Simulation of a small molecule analogue of a lithium ionomer in an external electric field Sara M. Waters, John D. McCoy, Amalie L. Frischknecht, and Jonathan R. Brown Citation: The Journal of Chemical Physics 140, 014902 (2014); doi: 10.1063/1.4855715 View online: http://dx.doi.org/10.1063/1.4855715 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/140/1?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Ultrafast coherent control of giant oscillating molecular dipoles in the presence of static electric fields J. Chem. Phys. 139, 084306 (2013); 10.1063/1.4818878 Electropumping of water with rotating electric fields J. Chem. Phys. 138, 154712 (2013); 10.1063/1.4801033 Polarization and interactions of colloidal particles in ac electric fields J. Chem. Phys. 129, 064513 (2008); 10.1063/1.2969103 Orientation of dipole molecules and clusters upon adiabatic entry into an external field J. Chem. Phys. 129, 024101 (2008); 10.1063/1.2946712 Manipulation of slow molecular beams by static external fields J. Chem. Phys. 125, 133501 (2006); 10.1063/1.2202829

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THE JOURNAL OF CHEMICAL PHYSICS 140, 014902 (2014)

Simulation of a small molecule analogue of a lithium ionomer in an external electric field Sara M. Waters,1 John D. McCoy,1,a) Amalie L. Frischknecht,2 and Jonathan R. Brown1,b) 1

Department of Materials Engineering, New Mexico Institute of Mining and Technology, Socorro, New Mexico 87801, USA 2 Sandia National Laboratories, Albuquerque, New Mexico 87185, USA

(Received 17 October 2013; accepted 9 December 2013; published online 6 January 2014) We have investigated the ion dynamics in lithium-neutralized 2-pentylheptanoic acid, a small molecule analogue of a precise poly(ethylene-co-acrylic acid) lithium ionomer. Atomistic molecular dynamics simulations were performed in an external electric field. The electric field causes alignment of the ionic aggregates along the field direction. The energetic response of the system to an imposed oscillating electric field for a wide range of frequencies was tracked by monitoring the coulombic contribution to the energy. The susceptibility found in this manner is a component of the dielectric susceptibility typically measured experimentally. A dynamic transition is found and the frequency associated with this transition varies with temperature in an Arrhenius manner. The transition is observed to be associated with rearrangements of the ionic aggregates. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4855715] I. INTRODUCTION

Ion-containing polymers form an important class of high performance materials.1–3 In particular, ionomers, which are copolymers containing relatively small fractions of functionalized ionic groups, have useful mechanical and electrical properties. A well-known commercialization is Dupont’s Surlyn, which, for example, has found extensive application as golf ball covers.4 The enhanced mechanical properties are due to aggregation of the ionic groups, which act as thermoplastic cross-links. In addition, these polymers display the intriguing property of being “self-healing,” and, for example, are able to seal over small punctures.5–7 Melt ionomers are also of interest for their possible application as electrolytes in batteries.8 Because ionomers are single-ion-conductors, they have the potential for high efficiency as electrolytes and also have the advantage of being chemically stable and nonflammable. However, current ionomers do not have sufficient ionic conductivity for use as electrolytes, presumably due to the ionic aggregates. For a general review of ionomer properties see Refs. 9 and 10. The detailed nature of the ionic aggregates and their effects on ion transport are difficult to quantify experimentally. Recently, progress has been made in understanding the ionic aggregates in a series of precise poly(ethylene-co-acrylic acid) (PEAA or EAA) copolymers and ionomers, synthesized by acyclic diene metathesis (ADMET) polymerization.11 These materials are especially helpful for fundamental studies because the acid groups occur along the polymer backbone separated by a precise spacing of 9, 15, or 21 carbons between each acid. These polymers are similar, both chemia) Author to whom correspondence should be addressed. Electronic mail:

[email protected]

b) Present address: William G. Lowrie Department of Chemical and

Biomolecular Engineering, The Ohio State University, Columbus, Ohio 43210, USA. 0021-9606/2014/140(1)/014902/9/$30.00

cally and physically, to ethylene-methacrylic acid copolymers (EMAA) of which Surlyn is an example. In addition Dupont produces12 un-neutralized resins of EMAA (and EAA) under the name of Nucrel. As one would expect, these products are random copolymers. For the precise EAA ionomers, the exact spacing between acid groups leads to more ordered systems and also allows easier comparison between experiment and simulations. In the melt, both PEAA copolymers and PEAA ionomers formed by neutralizing the acid groups with a metallic cation (Li+ , Na+ , Cs+ , or Zn+2 ) form strong ionic aggregates. The existence of the aggregates in ionomers shows up in X-ray scattering as a broad low wave-vector peak, known as the ionomer peak.1 Winey and co-workers have extensively studied the ionic aggregate morphology in PEAA ionomers neutralized with Zn, using both X-ray scattering and STEM imaging.13 They found roughly spherical ionic clusters, with core sizes of about 0.5 nm, as verified by fits to the X-ray scattering and also by direct imaging. On the other hand, a series of coarse-grained14–16 and atomistic17, 18 molecular dynamics (MD) simulations of PEAA ionomers have shown that there is a wide variety of possible ionic aggregate morphologies, depending on the cation type, the spacer length between acid groups in the polymer backbone, and the degree of neutralization. The simulated scattering structure factor is in good agreement with available experimental data across a broad range of conditions. In most of the systems studied, the aggregates tend to be long and “stringy” rather than spherical. At lower cation concentrations and larger spacer lengths, the aggregates tend to be isolated from one another and more or less spherical. As the spacer length decreases and/or the neutralization increases, the aggregates start to form extended, branched networks that can percolate through the simulation box.17, 18 The detailed correlation between ionic aggregate morphology and ion dynamics is still an open question. Coarsegrained MD simulations showed that counterion diffusion

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is faster in ionomers with percolated morphologies than in those with isolated ionic aggregates.16 In general, the ionic associations in ionomers lead to long relaxation times and high viscosities. The various different relaxation processes involved in ionomer dynamics have been explored most thoroughly using broadband dielectric relaxation spectroscopy (DRS). Since dielectric spectroscopy typically shows many features seen in dynamic mechanical spectroscopic measurements, dielectric measurements have the advantage of being closely tied to the mechanical properties important in many commercial applications. Additionally, DRS allows extraction of the fraction of free ions and their mobility.19 The relationship between morphology (as determined by Xray scattering and STEM imaging), and dynamics (as determined by DRS) was recently studied for a series of sulfonated polystyrene (SPS) ionomers by Runt, Winey, and co-workers.20–22 Similar studies, including quasi-elastic neutron scattering, have also been carried out for sulfonated poly(ethylene oxide) ionomers.23–26 These studies generally show a strong correlation between polymer segmental motion and ion mobility. In Zn-neutralized SPS ionomers, multiple segmental relaxation processes are observed, corresponding to restricted motions of the chains near ionic aggregates and relatively unrestricted motions in regions far from any aggregates.21 In SPS ionomers with monovalent cations, only a single broad segmental relaxation process was observed.22 Morphology and dynamics are coupled in the PEO-based ionomers; counterion and polymer segmental mobility increase with distance from long stringy aggregates found in these systems.26 Similar dynamical measurements have yet to be performed on the precise EAA ionomers described above. In this paper, we take a first step towards understanding relaxation processes in these materials using atomistic molecular dynamics simulations. The primary purpose of the current paper is to develop simulation methodology that will permit the molecular origins of dielectric response to be better understood. In particular, although the frequency range that is probed in the simulations is higher than is accessible experimentally, when combined with complementary experimental and simulation studies these methods would permit a more full understanding of ionomeric dielectric response. In addition, through time-temperature mappings, high temperature, high frequency simulation results can be used to make low temperature, low frequency predictions, at least roughly. Important quantities in such a mapping are the activation energies, EA , to different modes of motion. Due to the very slow dynamics of high molecular weight PEAA ionomers, we focus here on a melt of a short analogue of a PEAA ionomer, namely 2-pentylheptanoic acid, fully neutralized with lithium. These analogues, or oligomers, are not entangled and contain only one acid group, so we anticipate their dynamics will be enhanced compared to the corresponding polymer. In addition, by restricting the temperature range to be above the melting point of polyethylene, we avoid complications due to the semi-crystallinity sometimes observed in these materials. To the best of our knowledge, these are the first MD simulations of a dense molecular liquid of this kind in an AC electric field.

J. Chem. Phys. 140, 014902 (2014)

The remainder of the paper is as follows. A description of the model and methodology is presented in Sec. II. Section III reports our structure results; Sec. IV reports our dynamic results; and Sec. V contains a brief discussion.

II. MODEL AND METHODOLOGY

In the present study, we have modeled a melt of EAA oligomers. Each oligomer has one acid group that has been neutralized with a lithium counterion. The structure of the oligomer is shown in Fig. 1. The oligomers were 11 carbons long with a lithium neutralized acrylic acid (AA) group bonded to the sixth carbon (yielding a 18 EAA-mol.% or 36 wt.%27 ). The carbons at the chain ends were “end-capped” with an additional hydrogen to give terminal –CH3 groups. This molecule was chosen for study because it is the precursor molecule to the poly(ethylene-co-acrylic acid) (PEAA) copolymer with EAA groups separated by precisely nine carbons synthesized by Baughman et al.11 and studied in our previous work.17, 18 This oligomeric system has similarities and differences with EAA polymers. In both cases, the ion clusters form in a similar manner resulting in small angle ionomer peaks in the X-ray structure factors. In addition, the high frequency dynamics associated with the ion clusters are expected to be similar. On the other hand, the oligomers have only a single acid group per molecule and, consequently, are incapable of forming an ionically crosslinked network with polymers bridging between ion clusters. As a result, the long time (low frequency) dynamics of the oligomers and polymers are not expected to be similar. Molecular Dynamics (MD) simulations were performed with the LAMMPS open-source software28 using the Towhee Monte Carlo software29 to generate the initial molecular coordinates. The initial system studied consisted of 64 oligomers (2432 atoms including 64 lithium ions). This was later increased by a factor of 8 to a total of 512 oligomers (19 456 atoms including 512 lithium ions) and, unless otherwise specified, reported results will be for the larger, 512 oligomer system. The simulations used the fully atomistic OPLS-AA force field,29, 30 which includes bond, angle, dihedral, and nonbond Lennard-Jones interactions. Partial charges on each atom are also defined in the OPLS-AA force field, with Coulomb interactions given by UC = qi qj /(4π ε0 rij ), where qi is the partial charge on atom i, ε0 is the permittivity of free space, and rij is the distance between atoms i and j. The particle-particle particle-mesh (PPPM) solver was used for the electrostatic interactions. In all simulations a multipletime step integrator (RESPA) was used, so that forces were calculated every time step for bonded interactions, every two

FIG. 1. Structure of the oligomer studied, 2-pentylheptanoic acid.

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time steps for three- and four-body forces, every four time steps for nonbonded interactions, and every eight time steps for long-range electrostatic interactions. The system was initially constructed by placing the oligomers and lithium ions in random locations on a simple cubic lattice at a low density. The system was then isotropically compressed, resulting in a cubic box of 28.0 Å per side after 100.0 ps, at a temperature of 120 ◦ C. A 1.0 fs timestep (the maximum that could resolve the motion of the hydrogen atoms) was employed. Further equilibration resulted from a 1.0 ns run at a constant pressure of 1.0 atm and 120 ◦ C. Finally, the system was equilibrated at this constant volume (box length = 29.5 Å) and temperature for 50.0 ns. Equilibration was confirmed by a lack of drift in the total and potential energies over the final 10.0 ns. After the larger, 512 oligomer, system was constructed by replicating the 64 oligomer system 8 times, additional equilibration was conducted at this new size under a constant pressure of 1.0 atmosphere and temperature of 220 ◦ C for a runtime of 5.0 ns with a timestep of 0.5 fs. The resulting simulation box length was 61.28 Å corresponding to a density of 0.0845 atoms/Å3 . Both this timestep and box size were held fixed throughout the remainder of the study. Notice that the larger system size was necessary to capture the clustercluster correlations which occur at roughly 20 Å, dictating a box length greater than 40 Å. We found that when the electric field is applied, the ions heat rapidly. To prevent this problem, we applied the NVE ensemble to the lithium ions, and the NVT ensemble to all the other atoms using the Nosé-Hoover thermostat with a damping constant of 100 timesteps (50 fs). In order to drive ionic motion, static and oscillating electric fields F in the z-direction were employed by adding a force qF to each atom. Static electric field strengths were selected ranging up to 0.5 V/Å. Typical production runs were 10–20 ns. These simulations provided information for the low frequency limit of the oscillating field studies discussed below. Sinusoidal oscillating electric fields were used to probe the dynamic response of the ions following a methodology similar to that employed in our work on the dynamic heat capacity.31, 32 These runs were started from the zero-field equilibrated system and only the steady state response was used in the calculations. Although trial runs of varying field strengths were performed, the bulk of the runs were for a field of 0.25 V/Å, which was a compromise between a field strong enough to induce a response distinct from thermal noise and a field not strong enough to induce large nonlinear effects. The imposed oscillating electric fields, F, followed the form: F = F0 sin (2π t/τ ) ,

roughly, a logarithmic spacing. Reported values of frequencies in this paper are angular frequency ω, which is ω = 2π /τ . The direct interaction of the electric field with the ionomer is primarily with the ions (lithium and oxygen). The system response is primarily a polarization and eventual conduction of the ions. We track the system response through the coulombic energy EC which we assume to take the form: EC = C0 + C1 P F,

(2)

where C0 is the non-oscillatory contribution to the energy, C1 is a proportionality constant for the oscillatory contribution, F is the imposed electric field, and P is the polarization. The energy EC does not include the energy stored in the electric field itself, and, consequently, only the polarization contribution to the displacement field contributes here. It is assumed that the variation of EC is proportional to the electronic work done on the system. Because the coulombic contribution to the energy is not the only energy in the system and because the system is thermostated (permitting dissipative effects), the variation in the coulombic energy is not in general directly equal to the electronic work. However, we claim that for small electric fields, this variation should be linear in the electronic work. In this linear response region, the polarization is proportional to the electric field although phase lagged. Consequently, we assume that the coulombic energy per mole can be expressed as EC = C0 + χ F02 sin ((2π t/τ ) − δ) sin (2π t/τ ) ,

(3)

where we have defined χ as the magnitude of the coulombic susceptibility expressed on a per-mole basis (“moles” refers to total moles of atoms in the simulation). This would be the magnitude of the dielectric susceptibility times half the permittivity of free space if the total energy were tracked (i.e., 1 ε χ ). The storage (or in-phase) contribution, χ  , is given 2 0 D by χ  = cos(δ)χ and the loss (or out-of-phase) contribution, χ  , is given by χ  = sin(δ)χ . It will be shown in the next few figures that Eq. (3) accurately describes the simulation results. In Figs. 2 and 3 for a temperature of 250 ◦ C, the coulombic energy is shown as a function of time and of field. In Fig. 2, the coulombic energy for zero external field is shown along with that of a static, non-oscillatory field of F = 0.25 V/Å and of oscillatory fields of a high (8.98 × 1012 rad/s) and

(1)

where F0 is the field amplitude, t is the time, and τ is the period. Oscillating electric field simulations were conducted over a wide range of temperatures: 160, 190, 220, 250, 280, 310, 340, 370, 400, 450, and 500 ◦ C. All temperatures were well above the glass transition temperature of 22 ◦ C for the analogous PEAA copolymer.11 Run times for these simulations were 10–50 ns depending on the oscillatory period. The periods examined ranged from τ = 500 fs to 23 ns on,

FIG. 2. The raw coulombic energy for runs at 250 ◦ C with electric fields of 0 V/Å (blue), 0.25 V/Å (red), and an oscillating electric field (green) with F0 = 0.25 V/Å and frequencies of (a) 8.98 × 1012 rad/s and (b) 1.53 × 1010 rad/s.

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Average Coulombic Energy (Mcal/mol)

(a)

(b) -62.2

-63.80

C

low frequency (1.53 × 1010 rad/s), both with amplitude F0 = 0.25 V/Å. It is seen that the static electric field increases the potential energy by about 2.5 Mcal/mole from a zero field value of −64 Mcal/mole. Because the low frequency (large period) limit of the oscillatory response would correspond to switching between static fields of F = 0 and F = 0.25 V/Å, one would anticipate an EC amplitude of half the increase in the change in the columbic energy seen in the static field cases (i.e., an amplitude of χ F0 2 = 1.25 Mcal/mole). In addition, the low frequency limit of C0 is the zero field value of −64 Mcal/mole and the low frequency limit of χ is (1.25/0.252 ) = 20 Mcal-Å2 /mole-V2 . When the field oscillates at high frequency, the system does not have time to respond and χ approaches zero while C0 approaches the low frequency limit. As the frequency decreases, C0 is seen to increase slightly before returning to its low frequency limit. In particular, notice that in Fig. 2(b) the minimum of the system response to the oscillatory field is higher than in the zero field case. The degree to which the linear response functional form captures the system response can be seen in the LissajousBowditch-like plots shown in Fig. 3. The lines are fits to the functional form of Eq. (3) while the points are the simulation results of Fig. 2 averaged over several hundred periods. The minimum value of the high frequency case (Fig. 3(a)) is seen to be −63.96 Mcal/mole and the maximum to be −63.77 Mcal/mole leading to χ  ∼ |χ | = 3.0 Mcal-Å2 /mole-V2 . The low frequency case (Fig. 3(b)) has a considerably higher minimum (−63.20 Mcal/mole) and maximum (−62.15 Mcal/mole) and a larger χ  ∼ |χ | = 16.8 McalÅ2 /mole-V2 . The system response is clearly phase lagged. If it were not, the energy would be a single valued function of the electric field and the curves would be simple parabolas rather than the wing-shaped functions observed. It should also be noticed that the low frequency example in Fig. 3(b) displays some flattening at low energy which is evidence of the onset of nonlinear effects. At this field strength, these nonlinear effects are minor; however, it is anticipated that higher field strengths would require a more complicated analysis to capture the richer system response.32 In Fig. 4, the effect of field amplitude on the susceptibility is shown. The amplitude of the variation in the coulombic

Amplitude of E Variation (kcal/mole)

014902-4

250

200

150

100

50

0 0

0.02 0.04 0.06 2 2 (Amplitude of Electric Field) (V/Å)

FIG. 4. Variation of the oscillatory component of the coulombic energy vs. F0 2 , at T = 250 ◦ C. From lower to upper lines, the frequencies are: 1260, 393, and 153 × 1010 rad/s.

energy (expected to be χ F0 2 in the linear response regime) is plotted as a function of F0 2 . As expected the amplitude is proportional to F0 2 , which, from Eq. (3), implies that the susceptibility χ is independent of F0 in the range studied. At low frequencies the behavior is not as linear indicating the onset of nonlinear effects. Larger electric fields would induce distinct nonlinear effects. The use of an electric field, F, is commonly used to probe ionic dynamics and the system response is often quantified in terms of the current density, J, instead of the polarization: J = σ (ω)F where σ (ω) is the frequency dependent conductivity. The dielectric susceptibility (times vacuum permeability) is related33 to the conductivity by σ (ω) − σ (0) ∝ iωχ D where χ D = (ε(ω) − ε0 )/(ε∞ − ε0 ), ε(ω) is the permittivity, ε0 is the zero frequency permittivity, ε∞ is the infinite frequency permittivity. In either case, non-integral power-law dependencies on frequency are often observed in amorphous materials.33 Recall that we have defined χ to be the coulombic contribution to χ D ε0 /2 on a per mole basis. III. RESULTS – STRUCTURE

As discussed in the Introduction, the lithium and carboxylic ions form ionic aggregates. A snapshot from our MD simulations at 220 ◦ C in the absence of an electric field is shown in Fig. 5(a). Here we show just the oxygen atoms in

-62.4 -63.85

-62.6

-63.90

-62.8 -63.0

-63.95

-63.2 -0.2 -0.1

0

0.1

0.2 -0.2 -0.1 Electric Field (V/Å)

0

0.1

0.2

(a) FIG. 3. The coulombic energy as a function of the electric field for runs with F0 = 0.25 V/Å, T = 250 ◦ C, and frequencies of 8.98 × 1012 rad/s (left) and 1.53 × 1010 rad/s (right). Dots indicate averaged data points and lines indicate the coulombic energy fit to Eq. (3). The fit parameters are: (a) C0 = −63.94 Mcal/mol, C1 = 0.189 Mcal/mol, and δ = 0.61622 and (b) C0 = −63.136 Mcal/mol, C1 = 1.0093 Mcal/mol, and δ = 0.51897.

(b)

Z FIG. 5. Snapshots from the simulations showing oxygen (red) and lithium (yellow) atoms, for (a) F = 0 at 220 ◦ C and (b) F = 0.25 V/Å at 280 ◦ C.

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10

Li-Li

10 Total

Li-O

10

Li-Li 8

8

6

6

6

4

4

2

2

0

0

g(r)

8

oligomer polymer

4 2

Li-O

O-O

8

8

6

6

4

4

2

2

0

0

5

10

15

20

25

0

5

10

15

20

25

0

r (Å)

FIG. 6. The total and partial g(r)s for three different atom combinations at 220 ◦ C with no electric field present. The heights of the first peaks are: total (8.9); Li-Li (22.1); O-O (26.3); Li-O (69.3).

red and the lithium atoms in yellow. The ions form extended, “stringy” aggregates with an alternating lithium-oxygen motif. The structure of these aggregates is very similar to those found in recent MD simulations of PEAA ionomers neutralized with lithium,17 which had four acid groups per chain as opposed to the single functional group in the oligomers studied here. Visual inspection shows that there are no free lithium ions over the course of our simulations. The local structure can be quantified through the pair correlation functions gij (r), where i and j denote the different species types. These pair correlations do not distinguish between sites on the same and on different chains. Results for the equilibrium system with no electric field are shown in Fig. 6 for pair correlations between Li-Li, Li-O, O-O, and the total where the total g(r) is a weighted average over the various species. The full height of the first peak is not shown so that the other peaks, and, in particular, the small broad peak at about 18 Å, can be seen more clearly. It is this peak at larger distances that is primarily responsible for the ionomer peak seen in scattering studies; this peak corresponds to correlations between the ionic aggregates. The total g(r) is dominated by the carbon-carbon and the carbon-hydrogen interactions because they greatly outnumber the lithium and oxygen atoms. Consequently, the total g(r) does not distinctly show the ionomer peak and it is the differential weighting of the scattering from the different atom types that permits the effect of ionomer structures to be seen in the structure factor. The local structure of the ionic aggregates, as primarily revealed by the Li-Li and Li-O g(r)’s, is very similar to that seen in the fully neutralized polymeric system, as shown in Fig. 7 (see also Fig. S6 in the supplementary material for Ref. 17). In particular, the short ranged (10 Å), the polymeric peak occurs at slightly smaller distance. The structure factor is proportional to scattering intensity and, consequently, is often used for direct comparison with experiment. The structure factor, S(k), can be determined from the pair correlation functions by a Fourier transform:  ci cj fi fj  ci fi2 + 4πρ S(k) = k i i,j  (4) × [r sin(kr)(gij (r) − 1)]dr, where this includes the self-scattering term. Here the ci are the concentrations of the atom types, ρ is the total number density, k is the wavevector, and the atomic form factors, fi , are taken from the work of Waasmaier and Kirfel.34 The structure factor corresponding to the g(r) shown in Fig. 6 is shown in Fig. 8. S(k) is seen to have a pronounced peak at 4 nm−1 . This is the ionomer peak that is associated with the ionic aggregates. The peak in the total S(k) near 13 nm−1 is related to the amorphous halo seen in scattering of polyethylene and PE-based ionomers. This peak comes from scattering between the aliphatic chains and other short-range interactions. The amorphous halo peak here is very similar to the amorphous halo seen in the fully neutralized polymeric system, while the ionomer peak occurs at a lower wavevector in the oligomeric system.35 This indicates that the ionic aggregates are spaced somewhat farther apart in the oligomeric 150 100

S(k)

g(r)

0

50 0

0

5

10 -1 k (nm )

15

20

FIG. 8. The total structure factor at 220 ◦ C (corresponding to the g(r) in Fig. 6) and with no applied electric field.

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160C 1.26x10

13

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rad/sec

160C 2.24x10 rad/sec

Y

160C 1.26x10

13

9

rad/sec

160C 2.24x10 rad/sec

Z

500C 1.26x10

13

rad/sec

9

X

500C 1.26x10

500C 2.24x10 rad/sec

FIG. 9. Lithium positions in the x-y plane for a single snap-shot for a high and low temperature (500 ◦ C and 160 ◦ C) and a high and low frequency (1.26 × 1013 rad/s and 2.24 × 109 rad/s) (the electric field is in the z-direction).

13

rad/sec

9

500C 2.24x10 rad/sec

X

FIG. 10. Lithium positions in the x-z plane for the same conditions as in Fig. 9.

IV. RESULTS – DYNAMICS

We first briefly discuss the effect of a static electric field F on the ion dynamics. Fig. 11 shows the mean-squared displacement (MSD) of the lithium ions for various values of F, at an intermediate temperature of 280 ◦ C. Fig. 11(a) shows the MSD in the field direction, while Fig. 11(b) shows the MSD in the direction perpendicular to the field. Clearly, increasingly strong fields drive increasing motion of the lithium ions. At zero field, the lithium ions move about 8 Å over 6 ns, far enough to have exchanged with a few of their nearest neighbors in an ionic aggregate (note a distance of 8 Å corresponds to the trough between the 3rd and 4th peaks in the Li-Li g(r) in Fig. 5). A small applied field of F = 0.01 V/Å does not noticeably enhance the lithium ion motion which remains highly sub-diffusive over the timescales simulated. For the two stronger fields, the ions move much faster. In steady state the MSD perpendicular to the field should be diffusive and therefore have a slope of 1, whereas the MSD parallel to the field should show displacements that are linear in the field 8

10 7 10 6 10 5 10 4 10 1000 100 10 1

2

MSD (Å )

system than in the polymeric system. Since there is only one anion per molecule in our system, the spacing between aggregates is not limited by the chain length, since chains do not “bridge” between aggregates as they do in the ionomers.14 The ionomer peak is also higher for the oligomers, indicating a larger degree of ordering than in the polymeric system. Having only one ionic group per chain leads to less disorder due to chain looping and bridging, and allows the ions to form more ordered aggregates. As might be expected, an applied electric field acts to align the aggregates in the direction of the field (z). Fig. 5(b) shows a snapshot of the aggregates in a static electric field of F0 = 0.25 V/Å at 280 ◦ C. The stringy aggregates are predominantly aligned along the z-direction, although there are significant excursions of aggregates into the perpendicular directions along with some branches linking different aggregates. The alignment of the aggregates persists in an oscillating electric field. To see this, we plot the x-y positions of each lithium ion in the entire simulation box for a single snapshot in Fig. 9, and the x-z positions of the lithium ions in Fig. 10. In Fig. 9, because the z-axis (which is also the direction of the electric field) is collapsed with all ions plotted as though at z = 0, the aggregate network appears denser than is really the case. For low frequency fields, the projection of the ions on to the x-y plane appears more tightly packed and, for the high temperature case, the ionic aggregates appear to be distributed on a regular grid. Viewing these snapshots in the x-z plane (Fig. 10), it is seen that the presence of a low frequency electric field has the effect of aligning the ion strands in the direction of the field. This is particularly pronounced in the high temperature case. Interestingly, the high frequency field has a much smaller effect on the ion alignment.

(a)

0.1

(b)

1

0.1 time (ns)

1

10

FIG. 11. MSD for the lithium ions at 280 ◦ C (a) in the field direction and (b) perpendicular to the field direction, as a function of time for F = 0 (blue dashed curves), 0.01 V/Å (tan curve), 0.25 V/Å (purple curve), and 0.5 V/Å (green curve). The dashed black lines have slopes of (a) 2 and (b) 1.

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25.5 ln(Onset Frequency) (rad/s)

7

10

log (χ')

6 5 4 3

9

10

11

12

13

log (frequency) (rad/s)

25.0

24.5

24.0

23.5 1.2

1.3

10

1.8

in ionomers and other conductive amorphous materials.33 For high temperatures, a glass-like transition is observed in the storage modulus. The frequency associated with this transition decreases as the temperature decreases. The location of the transition roughly follows an Arrhenius form, ln(ω) ∼ (−EA /RT) where EA is the activation energy and R is the gas constant. The Arrhenius fit to these transition frequencies is shown in Fig. 13 and yields an activation energy of EA = 6.01 kcal/mole (25.2 kJ/mole), which predicts that at room temperature the transition would occur at 108 rad/s. For low temperatures the transition frequency in our work occurs outside of the simulation range and only a power law is seen. In Fig. 14, the loss susceptibility is plotted in a manner similar to that of the storage susceptibility in Fig. 12. Here there is evidence of two transitions. More careful analysis of the storage susceptibility would, no doubt, show this as well. Both peaks move to the left as the temperature is decreased in keeping with the behavior of the transition seen in the storage susceptibility. As mentioned in the Introduction, there are no experimental DRS studies of precise EAA ionomers in the literature. Several previous DRS studies of the related EMAA ionomers37–39 and others40 were interpreted in terms of the Eisenberg-Hird-Moore (EHM) model41 of ionic multiplets

7

6

10

( z = z0 + vz t where vz is the velocity) so the MSD should have a slope of 2 on a log-log plot. For the data in Fig. 11, at F = 0.25 V/Å the lithium ion MSDs approach slopes of 0.94 and 1.8 for the perpendicular and parallel directions, respectively, while at F = 0.5 V/Å they approach slopes of 0.94 and 2.0. These stronger applied fields thus mobilize the ions enough that they nearly reach the diffusive regime perpendicular to the field after only 8 ns or so. In the rest of the paper we will focus on results with electric field amplitudes of F = 0.25 V/Å; in a static field of this amplitude the lithium ions move on the order of 34 Å over 6 ns perpendicular to the field, more than half the simulation box size, and about 400 Å in the field direction over 6 ns. Visual examination of the simulations reveals that even though the lithium ions move a considerable distance during the simulation, they are always found in an ionic aggregate, rather than as an isolated ion or ion pair. This is consistent with the ion dynamics observed in recent coarse-grained simulations of ionomers.16 The main result of this study is the dynamic response of the system to an imposed oscillatory electric field. Because the columbic energy is tracked, it is primarily the ion dynamics which are studied, although, of course, other aspects of the systems could be tracked (such as the Lennard-Jones energy) which would result in a quantity more sensitive to other aspects of the system dynamics (such as carbon or hydrogen motion). In Fig. 12 the storage contribution to the susceptibility is plotted over 4.5 orders of magnitude in frequency. Each curve is the result for a different temperature and the successive curves have been shifted vertically by 0.3 for clarity. The curves themselves are reasonable forms for this type of plot, but because the Kramers–Kronig relations were not enforced in the fitting of the storage and loss curves, these curves are intended as guides to the eye and not as a detailed analysis.36 Lower curves are at higher temperatures. Some trends are obvious. The high frequency limiting behavior of the storage contribution is a power-law with a power of negative 0.24 (a slope of −0.24 as plotted here). This terminal power law behavior of a fractional power is common

1.7

FIG. 13. The natural logarithm of the onset frequency (calculated from the low frequency edge of the location of the “step” from low to high frequency behavior) vs. inverse temperature. The slope of 3030 K−1 implies an activation energy of 6.01 kcal/mole.

log (χ'')

FIG. 12. The storage dielectric susceptibility for various temperatures (electric field amplitude of 0.25 V/Å). Successive curves are shifted by 0.3 on the y-axis (bottom curve unshifted). Temperatures from top to bottom: 160 ◦ C, 190 ◦ C, 220 ◦ C, 250 ◦ C, 280 ◦ C, 310 ◦ C, 340 ◦ C, 370 ◦ C, 400 ◦ C, 450 ◦ C, and 500 ◦ C. Susceptibility is in units of kcal Å2 /(mol V2 ).

1.4 1.5 1.6 1000/T (K)

5

4

3 9

10

11

12

13

log(frequency) (rad/s) FIG. 14. The loss dielectric susceptibility for various temperatures. Successive curves are shifted by 0.3 as in Fig. 12. Symbols and units as in Fig. 12.

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and clusters in those ionomers. Recent experimental and simulation work on the morphology of precise PEAA ionomers is inconsistent with the EHM model; in particular, the ionic aggregates are distributed with liquid-like order and do not show a separate phase-separated state.13, 15, 17, 18 A further difficulty in comparing our work to experiment is that our simulations were carried out at both higher temperatures and frequencies than most experimental DRS studies. Previous dielectric studies by Yano and co-workers,37–39 and Kutsumizu et al.42, 43 on EMAA polymers show several relaxation peaks depending on details of the system (percent neutralization, cation type, temperature). In these studies, the high temperature (corresponding to low frequency) α-peak is attributed to a “glass-rubber” transition of the ionic clusters. Typical activation energies in these studies of polymeric materials are around 50 kJ/mol for the β process, and 170–300 kJ/mol for the α process, both higher than the 25 kJ/mol found here for the oligomer. In addition, the experimentally observed processes occur at lower temperatures (300–350 K) than our simulations and at lower frequencies (1–100 kHz) than those in our simulations. Reference 44 provides an excellent review of experiments pertaining to ionomer structure including studies of dielectric and mechanical response functions. Because we are tracking the coulombic energy, the transitions we see must be related to ionic motion. We know that the field aligns the ionic aggregates, and also that a static field strength of 0.25 V/Å as applied here is sufficient to drive significant motion of the lithium ions. Visual inspection of snapshots from the simulations at 280 ◦ C reveals that at frequencies higher than the peak in the loss modulus, the ionic aggregates remain intact throughout the oscillatory simulation. In other words, ions remain in the same aggregates that they start in. By contrast, at frequencies near the peak in the loss modulus, the ionic aggregates rearrange through collisions and break-ups, and the lithium ions move considerably within the aggregates, by exchanging with their neighbors. The motion is very similar to the ion dynamics observed in coarse-grained simulations (in the absence of a field), in which counterions moved by cluster rearrangements.14–16 At the higher temperatures, there are two peaks in the loss modulus. Inspection of snapshots at 500 ◦ C shows that the lower frequency peak is still due to ionic aggregate rearrangements. At frequencies corresponding to the higher frequency peak, around 1011 rad/s, the ionic aggregates do not rearrange and are highly aligned and ordered along the field direction. There is however more local motion than at even higher frequencies. This high frequency loss peak must thus be due to local ionic motions, whereas the lower frequency loss peak is associated with ionic aggregate rearrangements. The loss peaks should continue to shift to lower frequencies with lower temperatures, and we expect the ionic aggregate motion seen here could be experimentally observable in similar oligomeric systems. In longer-chain ionomer melts, the dynamics will be slower due to the much longer relaxation times of the ionomer chains, particularly if they are entangled. However, as we have shown, the ionic aggregate structure in the equivalent precise ionomer melts is very similar to that seen for the small

J. Chem. Phys. 140, 014902 (2014)

molecule analogue simulated here, so we anticipate that the relaxation mechanisms associated with ionic aggregate rearrangements are likely to also exist in longer-chain ionomers. The dynamics found here may provide insight to future DRS experiments on similar materials, while atomistic simulations of the dynamics in ionomer melts remain a challenge for the future. V. CONCLUSIONS

In this paper we have demonstrated that the dynamics of ions in an ionomer analogue, lithium-neutralized 2pentylheptanoic acid, can be profitably studied in MD simulations by tracking the system’s response to an imposed electric field. In a simulation there are many components of the system energy that can be used for this purpose. We chose to employ the coulombic energy because we believed that it would have the most pronounced variation in response to the field. In analyzing the results we assumed that the coulombic energy varies with the electrical work done on the system and that the polarization follows the electric field in a phaselagged manner. This was demonstrated to hold for our simulations from Lissajous-Bowditch plots of the coulombic energy vs. the electric field. We also demonstrated that the variation of the coulombic energy is proportional to the amplitude of the electric field squared as would be anticipated for linear response. We note that nonlinear effects are easily observed in this system. We have not explored such effects in detail, but instead have gone to some pains to stay in the linear response regime. In the absence of an electric field, the melt of oligomers forms stringy ionic aggregates, with a local structure very similar to the aggregates formed by an analogous ionomer.17 Both static and oscillating electric fields induce alignment of the aggregates along the field direction. The fields also induce much greater ion mobility than in the absence of a field. A large number of simulations were performed in oscillating fields at various frequencies and for a range of temperatures. The storage component of the susceptibility is found to show a distinct “transition.” The onset frequency of this transition as a function of temperature is well fit by an Arrhenius relation. The loss modulus shows two peaks at higher temperatures. The first peak is associated with ionic aggregate rearrangements, while the second peak is associated with more local ion motions. These results demonstrate that MD simulations in an applied AC field can be used to identify molecular relaxation processes in ionomer-like materials. ACKNOWLEDGMENTS

This work was supported by the Laboratory Directed Research and Development program at Sandia National Laboratories, a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy’s National Nuclear Security Administration under Contract No. DE-AC04-94AL85000. We thank Dan Bolintineanu for assistance with the structure analysis, and Dan Bolintineanu and Mark Stevens for helpful discussions.

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