Diffusive Flux as a New Metric for IonConducting Soft Materials Nikki H. LaFemina,† Quan Chen,‡ Karl T. Mueller,†,§ and Ralph H. Colby*,‡ †
Department of Chemistry and ‡Department of Materials Science and Engineering, Pennsylvania State University, University Park, Pennsylvania 16802, United States § Physical and Computational Sciences Directorate, Pacific Northwest National Laboratory, P.O. Box 999, Richland, Washington 99352, United States ABSTRACT: 7Li NMR diffusion measurements for single-ion conducting ionomers with strong solvation for lithium reveal that diffusion is considerably faster than expected from ionic conductivity measurements, suggesting that neutral ion pairs dominate lithium transport in this class of materials. Ion aggregation is controlled by the overlap parameter for ion pair polarizability volumes; at ion contents higher than the polarizability volume overlap threshold, ion pairs aggregate strongly, making the dielectric constant saturate and the glass transition temperature increase more rapidly with ion content. Neutral ion pairs hop from one aggregate to a neighboring one by polymer segmental motion. Only when the ion pair has an extra lithium (i.e., a positive triple ion) do such hops contribute to ionic conductivity.
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static dielectric constant of the polymer actually forces stronger ion aggregation,7 because the polarizability volumes of ion pairs then overlap more. This conundrum appears to be ubiquitous to this class of materials, and we close with some suggested paths forward. Owing to the strong potential of single-ion conductors, our team has focused on the design and synthesis of anionic ionomers with polar solvating comonomers6,7,9 and study their ion aggregation,9,16,17 dielectric constant,17,21,22 ionic conductivity,6,7,9,21,22 and polymer dynamics.23−26 We started with ionomers based on PEO but tried many higher dielectric constant polymers7 without success. PEO-ionomers, depicted in Figure 1, with the dielectric constant 1, while at lower ion content, scattering does not show any sign of ion aggregates. The Nernst−Einstein equation31 relates diffusion coefficient, DNMR, and ionic conductivity, σDC. Failure of the Nernst− Einstein equation is described by a Haven ratio,32−34 HR, different from unity (e is the elementary charge). p D NMR e HR ≡ 0 σDCkT (2)
Figure 2. Ion pairs with their polarizability volumes, Vp, shown as the larger circles. (a) At low ion content, polarizability volumes are dilute and ion pairs rarely interact, allowing the Onsager equation to quantitatively describe their static dielectric constant. (b) At the ion content where polarizability volumes just overlap, the pairs start to interact with each other and the static dielectric constant reaches its maximum value.28 (c) Above the overlap threshold for polarizability volumes, some ion pairs aggregate,9 while the isolated ion pairs remain at their polarizability volume overlap concentration.
Combining 7Li NMR pulsed field gradient stimulated echo measurements of the diffusion coefficient18 with measured ionic conductivity results in Haven ratios considerably larger than unity (Figure 4). The ions arrange themselves in many states: free Li cations, Li-sulfonate isolated pairs, positive triple ions that have two Li associated with one sulfonate, negative triple ions that have two sulfonates associated with one Li, and quadrupoles that have two of each type of ion; and if p0Vp > 1, there are a variety of larger aggregates. Of these, any that involve two or more sulfonates are temporarily cross-linked and unable to move by segmental motion because each sulfonate is covalently attached to the polymer. In contrast, anything involving only one sulfonate (ion pairs and positive triple ions) can move with the segmental motion of the polymer.1,21,22 The dielectric constant of our ionomers is reasonably high, suggesting that many of the ions are in the isolated pair state. Those ion pairs move lithium around by segmental motion but cannot contribute to ionic conductivity because ion pairs have zero net charge. Consequently, the only explanation for the Haven ratio being considerably larger than unity is then that
Dielectric spectroscopy can measure the static dielectric constant of liquids at low frequency. The in-phase component of the dielectric constant is never independent of frequency for ionomers; ionic relaxations are still actively raising the dielectric constant on the time scale where electrode polarization (EP) starts. The static dielectric constant can be extracted in three independent ways that effectively separate EP from the material response, and these three agree well with each other.22 As Figure 3 shows for these PEO-Li ionomers, the dielectric constant increases with ion content while p0Vp < 1, as expected by eq 1. However, once p0Vp exceeds unity, ion pairs are forced to interact strongly and they simply aggregate to maintain the number density of ion pairs that existed at the overlap point where p0Vp = 1, with the dielectric constant independent of ion content9 once p0Vp > 1. This saturation leaves the same number density of ion pairs in the medium outside the aggregates, but the ion aggregates raise the glass transition temperature, Tg, 1180
DOI: 10.1021/acsenergylett.6b00545 ACS Energy Lett. 2016, 1, 1179−1183
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ACS Energy Letters
DNMR; p0DNMR is the diffusive flux of lithium ions. As evident from eq 2, the Haven ratios in Figure 4 with 4 < HR < 400 mean that this new p0DNMR diffusive flux metric will be 4−400 times greater than expected from ionic conductivity using the Nernst−Einstein equation. The p0DNMR diffusive flux determines the transit rates for Li motion across the electrolyte between the electrodes of a lithium battery, not σDC. Figure 5
Figure 4. Haven ratios, HR (eq 2), for PEO-sulfonate ionomers. The Haven ratios are all considerably larger than unity (4 < HR < 400), suggesting that diffusion of Li is rapid compared to what would be expected from ionic conductivity using the Nernst− Einstein equation. At low ion content (Figure 3a with p0Vp < 1), HR increases with temperature because solvation entropy favors ion pairs over positive triple ions,14 and HR decreases as ion content is increased because εs increases (Figure 2) to allow a higher fraction of rare positive triple ions. At high ion content (black squares in panel b with p0Vp ≈ 1.5), ion aggregates coexist with ion pairs9 and solvation entropy then favors aggregates, making HR very large and strongly decreasing as pairs aggregate more as temperature is raised. Near the polarizability volume overlap (p0Vp ≈ 1.0, blue triangles in Figure 4b), HR ≈ 4 is the smallest value and almost independent of temperature, presumably resulting from these two solvation entropy effects working against each other.
Figure 5. Diffusive flux of Li in PEO-sulfonate ionomers. (a) While ionic conductivity fits the Vogel−Fulcher−Tamman equation that strongly curves in this plot at lower temperatures, the new diffusive flux metric is Arrhenius, with activation energy 55 ± 3 kJ/mol, independent of ion content. The PGSE NMR measurements are limited to quite high temperatures, but closer to Tg, the diffusive flux is expected to be non-Arrhenius. (b) The diffusive flux at 60 °C displays a maximum at intermediate ion content (black squares), coinciding with overlap of polarizability volumes (Vp = 1.6 nm3), beyond which the glass transition temperature (blue circles) increases more strongly.
most of the lithium diffusion is coming from segmental motion of isolated pairs. Individual pairs can leave one ion aggregate and move by the segmental motion of the polymer to a nearby different ion aggregate. The only means for a net charge to move is then the rare event that the ion pair has a second Li, the positive triple ion.14,19,20 The segmental motion of positive triple ions contributes to ionic conductivity (as would the motion of free Li but in a sea of ion pairs free Li rapidly forms positive triple ions), while the segmental motion of both positive triple ions and ion pairs contributes to Li diffusion. With most ions in the isolated ion pair state (at least for p0Vp < 1, the PEO600-6, 11, and 17% Li ionomers) this leads to the large Haven ratios observed. Because many Li are moving in the neutral ion pair state and not contributing to ionic conductivity, it seems that σDC is the wrong metric for quantifying motion of lithium in this class of materials. A far superior metric is simply the product of the total cation number density, p0, and their diffusion coefficient,
shows the Arrhenius temperature dependence of p0DNMR diffusive flux for the PEO-sulfonate ionomers. Ionomers with intermediate ion content with p0Vp ≈ 1.0 have the largest diffusive flux. The optimal p0DNMR diffusive flux occurs at an ion content below the point where polarizability volumes overlap. For our Li ionomers, because Vp = 1.6 nm3, the highest diffusive flux is at p0 = 0.43 where p0Vp = 0.7. At lower ion contents, p0DNMR logically increases with ion content because the glass transition temperature is not yet increasing rapidly with ion content.9,22 Beyond overlap of ion pairs, the p0DNMR diffusive flux decreases with ion content because the glass transition temperature, Tg, is increasing rapidly as ions aggregate more.9 1181
DOI: 10.1021/acsenergylett.6b00545 ACS Energy Lett. 2016, 1, 1179−1183
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ACS Energy Letters
losing much entropy. Ideally, that might be combined with some new functional group that can solvate cations without being a hydrogen bond acceptor, although the only ones we know about involve N or O (with electron lone pairs) that do not have protons, and these always do accept hydrogen bonds also. One might also try designing ion pair separators that have anion solvators at one end and cation solvators at the other end, connected by a segment that is sufficiently rigid whose separation distance could be varied on the nanometer scale to optimize diffusive flux of Li.
For aggregated samples at high ion content, a second mechanism of ion motion emerges beyond the simple “hopping” of ion pairs. Dielectric spectroscopy measures an ionic segmental relaxation that grows in intensity with ion content below polarizability volume overlap, with a peak at time scale τα2, reflecting an average time for ion hops. Small-angle Xray scattering (SAXS) measures the spacing between aggregates, d, and the expectation of the ion hopping models such as Ratner’s1 is that DNMR = d2/(6τα2). Figure 6 shows that
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. ORCID
Quan Chen: 0000-0002-7771-5050 Ralph H. Colby: 0000-0002-5492-6189 Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS We gratefully acknowledge the Department of Energy, Office of Basic Energy Sciences, under Grant DEFG02-07ER46409, Conduction Mechanisms and Structure of Ionomeric SingleIon Conductors, which funded a five-faculty multistudent team for seven years. We also acknowledge the other three faculty Janna Maranas, James Runt, and Karen Wineyfor many years of collaboration and most of all the other Ph.D. students: Shichen Dou (2007), Wenqin Wang (2010 UPenn), Shih-Wa Wang (2011), Alicia Castagna (2012), U Hyeok Choi (2012), Kan-Ju Lin (2012), David Roach (2012), Kokonad Sinha (2012), Greg Tudryn (2012), Siwei Liang (2013), Hanqing Masser (2013), Wenjuan Liu (2014), Michael O’Reilly (2014 UPenn), Huai-Suen Shiau (2014), Helen Jing-Han Wang, (2015), and Joshua Bartels (2015).
Figure 6. Test of the hopping model for Li+ diffusion based on motion of ion pairs on time scale τα2 measured by dielectric spectroscopy,22 moving the spacing between aggregates, d, measured by SAXS.16 Note that the diffusion measurements by PGSE NMR18 need ions to move distances on the order of micrometers on time scales on the order of 300 ms to reach fully diffusive conditions which are huge compared with the interaggregate spacing, with hopping distance, d, roughly 4 nm and hopping times on the order of 100 μs.
relation is in fact observed for samples below the polarizability volume overlap threshold, meaning that ion motion becomes diffusive on the scale of these hops. However, with strong ion aggregation in the highest ion content sample (black squares in Figure 6 with p0Vp ≈ 1.5), the diffusion of Li is larger than expected from the hopping model alone, and our interpretation of that is related to the fact that with lots of PEO present, the large ion aggregates are ion chains that allow Li to bind to many ether oxygens. Li+ ions are free to exchange in these chainlike aggregates, as seen in simulations,35−37 and this provides a boost to the measured diffusion coefficient, particularly evident at low temperatures and high ion contents, where the segmental motion hopping becomes quite slow. The intuition of raising the dielectric constant of the surrounding nonionic polymer to soften ionic interactions by using cyclic carbonate instead of ethylene oxide in polysiloxane ionomers actually resulted in an increase in the polarizability volume of ion pairs, the ion pairs interacting more and aggregating at lower ion contents, raising Tg and lowering both ionic conductivity and the p0DNMR diffusive flux.7 That finding suggests that it will not be easy to boost the diffusive flux of Li in polar anionic ionomers; the design assumption of raising the dielectric constant failed, and consequently, this problem requires more clever solutions. One such solution that remains unexplored would be to design self-solvating anions surrounded by −NH, −NH2, and −OH groups that are sufficiently close to the polymer-attached anion that they can interact without
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