Design Rules for Highly Conductive Polymeric Ionic Liquids from

Article Cite This: Macromolecules XXXX, XXX, XXX−XXX

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Design Rules for Highly Conductive Polymeric Ionic Liquids from Molecular Dynamics Simulations Yizi Cheng,† Junhong Yang,† Jui-Hsiang Hung,† Tarak K. Patra,*,† and David S. Simmons*,‡ †

Department of Polymer Engineering, University of Akron, 250 South Forge St., Akron, Ohio 44325, United States Department of Chemical and Biomedical Engineering, University of South Florida, Tampa, Florida 33612, United States

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ABSTRACT: Polymeric ionic liquids (PILs) are of considerable interest as next-generation battery materials due to their potential to combine the solid-state stability of polymers with the high ion conductivities of ionic liquids. However, polymerization of ionic liquids to form a polymer generally leads to a suppression in ion transport rates that has proven to be a major barrier to the realization of commercially viable PIL solid electrolytes. Here we employ a combination of all atom and coarse-grained molecular dynamics simulations to identify strategies by which ion conductivity can be maximized by maximizing both PIL segmental relaxation rates and the extent of ion transport decoupling from chain dynamics. Results indicate that combined ion size correlates well with PIL glass transition temperatures and segmental dynamics but that ion/polymer decoupling is controlled primarily by the size of the free ion. We also find that ion aggregation promotes both reduced glass transition temperatures and enhanced ion/polymer decoupling. These results suggest that PIL ion mobility can be improved by combining ultralarge bound ions with very small free ions and with chemistries that promote ion aggregation.

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INTRODUCTION

This strategy of Tg minimization in PILs faces an apparent limit: with few exceptions, polymers have a higher Tg and slower relaxation dynamics than chemically similar small molecules. This trend appears to hold even in the limit of very large ion pair volumes.5 It is presently unclear how this can be ameliorated. Moreover, the strong temperature dependence of relaxation time in many polymers13,14 can lead to a limited temperature range in which these systems exhibit acceptable ion transport rates. An alternate strategy therefore aims to decouple ion transport from the dynamics of the matrix polymer,15 such that ion transport can occur more rapidly and in a less temperature-dependent manner than polymer segmental relaxation. One strategy for decoupling ion/polymer dynamics is the selection of chemistries leading to aggregation of ionic liquid moieties into percolating nanodomains within the polymer matrix.16 By sequestering free ions away from chain backbones, this has the potential to allow ion transport at rates higher than would be permitted by slow chain backbone motion, provided that the aggregated domains percolate.17 Similar decoupling of dynamics between domains of size on the order of tens of nanometers is seen in a range of nanostructured polymers.18−21 Prior work has identified this as a potential route to high conductivity,4 but its effect on ion transport is nonuniform and poorly understood.22−24 Indeed, there is mixed evidence on

Polymeric ionic liquids are of interest as next-generation iontransport materials because of their potential to combine the thermodynamic and mechanical stability of polymers with the ion transport facility of ionic liquids.1−3 However, these materials face a significant limitation: ion mobility in nonaqueous polymeric ionic liquids is commonly closely coupled to polymer segmental dynamics.4−6 These dynamics tend to be significantly slower (typically by several orders of magnitude)7 than those of pure small-molecule ionic liquids, typically driving a 10−1000-fold suppression in ionic conductivity upon incorporation into a polymer.8 Realizing commercially viable ionic-liquid-containing polymers will require substantial increases in conductivity in these systems beyond the current state of the art while maintaining excellent mechanical properties.9 The suppression in segmental dynamics with polymerization is primarily driven by an enhancement in the glass transition temperature Tg.5 For this reason, it is generally recognized that reducing Tg should be a key goal in ion-conducting polymers. This presents a major challenge given the lack of a generally accepted predictive theory relating molecular structure to glass formation behavior.10,11 Recently it has been suggested that maximization of ion pair volume may provide a general route to Tg suppression in these systems, providing a potential design rule.5 However, this proposed correlation between Tg and ion pair volume appears to be nonuniversal,12 and realizing predictive design of these materials’ Tg’s will require further molecular mechanistic insight. © XXXX American Chemical Society

Received: March 16, 2018 Revised: August 2, 2018

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DOI: 10.1021/acs.macromol.8b00572 Macromolecules XXXX, XXX, XXX−XXX

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scale relaxation times disappear in model systems,45 while remaining computationally tractable. Atomistic Simulations. We first perform molecular dynamics simulations of a matrix of imidazolium-based ionic liquids and polymeric ionic liquids modeled at an all-atom level. Three ionic liquid cations are simulated, as shown in Figure 1: ethyl-3-methylimidazolium (C2mim+), 1-butyl-3-

the question of whether ion aggregation promotes higher or lower ion conductivity in these systems.22−24 Here we employ a combination of all-atom and coarsegrained molecular dynamics simulations to establish molecular design rules for polymeric ionic liquids exhibiting high ion conductivity via the combination of a low glass transition temperature and high decoupling. At an all-atom level, we probe a matrix of ionic liquids and polymeric ionic liquids to obtain insight into the relationship between Tg, decoupling, molecular size, and high-temperature activation behavior. We then employ bead−spring molecular dynamics simulations to broaden the range of coarse molecular properties explored to include the relative sizes of bound and free ions as well as the presence or absence of ion aggregation. Results suggest that lower-Tg PILs often exhibit less decoupling but that it may be possible to disrupt this unfavorable correlation via the introduction of ion aggregation and highly asymmetrically sized ion pairs. These findings suggest the possibility of polymeric ionic liquids combining low Tg with highly decoupled ion mobility, with the potential to realize PILs with substantially enhanced ion conductivity.

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SIMULATION METHODS We perform molecular dynamics simulations of both all-atom and coarse-grained polymeric ionic liquids within the LAMMPS molecular dynamics package.25 The models employed at each of these levels of atomistic detail are described below. In all cases we use a common simulation procedure to probe polymer and ion dynamics over a range of temperature. Specifically, beginning with a randomly generated configuration, we subject each system to a long equilibration at a high initial temperature. We then subject the system to a fast thermal quench. Each of many configurations saved during this quench is then subjected to a long isothermal isobaric annealing period prior to data collection. The duration of this annealing period is varied with temperature to ensure that it is at least 10 times the material’s segmental relaxation time at each temperature, as defined below. This isothermal annealing protocol ensures that segmental dynamics and transport reflect equilibrium values, consistent with prior simulation studies employing this type of approach.18,20,21,26−34 Data are then collected in the isothermal isobaric ensemble with runs beginning from these equilibrated configurations, with data collection periods of roughly 10 times the segmental relaxation time. The relaxation time data for all systems are reported in the Results section; with this protocol, in each case the equilibration period at each temperature is at least 10 times the reported relaxation time, and the data collection period is of a similar duration. This procedure is similar to quench-andanneal procedures widely employed to simulate glass formation in prior work,26,27,35−43 and it is designed to efficiently mimic the results of an experimental melt quench toward the glassy state. In all cases, short-ranged electrostatic interactions are computed directly, and long-ranged electrostatic interactions are computed via the particle−particle particle mesh (pppm) solver44 as implemented in LAMMPS. Simulations are performed at constant pressure, with constant temperature and pressure maintained via the Nosé−Hoover thermostat and barostat, as implemented in LAMMPS. System sizes are chosen to contain at least 400 ions, which prior work has indicated is above the size range in which finite size effects on segmental-

Figure 1. Chemistries of imidazolium-based ILs and PILs simulated in this work with their counterions.

methylimidazolium (C 4 mim + ), and 1-hexyl-3-methylimidazolium (C6mim+). Each of these cations is simulated in the presence of three different anions: chlorine (Cl−), hexafluorophosphate (PF6−), and bis(trifluoromethylsulfonyl)imide (Tf2N−). Vinyl polymeric analogues of each cation are additionally simulated as shown in Figure 1 (designated as V2mim, V4mim, and V6mim, respectively), with the same three anions probed in each case. We thus simulate a total of 18 systems at an all-atom level. Each IL system contains 200 ion pairs. In the case of PILs, the cation is polymerized with degree of polymerization 30, with a total of 180 ion pairs simulated. Representative images of simulated ILs and PILs are shown in Figure 2. Simulations of ionic liquids employ an extension of the OPLS-AA/AMBER force field to model imidazolium ionic liquids developed by Lopes and Padua, and additional details of the force fields employed can be found there.46−48 These force fields were developed based upon structural data obtained from ab initio quantum simulations and employ the same functional forms employed in the OPLS all-atom force field. For polymeric ionic liquids, this ionic liquid force field is therefore combined with a force field for a carbon−carbon backbone in the standard OPLS model to enable modeling of the polymeric chain. It is likely that further refinement of this polymeric force field could improve quantitative predictions, but here we focus on semiquantitative to quantitative trends in the reasonable (if potentially imperfect) realistic chemical representations obtained via this tabulation method. As shown in the Results section, resulting trends in Tg are in good agreement with experiment, validating this approach for this purpose. The van der Waals interaction between a pair of atoms are truncated and shifted to zero at a separation distance of 12 Å. The electrostatic interactions between the atoms are directly calculated for the interparticle distance up to 12 Å. Simulations B


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Figure 2. Simulation images of C4mim (top row) and V4mim (bottom row) with Cl (left column), PF6 (middle column), or Tf2N (right column) counterions. For ionic liquids (top row) the entire simulation box is shown. For polymeric ionic liquids (bottom row), a polymer chain and selected nearby counterions are shown with. Cations and polymers are shown in blue and anions in orange.

employ the Verlet time integration scheme with a 1 fs time step, at a pressure P = 0. Initial configurations are generated using Avogadro49 and Moltemplate.50 Coarse-Grained Simulations. To probe a broader range of PIL molecular properties than can be accessed via available atomistic force fields, we next perform coarse-grained molecular dynamics simulations generalized from the bead− spring polymer model of Kremer and Grest.51 The present extension of this model to polymeric ionic liquids is inspired by recent work in which the Kremer−Grest model was extended to probe the structure of ionomers.52,53 This type of extension was also shown to capture key aspects of ionomer glass formation and dynamics.20,21 Here we simulate bead−spring side-chain polymeric ionic liquids in which each backbone repeat unit possesses an attached side chain terminated in a covalently bound cation. These cations are neutralized by free anions. The structure of this model is inspired by experimental polymeric ionic liquids in which an ionic moiety is attached to a polymer backbone by an alkyl spacer.54 Simulated chains are composed of 15 backbone beads, with a two-bead spacer separating each backbone bead from the attached ion. Each system contains 30 polymer chains neutralized by associated counterions. The van der Waals interactions between nonbonded beads of types i and j are modeled via the 12−6 LJ potential ÄÅ É ÅÅi σij y12 i σij y6ÑÑÑ Å j z j z Å VLJ, ij(r ) = 4εijÅÅjj zz − jj zz ÑÑÑÑ, r < rcut ÅÅk r { k r { ÑÑÑÖ (1) ÅÇ

Figure 3. Schematic representation of model CG polymeric ionic liquids and counterions.

and monomer) which renders the interaction fully repulsive. Nonaggregating ion systems are modeled by setting this cutoff to 2.5 σij, which incorporates the attractive tail, and all other interactions employ rcut = 2.5 σij. Charged beads additionally interact via a Coulombic potential qiqj Ecoul = 4πε0εrr (2) where qi and qj correspond to charges of ions i and j, respectively, ε0 is the permittivity of free space, and εr = 1 is the relative dielectric constant of the medium. Coulombic interactions within a distance of r = 5σ are calculated directly. Simulations are performed at pressure P = 1 in reduced LJ units. Simulation Analysis. Structural relaxation dynamics are quantified via the self-part of the intermediate scattering function, given by

with ε and σ setting the energy scale and range for nonbonded pairwise interactions. Bead sizes σ of free and bound ions are varied over a range of 0.5−2.5 and described in the Results section. In all cases we employ a simple averaging mixing rule for σ for unlike interactions between beads of type i and j; i.e., σij = (σii + σij)/2. All interactions employ ε = 1. We simulate two general scenarios for ion structureone in which ions aggregate and one in which they do not. In the former case, we set rcut for interactions between ions and neutral beads to be 21/6σij (where σij is the value of σ for that combination of ion

N

Fs(q, t ) = C

1 ∑ ⟨exp[−iq·(rj(t ) − rj(0))]⟩ N j

(3)


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Macromolecules where q is the wave vector, rj(t) is the position of particle j at time t, and N is the number of particles in the system. The behavior of Fs(q,t) at a scalar wavenumber q is ultimately computed by averaging over wavevectors q corresponding to that wavenumber. The relaxation time at a given temperature is defined by convention, as in other recent simulation work, as the time at which this function decays to a value of 0.2, using a fit to a stretched exponential decay function for smoothing and interpolation.28,29,55 Determination of the glass transition temperature Tg from simulation is generally complicated by the large separation of time scales between those accessible to molecular simulation and the 100 s conventional time scale of the experimental laboratory Tg. Here, we employ a strategy employed in other recent simulation work in which a fit to simulation-accessible relaxation time data is extrapolated to the 100 s time scale of the experimental glass transition. We specifically perform this extrapolation via a fit to the relaxation time temperature dependence to the functional form of Schmidtke et al.56 Evidence suggests that this functional form provides an improved fit to diverse relaxation time data56 than does the more commonly employed Vogel−Fulcher−Tammann (VFT)57,58 form. For each system, we compute a Nernst−Einstein ion conductivity as a function of temperature via the equation σ=

∑ i

2 pq D i i i

kBT

(4)

where pi, qi, and Di correspond to the concentration, charge, and diffusion coefficient of the ith ion species. We then define a molar Nernst−Einstein conductivity as σ Λ= (5) c

Figure 4. For a representative all atom simulation, V4mim/PF6, at its lowest temperature simulated (789 K): (a) mean-square displacement vs time for free anions (orange diamonds) and bound cations (blue circles) on a log−log scale and the differential d log ⟨r2⟩/d log(t) of these data for the anion (b) and the cation (c).

where c is the mole fraction of ions. We emphasize that eq 4 is only strictly valid in the limit in which motion of anions and cations is uncorrelated. This approximation is much better for PILs than ILs, where codiffusion or counterdiffusion of cations and anions can lead to substantial deviations from this equation. However, we do emphasize that even in PILs, prior work has shown that there can be some deviations of the ion conductivity from its Nernst−Einstein value as a result of polymer chain motion.59 The diffusion coefficient of a given ion species is measured from the diffusive regime of mean-square displacement for that species. The strategy of collecting data for 10 times the mean segmental relaxation time of the system at all temperatures generally provides access to the diffusive regime for free ions. Figures 4a and 5a show representative log mean-square displacement vs log time data for anions and cations for allatom and coarse-grained simulations, respectively. As can be seen in these figures, over the course of the simulations the anion traverses a distance of approximately 1000 Å in the AA simulation and near 100 LJ distance units in the coarse-grained simulation, far larger than the box size. Differentiation of these data in parts b and c of these figures shows that the exponent α in the relation ⟨r2⟩ ∼ tα goes to one at the longest times probed for the free ion, confirming realization of the diffusive regime. In contrast, for the bound ion α plateaus to approximately 0.5 at the longest times probed, indicating subdiffusive Rouse mode motion for the bound ions (for which diffusion constants are not reported). The diffusion constant is thus

computed for free ions only via the relation D = ⟨r2⟩/6t at these long, diffusive, time scales. This procedure, like determination of relaxation times, was automated to enable high-throughput data analysis and manually spot-checked. In this study, each simulated system is simulated a single time at each temperature to maintain computational tractability over long simulation times and numerous systems. Prior studies have reported typical uncertainty intervals for dynamical and static quantities obtained via quench and anneal procedures comparable to those used here. For example, in our prior study,28 the relative standard deviation in the Tg determined across multiple runs of a single system composed of a single 256 repeat unit polymer chain was determined to be 0.3% of Tg or less.60 Data collection periods in that study were 10 times longer, but the system was about a factor of 1.5−2 smaller. Because the properties report average over both particle number and time, n−1/2 central limit theorem scaling predicts relative Tg run-to-run variations in the range of 0.75% or less in this study. This is much smaller than the data points in figures reporting Tg. This small system-to-system variation in Tg implies small system-to-system variation in relaxation times at each temperature, since the former is extracted from the latter. Other single-particle dynamical properties reported in this study can be expected to exhibit similarly low system-tosystem variation because they reflect similar statistical sampling over similarly local dynamics and short time scales. Uncertainties in reported thermodynamic quantities such as volume are generally even smaller because they sample very D


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Figure 6. Plot of relaxation time vs inverse temperature (a) and vs Tgnormalized inverse temperature (b) for systems indicated in the figure (a) caption. ILs are denoted with hollow points and PILs with filled points. Solid lines in part b are fits of each data set to the Cooperative model of Schmidtke et al.

Figure 5. For a representative simulation of the CG PIL with attractive ion/polymer interactions and σ = 1 for both ions at its lowest temperature simulated (0.482): (a) mean-square displacement vs time for free anions (orange diamonds) and bound catioins (blue circles) on a log−log scale and the differential d log ⟨r2⟩/d log(t) of these data for the anion (b) and the cation (c).

these correlations. In particular, PILs are found to generally exhibit higher Tg than ILs and to exhibit a negative correlation between Tg and ion pair volume. Are the present simulation results consistent with the proposition of a class-specific correlation between ion pair volume and Tg? For simulated PILs, a linear dependence on Vm is found to account for 79% of the variation in Tg between systems (coefficient of determination R2 = 0.79 for a linear relationship between Tg and Vm). This finding is reasonably consistent with the trend identified by Choi et al. for PILs over this ion pair volume range. For the nonpolymeric ionic liquids, variation in Vm is found to account for only 39% of the variation in Tg, again based upon a linear model for the relationship between these variables. This general finding of much a stronger correlation of Tg with Vm for PILs than ILs is insensitive to the choice of time scale used to defined Tg; if we instead define a Tg without appreciable extrapolation on a computational time scale of 1 ns, R2 for this relationship is found to be 0.90 for PILs and 0.39 for ILs. Evidently, simulation results point toward a significant correlation between Vm and Tg for PILs but a poor correlation between these quantities for ILs. These findings appear to be consistent with the experimental data aggregated by Choi et al. Specifically, experimental data for PILs digitally extracted from Figure 1 of that paper exhibit an appreciable correlation between Tg and Vm (R2 for a thirdorder polynomial dependence of Tg on Vm is 0.94, with values of 0.91 for a second order or 0.94 for a fourth-order polynomial fit). However, digitized data for the nonpolymerizable ionic liquids, taken alone, do not exhibit any such correlation (R2 = 0.00 for a linear relationship between Tg and

well in simulation (i.e., they decorrelate at very high frequencies, such that time averaging yields a very high number of effective independent measurements).

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RESULTS All-Atom Simulations. Segmental Relaxation Behavior and the Glass Transition. As shown by Figure 6a, the systems simulated at an all-atom level exhibit a wide range of relaxation time τ temperature dependences, with PILs generally relaxing more slowly than ILs at a given temperature. This is consistent with the observation that Tg’s are typically enhanced and segmental dynamics correspondingly suppressed with increasing molecular weight. If the temperature scale for each system is normalized by its extrapolated experimental time scale Tg, the resulting Angell plot61 (Figure 6b) reveals a substantial range of fragilities of glass formation, indicating that these simulated systems capture a diversity of both Tg and breadth of the glass transition. As discussed in the Introduction, recent work by Colby and co-workers has suggested that polymeric ionic liquids, polymerizable ionic liquid monomers, and nonpolymerizable ionic liquids and salts each obey a distinct correlation between Tg and ion pair volume. As shown by Figure 7a, glass transition temperatures, extrapolated to an experimental 100 s time scale via the functional form of Schmidke et al.,56 exhibit reasonable agreement with the data employed by Choi et al. to argue for E


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ion pair volume for PILs. At the same time, their model reflects the empirical fact that many factors other than ion pair volume, including van der Waals cohesive energy, contribute to Tg and therefore weaken the strength of this correlation. Our findings appear to be qualitatively consistent with this scenario. Moreover, if this interpretation is correct, then these effects should be observable in dynamics even at very high temperatures, far from Tg. In particular, this effect should be observable in the high-temperature Arrhenius activation regime that is typically found above the onset temperature TA of glass formation. In this regime, the relaxation time temperature dependence is described by the Arrhenius form: ÄÅ ÉÑ ÅE Ñ τ = τ0 expÅÅÅÅ a ÑÑÑÑ (6) ÅÅÇ kT ÑÑÖ where Ea is the high-temperature activation energy. To test the above proposition, we therefore quantify the high-temperature activation energy of each system by fitting its high-temperature Arrhenius regime to this functional form to extract Ea. As shown in Figure 8a, Ea is related to ion pair volume in a similar manner as is Tg: Ea exhibits a strong negative

Figure 7. (a) Glass transition temperature of systems simulated this work (large points) and experimental data extracted from Choi et al.5 (small points), plotted vs ion pair volume. (b) Glass transition of simulated systems only plotted against ion pair volume, with separate linear fits to PILs and ILs.

Vm, and R2 < 0.2 for any polynomial relationship up to sixth order between these quantities). Choi et al. argued that a correlation could nevertheless be identified by grouping nonpolymerizable ILs with alkali inorganic salts but excluding polymerizable IL monomers. However, the basis for this choice of groupings is unclear, and the present simulations therefore appear to be consistent with the direct experimental observation of a poor correlation between Tg and Vm for ILs taken alone. Why does Tg exhibit this correlation with ion pair volume for polymeric ionic liquids but not small-molecule ionic liquids? The simplest explanation accounting for both of these observations is that systems with larger ion pairs possess weaker electrostatic repulsions between ions within the same chain. Because intrachain electrostatic repulsions increase chain stiffness, and chain stiffness correlates with higher Tg and thus slower chain dynamics,13 this mechanism naturally anticipates an inverse correlation between ion size and glass transition temperature in polymers; on the other hand, these long-range electrostatic molecular stiffness effects are naturally absent in small molecules. Indeed, in recent work, Sokolov and co-workers proposed a simple model for the glass transition temperature in ionic liquids and polymeric ionic liquids that captures these general trends.12 Within this model, a term involving chain stiffness, which is present for PILs but not for ionic liquids, plays a major role in the appreciable anticorrelation between Tg and

Figure 8. (a) High-temperature activation barrier plotted vs ion pair volume for simulated systems, with the symbols corresponding to those used in Figure 7. (b) Tg plotted vs activation barrier for these systems.

correlation with Vm for PILs (R2 = 0.88 for a linear relationship) but is nearly uncorrelated (R2 = 0.17) for ILs. The similarity of these interrelations suggests that the hightemperature activation energy Ea may be a more direct determinant of Tg than is Vm. Indeed, as shown by Figure 8b, the PILs exhibit an appreciable correlation between Tg and high-temperature activation energy. Ion Mobility and Decoupling. We now consider the question of how the above trends in segmental dynamics and glass formation are reflected in ion mobility. We begin by F


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ity63−65 and a power-law relationship observed between different polymer chain normal modes.66 Within relationships of this form, the degree of decoupling can be quantified by the parameter ε: when this parameter is 0, the two dynamical processes are coupled; values greater than 0 indicate an increasing degree of decoupling. To quantify the decoupling relationship between conductivity and segmental dynamics, we therefore begin in Figure 10 by plotting log conductivity vs log inverse relaxation

computing effective molar conductivity for each system via the approach described in the Methods section. We emphasize that the true conductivity in ionic liquid systems will often be lower than this value due to the potential for correlated transport of oppositely charged ions. In PILs, however, which are single-ion conductors, this problem tends to be less significant. As shown by Figure 9a, these systems exhibit a wide range of ion mobility

Figure 10. Molecular conductivity as a function of relaxation time in log scale for ILs and PILs.

time for each system under consideration. By fitting data at temperatures within the glass formation range to eq 7, we then extract a value of ε for each system. Results indicate that PILs generally exhibit greater decoupling than ILs, with values of ε ranging from 0.13 to 0.42 for the former and 0.00 to 0.14 for the latter. This finding is qualitatively consistent with decoupling observed between ion diffusion and segmental dynamics in simulations of PILs by Ganesan and co-workers.15,67 Because greater decoupling indicates that ion dynamics slow down less severely upon cooling than do segmental dynamics, this effect can somewhat offset the tendency toward a higher Tg in PILs than ILs. It is therefore of natural interest to understand what controls the value of ε and how it can be maximized. Prior work has indicated that ε may be correlated to the fragility of glass formation,68−70 which is a measure of the thermal abruptness, or degree of deviation from Arrheniusness, of the transition. Theis quantity is commonly quantified via the kinetic fragility index m, which is defined as

Figure 9. Nernst−Einstein molar conductivity vs (a) inverse temperature and (b) Tg-normalized inverse temperature for AA PILs and ILs, with symbols shown in the legend.

temperature dependences. Consistent with the general slowdown in segmental relaxation in PILs relative to ILs shown in Figure 6a, PILs tend to exhibit lower conductivity than ILs at any given temperature. As shown by Figure 9b, normalizing the temperature scale by Tg in this plot reduces the variability of the conductivity behavior but does not fully eliminate it. This is a consequence of two effects: system to system variability in the fragilityor non-Arrheniusnessof glass formation and a variable degree of decoupling of ion mobility from segmental dynamics. Systems exhibiting decoupling commonly exhibit a fractional power law relationship known in this context as the Walden rule62 Λτ1 − ε = constant

m=

d log τ d(Tg /T )

T = Tg

(8)

In addition to experimental evidence, the physical reasoning behind this hypothesis is as follows: decoupling in the glass formation range is often explained as resulting from differences in averaging over an emergent dynamic heterogeneity; more fragile glass formers are commonly expected to be more dynamically heterogeneous. However, each of these propositions are themselves unproven, and for this reason the general validity of a correlation between ε and m is unsettled. Indeed, recent work reported a poor correlation between these quantities.71 The data in Figure 11a for these simulations are consistent with this latter findingthis figure illustrates that ε

(7)

where ε is the decoupling exponent. This relationship is equivalent to empirically observed power-law relatonships between other decoupled dynamical properties in the glass formation range, including the fractional Stokes−Einstein relationship between diffusion and viscosG


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Figure 12. Decoupling exponent plotted vs high-temperature activation barriers for AA systems, with symbols the same those used in Figure 7.

high-temperature activation barrier and low decoupling exponent, such that they are reasonably consistent with the trend identified in the polymeric ionic liquids. Ultimately, this work suggests that for PILs ion pair volume is an important controlling parameter for both Tg and decoupling. Lower total ion pair volumes tend to correlate with both higher Tg’s and higher degrees of decoupling. Physically, this is consistent with the scenario proposed above: lower ion pair volumes lead to larger intrachain electrostatic repulsions, larger chain stiffness, higher high-temperature activation barriers, and higher Tg’s. At the same time, this scenario naturally increases the size differential between the chain Kuhn segment and the free ion, which is intuitively consistent with greater decoupling. This suggests that while increased decoupling may somewhat offset the enhancement in Tg with decreased ion pair volume, overall ion pair volume is not a particularly useful variable in achieving intentionally enhanced decoupling of ion mobility and polymer mobility, since it will be accompanied by higher Tg. Coarse-Grained Simulations. We now consider the question of whether Tg and ε can be more strongly decorrelated to achieve PILs with both low Tg and large decoupling. As described in the Methods section, we specifically focus on two issues: the relative size of the free and bound ions and the question of whether the ions are miscible with the chain backbone or self-assemble into distinct ion domains. As described in the Methods section, we probe the former issue by simulating two matrices of ion sizes. In the first, the bound ion size is held equal to that of the chain monomer bead size, and the size of the free ion is varied from half to double this value. In the second, the size of the free ion is held equal to that of the chain monomer bead size and the size of the bound ion is varied from half to double this value. For each of these simulation, we consider two limits of interaction: one in which ion−polymer interactions include the attractive tail of the 12−6 LJ potential modeling van der Waals interactions and one in which these interactions are truncated such that they are fully repulsive. Results for this coarse-grained model are reported in dimensionless LJ units. While the mapping of these units to real chemistries is system-specific (and may in any case not be fixed across thermodynamic conditions and time scales), a reasonable rough mapping for the purposes of physical comparison is 1 LJ distance unit σLJ to ∼1 nm,72−74 1 LJ time unit τLJ to ∼1 ps,75 and 1 LJ temperature unit to 1000 K. However, we emphasize that this coarse model should not be viewed as mapping in a direct way to any specific chemistry but

Figure 11. Ion/polymer decoupling exponent plotted vs (a) kinetic fragility index, (b) glass transition temperature, and (c), ion pair volume. Symbols are those used in Figure 7.

and m do not exhibit a strong general correlation within either the PILs or ILs probed in this study. We similarly test for any correlation between ε and Tg itself. As shown in Figure 11b, these two quantities are found to be poorly correlated in IL and PILs, considered separately. When the two classes of material are combined, a modest correlation appears to emerge; however, this is simply a consequence of the fact that PILs tend to have both higher Tg’s and higher values of ε than IL’s and not of some more general correlation. Given that ion pair volume appears to play an important role in determining Tg, it is reasonable to ask whether this quantity also influences decoupling. As shown in Figure 11c, the decoupling parameter is indeed strongly correlated to ion pair volume when PILs and ILs are treated separately (R2 ∼ 0.8 for a linear relationship between ε and Vm for each data set). However, for equivalent ion pair volumes PILs exhibit much more decoupling than ILs, such that the two sets of systems evidently do not follow the same correlation. This is similar to the outcomes reported above for Tg vs Vm and Ea vs Vm. Finaly, we test the question of whether greater decoupling is presaged by greater values of the high-tempertature activation energy. As shown by Figure 12, for PILs there is a strong correlation between high-temperature activation barrier and the decoupling exponent. Among ILs this correlation is largely absent, but as a group the ILs generally possess both a low H


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Figure 13. MD snapshot of coarse-grained PIL systems for nonaggregating ions (left) and aggregating ions (right). In the upper row, deep gray beads are backbone beads, light gray beads are side-chain beads, blue beads are bound ions, and purple beads are free ions. In the lower row of images, only ions are shown to enable better visualization of their dispersion state.

Segmental Relaxation Behavior and Glass Formation. As can be seen in Figure 15, these coarse-grained simulations exhibit a super-Arrhenius temperature dependence of chain dynamics in a qualitatively similar manner to the all-atom simulations. Moreover, the systems in which ion−polymer interactions are poor and ions aggregate into distinct domains exhibit systematically lower relaxation times (faster dynamics) than those in which ion−chain interactions are favorable. This is a natural consequence of the tendency of Tg to scale with cohesive energy: weaker interactions between any pair of components will tend to reduce cohesive energy and favor a lower Tg. Moreover, work by our group and others has indicated that interfaces between domains with unfavorable interactions, such as those between the polymer and ion domains in these systems, tend to accelerate nearby molecular dynamics.18,28,30,77,78 Indeed, this scenario is supported by examination of typical relaxation functions for aggregated and nonaggregated ion systems. As exemplified by Figure 14, the relaxation functions for the aggregated-ion systems exhibit a much higher degree of stretching than those corresponding to their homogeneous analogues. A higher degree of stretching is

instead as capturing broad qualitative to semiquantitative physical trends. The structural outcome of these two classes of interaction potential can be seen in Figure 13 (rendered in VMD76): favorable ion−polymer interactions lead to a reasonably homogeneous distribution of ion in the systems, while repulsive interactions lead to the formation of distinct ion aggregates. In these simulations these ion domains percolate across the box. However, we emphasize that this may be a consequence of the relatively small box sizes employed in this work to allow computationally tractable access to the supercooled regime of dynamics. Much larger box-size simulations would be necessary to determine whether domain connectivity would persist over large length scales. The focus of this section is therefore on local mobility and conductivity of ions; good ion conductivity over macroscopic length scales will in practice be additionally contingent upon long-range percolation of these domains. The present findings should therefore be viewed as pointing toward necessary but not necessarily sufficient conditions for enhanced ion conductivity. I


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Figure 14. Self-intermediate scattering functions at the indicated temperatures for sample nonaggregating (top) and aggregating (bottom) coarse-grained models.

Figure 15. Polymer segmental relaxation times τp vs inverse temperature for coarse-grained PILs in which the size of the free ions (top) or bound ions (bottom) is varied according to the legend. Filled symbols are nonaggregating ion systems, while open symbols are aggregating ion systems.

a signature of a broader distribution of underlying relaxation times, consistent with the more diverse array of local dynamical environments naturally spanning an interface between compositionally distinct local environments. How do trends in the dynamics of these coarse-grained polymers compare to those observed in the all atom systems? We first examine the relationship between volume and glass transition temperature in these systems, shown in Figure 16a. Because these chains all have the same structure, we employ molar volume as an proportionate measure of ion-pair volume. None of the all-atom PILS simulated here are observed to form pronounced aggregated structures, and we thus consider the nonaggregating coarse PILs to provide the best basis for comparison to the all-atom systems. Unlike for the all-atom systems, the CG models exhibit a nonmonotonic dependence of Tg on volume. However, the left branch of this dependence corresponds to systems in which the mean ion size is less than the chain bead size. Real PILs generally employ bulky ionic moieties, such that this scenario is not representative of the systems simulated above. Restricting the comparison to nonagreggating CG model systems with mean ion diameter greater than the neutral bead size, one observes a negative correlation of Tg with volume, consistent with AA simulation and experiment. Data for AA PILs also revealed a correlation between hightemperature activation barrier and ion pair volume. As shown in Figure 16b, a similar correlation is observed in the CG models. However, the aggregating and nonaggregating ions obey two distinct correlations, with the aggregating ion systems exhibiting lower high-temperature activation barriers for the equivalent molar volumes. This distinction makes sense, as the introduction of internal interfaces tends to accellerate dynamics in a manner that is not simply correlated to local

density.26,79 In addition to any impacts of aggregation on ion decoupling, formation of distinct ionic aggregates thus appears to favor rapid ion transport by simply lowering barriers to relaxation in the fluid. Ion Transport and Decoupling. Evidently, the coarsegrained models considered here provide access to a range of relaxation behavior qualitatively comparable to the chemically realistics systems, while also enabling exploration of a broader range of potential properties. We now consider how these features impact ion transport and ion/polymer mobility decoupling. As shown by Figure 17, ion mobilities obey a spectrum of super-Arrhenius temperature dependencies as in the all-atom systems. Moreover, ion dynamics in aggregatedion systems are systematically faster and more Arrhenius than in equivalent nonaggregated-ion systems. It is also apparent that ion mobility is less sensitive to bound ion size than to free ion size. To quantify the extent to which these trends are simply driven by the analogous trend in polymer dynamics reflected in Figure 15, we replot these data vs the polymer segmental relaxation time in Figure 18. As can be seen here, these systems exhibit a range of degrees of decoupling between ion and chain dynamics, with aggregated-ion systems generally exhibiting greater decoupling than the equivalent nonaggregated system. Again, ion transport rates are seen to vary less with variation in the bound ion size than variation in the free ion size. We again quantify these trends in decoupling by computing the decoupling exponent for ion transport in each system based upon fits of the data in Figure 18. As shown by Figure J


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Figure 17. Nernst−Einstein ion conductivities vs inverse temperature for CG PILs in which (a) free ion size is varied and (b) bound ion size is varied. Symbols are those used in Figure 14.

Figure 16. (a) Glass transition temperatures of CG PILs plotted vs molar volume and (b) molar volume plotted vs high-temperature activation barrier. Symbols for standard CG PIL systems are shown in the legend, with hollow symbols denoting aggregating ion systems and filled symbols matching nonaggregating ion systems. Purple circles denote a system with bound ion size of 2.0 and free ion size of 0.5.

19, consistent with the all atom polymeric ionic liquids, the coarse-grained PILs exhibit a poor correlation between the ion/polymer decoupling exponent and either Tg or the kinetic fragility index. A clearer dependence of decoupling on molecular properties can be extracted by plotting in Figure 20a the decoupling exponent versus the ion pair volume for each of the four data sets formed by the matrix of aggregated vs nonaggregated ions and varied bound ion size vs varied free ion size. Similarly, in Figure 20b we plot these same data against the average of the anion and cation diameter parameter σ+−. While there is no general relationship between these quantities that spans all systems, a focus on individual sets of data in which only a single molecular property is varied reveals several design rules for maximization of ion/polymer decoupling. First, the decoupling exponent ε is modestly sensitive to bound ion size (red symbols in Figure 20) but strongly sensitive to free ion size (blue symbols in Figure 20), with ε exhibiting a linear inverse correlation with free ion size. This is a strong effect; for example, in nonaggregated-ion systems, reduction from ions of size 50% larger than to 50% smaller than the chain repeat unit increases ε from less than 0.1 to 0.5. Second, these figures indicate that ion aggregation appreciably increases ion/ polymer decoupling. These two effects are synergistic, with

Figure 18. A log−log plot of Nernst−Einstein ion conductivity vs inverse segmental relaxation time for CG PILs in which (a) free ion size is varied and (b) bound ion size is varied. Symbols are those used in Figure 14.

ion aggregation exhibiting an enhanced effect in the limit of smaller ions. These observations suggest the following strategy toward the maximization of ion mobility. First, the correlation identified by Colby appears to operate based on combined ion size, without regard to the question of the relative size of the free K


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and bound ion. On the other hand, decoupling exhibits a stronger sensitivity to free ion size than bound ion size. This suggests that PIL systems with large bound ions and small free ions have the potential to combine low Tg with high ion/ polymer decoupling. If so, such systems can be expected to exhibit ion mobilities that are both higher and more temperature stable (i.e., more Arrhenius) than standard PILs. To test this, we simulate aggregated and nonaggregating coarse-grained systems in which we combine a large bound ion (twice the repeat unit size) with a small free ion (half the repeat unit size). As shown by Figure 16a, these systems indeed exhibit relatively low glass transition temperatures due to their large average ion pair volume. At the same time, as shown by Figure 20, they exhibit exceptionally large decoupling exponents (purple symbols in both figures), comparable to or exceeding CG systems that possess a small (σ = 0.5) free ion but an average sized (σ = 1.0) bound ion. Evidently, free ion size indeed dominates over bound ion size in determining decoupling behavior, and large ion-size mismatches may additionally favor decoupling. The Nernst−Einstein conductivity temperature dependences for these aggregating and nonaggregating asymmetric ion systems are shown in Figure 21. As anticipated above, each

Figure 19. Decoupling exponent plotted vs glass transition temperature (a) and kinetic fragility index (b) for CG PIL systems. Lines are linear fits to the data, with R2 = 0.366 for (a) and 0.058 in (b).

Figure 21. Nernst−Einstein conductivity vs inverse temperature for the small-free-ion/large-bound-ion systems simulated in the work (shown in purple) in comparison to the same data for systems with varying free ion size.

of these systems (shown in purple symbols) exhibits the highest absolute Nernst−Einstein ion conductivity for its class (aggregating and nonaggregating) of PIL. The aggregating version of this asymmetrically sized ion system exhibits the highest and least temperature sensitive Nernst−Einstein conductivity of all systems considered in this study, with NE conductivities more than 5 orders of magnitude higher than the worst ion conductors considered at low temperature. These results suggest that experimental realization of a polymeric ionic liquid with extremely large bound ions, small free ions, and percolating aggregated ionic domains has the potential to realize exceptionally high ion conductivities.

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CONCLUSIONS Here we have probed the relationships between segmental relaxation, ion mobilities, and glass formation behavior in both all atom and bead−spring simulated ionic liquids and polymeric ionic liquids. We have specifically focused on the

Figure 20. Decoupling exponent for coarse-grained PILs vs (a) ion pair volume and (b) mean diameter of cation/anion pair. Symbols are the same as those in Figure 19.

L


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Nernst−Einstein conductivity to probe the relationship of ion mobility, neglecting correlated ion motions, to these other quantities. Results from all-atom molecular dynamics simulations suggest that both Tg and ion/polymer decoupling tend to increase with decreasing combined ion volume in polymeric ionic liquids. Similar trends are not seen, however, within nonpolymeric ionic liquids, implicating chain stiffening and long-range connectivity in this trend. This hypothesis is consistent with the observation that the observed trends in Tg are strongly correlated to equivalent trends in the hightemperature activation behavior at temperatures well above the glass formation range. This finding suggests that these trends in dynamics are embedded in relatively simple high-temperature activation behavior and simply propagate down into the glass formation range. This physical picture is consistent with a recent scaling model proposed by Bocharova et al.12 Future work should therefore focus on understanding more fully the relationship between molecular structure and hightemperature activation energy in these materials, including issues such as dependence on imidazolium alkyl side-chain length. At present, there is no widely accepted theory for the dependence of relaxation times on precise chemical structures even in this high temperature range. A better understanding of the role of correlated ion motions in the dynamics of these systems and of their potential effects on the trends observed here in single-ion mobility is also needed and has been a subject of other recent works.15,59,67 These broad trends related to segmental dynamics and glass formation may also be convoluted in real chemistries by chemically specific effects involving, for example, ion pairing and correlated ion motion. The present findings provide a framework for future work probing the substantial variation around many of the trends identified here as well as for probing the dependence of glass formation and ion aggregation in specific chemical structure. Finally, results from coarse-grained simulations suggest a new strategy for maximizing ion mobility by both minimizing Tg and maximizing ion/polymer decoupling. Specifically, these simulations suggest that unlike Tg, decoupling is preferentially sensitive to the size of the f ree ion. This suggests that combination of a very large bound ion with a small free ion can be expected to maximize ion mobility in polymeric ionic liquids. Moreover, we find that ion aggregation tends to both lower Tg and increase decoupling. Provided that these domains percolate, use of ions exhibiting strong chemical incompatibility with the chain backbone can thus be expected to maximize ion mobility. An initial test in this coarse-grained model suggests that the combination of these strategieslarge bound ions, small free ions, and aggregating ion chemistries may provide a path to ultrahigh-ion-conductivity polymeric ionic liquids for next-generation energy applications.

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Y.C. and J.Y. contributed equally. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This material is based upon work supported by the National Science Foundation under Grant DMR1554920. The authors acknowledge the W. M. Keck Foundation for their generous support of this work.

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*(D.S.S.) E-mail [email protected]. *(T.K.P.) E-mail [email protected]. ORCID

David S. Simmons: 0000-0002-1436-9269 Present Address

T.K.P.: Center for Nanoscale Materials, Argonne National Laboratory, Argonne, IL 60439. M


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