Letter pubs.acs.org/macroletters
Using Time−Temperature Superposition for Determining Dielectric Loss in Functionalized Polyethylenes Mayank Misra and Sanat K. Kumar* Department of Chemical Engineering, Columbia University, 500 West 120th Street, New York, New York 10027, United States S Supporting Information *
ABSTRACT: Using Molecular Dynamics simulations, we probe the effect of various pendant polar groups on the dielectric loss of polyethylene copolymers. The dielectric loss was computed using the autocorrelation function of the total dipole moment of a completely relaxed PE−X sample. Since this calculation is computationally expensive (wall time ≥ 4200 h), we explore the use of the time−temperature superposition (tTS) principle to make it more tractable. An important point is that short time MD simulations do not allow the dipole autocorrelation function to decay completely to zero. However, we find that the tTS method performed well in determining dielectric losses in the system as long as these unrelaxed components are not included in the calculation. This methodology, which provides us with a significantly faster and reliable pathway for calculation of dielectric loss, allows us to identify the role of polar side groups on the behavior of nonpolar polymeric dielectrics. olymeric dielectric films are finding increased usage in power electronics due to their ease of processability, low weight, and self-healing ability.1,2 The state-of-the-art in polymeric capacitor films is biaxially oriented polypropylene (BOPP),3 which has an exceptionally high electrical breakdown strength (EBD > 700 MV/m) and low dielectric loss (ε″ > 0.1%), but a small static dielectric permittivity (ε′ = 2.2). Since the energy density of the material is ∝ ε′E2BD, an attractive scheme for improving material properties is by increasing the static dielectric permittivity while preserving the low loss. While one way to increase the permittivity is by the addition of polarizable groups to PP, the slow dynamics of most polar groups also produces increased dielectric loss in the range of frequencies relevant to power electronics. In our recent work,4 we have explored the role of different chemically bound functional groups for increasing the ε′ of polyethylene (PE). We further note that recent experiments, which have shown that the addition of a small number of polar hydroxyl (−OH) groups increases the static dielectric permittivity of a nonpolar polymer,5−7 however, demonstrates that these groups do not significantly increase the dielectric loss in the frequency ranges of interest. Consequently, the role of polar groups on dielectric loss is not clear. Here, we utilize Molecular Dynamics (MD) simulations to estimate the role of different polar groups attached to a polyolefin backbone on the dielectric loss. While MD is an attractive approach for the exploration of the dipole relaxation of the chain, the relatively modest upper limit of accessible time-scales (typically smaller than 1 μs) is the limiting factor in the context of experiments, where time scales of seconds are more the norm.8,9 While coarse-grained models could be useful in this context,10 these models do not capture the local structural relaxation process (like hydrogen bond
P
© XXXX American Chemical Society
relaxation), which could profoundly impact loss mechanisms. Instead, here we employ the time−temperature superposition (tTS) principle to determine the local dynamics of semicrystalline functionalized PE over broad time scales. The benefit of using tTS is that we can capture the complete segmental relaxation of the polar chain moieties of interest across experimentally relevant frequencies. An important caveat here is that the applicability of tTS requires that there is a single dominant (slow) relaxation process, which in this case is postulated to be the relaxation of the functional group. We expect this approximation to be applicable for temperatures well above the glass transition temperature, Tg, of the polymer; at lower temperatures, we expect complications from the slowing down of segmental dynamics and the “splitting” of this relaxation into α and β processes, invalidating the assumption of time scale separation.11−14 The reorientation of molecular dipoles is a relatively slow process in comparison to electronic transitions or molecular vibrations, which have frequencies generally above 1012 Hz. We thus assume a separation of time scales and postulate that the loss processes relevant to dielectric applications of the functionalized polymers is primarily due to these slow relaxations of the functional groups, which, in addition, have much higher dipole moments than the surrounding nonpolar polymer backbone. The theory of relaxation behavior was pioneered by Debye. It begins with relaxation processes Received: December 28, 2016 Accepted: February 2, 2017
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DOI: 10.1021/acsmacrolett.6b00978 ACS Macro Lett. 2017, 6, 200−204
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ACS Macro Letters described by the normalized dipole correlation function, Φ(t) (DCF):
Φ(t ) =
M(0) ·M(t ) M(0) ·M(0)
In the regime of linear response, the application of Kubo’s general theory for statistical mechanics of linear dissipative systems implies that dynamic properties can be expressed in terms of quantities relevant to the dielectric in the absence of the field.15 The complex dielectric permittivity ε*(iω) = ε′(ω) − iε″(ω) is obtained following the superposition relation: ε*(iω) − ε∞ = ε0 − ε∞ = 1 − iω
∫0 ∫0
∞
⎡ dΦ(t ) ⎤ dt[exp(−iωt )]⎢ − ⎥ ⎣ dt ⎦
Figure 1. Dipole correlation function of 4.2 mol % PE−OH after 20, 100, 200, 500, and 2000 ns.
∞
dt[exp( −iωt )]Φ(t )
this time scale. A hydroxyl is defined as being hydrogen bonded7 when there are two −OH groups within a cutoff distance (0.3 nm) and a cutoff angle (30°). For each hydroxyl in a given 20 ns time interval, we calculate the fraction of time it is hydrogen bonded. Following that, we calculate the average fraction of time hydroxyls are hydrogen bonded by averaging over all the hydroxyls in the system. This calculated time for each time interval split is compared to the Φm over that time scale (Figure 2). We find a linear correlation with a high
where ε0 and ε∞ are the limiting low and high-frequency permittivities, respectively. Consequently, to assess the dielectric loss of polyethylene substituted with polar groups we calculate the dipole correlation function using Molecular Dynamics. We simulate the functionalized polyolefin chain by using the optimized potential for liquid simulations, which is an experimentally validated all atom force field (OPLS-AA).16,17 Bond stretches and angle bendings are represented using harmonic potentials, and dihedral motions are governed by the OPLS-AA triple cosine series potential. van der Waals interactions between the different chain moieties are described by standard Lennard-Jones 12−6 (LJ) potentials. The parameters for interactions between unlike pair of atoms i and j are calculated using mixing rules, that is, σij = σiiσii and ϵij = ϵiiϵii , where σ is the LJ size parameter and ϵ is the LJ well-depth parameter. Note that the symbol ϵ used here for potential depth of LJ potential is different from ε, which is used for representing dielectric permittivity. Electrostatic interactions between two atoms, each with a partial charge of qi, are modeled by Uϵ(r ) = ∑i ∑j
qiqj
4πεor
, where r is their separation
distance and ε0 is the permittivity of free space. To speed up the simulations, we exploit general-purpose graphical processing units (GPGPU) to accelerate the van der Waals and longrange Coulombic calculations, 1 8 , 1 9 implemented in LAMMPS.20 We generate the conformation of a single polyethylene chain with 1000 backbone carbon atoms at random while respecting bond length constraints. This chain is equilibrated at 500 K and 1 bar, after which we randomly replace 4.2 mol % H atoms of the CH2 with −OH groups to generate the amorphous PE− OH starting state. The system is continuously cooled from 500 to 300 K at a rate of 2.5 K/ns under isobaric conditions, and then its volume is allowed to stabilize at 300 K for 20 ns. Thereafter, it is equilibrated in the canonical ensemble for 10 ns, and the net dipole moment is sampled for various periods of time (20−2000 ns in a series of simulations) for the calculation of Φ(t), Figure 1. The data obtained from the short time scale simulations show nonzero long time “plateau” (Φm) values for the dipole correlation function,8 but this plateau value decreases with increasing simulation times. Apparently, the system requires more than 1000 ns for the dipole correlation function to reach zero. We argue that this plateau value is determined by the fraction of −OH groups that remain hydrogen bonded during
Figure 2. Correlation between the average value of dipole correlation, Φm, and average fraction of time hydroxyls are hydrogen bonded in a 20 ns split.
Pearson correlation (ρΦm,δhb = 0.95), showing that hydrogen bonding is playing a critical role in the dipole relaxation of these systems. Apparently, we have to gather more data (run longer) to allow these hydrogen bonds to establish bonding with other partners. We note that the calculation of the full, equilibrated dipole correlation function of 4.2 mol % PE−OH requires simulation times on the order of microseconds. To reduce this computational time, we explore the notion of time−temperature supposition (tTS). The semicrystalline system at 300 K is heated to 320 K and equilibrated for 5 ns under NVT conditions. Thereafter, a production run of 20 ns is performed to calculate DCF under NVT conditions. Predictably, the dipoles relaxed further at 320 K. The same procedure is then implemented for various temperatures until the dipoles completely relax at 500 K. The DCF for 320, 400, and 500 K were shifted to create the master curve for 300 K (Figure 3). It 201
DOI: 10.1021/acsmacrolett.6b00978 ACS Macro Lett. 2017, 6, 200−204
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Figure 3. (a) Master curve of dipole correlation function of 4.2 mol % PE−OH at 300 K using tTS (b) compared with 2000 ns simulation where aT is the shift factor.
is of our particular interest to note that these short time simulations yield “plateaus” at long time scales. We use the “decaying” part of the correlation function but ignore these plateaus (which correspond to the slowest relaxing H-bonds in each specimen) to create the master curve. Similarly, the fastest part of the relaxation function, which corresponds to librational motions, represents a different time scale process which is thus not included in the tTS procedure. The master curve obtained in this fashion, which accounts for the segmental relaxations, is compared to the single long simulation (Figure 3b) and they agree with each other. To examine the methods quantitatively, we fit the two obtained dipole correlation functions to stretched exponential functions, Φ(t) = exp[−(t/τ)β]. The average relaxation time for a single long run and from tTS are calculated to be 2.19 ± 0.2 and 2.72 ± 0.5 ns, respectively. Importantly, while the two flavors of simulation provide the same relaxation behavior, however, the computational time required for tTS is 2 orders of magnitude less. We use this tTS method to find the effect of four different types of polar groups on the characteristic relaxation time of substituted polyethylene. To generate PE−X, where X is − NH2,−SH, and PE−NO2, we follow the same protocol as for PE−OH. We randomly replace 4.2 mol % H atoms of the CH2 with the −X polar group to generate the amorphous PE−X, followed by isobaric cooling at a rate of 2.5 K/ns. The systems are then equilibrated followed by the production run for calculating the correlation function. The dipoles in PE−NH2 system relaxes completely at 500 K, while PE−NO2 and PE− SH systems were heated to 600 K for the dipoles to relax completely. Figure 4a compares the dipole correlation functions (obtained from tTS, see Supporting Information) of various
Figure 4. Dielectric correlation function for (a) various PE−X systems obtained from time temperature superposition and (b) the calculated dielectric loss from the master curves.
PE−X systems. To calculate the dielectric loss and average relaxation time, we fit the obtained correlation function to a stretched exponential function. The parameter used for the fits are shown in Table 1. The average relaxation time is calculated Table 1. Dipole Moment and Streched Exponential Fitting Parameters for the DCF for Various Systems dipole moment (D) PE−OH PE−NH2 PE−SH PE−NO2
2.05 1.65 1.82 2.57
τ
τ (ns) 2.0 9.7 1.1 2.6
× × × ×
β
10−1 10−3 10−1 103
0.17 0.49 0.35 0.23
τavg (ns) 2.19 2.04 5.48 1.05
× × × ×
100 10−2 10−1 105
( ). The average relaxation is the fastest for
using τavg = β Γ
1 β
PE−NH2 closely followed by PE−SH and PE−OH, while PE− NO2 is orders of magnitude higher. It is important to recognize this big difference between PE−OH, which relaxes in nanoseconds, while PE−NO2 requires several microseconds; the latter is inaccessible using regular all-atom molecular dynamics simulation and is only achieved through the use of the tTS approach that is proposed in this work. As described earlier, the fitted stretched exponential function is used to calculate the dielectric loss; the resulting character202
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PE−NO2. We see that the relaxation time (inverse of the loss peak) of the nitro group for the samples are similar providing a second validation of our tTS based calculation. We acknowledge that the structure of PNO3EMA is slightly different from PE−NO2 due to the presence of a longer side chain, and to investigate the effect of the side group length we calculate the loss due to the addition of spacer groups in PE−OH. We use the PE−OH at 500 K as the initial structure and add C4H8 as a spacer between the backbone and − OH and equilibrated for 2 ns to create a melt structure of PEC−OH. To calculate the dielectric loss, we follow the same procedure as for PE−OH. The obtained dielectric loss for PEC−OH is found to be comparable to PE−OH (Figure 5b). These results are compelling though more experimental results are necessary to give us confidence that this methodology is a reliable means of simulations for practically interesting systems. The dielectric loss has been associated with the size and the dipole orientation for small molecules.22 To examine the applicability of this idea we calculate the dipole moment of each polar group (CH2−X) from the trajectories of the simulations (Table 1). The relaxation time is found to exponentially grow with the total dipole moment of the system, thus, suggesting that a possible path to designing materials with lower loss is by keeping the dipole moment of an individual polar group small but increasing their concentration, while they are still spatially well-distributed. In summary, inspired by previous experimental and simulation results for PE−OH,4,7 we investigated the effect of other polar groups on the relaxation of polyethylene. To fully explore the relaxation behavior of these substituted PEs we ran a simulation to microseconds. We also developed a tTS based scheme for capturing the same relaxation process, but with two orders of magnitude lower computational time requirements than a long simulation at a single temperature. We find that the two flavors of MD simulations give the same relaxation behavior thus providing us with a significantly faster and accurate method for calculation of dielectric properties. We extend this technique to predict the dielectric loss of PE−NH2, PE−SH, and PE−NO2.
istic time for local relaxations determines the location of the dielectric loss peak. Figure 4b shows the calculated dielectric loss for PE−OH, PE−NH2, PE−SH, and PE−NO2 obtained from the master curves in Figure 4a. We observe that the loss peaks for PE−NH2 and PE−SH are at 1011 and 1010 Hz, respectively, and the loss rapidly drops off to insignificant values on reaching relevant operational frequencies of MHz. Although PE−OH has peak dielectric loss at 108 Hz, which is orders of magnitude higher than PE−NH2 and PE−SH, the loss drops to less than 1% at the frequency of interest, namely, 105−106 Hz. The PE−NO2 has a loss peak very close to relevant operational frequencies, thus making it inappropriate for applications. It is hard to validate the high dielectric loss prediction of PE−NO2 due to the limited experimental exploration of this class of polymers. Borisova21 compared the mobility of sidechain nitro groups in a class of polyolefin-based polymers. PVNO3 is most similar to the PE−NO2 among the polymers studied experimentally. At 300 K and 1 kHz, the dielectric loss for PE−NO2 is predicted to be 0.03, while experiments on PVNO3 find 0.08, which is a reasonable agreement. Due to lack of experimental data, we cannot compare the dielectric loss at other frequencies. However, the frequency dependence of dielectric loss is available for a different polymer PNO3EMA. Figure 5a shows the experimental dielectric loss of PNO3EMA21 along with the predicted dielectric loss for the
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ASSOCIATED CONTENT
* Supporting Information S
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acsmacrolett.6b00978. Dipole correlation function of 4.2 mol %, PE−NH2, PE− SH, and PE−NO2 at various temperatures (PDF).
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. ORCID
Mayank Misra: 0000-0002-2700-1228 Sanat K. Kumar: 0000-0002-6690-2221 Notes
The authors declare no competing financial interest.
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Figure 5. (a) Comparison of dielectric loss of 4.2 mol % PE−NO2 with experimentally calculated dielectric loss of PVNO3 (green circle) and PNO3EMA (blue line).21 (b) Calculated dielectric loss of 4.2 mol % PE−OH with (blue line) and without (purple line) C4H8 as a spacer between the backbone and side group.
ACKNOWLEDGMENTS The authors acknowledge financial support for this work through a Multidisciplinary University Research Initiative (MURI) grant from the Office of Naval Research under 203
DOI: 10.1021/acsmacrolett.6b00978 ACS Macro Lett. 2017, 6, 200−204
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ACS Macro Letters Contract No. N00014-10-1-0944. The simulations presented in this paper were performed on Yeti HPC cluster system maintained by the Columbia University Information Technology Center.
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REFERENCES
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