphous polymers (i.e., a-6 PMMA, a-PS, and a-PVC) using LD and GKMA. The results show that in all three 7 amorphous polymers studied, locons are the predominant mode type, comprising more 8 than 90% of the modes. GKMA results showed that even though locons in previously 9 studied systems have exhibited negligible/small contributions, here, since they are the 10 predominant mode type, their contributions comprise more than 80% of the TC. This 11 shows that locons can play an important role in amorphous polymers, although their 12 contributions are much smaller on a per mode basis, by comparison to delocalized modes. 13 Deeper investigation of the character of locon modes showed that 10-30% still involve 14 some participation of atoms over a large region. Such modes are still locons in the sense 15 that their energy is concentrated on a small subset of atoms, but this energy can still be 16 spread over a spatial region comparable to the size of the supercell (i.e., > 10A). The 17 introduction of a new descriptor, termed the MSE facilitated quantification of the length 18 scale over which such modes are spread.

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Further modal analysis showed that a major fraction of the localized modes are localized 21 onto a single polymer chain, as quantified by the value 2 . This is presumed to be caused 22 by the inhomogeneity in bond strength between intra and inter chain interactions. 23 Calculations of 2 for the different polymers showed that for more than half of the localized 24 modes, greater than half of their energy was confined to a single polymer chain. The 25 minority of locons that are nested onto multiple chains, however, have a larger average 26 TC than the main type of locons that are localized onto a single chain, suggesting that 27 these modes serve as bridges across the chain-chain boundaries through van der Waals 28 interaction. Interestingly, this seems to be an important heat transport channel for 29 amorphous polymers, even though it happens through locons.

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Furthermore, the localized nature of the modes in amorphous polymers offers an 32 explanation for why their TC is about an order of magnitude smaller than that of 33 amorphous semiconductors. Notably, since localized mode contributions are in general 34 smaller than that of delocalized modes, having a material with predominantly locons 35 translates to an expectation of low TC.