Structural, Ionic, and Electronic Properties of Solid-State Phthalimide-Containing Polymers for All-Organic Batteries

Redox-active polymers serving as the active materials in solid-state electrodes offer a promising path toward realizing all-organic batteries. While both cathodic and anodic redox-active polymers are needed, the diversity of the available anodic materials is limited. Here, we predict solid-state structural, ionic, and electronic properties of anodic, phthalimide-containing polymers using a multiscale approach that combines atomistic molecular dynamics, electronic structure calculations, and machine learning surrogate models. Importantly, by combining information from each of these scales, we are able to bridge the gap between bottom-up molecular characteristics and macroscopic properties such as apparent diffusion coefficients of electron transport (Dapp). We investigate the impact of different polymer backbones and of two critical factors during battery operation: state of charge and polymer swelling. Our findings reveal that the state of charge significantly influences solid-state packing and the thermophysical properties of the polymers, which, in turn, affect ionic and electronic transport. A combination of molecular-level properties (such as the reorganization energy) and condensed-phase properties (such as effective electron hopping distances) determine the predicted ranking of electron transport capabilities of the polymers. We predict Dapp for the phthalimide-based polymers and for a reference nitroxide radical-based polymer, finding a 3 orders of magnitude increase in Dapp (≈10–6 cm2 s–1) with respect to the reference. This study underscores the promise of phthalimide-containing polymers as highly capable redox-active polymers for anodic materials in all-organic batteries, due to their exceptional predicted electron transport capabilities.

To determine T g , we used the same procedure as in Ref. S1. Namely, we fit the densitytemperature data via two independent linear fits to find the melt and glass regions.To select the best fits, we did the following: starting from a glass fit starting from the fourth datapoint from the lowest T and a melt fit starting from the fourth datapoint from the highest T , we scanned through all combinations of glass and melt fits obtained by progressively increasing the number of datapoints to be included in the fits.The best fits were selected by choosing the ones that maximize the sum of the R 2 coefficients of the linear fits; T g was then obtained as temperature at which the two linear fits cross (e.g.,, see Figure S3).

S-3
100 200 300 400 500 600 700 800 900 Temperature (K) Figure S3: Example T g fits.These three cooling trajectories are for the PMAP system at 0% state of charge and 5% swelling. S-4 2 Polymer-Electrolyte Systems Molecular composition of the systems.Table S1 reports all the molecules contained in the systems represented in Figure 1 of the main text for the various polymers.Procedure to determine the composition of the systems at different swelling ratios.We determined the composition of the systems based on the pure systems, the dry polymer and the electrolyte solution.We prepare a solution of 0.5 M by computing the number of ions needed for that concentration in a given volume, then using the Gromacs tool gmx insert-molecules to fill the given volume with solvent molecules.We then perform a short (2 ns) equilibration that leads to a negligible box volume adjustment.In parallel, we equilibrate the pure polymer systems, i.e., systems containing only 100 30-mer chains of the polymer, following the same protocol as shown in Figure S4.Based on the volume of the equilibrated pure polymers, we compute the volume that corresponds to a 5%, 10%, or 20% increase in volume.The computed volume increase is the volume of electrolyte solution that we add to realize the 5%-, 10%-, or 20%-swollen systems, respectively.What one actually needs is the number of molecules of electrolyte solution that corresponds to the 5%, 10%, or 20% volume.For example, the pure DME:TBAPF6 electrolyte solution that we equilibrated contains 1037 DME molecules, 70 TBA + , and 70 PF 6 − ions.Meanwhile, an equilibrated box of 100 30-mers of PEPP has a volume of ≈843 nm 3 ; 5% of this volume is ≈42 nm 3 .Given that the solvent box volume is ≈220 nm 3 , ≈42 nm 3 of it will contain: which are the numbers that are reported in Table S1 for the PEPP:DME:TBAPF 6 system.
We use this composition-100 30-mers, 197 DME molecules, 13 TBA + , and 13 PF 6 − -and run the protocol of Figure S4.The final volume of the equilibrated PEPP:DME:TBAPF 6 system is found to be ≈875 nm 3 .This means that the effective swelling for the system with respect to the dry polymer is ≈4%, and that the system shows negative deviation from ideal mixing behavior.The effective swellings for PEPP's 10%-and 20%-swollen systems are found to be ≈8% and ≈16%, respectively.For all the systems the effective swellings are in the range 3.5-4%, 7-8%, and 14-16%, for the 5%, 10%, and 20% cases, respectively.

S-6
Figure S4: Atomistic MD simulation protocol.After the system is setup with Polyply, S2 we equilibrate it well in the melt regime (900 K), after which configurations are taken out starting from 150 ns every 50 ns.These configurations are cooled down to 100K with a 10 K/ns rate.From the cooling trajectory, snapshots are taken at temperatures T = 0.8 • T g , T = 1.2 • T g , and T = 300 K that are relaxed at those temperatures for 100 ns.Strucutral, ionic, and electronic characterizations are conducted on the last 50 ns of the "relaxation" trajectories.(top-left).This map needs to be normalized to account for the increasing considered volume with increasing r ij distances, and hence the higher chances of finding a neighbor with increasing r ij distances.In 3D, this normalization is the volume of the spherical shell, ρ4πr 2 dr; this normalization results in the map shown in the top-right.Finally, in order to compare 2D maps obtained from different systems, which can contain different amounts of phthalimide pairs, we normalize by the total number of ij pairs (bottom-right).) PMAP; SoC = 0, 20, 60%; solv% = 5% PMAP; SoC = 0, 20, 60%; solv% = 10% PMAP; SoC = 0, 20, 60%; solv% = 20% PEPP; SoC = 0, 20, 60%; solv% = 5% PEPP; SoC = 0, 20, 60%; solv% = 10% PEPP; SoC = 0, 20, 60%; solv% = 20% PVBP; SoC = 0, 20, 60%; solv% = 5% PVBP; SoC = 0, 20, 60%; solv% = 10% PVBP; SoC = 0, 20, 60%; solv% = 20% Figure S13: TBA + ion diffusion coefficient, D TBA , as a function of temperature, for the different states of charge and swelling %.Data for each system are the ones at T = 1.2 • T g reported in Figure 4A, but now plotted as a function of 1000/(T − T g ).D TBA coefficients decrease as the distance from T g decreases.Moreover, consistently with Figure 4A, the plot shows that D TBA coefficients generally increase with increasing swelling %, while they decrease with increasing state of charge.Note that, in each 3-datapoint series, the left-hand side datapoint is SoC= 0%, the middle one SoC= 20%, and the right-hand side one is SoC= 60%.
where ε 1 and ε 2 are the relative permittivities of the polymer (assumed to be ε polymer = 3 based on the experimental value for nitroxide radical-based polymers) S4 and the solvent (ε DME = 7.2 and ε H 2 O = 80.1), S5

Figure S14 :
Figure S14: Log-log plot of the mean-square displacement (MSD) of TBA + .(left) PMAP, (center) PEPP, (right) PVBP; (top) 5%, (middle) 10%, (bottom) 20% swelling.T = 1.2 • T g .Each plot reports three MSDs, one for each state of charge (0% in blue, 20% in gray, 60% in yellow).Each of the MSD curve is an average (thick line) of 5 50-ns MSD curves (thin lines) taken from a 100 ns trajectory at different starting times.The slopes of the linear fits of the MSD average curve at the longest simulated times are reported in the legend.A guideline for the unit slope (diffusive regime) is also shown in each plot.The slopes (scaling exponents) are on average around 0.7 to around 0.4, with the systems with 0% state of charge being on average the closest to the diffusive regime, and the systems with 60% state of charge being the farthest.The Einstein relation is used to extract the diffusion coefficients in the highlighted region (the extracted diffusion coefficients are reported in Figure4of the main text).

Figure S15 :
Figure S15: Orbital overlap distributions as a function of state of charge for the different polymers.Orbital overlap distributions corresponding to the electronic coupling distributions of Figure 5 of the main text.Mean of the overlap distributions, ⟨logS ij ⟩, obtained as the mode of the skew Gaussian distribution fits, are reported in the legends.Electrolyte solution volume % = 10%.T = 300 K.

Figure S16 :Figure S17 :
Figure S16: Correlation between electronic couplings and orbital overlaps.Correlation between DIPRO S3 electronic couplings and orbital overalps.(left) N-methylphthalimide. (right) TEMPO.The linear fit parameters and R 2 score can be found in the legends.

Figure S18 :Figure S19 :Figure S20 :
Figure S18: Neural network predicted overlaps vs. reference values for three samples.(left) PMAP, (center) PEPP, (right) PVBP.The distribution predicted by the neural network surrogate model is in blue, while the reference distribution of explicitly computed orbital overlaps is in red.Distributions are normalized.

Figure S21 :
Figure S21: Computed K T as a function of electronic coupling threshold.Comparison of computed K T as a function of chosen threshold for the electronic couplings.From top to bottom thresholds are: 10, 50, 200, and 400 meV.Only when the threshold reaches the (rather high) value of 400 meV, some of the 60% results change qualitatively (bottom row).

Figure S22 :
Figure S22: Number of networks as a function of electronic coupling threshold.Comparison of computed number of networks as a function of chosen threshold for the electronic couplings.From top to bottom thresholds are: 10, 100, 200, and 400 meV.

Figure S23 :
Figure S23: Computed largest network size as a function of electronic coupling threshold.Comparison of computed largest network size as a function of chosen threshold for the electronic couplings.From top to bottom thresholds are: 10, 100, 200, and 400 meV.

Figure S27 :
Figure S27: Reorganization energies.(top-left) Absolute values of λ for the different polymers and for N-methyl-phthalimide (NMePh).(top-right) Differences between the λ of NMePh, λ NMePh and the λ values for the different polymers.Values computed at the ωB97X-D/def2-SV(P) level.(bottom) λ out as a function of the relative permittivity of the environment, ε r .The latter is computed using the following Debye model: εr−1 εr+2 = Φ 1

a
swelling [NO] [TEABF 4 ] D app k hop Value not reported.

Figure S29 :
Figure S29: D app for PTMA.(A) Rendering of the PTMA:H 2 O:TEABF 4 system swollen at 10% swelling.(B) TEMPO interdistance r ij vs. overlap heatmap for PTMA.The center-to-center distance at electron transfer, δ, is computed as the orbital overlap-weighted mean distance: δ= n V ij x i / n V ij ;shown is the map for PTMA:H 2 O:TEABF 4 at 10% swelling; see TableS4for the obtained δ values.(C) Electronic coupling distribution for the PTMA:H 2 O:TEABF 4 at 10% swelling.The mean electronic couplings, ⟨V ij ⟩, obtained as the mode of the skew Gaussian distribution fit, is reported in the legend.T = 300 K.

Table S1 :
Details for the various systems.Number of solvent molecules and ions that compose each system.S.o.C. stands for "state of charge".
and Φ 1 and Φ 2 are the volume fractions of the polymer and solvent, respectively.Additional results for the apparent diffusion coefficient, D app
a Value not reported.