Letter pubs.acs.org/macroletters
Understanding the Ionic Conduction in Dielectric Polymers at High Electric Fields Using Molecular Dynamics Simulations Yanhui Huang,*,† Joakim Jam ̈ beck,‡ and Mikael Unge‡ †
Department of Materials Science and Engineering, Rensselaer Polytechnic Institute, 110 Eighth Street, Troy, New York 12180, United States ‡ ABB AB, Corporate Research, SE 721 78, Västerås, Sweden S Supporting Information *
ABSTRACT: Applying molecular dynamics, we have studied ionic transport in polyethylene at moderate and high electric fields. The ion mobility of a variety of species is calculated and compared with existing theories. It reveals that ion mobility starts to deviate from the Einstein relation and increase roughly linearly with field beyond 100 MV/m, which results in a superlinear increase of the ionic current at high field that is consistent with the experimental results. At high field, we argue that the accelerated ion molecule is able to facilitate the conformation change of the surrounding polymers through elastic scattering, which in turn accommodates its passage. The extent of the mobility enhancement depends on the mass and size of the ion molecule. The ion transport is a thermally activated process, but has smaller activation energy than diffusion due to the additional energy provided by the field. polyethylene, including Na+, H3O+, H2O+, O2−, OH−, CH3+ (polymer dissociates), cumylalcohol+ (cross-link byproduct), and anthracene+ (additive voltage stabilizer). Cumylalcohol and anthracene are large molecules that contain 9 and 14 carbon atoms respectively and their chemical structures are shown in Figure S1 in the Supporting Information. The mobility values are derived and compared with existing theories. The molecular dynamics simulation is performed using classical Verlet algorithm7 in LAMMPS.8,9 An OPLS-AA force field optimized for long hydrocarbons with a time step of 1 fs was used.10,11 An amorphous polyethylene system containing 16 linear polymer chains with 50 CH2 groups each were prepared by compressing the chain under high pressure followed by annealing and equilibration. A partial charge is assigned to each atom to capture the columbic interaction. The partial charge for polyethylene is taken directly from the force field.10,11 Additional DFT calculations were performed to derive the partial charges on the ion molecules. The diffusion constant and drift velocity is analyzed from the mean-squaredisplacement of the ion molecule during a total simulation time
Dielectric polymers can contain an appreciable amount of impurities, including adsorbed gas or water molecules, inorganic salts picked up from the environment, deliberately added chemical agents, dissociated molecular segments, and so on.1 These small molecules or atoms can accept or lose an electron to become charged and drift under electric field, contributing to an ionic current. The mobility and transport behavior of these ionic species, however, are rarely reported due to the difficulties in distinguishing the ionic current from its electronic counterpart. One conventional method is to deduce the ion mobility from the diffusion constant based on the Einstein mobility relation,2
μ=
Dq kBT
(1)
where μ is the charge mobility, D is the diffusion constant, q is the charge quantity, kB is the Boltzmann constant, and T is the temperature. But the validity of this relation on ionic conduction in polymer dielectrics is not well scrutinized. There has been impressive progress in recent years using computer simulations to aid dielectric material design and understand microscopic mechanisms.3−6 In this work, we applied molecular dynamics simulation to study the transport behavior of a variety of ionic species that are typically present in © XXXX American Chemical Society
Received: January 24, 2017 Accepted: May 5, 2017
571
DOI: 10.1021/acsmacrolett.7b00054 ACS Macro Lett. 2017, 6, 571−574
Letter
ACS Macro Letters of 10 ns. The details of the simulation method and analysis are described in the Supporting Information. The diffusion of the ion molecule is well described by the random walk model and the diffusion constant and drift velocity can be well calculated from the mean square displacement (MSD). The relevant results and discussion are included in the Supporting Information. The computed diffusion constant of O2− is 8.3 × 10−10 m2/s, which falls into the experimental range of 10−10 m2/s.12,13 The drift velocity of O2− is found to be a constant at a time scale greater than 500 ps, and the value is plotted in Figure 1, which reveals a
Figure 2. Electrical mobility of a variety of ions as a function of field. The lines are linear fit and only appear curved as the mobility is plotted in a log scale.
polymer chain constraints, resulting in a long relaxation time (normally at MHz). The relative ionic dielectric constant due to dipole reorientation only increased by 0.017 over 10 ns at 1000 MV/m. The polarization effect should not impact ion drift significantly. The reason for the electrical mobility enhancement and deviation from the Einstein relation at high field can be understood from the energy perspective. The drifting ion is constantly scattered by polymer segments, and the mean free path is down to ∼2 Å.15 So, at small and moderate electric fields, the energy gained by the ion from the electric field in one scattering event is small compared to the thermal energy of kBT = 0.026 eV at room temperature, and the impact on the thermal motion is negligible. The field for the energy gain to reach kBT is ∼100 MV/m. At this field and beyond, the accelerated ion can impart appreciable energy to the polymer segments upon collision and brings the system out of equilibrium. Einstein relation, however, is based on the assumption that the system is in equilibrium and the thermal energy follows the Maxwell− Boltzmann statistics. So, the relation breaks down at high electric fields. Another theory that accounts for the field effect in ionic conduction in solids was developed originally for the ions or vacancies moving in crystals.16 The carrier is assumed to be located in a potential well and the electric field assists the escape of the carrier by reducing the potential barrier. A sinh law of the mobility is derived based on this model,16
Figure 1. Velocity (square) and mobility (circle) of the O2− as a function of the field. The mobility value derived from Einstein relation is also marked in the plot for comparison.
superlinear increase with the field. Correspondingly, the mobility values only follow the Einstein relation at low fields and start to increase nearly linearly at higher fields. Similar behaviors are also found for other molecules and results are summarized in Table 1 and plotted in Figure 2. The error bars were calculated based on the block average method. Note that values for high fields and small molecules are generally more accurate as the molecule can visit more places in the cell within the limited simulation time. And the data fluctuation at low field is more likely due to statistical errors. For H2O+, there appears a sudden “drop” of mobility at low field compared to the Einstein relation and this is beyond the error bar scale. Whether this is a real effect or a statistical error remains to be further examined. The data show that the mobility generally decreases with the molecule size, and all molecules show a deviation from the Einstein relation at higher field. The onset field of 50−100 MV/m for superlinear velocity increase is consistent with the experimental data14 of superlinear current increase in polyethylene. Though a partial charge is assigned to the C and H atom in the polymer, the dipole alignment with the field does not appreciably change the polymer morphology. The dipole of C− H bond is very small and it cannot rotate freely due to the
μ=
⎛ −ΔG ⎞ ⎛ Eqa ⎞ 2av0 exp⎜ ⎟ ⎟sinh⎜ E ⎝ kBT ⎠ ⎝ 2kBT ⎠
(2)
Here a is the width of the potential well, v0 is the phonon frequency, and ΔG is the potential barrier. The model is based on a very crude physical picture and cannot be directly related to classical quantum mechanics. For instance, the model does not state explicitly how the potential barrier is reduced by the field, but the actual kinetic process does matter and underlines
Table 1. Computed Electrical Mobility of Ion Molecules as a Function of Field mobility (10−8 m2/V s) E (MV/m)
Na+
CH3+
H3O+
H2O+
O2−
OH−
cumyl-alcohol+
anthracene+
Dq/kT 50 100 500 1000
12 ± 3.2 8.9 ± 1.2 11 ± 0.9 17 ± 0.4 32 ± 0.6
3.9 ± 0.9 4.1 ± 1.3 5 ± 0.4 7.6 ± 0.4 1.4 ± 0.7
7.5 ± 1.4 5.6 ± 1.3 5.8 ± 0.8 10 ± 0.3 17 ± 0.8
7.1 ± 2.5 2.7 ± 0.9 3.5 ± 0.7 6.2 ± 0.8 13 ± 0.4
3.2 ± 0.6 3.8 ± 1.0 5.3 ± 0.8 9.7 ± 0.3 16 ± 0.6
4.9 ± 1.1 5.2 ± 0.7 5.2 ± 0.8 11 ± 0.7 17 ± 5.6
0.45 ± 0.11 0.99 ± 0.53 0.61 ± 0.17 1.2 ± 0.12 1.8 ± 0.08
0.64 ± 0.39 NA 0.54 ± 0.17 0.9 ± 0.12 1.1 ± 0.10
572
DOI: 10.1021/acsmacrolett.7b00054 ACS Macro Lett. 2017, 6, 571−574
Letter
ACS Macro Letters
process and the data fit into an Arrhenius type of behavior of μ = A exp(−Ea/kBT), with Ea being the activation energy and A being the coefficient. This is shown in Figure 4. The activation
the quantitative relationship. For the case of amorphous polymers, the model is only qualitatively correct and the quantitative prediction is off: the formula reduces to a field independent form at low field that is consistent with the Einstein mobility relation, while at high field, though the model has predicted an increase of mobility with field, a relationship of μ ∝ sinh(E)/E is not observed in the simulation, in which the mobility increases rather linearly with field, in contrast to the higher order increase predicted by the model (sinh(x)/x = 1 + x2/6 + O(x4)). Here in amorphous polymers, we argue that the electric field reduces the energy barrier in a way of facilitating the conformation change of interacting polymer chains to accommodate the ion passage, which is achieved through accelerated ion bombardment. It is challenging from statistical analysis to prove that there is a significant change in the dihedral angle of the interacting polymers under high field. But this is not surprising because even with the help of the large electric field, the chance of successful ion passing is still very limited, less than 1 per 1000 trials. As the conformation of polymers also fluctuates thermally and that any change is transient and only last for a few tens of femtoseconds, seeking statistical significance in this sense could be very difficult. But we did find some indirect evidence that support this argument, which are discussed in the following text. By this hypothesis, the energy transferred from the ion molecule to the polymer not only depends on the field but is also affected by the mass and size of the molecule. It can be easily derived from momentum and energy conservation that the polymer segment gains most energy when the impact ion has the same mass as it. Here we define a dimensionless number η = (dμ/dE)/μ0 to characterize the extent of enhancement by field on ion mobility, with dμ/dE being the slope and μ0 being the y-axis intercept of the fitted line of μ − E plot. The results are summarized in Figure 3. For small ions
Figure 4. Log of O2− mobility as a function of temperature at different electric fields. The zero electric field data is calculated from the diffusion constant by the Einstein relation.
energy represents the energy required to change the polymer conformation to accommodate the ion molecule passage. It is worth noting that the activation energy decreases with the electric field, as the field provides additional energy to help overcome the energy barrier. As shown in Figure 5,
Figure 5. Activation energy of O2− and cumylalcohol+ as a function of field.
cumylalcohol exhibits greater activation energy than O2− due to its large size and the extent of reduction by field is also smaller. This supports the reasoning above that polymers interacting with large molecules receive less energy. Figure 5 also shows that for both molecules, the marginal reduction in activation energy decreases with the field magnitude, and this is contradictory to the trend predicted by eq 2. A possible reason is that, at low field, the molecule bombards the polymer multiple times before crossing over, and the total energy imparted is an accumulated result. To exclude possible artifacts associated with the system size and the use of thermostat, the simulations of O2− and cumylalcohol+ are also performed on a much larger system that contain 4× the number of polymers with 10× longer chain length under NVE condition, but no qualitative differences were found (see Figure S6 in the Supporting Information). Employing longer chains decreases the diffusion constant and mobility of the molecular ions by some extent, but the influence is small for small ions like O2−. The activation energy of O2−
Figure 3. Dimensionless number η that characterizes the impact of electric field on the ion mobility for different ions. CA+ stands for cumylalcohol+ and An+ stands for anthracene+.
with similar sizes, O2− show largest η due to its large mass of 32 g/mol, compared to the mass of 56 g/mol (4 CH2 units) of the polymer segment that comes into interaction. For large ion molecules like cumylalcohol and anthracene, multiple polymer segments can interact with the molecule simultaneously, so the mean free path is smaller and the transferred kinetic energy is diluted by multiple chains, which together yield small η for these molecules. The temperature dependence of ion mobility is also analyzed. Either diffusion or conduction show as a thermally activated 573
DOI: 10.1021/acsmacrolett.7b00054 ACS Macro Lett. 2017, 6, 571−574
ACS Macro Letters
■
diffusion constant increases to 0.32 eV in the large system, but both values are close to the experimental data of 0.25 eV.17 Any simulation work has its limitations when compared to experimental reality, and this work is no exception. Modeling dynamics in a polymeric system is often difficult due to the extremely long relaxation times and that the fully relaxed state is difficult to achieve. To expedite the computation, in this study, we rely on a relatively small system with short chains that can lead to an overestimation on diffusion constant and mobility values due to the enhanced chain mobility. For the large system validation, the polymer matrix may be less relaxed and the polymer motion may convolute with the ion drift. But these effects are expected to happen in an equal manner for all systems. Therefore, though the absolute numbers may be affected, the relative values should not be, that is, the trend is intact and the arguments and analysis made above should still be valid. In summary, molecular dynamics simulations are performed to study the ionic charge transport in polyethylene at an increasing field up to 1000 MV/m. A variety of ionic species were investigated, including water, dissociated polymer segments, cross-link byproducts, and so on. The results reveal that the ion mobility starts to deviate from the Einstein mobility relation and increases linearly with field beyond 100 MV/m, a point when the electric field is sufficient to accelerate the ion to an energy ≥kBT and bring the system out of thermal equilibrium, rendering the Einstein relation inapplicable. The conventional ionic conduction model based on the picture of ion escaping from a potential well in a regular lattice addresses the field effect, but fails to accurately describe the quantitative relationship between mobility, field, and temperature for this case. A tentative explanation is that the change of interacting polymer conformation is facilitated through constant high energy ion bombardment that, in turn, accommodates the ion passage. The effect of electric field on the ion mobility then depends on how much energy the local polymer segments can gain from the field through elastic scattering, and a relationship with the mass and size of the ion molecule is indeed found. Small ions with matching masses to the polymer segments show most significant increase of mobility, while large molecules like anthracene exhibit small increase. In addition, the high field mobility is less sensitive to temperature compared to pure diffusion, as the field reduces thermal energy required to overcome the activation energy barrier.
■
REFERENCES
(1) Teyssedre, G.; Laurent, C. Charge Transport Modeling in Insulating Polymers: From Molecular to Macroscopic Scale. IEEE Trans. Dielectr. Electr. Insul. 2005, 12 (5), 857−875. (2) Einstein, A. Ü ber Die Von Der Molekularkinetischen Theorie Der Wärme Geforderte Bewegung Von in Ruhenden Flüssigkeiten Suspendierten Teilchen. Ann. Phys. 1905, 322 (8), 549−560. (3) Le Roy, S.; Baudoin, F.; Griseri, V.; Laurent, C.; Teyssedre, G. Space Charge Modeling in Electron-Beam Irradiated Polyethylene: Fitting Model and Experiments. J. Appl. Phys. 2012, 112 (2), 023704. (4) Misra, M.; Mannodi-Kanakkithodi, A.; Chung, T. C.; Ramprasad, R.; Kumar, S. K. Critical Role of Morphology on the Dielectric Constant of Semicrystalline Polyolefins. J. Chem. Phys. 2016, 144 (23), 234905. (5) Webb, M. A.; Savoie, B. M.; Wang, Z.-G.; Miller Iii, T. F. Chemically Specific Dynamic Bond Percolation Model for Ion Transport in Polymer Electrolytes. Macromolecules 2015, 48 (19), 7346−7358. (6) Sato, M.; Kumada, A.; Hidaka, K.; Hirano, T.; Sato, F. Quantum Chemical Calculation of Hole Transport Properties in Crystalline Polyethylene. IEEE Trans. Dielectr. Electr. Insul. 2016, 23 (5), 3045− 3052. (7) Van Gunsteren, W.; Berendsen, H. Algorithms for Macromolecular Dynamics and Constraint Dynamics. Mol. Phys. 1977, 34 (5), 1311−1327. (8) Plimpton, S. Fast Parallel Algorithms for Short-Range Molecular Dynamics. J. Comput. Phys. 1995, 117 (1), 1−19. (9) http://lammps.sandia.gov. (10) Siu, S. W.; Pluhackova, K.; Böckmann, R. A. Optimization of the Opls-Aa Force Field for Long Hydrocarbons. J. Chem. Theory Comput. 2012, 8 (4), 1459−1470. (11) Kaminski, G. A.; Friesner, R. A.; Tirado-Rives, J.; Jorgensen, W. L. Evaluation and Reparametrization of the OPLS-AA Force Field for Proteins via Comparison with Accurate Quantum Chemical Calculations on Peptides. J. Phys. Chem. B 2001, 105 (28), 6474− 6487. (12) Zeman, S.; Kubik, L. Evaluation of Oxygen Permeability of Polyethylene Films. Tech. Sci. 2012, 15, 331−345. (13) Evans, N.; Quinton, T. Permeability and Diffusion Coefficient of Oxygen in Membranes for Oxygen Electrodes. Respir. Physiol. 1978, 35 (1), 89−99. (14) Montanari, G.; Mazzanti, G.; Palmieri, F.; Motori, A.; Perego, G.; Serra, S. Space-Charge Trapping and Conduction in LDPE, HDPE and XLPE. J. Phys. D: Appl. Phys. 2001, 34 (18), 2902. (15) Serra, S.; Tosatti, E.; Iarlori, S.; Scandolo, S.; Santoro, G. Interchain Electron States in Polyethylene. Phys. Rev. B: Condens. Matter Mater. Phys. 2000, 62 (7), 4389. (16) Dissado, L. A.; Fothergill, J. C. Electrical Degradation and Breakdown in Polymers; The Institution of Engineering and Technology: London, U.K., 1992. (17) Kurek, M.; Klepac, D.; Ščetar, M.; Galić, K.; Valić, S.; Liu, Y.; Yang, W. Gas Barrier and Morphology Characteristics of Linear LowDensity Polyethylene and Two Different Polypropylene Films. Polym. Bull. 2011, 67 (7), 1293−1309.
ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acsmacrolett.7b00054.
■
Letter
Supplementary figures, tables, and detailed description of the simulation and analysis methods (PDF).
AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. ORCID
Yanhui Huang: 0000-0002-1807-2737 Notes
The authors declare no competing financial interest. 574
DOI: 10.1021/acsmacrolett.7b00054 ACS Macro Lett. 2017, 6, 571−574
