Letter Cite This: ACS Macro Lett. 2018, 7, 734−738
pubs.acs.org/macroletters
Globally Suppressed Dynamics in Ion-Doped Polymers Michael A. Webb, Umi Yamamoto, Brett M. Savoie, Zhen-Gang Wang, and Thomas F. Miller, III* Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, California 91125, United States S Supporting Information *
ABSTRACT: We investigate how ion−polymer complexation suppresses molecular motion in conventional polymer electrolytes using molecular dynamics (MD) simulations of lithium hexafluorophosphate in poly(ethylene oxide) and a modified Rouse model. The employed model utilizes an inhomogeneous friction distribution to describe ion−polymer interactions and provides an effective way to examine how ion−polymer interactions affect polymer motion. By characterizing the subdiffusive Li+ transport and polymer relaxation times at several salt concentrations, we observe that increases in local friction due to ion-polymer complexation are significantly smaller than previously assumed. We find that a Rouse-based model that only considers local increases in friction cannot simultaneously capture the magnitude of increased polymer relaxation times and the apparent power-law exponent for Li+ subdiffusion observed in MD simulations. This incompatibility is reconciled by augmenting the modified Rouse model with a term that increases the global friction with the square of the salt concentration; this significantly improves the agreement between the model and MD, indicating the importance of ion−ion interactions and distributions on ion/polymer mobility.
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dilute concentration (a ratio of 1:20 Li+/O), the relaxation times for polymer segments that bind Li+ differ by only 2-fold from relaxation times for unbound segments.14 Because polymer relaxation times scale linearly with friction in the Rouse model, this 2-fold difference in relaxation times between complexed and uncomplexed segments contradicts the 8-fold difference in friction between complexed segments and the neat polymer. This disparity has not been explicitly examined in the dilute-ion regime where the effect, relative to the neat polymer, is most naturally quantified. Questions thus remain as to how local and how strong the effect of ion−polymer complexation is on polymer mobility and how this changes with salt concentration. In this Letter, we demonstrate the suppression of ion and polymer motion at increasing salt concentrations of LiPF6 in PEO using MD simulations and analyze the origin of this phenomena using the Rouse model formalism.28 We modify the Rouse model to feature an inhomogeneous friction distribution for polymer beads that mimics the effect of ion− polymer complexation. While similar to a previous simulationbased model,26 the inhomogeneous friction Rouse model (IFRM) enables efficient calculations over a range of concentrations with controlled assumptions. Using results of MD simulations and the IFRM, we show that representing ion−polymer interactions only as local increases in friction, which is the prevailing view, cannot simultaneously explain
olymer electrolytes, which are typically comprised of a polymer melt like poly(ethylene oxide) (PEO) and a dissociating salt like lithium hexafluorophosphate (LiPF6), represent relatively simple systems for which ion−polymer interactions govern a range of observed phenomena; nonetheless, key features of these interactions remain poorly understood. While significant experimental1−8 and theoretical9−14 research indicates that the major mechanisms of ion transport in conventional polymer electrolytes, such as ion−polymer codiffusion13 and solvation-site hopping,15−20 critically depend on the polymer dynamics, a confounding issue is that the ionpolymer association itself induces changes to the polymer properties that affect the Li+ diffusivity. This is demonstrated via experimental observations of increased Tg and decreased molar conductivity with increased salt concentration21−23 and via computational observations that polymer segments coordinating Li+ exhibit suppressed mean-squared displacement and slower torsional motions compared to noncoordinating segments.3,24,25 The basic picture that has emerged is that polymer motion is locally slowed nearby the Li+. An intuitive explanation for the slowdown is that the ioncoordinating polymer segments feel an effectively higher friction. Maitra and Heuer used Rouse theory and Brownian dynamics simulation to show that increasing the friction of complexed polymer segments 8-fold relative to a neat polymer leads to good agreement with molecular dynamics (MD) simulations;26 other Rouse-based models for Li+ diffusion employ phenomenologically increased relaxation times, which is conceptually similar to increasing the local friction of coordinating polymer segments.13,14,27 Interestingly, at a fairly © XXXX American Chemical Society
Received: March 30, 2018 Accepted: June 4, 2018
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DOI: 10.1021/acsmacrolett.8b00237 ACS Macro Lett. 2018, 7, 734−738
Letter
ACS Macro Letters both the subdiffusive Li+ transport and polymer relaxation across length scales; nonlocal friction effects are found to be necessary. We begin with simulations of neat PEO (as a reference for unperturbed polymer dynamics) and PEO/LiPF6 at four salt concentrations, Li+/O = 1:640 (dilute), 1:40, 1:20, and 1:12; simulation details are in the Supporting Information (SI). Figure 1a compares the Li+ MSD at the various concentrations
curves with increasing concentration in Figure 1b is consistent with longer polymer relaxation times at higher concentrations; the nonmonotonic behavior is likely due to a transition from the free Rouse regime (MSD ∼ t0.5) to an entanglement-like regime (∼t0.25). However, the significant enhancement in the PLE for the dilute Li+ (red) compared to that for the oxygen atoms in the neat polymer (black) is unexpected. Specifically, the subdiffusive PLE for the Li+ is expected to align with that of the monomer segments, which is MSD ∼ t0.5 for Rouse theory28,29 or ∼t0.6 for atomistic simulations.30 The observed enhancement occurs for times between the segmental relaxation time and the time scale for Li+ exchange between polymer segments (Figure 1c) determined from the decorrelation of the Li+ coordination function, h(t ) =
∑i ∈ δi(t ) ∑i ∈ δi(0)
(1)
where δi(t) equals unity if oxygen i is within 2.5 Å of a tagged Li+ at time t and zero, otherwise, and denotes the set of oxygens with δi = 1 at t = 0. This apparent departure from Rouse-like behavior in the dilute Li+ limit has not been previously noted, possibly because typical salt concentrations for simulation studies (i.e., 1:20 or 1:12 for Li+/O) approach a power-law scaling that matches expectations. Additional simulations that enforce intrachain or interchain coordination of the Li+ confirm that the major trends in Figure 1a,b are signatures of coupled ion-polymer motion (Figure S1). To quantify changes in polymer dynamics, we compute Rouse-mode relaxation times for the atomistic polymers after mapping to discrete chains with N = 320 (SI, Section 2). The results (Figure 1d) reveal that the polymer dynamics in the MD simulations are slowed across all length scales at all finite salt concentrations. The relaxation time τp′ roughly corresponds to relaxation of subchains of the polymer consisting of N/(p − 1) segments, such that the lowest p relates to large-scale motions on the size of the whole or large sections of the chain, while the highest p indicates local motions on the scale of monomers. Computation of the internal monomer mean-square distance for the polymer chain comprising the melt (Figure S4) reveals that subchains up to 30 monomers exhibit Gaussian, randomwalk statistics, so our discussion is most relevant for Rouse modes with p ≤ 320/15 + 1 ≈ 22 since subchains of this size are well equilibrated. To focus on changes with salt concentration, a long chain is used to suppress contributions to the Li+ diffusivity due to global chain diffusion; although entanglements could possibly affect local polymer dynamics, the same chain length is used in all simulations so relative slowdowns can be quantified. In particular, the reported relaxation times are normalized relative to that of the shortest length scale mode in the neat polymer, τN′(0). As the concentration increases, the relaxation times are noticeably increased, indicating that the polymer dynamics are slowed across a considerable range of length scales. Interestingly, this occurs even at dilute concentrations (Li+/O = 1:40, 1:20), for which the majority of the polymer segments are not directly coordinating Li+ (Figure 1e). We now investigate whether observations from the MD simulations manifest in a Rouse model with locally increased friction. In particular, we calculate properties of a Rouse chain with N = 320, where the friction in the Brownian motion of a subset of beads is increased to mimic the effect of Li+ complexation. Formulating the equations of motion in matrix
Figure 1. Analysis of Li+ and polymer dynamics in PEO. (a) Meansquared displacement and (b) the apparent power-law exponent, for Li+ at various salt concentrations and for oxygen atoms in the neat polymer. (c) Li+ coordination correlation function. In (a, b), the thicker lines indicate times between the local segmental relaxation time τ′N determined in (d) and a characteristic hopping time τhop determined from (c). In (c), the vertical lines show τhop for each concentration, which is determined as the time for the coordination correlation function h(t) to decay to 0.5. (d) Rouse-mode relaxation times relative to the shortest relaxation time in the neat polymer; the slanted dashed line provides the expected scaling from Rouse theory, and the bracket highlights the ratio between relaxation times for the neat polymer and the polymer electrolyte with Li+/O = 1:12 at p ≈ 100. Data for modes with p ≲ 22 have an overlay since those modes exhibit deviations from random-walk statistics (SI, Section 4). (e) For characteristic MD snapshots at each concentration, a depiction of polymer segments (shown sequentially from top-to-bottom) coordinating Li+.
to the MSD of oxygens in the neat polymer. As expected, Li+ is slower than oxygen in neat PEO at intermediate times due to strong Li+−O interactions that retard the ion−polymer coupled motion,20 and the MSD decreases with increasing salt concentration. Figure 1b indicates the subdiffusive nature of the transport by tracking the apparent (time-dependent) power-law exponent (PLE) of the MSDs in Figure 1a; the subdiffusive transport is relevant here because it corresponds to local motion with the polymer. The Li+ MSD in the dilute regime shows an increase in the PLE relative to oxygen atoms in the neat polymer, and the data at finite concentrations is qualitatively similar, except that the observed maximum slightly decreases and occurs at later times. The rightward shift in the 735
DOI: 10.1021/acsmacrolett.8b00237 ACS Macro Lett. 2018, 7, 734−738
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ACS Macro Letters form and diagonalizing yields a solution for the MSD of the jth bead as ⟨(R j(t )̃ − R j(0))2 ⟩ = 6DCM t ̃ + 2
the neat-polymer MSD PLEs, the local friction should be increased to 3ζ0, which is much smaller than the factor-of-eight reported in a study using finite salt concentrations.26 Here, increasing the friction to 8ζ0 leads to drastic deviations for the MSD PLE relative to the neat polymer that is not reflected by the dilute Li+ MD results. However, the IFRM does not reproduce MD results at finite salt concentrations when local increases in friction are limited to 3ζ0. This is clearly demonstrated in Figure 2c, which provides the Rouse relaxation-time analysis for the IFRM at the same salt concentrations as used in the MD simulations. For these calculations, groups of three consecutive beads are randomly chosen to have their friction increased to 3ζ0 until a desired salt concentration is obtained; no bead is chosen more than once during random selection to avoid unphysical overlap in terms of Li+ coordination, and the local friction increase to 3ζ0 is used to preserve the correct apparent PLE in the dilute regime. Akin to the MD results in Figure 1e, Figure 2c demonstrates an overall increase in polymer relaxation times with increasing salt concentration. However, the polymer relaxation times for any given mode are well within a factor-ofthree in Figure 2c, whereas the MD results exhibit relaxation times that are an order-of-magnitude longer at the highest concentration compared to the neat polymer. The results of Figure 2a−c demonstrate that the IFRM cannot simultaneously reproduce results across salt concentrations. To reproduce behavior in the dilute Li+ limit, only a 3fold increase in local friction is warranted, but this drastically underestimates polymer relaxation times at finite salt concentrations. Conversely, to reproduce segmental relaxation times at finite salt concentrations, an order-of-magnitude increase is required, but this leads to poor agreement for the PLE in the dilute Li+ regime. These results highlight a flaw in assuming that increases in friction are localized to coordinating polymer segments. To quantify the suppression of polymer motion due to Li+− polymer interactions, we characterize the relative mobility of polymer atoms in Figure 3 using the mean square fluctuation
∑ Oji2λi−1(1 − e−λ t )̃ i
λi ≠ 0
(2)
where t ̃ = t/̃ (3π2τN) is a dimensionless time, λi is the ith eigenvalue of the typical Rouse interaction matrix,28,31 and the Oji are elements of the matrix of eigenvectors. All beads are initially assigned a friction of ζ0, making the friction homogeneous across the Rouse chain and reducing the results of eq 2 to the typical Rouse model. Since the neat polymer dynamics are well represented by the typical Rouse model when all beads have friction ζ0, eq 2 provides a way to mimic ion-polymer complexation by increasing the friction for a subset of beads; we call this the inhomogeneous friction Rouse model (IFRM). We first examine the case where the Li+−polymer interactions result in friction increases that are localized to the coordinating polymer segments. Since six oxygens typically coordinate the Li+, the friction of three beads at the center of the chain is increased to represent the dilute Li+ condition; the central bead of the chain is monitored to mitigate end-effects. The IFRM captures key features of the MD results in the dilute Li+ regime. First, Figure 2a shows that locally increasing
Figure 2. Polymer dynamics for the inhomogeneous friction Rouse model (IFRM). (a, b) For dilute salt conditions, (a) the mean-squared displacement and (b) apparent power-law exponent for a central bead with increased friction relative to the remainder of the chain. For finite salt conditions, (c) Rouse-mode relaxation times for the IFRM relative to τ(0) N , the shortest relaxation time in the homogeneous friction case. Each line reflects an average for 10 different chains.
Figure 3. Li+ perturbation on the polymer dynamics as the meansquared fluctuation (MSF) of polymer atoms (for a 50 ps time interval) as a function of distance from the nearest Li+; the horizontal line indicates the MSF for the neat-polymer melt, and the vertical line indicates the radius of influence of a Li+ on surrounding polymer atoms.
the friction suppresses the MSD at short to intermediate times relative to the homogeneous friction case with ζ0, akin to the MD results comparing the dilute Li+ and neat-polymer MSDs (Figures 1a and S1b). Second, Figure 2b shows that locally increasing the friction enhances the PLE for the MSD relative to the homogeneous friction case, akin to the MD results comparing the complexed Li+ under dilute salt conditions and neat-polymer segments (Figures 1b and S1c). The departure from a Rouse-like MSD PLE of t0.5 thus emerges naturally using the IFRM. To match the deviation between the dilute Li+ and
(MSF) of polymer atoms over a fixed time interval. The dilute Li+ simulations demonstrate that polymer atoms exhibit reduced mobility when closer to Li+ but converge to the neat-polymer MSF at farther distances. Because no other ions are present, the slowdown is solely attributed to interaction with the lone Li+, which has a radius of influence r+* ≈ 6.5 Å, since data points at greater separations are essentially at the 736
DOI: 10.1021/acsmacrolett.8b00237 ACS Macro Lett. 2018, 7, 734−738
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ACS Macro Letters neat-polymer value. At finite salt concentrations, polymer atoms remain slowest near a Li+ but do not converge to the neat-polymer MSF. While some slowdown is expected due to simultaneous interaction with multiple ions, polymer atoms are collectively slower up to 2r*+ from any Li+, suggesting that still more complicated interactions are at the heart of the globally suppressed dynamics. While the effect of modifying friction on the polymer relaxation times using the IFRM is roughly linear with salt concentration, simple and empirical descriptions for the viscosity of electrolyte solutions often have other functional dependences.32−36 For example, the extended Jones-Dole equation33,34 computes the viscosity η using η = 1 + Acs1/2 + Bcs + Dcs2 η0 (3)
relaxation times, additionally increasing the friction of beads with the x2s term in eq 4 enables simultaneous description of both the relative increase in polymer relaxation times across all length scales and the subdiffusive PLE for Li+ across salt concentrations. These results convincingly demonstrate that interactions apart from local ion-polymer complexation are required to rationalize the increase in bead friction. Predictions based on eq 4 are also in excellent agreement with recent experimental work from Mongcopa et al., which used quasi-elastic neutron scattering to obtain the monomeric friction as a function of salt concentration.37 While their data was fit to an exponential function, the data is also well fit by eq 4, which is effectively a quadratic function of xs, and yields physically reasonable fitting coefficients (SI, Section 5). The work presented here provides a unified Rouse-based description of how ion-polymer complexation manifests as increased polymer friction across the full range of relevant salt concentrations. Whereas previous work emphasized that friction was locally increased by an order-of-magnitude, we show that this leads to substantial deviation from MD results for subdiffusive motion in the dilute Li+ regime, and a 3-fold increase is more plausible. However, this 3-fold increase for complexed polymer segments severely underestimates polymer relaxation times at finite concentrations, and interestingly, MD simulations show suppressed polymer motion even when polymer atoms are farther than 10 Å from any Li+. By analogy with strong electrolyte solutions, we include a term in the beadfriction that phenomenologically accounts for neglected effects such as ion−ion interactions and associative effects. While such effects might be captured by introducing length- and timedependent friction contributions to the Rouse matrix using a Generalized Langevin Equation Rouse model,38,39 this approach brings additional complexity and multiple parameters. Here, by using one empirical parameter, the simply modified IFRM achieves good agreement with MD simulations and recent experimental data. These results demonstrate the importance of factors other than direct ion-polymer complexation in the suppression of dynamics in polymer electrolytes and invite further study to understand the origins of this phenomena.
where A, B, and D are coefficients, η0 is the neat solvent viscosity, and cs is the salt concentration. The term Ac1/2 s is only important at small concentrations and relates to ion−ion interactions, the term Bcs relates to ion−solvent interactions, while the term Dc2s is less understood but attributed to ion−ion associative interactions and higher-order terms of long-range Debye−Hückel Coulombic forces.34 Inspired by this phenomenology, we modify the IFRM to compute the friction of the ith bead as ζi = ζ0 + 2ζ0δi + κxs2
(4)
where ζ0 is the friction of a neat-polymer bead, xs is the mole r fraction of added salt (for r = Li+/O, xs = 3 + r in PEO), δi is a reporter function equal to one if the ith bead is coordinating Li+ and zero otherwise, and κ is an empirical parameter that accounts for neglected phenomena at higher concentrations, such as ion−ion interactions and associative effects. The x1/2 s term is intentionally neglected in eq 4 due to good agreement at low concentrations. Meanwhile, x2s increases the friction of all beads, such that the center-of-mass diffusion coefficient (p = 0 mode) should decrease in proportion to Nκx2s . The Rouse-mode relaxation times computed using eq 4 are shown in Figure 4 and demonstrate remarkable agreement with
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ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acsmacrolett.8b00237. Simulation details; Analysis of dynamics during constant coordination; Rouse-mode analysis details; Analysis of internal MSD between monomers; and Experimental comparison (PDF).
Figure 4. Relaxation times from the inhomogeneous friction Rouse model using eq 4 to modify the bead friction.
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the MD results in Figure 1e for the concentration-dependence of Rouse-mode relaxation times. In lieu of a theoretical description, κ is empirically set to 400 to approximately match the IFRM and MD results for relative relaxation times between the neat polymer and Li+/O = 40:1 at p = 100, an arbitrary reference. Beyond this single matching point, the relative relaxation times at all concentrations resemble those in Figure 1e and provide a much better description than the results in Figure 2c. Whereas increasing the friction locally according to ion-polymer interactions achieved just a 3-fold difference in the
AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. ORCID
Michael A. Webb: 0000-0002-7420-4474 Brett M. Savoie: 0000-0002-7039-4039 Zhen-Gang Wang: 0000-0002-3361-6114 Thomas F. Miller III: 0000-0002-1882-5380 Notes
The authors declare no competing financial interest. 737
DOI: 10.1021/acsmacrolett.8b00237 ACS Macro Lett. 2018, 7, 734−738
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ACS Macro Letters
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ACKNOWLEDGMENTS This research was supported by the National Science Foundation under DMREF Award Number NSF-CHE1335486. M.A.W. also acknowledges support from the Resnick Sustainability Institute. This research used resources of the Oak Ridge Leadership Computing Facility at the Oak Ridge National Laboratory, which is supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC05-00OR22725.
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